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[ "Towards Computational Awareness in Autonomous Robots: An Empirical Study of Computational Kernels", "Towards Computational Awareness in Autonomous Robots: An Empirical Study of Computational Kernels" ]
[ "Ashrarul H Sifat ", "Burhanuddin Bharmal ", "Haibo Zeng ", "Jia-Bin Huang ", "· Changhee ", "Jung ", "Ryan K Williams " ]
[]
[]
The potential impact of autonomous robots on everyday life is evident in emerging applications such as precision agriculture, search and rescue, and infrastructure inspection. However, such applications necessitate operation in unknown and unstructured environments with a broad and sophisticated set of objectives, all under strict computation and power limitations. We therefore argue that the computational kernels enabling robotic autonomy must be scheduled and optimized to guarantee timely and correct behavior, while allowing for reconfiguration of scheduling parameters at runtime. In this paper, we consider a necessary first step towards this goal of computational awareness in autonomous robots: an empirical study of a base set of computational kernels from the resource management perspective. Specifically, we conduct a datadriven study of the timing, power, and memory performance of kernels for localization and mapping, path planning, task allocation, depth estimation, and optical flow, across three embedded computing platforms. We profile and analyze these kernels to provide insight into scheduling and dynamic resource management for computation-aware autonomous robots. Notably, our results show that there is a correlation of kernel performance with a robot's operational environment, justifying the notion of computation-aware robots and why our work is a crucial step towards this goal.
10.1007/s40747-023-01059-7
[ "https://arxiv.org/pdf/2112.10303v1.pdf" ]
245,334,489
2112.10303
e05c6ddcd2ec085501fff0bd9b238effc2568d39
Towards Computational Awareness in Autonomous Robots: An Empirical Study of Computational Kernels Ashrarul H Sifat Burhanuddin Bharmal Haibo Zeng Jia-Bin Huang · Changhee Jung Ryan K Williams Towards Computational Awareness in Autonomous Robots: An Empirical Study of Computational Kernels Received: November 17, 2021 / Accepted: TBD The potential impact of autonomous robots on everyday life is evident in emerging applications such as precision agriculture, search and rescue, and infrastructure inspection. However, such applications necessitate operation in unknown and unstructured environments with a broad and sophisticated set of objectives, all under strict computation and power limitations. We therefore argue that the computational kernels enabling robotic autonomy must be scheduled and optimized to guarantee timely and correct behavior, while allowing for reconfiguration of scheduling parameters at runtime. In this paper, we consider a necessary first step towards this goal of computational awareness in autonomous robots: an empirical study of a base set of computational kernels from the resource management perspective. Specifically, we conduct a datadriven study of the timing, power, and memory performance of kernels for localization and mapping, path planning, task allocation, depth estimation, and optical flow, across three embedded computing platforms. We profile and analyze these kernels to provide insight into scheduling and dynamic resource management for computation-aware autonomous robots. Notably, our results show that there is a correlation of kernel performance with a robot's operational environment, justifying the notion of computation-aware robots and why our work is a crucial step towards this goal. Introduction Robots are demonstrating the ability to tackle real-world problems if the problems are well-defined and composed of a reasonably small, fixed set of high-level tasks, e.g., welding on an assembly line [1]. However, our understanding remains limited for autonomous robots that solve complex problems composed of a wide range of difficult computational tasks. Consider for instance a search and rescue problem where autonomous robots must actively collaborate with human searchers in complex, uncertain environments (e.g., wilderness or disaster sites) [2,3]. In such a problem, an autonomous robot will be expected to execute (at least) the following base set of computational kernels: simultaneous mapping and localization [4]; path planning, navigation and control [5,6,7,8,9,10,11]; high-level decision-making such as task allocation [12,13,14,15]; and a range of computer vision (CV) tasks such as optical flow [16] and depth estimation [17]. Furthermore, the remoteness of the operational environment necessitates on-board computation by the robots, placing strict limits on computation and power. Finally, as the robots work in concert with humans, there must exist guarantees on robot behavior, and thus computation, to ensure safety. We argue that the search and rescue example motivates the need for robots that are computation-aware, allowing the computational kernels that support autonomy to be scheduled and optimized on embedded hardware in a manner that adapts to changing objectives and uncertain environments, while guaranteeing the timeliness and correctness of autonomous behavior. The holistic overview of such a system, depicted in Figure 1 is proposed. Towards this vision of computational awareness in autonomous robots, we must first understand the computational requirements for the base set of kernels that support robotic autonomy (listed above). In the current literature, there is a deep understanding of the aforementioned computational kernels from a theoretical point of view, for example yielding insights into worst-case computational complexity [18,19,20]. However, given a particular computational platform (such as the NVIDIA Xavier [21] studied in this work), there is very little existing work on the relationship between the high-level tasks an autonomous robot performs and the underlying computational requirements for performing such tasks in a timely manner. Additionally, there have been recent efforts to integrate both hard and soft real-time scheduling-capable kernels with the Robotic Operating System (ROS) [22], [23], which can form the basis for guaranteeing the timeliness and correctness of autonomous behaviors, and with ROS 2.0, real-time scheduling support specialized for robotic applications is in sight [24]. It is therefore critical that we build an understanding of computational kernels for autonomous robots so that such capabilities can be fully exploited in the near future. Thus, this paper performs an empirical study of the computational requirements of a base set of Fig. 2 The list of representative computational kernels and the proposed workflow for achieving computational awareness in autonomous robots with objectives achieved in this paper. kernels depicted in Figure 2, which is the first step for the learning and regression in our proposed workflow for achieving computational awareness in autonomous robots. Related Work and Contribution. While there is very little existing work on computational awareness in autonomous robots, we can point to a few recent efforts that are related to this paper. A recent work focuses on benchmarking connected and autonomous vehicles (CAVs) applications from the quality of service (QoS) perspective [25]. The benchmarking done in this work is solely concentrated on CAVs and derive insights such as the need for heterogeneous architecture, high memory bandwidth and optimized cache architecture, which we also recognise in our systems. However, it lacks the in-depth analysis of each computational kernel that might need to run in parallel in a mobile robot unlike the CAVs and can have a different wide range of parameters to consider. Similarly, a specialized benchmark for mobile vision algorithms is presented in [26], which provides a useful micro-architectural analysis of vision kernels. However, it also lacks the insight into the bigger spectrum of robotic computational kernels as well as environmental correlation and kernel specific parametric study. Kato et al. [27] present Autoware on board, to integrate the Autoware open source software for autonomous driving [28] on the Nvidia Drive Px2 and subsequently measure the execution time performance. However, this work lacks sufficient understanding on how the computation correlates with the driving environment. Zhao et al. [29] perform a field study for autonomous vehicles (AV) to gain insights on how the computing systems should be designed. This work looks into the sequential operation of the AV computational kernels from a safety perspective and present a innovative safety criteria for AV. Both of these works [27,29] are solely concentrated on AV which have different constraints than mobile robots such as an Unmanned Aerial Vehicle (UAV). Finally, Carlone and Karaman [30] consider the visual-inertial navigation problem, and propose anticipation with environmental cues to build a simplified model on robot dynamics and produce a computationally efficient system. However, it is an analysis of just one kernel, naturally not enough to implement a general framework such as our proposed solution. All in all these literature give valuable insight into need for computational awareness and specific studies but not the comprehensive analysis for autonomous robotic platforms. PRELIMINARIES Autonomous robots continue to suffer from strict limits on computation and power, especially for small aerial vehicles. From a resource management perspective, we therefore focus on profiling that can inform two primary techniques: (1) thread-level scheduling that prioritizes among multiple computational kernels competing for shared computational resources such as CPU, GPU, and memory [31,32,33,34]; and (2) the Dynamic Voltage and Frequency Scaling (DVFS) mechanism on modern processors that allows for finegrained frequency scaling and power management [35]. We aim to analyze each computational kernel for sustainability and predictability by showing their worst case behavior, average behavior, and the variation from them. For an autonomous robot, we argue that a static, single-valued worst case execution time (WCET) is too pessimistic as the robot's computational needs are highly dynamic and potentially dependent on the environment and goal. In order to take advantage of the above techniques in a robotics setting, we profile the utilization, timing, memory, and power consumption properties of a base set of computational kernels that support autonomy. There are several different parameters to consider when selecting the most appropriate scheduling algorithm. Figure 3 depicts the general operation of a scheduler, with definitions for the parameters we aim to measure given below. -CPU Utilization: The ratio of a CPU's load and the computational capacity of the CPU. -Throughput: The total number of processes completed per unit time, i.e., the total amount of work done in a unit of time. -Turnaround/Response Time: The amount of time taken to execute a particular process/thread. -User Time: The amount of time a process/thread spends in the user space of the host operating system. This is also referred to as CPU execution time. -System Time: The amount of time a process spends in kernel level instructions such as interrupts, I/O calls, etc. -Activation Period : The interval or period after which each periodic kernel is repeated and therefore dispatched to the CPU scheduler. In general, CPU utilization and throughput are maximized and other factors are minimized to achieve performance optimization. While the response time is varied widely across different systems and scheduling policies, the execution time is tied most closely to the specifics of hardware and the computational kernel itself. Therefore, in this paper we focus our profiling on the execution time in user space and kernel space as a means of characterizing the requirements for our computational kernels. At the same time, thread-level execution time breakdown and CPU-affinity are also a necessary part of scheduling. Therefore, among the other parameters of analysis we will provide the number of processes per computational kernel, number of threads per process, average utilization of threads, memory utilization, and power consumption. We also investigate relevant parameters for GPU-intensive vision kernels, including instructions per cycle (IPC), shared utilization, various (xyz) functional unit utilization, achieved occupancy, and various memory load throughput. As the first step towards optimizing and guaranteeing the timeliness of computation for an autonomous robot, we believe these parameters are of vital importance to understand. GPU performance parameters also vary significantly across different architectures. The definitions of the parameters we measure are given below: Finally, it is worth noting that the computational complexity of the tasks carried out by an autonomous robot can be used to mathematically formulate the execution time required for each CPU process. However, the complex nature of CPU execution cycles combined with the hardware and kernel related variables and dependencies makes it infeasible to predict computational needs merely from a mathematical formulation. Hence, we begin in this paper with profiling of our computational kernels in various settings across popular computational platforms for robotics. Only then is it possible to answer the pressing questions of combining computational kernels, implementing appropriate scheduling policies, and determining the rates at which we can execute the kernels. Computational Kernels This section provides a brief overview of the concept and operations of the computational kernels we profile. Note that each computational kernel may be broken into multiple threads or even processes, as detailed in Section 4. Simultaneous Localization and Mapping Simultaneous Localization and Mapping (SLAM) is an age-old problem in robotics which still attracts significant interest from the robotics community. Indeed, SLAM enables a robot to build an understanding of its environment over time and is thus foundational to robotics applications (see the left hand side of Figure 2 for a depiction of typical SLAM output). In this work, we select two point cloud-based methods for SLAM to study: Gmapping [36] and Lidar Odometry and Mapping (LOAM) [37], which are commonly applied in robotics. We select two versions of SLAM to demonstrate two related concepts: (1) computational kernels for a given task can vary significantly in performance; and (2) different kernel versions may be used interchangeably based on computational/power limitations and robot objectives. As one of the most popular computational kernels for outdoor SLAM, Gmapping consists of the Rao-Blackwellized particle filer to learn grid maps from laser range data. This approach uses a particle filter in which each particle carries an individual map of the environment. It is one of the pioneering approaches to SLAM [4], with a tractable computational complexity of O(N M ), where N is the number of particles and M is the size of the corresponding grid map. We also profile LOAM which produces real-time 3D maps from Lidar point cloud data, which are generally of a higher quality than Gmapping. LOAM has four parallel process nodes running simultaneously to update the mapping and odometry data and produces updates on maps in real-time (see Figure 4). Remark 1 It is important to note that the generation of maps and localization solutions is highly dependent on the features of the environment. We will demonstrate that this phenomenon can be seen directly in the computational profiling for SLAM, where we contrast between indoor and outdoor environments in terms of execution times and CPU usage. Path Planning Path planning is another critical computational kernel for autonomous robots (see Figure 2 left for a depiction of typical planner output). In this paper, we focus on planning methods based on Rapidly-Exploring Random Trees (RRTs) [38], as they are very popular and there are numerous RRT variants for various application scenarios. We profile a 2D RRT method which utilizes R-tree [39] to improve performance by avoiding point-wise collision and distance checking. When profiling the RRT kernel, we generate an environment with random obstacles to represent dynamic and unknown environments that the robot is presumed to traverse. Importantly, the execution time of RRT will depend on the size of the environment, the number of planning iterations allowed, the goal location to reach, obstacles, and the step or sampling size of the planner. In this work, we investigate the effect of step/sampling size (R), environment size, and obstacle density on RRT computational requirements. Task Allocation Single robots and multi-robot teams often have multiple high-level tasks to complete over a mission, with the tasks potentially streaming in or changing in nature over time (see the left hand side of Figure 2 for a depiction of task allocation). To model the autonomous decision-making necessary to allocate robot resources to tasks optimally, we profile task allocation problems which are difficult combinatorial optimization problems with resource constraints. In this paper, we benchmark and analyze a novel task allocation problem that aims to allocate a set of robots' functionalities to a mission such that certain functionality combinations are satisfied. A combinatorial optimization problem with matroid constraints is solved using a greedy algorithm, yielding a solution with bounded suboptimality [40,12]. The parameters of interest for this task allocation kernel are the robot cardinality, N , requirement cardinality R, and functionality cardinality F . By changing these parameters, we can alter the complexity of the task allocation problem and profile its computational requirements. Computer Vision Kernels To support a robot navigating an environment safely while avoiding dynamically moving objects (such as pedestrians), we study two additional kernels for computer vision tasks (see Figure 2 left for a depiction of the typical input/output). Specifically, in this work, we focus on 1) optical flow (for estimating dense motion between consecutive frames) and 2) monocular depth estimation (for predicting dense scene depth). In each kernel, we choose the state-of-the-art models of PWC-Net [16] for optical flow and MiDaSv2 [17] for depth estimation. For both models, we use the publicly available pre-trained models in PyTorch for our empirical analysis. Empirical Analysis Mobile Robot Platform As our motivation in this work is to deploy computation-aware robots to solve real-world problems, we aimed to exploit real-world data for profiling where possible. Sensor data collection for several computational kernels was conducted using the Jackal Unmanned Ground Vehicle (UGV) platform from Clearpath robotics [41]. The UGV was equipped with the Velodyne VLP-16 LIDAR and a Go-Pro Fusion 360 camera, with the sensors mounted as seen in Figure 5. Computational Platforms We consider three representative hardware platforms for robots. Their technical specifications are summarized in Table 1. Our first platform is the NVIDIA Jetson AGX Xavier [21] which has significant computational capacity for autonomous robots in a compact form factor. Additionally, the Xavier has a heterogeneous architecture based on a System-on-Chip (SoC) that supports both CPU and GPU computation. The second computational platform we consider is the NVIDIA Jetson Xavier NX [42] which is the state-of-the-art embedded platform from NVIDIA that is suited for drones and smaller robots with strict resource constraints. Both Xavier devices have shared memory between the CPU and GPU which along with their architecture makes them extremely power efficient. Finally, we profile our kernels on the Zotac Mini PC system with specifications similar to a desktop computer, but with a form factor that allows integration into a larger robot with high power availability. From the software perspective, the Linux environment is the most popular open source operating system used for robotics worldwide. Linux coupled with ROS is the go-to system for robotics researchers. Thus, we profile our platforms/kernels in Ubuntu 18.04 with ROS middleware (ROS Melodic). The kernel version of Linux for the Xavier platforms is 4.9.140-tegra which is the latest ARM-compatible version of the vanilla Linux kernel. For the Zotac platform, we use the Linux kernel 5.4. Profiling Data Acquisition For CPU execution time, utilization, and shared memory usage, we primarily use psutil [43]. Psutil is a python tool that implements many basic Linux command-line operations such as ps and uptime, among others. However, for the task allocation kernel, CPU execution time is calculated using the times Linux kernel api, CPU utilization is determined using mpstat, and the vmstat command provides memory usage. The process and threads details for kernels were captured using NVIDIA Nsight Systems whereas GPU statistics were recorded using the NVIDIA Profiler tool [44] for NVIDIA Xaviers. For the Zotac platform, GPU memory usage, GPU utilization, and GPU power were recorded using the gpustat command which is based on the Nvidia-smi interface. For power measurements, on-board INA power monitors in the NVIDIA platforms are utilized which allow power monitoring at runtime with an accuracy of 95%. For the Zotac platform, we calculate CPU power usage with the energyusage python command which calculates the power via the RAPL (Running Average Power Limit) interfaces found on Intel processors [45]. The objective of this paper is to gain insight into the computational aspects of crucial robotic kernels. Therefore, we use these well known and rigorous tools developed over a long time and reputation, so that our results are not questionable. Developing custom tools is not in the scope of this paper, but can certainly be pursued when the optimization problem is narrowly defined such as in RESCH [46]. Profiling Results In this section, we present our profiling results with only limited initial insights. The implications and usefulness of these results are discussed in detail in Section 5. As a reference, we provide a recorded video of the data collection experimentation at https://youtu.be/LvapXWqEAqU. SLAM The LOAM SLAM benchmark was carried out with the real-world data collected as described previously. For the data presented in this paper, the same loop in a large, mostly outdoor area was traversed three times as is seen in Figure 6. During this traversal, the first completion of the loop did not achieve loop closure in the SLAM process (feature reconciliation that improves maps as more data is collected), whereas the second and third loop completion did see loop closure. As a result, it is visible that the execution time decreases rapidly between the second and third loop which is visible in the computational profile ( Figure 7). Moreover, there was an average of 30% drop in execution time when passing through indoor confined spaces such as a tunnel. Finally, in Figure 7a we see that the RAM utilization increases with the traversal of the environment. Based on the above discussion, we can deduce that repeated environment traversal, environment complexity (indoor vs. outdoor), and mission length correlate strongly with the computational requirements for SLAM. Remark 2 Regarding the explanation of the memory consumption, it is highly unlikely that it has a memory leak, at least from primary inspection of the code as memory allocation was not used in it. LOAM SLAM [37] is a very well cited and high quality LIDAR based 3D SLAM technique which has high load on the processor. It has 16 threads (4x4) and 4 dominant threads from the four processes. Moreover, the scope of this paper is not to look into particular coding issues but to identify computational and environmental aspects in robotics kernels. As we do not see this attribute in the 2D SLAM discussed later, the memory buildup is likely a characteristic of 3D high definition LOAM SLAM. However, further micro-architectural analysis into this kernel would be necessary to pin down this attribute and possibly find a remedy for this. Table 2 summarizes the different processes, thread counts, and percentages of execution times for the LOAM kernel in the Xavier platform. To specify more clearly, this data in this table are generated using the nvidia profiler and its definitions are a little different than the ones presented in section II. Here, The % utilization of dominant thread denotes % utilization of a single core, the % of Execution time denotes the percentage of the particular process's execution time compared to the total execution times of these four process that make up the LOAM kernel and the average execution time is just the average of that process. SLAM is highly dependent on sensor data and thus the ROS middleware is used to enable smooth message passing between the different SLAM processes. The Laser Mapping and Odometry processes contribute towards the map building for LOAM and thus require the most resources combined. On the other hand, the Scan Registration process only contributes towards point cloud data processing which is a constant cycle and therefore has very little variation in data. Another vital aspect is that all of these processes have one dominant thread for scheduling purposes as the other threads can be regarded as overhead and therefore makes it convenient from a scheduling perspective. From the tracing profiles in Nsight Systems, it is visible that these four processes run in parallel in different CPU cores while the ROS middleware ensures the message passing between them. Because of ROS's lack of Quality of service (QoS) in communication and real-time support, some momentary drops in packets are observed when the data of CPU performance were collected during the kernel's operation. Figures 8 and 9 depict the LOAM performance metrics in the Xavier NX and Zotac platforms, respectively. It is visible that the execution times are mostly the same for Xavier NX but the CPU and memory utilization are significantly higher which is expected behavior. However, in the Zotac platform the execution times are significantly lower than the Xavier platforms which is also expected considering its higher frequency and higher CPU and memory capacity. Moving to the Gmapping SLAM kernel, which also uses our real-world data, it is evident from the profile in Figure 10 that the variation is far less when compared to LOAM and it has a lower utilization and execution time for the Xavier AGX platform. However, due to the powerful computational capabilities of Zotac, we don't see this difference between LOAM and Gmapping in Zotac. The RAM usage of this kernel is also much more stable. However, the quality of maps produced by Gmapping is significantly lower than LOAM, which is a critical trade off. Indeed, in the authors' experience the maps produced by Gmapping are often insufficient for robot navigation, especially in large and/or complex environments. The summary of processes and threads of Gmapping in Table 3, (same interpretation as Table 2) also conveys the relative simplicity of Gmapping, being mostly a single process kernel as the other processes are just static transform broadcasters and contribute very little to the computational load. For the other two platforms, the behaviour and utilization of Gmapping SLAM is very similar and therefore we omit the detailed distribution plots for brevity. A fundamental insight is that the Gmapping is a much simplistic version of SLAM and also 2D. Therefore it is expected that it has very small variation across different platforms but prior to this study we have not found any findings that proves it with real world dataset. This finding is quite useful for users when planning for deployment. To conclude, Figure 11 presents the comparison of execution times for the two SLAM kernels over our three computational platforms. RRT-based Path Planning The RRT path planner was executed as a standalone python3 code using the libraries from [47]. As a standalone python code, only a single thread was monitored in all the platforms. For an RRT planner, as specified in Section III, step/sampling size (R), environment size, and obstacle density were varied for each of the computational platforms. The resulting computational profiles in Figure 12 show the effect of these parameters on the execution times. Typically, the sampling or step size must be determined arbitrarily based on prior experience in related use cases. Generally, a lower sampling size will result in higher execution times but a larger sampling size can also generate infeasible paths as small environment features (such as obstacles) can be overlooked. Moreover, as the density of obstacles and size of environment increase, the execution times also increase. A lower sampling size in a large area, such as a square kilometer, is not desirable to find paths in a timely manner. However, with the obstacle density increasing, higher sampling sizes also tend to generate infeasible paths. Thus, knowledge of the average and worst case execution times in such scenarios are a must to plan paths in a timely manner for autonomous robots in complex and/or dynamic environments. Obstacle Density is a crucial environmental parameter for path planning and its relation to response times are rarely studied in literature. Fig. 11 Boxplots for execution times for SLAM kernels for different hardware platforms. The x-axis denotes the obstacles density that is the density of obstacles per sqm Task Allocation The task allocation kernel is written in C code by the authors following the algorithms from the task allocation problem in [40]. As a standalone C code, this kernel was also monitored as a single thread in all the platforms. As stated in Section III, we vary the robot cardinality, N , the requirement cardinality R, and functionality cardinality F to generate a set of task allocation problems with varying difficulty. Specifically, our test cases are built by setting N × average(F ) = F and R = F/2 . We choose N from 5 to 100. Figure 13 depicts the execution time variations due to these parameter changes. In addition, Figure 14 validates the polynomial complexity of the kernel, originally proved in [40]. The useful insight for a single threaded kernel is quite straightforward from this data. The More power and frequency for each cores the platform has, the faster is completes the tasks. That's why we see the execution times get lower from Nx to AGX to Zotac. Optical Flow and Monocular Depth The CV kernels were not operated with a particular activation period but on a typical while loop with no timing constraints. Therefore, we also provide the turnaround time information for these kernels. Both kernels are also run standalone utilizing libraries from the Github repositories [17,16] and take as input vision data from our real-world dataset. Figures 15 and 16 show the computation and power profiles for the monocular depth estimation kernel in maximum and low power modes respectively. The first insight is the high usage of the shared CPU/GPU memory on the Xavier platform, which is highly utilized and almost completely consumed for Pwc-Net. The high utilization of RAM is due to fact that memory is shared between the CPU and GPU via a SoC in the Xavier platforms and these kernels are using GPU intensive PyTorch libraries. Also there is a significant increase in the execution times and utilization of CPU in low power mode. The interesting aspect from Figure 15 and 16 is that there are spikes of CPU usage in both of the power modes. This behavior is caused by the insertion of two out of context images in the dataset of our environment. The regular images that represent the steady parts of the graphs are from our own collected data in which the environment is gradually changing as the robot moves. However, when it faces an out-of-context or new environmental feature set, it spends a lot of time processing the change in the CPU. There is also evidence of a CPU bottleneck issue after analyzing this portion of execution in the NVIDIA Nsight system and thus there is an opportunity of optimizing the kernel for usage on embedded heterogeneous platforms like Xavier. The optical flow estimation kernel is conducted in a similar manner as the depth estimation kernel without the insertion of out-of-context images. It is fed with a set of sequential images from the same path the SLAM kernels were conducted on. The profiling information is visualized in Figure 17 and 18. Here, Fig. 12 Boxplots of execution or cpu user times of RRT path planning experiments there is a slight variation in computational loads, which is attributed to changing scenarios in the dataset. There is a shift to CPU computation from GPU during those instances, which contributes to the increase in CPU execution time which is more evident in the low power mode. Moreover, the GPU resources are also utilized at a minimum level which is evident from the utilization summary of several GPU parameters. The achieved Occupancy for MiDaS and Pwc-Net are respectively 25.28% and 20.77% and all the functional unit utilization are low or idle (0-2) when analyzed using the Nvidia Profiler. The Warp Execution Efficiency for MiDaS and Pwc-Net were observed to be 96 and 95% whereas the Shared Utilization were low or mid level (0-6) for both cases. These data were observed for 61 and 35 kernel threads for MiDaS and Pwc-Net respectively and given as a summary or average here. For the Zotac platform, where we have to measure the GPU parameters using Nvidia-smi interface, we observe low utilization of the GPU resources. For Midas in Zotac, the GPU utilization is 35.5 with GPU memory utilization at 15.35775 % and a much high GPU We depict the distribution of the different parameters over the dataset of both of these CV kernels in Figures 15, 16, 17 and 18 for the Xavier AGX platform only. For the other platforms, we observe similar spikes in computation for the out-of-context images for depth estimation. For the optical flow estimation, the execution times are also very similar. However, the resource utilization for the Zotac is quite low compared to the Xaviers and the data as well and cause of this is discussed in the next section. We again omit the similar figures of distribution of execution time and resource utilization for Xavier NX and Zotac for brevity. Finally we present in Figure 19 the boxplots for the computer vision kernels across our computational platforms to represent these comparisons. It is visible that in the Zotac platform, the execution times are higher whereas the overall response times are proportionally lower. This is attributed to the fact that the higher CPU core count of Zotac yields more parallelism of the vision kernels in Zotac compared to the Xaviers and thus overcomes the CPU bottleneck issue in the Xaviers to some extent. Remark 3 The data collection experimentation and also corresponding images related to SLAM and MiDaS are depicted in the video https://youtu.be/LvapXWqEAqU. This gives a clearer picture of the environment the experiments were conducted in, including both outdoor and indoor scenarios. Discussions Our profiling efforts yield several significant insights that we argue are fundamental to our goal of guaranteeing the timeliness and correctness of complex autonomous behaviors in robotic systems. First, from all the data presented earlier as well as the summary in Table 4 it is quite evident that the operation of these kernels simultaneously, which is necessary for complex robotic applications, will degrade performance and yield missed deadlines (which are equal to the activation periods), especially in low power scenarios. While the overall utilization is not merely a summation of the individual utilizations, the surprising changes in response times in certain scenario can generate these deadline misses. A change in activation periods and appropriate scheduling as well as prior knowledge of these changes in response/execution times will be necessary to operate a combination of vision and CPU-bound kernels, which if done naively may yield catastrophic impacts on performance and safety of autonomous behaviors. This motivates our ongoing efforts to develop a dynamic scheduling parameter adjustment for autonomous robots based on the data presented in this work. Fig 20 refers to an example application using the computational kernels that can be scheduled in a Directed Acyclic Graph (DAG) scheduling framework such as implemented in ROSCH [48]. In any such DAG scheduler, the comprehensive knowledge of individual execution times are a must for designing a dynamic computationally aware scheduler [49,50]. Next, the LOAM kernel shows a spatial correlation in the event of loop closure and revisiting environmental cues with a drop of the execution time and other related parameters. However, the memory usage continues growing with the traversal of the environment. In about 30 minutes of traversal, the RAM usage grows from 6% to 12% in the Xavier AGX platform, with similar growth in Xavier NX and Zotac. On the other hand, the Gmapping SLAM kernel shows a quite stable behavior, but with significantly lower map quality. Interestingly, in low power mode the LOAM kernel is inoperable at the default activation period of 0.1 seconds as 100% utilization is seen, resulting in missed execution deadlines. These SLAM insights allow for unique opportunities such as dynamic scheduling parameter adjustments that anticipate loop closure events, or robot mission planners that balance location revisitation with the availability of computational resources. Shifting to the task allocation and RRT path planning kernels, we see an even greater variation in both average and WCET as evident from the boxplots in Figures 12 and 13, with these effects being directly dependent on a robot's operational environment and/or objectives. Once again such insights open up dynamic resource management opportunities, such as controlling the quality of planned paths based on computational resources and safety margins, or mission planners that anticipate task loads in an environment and act to mitigate impacts on kernel scheduling. The single threaded processes such as RRT and TA can have differences in execution times between the AGX and NX platforms because the NX is more conservative in power consumption than the AGX. This can affect the execution of kernels specially for higher periods of executions. That is evident in Fig. 12. Moreover, we can utilize the GPU for optimization of RRT and TA by running the critical sections on the GPU. Focusing now on the CV kernels, significant variation in resource utilization is observed with changing Xavier power modes. For example, the execution time increases 3 fold both in the Midas depth estimation and optical flow estimation. The vision kernels also show a classic example of GPU starvation and CPU bottleneck in Xavier hardware. Parts of the algorithm are sequential with no parallel operation implemented, in particular, portions dedicated to preprocessing the images before calling the PyTorch libraries. Most of the GPU resources are underutilized while occupying a huge amount of shared memory space. This usage of a huge amount of memory calls for a conflict resolution in CPU and GPU memory access, which is investigated in very recent literature [32], [33], [34], and can be put into action using the data from this paper. In the Zotac system, which has a discrete GPU unlike the Xaviers, the CV kernels show much better performance and lower CPU bottleneck issue, however the Zotac platform is not viable for aerial robots. This final point indicates that for vision kernels there may be a strong case for offloading computation to more capable agents (e.g., from an aerial robot to a ground robot or the cloud). Although it bridges the gap between theoretical and practical usage of robotic computational kernels, some of the results in this paper are quite predictable. However, the Pwc-Net results in fig. 19 depicts quite the unpredictable behaviour. It reflects that only using powerful hardware with discrete GPU cannot yield better response times for all the pytorch enabled computation and some DNN implementations are more compatible to the embedded hardware like Xaviers than the others. It however uses much less RAM in Zotac due to its discrete GPU memory. A final insight from the summary in Table 4 is that the memory conflict for concurrent operation of the kernels may also influence the schedulability and predictability of operation similar to the typical CPU resource management schemes. Remark 4 An important note is the presence of noise in our data, stemming primarily from our data sampling mechanism and file writing operations for profiling result collection. The use of an average differential mechanism keeps the variance of that noise present in data. However, we have also profiled the noise and account for its average value. Specifically, we can quantify that the error margin of our presented data is 2.36% for CPU utilization, 0.015 % for RAM utilization and 6.9% for CPU power. The execution times are devoid of this error. Finally we also derive additional insights into scheduling, optimization and resource management for future deployment of these kernels simultaneously. a) GPU-intensive kernels: The DNN based algorithms have various ways of optimization depending on the hardware such as utilizing the vision accelerators in the Xaviers as well as generalized approximation techniques such as matrix factorization, compression. Moreover, GPU intensive kernels also have environmental correlation which can be modeled and further studied with empirical data such as the other kernels in this paper. b) CPU-intensive kernels:The analysis in this paper utilizes the default CFS scheduler of the Linux kernel. Specifically, the parameters which could be dynamically adjusted for scheduling these computational kernels via Linux schedulers are priorities, scheduler tick, vruntime, deadlines and the scheduling mechanism itself. Therefore, our followup work is on analyzing real time schedulers such as FIFO/RR/EDF and in that case we will also define hyperperiod of tasksets and target latencies and also analyse the effects of these schedulers on the same para,meters as this paper but with a combinatorial implementation of the computational kernels. Such experiments will generate a learned computational map which can be used for our ultimate objective of computation aware autonomy, as specified in Fig 1 There are several scheduling algorithms that can be utilized such as deadline based hard real time, soft real time and novel environment aware schedulers. Usage of RT-patches and RTOS, CPU isolation, memory locking are also some of the possible options. The applicability of these will depend heavily on the mission objectives. c) Combining a and b: Moreover, the Jetson Xavier architecture shares the RAM between the CPU and GPU, therefore the GPU intensive kernels have a much higher RAM utilization. This could pose an issue when used combinatorially with other kernels and some literature which address these memory access issues in heterogeneous architectures are cited in the discussions [23][24][25]. Conclusion Autonomous systems cannot treat computation as a black box anymore as has been done in most of the systems deployed in present time. Our study provides a detailed preliminary insight into the means of achieving the timeliness guarantee. This paper aims to gain deeper insights into the robotics computational kernels in real environments. To limit the experiments' extent to meaningful and interpretable content, we choose a set of representative computational kernels needed for broader autonomy. Most of them are from existing libraries except the Task Allocation kernel (which was developed by the authors due to unavailability of C code implementation). We rigorously analyze the effects of environmental parameters such as depth and motion features, complexities of indoor/outdoor, size and obstacle densities. While the results were mostly not surprising, our quantified profiles fill the gap of current literature for timely and safe autonomy. The work on environmental correlation of SLAM and RRT path planning kernels are of vital importance for practical applications. Moreover, for the application of a DAG scheduling comprised of these computational kernels, this individual execution times measurements are considered as input data for he schedulers. We also use existing and well renowned tools such as psutil so that the methodologies are not questionable. We have not changed the many different features on the hardware platforms, but still from the results, it can be deduced that the number of cores, maximum power consumption, operating frequency and also architecture difference such as integrated or discrete GPU all have an effect on the performance. Fig. 1 1Framework for computationally aware autonomy for timely and resilient systems. - Achieved Occupancy: Ratio of the average active warps per active cycle to the maximum number of warps supported on a GPU. xyz FU Utilization: The utilization level of the GPU xyz functional units on a scale of 0 to 10. Here xyz depends on the hardware platforms. Typically xyz = alu, cf, special etc. Fig. 3 3CPU scheduling parameters and operation. -Warp Execution Efficiency: Ratio of the average active threads per warp to the maximum number of threads per warp supported on a GPU. -Shared Utilization: The utilization level of shared memory relative to peak utilization on a scale of 0 to 10. -GPU Utilization: Percent of time over the past sample period during which one or more kernels was executing on the GPU. (Only for Zotac platform using Nvidia-smi interface) -GPU Memory Usage: Total memory allocated by active contexts. (Only for Zotac platform using Nvidiasmi interface). The GPU memory utilization can be derived by scaling this value (%) to the GPU memory capacity. Fig. 4 4Block Diagram of Loam Software System[37]. Fig. 5 5Platform for data collection. Velodyne LIDAR and Go-Pro Fusion 360 camera are mounted on a Jackal UGV. Fig. 6 6Geographical locations of the data points forFigure 7. Depicted is the execution times for LOAM in the Xavier AGX platform. The colors indicate the execution time required at a particular location (red is high, green is low). Fig. 7 7Computational profile data of LOAM SLAM in Xavier AGX platform. (a) Execution time which includes User Time and System Times for an activation period of 0.1 sec. In the Right axis is RAM utilization as a percentage of used RAM to the total available RAM. (b) CPU Utilization and Power measurements. (c) Indoor and outdoor environment corresponding to data points circled in (a) Fig. 8 8Computational profile data of LOAM SLAM in Xavier NX. (a) Execution time which includes User Time and System Times for an activation period of 0.1 sec. (b) CPU Utilization and Power measurements. Fig. 9 9Computational profile data of LOAM SLAM in Zotac. (a) Execution time which includes User Time and System Times for an activation period of 0.1 sec. (b) CPU Utilization and Power measurements. Fig. 10 10Computational profile data of Gmapping SLAM. (a) Execution time (User Time and System Times) for an activation period of 0.1 sec and corresponding RAM utilization. (b) CPU Utilization and Power measurements. Fig. 13 13Boxplots for execution or cpu user times of multi robot task allocation experiments power usage at 72 W. For Pwc-Net in Zotac, the GPU utilization is 16.4 %, memory utilization at 16.9 %, and GPU power usage at 46 W. Fig. 14 14Complexity of task Allocation kernels Fig. 15 15Midas depth estimate kernel profile in Xavier AGX. (left) Turnaround, Execution (User and System) and RAM utilization of Midas. (right) CPU utilization and CPU, GPU, SoC power consumption profiles. Fig. 16 16Midas depth estimate kernel profile at low (15W) power mode of Xavier AGX. (left) Turnaround, Execution (User and System) and RAM utilization of Midas. (right) CPU utilization and CPU, GPU, SoC power consumption profiles. Fig. 17 17Pwc-net optical flow estimation kernel profile in Xavier AGX. (left) Turnaround, Execution (User and System) and RAM utilization of Midas. (right) CPU utilization and CPU, GPU, SoC power consumption profiles. Fig. 18 18Pwc-net optical flow estimation kernel profile at low (15W) power mode in Xavier AGX. (left) Turnaround, Execution (User and System) and RAM utilization of Midas. (right) CPU utilization and CPU, GPU, SoC power consumption profiles. Fig. 19(a) Execution times. (b) Response/Turnaround times for vision kernels. Max=maximum power. Fig. 20 20An example DAG comprised of the computational kernels presented in this paper. Table 1 1Computational Platforms specifications.Device CPU GPU Memory Power Range Dimension Architecture cores Maximum Frequency cache Architecture Cores (mm) Nvidia Jetson ARM v8.2 64-bit 8 2.26GHz 8MB L2 Volta 512 16GB 10W -72W 105×105×65 Xavier AGX + 4MB L3 (Integrated) Nvidia Jetson Nvidia Carmel 6 2.26GHz 6MB L2 Volta 384 8GB 10W -15W 103×90.5×34.66 Xavier NX ARM v8.2 64-bit + 4MB L3 (Integrated) Zotac Intel Core 6×2 4.5Ghz 256KB L2 Turing 2304 32GB 330W 210×203×62.2 EN72070V i7-9750H + 12MB L3 (Discrete, 8GB) Table 2 2Process/Thread summary for LOAM in Xavier AGX platform.Process LaserMapping LaserOdometry Transform Multi-Scan Maintenance Registration Number of Threads 4 4 4 4 % Utilization of dominant thread 98.97 97.72 60.7 98.07 % of Execution Time 32.8 28.8 0.17 38.13 Avg. Execution Time 0.08448 0.0736 0.0003 0.0947 Variance in Exec Time 0.00106719 0.0007106 2.91E-6 0.000108 Table 3 3Process and Thread's profiling summary of SLAM Gmapping in Xavier AGX platformProcess Gmapping Static Transform Base Link-Laser Odom Map main node Publisher Transform Broadcaster Number of Threads 4 4 4 4 Number of Threads 8 4 4 4 % Util. of dominant thread 32.82 87.87 87.96 88.27 % of Execution Time 97.5 0.8 0.8 0.7 Avg. Execution Time 0.10231 0.00156 0.0014 0.00094 Variance in Exec Time 0.00385 0.1.5E-5 1.4E-5 9.5E-6 Table 4 4Summary of Benchmarks and the Profiling ResultsPlatform Kernels Power CPU RAM Mode Util (%) Util(%) LOAM SLAM Max 32 5-20 Nvidia Gmapping SLAM Max 22 12 Xavier RRT Path planning Max 13.2 17.1 AGX Task Allocation Max 14 17.7 MiDaS depth est. Max 12.2 56 15W 26 55 Pwc-Net Optical Flow est. Max 13.4 86.7 15W 27.8 40 LOAM SLAM Max 57 10-22 Nvidia Gmapping SLAM Max 31 32 Xavier RRT Path planning Max 18 24.5 NX Task Allocation Max 20 23.5 MiDaS depth est. 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[ "https://github.com/Autoware-AI/autoware.ai", "https://github.com/torvalds/linux", "https://github.com/giampaolo/psutil", "https://github.com/motion-planning/rrt-algorithms.git." ]
[ "Learning Safe, Generalizable Perception-based Hybrid Control with Certificates", "Learning Safe, Generalizable Perception-based Hybrid Control with Certificates" ]
[ "Charles Dawson ", "Bethany Lowenkamp ", "Dylan Goff ", "Chuchu Fan " ]
[]
[]
Many robotic tasks require high-dimensional sensors such as cameras and Lidar to navigate complex environments, but developing certifiably safe feedback controllers around these sensors remains a challenging open problem, particularly when learning is involved. Previous works have proved the safety of perception-feedback controllers by separating the perception and control subsystems and making strong assumptions on the abilities of the perception subsystem. In this work, we introduce a novel learning-enabled perception-feedback hybrid controller, where we use Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) to show the safety and liveness of a full-stack perception-feedback controller. We use neural networks to learn a CBF and CLF for the full-stack system directly in the observation space of the robot, without the need to assume a separate perception-based state estimator. Our hybrid controller, called LOCUS (Learning-enabled Observationfeedback Control Using Switching), can safely navigate unknown environments, consistently reach its goal, and generalizes safely to environments outside of the training dataset. We demonstrate LOCUS in experiments both in simulation and in hardware, where it successfully navigates a changing environment using feedback from a Lidar sensor.
10.1109/lra.2022.3141657
[ "https://arxiv.org/pdf/2201.00932v1.pdf" ]
245,668,998
2201.00932
3645492df4ac4463ddefaa37d055d274a06680e1
Learning Safe, Generalizable Perception-based Hybrid Control with Certificates Charles Dawson Bethany Lowenkamp Dylan Goff Chuchu Fan Learning Safe, Generalizable Perception-based Hybrid Control with Certificates IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2021 1Index Terms-perception-based controlsafe controlcontrol barrier functionscertificate learning Many robotic tasks require high-dimensional sensors such as cameras and Lidar to navigate complex environments, but developing certifiably safe feedback controllers around these sensors remains a challenging open problem, particularly when learning is involved. Previous works have proved the safety of perception-feedback controllers by separating the perception and control subsystems and making strong assumptions on the abilities of the perception subsystem. In this work, we introduce a novel learning-enabled perception-feedback hybrid controller, where we use Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) to show the safety and liveness of a full-stack perception-feedback controller. We use neural networks to learn a CBF and CLF for the full-stack system directly in the observation space of the robot, without the need to assume a separate perception-based state estimator. Our hybrid controller, called LOCUS (Learning-enabled Observationfeedback Control Using Switching), can safely navigate unknown environments, consistently reach its goal, and generalizes safely to environments outside of the training dataset. We demonstrate LOCUS in experiments both in simulation and in hardware, where it successfully navigates a changing environment using feedback from a Lidar sensor. I. INTRODUCTION U SING visual input to control autonomous systems without compromising safety or soundness is a challenging problem in robotics. Traditional methods from control theory provide powerful tools for safety and performance analysis but lack the expressiveness to deal with rich sensing models such as vision or Lidar. On the other hand, learning-based methods have been used successfully on visual-feedback control tasks including autonomous driving [1] and aircraft taxiing [2], but ensuring the safety of these controllers remains an open question. Most existing work analyzing the safety of learned visionbased controllers has been done post hoc, where the controller Manuscript received: September 9, 2021; Revised November 23, 2021; Accepted December 30, 2021. This paper was recommended for publication by Editor Jens Kober upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the NASA University Leadership Initiative (grant #80NSSC20M0163) and the Defense Science and Technology Agency in Singapore, but this article solely reflects the opinions and conclusions of its authors and not any NASA entity, DSTA Singapore, or the Singapore Government. CD is supported by the NSF GRFP under Grant No. 1745302. 1 The robot switches between goal-seeking and exploratory modes, but is guaranteed to maintain safety (using the barrier function) and eventually reach the goal if a safe path exists (using the Lyapunov function). is synthesized and then independently verified. Works such as [3] assess safety through adversarial testing, searching for an input sequence that causes a learned vision-based controller to fail. Others such as [4] learn a generative model to predict observations from states, then conduct a reachability analysis to check the safety of the concatenated generator-controller network. In both cases, controller safety and controller synthesis are treated as two separate issues, rather than using safety considerations to inform the synthesis process. Integrating safety into the control synthesis process has born impressive results in the state-feedback regime, in the form of certificate learning for control. These approaches draw on control theoretic certificates such as Lyapunov functions [5], [6], barrier functions [7], [8], and contraction metrics [9], [10] that prove properties such as stability, forward invariance, and differential tracking stability of dynamical systems. Recent works have demonstrated that learning these certificates alongside a state-feedback control policy can enable provably safe controllers for walking [11], stable flight in turbulent conditions [9], and decentralized multi-agent control [12]. Several works have attempted to extend these approaches to observation-feedback control synthesis [13]- [15], but these works make a number of strong assumptions. For instance, [14] and [15] assume that the robot's state can be inferred with bounded error from a single observation, limiting them to "third-person" camera perspectives where each observation completely captures the state of the robot and its environment. As a result, these approaches do not readily apply when the vision system (either a camera or Lidar scanner) is mounted to the robot and cannot observe the entire environment. Further, these approaches assume that the environment (including the locations of any obstacles) is known prior to deployment, limiting the ability of these controllers to generalize to new environments. We relax these assumptions to permit deployment on a robot in previously-unseen environments, although we retain some assumptions on the sensor model (see II-A). In this work, our main contribution is to combine techniques from machine learning and hybrid control theory to learn safe observation-feedback controllers that generalize to previously-unseen environments. In particular, we combine learned observation-based barrier and Lyapunov functions with a novel certificate-based hybrid control architecture. We provide proofs that our hybrid controller will remain safe and reach its goal without deadlock when provided with valid certificate functions, and we use neural networks to learn certificates that are valid with high probability. To our knowledge, this represents the first attempt to provide safety and goal-reaching guarantees for a perception-feedback controller that considers the full autonomous navigation stack (rather than sand-boxing the perception system with assumptions on error rates and considering the controller separately). We validate our approach, called LOCUS (Learning-enabled Observation-feedback Control Using Switching) both in simulation and in hardware, where we demonstrate a mobile robot navigating an unseen environment with Lidar measurements using our learned hybrid controller. Experiments show that our learned controller can be run 6 times faster than MPC, reaches the goal almost twice as quickly as end-to-end reinforcement learning policies, and maintains a perfect safety rate in previously-unseen environments. II. PROBLEM STATEMENT Our goal is to synthesize an observation-feedback controller for a robot with nonlinear dynamics navigating an unknown environment. The robot perceives its environment using a Lidar sensor, as in Fig. 1, and it must navigate to a goal point while avoiding collisions. In contrast with higher-level approaches that combine SLAM with a global path planner, or robust planning approaches like [16], we restrict our focus to real-time controllers with feedback from local observations. Our observation-feedback controller can be combined with SLAM and planning modules to track waypoints along a path, but it can also be used without those components, or when those components fail, without compromising safety. Formally, we consider a robot with nonlinear discrete-time dynamics x t+1 = f (x t , u t ), where x ∈ X ⊆ R n and u ∈ U ⊆ R m , and observation model o t = o(x t ) ∈ O ⊆ R p (denote by o i t the i-th element of this observation). In particular, we focus on a Lidar observation model that measures the locations where n rays evenly-spaced rays originating at the robot first make contact with an object in the environment. In this model, O = R 2nrays , and the points are expressed in robot-centric coordinates. Given a goal location x g , we seek a control policy u t = π(o t ) that drives x t → x g as t → ∞. Moreover, we require that u t avoids colliding with obstacles in the environment, defined as occurring when min i=0,...,nrays ||o i t || ≤ d c , where d c > 0 is a user-defined margin (i.e. collision occurs when the robot gets too close to any obstacle in the environment). We make a number of assumptions about the robot's dynamics and capabilities, as well as on the structure of the environment. Some of these assumptions are standard, while others are particular to our approach, as outlined in Section III. A. Assumptions 1) Dynamics & State Estimation: Our approach is applicable in two regimes. If a (potentially noisy) state estimate is available, we make no assumptions on the dynamics other than controllability (a standard assumption). If no state estimate is available (other than a measurement of range ρ and bearing φ to the goal point), then we assume that the robot's dynamics are controllable and approximately local, by which we mean the state update has the form x t+1 − x t ≈ f ∆ (u t ). This means that the change in state (expressed in a local frame) can be approximated using only the control input (i.e. without a state estimate). This second case is motivated by our choice of hardware platform (a mobile robot with Dubins car dynamics, which satisfy this assumption and thus do not require any state estimation), but we include both cases for completeness. 2) Observation Model: We assume that the robot observes its environment using a 2D, finite-resolution, finite-range Lidar sensor. This sensor measures the xy location in the robot frame of the first point of contact with an obstacle along each of n rays evenly-spaced rays. If no contact point is detected within maximum range d o , the measurement saturates at that range. This model distinguishes us from less-realistic overhead camera models in [15], but it is important to note that we do not explicitly model sensor noise (we hope to extend our theoretical analysis to cover sensor noise in a future work). 3) Environment: We assume only that there are no "hedgehog" obstacles with spikes thinner than the gap between adjacent Lidar beams and that all obstacles are bounded. III. OVERVIEW To solve this safe navigation problem, we propose the controller architecture in Fig. 2. The core of this controller is a Control Barrier Function (CBF) defined on the space of observations, which enables safe observation-based control by guaranteeing that the robot will not collide with any obstacles. Unlike other CBF-based approaches, which define the CBF in terms of the state-space and require accurate state estimates and knowledge of the environment [7], our approach defines the CBF directly in observation-space. We also use a Control Lyapunov Function (CLF) to guide the robot towards its goal. In an offline phase, we use neural networks to learn the observation-space CBF and CLF. In an online phase, we use the learned CBF and CLF in a hybrid controller that switches between goal-seeking and exploratory behavior to safely navigate an a priori unknown environment. To ensure that the CBF and CLF constraints are respected during the online phase, the controller uses its knowledge of system dynamics to approximate future observations with a onestep horizon, using this "approximate lookahead" to select optimal control actions given the CBF/CLF constraints. Section IV describes an extension of CBF theory to include control from observations, and Section V describes the hybrid goal-seeking/exploratory controller, including the approximate lookahead strategy. Section VI describes our approach to learning observation-space CBFs and CLFs using neural networks. Section VII presents experiments, both in simulation and in hardware, demonstrating the performance of our approach. IV. ENSURING SAFETY AND CONVERGENCE WITH OBSERVATION-SPACE CERTIFICATES As shown in Fig. 2, at the foundation of our approach is a pair of certificate functions: a Control Barrier Function (CBF) that allows the robot to detect and avoid obstacles, and a Control Lyapunov Function (CLF) that guides the robot towards its goal. Existing approaches to designing CBF/CLFbased controllers have defined these certificates as functions of the robot's state [7], [12], [14], [17]; however, this dependence on state means that state-based certificates, particularly CBFs, have difficulty generalizing to new environments. For example, a state-based CBF might encode regions of the state space corresponding to obstacles, but if the obstacles move then the entire CBF is invalidated. A more flexible implementation is to define these certificates as functions of observations (e.g. Lidar returns and the range and bearing to the goal). Instead of encoding which regions of the state space are "inside" obstacles, an observation-based CBF can encode which observations indicate that the robot is unsafe (e.g. when the minimum distance measured via Lidar is less than some threshold). In this paper, we extend traditional state-based CBFs to observation-based CBFs (oCBFs) defined as functions of observations . Formally, oCBFs are scalar functions h : O → R such that for some 0 ≤ α h < 1 o ∈ O saf e =⇒ h(o) ≤ 0 (1) o ∈ O unsaf e =⇒ h(o) ≥ 0 (2) ∀x ∈ X ∃u ∈ U s.t. h [o(f (x t , u))] − α h h[o(x t )] ≤ 0 (3) Similarly, an observation-based CLF (oCLF) can be defined as a scalar function V : O → R such that for some 0 ≤ α V < 1 V (o) ≥ 0; o ∈ O goal ⇐⇒ V (o) = 0 (4) ∀x ∈ X ∃u ∈ U s.t. V [o(f (x t , u))] − α V V [o(x t )] ≤ 0 (5) Of course, these definitions require that the space of observations O is rich enough to both differentiate between safe and unsafe sets O saf e and O unsaf e (e.g. via Lidar measurements) and detect when the robot has reached the goal in O goal (e.g. by measuring range to the goal). In some ways, this extension is trivial, since every oCBF h : O → R (resp. oCLF V : O → R) defines a state-space CBF h • o : X → R (resp. CLF V • o : X → R) , and so oCBFs and oCLFs inherit the safety and convergence guarantees of their state-space equivalents. In particular, as long as a policy selects actions from the set K CBF = {u : h [o(f (x, u))] − α h h[o(x) ] ≤ 0}, the system will remain safe (or, if started in the unsafe region, move to the safe region). Similarly, as long as a policy selects actions from the set K CLF = {u : V [o(f (x, u))] − α h V [o(x)] ≤ 0} , the system will converge to the goal. In addition to inheriting these safety and convergence guarantees, oCBFs and oCLFs also inherit the two main drawbacks of CBFs and CLFs. First, it is very difficult to construct h and V by hand to satisfy conditions (3) and (5). Second, if the intersection K CBF ∩ K CLF empty, then traditional CBF/CLF controllers have to choose between preserving safety or convergence. As such, the novelty of our approach is not in the straightforward extension to observation-based certificates, but in our solutions to these two drawbacks. To address the first drawback, we use neural networks to learn an oCBF and oCLF, allowing our approach to generalize to previouslyunseen environments; to solve the second issue, we use a novel hybrid control architecture to prove the safety and convergence of the resulting learning-enabled controller. We discuss each of these advances in the next two sections; we begin by discussing the hybrid control architecture, then describe how to learn oCBFs and oCLFs for use with that controller. V. HYBRID CBF/CLF CONTROL POLICY oCBF and oCLF certificates guarantee safety and stability by defining the sets of allowable control inputs K CBF and K CLF . As long as the input lies within K CBF the robot will remain safe, and as long as it remains within K CLF it will converge to its goal. However, there is no guarantee that these sets will have a non-empty intersection. We can easily construct "bug-trap" environment like that in Fig. 3 where following the oCLF would lead the robot into an obstacle, violating the oCBF. In these situations, it is common to relax the oCLF condition, allowing the robot to remain safe even if it can no longer reach the goal [11]. Although this relaxation preserves safety, it can lead to deadlock when the robot becomes stuck and unable to reach its goal. To avoid deadlock (i.e. ensure liveness), we propose a hybrid controller with two modes, Goal-seeking and Exploratory, as in Fig. 3. We will first intuitively explain the operation of these two modes, then provide a formal description of each mode and a proof of the safety and liveness of this controller. Both modes enforce the oCBF condition u ∈ K CBF at all times, but they differ in their use of the oCLF. The goal seeking mode enforces the oCLF condition along with the oCBF condition, u ∈ K CBF ∩K CLF , maintaining safety while approaching the goal. When it is no longer feasible to satisfy both the oCLF and oCBF conditions, the robot switches to the exploratory mode, where it ignores the oCLF condition and executes a random walk around a level set of the oCBF (this can be seen as randomly exploring the region near the surface of the obstacle that caused it to become stuck, inspired by classic "right-hand rules" for maze navigation but adapted for a nonlinear dynamical system). The robot switches back to the goal-seeking mode when it encounters a region where the oCLF has decreased below the value where the robot became stuck (indicating that the robot is now closer to the goal than it was when beginning its exploration). This process is visualized in Fig. 3. Note that this controller is substantially different from the vision-based hybrid controller in [15]; ours switches between seeking the goal and exploring an unknown environment, while [15] switches between different controllers hand-designed for navigating a known environment. Formally, the robot's behavior is given by solving an optimal control problem with a one-step horizon (in the goal-seeking mode) or executing a constrained stochastic policy (in the exploratory mode). As opposed to a traditional MPC policy, which must consider a multi-step horizon, the oCBF and oCLF encode all the information needed to ensure long-term safety and convergence into a single step. Crucially, the oCBF and oCLF rely only on local information: Lidar observations and range and bearing relative to the goal. In the goal-seeking mode, the robot solves the optimization: u = arg min u λ 1 ||u|| (6a) s.t. V t+1 − α V V t ≤ 0 (6b) h t+1 − α h h t ≤ 0(6c) Constraint (6b) ensures that the controller progresses towards the origin, while constraint (6c) ensures that the controller remains safe. In practice, we solve an unconstrained analogue of this problem using a penalty method, incorporating the oCLF and oCBF constraints into the objective with a penalty coefficients λ 2 and λ 3 , where λ 1 = 0.01, λ 2 = 1.0, and λ 3 = 10 3 such that λ 3 λ 1 λ 2 . We find that this choice of penalty coefficients results in good performance in practice by effectively filtering out any solutions that do not satisfy the constraints (6b) and (6c). We also note that even if this approach fails to find a feasible solution, the controller can immediately detect this fault, raise an error, and switch to a fail-safe mode. The robot transitions from the goal-seeking mode to the exploratory mode when the oCLF constraint in (6a) becomes infeasible, indicating that the robot is stuck and must explore its environment to find a new path to the goal. Once in the exploratory mode, the robot follows a stochastic policy by discretizing the input space and sampling from the (appropriately normalized) distribution Pr(u) ∝      0 h t+1 − (1 − α h )h t ≥ 0 ∨ |h t+1 − h 0 | ≥ h 1/exp (λ 1 [h t+1 − (1 − α h )h t ] + otherwise + λ 2 [|h t+1 − h 0 | − h ] + + λ 3 v 2(7) where h 0 is the value of the oCBF recorded upon entering the exploratory mode, h is the width of the region around the obstacle that the policy should explore, v is the forward velocity implied by control input u (included to encourage exploration), [x] + = max(x, 0.001x) is the leaky rectified linear unit, and λ i are coefficients specifying the relative importance of each term. In our experiments, λ 1 = λ 2 = 10 3 , and λ 3 = −0.1 to encourage faster exploration. The robot switches from the exploratory to the goal-seeking mode when it reaches a state where the oCLF value decreases below α V V 0 , where V 0 is the value of the oCLF observed upon most recently switching to the exploratory mode. Both the optimization problem (6a) and the stochastic policy (7) are evaluated by discretizing the input space and using the approximate onestep lookahead strategy discussed in Section V-A; in practice, both policies can be evaluated quickly by parallelizing the search over the input space. Since the oCBF constraint is enforced in both the goalseeking and exploratory modes, the hybrid controller will be safe regardless of which mode it is in. As a remark, the discrete-time formulation of oCBFs allows the robot to tolerate changes in its environment, both continuous (moving an obstacle), and discontinuous (adding an obstacle). As long as the change does not flip the sign of h (in essence, as long as the new obstacle position is far enough away that the robot does not instantly become unsafe), the oCBF controller will recover and avoid the obstacle. If the new obstacle makes the robot unsafe, the oCBF can immediately detect that fault. It remains to prove that this controller will eventually reach the goal. We will do this by combining the following lemmas. Lemma V.1. While in the goal-seeking mode, the robot converges to the goal. Proof. This property will hold as long as constraint (6b) holds for the optimal solution of (6a). Since the robot transitions out of the goal-seeking mode when this constraint is violated, this condition must hold whenever the robot is in this mode. Lemma V.2. As long as there exists a path from the robot's current state to the goal such that the oCBF condition holds along that path (i.e. as long as the goal is reachable), the robot will eventually exit the exploratory mode, at which point the value of the oCLF will have decreased by a factor of α V relative to its value entering the exploratory mode. Proof. Let x 0 denote the point where the system enters the exploratory mode, and let h 0 (resp. V 0 ) denote the value of the oCBF (resp. oCLF) observed at x 0 . In the absence of any oCBF, the robot would be able to continue from x 0 along a trajectory towards the goalx =x 1 ,x 2 , . . . such that V t+1 ≤ α V V t at each point. There are two cases: either the goal lies inside the h 0 -superlevel set tangent to x 0 (the goal lies inside an obstacle represented by the oCBF) or it does not. In the former case, no safe path exists to the goal (violating the assumption of this lemma). In the later case, since we assume contours of the oCBF to be closed,x will eventually leave the h 0 -superlevel set bordered by x 0 . As a result, there exists a point along this hypothetical oCLFonly trajectoryx T that lies near the h 0 -level set of the oCBF and V (x T ) ≤ α V V (x 0 ). Since the system is assumed to be controllable and contours of the oCBF closed, the stochastic policy in (7) will eventually explore all states in the region around the oCBF h 0 -level set. Thus, the stochastic exploration policy will eventually approachx T (after circling around the h 0 -level set), at which point the hybrid controller will switch to the goal-seeking mode. Theorem V.3. The hybrid goal-seeking/exploration controller in Fig. 3 will eventually reach the goal. Proof. Denote by [t i , T i ] the period of time the robot spends in the goal-reaching mode on the i-th time it enters that mode, and consider the mode-switching behavior of the robot over an infinitely long horizon. Since Lemma V.2 shows that the system will always exit the exploratory mode, either there are infinitely many episodes in which the robot is in the goal-seeking mode (i.e. i = 0, 1, . . . , ∞), or the robot eventually reaches the goal-seeking mode and stays there (i.e. i = 0, 1, . . . , N g and T Ng = ∞). We present the proof for the first case; the second case follows similarly. Consider the trace of oCLF values V (t 0 ), . . . , V (T 0 ), V (t 1 ), . . . , V (T i ), . . . , V (t i+1 ), . . . (8) and let t denote an index into this sequence (i.e. the cumulative time spent in the goal-reaching mode). We will show that this sequence is upper-bounded by the geometric sequence α t V V (t 0 ). Within any episode [t i , T i ], the sequence V (t i ), . . . , V (T i ) is upper-bounded by V (t) ≤ α (t−ti) V V (t i ) by the properties of the oCLF. Further, by Lemma V.2 we have the relation V (t i+1 ) ≤ α i V (T i ). As a result, we see that each term of the concatenated sequence (8) obeys V (t + 1) ≤ α t V V (t ) . As a result, as t → ∞, V (t ) → 0. By the properties of the oCLF, this implies that the robot's state approaches the goal as the cumulative time spent in the goal-reaching state increases (regardless of intervening mode switches). Since Lemma V.2 implies that cumulative goalreaching time goes to infinity with cumulative overall time (and our system operates in discrete time), this completes the proof that our hybrid controller converges to the goal. A. Approximate one-step lookahead The control policies (6a) and (7) require a prediction of h t+1 and V t+1 after executing a possible control action u. In turn, predicting h t+1 and V t+1 requires predicting the future observations o t+1 and future range and bearing (ρ t+1 , φ t+1 ) to the goal. To generate these predictions, we use prior knowledge of Fig. 4. Future observations can be approximated by translating and rotating past observations. By considering multiple possible futures, the robot can select an action according to one-step lookahead policies like (6a) and (7). the robot's dynamics to predict 1 ∆x = x t+1 −x t . This change in state implies a change in the robot's position and orientation in the 2D workspace, defining a transformation T u ∈ SE(2), describing the future position in the robot's current local frame. This transformation can be used to predict future observations, subject to errors introduced by discontinuities in the environment: o i t+1 = T −1 u o i t (Lidar update) (9) g t+1 = T −1 u ρ t cos φ sin φ T (Goal update) (10) ρ t+1 = ||g t+1 ||; φ t+1 = ∠g t+1 (Range and bearing) (11) The predicted o t+1 , ρ t+1 , and φ t+1 can be used to evaluate h t+1 and V t+1 in the goal-seeking and exploratory control policies. Of course, the approximate nature of this update means that the safety and convergence properties of the oCBF and oCLF are no longer guaranteed to apply, but this issue can be mitigated in practice by adding a margin to the oCBF/oCLF conditions (e.g. [14]). Following our discussion of how to learn an oCBF and oCLF in the next section, we will use empirical results from simulation and hardware to demonstrate that this hybrid lookahead controller exhibits safe and convergent behavior in a range of environments, despite its use of approximations. V t+1 −(1−α V )V t ≤ −γ V in (6a) with γ V ≥ 0, as in VI. LEARNING OBSERVATION-SPACE CBFS Recent works have successfully applied neural networks to learn state-space CBFs [8], [12] and CLFs [18]- [21]. These approaches sample points from the state space, label those points as either safe or unsafe, and then train a neural network to minimize the violation of the state-dependent equivalents of conditions (1)-(3) (for CBFs) and (4)-(5) (for CLFs) on all points on the training set. To evaluate conditions (3) and (5), these approaches either use a fixed control policy [8], [18], [19] or learn a single control policy [12], [20], [21]. These approaches are useful when the safe and unsafe regions of the state space are well-defined and static, but they essentially memorize which regions of the state space are safe and which are unsafe. As a result, they do not generalize when the safe/unsafe regions change, for example when moving between environments with different arrangements of obstacles. There are two key insights in our approach to learning oCBFs and oCLFs, as contrasted with prior approaches to learning CBFs and CLFs. First, by defining these certificates in terms of observations allows the learned certificates -especially the oCBF -to generalize much more easily to new environments. Second, we train our oCBF and oCLF to not simply to satisfy conditions (3) and (5) for a single example controller but to ensure the feasibility of the exploratory control policy (6a). This task-specific learning approach allows us to learn an oCLF and oCBF that are directly relevant to the task at hand, as well as improve generalization by not committing to a single control policy. In our approach, the oCLF is defined as V (ρ, sin φ, cos φ) = V ω (ρ, sin φ, cos φ) + ρ 2 + (1 − cos φ)/2, where V ω is a neural network with parameters ω (two hidden layers, 48 ReLU units each, one output dimension), ρ and φ are the range and heading to the goal in the local frame, respectively, and the last term encodes the prior belief that the oCLF should correlate with distance from the goal. Because the Lidar observations o are symmetric under permutation, we must adopt a different structure for learning an oCBF that reflects this symmetry. Taking inspiration from [12], we define a permutation-invariant encoder e(o) = max i e θ (o i ), which passes every point in the Lidar observation through an neural network e θ with parameters θ (two hidden layers, 48 ReLU units each, 48 output dimensions). e θ lifts each two-dimensional point to a 48-dimensional space, and we then take the element-wise maximum among all observed points. We then construct the oCBF as h(o) = h σ (e(o)) − min i ||o i || + d c , where h σ is a neural network with parameters σ (two hidden layers, 48 ReLU units each, one output dimension) and the minimum is taken over all points in the Lidar observation o. The final two distance terms impose the prior belief that the oCBF should correlate with the distance to the nearest obstacle, and we use an additional learned term to ensure that h satisfies (3). To train these neural networks, we sample N points uniformly from the state space, use a simulated Lidar model to compute the observation o at each point, then use stochastic gradient descent to minimize the loss L = 1 N N i=0 (a 1 ReLU( h + h(o i ))1 saf e +a 2 ReLU( h − h(o i ))1 unsaf e + a 3 L g ) , where a 1 = a 2 = 100, a 3 = 1, 1 saf e and 1 unsaf e are indicator functions for the safe and unsafe sets, and L g (o) is the optimal cost of the relaxed goal-seeking control policy (6a). We also find that adding a regularization term for the L2-norm of h σ and V ω helps improve training performance. Loss curves for the training and validation datasets are shown in Fig. 5, covering 72 training epochs. A. Verification of learned certificates In Section V, we prove that if h and V are valid oCBFs and oCLFs, then our hybrid controller is guaranteed to reach the goal and remain safe. However, because we learn the oCBF and oCLF using neural networks trained on a finite number of sampled points, we do not have an explicit guarantee that the learned h and V will be valid. Related works in certificate learning rely on probabilistic verification methods such as those discussed in [12] and [22]. These methods draw on results from statistical learning theory to show that if a learned certificate is valid (with some margin) on a large set of sample points, then there is a high probability that it will be valid in general. Even if a certificate is invalid in a small subset of the state space, almost-Lyapunov theory [23] suggests that global safety and stability guarantees may still exist. To check the validity of our learned certificates, we sample 100,000 points from the state space and check whether constraint (6c) is feasible at each point; we found that this constraint is feasible at each tested point. However, we acknowledge that this sampling analysis falls short of a formal proof, and factors such as sensor noise and inaccuracies in the approximate lookahead model mean that the formal guarantees from Section V may not hold in practice. To demonstrate empirically that our learned certificates yield a safe, performant controller in practice, the next section presents experimental results showing that our controller remains safe and reaches the goal in a wide range of simulated and hardware environments. In our future work, we hope to extend our theoretical analysis to explain this strong empirical performance. VII. EXPERIMENTS We validate our hybrid lookahead controller in two ways. First, we compare its performance against model-predictive control (MPC), reinforcement learning (RL), and state-based CBF/CLF baselines in simulation with a Dubins car model, demonstrating that our controller exhibits improved performance and generalizes well beyond the training environment. Second, we deploy our controller in hardware to navigate a mobile robot through a changing environment. A. Generalization beyond training environment To assess controllers' ability to generalize beyond the environments in which they were trained, both our oCBF/oCLF controller, an end-to-end RL controller, and a state-based CBF/CLF controller were trained in a static environment with 8 randomly placed obstacles. For the RL agent, we use a neural-network policy with the same inputs as our controller, trained in an end-to-end manner using proximal policy optimization (PPO) as implemented in the OpenAI Safe RL benchmarks [24]. The RL agent was trained with a large reward for reaching the goal, a smaller dense reward equal to −ρ 2 + (1 − cos φ)/2, and a large cost for colliding with objects. The state-based CBF/CLF method was trained identically to our own method except with h(x) = h σ (x). The MPC method is model-based and does not need training; it constructs a convex approximation of the locally-observable free space (inspired by [25]), plans a path towards the goal within that region, and executes the first 0.1 s of that path. When it becomes stuck, MPC attempts to navigate around the boundaries of obstacles. All controllers were run at 10 Hz, and numerical simulations of continuous-time Dubins car dynamics occurred at 100 Hz. Example trajectories for each controller navigating a Dubins car through a randomized 2D environment are shown in Fig. 6, and Fig. 7 shows the collision and goal-reaching rates for each controller across 500 such randomized environments (along with the average time required to reach the goal when successful). These data show that our proposed approach achieves a 100% safety rate while reaching the goal within 10 s in 93.2% of trials (less than 100% due to obstacles occluding the goal in some cases). Our method significantly outperforms the other learning-based methods (both of which have lower safety and goal-reaching rates due to difficulty generalizing beyond the training environment). MPC achieves slightly worse safety and goal-reaching rates than our approach: 99.6% and 90.4%, respectively. Typical behaviors for each controller are shown in Fig. 6: the state-based learned CBF does not adapt to the new environment and is unsafe, the PPO policy is not precise enough to pass through the gap to reach the goal (although it remains safe). Only our approach and MPC consistently reach the goal safely, although our approach achieves a slightly higher goal-reaching rate and is much less computationally intensive (18 ms of online computational time for our method vs. 102 ms for MPC). B. Hardware demonstration To validate our controller in the real world, we deployed our controller on a TurtleBot 3 mobile robot. We model this platform as a Dubins car with control inputs for forward and angular velocity. The TurtleBot is equipped with an HLS-LFCD2 2D laser scanner with 1 • resolution in a 360 • range, which we downsample to 32 equally-spaced measurements. Lidar scans are preprocessed to convert from polar to local Cartesian coordinates before being passed to the learned encoder. Range and bearing to the goal are estimated Fig. 7. Goal-reaching rate, safety rate, and time-to-goal for oCBF/oCLF, state-based CBF/CLF, MPC, and PPO on waypoint tracking, averaged over 500 randomized environments. Trial runs were limited to a maximum of 10 s, and time-to-goal was averaged only over trials that reached the goal. using odometry, demonstrating robustness to imperfect state estimation. The controller is implemented with a zero-order hold at 10 Hz, but Lidar data are only available at 5 Hz (stale scans were used when new data were not available). Although we do not explicitly model sensor uncertainty, we observed both small-magnitude noise and spurious detections in our Lidar data (these effects can be seen in the supplementary video), demonstrating that our controller can handle the noise resulting from real sensors. We compare our method against the TurtleBot's built-in SLAM module in combination with an online dynamic window path planner. Fig. 8 shows our controller successfully avoiding both fixed obstacles and an obstacle thrown into its path midexperiment. Video footage of this experiments (and others in different environments) is included in the supplementary materials. Our controller successfully escapes the initial trap and avoids the thrown obstacle to reach the goal, while the combined SLAM/planning system becomes stuck when the new obstacle is added. The planner eventually becomes unstuck (after rotating the robot in place to re-create the map), but this issue demonstrates the difficulty of guaranteeing liveness for an online SLAM and planning system. In contrast, our controller is entirely reactive, allowing it to avoid the new obstacle without pausing, providing a significant advantage compared to model-based online planning methods. VIII. CONCLUSIONS AND FUTURE WORK Safely integrating rich sensor models into autonomous control systems is a challenging problem at the intersection of robotics, control theory, and machine learning. In this work, we explore one direction for constructing safe, generalizable perception-based controllers. We use neural networks to learn an observation-based control barrier function, which we combine with a hybrid control architecture that allows us to prove the safety and liveness of the resulting learning-enabled controller. We demonstrate in simulation that this architecture outperforms other learning-and model-based control systems (including end-to-end learning using RL), and we successfully deploy our controller in hardware. In addition to these successes, it is important to highlight the drawbacks of our approach, which point to interesting areas of future work. In particular, we rely on an approximate model of the Lidar sensor to predict observations one step into the future. This reliance prevents us from easily extending our approach from Lidar to image data, as it is much more difficult to construct an approximate model for image updates. In future work, we hope to replace this model-based update with a generative model learned from data. Using a learned, generative update might also allow us to relax the assumption of local dynamics, presenting a promising line of future study. Fig. 1 . 1Our controller combines a learned Control Barrier Function (based on Lidar sensor data, green) and a learned Control Lyapunov Function (based on range and bearing relative to the goal, blue) with a hybrid control architecture. Fig. 2 . 2Block diagram of controller architecture. Fig. 3 . 3The hybrid control scheme allows the robot to escape potential deadlocks, as in this bug-trap environment. Fig. 5 . 5Loss L on a dataset of 10,000 points (10% reserved for validation). Fig. 6 . 6Plot of trajectories for observation-space CBF/CLF, state-based CBF/CLF, MPC, and end-to-end RL policies in random environment. Fig. 8 . 8(Left) Our controller navigating around static and dynamic obstacles, shown as a composite image. (Right) A SLAM module paired with an online global planner gets stuck when an unexpected object is added to the scene. A video of these experiments is included in the supplementary materials. 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Gao, "Neural Lyapunov Control," in Advances in Neural Information Processing Systems, vol. 32, 2019, pp. 3245-3254. [Online]. Available: https://github.com/YaChienChang/ Neural-Lyapunov-Control Stabilizing neural control using self-learned almost lyapunov critics. Y.-C Chang, S Gao, 2021 IEEE International Conference on Robotics and Automation, ICRA, 2021. Y.-C. Chang and S. Gao, "Stabilizing neural control using self-learned almost lyapunov critics," in 2021 IEEE International Conference on Robotics and Automation, ICRA, 2021, pp. 51-57. Learning stability certificates from data. N M Boffi, S Tu, N Matni, J J E Slotine, V Sindhwani, Conference on Robot Learning. arXiv. N. M. Boffi, S. Tu, N. Matni, J. J. E. Slotine, and V. Sindhwani, "Learning stability certificates from data," in Conference on Robot Learning. arXiv, aug 2020. [Online]. Available: http://arxiv.org/abs/ 2008.05952 Almost Lyapunov functions for nonlinear systems. S Liu, D Liberzon, V Zharnitsky, Automatica. 113108758S. Liu, D. Liberzon, and V. Zharnitsky, "Almost Lyapunov functions for nonlinear systems," Automatica, vol. 113, p. 108758, mar 2020. Benchmarking Safe Exploration in Deep Reinforcement Learning. A Ray, J Achiam, D Amodei, A. Ray, J. Achiam, and D. Amodei, "Benchmarking Safe Exploration in Deep Reinforcement Learning," 2019. Computing Large Convex Regions of Obstacle-Free Space Through Semidefinite Programming. R Deits, R Tedrake, Springer International PublishingR. Deits and R. Tedrake, Computing Large Convex Regions of Obstacle-Free Space Through Semidefinite Programming. Cham: Springer International Publishing, 2015, pp. 109-124. [Online]. . 10.1007/978-3-319-16595-0_7Available: https://doi.org/10.1007/978-3-319-16595-0 7
[ "https://github.com/YaChienChang/" ]
[ "Topological censorship and chronology protection", "Topological censorship and chronology protection" ]
[ "John L Friedman \nDepartment of Physics\nUniversity of Wisconsin-Milwaukee\nPO Box 413MilwaukeeWisconsin, US\n", "Atsushi Higuchi \nDepartment of Mathematics\nUniversity of York\nYO10 5DDHeslingtonYorkUK\n" ]
[ "Department of Physics\nUniversity of Wisconsin-Milwaukee\nPO Box 413MilwaukeeWisconsin, US", "Department of Mathematics\nUniversity of York\nYO10 5DDHeslingtonYorkUK" ]
[]
Over the past two decades, substantial efforts have been made to understand the way in which physics enforces the ordinary topology and causal structure that we observe, from subnuclear to cosmological scales. We review the status of topological censorship and the topology of event horizons; chronology protection in classical and semiclassical gravity; and related progress in establishing quantum energy inequalities. This article is dedicated to Rafael Sorkin, whose friendship and tutoring from third grade onward is responsible for one of us (JF) having spent his adult life in physics and whose work has inspired both of us.
10.1002/andp.200510172
[ "https://arxiv.org/pdf/0801.0735v2.pdf" ]
119,294,274
0801.0735
a6db5933b2972c58d2c4f2d3e4789822e7b1f8a7
Topological censorship and chronology protection 3 Jun 2008 John L Friedman Department of Physics University of Wisconsin-Milwaukee PO Box 413MilwaukeeWisconsin, US Atsushi Higuchi Department of Mathematics University of York YO10 5DDHeslingtonYorkUK Topological censorship and chronology protection 3 Jun 2008 Over the past two decades, substantial efforts have been made to understand the way in which physics enforces the ordinary topology and causal structure that we observe, from subnuclear to cosmological scales. We review the status of topological censorship and the topology of event horizons; chronology protection in classical and semiclassical gravity; and related progress in establishing quantum energy inequalities. This article is dedicated to Rafael Sorkin, whose friendship and tutoring from third grade onward is responsible for one of us (JF) having spent his adult life in physics and whose work has inspired both of us. I. INTRODUCTION In addition to the gravitational waves and black holes that reside in our universe, vacuum solutions to the classical Einstein equation generically exhibit white holes and structures with noneuclidean topology, and there is a generically large space of time-nonorientable solutions. There are also vacuum solutions (and positive energy solutions) with closed timelike curves, though how generic these solutions are remains an open question. The absence of white holes is a central mystery, tied to the thermodynamic arrow of time. The absence of macroscopic topological structures and of macroscopic closed timelike curves (CTCs) is a similarly central feature of our experience. Where have they gone? Why is the topology and the causal structure of spacetime "ordinary" on a macroscopic scale, when what we call ordinary is extraordinary in the space of solutions? Two entirely different answers are consistent with our knowledge. The first is simply that the classical theory has a much broader set of solutions than the correct theory of quantum gravity. It is not implausible that causal structure enters in a fundamental way in quantum gravity and that classical spacetimes with closed timelike curves approximate no quantum states of the spacetime geometry. Less likely to us is the analogous explanation for trivial spacetime topology, that the correct theory of quantum gravity allows only euclidean topology and forbids topology change. A second possible answer is provided by topological censorship and chronology protection: One supposes that quantum gravity allows microscopic (near Planck-size or near string-size) structures that have nontrivial topology and/or violate causality; and one shows that classical general relativity and the character of macroscopic matter (described by classical or semiclassical fields) forbid exotic structures that are macroscopic in their spatial and temporal size. This brief review emphasizes a few related areas of recent work and is not intended to be comprehensive in any sense. A monograph by Visser [1] reviews much of the work prior to 1995 and supplies a comprehensive bibliography; informal reviews are given by Thorne [2] and Gott [3] (whose bibliography has brief, useful descriptions of each reference). For a recent review of work on energy inequalities, see,for example, Roman [9], and an earlier popular article by Ford and Roman [10]. Earlier reviews by one of the present authors are Refs. [11,12]. For a more detailed (and more technical) review of the Cauchy problem on spacetimes with CTCs and on Lorentzian universes-from-nothing, see Ref. [13]. Becasue most of chronology protection is concerned with isolated regions of CTCs, and because of space limitations, the extensive work on Gott spacetimes is not covered here and no attempt is made to provide a comprehensive bibliography. Interested readers should balance Gott's view [3] with articles constraining CTC formation in 3-dimensional spacetimes, beginning for example, with work by Deser, Jackiw and 't Hooft [4], Cutler [5], Carroll,Fahri and Guth [7] (and references therein), and Tiglio [8]. II. TOPOLOGICAL CENSORSHIP A. Expectation of nontrivial topology Beyond the wormholes that science fiction has made a part of popular culture lie an infinite variety of topological structures in three dimensions, a countably infinite set of prime 3-manifolds. Witt [14] showed that all 3-manifolds (prime and composite) occur as spacelike hypersurfaces of vacuum spacetimes -solutions to the vacuum Einstein equations. This is not enough to show that they arise as isolated structures: A flat universe with the topology of a 3-torus satisfies the field equations, but its existence does not imply that one can find a vacuum geometry for which the universe looks everywhere like a 3-sphere except in an isolated region. Isolated systems are ordinarily modeled as asymptotically flat spacetimes; and a recent result by Isenberg et al. [15] shows that all 3-manifolds do in fact occur as isolated structures, as spacelike hypersurfaces of asymptotically flat, vacuum spacetimes. Whether topological structures can arise from a spacetime that is initially topologically trivial is a more difficult question. Given any two 3-manifolds, S 1 and S 2 , one can always find a spacetime (a 4-manifold with Lorentzian metric) that joins them and for which they are each spacelike: There is a spacetime whose spacelike boundary is the disjoint union of S 1 and S 2 [16,17]. (A strengthened version of this and a review of earlier work on classical topology change is given by Borde [18]). But one pays a price for topology change. A theorem due to Geroch [19] shows that a spacetime whose boundary comprises two disjoint spacelike 3-manifolds always has closed timelike curves. (See [18] for a version appropriate to asymptotically flat spacetimes.) Topology-changing spacetimes must also have negative energy. Much of the recent work involving restrictions on spacetime topology and on causal structure relies on the null energy condition (NEC): An energy-momentum tensor T αβ satisfies the null energy condition if, for any null vector k α , T αβ k α k β ≥ 0. In particular, Tipler shows that topology-changing spacetimes violate the null energy condition [20]. Theorems involving the null energy condition use it to infer increasing convergence of light rays. Their proofs require only that the average value of T αβ k α k β along null geodesics γ be nonnegative. This weaker requirement, the averaged null energy condition (ANEC), has the form γ dλ T αβ k α k β > 0, (2.1) with λ an affine parameter along the null geodesic and k α the corresponding null tangent vector. Because classical matter satisfies the null energy condition, it seems unlikely that one can create topological structures with a time evolution that is nearly classical. In path-integral approaches to quantum gravity, however, the amplitude for a transition from one 3-geometry to another is ordinarily written as a sum over all interpolating 4-geometries. Whether one works in a Euclidean or Lorentzian framework, topology change is thus permitted, if only at the Planck scale. One can then ask whether small-scale topological structures can grow to macroscopic size and persist for macroscopically long times. The meaning of topological censorship is that they cannot: Isolated topological structures with positive energy collapse, and they do so quickly enough that light cannot traverse them. B. Gannon's Singularity Theorem The first result of this kind was due to Dennis Gannon [21], a similar result obtained independently by Lee [22]. Theorem 1 Let M, g be an asymptotically flat spacetime, obeying the null energy condition, T αβ k α k β ≥ 0. (2.2) If M, g has a nonsimply connected Cauchy surface, then it is geodesically incomplete. Given the recently proven Poincaré conjecture [23,24,25], Gannon's theorem implies that if the topology of the Cauchy surface S is not trivial (i.e., if adding a point at infinty to S yields any closed 3-manifold other than S 3 ), then the spacetime is geodesically incomplete. Requiring that the system be isolated is essential to the theorem. Topological structures large enough to expand with the Hubble expansion are not ruled out, and the topology of the universe is unrestricted. Each of the countably many hyperbolic, spherical, and flat 3-manifolds is consistent with the homogeneity and isotropy of the observed universe. In fact, only with the high angular resolution of the recent microwave anisotropy probes has it been possible to show that the apparent size of the visible universe is not an illusion arising from light traversing several times a space whose size is a fraction the Hubble length [26] (for a review, see Levin [27]). 1 The geodesic incompleteness proved by the theorem is the way most singularity theorems are stated. It is expected in this context to imply gravitational collapse, leading (within the classical theory) to unbounded curvature within a black hole: Geodesics would then be incomplete because they reach the singularity (that is, scalars locally constructed from the curvature grow without bound) within finite affine parameter length. The topological censorship theorem reinforces this expectation and confirms part of it. C. Topological Censorship The conjecture can be stated in two equivalent forms [29,30]. Recall that to define a black hole -a region from which light cannot escape -one first makes precise the notion of a light ray reaching infinity by attaching to spacetime a boundary, future null infinity (I + ) [31]. Light that is not trapped is then light that reaches future null infinity, and a black hole is the region from which no future-directed null geodesic reaches infinity. Past directed null geodesics that are not trapped similarly reach past null infinity. (Unless the spacetime has a white hole, no past-directed light ray will be trapped). An observer that can communicate with the outside world is one whose past and future directed null rays reach null infinity. She is then outside all black (and white) holes, in the domain of outer communication. We also need the term causal curve, a curve whose tangent is everywhere timelike or null. (Friedman, Schleich, Witt). Let M, g be an asymptotically flat, globally hyperbolic spacetime satisfying the averaged null energy condition. Then every causal curve from past null infinity to future null infinity can be deformed to a curve near infinity. (More precisely, each causal curve can be deformed with its endpoints fixed at I to a curve that lies in a simply connected neighborhood of I.) Theorem 2 Topological Censorship Version A That is, no causal curve can thread the topology. The theorem implies that no observer who remains outside all black holes (and who did not emerge from a white hole) can send a signal that will probe the spacetime topology. Topological censorship can be regarded as a statement about the topology of the domain of outer communications: The region outside all black (and white) holes is topologically trivial. (Galloway). Let M, g be an asymptotically flat spacetime obeying the averaged null energy condition, and suppose the domain of outer communication is globally hyperbolic. Then the domain of outer communications is simply connected. Theorem 3 Topological Censorship Version B Again, given the Poincaré conjecture, simply connected is equivalent to topologically trivial. Here this follows from the fact that a globally hyperbolic spacetime has topology S × R. Topological censorship implies that S is simply connected, and the Poincaré conjecture then implies that S has trivial topology, whence S × R has trivial topology. The proof of Version A that is simplest to outline relies on an argument used by Penrose, Sorkin and Woolgar [32] in their proof of a positive mass theorem. Suppose a causal curve γ joining I − to I + is not homotopic to an asymptotic curve. Denote by λ ± the generators of I ± that contain the endpoints of γ. Partially order all curves from λ − to λ + , writing γ 2 ≥ γ 1 , if γ 2 is faster than γ 1 . That is, γ 2 ≥ γ 1 if γ 2 leaves λ − later than γ (or at the same time) and if γ 2 reaches λ + earlier (or at the same time). Then a fastest curve γ ∞ in the homotopy class of γ is a null geodesic without conjugate points. But the Raychandhuri equation together with the null energy condition implies that null geodesics have conjugate points in finite affine parameter length, a contradiction. D. Implications for black holes Chruściel and Wald showed that, in form B (Theorem 3), topological censorship implies that stationary black holes have spherical topology [33]. The result itself is due initially to Hawking [67], with a proof that assumes analyticity. The proof based on topological censorship strengthens the theorem: It does not require analyticity (or smoothness) and it uses the weak energy condition. The proof given in Ref. [33] was written before Galloway's version B appeared, and by reading Galloway first, one can avoid some of the detailed arguments in [33]. The proof uses the fact that the domain of outer communication is simply connected to show that its boundary is homeomorphic to the boundary of a compact three-manifold whose interior is simply connected. This is enough to prove the theorem, because a standard result for 3-manifolds asserts that such a boundary is a disjoint union of spheres. The result has been amplified in several directions [34,35,36,37,38,39]. In particular, one can dispense with stationarity to show spherical topology for slices of the horizon by Cauchy surfaces that lie to the future of a slice of past null infinity [35,36] or whose topology is unchanging [34]. At first sight, it would seem that black holes must always have spherical topology, because the event horizon is the boundary of the simply connected domain of outer communication. This would match the intuitive picture of their formation and the familiar pair-of-pants coalescence of two black holes. A typical slicing of the horizon begins with two disjoint points that expand to two disjoint spheres; coalescence begins with the intersection of the two spheres in a single point; and the final slices are single spheres. In fact, however, as Hughes et al. first found numerically [40], one can find examples of collapse in which the intersection of a spacelike hypersurface with the horizon is toroidal. Examples of horizons with slices with higher genus are not difficult to construct, and work by Siino [41] shows that slicings with arbitrary genus can be constructed when the horizon has caustics. The reason is that the past endpoints of the horizon's null generators form an acausal set; and it is the intersection of wiggly spacelike slices with this acausal set that can have arbitrarily high genus. It nevertheless appears that there is always an alternative slicing of the horizon in which black holes are spherical. That is, Siino proves the following result: Let M, g be a strongly causal spacetime with an event horizon that is a smooth S 2 to the future of some spacelike hypersurface. Then the domain of outer communications of M, g can be foliated by spacelike slices, each of which intersect the future horizon in a union of disjoint points and of spheres that are either disjoint or that intersect in points. The assumption of a final smooth S 2 is related to the unproved cosmic censorship conjecture, and one would like to replace that assumption by a positive energy condition. Underlying the conjecture is the way a toroidal black hole adheres to topological censorship: The torus closes before light can traverse it, and in the examples we know of, that constraint appears to allow a foliation in which black holes are spherical. A simple example, suggested (in one lower dimension) by Greg Galloway, is similar to one given by Shapiro, Teukolsky and Winicour [42] (see also [43]). It is a null surface in Minkowski space that allows an initially toroidal slicing. (One can construct an artificial spacetime for which this null surface is the event horizon, but the construction is unrelated to the nature of the null surface and its slicings.) The surface is generated by the future-directed outward null rays from the spacelike disk t = 0, z = 0, x 2 + y 2 ≤ 1, with t, x, y, z standard Minkowski coordinates. The surface is rotationally symmetric about the z-axis. A spacelike surface that cuts through the null rays and then the disk, before dipping below the disk gives a toroidal slice. And spacelike surfaces that lie below the edge of the disk and go upward at the center of the disk give spherical slices. E. Lorentzian universes from nothing Universes whose spatial slices are compact 3-manifolds are finite in space, but have no boundary. One can similarly constuct 4-manifolds with no past boundary that are finite in time -universes from nothing. In a Euclidean framework for quantum gravity, manifolds of this kind arise in the Hartle-Hawking wavefunction of the universe. An example with a Lorentzian metric and with CTCs is given by Gott and Li [44]; but CTCs are not an essential feature of Lorentzian universes from nothing. For a large class of these spacetimes, one can always choose metrics without CTCs; time nonorientability is then their only causal pathology. They are the only examples of topology change in which one has a smooth, nondegenerate Lorentzian metric without closed timelike curves. The double-covering space of these spacetimes is globally hyperbolic, and that fact implies the existence of generalized Cauchy surfaces. A simple example of such a spacetime is a Möbius strip, whose median circle is taken to be a spacelike hypersurface. Initial data on this median circle consists of a specification of a field and its gradient, and the initial value problem is well defined when the metric has no CTCs. The Möbius strip can be constructed from a cylinder with coordinates t, φ, by the antipodal identification t → −t, φ → φ + π. In four dimensions, antipodally-identified de Sitter space is an example that is locally indistinguishable from ordinary de Sitter. An initial value surface in this case is the antipodally identified 3-sphere at t = 0, and some initial value surface passes through each point of the spacetime. More generally, one can construct a Lorentzian universe with no past boundary from any compact 3-manifold S that admits a diffeomorphism I : S → S by an analogous antipodal identification of S × R. If one begins with antipodally symmetric initial data for the Einstein equations on S, then the resulting solution to the field equations on S × R induces a metric on the identified space that is also a solution [45,46]. Because classical physics on Lorentzian universes from nothing has no pathology, it is natural to ask if there is any reason why these spacetimes are forbidden if one incorporates quantum physics. Kay [47] proposes to impose the standard canonical commutation relations in a neighborhood of any point with respect to one time orientation (the "F-locality condition"). This condition rules out time non-orientable spacetime in an obvious manner. Gibbons [48] also rules out conventional quantum field theory based on a complex Hilbert space in some time non-orientable spacetimes, including antipodally identified de Sitter spacetime. Refs. [45,46] investigate whether quantum field theory in Lorentzian universes from nothing could be defined in globally hyperbolic neighborhoods with suitable overlap conditions on the intersections. It turns out that there are no difficulties in defining an algebra of fields in this manner. It is also possible to construct states satisfying the positivity conditions in each of the neighborhoods if the union of two neighborhoods is always time orientable. This restriction on the neighborhoods, however, is rather artificial because there will always be sets of points in the spacetime for which no correlation function is defined. If one then allows the union of two neighborhoods on which field theory is defined to be time non-orientable, one can show that there are no physically reasonable states satisfying the postivity condition in each neighborhood. One may be able to construct a consistent field theory in these spacetimes using path-integral quantization, but this possibility has not been explored. III. CHRONOLOGY PROTECTION A. Overcoming the grandfather paradox: Existence of solutions for generic data in a class of spacetimes with CTCs Closed timelike curves were traditionally regarded as unphysical because of the grandfather paradox, the fact that a time machine would allow one to go back in time and do away with her grandfather. More precisely, it was thought that most initial data on a spacelike surface would fail to have a consistent evolution: Locally evolving the data would not lead to a global solution, because the local solution would be inconsistent with the data, once the evolution returned to the initial surface. A simple example of a spacetime for which generic initial data has no solution is the cylinder obtained from the slab of Minkowksi space between t = −1 and t = 1 by identifying the points −1, x, y, z and 1, x, y, z. Data on the surface t = 0 for, say, a massless scalar field ψ can be locally evolved around the cylinder (to t = 1 and then from the identified surface t = −1 back to t = 0). The locally evolved solution, however, returns to the surface with a value of the field that disagrees with its initial value unless the initial data is chosen to give a solution periodic with period T = 2. One of the surprises that spurred interest in spacetimes with CTCs is the existence of a different class of spacetimes for which solutions exist for free fields with arbitrary initial data, for interacting classical particles (billiard balls), and perhaps for interacting fields as well. Whether physics is consistent with closed timelike curves, however, does not rely on consistency of classical physics. A path-integral assigns probability amplitudes to histories, whether or not there are classical histories. And in the billiard-ball case, where classical solutions exist but are not unique, a path-integral dominated by classical solutions would simply assign large amplitudes to the alternative classical histories. We review the billiard-ball example and interacting field arguments briefly in the next section, after we have introduced another classical obstacle to closed timelike curves (an instability of the Cauchy horizon) that these examples overcome. In this section we consider free fields. Two-dimensional spacetimes built from Minkowski space that avoid the grandfather paradox are easily constructed. An example, similar to spaces discussed by Geroch and Horowitz [49] and by Politzer [50], is constructed by removing two parallel timelike slits from Minkowski space and gluing the edges of the slits. Corresponding points on the two inner edges are identified by the translation T shown in Fig. 1; and corresponding points on the two outer edges are similarly identified. CTCs join identified points of the inner edges, from Q on the left to T (Q) on the right. The hypersurface S, lying to the past of the CTCs, is an obvious candidate for a generalized Cauchy surface of M, g. Definition. A generalized Cauchy surface S is an achronal hypersurface of M for which the initial value problem for the scalar wave equation is well-defined: Any smooth data in L 2 (S) with finite energy, for a scalar field Φ, has a unique solution Φ on M . In fact, it is easy to see that initial data in L 2 (S) leads to a solution in L 2 (M ). In the past of the CTCs (in the past of the Cauchy horizon), solutions to the massless wave equation can be written as the sum f (t − x) + g(t + x) of right-moving and left-moving solutions. To obtain a solution in the spacetime M , one simply propagates left moving data that encounters the slit in the obvious way. For example, if a left-moving wave enters the left slit at Q, it emerges unaltered from the right slit at T (Q). The solution is unique. But it is discontinuous along future-directed null rays that extend from the endpoints of the slits, because the result of the wave propagation is to piece together solutions from disjoint parts of the initial data surface (see Friedman and Morris [51]). Analysis of the initial value problem for a related spacetime with spacelike slits is given by Goldwirth et al. [52]. Although, in our 2-dimensional example, the solution has discontinuities, in four dimensions one can construct spacetimes for which smooth, unique solutions to the scalar wave equation exist for all data on a generalized Cauchy surface. The examples for which these statements are proved are time-independent spacetimes [51,54], in which CTCs are always present. The spacetimes are asymptotically flat, and one can define future and past null infinity. In Minkowski space past null infinity is a generalized Cauchy surface for massless wave equations, and the theorems show that it is also generalized Cauchy surface for a class of spacetimes with CTCs. The spacetime considered in Ref. [51,55] has a wormhole that joins points at one time to points at an earlier time. Recall that a wormhole is constructed from R 3 by removing two balls and identifying their spherical boundaries, Σ I and Σ II , as shown in Fig. 2. The history of each sphere is a cylinder in spacetime, and one constructs a spacetime with CTCs by removing two solid cylinders from Minkowski space and identifying their boundaries C I and C II after a time translation, so that the sphere at time t + T is identified with a sphere at an earlier time t, as shown in Fig. 3. Thus a particle entering the wormhole mouth at T (Σ I ) emerges from the mouth Σ I at an earlier time. The existence proof relies on a spectral decomposition of the field. Technical difficulties arise from the fact that the boundary conditions are frequency dependent, and solutions with different frequencies are not orthogonal. As a result, one cannot use the spectral theorem, and a separate proof of convergence of the harmonic decomposition is required. Recent work by Bugdayci [53] constructs a solution for the case of a metric that is flat everywhere outside the identified cylinders, as a multiple scattering series. More recently, Bachelot [54] proved a similar existence theorem and a strong uniqueness theorem for another family of stationary, four-dimensional spacetimes that are flat outside a spatially compact region. These spacetimes have Euclidean topology and their dischronal regions have topology (solid torus) ×R. The metric is axisymmetric, with one free function a that describes the tipping of the light cones in the direction of the rotational Killing vector ∂ φ . B. Classical chronology protection Parallel slits do not exhibit a property of almost all other two-dimensional spacetimes whose CTCs lie to one side of a Cauchy horizon: a classical instability of the Cauchy horizon. Once the slits are not parallel (once they are, in effect, walls in relative motion), the instability arises. The paradigm spacetime for this instability is Misner space [31,56,57]. Misner space can be obtained from a 1-dimensional room whose walls are moving toward each other at relative speed v, by identifying left and right walls at the same proper time read by clocks on each wall. The resulting space is the piece of Minkowski space between two timelike lines (the walls), with the lines identified by the boost that maps one line to the other. In the diagram below, Fig. 4, numbers label readings of a single clock, shown at identified points of the left and right walls. A light ray beginning at the left wall at t = 0 is boosted each time it traverses the space, in the same way that light is boosted when reflected by a moving mirror. As it loops around the space, the light ray approaches a closed null geodesic through the clock at Identified points are spacelike separated for t < 4. For t = 4 the identified clocks are separated by a null geodesic that marks the boundary of the globally hyperbolic spacetime to its past and the dischronal region above it. Through each point of the dischronal region passes a closed timelike curve; an example is the dashed line joining the clock images at t = 5. The divergence of solutions to the wave equation is clear in the geometrical optics limit. A family of light rays that loop about the space are boosted at each loop, their frequencies increased by the blueshift factor [(1 + v)/(1 − v)] 1/2 . Their energy density, measured in the frame of any inertial observer, diverges as they approach the Cauchy horizon. Each looping light ray is an incomplete null geodesic: It reaches the horizon in finite affine parameter length, because each boost decreases the affine parameter by the factor [(1 + v)/(1 − v)] 1/2 . In discussing Gannon's singularity theorem for nontrivial topology, we noted that, in the context of gravitational collapse, geodesic incompleteness is thought generically to imply a curvature singularity. Here, however, the spacetime is smooth, and the incomplete geodesic is unpleasant only because it leads to an instability of the Cauchy horizon. This behavior is not unique to Misner space or to two dimensions: A theorem due to Tipler [20] shows that geodesic incompleteness is generic in spacetimes like Misner space in which CTCs are "created" -spacetimes whose dischronal region lies to the future of a spacelike hypersurface. The nature of the geodesic incompleteness is clarified by Hawking[67], at least in the case when the null generators of the Cauchy horizon have the character of a fountain, all springing from a single closed null geodesic (past-directed generators appoach the closed null geodesic). The boosted light rays that give rise to the Misner space instability will characterize these more general Cauchy horizons in four dimensional spacetimes. In four dimensions, however, the instability competes with the spreading of the waves. In the geometrical optics limit, the boosting will not lead to a divergent energy density if the area of a beam increases as the beam loops by a factor greater than the increase in energy density due to the boost in frequency. That is, if the ratio A n+1 /A n of beam areas at successive loops is greater than (ω n+1 /ω n ) 2 , the boosted frequency will not lead to a divergence of the energy at the Cauchy horizon. Hawking's analysis follows an example given by Morris, Thorne and Yurtsever [58,62] of a wormhole whose mouths move toward each other in a way that initially mimics the Misner-space walls, as shown in Fig. 5. For the case of a spacetime flat outside the wormhole mouths (the identified spheres), each time a beam of light traverses the wormhole, its frequency is boosted and its area increases by the approximate factor (d/R) 2 , with d the distance between wormholes, R the radius of the throat. In this example, the horizon generators do have the fountain behavior described above. The fountains assumed by Hawking, however, are probably not generic for the kind of Cauchy horizons he considers (compactly generated, noncompact). (A Cauchy horizon is compactly generated if all its past-directed null generators enter and remain in a compact region.) Chruściel and Isenberg [59] give examples of compactly generated noncompact Cauchy horizons whose generators do not have fountainlike behavior (for which no closed null geodesic is an attractor); they show that fountains are not generic for compact Cauchy horizons and argue that they are not generic for the compactly generated, noncompact case. Because the evolution of fields on the wormhole spacetime does not lead to an instability of the Cauchy horizon, it appears that asymptotically flat spacelike hypersurfaces to the past of the Cauchy horizon (e.g., a t = constant surface of Minkowski space through τ = 0 in Fig. 5) are generalized Cauchy surfaces. One can formally construct a solution as a multiple scattering series, and, for sufficiently small ratio R/d, we expect the series to converge. For interacting particles, modeled as billiard balls, Echeverria et al. [60] looked at the simpler time-independent wormhole spacetime of Fig. 3, in which billiard balls entering the right mouth of the wormhole at time t exit from the left mouth at time t − τ . Consider a ball that is aimed at the right wormhole along a timelike line that intersects itself. Initial data for the ball has a local solution that, when extended has the exiting billiard ball aimed to strike its earlier self, apparently preventing a solution. In fact, however, there appear always to be glancing blow solutions for arbitrary initial position and velocity of the incoming ball. In these solutions, the incoming ball is hit by its earlier self in just the right way that it enters the wormhole and emerges aimed to strike itself that glancing blow. 2 With both the grandfather paradox and the classical Cauchy horizon instability overcome, what prevents the classical formation of spacetimes with CTCs? In the wormhole example, the null energy condition must be violated, if one is to keep the wormholes open long enough for light to traverse them, and hence long enough for closed timelike curves to traverse them. Hawking shows that this is generically true [67]. Classical chronology protection. Let M, g be a spacetime with a compactly generated Cauchy horizon. Assume that the Cauchy horizon is the boundary of the domain of dependence of a noncompact partial Cauchy surface S. Then the null energy condition is violated. The Cauchy horizon is expected to be compactly generated in a nonsingular, asymptotically flat spacetime for which the Cauchy horizon bounds the domain of dependence of an asymptotically flat spacelike hypersurface S. The argument is then: (i) Generators of a future Cauchy horizon are null geodesic segments with no past endpoints. (ii) Because they enter a compact region, some of the generators must converge to the past. (iii) Positive convergence for a past-directed generator, together with the Raychaudhuri equation, implies that the null energy condition must be violated on the future horizon. Otherwise the generators would have past endpoints. If one allows the spacetime to be singular to the future of S, as in recent examples by Ori [63], it is not difficult to show that one can find chronology-violating asymptotically flat spacetimes satisfying the null energy condition. In this case the Cauchy horizon is singular and thus not compact (past directed null geodesics run off the manifold in finite affine parameter length). In Ori's most recent example, a spacetime whose only matter is a compact region of dust has a Cauchy horizon with closed null geodesics that is nonsingular in a neighborhood of these geodesics. Because quantum fields fail to satisfy the null energy condition, a number of attempts have been made to circumvent the classical chronology protection theorem and topological censorship by finding a way to have negative energy regions that persist for long times; or to have solutions with CTCs or nontrivial topology that require only small amounts or small regions of negative energy. In our opinion, however, the recent work on quantum energy inequalities, discussed in Sect. IV below, has set increasingly stringent constraints on violations of the energy conditions. We think it likely that macroscopic violations of either topological censorship or classical chronology protection are inconsistent with the fundamental properties of semiclassical quantum fields. C. Quantum chronology protection Various pathologies have been found in quantum field theory in spacetimes with dischronal regions. A quantum instability in spacetimes with a Cauchy horizon may prevent the formation of CTCs on scales large compared to the Planck scale; or, together with a loss of unitarity for interacting fields on spacetimes with CTCs, it may indicate that chronology violations do not occur in the fundamental theory. We begin with a brief heuristic summary of the quantum instability and then present a more technical description that includes more recent results. Quantum instability The classical instability of Misner space has a quantum counterpart that is present in cases where classical fields remain finite. The quantum instability and the loss of unitarity are each related to propagation of fields around closed null or timelike curves. In computing the energy density of a quantum field in the vacuum, one must renormalize the field to produce a finite result, subtracting off a divergent zero-point energy of the vacuum. One can, for example, impose a short-distance cutoff, subtract a spacetime-independent term that would be the zero-point energy of a flat spacetime with a short-distance cutoff, and then take the continuum limit (as the cutoff goes to zero). The subtraction eliminates the divergence in the propagator from nearby points that are separated by a null geodesic. When there are closed null geodesics, however, more than one null geodesic connects nearby points, and additional divergent contributions arise from vacuum fluctuations that propagate around these closed null curves. The result is that, as one approaches a Cauchy horizon, the finite, renormalized energy density can grow without bound. A divergence of the renormalized energy-momentum tensor at a Cauchy horizon, T ren αβ , was examined by various authors for free scalar field theory in several chronology violating spacetimes [64,65,66,67,68]. These results, together with the Hawking's energy-condition violation theorem of the previous section (classical chronology protection) led Hawking to propose his Chronology Protection Conjecture: The laws of physics prevent closed timelike curves from appearing. More precisely, Hawking conjectured that the laws of physics prevent local creation of closed timelike curves, as characterized by the existence of a compactly generated Cauchy horizon. In several examples, however, the value of T ren αβ (x) remains finite as the point x approaches the Cauchy horizon [69,70,71,72]. Subsequently Kay, Radzikowski and Wald (KRW) [73] showed that the twopoint function, from which the energy-momentum tensor is obtained, is singular, in the sense which will be explained below, for a free scalar field at some points on a compactly generated Cauchy horizon. Thus, although T ren αβ may be bounded as the point approaches the Cauchy horizon, it is not well-defined on the horizon. The KRW theorem uses the fact that for a Hadamard state the two-point function φ(x)φ(x ′ ) is divergent if and only if the two points x and x ′ can be connected by a null geodesic [74]. There is now a wide consensus that a physically reasonable state must be a Hadamard state (see, e.g. Ref. [75] for use of the Hadamard condition). This condition roughly states that the light-cone singularity structure must be the same as for the vacuum state in Minkowski spacetime. In a globally hyperbolic spacetime every point x has a convex normal neighborhood N x [31] small enough that no null geodesic leaves and reenters it. This implies for each point x in a globally hyperbolic spacetime the following property: For a small enough convex normal neighborhood N x and for any two points y, y ′ ∈ N x , the two-point function φ(y)φ(y ′ ) is divergent if and only if y and y ′ are connected by a null geodesic inside N x . Because this property is necessary to define T ren αβ (x), we say that the two-point function is singular at x if the property is not satisfied. Now, for a compactly generated Cauchy horizon, every past-directed null geodesic generator λ stays in a compact region. There must then be a point x such that, given any neighborhood N x , the geodesic λ passes through N x infinitely many times. This implies that for any convex normal neighborhood N x of x there are points y and y ′ in N x such that the two-point function φ(y)φ(y ′ ) is divergent. Hence we have the following theorem due to KRW: Theorem 4 The two-point function of a free scalar field is singular on certain points on a compactly generated Cauchy horizon. For explicit verification of the KRW theorem in examples with vanishing T ren αβ , see Ref. [76]. Although KRW proved this theorem for a scalar field, it would be straightforward to generalize their result to non-interacting fields of any spin. Because of examples in which the stress tensor does not diverge, and because the theorem does not imply that the strong quantum instability of the earlier examples is generic, one can argue that the quantum singularity is too weak to enforce chronology protection. Kim and Thorne [66] had already entertained that argument in their strong-instability example, finally suggesting that the issue can be decided only within a theory of full quantum gravity. Visser [77,78] similarly argues that the above theorem should be interpreted as the statement that quantum field theory in a background spacetime is unreliable on the Cauchy horizon, and that quantum gravity is needed to determine what really happens. The theorem does, however, indicate a drastic difference between the behavior of physical fields and spacetime on observed scales and their behavior on the smallest scales if there is to be chronology violation. Kay's F-locality condition mentioned before requires (among other things) that quantum scalar fields in non-globally hyperbolic spacetime commute for any two spacelike separated points in some neighborhood N x of any point x. The work of KRW showed that this condition cannot be satisfied in spacetimes with a compactly generated Cauchy horizon. Interestingly, it can be satisfied on the timelike cylinder obtained as the quotient of Minkowski space by a time translation [47,79]. However, this was shown not to be the case for a massive scalar field in a two-dimensional spacelike cylinder with a generic metric [80]. Loss of unitarity In the classical billiard-ball models mentioned above, it was found that the Cauchy problem is not well defined because there is often more than one solution for a given set of initial data [58,60]. In the corresponding quantum theory, however, it was found that the solution of the wave equation is unique. Although this observation gave some hope that the quantum theory of interacting systems with CTCs might be consistent, it turns out that unitarity is lost for interacting quantum fields in spacetimes with CTCs. Ways to recover a consistent quantum theory have been suggested, but all have features that seem undesirable. Unitarity of the scattering matrix S f i = δ f i − iT f i in quantum field theory can be expressed as 2 Im T f i = − n T nf T ni ,(3.1) As for free field theories, it was shown in Ref. [81] that, if the Cauchy problem is well defined for classical field equations, then these unitarity relations are satisfied. Thus, for example, the massless scalar field theory in the spacetime with a chronology violating wormhole studied by Friedman and Morris [51] is unitary. The unitarity relations (3.1) were studied for the λ ϕ 4 theory in the above-mentioned wormhole spacetime by Friedman, Papastamatiou and Simon [82] and in Gott spacetime by Boulware [83]. They defined the perturbation theory in a path-integral framework and found that the Feynman propagator i∆ F (x, y) has an extra imaginary part E(x, y): i∆ F (x, y) = θ(x 0 − y 0 )D(x, y) + θ(y 0 − x 0 )D(x, y) + E(x, y) . (3.2) At first order in λ the T f i in (3.1) with the initial and final states both being one-particle states corresponds to a tadpole diagram. The imaginary part of this diagram is essentially given by E(x, x), which is non-zero. This implies violation of the relation (3.1) because the right-hand side starts at order λ 2 . In an unpublished work Klinkhammer and Thorne showed using the WKB approximation that the quantum theory of the billiard system with CTCs is non-unitary. This non-unitarity arises due to the fact that the classical system allows a multitude of solutions for a given set of initial data and that the number of solutions depends on the initial data. Simple quantum mechanical models which mimic chronology violating interacting field theory were studied by Politzer [50]. He confirmed perturbative non-unitarity in a billiard model. He also studied some exactly solvable models exhibiting nonunitarity [84]. Fewster, Higuchi and Wells [79] studied a generalization of Politzer's model by solving the differential equation satisfied by the Heisenberg operators. They found that the canonical (anti-)commutation relations are not preserved in time, and, consequently, that the theory is non-unitary. They also compared their method and the path-integral quantization and found that the two methods give different results. This conclusion seems to be related to the observation [85] that the Schrödinger and Heisenberg pictures do not agree in non-unitary quantum mechanics. Loss of unitarity poses difficulties for the conventional Copenhagen interpretation of quantum mechanics. Suppose, for example, one makes a measurement in a region spacelike separated from the CTCs. One would expect that the result would be unaffected by the chronology violation. However, the measurement could be interpreted to have occurred either before or after the CTCs, and the probability assignment would depend on which interpretation was taken because of non-unitarity [86]. There have been a few proposals for eliminating non-unitarity or for finding a probability interpretation that accepts loss of unitarity: Adopting only the unitary part of the evolution operator [87] and making the non-unitary operator a restriction of a larger norm-preserving operator [85]; but the former would make the evolution of states highly nonlinear, and the latter necessitates the use of negative-norm states. Friedman et al. [82] and Hartle [88] have advocated the sum-over-histories approach to the interpretation of quantum mechanics. This gives a prescription for computing probabilities, but probabilites in a globally hyperbolic past of any CTCs are affected by the existence of CTCs in the future. Hawking [89] has advocated the use of the superscattering operator, i.e. the linear mapping from the initial to final density matrices. However, Cassidy [90] has found that the initial pure state evolves nonlinearly into a mixed state. Also, non-unitarity in the models studied in Ref. [79] is such that evolution cannot even be described by a superscattering matrix. Let us conclude this subsection by describing the work of Deutsch on the grandfather paradox in the context of quantum information theory [91]. Let the initial state containing the grandfather of the killer be |ψ and letρ be the density matrix resulting from the evolution of |ψ ψ| through the CTC region, which may or may not contain the killer. (Deutsch asserts that a generic pure state would inevitably evolve to a mixed state, violating unitarity, if there was a closed timelike curve.) The state |ψ ψ| in a Hilbert space H 1 and its future self,ρ, in a Hilbert space H 2 form a tensor product state |ψ ψ| ⊗ρ in H 1 ⊗ H 2 when they encounter one another. The interaction of the two parts is described by a unitary matrix U on H 1 ⊗ H 2 . After the interaction, the Hilbert space H 2 continues to the future and H 1 goes back to the past. The state going back to the past is obtained by tracing out the resulting state over H 2 and must equalρ. Thus, the consistency condition is Tr 2 U (|ψ ψ| ⊗ρ)U † =ρ. Deutsch showed that there are solutions to this equation for any initial state |ψ in simple examples. However, as we have seen, the evolution in chronology violating spacetimes cannot be described in general by a linear superscattering operator as envisaged in these models. IV. QUANTUM ENERGY INEQUALITIES A. Introduction In quantum field theory, the weak energy condition is violated by an energy-density operator whose vacuum expectation value is unbounded from below in Minkowski space, even for a free scalar field. The energy condition is recovered in the classical limit, because the terms responsible for its violation oscillate in time and a classical field measures an average of the fluctuating quantum field. This averaging argument leads in two related ways to energy conditions satisfied by quantum fields: It suggests that the averaged null energy condition, (2.1), may still be valid; the ANEC is sufficient for establishing chronology protection and topological censorship (as well as some singularity theorems [99,100,101]). And it suggests that a weighted average of the energy density along a timelike curve may be bounded from below. Indeed, in a Minkowski-space context, Ford and Roman found such bounds [94,95,96], commonly called quantum (weak) energy inequalities (QEIs). In the next subsection we briefly discuss some work related to the ANEC in quantum field theory and then review recent developments in QEIs. (See Refs. [9,97] for more comprehensive reviews of the QEIs.) First, however, we present a simple example of the violation of the weak energy condition by a free scalar field in Minkowksi space. The example relates the violation to the subtraction of an infinite constant from a formally positive energy-density operator T 00 ; and it exhibits the oscillation in time that leads to averaged inequalities. The classical energy density T 00 for a real massless scalar field ϕ in four dimensions is T 00 = 1 2 φ 2 + (∇ϕ) 2 . (4.1) To define the renormalized energy-density operator, one subtracts the infinite vacuum energy density, 0|T 00 |0 . This renormalized operator is no longer positive-definite, although the energy, its spatial integral, is. One therefore expects states for which the expectation value of the renormalized energy-density operator is negative, and this is indeed the case [92,93], as seen in the following simple example. Let a † (p) be the creation operator for the scalar particle with momentum p and consider the normalized superposition of the vacuum state and a two-particle state, |ψ = cos α|0 + sin α √ 2 d 3 p (2π) 3 2p f (p)a † (p) 2 |0 (4.2) with p ≡ p and d 3 p (2π) 3 2p [f (p)] 2 = 1 ,(4.3) where the function f (p) has been assumed to be real for simplicity. For example, if the function f (p) is chosen to be peaked about p = p by letting f (p) = (12π 2 p/δ 3 ) 1/2 if p − p < δ and zero otherwise, then one finds the expectation value of the renormalized energy-momentum tensor at the origin for δ ≪ p as follows: ψ|T ren 00 (t, 0)|ψ ≈ pδ 3 6π 2 sin α − √ 2 cos α cos 2pt + 2 sin α . (4.4) This quantity is negative at t = 0, say, if sin α(cos α − √ 2 sin α) > 0, and it is unbounded from below as a function of p. The NEC is also violated because for any null vector t α in the direction perpendicular to p, one finds ψ|T ren αβ |ψ t α t β ≈ ψ|T ren 00 |ψ (t 0 ) 2 . As anticipated, the violation of the weak energy condition is associated with the oscillating term in Eq. (4.4). Our review reflects the view that overcoming the obstacles to forming CTCs and wormholes has become increasing difficult. The opposite view is taken in a review with extensive references by Lemos et al. [98]. B. Results and implications The NEC can be replaced by the ANEC (or a condition similar to it) in proving some singularity theorems [99,100,101] as well as chronology protection and topological censorship. It has been shown that the ANEC holds in free quantum scalar field theory in Minkowski space and in two-dimensional curved spacetime under certain assumptions [102,103,107]. However, Klinkhammer [103] has pointed out that the ANEC does not hold if one compactifies any of the space dimensions in Minkowski space. If, for example, one identifies the space coordinate x with x + L in two-dimensional Minkowski space, then the energy momentum tensor for the massless scalar field in the vacuum state is non-zero due to the Casimir effect and given by [104,105,106] 0|T 00 ren |0 = 0|T ren 11 |0 = −π/6L 2 and 0|T ren 01 |0 = 0. Hence for k 0 = ±k 1 = 1, one has 0|T ren αβ |0 k α k β = −π/3L 2 = const, and the ANEC is necessarily violated. Wald and Yurtsever also point out that the ANEC is violated in generic spacetimes for the minimally-coupled massless scalar field theory [107]. (However, it would be physically wrong to conclude that the Casimir effect with perfectly reflecting mirrors violates the ANEC. See, e.g. [108].) There have been some attempts to rescue the ANEC. Yurtsever proposed that the integral of ψ|T ren αβ |ψ k α k β along the null geodesic, to which k α is tangent, may be bounded below by a state-independent constant, and he shows that, if true, it can be used to rule out some wormhole spacetimes [109]. Flanagan and Wald investigated the ANEC smeared over Planck scale in spacetimes close to Minkowski space, imposing the semi-classical Einstein equations and showed that it holds for pure and mixed states if the curvature scale is much larger than the Planck scale and if incoming gravitational waves do not dominate the spacetime curvature [110]. For a minimally coupled scalar field, the classical energy-density ρ ≡ T αβ t α t β , where t α is a timelike vector of unit length, is positive definite, and can be given in the following form: ρ(x) = j P (j) ϕ(x) 2 , (4.5) where P (j) is a differential operator with smooth coefficients. Let us define the energy-density operator, in the corresponding quantum theory, normal-ordered with respect to a reference state |ψ 0 as follows: :ρ(x): ≡ j P (j)φ (x) 2 − ψ 0 | P (j)φ (x) 2 |ψ 0 . (4.6) Although each of these terms is infinite, the difference is finite and uniquely determined. This operator differs from the renormalized energy-density operator by a smooth c-number function which depends only on spacetime properties. Now, let x = γ(t) be a timelike curve and g(t) be a smooth real function satisfying +∞ −∞ g 2 (t) dt = 1. A QEI takes the following form in general: +∞ −∞ dt g 2 (t) ψ|ρ[γ(t)]|ψ ≥ −C(γ, g) (4.7) for any Hadamard state |ψ , where C(γ, g) is a positive number independent of the state |ψ . One of the first QEIs was for the minimally-coupled massless scalar field in four-dimensional Minkowski space with the Lorentzian sampling function. It has the form [94,95,96] +∞ −∞ dt ψ| :ρ(t, x): |ψ τ π(t 2 + τ 2 ) ≥ − 3 32π 2 τ 4 . (4.8) The QEIs appear to place stringent constraints on spacetimes with nontrivial topology, spacetimes with CTCs, as well as on warp-drive [111] spacetimes. To make these constraints rigorous, however, inequalities in general spacetimes are required; and QEIs with compactly-supported sampling functions are more useful in these applications. Flanagan [112] derived QEIs for minimally-coupled massless scalar field in general two-dimensional curved spacetime with general smooth sampling functions. (The bounds given by Flanagan's QEIs are optimal.) Fewster was able to establish QEIs in general globally hyperbolic spacetime in any dimensions using a general smooth sampling function for minimally-couple scalar field of arbitrary mass [113]. We present his inequality as a representative of the QEIs. Let us consider, for simplicity, a timelike curve given by x = 0 for a given coordinate system in a globally hyperbolic spacetime and write the minimally-coupled scalar field on this curve,φ(t, x = 0), simply asφ(t). Fewster's inequality relies on the following lemma, which is an immediate consequence of the work by Radzikowski [74], which gives the Hadamard condition in the language of microlocal analysis: Lemma 1 Let P be a differential operator with smooth coefficients. Define the double Fourier transform of the point-separated two-point function for Pφ on a Hadamard state |ψ 0 with a smooth compactly-supported sampling function g(t) bŷ ∆ P g (k 1 , k 2 ) ≡ +∞ −∞ dt 1 +∞ −∞ dt 2 e i(−k1t1+k2t2) g(t 1 )g(t 2 ) ψ 0 |[Pφ(t 1 )][Pφ(t 2 )]|ψ 0 . (4.9) Then∆ P g (k 1 , k 2 ) tends to zero faster than any polynomial as k 1 → +∞ or k 2 → +∞. The physical interpretation of this lemma is that a Hadamard state is annihilated by the "positive-frequency" part +∞ −∞ dt e iktφ (t) in the limit k → +∞. The following result [97], a generalization of the argument used in [114], essentially gives the QEI of Ref. [113]: Lemma 2 Define the following normal-ordered product: :[Pφ(x)] 2 : ≡ [Pφ(t)] 2 − ψ 0 |[Pφ(t)] 2 |ψ 0 . (4.10) Then, the expectation value of :[Pφ(x)] 2 : averaged over the timelike curve x = 0 with the sampling function g 2 (t) satisfies +∞ −∞ dt g 2 (t) ψ| :[Pφ(t)] 2 : |ψ ≥ − 1 π ∞ 0 dk∆ P g (k, k) . (4.11) The right-hand side is finite by Lemma 1. Proof. The left-hand side, which we denote by A, can be written as A = +∞ −∞ dk 2π +∞ −∞ dt 1 +∞ −∞ dt 2 g(t 1 )g(t 2 ) × { ψ|P ϕ(t 1 )P ϕ(t 2 )|ψ − ψ 0 |P ϕ(t 1 )P ϕ(t 2 )|ψ 0 } e −ik(t1−t2) . (4.12) The expression inside the curly brackets is invariant under the interchange t 1 ↔ t 2 because the commutator [ϕ(t 1 ), ϕ(t 2 )] is state-independent. Hence, one can restrict k to be positive and multiply the integral by two. After this operation, we see that the first term is of the form ∞ 0 dk ψ|O † (k)O(k)|ψ ≥ 0 and the second term is exactly the right-hand side of the inequality (4.11). The Fewster inequality follows immediately from this lemma and Eq. (4.6). Although most of the work on QEIs involves minimally-coupled scalar fields, corresponding inequalities have also been derived for spin-one fields including the electromagnetic field [115,116] and for spinor fields [117,118,119]. The QEIs impose severe restrictions on wormhole spacetimes [96]. Since it is difficult to calculate the QEI bounds in curved spacetime (see Ref. [120,121] for examples of explicit bounds in de Sitter and other spacetimes), these analyses typically use the QEIs in Minkowski spacetime, e.g. Eq. (4.8), with sampling times shorter than the shortest curvature scale involved in the problem to justify their use. QEIs are used in this manner to justify the expectation that the negative energy density, which are necessary for a traversable wormhole to exist as we have seen, cannot be larger than C 2 /r 4 m , where r m is the smallest scale in the problem and C is a number which is typically taken to be 10 2 or so to err on the conservative side. Then the argument to put restrictions on wormholes goes very roughly as follows. For a wormhole of size r 0 with typical curvature scale 1/r 2 0 at the throat, the Einstein equations there tell us that G/r 2 0 C /r 4 m . Thus, r 2 m /r 0 Cℓ p , where ℓ p = √ G = 1.6 × 10 −33 cm is the Planck length. If the smallest length scale r m is comparable to the wormhole size r 0 , then the wormhole cannot be much larger than the Planck length. To circumvent this conclusion, one needs to have r m ≪ r 0 , typically by concentrating the negative energy in a very narrow region near the throat. Thus, it appears very difficult to construct (theoretically) a traversable wormhole, and that to do so would require fine-tuning of parameters. (See Refs. [122,123] for recent wormhole models which attempt to evade the argument of Ref. [96]. The apparent evasion of the constraints relies on evaluating the energy in one frame, and frame-independent statement of the QEIs reinstates the constraint [124,125].) A similar argument was used in Ref. [126] to demonstrate that the warp-drive spacetime as given in Ref. [111] is incompatible with ordinary quantum field theory. This spacetime contains a bubble of nearly flat spacetime which could move faster than light. (Olum shows that superluminal travel of this sort, defined suitably, must involve violation of WEC [127].) A spaceship could sit inside this bubble and thus move superluminally with respect to the metric outside the bubble. However, there is a sphere with a finite thickness where the energy must be negative to satisfy the Einstein equations. It turns out that the smallest scale involved is the thickness ∆ of the sphere of negative energy. The situation is then similar to that of a wormhole with r m ∼ r 0 . The negative energy must be concentrated on a sphere of thickness at most a few orders of magnitude larger than ℓ p , and the energy density there must be only a few orders of magnitude smaller than /ℓ 4 p . The total negative energy needed is estimated to be much larger than the (positive) total mass of the visible universe for a bubble size large enough to fit in a spaceship. C. Future problems As we have seen, the ANEC is known to be violated even in ordinary quantum field theory in curved spacetime. It would be interesting to see if Yurtsever's proposal of bounding the ANEC integral from below can be realized in a useful way. It is also interesting to find out whether it continues to be satisfied in spacetimes which are not close to Minkowski space if one enforces the semi-classical Einstein equations. It would also be useful to investigate the validity of the ANEC for interacting field theories (in Minkowski space to start with). Interestingly, Verch has shown that the ANEC holds in two-dimensional Minkowski spacetime for any interacting field theory with a mass gap [128]. QEIs have so far been established primarily for free fields. Recently, Fewster and Hollands [129] have shown that there are QEIs in two-dimensional conformal field theories in Minkowski space. It would be interesting to investigate QEIs in more general interacting field theories, considering the recent work in which a static negative energy region was constructed using a 2+1 dimensional interacting field theory [130]. One should also investigate a possible role the QEIs could play in recently proposed models of dark energy with the negative pressure exceeding the energy density [131,132,133]. The QEIs place bounds only on the difference of the averaged energy density of the given state and that of the reference state. Strictly speaking, they have nothing to say about the averaged energy density of a single state, and they therefore do not restrict Casimir-type energy. The average energy density of the reference state was implicitly asssumed to be at most of order /r 4 m , where r m was the smallest characteristic scale. This assumption is reasonable, but attempts to rigorously justify it are still in their infancy. It is possible to apply the QEIs to find bounds on Casimir-type energies [134], and "absolute QEIs", which do not need reference states, have been obtained recently [135,136]. FIG. 1 : 1A simple spacetime with CTCs and a generalized Cauchy surface S is shown in this figure. (The precise definition of a generalized Cauchy surface is given below.) Two parallel segments of equal length are removed from Minkowski space, two disjoint edges are joined to the left and right sides of each slit, and edge points related by the timelike translation T are then identified. FIG. 3 : 3The sphere labeled Σ1 is identified with the timelike separated sphere at T (ΣI ) FIG. 4: Misner space is the region between the two identified timelike lines. A solid null ray loops around the space, approaching a closed null geodesic marked CNC. Closed timelike curves (one labeled CTC) pass through each point above the closed null geodesic.t = 4. FIG. 5 : 5The wormhole spacetime of Morris et al. is shown in this figure. The two mouths at the same proper time are identified. FIG. 2: A wormhole is constructed by removing two balls and identifying their spherical boundaries ΣI and ΣII after a reflection: Points labeled by the same letter, with subscripts I and II, are identified.P I P II R I R II Q I Q II Σ I Σ II Theorem 5The expectation value of the normal-ordered energy density given by Eq. (4.6) on a Hadamard state |ψ satisfies the following inequality: dt g 2 (t) ψ| :ρ(t, x = 0): |ψ ≥ − 1 π+∞ −∞ j ∞ 0 dk∆ P (j) g (k, k) . (4.13) The geometries of the early and present universe are apparently deSitter, and work by Witt and Morrow[28] shows that every 3-manifold occurs as a slice of a deSitter 4-geometry -though not in general a slice consistent with our universe. By adding an additional degree of freedom, however, one can apparently obtain systems that do not admit classical solutions[11,61]. 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[ "S Fichet \nInternational Institute of Physics\nUFRN\nOdilon Gomes de Lima\n1722 -Capim Macio -59078-400 -Natal-RNAvBrazil\n", "B Herrmann \nLAPTh\nUniversité de Savoie\nCNRS\n\n", "Y Stoll \nLAPTh\nUniversité de Savoie\nCNRS\n\n", "\nChemin de Bellevue\nBP 110F-74941Annecy-le-VieuxFrance\n" ]
[ "International Institute of Physics\nUFRN\nOdilon Gomes de Lima\n1722 -Capim Macio -59078-400 -Natal-RNAvBrazil", "LAPTh\nUniversité de Savoie\nCNRS\n", "LAPTh\nUniversité de Savoie\nCNRS\n", "Chemin de Bellevue\nBP 110F-74941Annecy-le-VieuxFrance" ]
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We point out that the hypothesis of a SU (5)-like supersymmetric Grand Unified Theory (GUT) implies a generic relation within the flavour structure of up-type squarks. Contrary to other wellknown SU (5) relations between the down-quark and charged lepton sectors, this relation remains exact in the presence of any corrections and extra operators. Moreover it remains valid to a good precision at the electroweak scale, and opens thus new possibilities for testing SU (5)-like GUTs. We derive the low-energy effective theory of observable light up-type squarks, that also constitutes a useful tool for squark phenomenology. We use this effective theory to determine how to test SU (5) relations at the LHC. Focussing on scenarios with light stops, compatible with Natural SUSY, it appears that simple tests involving ratios of event rates are sufficient to test the hypothesis of a SU (5)-like GUT theory. The techniques of charm-tagging and top-polarimetry are a crucial ingredient of these tests.PACS numbers:
10.1016/j.physletb.2015.01.013
[ "https://arxiv.org/pdf/1403.3397v4.pdf" ]
119,218,599
1403.3397
2ef9c093e9cbf6c7e2cf971ddc959a05d35e4eaa
13 Mar 2014 (Dated: March 17, 2014) S Fichet International Institute of Physics UFRN Odilon Gomes de Lima 1722 -Capim Macio -59078-400 -Natal-RNAvBrazil B Herrmann LAPTh Université de Savoie CNRS Y Stoll LAPTh Université de Savoie CNRS Chemin de Bellevue BP 110F-74941Annecy-le-VieuxFrance 13 Mar 2014 (Dated: March 17, 2014)A new flavour imprint of SU(5)-like Grand Unification and its LHC signatures We point out that the hypothesis of a SU (5)-like supersymmetric Grand Unified Theory (GUT) implies a generic relation within the flavour structure of up-type squarks. Contrary to other wellknown SU (5) relations between the down-quark and charged lepton sectors, this relation remains exact in the presence of any corrections and extra operators. Moreover it remains valid to a good precision at the electroweak scale, and opens thus new possibilities for testing SU (5)-like GUTs. We derive the low-energy effective theory of observable light up-type squarks, that also constitutes a useful tool for squark phenomenology. We use this effective theory to determine how to test SU (5) relations at the LHC. Focussing on scenarios with light stops, compatible with Natural SUSY, it appears that simple tests involving ratios of event rates are sufficient to test the hypothesis of a SU (5)-like GUT theory. The techniques of charm-tagging and top-polarimetry are a crucial ingredient of these tests.PACS numbers: INTRODUCTION One of the most fascinating features of the Standard Model of Particles (SM) is that the matter fields fit into complete representations 10 and5 of the SU (5) gauge group, as 10 = (Q, U, E),5 = (L, D) [1]. This suggests that the SM is the low-energy effective theory (EFT) of a SU (5)-like Grand Unified Theory (GUT) -either SU (5)symmetric or containing SU (5) as a subgroup (see [2] and the reviews [3]). Many classes of SU (5)-like GUTs exist, with a variety of low-energy features. [17] From the viewpoint of testing whether or not Nature is microscopically SU (5)-symmetric, this model-dependence is an irreducible theoretical uncertainty. It is thus highly challenging to find how to test the SU (5) hypothesis in a way as model-independently as possible. In this Letter we point out a so-far unexplored SU (5) property confined to the up-squark sector, that is much less sensitive to model-dependence than previous, wellstudied properties. This property arises in the flavour structure of up squarks, and its persistence at the weak scale is closely checked. We then derive the up-squark effective theory, applicable whenever unobserved squarks are assumed to be heavy. We use it to determine the most direct way to test the SU (5) hypothesis using our new relations, focussing on the light-stops scenario -that is favoured by LHC and consistent with Natural SUSY. A NEW RELATION IN THE UP-SQUARK SECTOR Besides gauge-coupling unification at the GUT scale, matter field unification implies the famous relation y d = y t ℓ(1) between down quark and lepton Yukawa couplings. This relation is exact up to GUT scale threshold corrections that arise from integrating out heavy GUT states. In addition, the renormalization group (RG) flow down to the weak scale has to be taken into account, such that y d − y ℓ unification is not straightforward and has generated a lot of litterature (see [5] and many subsequent works). Grand Unified Theories are closely linked to supersymmetry (SUSY), which strikingly favours the gauge coupling unification in its most simple realizations (MSSM, NMSSM) as well as in more evolved hypothesis (e.g. extra-dimensions, extra SU (5)-like matter [6]). In SUSY SU (5)-like GUTs, the two Higgs supermultiplets, denoted H 1 , H 2 (≡ H d , H u ), need to be embedded in a 5 and a5 representation respectively, denoted H 1 and H 2 . Interactions between matter and Higgses are given by the superpotential W = λ ij 1 H 1 10 i5j + λ ij 2 H 2 10 i 10 j .(2) Below the GUT scale, assuming that unwanted Higgs triplets are heavy due to some splitting mechanism [3,7], the superpotential reads W = y ij u H 2 Q i U j + y ij d H 1 Q i D j + y ij ℓ H 1 L i E j .(3) where y d = y t ℓ = λ 1 . Furthermore, SUSY needs to be broken around the TeV scale, and the Lagrangian also contains the SUSY-breaking scalar trilinear terms a u,d,ℓ , L soft ⊃ a u h uqũ + a d h dqd + a l h dlẽ ,(4) where h u,d are the two Higgs doublets andq,ũ,d,l,ẽ are the squarks and sleptons. The Lagrangian also contains five scalar mass terms denoted by m 2 Q,U,D,L,E (see, e.g., [8]). With the assumption that the source of SUSY breaking is SU (5)-symmetric, the above terms satisfy the relations a d = a t ℓ , m 2 Q = m 2 U = m 2 E , m 2 L = m 2 D(5) at the GUT scale, which are exact up to GUT threshold corrections. The correlations that the SU (5) relations (5) induce between the quark and lepton sectors have been extensively studied within specific GUT scenarios [10]. Such correlations, although certainly interesting in specific models, can be hardly used as a generic test of the SU (5) hypothesis, as RG corrections received by quark and leptons are fundamentally different. We now point out the existence of relation implied by the SU (5) matter embedding, which seems unnoticed so far. The 10 i 10 j term in the superpotential (2) is symmetric, such that only the symmetric part of λ ij 2 is selected. This leads to a symmetric top Yukawa coupling at the GUT scale, y u = y t u .(6) Moreover, in presence of SUSY, this enforces that the trilinear coupling is also symmetric, a u = a t u .(7) These relations are confined within the up-(s)quark flavour space, and can thus be expected to be more stable against quantum corrections than the quark-lepton relations. Moreover, these relations remain rigourously exact in presence of any GUT threshold corrections, because the up-squark self-energy gets in any case contracted with 10 i 10 j and is therefore always symmetric. [18] For a non-SUSY theory, Eq. (6) does not seem particularly exploitable. In this case the only physical parameters are mass eigenvalues and CKM angles, and these would not be enough to find out whether or not y u is symmetric. The situation becomes different once one considers broken SUSY, because more degrees of freedom of the Yukawa matrices are probed by the superpartners. THE UP-SQUARK MASS MATRIX Let us see how our new SU (5) relations translate into observable properties. As the relations (6) and (7) hold within the up-squark sector, we need to scrutinize the up-squark mass term L ⊃ũ † M 2 uũ , wherẽ u = (ũ L ,c L ,t L ,ũ R ,c R ,t R ) t contains the six up-squarks states. In the super-CKM basis, defined such that the Yukawa matrices are diagonal, the up-squark mass matrix has the form M 2 u = m 2 Q + O(v 2 )1 3 vu √ 2â u + O(vM )1 3 vu √ 2â t u + O(vM )1 3m 2 U + O(v 2 )1 3 ,(8) where M denotes the SUSY scale. In general, â u = W † a u V ,m 2 U = V † m 2 U V ,m 2 Q = W † m 2 Q W , with y u = V †ŷ u W ,ŷ u diagonal. The mass matrix M 2 u involves only a u , m 2 Q and m 2 U , so that we do not have to consider the other SUSY-breaking terms at this stage. As the observables we are going to study are not sensitive to CP violation, we can take these parameters to be real without loss of generality. The SU (5) relation (6) implies V * u = W u . Dropping the conjugates as all parameters are real, it appears that the SU (5) relations are satisfied in the super-CKM basis, a u =â t u ,m 2 Q ≈m 2 U . The mass matrix M 2 u involves only a u , m 2 Q and m 2 U , so that we do not have to consider the other SUSY-breaking terms at this stage. As the observables we are going to study are not sensitive to CP violation, we can take these parameters to be real without loss of generality. Therefore the up-squark mass matrix possesses a peculiar structure above the GUT scale. However below the GUT scale this structure is potentially spoiled by the non-SU (5) quantum corrections. It is thus necessary to evaluate how stable the relations a u = a t u and m 2 Q ≈ m 2 U remain upon the RG flow. This can be qualitatively verified using the two-loop RG equations of MSSM-like theories ( [11], see also [10]). The only sizeable discrepancy appears between the low-energy m 2 Q 33 and m 2 U 33 . The other discrepancies are overall negligible with respect to other theoretical and experimental sources of uncertainty. This conclusion is confirmed by the numerical evaluations made in the next Section. We parametrize the complete up-squark mass matrix as M 2 u =                 .(9) The above low-energy SU (5) relations, a u = a t u and m 2 Q ≈ m 2 U , then translate into m 2 15 ≈ m 2 24 , m 2 16 ≈ m 2 34 , m 2 26 ≈ m 2 35 ,(10)m 2 12 ≈ m 2 45 , m 2 13 ≈ m 2 46 , m 2 23 ≈ m 2 56 ,(11)m 2 11 ≈ m 2 44 , m 2 22 ≈ m 2 55 .(12) UP-SQUARK EFFECTIVE THEORY Although the pattern of squark masses is arbitrary in full generality, a likely situation is that the masses exhibit some hierarchy. This is favoured from naturalness considerations, from LHC bounds, as well as from certain classes of models. In such a situation the physics of the light squarks can be conveniently captured into a low-energy effective theory, where heavy squarks are integrated out. Let us reorganize the up-squark mass term such that L ⊃ũ † M 2 uũ ≡ Φ † M 2 Φ = φ † , φ † M 2M 2 M 2 † M 2 φ φ ,(13) whereφ contains the heavy states and φ the light ones. The relevant piece of the corresponding Lagrangian has the general form L ⊃ DΦ 2 − Φ † M 2 Φ + Oφ +Ôφ + h.c. ,(14) where O andÔ represent the interactions with other fields, that are potentially exploited to probe the upsquark sector. Assuming that the eigenvalues ofM 2 are large with respect to the energy at which one probes the theory, the heavy squarksφ can be integrated out, [19] leaving the low-energy Lagrangian of light squarks, L eff = Dφ 2 + O−Ô M −2 −M −4 ∂ 2 M 2 − O 2M 2 †M −4M 2 φ + h.c. −φ † M 2 −M 2 †M −2M 2 − 1 2 M 2 †M −4M 2 , M 2 φ .(15) In this effective Lagrangian we keep only the leading and the subleading terms of the E 2M −2 expansion relevant for our purposes. Here, E denotes the energy scale. It contains in principle higher dimensional couplings and derivative terms, which are either subleading or irrelevant for the observables we are going to consider, and are thus neglected. To obtain Eq. (15), one has to use the field redefinition Φ → (1 − 1 2M 2 †M −4M 2 )Φ in order to canonically normalize the light squarks. The { , } is the anti-commutator. The imprint of the heavy up-squarks in the light upsquarks Lagrangian (15) appears as corrections to the light up-squark masses and couplings. Physically, these corrections have to be understood both as tree-level exchange of heavy up-squarks, and as the first terms of the expansion with respect to the small parameters that describe mixing of heavy and light squarks. We emphasize that, although this effective theory approach might bear some resemblance with the "mass insertion approximation" (MIA) [15], the two approaches are fundamentally different. The MIA is an expansion in the limit of small off-diagonal elements of the mass matrix M 2 , i.e. in terms of small parameters M 2 i =j / tr{M 2 }. In contrast, the expansion parameter of the effective theory is E 2M −2 , and M 2 can have arbitrary off-diagonal entries. From Eq. (15) we see that flavour-violating couplings of the light squarks enter at first order and are controled byM −2M 2 . The flavour respecting couplings will be instead modified at the second order. The light mass matrix M 2 receives a correction independent of M 2 at first order, and corrections proportional to M 2 at second order. TESTING SU (5)-LIKE GUTS AT THE LHC: THE EXAMPLE OF TWO LIGHT-STOPS Armed with the up-squark effective Lagrangian, we are now ready to test the low-energy SU (5) relations a u ≈ a t u and m 2 Q ≈ m 2 U . In this Letter we focus on LHC physics, in the assumption that the lightest squarks are produced at the LHC, while the unobserved squarks are heavy such that the truncation of the effective Lagrangian (15) is valid. Note that the expansion parameter behaves as the mass squared, such that for masses of, e.g., m φ ∼ E ∼ 1 TeV and mφ ∼ 3 TeV, higher terms of the expansion are already suppressed by a factor of order 10. Supersymmetric scenarios with two light squarks that are mainly stops, φ = (t L ,t R ),φ = (ũ L ,c L ,ũ R ,c R ) are favoured by LHC data and are one of the features of the Natural SUSY framework [12]. For simplicity, we will in the following denote these two light squarks as "stops". We consider R-parity conserving scenarios. We assume that both stops are copiously produced through flavordiagonal processes. Indeed, gluon-initiated production of squark pairs will be the dominant production channel for squarks having a mass of about 1 TeV at the LHC with a center-of-momentum energy of 13 or 14 TeV. The effective Lagrangian of the stops is obtained by expanding Eq. (15), where we introduce the parameters m 2 11,44 ≡ Λ 2 1 , m 2 22,55 ≡ Λ 2 2 . We observe that the stop mass matrix depends at the leading order on m 2 33,66,36 . There is no relevant information to test the SU (5) hypothesis in this matrix, and the higher corrections to the mass matrix cannot be exploited either. The stop mass eigenstates are (t 1 ,t 2 ) t = R(θ) t (t L ,t R ) t , where R(θ) is a SO(2) rotation with the stop mixing angleθ. The stop mixing angle can be large and is a crucial feature of low-energy SUSY. [20] Knowing its value will not be necessary in the following SU (5) tests, although it will appear in intermediate steps. The case mt 1,2 > mW > mB As a first typical example, we consider the case where stops can decay both to the lightest neutralino, which is mostly bino-like,χ 0 1 ≈B, and to a mostly wino-like second-lightest neutralinoχ 0 2 ≈W . This mass hierarchy is inspired by the fact that in GUTs we have the approximate relation MB ≈ MW /2 at the weak scale. Let us focus on the flavour-violating couplings of the stops, that appear at first order in Eq. (15). The operators that couple the stops toB andW in Eq. (15) areÔ ∝ (u L , c L , −4 u R , −4 c R )B andÔ ∝ (u L , c L )W [13]. At first order in the effective Lagrangian (15), the flavour-violating couplings of the stops are proportional toB   m 2 13 Λ 2 1 u L + m 2 23 Λ 2 2 c L − 4 m 2 34 Λ 2 1 u R − 4 m 2 35 Λ 2 2 c R m 2 16 Λ 2 1 u L + m 2 26 Λ 2 2 c L − 4 m 2 46 Λ 2 1 u R − 4 m 2 56 Λ 2 2 c R   R(θ) t 1 t 2 ,(16)W   m 2 13 Λ 2 1 u L + m 2 23 Λ 2 2 c L m 2 16 Λ 2 1 u L + m 2 26 Λ 2 2 c L   R(θ) t 1 t 2 .(17) The low-energy SU (5) relations Eq. (10)-(12) directly relates these effective couplings. A numerical analysis shows that non-SU (5) discrepancies induced by the RG flow are typically of order 1%. The stop decay chains of interest aret 1,2 → qB andt 1,2 → qW → q Z/hB, where q = u, c observed as hard jets. These two decays, occurring respectively through U (1) Y and SU (2) L interactions, are observed separately as the final states are different. In the complete production-decay process, we ask for a flavour violating decay for one out of the two produced stops. Let us first assume that one simply counts the amount of flavour-violating events occuring in the decays toB andW , without disentangling between the nature of the jets nor between the original stops. It turns out that, whenever the SU (5) hypothesis is verified, and for arbitrary stop mixing angle, both decay rates N Y , N L are controlled by the same combination of parameters, according to Here, σt i denotes the inclusive cross section of the flavour-conserving production process pp →t it * i at LHC. It is thus, in principle, possible to test the SU (5) hypothesis using these simple decay rates. However estimating precisely the overall factors relating N L and N Y to the quantity (18) can be challenging because this requires to know the realistic cross-section including all the kinematic selections. This drawback can be avoided using charm-tagging techniques. The use of c-tagging allows to identify a fraction N c Y,L of the jets due to c-quarks, where N Y,L = N c Y,L + N c Y,L . The remaining fraction, N c Y,L includes the jets that cannot be identified as c-quarks, i.e. up-quarks and mistagged charm-jets. As a result, whenever the SU (5) hypothesis is fulfilled, the four decay rates N c Y,L and N c Y,L satisfy the relation N c Y N c L = N c Y N c L .(19) Note that this test of the SU (5) hypothesis does not require to know the stop-mixing angle, nor to have a precise estimation of the stop-production cross-sections. The case mW > mt 1,2 > mB As a second example, let us assume that the stops can only decay into the lightest neutralinoχ 0 1 ≈B. Using only the information from flavour-violating decays intõ B+jets, it turns out it is not possible to test the SU (5) hypothesis, even if one disentangles the decays oft 1 and t 2 , even if one disentangles the decays oft 1 andt 2 and knows the value of the stop mixing angle. For this case, let us consider the flavour-conserving interactions of the stops. The relevant operator coupling to the stops in Eq. (15) is O ∝ (t L , −4 t R )B. The decays of interest for our SU (5) testing purpose are thus t 1,2 →B t L,R . For the stops decaying into top quarks, techniques of top polarimetry [14] potentially provide a way to distinguish between decays into top of left and right chirality, t L and R . Using kinematic information it is also possible to disentanglet 1 andt 2 decays. Considering that one can both distinguish between the original stopst 1 ,t 2 and the outgoing t L , t R , we end up with four observable decay rates N 1,L , N 1,R , N 2,L , and N 2,R . At leading order, the matrix coupling the stops to O is unitary, OR(θ)(t 1 ,t 2 ) t . The four decay rates satisfy thus two non trivial relations, conveniently chosen as N 1,L N 1,R = 1 16 2 N 2,R N 2,L , 16 N 1,L σt 1 + N 2,L σt 2 = N 1,R σt 1 + N 2,R σt 2 . (20) Note that the various ratios of decay rates provide inequivalent measurements of the stop mixing angle. A third relation coming from the overall normalization also exists but is related to the total cross-section, that needs to be estimated precisely, and which is not necessary for our purpose of testing the SU (5) hypothesis. The relevant information arises instead at next-to-leading order in the distortion of the coupling matrix in the stop effective theory. Whenever the SU (5) hypothesis is true, the coupling takes the form . B t L , −4 t R 1 − a −b −b 1 − a R(θ) t 1 t 2(21) (23) The correction to the coupling matrix in Eq. (21) induces a slight overall decrease in the stop decay rate. One quick way to see this is that the determinant of the coupling matrix is smaller than unity. This slight global decrease would probably be most difficult to detect because it requires a very good knowledge of the production cross-section including all cuts. The crucial signature lies instead in the matrix structure. If the SU (5) hypothesis is true, the distortion is symmetric, which implies that (24) Instead, if the SU (5) hypothesis is not true, both relations are not satisfied. Again, this test uses only ratios of decay rates, and thus does not depend crucially on the overall normalization. SUMMARY In this Letter we point out the existence of SU (5)like GUT relations that have remained unexplored so far. These new SU (5) relations are insensitive to GUT threshold corrections and are confined to the flavour structure of the up-squark sector, which makes them more stable with respect to quantum corrections than the other well-known SU (5) relations. Due to these features, the new relations open new ways to test whether Nature is microscopically SU (5)-symmetric. We focus on tests at the LHC using unprecedented simple tests involving the properties of up-type squarks. We set up the effective theory for light up-type squarks, that is also a useful tool beyond our SU (5)-testing purpose. 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Nojiri, JHEP 1303, 105 (2013) [arXiv:1211.7261 [hep-ph]]. . G Belanger, R M Godbole, L Hartgring, I Niessen, arXiv:1212.3526JHEP. 1305167G. Belanger, R. M. Godbole, L. Hartgring and I. Niessen, JHEP 1305, 167 (2013) [arXiv:1212.3526]. . B Tweedie, arXiv:1401.3021hep-phB. Tweedie, arXiv:1401.3021 [hep-ph]. . F Gabbiani, A ; J S Masiero, S Hagelin, T Kelley, ; F Tanaka, E Gabbiani, A Gabrielli, L Masiero, Silvestrini, hep-ph/9604387Nucl. Phys. B. 322321Nucl. Phys. BF. Gabbiani and A. Masiero, Nucl. Phys. B 322, 235 (1989). J. S. Hagelin, S. Kelley and T. Tanaka, Nucl. Phys. B 415, 293 (1994). F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, Nucl. Phys. B 477, 321 (1996) [hep-ph/9604387]. . S Fichet, B Herrmann, Y Stoll, Work in progressS. Fichet, B. Herrmann, Y. Stoll, Work in progress. Note that some early models are excluded by proton decay. 4Note that some early models are excluded by proton de- cay [4]. Note yu, au are symmetric, but generally not hermitian. Here, we assume them to be real since no CP-violating observables will be studied. Note yu, au are symmetric, but generally not hermitian. Here, we assume them to be real since no CP-violating observables will be studied. We remind that it is not necessary to use eigenstates to perform this operation. We remind that it is not necessary to use eigenstates to perform this operation.
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[ "Development of a Risk-Free COVID-19 Screening Algorithm from Routine Blood Tests Using Ensemble Machine Learning", "Development of a Risk-Free COVID-19 Screening Algorithm from Routine Blood Tests Using Ensemble Machine Learning" ]
[ "Md Mohsin ", "Sarker Raihan ", "MdUddin Mohi ", "Khan ", "Laboni Akter \nDepartment of Electrical & Electronic Engineering\nDepartment of Electrical & Computer Engineering\nIslamic University of Technology\nBoardbazarGazipur\n\nUniversity of Toronto\nTorontoOntario\n", "Abdullah Bin [email protected] ", "Shams ", "Bangladesh ", "\nDepartment of Biomedical Engineering\nKhulna University of Engineering & Technology\nKhulnaBangladesh\n" ]
[ "Department of Electrical & Electronic Engineering\nDepartment of Electrical & Computer Engineering\nIslamic University of Technology\nBoardbazarGazipur", "University of Toronto\nTorontoOntario", "Department of Biomedical Engineering\nKhulna University of Engineering & Technology\nKhulnaBangladesh" ]
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The Reverse Transcription Polymerase Chain Reaction (RTPCR) test is the silver bullet diagnostic test to discern COVID infection. Rapid antigen detection is a screening test to identify COVID positive patients in little as 15 minutes, but has a lower sensitivity than the PCR tests. Besides having multiple standardized test kits, many people are getting infected and either recovering or dying even before the test due to the shortage and cost of kits, lack of indispensable specialists and labs, time-consuming result compared to bulk population especially in developing and underdeveloped countries. Intrigued by the parametric deviations in immunological and hematological profile of a COVID patient, this research work leveraged the concept of COVID-19 detection by proposing a risk-free and highly accurate Stacked Ensemble Machine Learning model to identify a COVID patient from communally available-widespread-cheap routine blood tests which gives a promising accuracy, precision, recall and F1-score of 100%. Analysis from R-curve also shows the preciseness of the risk-free model to be implemented. The proposed method has the potential for large scale ubiquitous low-cost screening application. This can add an extra layer of protection in keeping the number of infected cases to a minimum and control the pandemic by identifying asymptomatic or pre-symptomatic people early.
10.1201/9781003256083
[ "https://export.arxiv.org/pdf/2108.05660v3.pdf" ]
236,987,139
2108.05660
79665042ce44208528f43279b54215fe197189a4
Development of a Risk-Free COVID-19 Screening Algorithm from Routine Blood Tests Using Ensemble Machine Learning Md Mohsin Sarker Raihan MdUddin Mohi Khan Laboni Akter Department of Electrical & Electronic Engineering Department of Electrical & Computer Engineering Islamic University of Technology BoardbazarGazipur University of Toronto TorontoOntario Abdullah Bin [email protected] Shams Bangladesh Department of Biomedical Engineering Khulna University of Engineering & Technology KhulnaBangladesh Development of a Risk-Free COVID-19 Screening Algorithm from Routine Blood Tests Using Ensemble Machine Learning COVID19Machine LearningEnsemble LearningStacked Ensemble LearningCOVID detection from BloodCOVID19 Reproduction Numbers (R0)Hematology The Reverse Transcription Polymerase Chain Reaction (RTPCR) test is the silver bullet diagnostic test to discern COVID infection. Rapid antigen detection is a screening test to identify COVID positive patients in little as 15 minutes, but has a lower sensitivity than the PCR tests. Besides having multiple standardized test kits, many people are getting infected and either recovering or dying even before the test due to the shortage and cost of kits, lack of indispensable specialists and labs, time-consuming result compared to bulk population especially in developing and underdeveloped countries. Intrigued by the parametric deviations in immunological and hematological profile of a COVID patient, this research work leveraged the concept of COVID-19 detection by proposing a risk-free and highly accurate Stacked Ensemble Machine Learning model to identify a COVID patient from communally available-widespread-cheap routine blood tests which gives a promising accuracy, precision, recall and F1-score of 100%. Analysis from R-curve also shows the preciseness of the risk-free model to be implemented. The proposed method has the potential for large scale ubiquitous low-cost screening application. This can add an extra layer of protection in keeping the number of infected cases to a minimum and control the pandemic by identifying asymptomatic or pre-symptomatic people early. Introduction The pandemic by SARS-CoV-2 infection has claimed over 4 million lives around the world to this date. Since the outbreak, countries have expeditiously ramped up their capacity in full for testing and contract tracing, emanating a 195.545 million confirmed cases from 2.35 billion tests conducted worldwide (Schiffmann, 2021). The subatomic experiment accomplished using the converse polymerase chain response (PCR) method is the tool of choice, or the best quality standard, for detecting SARS-CoV-2 contamination. However, the test is time-consuming, necessitating the use of specialized hardware and substances, the collaboration of particular and qualified recruits designed for the sample assortment, and relying on the sufficient hereditary security of the RNA groupings chosen for tempering the preliminary (Vogels et al., 2020). Attempt to develop tests based on IgM/IgG antibodies also suffer from low sensitivity, specificity and costly reagents. To improve symptomatic capacities, the information science local area has suggested a few AI machine learning (ML) models. The majority of these models depend on processed tomography outputs or chest X-ray beams. Despite the detailed promising outcomes, a few concerns have been higher in regards to these and different works, particularly as to arrangements dependent on chest X-ray beams, which have been related with high number of negative outcomes. Then again, arrangements dependent on CT imaging are influenced by the attributes of this methodology: CTs are exorbitant, tedious, and need particular devices; accordingly, methodologies dependent on this imaging procedure cannot be applied sensibly for screening tests. Detection based on chest CT images and X-rays cannot be performed abundantly due to even higher cost & radiation exposure. Considering the detrimental effects of nCoV breeding inside host lung cells utilizing the Angiotensin-Converting Enzyme 2 (ACE2) found in ample amount on type-II alveolar cells and considering inflammation in lungs & respiratory tracts, recent studies have reported significant change in immunological & hematological parameters in the host blood stream. The predominant purpose of this study is to develop a novel machine learning technique, featuring diagnosis of COVID-19 subjects that have undergone routine pathological blood tests. This study proposes a Double Layer Stacked Ensemble Machine Learning model that can classify the individuals whether they are infected with nCoV or not with high accuracy. Despite focusing on higher accuracy, this study also gave in equal importance to achieve high precision & recall so that no individual is misclassified. The unconventionality of this study is that the research has improved the overall performance metrics of the proposed model by analyzing consequences of misclassification based on different Basic Reproduction Numbers (R 0 ) related to COVID-19 pandemic. The proposed model takes an attempt to improve COVID-19 patient classification ML algorithm so that a cheap and widely available COVID diagnostic method can be developed utilizing blood samples. Related Works AlJame et al. (AlJame et al., 2020) flourished ML model named ERLX for identifying COVID-19 from routine blood tests. There, two levels had been used wherein first level included Random Forest, Logistic Regression, and Extra Trees & XGBoost classifier in another level with 18 features. KNNImputer algorithm had been used to manage null values in the dataset, iForest used to eliminate outlier data, and SMOTE technique used to overcome an imbalanced data; feature importance is described by using the SHAP system. In that study, sample data was 5644 where 559 was confirmed COVID-19 circumstances. The model obtained the result with average accuracy of 99%, AUC of 99%, a sensitivity of 98%, and a specificity of 99%. Barbosa et al. (Barbosa et al., 2021) proposed a framework that helps Covid-19 determination dependent on blood testing. In this study, the laboratory parameters achieved 24 features from biochemical and the hemogram tests characterized to help clinical analysis. They used several types of machine learning techniques namely Random Forest, SVM, Bayesian Network, Naïve Bayes and achieved the accuracy 95.2%, sensitivity 96.8%, kappa index 90.3%, specificity 93.6%, and precision 93.8%. Kukar et al. (Kukar et al., 2021) represented a method to predict COVID-19 infection via ML dependent on clinically accessible blood test results. Using the Random Forest algorithm with 24 features, the COVID-19 screening achieved an in-depth clinical evaluation with accuracy of 91.67%. Methodology Blood test in healthcare is performed in order to ascertain the Biochemical & Physiological state of a person. The whole dataset containing various attributes comprising hemocyte count & hematochemical profile obtained from two types of blood-test, viz. Complete Blood Count (CBC) & Basic Metabolic Panel, is analyzed according to the block diagram portrayed in Figure- Dataset Collection The dataset used in this research was collected from Kaggle (Data4u, 2020) which includes 5644 samples with 111 attributes. Among these, 603 samples & only blood related 25 attributes were selected for this study. The attributes are: Hematocrit, Hemoglobin, Platelets, MPV, RBC, Lymphocytes, MCHC, Leukocytes, Basophils, MCH, Eosinophils, MCV, Monocytes, RDW, Neutrophils, Potassium, Creatinine, Sodium, Aspartate, Transaminase, INR, Albumin, Alanine, Transaminase, Proteina C reativa. Data Pre-processing Missing Data Handling Several techniques are available to counter missing values in a dataset. In this research work, the KNN imputer has been applied with k=5 number of neighbors to eliminate missing data contained in the dataset (AlJame et al., 2020). SMOTE Analysis An imbalanced dataset makes the classifier model highly biased towards the high frequency target class. To balance data, a well-known oversampling technique named 'Synthetic Minority Over-sampling Technique (SMOTE)' has been used in this study which does data augmentation by grabbing the concept of k-nearest neighbors. Augmented data is intelligently added to be used in the classification problem to improve the data distribution. SMOTE does not augment data freely but similar to the existing one (Akter & Akhter, 2020). Data Splitting The dataset was splitted into 70:30 ratio as training & test set. The stacked ensemble model was trained using 70% training data & overall system evaluation was performed using 30% test data. Feature Scaling In this study, standardization has been used as the feature scaling method. The esteems are located near the average by a unit standard deviation in such a scaling process. This senses that the feature's average turns into zero, then the distribution takes a unit standard deviation after using this technique (Sklearn.Standardscaler, v0.24.2). Stacked Ensemble Machine Learning This technique has widely been used since invention in which the 1 st level learners are trained & utilized to make primary prediction. The 1 st level predictions are combined as the training dataset for a stacked 2 nd level learner called meta-learner & the test dataset is then fed to 2 nd level learner in order to make final prediction (Géron, 2019). Machine Learning Algorithms This study maneuvers double layer stacked ensemble machine learning in order to precisely classify the subjects under Severe Acute Respiratory Syndrome (SARS) test caused by SARS-CoV-2 strain of coronavirus as COVID positive or negative based on the hematological profile containing 25 attributes of 1040 subjects. Four algorithms (K-Nearest Neighbors, Support Vector Machine, Random Forest, XGBoost) were chosen for the 1 st layer & AdaBoost was chosen as metalearner for the 2 nd layer of the proposed model based on high accuracy, precision, recall & F1-score. K-Nearest Neighbors (KNN) This algorithm measures the distance of the test datapoint from the training datapoints & classifies the test datapoint according to the class of the k-closest training data-points in the neighborhood (Shams et al., 2021). The hyperparameters this study used for KNN: n neighbors = 3, metric = minkowski, p = 2. Support Vector Machine (SVM) This algorithm draws the best decision boundary called 'hyperplane' at maximum distances from the target class in n-dimensional space so that new data points can certainly be categorized into the right classification later on (Shams et al., 2021). The hyperparameters this study used for SVM: kernel = rbf , random state = 0. Random Forest (RF) To overcome pitfalls (data overfitting) of Decision Tree (DT), RF produces a series of Decision Trees that have been trained using the 'bagging' technique wherein the general result is built from a multitude of DT learning models (Shams et al., 2021). The hyperparameters this study used for RF: n estimators = 49, criterion = entropy, random state = 0. XGBoost (XGB) It's an optimized distributed gradient boosting library that utilizes ensemble learning techniques. Parallel & distributed computing from an ensemble of various learning algorithms makes it faster & efficient (Gómez-Ríos et al., 2017). The hyperparameters this study used for XGB: n estimators = 83, max depth = 12, subsample = 0.7. AdaBoost (AdB) Boosting algorithms deal with both bias-variance trade-off while bagging algorithms control variance only. The tweak behind improved accuracy & performance of AdB is the combination of weighted sum of few weak-learning classifiers that were misclassifying some instances when implied individually (Gómez-Ríos et al., 2017). The hyperparameters this study used for AdB: n estimators = 67, learning rate = 1. Compute Statistical Metrics The following statistical metrics were being used to measure the performance of this proposed work: accuracy, precision, recall, and f1 score. Individual output is calculated using confusion matrix values such as true negative (TN), true positive (TP), false negative (FN), and false positive (FP) (Raihan et al., 2020). Result and Discussion The performance metrics of the detailed study are tabulated in Table- It's worth notifying that despite better performance, none of the algorithms in 1 st layer are reasonably precise to curb the community transmission of COVID-19. For illustration of the associated risks, let's consider the confusion matrix ( Figure-4.2(a)) of the best performing algorithm (RF) of the 1 st layer. In the 1 st layer RF, corresponding 'precision' is 99.37% & 'recall' is 95.15%. Note that precision unfolds the caliber of a model to classify positive values correctly & recall tells 'how often the model actually predicts the correct positive values' (Raihan et al., 2020). It's clear from the confusion matrix of Figure-4.2(a) that the number of people who are actually COVID negative but misclassified as positive (False Positive) is 8. If these FP individuals are now kept in Quarantine/Isolation, they will lose working hours ultimately hampering their monthly income & the country's GDP growth. In the worst-case scenario, if they are medicated for COVID-19, it's left for the reader to think about the consequence of the false medication if the individuals had other chronic diseases. Again, the dangerous & risky scenario of the confusion matrix of Figure-4.2(a), the number of people who are actually COVID positive but misclassified as negative (False Negative) is 1. This 'False Negative' classification is of utmost importance because these misclassified people are enough to spread COVID through community transmission if they move freely being assured of negative from COVID test which is wrong. According to Simple Infectious Disease Model (Gog et al., 2021), after the incubation period τ days, if an infected person, infects exactly R 0 new susceptible, then the number of new infected individuals only from him/her is given as: n E (t) = n E (0)R t/τ 0(1) Considering 1 initial infected (misclassified as negative) person & 7 days incubation period, the equation is graphically visualized in Figure- If the machine learning algorithms with high accuracy but unknown/less precision & recall of the reviewed literature (Table-4.2) keep performing FN misclassification due to less precision & recall, Figure-4.3 clearly says, the pandemic will remain a pandemic due to community transmission from the FN class. Therefore, the novelty of the idea & algorithm of this study is, it puts a shade on the risk factor associated with less precision & recall using R-curve plot & I max plot from SIR model using simplified-generalized equations. It is imperative to resolve the FN & FP classification for which this research work emphasized on improving precision & recall along with achieving high accuracy by feeding the output of the 4 selected algorithms of 1 st layer of the proposed model to the 2 nd (final) layer as input. In the 2 nd (final) layer called the meta-learner of the stacked ensemble model, after applying AdaBoost algorithm, the model achieved an improved performance (Table-4 Hence, according to the R-curve of Figure-4.3(b), the new infection cases by FN class will be zero irrespective of different R 0 values. Consequently, if the true positive individuals are held in isolation & all social norms are maintained, gradually the R 0 number will go below 1 & new infection cases will reduce like Figure-4.3(a) where R 0 < 1. Also due to zero FP class, the true COVID negative individuals will get relief from: being misclassified, losing working hours & from taking false medication in worst cases. Thus, the proposed highly accurate & risk-free Stacked Ensemble Machine Learning model is capable of classifying individuals as COVID-19 +ve or -ve based on Hematological profile of their Blood-samples. Conclusion To combat the pandemic, it is imperative to ramp up the number of tests conducted in a country with high accuracy and precision for reliable screening of infected patients in a short time. To supplement the conventional time-consuming and scarce RT-PCR test this study proposes a risk-free 'Stacked Ensemble Machine Learning' model to diagnose COVID-19 patients with relatively higher accuracy & precision from their Hematological data obtained from routine blood tests. This study characterized the limitations, risks associated with the models having unknown or less precision-recall in case of COVID-19 diagnosis exploiting the R-curve and maximum number of infections from simple SIR infectious disease model generalized for COVID-19 and hence harnessed the versatility-robustness of stacked ensemble model to resolve the problem. Nonetheless, this study keeps future scope of further development to address some challenges including usage of huge-diverse-high quality dataset, rigorous public testing and validation, incorporation of more attributes, and multiclass classification other than binary classification performed in this work. Figure 3 . 1 . 31Block diagram describing the proposed method for the classification of COVID-19 based on Blood Sample data. Figure 3 . 32.2.1. Data distribution before (inner-ring) & after (outer-ring) SMOTE analysis. Figure 4 . 1 . 41Performance Metrics (Graphical Visualization) of the Proposed Stacked Ensemble Model. Figure 4 . 2 . 42Confusion matrices of the best algorithm (a) Random Forest in 1 st layer (b) AdaBoost in 2 nd layer. 4.3(a) describing one month projection of new infection cases by only 1 person due to community transmission of COVID for various R 0 values. It will be seldom possible to curb the pandemic if the newly infected persons spread the disease again like Figure-4.3(c). Figure 4 . 3 . 43(a) R-curve: Effect of FN misclassification based on different R 0 values according to Simple Infectious Disease Model. (b) R-curve remains at zero if the individuals were properly classified. (c) Rapid contagious nature of infectious disease if R 0 = 2. (d) COVID-19 R 0 values of Bangladesh during 11 March 2021 to 7 April 2021. (e) Using R 0 values, rough idea of new infection cases as per simple SIR model. Now, to quantify the risk regarding infectious disease spread like Figure-4.3(c), let an area having 20,000 population, where initial infectives (misclassified as COVID negative), I 0 = 1 & remaining susceptible, S 0 = 19,999. According to the simple SIR model for COVID-19 (Murray, 2001): Contact ratio, q = R0 S0 & Maximum number of people that will be infected, I max = I 0 +S 0 − 1 q (1+ln(qS 0 )). Using recent COVID Reproduction Number (R 0 ) of Bangladesh Figure-4.3(d) (Arroyo-Marioli et al., 2021) & equations above, Figure-4.3(e) depicts approximation of maximum number of new infection cases. This study used a relatively small test set in the 1 st layer where the best algorithm gave FN I 0 = 1. In case of larger datasets of bigger area, this algorithm would contribute higher FN I 0 to the equation of I max & consequently Figure-4.3(e) would become worse. . 1 & 1Figure-4.1) where the AdB algorithm outperformed amongst others achieving all the performance metrics as 100%.Visually, the confusion matrix Figure-4.2(b) of the best algorithms (AdB) of the 2 nd layer depicts there is no FN & FP classification. Table 4 . 1 . 41Performance Metrics of the Proposed Stacked Ensemble Model.Layer-1 Layer-2 Performance Metrics KNN SVM XGB RF AdB Accuracy 84.62% 92.95% 95.19% 97.12% 100% Precision 98.73% 96.84% 97.47% 99.37% 100% Recall 77.23% 90.00% 93.33% 95.15% 100% F1 Score 86.67% 93.29% 95.36% 97.21% 100% At 1 st level classifier, analysis was carried out using different machine learning algorithms (LR, KNN, DT, ANN, SVM, XGB, RF, NB & AdB) in an effort to find out the best suited algorithms for the 1 st layer of the proposed model.Based on 'accuracy, precision, recall, F1-score' & giving priority on 'precision', four algorithms were chosen for the 1 st layer of the proposed model which are RF, XGB, SVM & KNN. RF, XGB, SVM & KNN algorithms showed the accuracy of 84.62%, 92.95%, 95.19% & 97.12%; the precision of 98.73%, 96.84%, 97.47% & 99.37%; the recall of 77.23%, 90%, 93.33% & 95.15%, and the F1-score of 86.67%, 93.29%, 95.36% & 97.21% respectively. 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[]
[ "Large-Scale System Identification Using a Randomized SVD", "Large-Scale System Identification Using a Randomized SVD" ]
[ "Han Wang \nColumbia University\n\n", "James Anderson \nColumbia University\n\n" ]
[ "Columbia University\n", "Columbia University\n" ]
[]
Learning a dynamical system from input/output data is a fundamental task in the control design pipeline. In the partially observed setting there are two components to identification: parameter estimation to learn the Markov parameters, and system realization to obtain a state space model. In both sub-problems it is implicitly assumed that standard numerical algorithms such as the singular value decomposition (SVD) can be easily and reliably computed. When trying to fit a high-dimensional model to data, for example in the cyberphysical system setting, even computing an SVD is intractable. In this work we show that an approximate matrix factorization obtained using randomized methods can replace the standard SVD in the realization algorithm while maintaining the non-asymptotic (in dataset size) performance and robustness guarantees of classical methods. Numerical examples illustrate that for large system models, this is the only method capable of producing a model.
10.23919/acc53348.2022.9867836
[ "https://arxiv.org/pdf/2109.02703v1.pdf" ]
237,431,420
2109.02703
d5b2fa2114ce525b8865677870f26d5d195d0513
Large-Scale System Identification Using a Randomized SVD 6 Sep 2021 Han Wang Columbia University James Anderson Columbia University Large-Scale System Identification Using a Randomized SVD 6 Sep 2021 Learning a dynamical system from input/output data is a fundamental task in the control design pipeline. In the partially observed setting there are two components to identification: parameter estimation to learn the Markov parameters, and system realization to obtain a state space model. In both sub-problems it is implicitly assumed that standard numerical algorithms such as the singular value decomposition (SVD) can be easily and reliably computed. When trying to fit a high-dimensional model to data, for example in the cyberphysical system setting, even computing an SVD is intractable. In this work we show that an approximate matrix factorization obtained using randomized methods can replace the standard SVD in the realization algorithm while maintaining the non-asymptotic (in dataset size) performance and robustness guarantees of classical methods. Numerical examples illustrate that for large system models, this is the only method capable of producing a model. Introduction System identification is the process of estimating parameters of a dynamical system from observed trajectories and input profiles. It is a fundamental component in the control design pipeline as many modern optimal and robust control synthesis methodologies rely on having access to a dynamical system model. Traditionally, system identification [1][2][3][4][5] was limited to asymptotic analysis, i.e., estimators were shown to be consistent under the assumption of infinite data. However, recent results [6][7][8][9][10] focus on the more challenging task of analyzing the finite sample setting. Theoretically, this type of analysis is more involved and requires tools from high dimensional probability and statistics. In this paper, we consider the problem of identifying a discrete-time, linear time-invariant (LTI) system parameterized by the matrices A ∈ R n×n , B ∈ R n×m , C ∈ R p×n and D ∈ R p×m , that evolves according to x t+1 = Ax t + Bu t + w t y t = Cx t + Du t + v t ,(1) from N < ∞ observations of the output signal {y i t } T t=0 and control signal {u i t } T t=0 of length T for i = 1, · · · , N . The vectors x t ∈ R n , w t ∈ R n , and v t ∈ R p in (1) denote the system state, process noise, and measurement noise at time t, respectively, the superscript denotes the output/input channel. In this setting, the problem is referred to as being partially observed. A more simplistic setting occurs when one has access to (noisy) state measurements, i.e., C = I, D = 0, and v t = 0 for all t. This is referred to as the fully observed problem. In the fully observed setting, estimates for (A, B) can be obtained by solving ordinary leastsquares (OLS) optimization problems. A series of recent papers [11,9,8,12,13] have derived non-asymptotic guarantees for ordinary least-squares (OLS) estimators. In the case of partially observed systems, which is conceptually more complicated than the fully observed case, OLS optimization can be used to estimate the Markov parameters associated with (1) from which the Ho-Kalman algorithm [14] can be employed to estimate the system parameters (A, B, C, D). The process of obtaining estimates of the system matrices from the Markov parameters is referred to as system realization which is the main focus of this paper. Using this framework, the authors of [8,6,[15][16][17][18] have derived non-asymptotic estimation error bounds for the system parameters which decay at a rate O( 1 √ N ). Note that these papers make different assumptions about the stability, system order, and the number of required trajectories to excite the unknown system. However, in contrast to the estimation error bounds, the computational complexity of system identification has received much less attention in the literature [19,20]. Due to the fact that the OLS problem is convex, and the computational bulk of the Ho-Kalman Algorithm is a singular value decomposition (SVD), it is taken for granted that system identification can be carried out at scale. As mentioned in [19], with the increase of system dimension, the computational and storage costs of general control algorithms quickly become prohibitively large. This challenge motivates us to design control algorithms that mitigate the "curse of dimensionality". In this paper, we aim to design an efficient and scalable system realization algorithm that can be deployed in the big data regime. From the view of computational complexity, the system identification methods proposed in [8,6,[15][16][17][18] are not scalable since the size of the Hankel matrix increases quadratically with the length of output signal T and cubically with the system state dimension n. The result is the singular value decomposition used in the Ho-Kalman Algorithm cannot be computed. This quadratic/cubic dependence on the problem size greatly limits its application in large scale system identification problems. Motivated by the limits of the scalability of numerical SVD computations, there has been a surge of work which has focussed on providing approximate, but more easily computable matrix factorizations. Thanks to advances in our understanding of random matrix theory and high dimensional probability (in particular, concentration of measure), randomized methods have been shown to provide an excellent balance between numerical implementation (in terms of storage requirements and computational cost) and accuracy of approximation (in theory and practice). Broadly speaking this field is referred to as randomized numerical linear algebra (RNLA), and we refer the reader to [21][22][23] and the references therein for an overview of the field. In particular, the machine learning and optimization communities have started to adopt RNLA methods into their work flows with great success, see for example [24,25]. The intuition is that randomized methods can produce efficient, unbiased approximations of nonrandom operations while being numerically efficient to implement by exploiting modern computational architectures such as parallelization and streaming. The RNLA framework we follow (see [26] for sketching-based alternatives, or [27] for random column sampling approaches) is a three step process [22]. First, sample the range of the target matrix by carrying out a sequence of matrix-vector multiplications, where the vector is an ensemble of random variables. Next, an approximate low-dimensional basis for the range is computed. Finally, an exact matrix factorization in the low dimensional subspace is computed. The performance of the randomized SVD (RSVD) has been studied in many works [21][22][23]25] and has found applications in large-scale problems across machine learning [28], statistics [29], and signal processing [30]. The main contribution of this work is a stochastic Ho-Kalman Algorithm, where the standard SVD (which constitutes the main computational bottleneck of the algorithm) is replaced with an RSVD algorithm, which trades off accuracy and robustness for speed. We show that the stochastic Ho-Kalman Algorithm achieves the same robustness guarantees as its deterministic, non-asymptotic version in expectation. However, it outperforms the deterministic algorithm in terms of speed/computational complexity, which is measured by the total number of required floating-point operations (flops) [ [31], §C.1.1]. Compared with O(pmn 3 ) flops required by the deterministic algorithm, the stochastic Ho-Kalman Algorithm only requires O(pmn 2 log n) flops. Preliminaries and Problem Formulation Given a matrix A ∈ C m×n , where C is the set of complex numbers. A denotes the spectral norm and A F denotes the Frobenius norm, i.e., A = σ 1 (A), where σ 1 is the maximum singular value of A, and A F = Trace(A * A), where A * denotes the Hermitian transpose of A. The multivariate normal distribution with mean µ and covariance matrix Σ is denoted by N (µ, Σ) . A matrix is said to be standard Gaussian if every entry is drawn independently from N (0, 1). Symbols marked with a tilde are associated to the stochastic Ho-Kalman algorithm, while those with hats are associated with the deterministic Ho-Kalman algorithm. Singular Value Decomposition The singular value decomposition of the matrix A ∈ C m×n , factors it as A = U ΣV * , where U ∈ C m×m and V ∈ C n×n are orthonormal matrices, and Σ is an m × n real diagonal matrix with entries σ 1 , σ 2 , · · · , σ n ordered such that σ 1 ≥ σ 2 ≥ · · · ≥ σ n ≥ 0. When A is real, so are U and V . The truncated SVD of A is given by U r Σ r V ⊤ r (r < min{m, n}), where the matrices U r and V r contain only the first r columns of U and V , and Σ r contains only the first r singular values from Σ. According to the Eckart-Young theorem [32], the best rank r approximation to A in the spectral norm or Frobenius norm is given by A [r] = r i=1 σ i u i v ⊤ i (2) where u i and v i denote the i th column of U and V , respectively. More precisely, minimize rank(X)≤r A − X = σ r+1 ,(3) and a minimizer is given by X ⋆ = A [r] . The expression (3) concisely sums up the scalability issue we are concerned with: on the left hand side is non-convex optimization problem with no polynomial-time solution; on the right is a singular value which for large m and/or n cannot be computed. In the sequel we shall see how randomized methods can use approximate factorization to resolve these issues. System Identification We consider the problem of identifying a linear system model defined by Eq (1) where u t i.i.d. ∼ N 0, σ 2 u I m , w t i.i.d. ∼ N 0, σ 2 w I n , and v t i.i.d. ∼ N 0, σ 2 v I p . We further assume that the initial state variable x 0 = 0 n (although the dimension, n, in unknown a priori ). Under these assumptions, we generate N pairs of trajectory, each of length T . We record the data as D N T = y i t , u i t : 1 ≤ i ≤ N, 0 ≤ t ≤ T − 1 , where i denotes i th trajectory and t denotes t th time-step in each trajectory. With the data D N T the system identification problem can be solved in two steps: 1. Estimation: Given D N T , estimate the first T Markov parameters of the system which are defined as G = D, CB, CAB, . . . , CA T −2 B ∈ R m×T p . Ideally, the estimation algorithm will produce finite sample bounds of the form G−Ĝ ≤ ǫ(N, T ). This is typically achieved by solving an OLS problem (see Appendix B). Realization: Given an estimated Markov parameter matrixĜ, produce state-space matrices (Â,B,Ĉ,D) with guarantees of the form A − ≤ ǫ A , B −B ≤ ǫ B , etc. This is most commonly done using the Ho-Kalman algorithm. In data collection, we refer the input/output trajectory (y t , u t ), t = 0, · · · , T − 1 as a rollout. There are two approaches to collect the data. The first method involves multiple rollouts [7,17,13,15], where the system is run and restarted with a new input signal N times. The second approach is the single rollout method [6,9,10,8,33], where an input signal is applied from time 0 to N × T − 1 without restart. As this paper only focuses on the realization step of the problem, we can use either of aforementioned methods for data collection. System realization via noise-free Markov matrix G With an estimated Markov matrix in hand, we wish to reconstruct the system parameters A, B, C and D. To achieve this goal, we employ the Ho-Kalman Algorithm [14]. We first consider the noise-free setting, i.e., G is known exactly. The main idea of the Ho-Kalman Algorithm is to construct and factorize a Hankel matrix derived from the G. Specifically, we generate a Hankel matrix as follows: H =        CB CAB . . . CA T 2 B CAB CA 2 B . . . CA T 2 +1 B CA 2 B CA 3 B . . . CA T 2 +2 B . . . . . . . . . . . . CA T 1 −1 B CA T 1 B . . . CA T 1 +T 2 −1 B        , where T = T 1 + T 2 + 1. We use H − (H + ) to denote the pT 1 × mT 2 Hankel matrix created by deleting the last (first) block column of H. We assume that 1. the system (1) is observable and controllable, and 2. n = rank(H) ≤ min{T 1 , T 2 }. Under these two assumptions, we can ensure that H and H − are of rank n. We note that H − can be factorized as H − =      C CA . . . CA T 1 −1      B AB . . . A T 2 −1 B := OQ, where O, Q denote the observability matrix and controllability matrix respectively. We can also factorize H − by computing its truncated SVD, i.e., H − = U Σ n V T = (U Σ n 1 2 )(Σ n 1 2 V T ). Therefore, the factorization of H − establishes O = U Σ 1 2 n , Q = Σ 1 2 n V T . And doing so, we can obtain the system parameter C by taking the first p rows of U Σ n 1 2 and the system parameter B by taking the first m columns of Σ n 1 2 V. Then A matrix can be obtained by A = O † H + Q † , where (·) † denotes the Moore-Penrose inverse. 1 Note that D is obtained without calculation since it is the first p × m submatrix of the Markov matrix G. It is worthwhile to mention that learning a state-space realization is a non-convex problem. There are multiple solutions yielding the same system input/output behavior and Markov matrix G. For example, if (A, B, C, D) is a state-space realization obtained from G, then (SAS −1 , SB, CS −1 , D) under a similarity transformation where S is any non-singular matrix is also a valid realization. System realization via noisy Markov parameter G In the setting with noise, the same algorithm is applied to the estimated Markov matrixĜ (see Appendix B for a simple method for obtainingĜ from D N T ), instead of the true matrix G. In this case the Ho-Kalman Algorithm will produce estimatesÂ,B,Ĉ andD. The explicit algorithm is shown as Alg 1 (deterministic). It was shown in [6] that the robustness of the Ho-kalman Algorithm implies the estimation error for A, B, C and D is bounded by O( 1 N 1/4 ), where N is the sample size of trajectories, i.e., max  − S −1 AS , B − S −1 B , Ĉ − CS ≤Ĉ G −Ĝ = O( 1 N 1/4 )(4) This result can be improved to O( 1 √ N ) from [8,18,16]. Note that the computational complexity of the Ho-Kalman Algorithm in Alg. 1 (deterministic) is dominated by the cost of computing the SVD (Step 7), which is O(pT 1 ×mT 2 ×n) when using the Krylov method (see e.g. [34,35]). Therefore, we want to use a small T to reduce the computational cost. However, to satisfy the second assumption that n = rank(H) ≤ min{T 1 , T 2 }, where T 1 + T 2 + 1 = T, the smallest T we can choose is 2n + 1 with T 1 = T 2 = n. In summary, the lowest achievable computational cost for SVD is O(n 3 ). Such dependency on the system dimension is prohibitive for large-scale systems (e.g. systems with n = 100 as we show in Section 5). Motivated by the drawbacks of the existing method, we aim to answer the following question: • Is there a system realization method which can significantly reduce the computational complexity without sacrificing robustness guarantees? The main result of this paper is to answer this question in the affirmative. By leveraging randomized numerical linear algebra techniques described in the next section, we design a stochastic version of the Ho-Kalman algorithm that is computationally efficient and produces competitive robustness guarantees. An informal version of our main results is given below: Theorem 1. (informal) The stochastic Ho-Kalman Algorithm reduces the computational com- plexity of the realization problem from O(pmn 3 ) to O(pmn 2 log n) when T 1 = T 2 = n 2 . The achievable robustness is the same as deterministic Ho-Kalman Algorithm. In the next section we introduce the randomized methods and their theoretical and numerical properties. We then incorporate them into the Ho-Kalman algorithm and analyze its performance. Algorithm 1 Stochastic/Deterministic Ho-Kalman Algorithm 1: Input: Length T , Estimated Markov parametersĜ, system order n, (T 1 , T 2 ) satisfying T 1 + T 2 + 1 = T 2: Outputs: State space realizationÂ,B,Ĉ,D 3: Generate a Hankel matrixĤ ∈ R pT 1 ×m(T 1 +1) fromĜ 4:Ĥ − =Ĥ(:, 1 : mT 2 ) ⊲ dim(Ĥ − ) = pT 1 × mT 2 5:Ĥ + =Ĥ(:, m + 1 : m(T 2 + 1)) 6: if Deterministic then 7:L =Ĥ − [n] ⊲ truncated SVD via (2) 8:Û ,Σ,V = SVD(L) 9: else if Stochastic then 10:Û ,Σ,V = RSVD(Ĥ − , n, l) 2 ⊲L =ÛΣV ≈L 11: end if 12:Ô =ÛΣ 1/2 ⊲ dim(Ô) = pT 1 × n 13:Q =Σ 1/2V * ⊲ dim(Q) = n × mT 2 14:Ĉ =Ô(1 : p, :),B =Q(:, 1 : m) 15: =Ô †Ĥ +Q † ,D =Ĝ(:, 1 : m) 16: Return ∈ R n×n ,B ∈ R n×m ,Ĉ ∈ R p×n ,D ∈ R p×m Randomized singular value decomposition The numerical computation of a singular value decomposition can be implemented in many ways. The structure of the matrix to be decomposed will likely play a role in determining which is the most efficient algorithm. We do not attempt to review methods here as the literature is vast. In the system realization problem, the Ho-Kalman Algorithm computes the SVD of H − , a dense truncated block Hankel matrix. To the best of our knowledge there are no specialized algorithms for this purpose. As such, we assume we are dealing with a general dense low rank matrix. A brief comparison between standard numerical methods and the randomized methods to be introduced next is given Section 3.2. The objective of the RSVD it to produce matrices U, Σ, V , such that for a given matrix A ∈ C m×n with rank(A) = r < min{m, n}, and tolerance ǫ > 0, the bound A − U ΣV * ≤ ǫ is satisfied where U and V have orthornormal columns and Σ ∈ R k×k is diagonal with k < r. Following [22], the RSVD of a matrix A with target rank k is computed in two stages (full implementation details are provided in Algorithm 2): 1. Find a matrix P ∈ R m×k with orthonormal columns such that the range of P captures as much of the range of A as possible. In other words, A ≈ P P * A. 2. Form the matrix M = P * A ∈ R k×n and apply the standard numerical linear algebra technique to compute the SVD of P . Step 1 is the range finding problem. This is where randomization enters picture. Let ω (i) be a standard Gaussian vector, and compute y (i) = Aω (i) . This can be viewed as a sample of Range(A). Repeating this process k times and concatenating samples into matrices we have Y = AΩ, an orthonormal basis for Y can then be computed using standard techniques, we use an economy QR decomposition. Again, we concatenate basis vectors q i into a matrix P . Because k is selected to be small, this process is computationally tractable. When A − P P * A is small, P P * A is a good rank-k approximation of A. In step 2, standard deterministic routines are called to compute the SVD of M . These routines are considered tractable as the the matrix M has dimension k × m where k is ideally much less than r. From the SVD of M , the matrices U, Σ, V can be easily constructed (lines 7-8 of RSVD). 9: Return U ∈ R m×k , S ∈ R k×k , V ∈ R n×k In practice, if the target rank is selected to be k, then one should sample the range of A k + l times where l is a small interger. Typically l = 10 is more than sufficient [22]. In RSVD, Ω is chosen to be a standard Gaussian matrix. Surprisingly, the computational bottleneck of RSVD is the matrix-vector multiplication in computing AΩ in step 4. To reduce the computational cost of this step, we can choose other types of random matrices such as the subsampled random Fourier transform (SRFT) matrix which reduces the flop count from O(mn(k + l) to O(mn log(k + l)) without incurring much loss in accuracy (we extend our results to this setting in Appendix A.2). It should be further noted that the computation of AΩ is trivially parallelizable. The orth function called on line 4 of RSVD computes an orthonormal basis for the range of its argument. This can be done in many ways, here we use an economy QR decomposition. Power Scheme for slowly decaying spectra When the input matrix A has a flat spectrum, RSVD tends to struggle to find a good approximate basis. To improve the accuracy a power iteration scheme is employed [34, p. 332]. Loosely, the power iteration are based on the observation that the singular vectors of A and (AA * ) q A are the same, while the singular values with magnitude less than one will rapidly shrink. In other words, it can reduce the effect of noise. More precisely, we apply RSVD to the matrix W = (AA * ) q A, and we have σ j (W ) = σ j (A) 2q+1 , j = 1, 2, · · · which shows that for σ j < 1, the power iteration will provide singular values that decay more rapidly while the singular vectors remain unchanged. This will provide a more accurate approximation, however it will require 2q + 1 times as many matrix vector multiplies. The following theorem provides a bound on the accuracy of the approximation that RSVD provides. We will make heavy use of this result in the sequel. Theorem 2. [22] Suppose that A is a real m×n matrix with singular values σ 1 ≥ σ 2 ≥ σ 3 ≥ · · · . Choose a target rank k ≥ 2 and an oversampling parameter l ≥ 2, where k + l ≤ min{m, n}. Then RSVD called on W = (AA * ) q A with target rank k, and oversampling parameter l produces an orthonormal approximate basis P which satisfies E (A − P P * A) ≤ 1 + k l − 1 + e √ k + l l · min{m, n} − k 1 2q+1 σ k+1 where e is Euler's number and E denotes expectation with respect to the random matrix Ω. Note that σ k+1 is the theoretically optimal value in the deterministic setting. Thus the price of randomization is the given by the contents of the square brackets with exponent 1 2q+1 . Adaptive randomized range finder. RSVD is implemented based on the assumption that the target rank k is know a priori. However, in practice, we may not know the true rank k in advance. Therefore, it is desirable to design an algorithm that can find a matrix P with as few orthonormal columns as possible such that (I − P P * ) A ≤ ε where ε denotes a given tolerance. The work in [22], based on results from [36] describes an adaptive randomized ranger finder that iteratively samples until the desired tolerance is obtained. It is worth noting that the CPU time requirements of RSVD and the adaptive version are essentially identical. Comparing RSVD with SVD computaton In this subsection, we briefly compare the cost of computing a full singular value decomposition with that of computing an approximate decomposition. This is a vast topic in numerical analysis and we refer the reader to [35,34,37,22,38] for implementation and complexity details. We assume that the matrix we wish to factor is dense with dimension m × n. Broadly speaking, there are two approaches to computing an SVD: direct methods based on a QR factorization, and iterative methods such as the Krylov family of algorithms. For computing an approximate SVD, one can use the the truncated method (2) which involves computing a full SVD (which may not be practical for large m, n), a rank-revealining QR decomposition can be used, and finally randomized methods can be used. In addition to the time complexity of the algorithm, it is also important to consider how many times the matrix is read into memory. For large matrices, the cost of loading the matrix into memory outweighs the computational cost. As such the number of passes over the matrix required by the algorithm needs to be considered. We summarize the complexity of these algorithms in Table 1. The parameter k indicates the target rank for approximate methods. Method Time complexity Passes Full SVD [39] O (mnmin(m, n)) min (m, n) Krylov [39] O (mnk) k Truncated [39] O (mnmin(m, n)) k RSVD [22] O (mn log k) 1 From Table 1, we observe that RSVD is the fastest SVD algorithm among the aforementioned, especially when m, n ≫ k. In term of space complexity, although they have the same dominant term, RSVD only requires one pass through the data while other methods require multiple passes, which is prohibitively expensive for huge matrices. When the power iteration is implemented in the RSVD algorithm, the number of passes changes to O(1). Main Results The stochastic Ho-Kalman algorithm we propose replaces the deterministic singular value decomposition and truncation (lines 7-8, Algorithm 1) with a single approximate randomized SVD (line 10, Algorithm 1) obtained using RSVD. Proofs of all results are deferred to the appendix. For the remainder of the paper, symbols with a tilde denote that they were obtained from the stochastic Ho-Kalman Algorithm, while symbols with a hat denote that they were obtained from the deterministic Ho-Kalman Algorithm. Finally, symbols with neither have been obtained from the ground truth Markov matrix G. From [6], we have the following perturbation bounds for the deterministic Ho-Kalman Algorithm: Lemma 3. [6] The matrices H,Ĥ and L,L satisfy the following perturbation bounds: • max H + −Ĥ + , H − −Ĥ − ≤ H −Ĥ ≤ min{T 1 , T 2 + 1} G −Ĝ . • L −L ≤ 2 H − −Ĥ − ≤ 2 min{T 1 , T 2 } G −Ĝ . We will now use Theorem 2 and Lemma 3 to provide average and deviation bounds on the performance of the stochastic Ho-Kalman algorithm. E L −L ≤ 2C 2 2 + n l − 1 + e √ n + l l C 1 G −Ĝ(5) where C 1 = min{pT 1 , mT 2 } − n, and C 2 = min{T 1 , T 2 }. Furthermore, if we exploit the power scheme with RSVD, then the right-hand side of (5) can be improved to 4C 2 1 + 1 2 n l − 1 + e √ n + l 2l C 1 1/(2q+1) G −Ĝ .(6) Proof. See appendix. From (5), we have that the perturbation bound is determined by the ratio between the target rank n and the oversampling parameter l. The error is large if l is small. In practice, it is sufficient to use l = 5 or l = 10. And there is rarely any advantage to select l > 10 [22]. In addition, from (6), we know that the bound will decrease if we increase the power parameter q. The effect of l in terms of running time and realization error is studied further in Section V where we observe that the stochastic Ho-Kalman algorithm is robust to the choice of l. In case the average perturbation as characterized by Lemma 4 doesn't feel like a helpful quantity, a deterministic error bound is also achievable: Lemma 5. (Deviation bound) Let the assumptions of Lemma 4 hold. Assume l ≥ 4 and let C 1 and C 2 be defined as in Lemma 4. Then we have L −L ≤ 2C 2 2 + 16 1 + n l − 1 + 8 √ n + l l + 1 C 1 G −Ĝ (7) with failure probability at most 3e −l . Moreover, L −L ≤ C 2 2 + 6 (n + l)l log l + 3 √ n + l C 1 G −Ĝ (8) with failure probability at most 3l −l . Proof. See appendix. Remark 1. Another way to implement the stochastic Ho-Kalman Algorithm is to use a structured random matrix like subsampled random Fourier transform, or SRFT to compute the RSVD. In contrast with Gaussian matrix, SRFTs have faster matrix-vector multiply times. As a result RSVD computation time decreases. We will present the bounds for SRFT matrix in the appendix. We are now ready to show the robustness of stochastic Ho-Kalman algorithm. The robustness result is valid up to a unitary transformation. Theorem 6. Suppose the system A, B, C, D is observable and controllable. Let O, Q be order-n controllability/observability matrices associated with G andÕ,Q be approximate order-n controllability/observability matrices (computed by RSVD) associated withĜ. Suppose σ min (L) > 0 and the following robustness condition is satisfied: E L −L ≤ σ min (L)/2. Then, there exists a unitary matrix S ∈ R n×n such that, E C −CS F ≤ E O −ÕS F ≤ E 5n L −L , E B − S * B F ≤ E Q − S * Q F ≤ E 5n L −L , andÃ, A satisfy E A − S * à S F ≤ C 3   E L −L σ min (L) ( H + + H + −Ĥ + ) + H + −Ĥ +   , where C 3 = 14 √ n σ min (L) . Proof. See appendix. As discussed in [6], H + −Ĥ + , E L −L are perturbation terms that can be bounded in terms of G −Ĝ via Lemma 3 and Lemma 4. Theorem 6 shows that the stochastic Ho-Kalman Algorithm has the same error bounds as its deterministic counterpart, which says the estimation errors for system matrix decrease as fast as O( 1 N 1/4 ). Our analysis framework can be easily extended to achieve the optimal error bounds O( 1 √ N ) mentioned in [33,8,18,16]. Numerical Experiments Stochastic versus deterministic Ho-Kalman Algorithm We begin the comparison between the stochastic and deterministic Ho-Kalman Algorithm on six randomly generated systems described by (1). For each system, its dimension (n, m, p) is shown in the second column in Table 2. Each entry of the system matrix is generated through a uniform distribution over a range of integers as follows: matrix A with random integers from 1 to 5, and matrices B, C, D with random integers from −2 to 2. The A matrix is re-scaled to make it Schur stable 3 , i.e., |λ max (A)| < 1. The standard deviations of the process and measurement noises are σ w = 1 and σ v = 0.5. The length of trajectory T is given in the second column in Table 2 with T 1 chosen to be the smallest integer not less than T /2 and T 2 = T − 1 − T 1 . The third column in Table 2 denotes the matrix dimension ofĤ − when we run Algorithm 1. We denote the true system as G(A, B, C, D) and the estimated system returned by the stochastic/deterministic Ho-Kalman algorithm asG(Ã,B,C,D)/Ĝ(Â,B,Ĉ,D). We will use G,G,Ĝ at times to reduce notational clutter. The realization error of the algorithm is measured by the normalized H ∞ error: G −G G∞ G H∞ / Ĝ −G H∞ G G∞ . The running time and the realization error of the deterministic and stochastic algorithms 4 are reported in Table 2 where the results for the stochastic algorithm are average over 10 independent trials. All experiments are done on a 2.6 GHz Intel Core i7 CPU. The reported running time in the stochastic setting is highly conservative: we did not parallelize the sampling (i.e., constructing AΩ in line 4 of Algorithm 2). Furthermore, as noted Table 2: Comparison between the stochastic and deterministic Ho-Kalman Algorithm. The running time is in seconds. The approximate SVD is computed using RSVD with oversampling parameter l = 10. To benchmark the algorithm performance, a naive implementation of RSVD is used; we do not use power iterations and do not make use of parallelization. Inf and N/A indicates that the deterministic algorithm fail to realize the system. earlier (and further described in the appendix), standard Gaussian matrices are theoretically "nice" to work with but structured random matrices, such as SRFT matrices which compute Y = AΩ via a subsampled FFT [40] will offer superior running times. We observe that the stochastic Ho-Kalman algorithm consistently leads to a dramatic speedup over the deterministic algorithm. The larger the system dimension is, the larger the run time gap is. It is worthwhile to mention that the deterministic Ho-Kalman Algorithm fails to provide a result in the 5 th and 6 th examples where the system state dimensions are above 100. Meanwhile, the stochastic algorithm runs successfully and takes a fraction of the time the deterministic algorithm took to solve a 60 state realization problem. The stochastic algorithm can easily be applied to much larger systems. However with no means of comparison to existing algorithms, and having established the theoretical properties of the algorithm, we do not pursue this avenue further here. Oversampling effects To illustrate the influence of oversampling (parameter l in RSVD), we run the stochastic Ho-Kalman Algorithm on the 4 th example (n = 100, m = 80, p = 50) in Table 2 and vary the oversampling parameter l from 1 to 10. In this experiment we use a power iteration parameter of q = 1. Running times (averaged over 10 runs) and realization errors (averaged over 10 runs) are shown via a boxplot in Figure 1a and graph in Figure 1b. In the box-plot, the central red mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. Outliers are denoted by "+". We observe that in Fig 1b, the realization error tends to be larger when a small oversampling number is used, although the change is slight. The observed behavior is consistent with the theoretical analysis of Lemma 4. We can also observe from Fig 1a that the computational time is insensitive to the oversampling parameters, as such taking larger values of l = 10 is advantageous. Power iteration effect We now investigate numerically how RSVD implemented with a power iteration impacts the performance of the stochastic Ho-Kalman Algorithm. Again we focus on example 4. Based on the results of the previous subsection, we fix the oversampling parameter in RSVD as l = 10, and sweep q from 1 to 4. The results are shown in Figures 1c and 1d. We observe that the realization error decreases as the power parameter q increases as indicated in Eq (6). In contrast to the oversampling parameter l, the runtime demonstrably increases with q at an empirically linear rate. This trend is expected and analyzed in [22]. The power iteration method is most effective for problems where the spectrum of the matrix being approximated decays slowly. In the noise free setting, rank(H − ) = n, where n = 100 in this example. In contrast the dimensions of H − are 12500 × 19920. When noise is introduced, H − becomes full rank and the spectral decay depends on σ w and σ v . For the values chosen, these results show that spectral decay appears to be sharp enough that the power iterations do not offer significant improvement in accuracy. However, as σ v and σ w increase, the effect will become more dramatic. Conclusion We have introduced a scalable algorithm for system realization based on introducing randomized numerical linear algebra techniques into the Ho-Kalman algorithm. Theoretically it has been shown that our algorithm provides non-asymptotic performance guarantees that are competitive with deterministic approaches. Furthermore, without any algorithm optimization, we have shown that the stochastic algorithm easily handles problem instances of a size significantly beyond what classical deterministic algorithms can handle. In forthcoming work, using sketching-based randomized methods, we have designed and analyzed a distributed second-order algorithm for solving OLS problems for the Markov parameter estimation problem. We are currently working on the derivation of end-to-end performance bounds for the full randomized system identification pipeline. Inequality (a) holds because x 1/(2q+1) is concave in x. Applying Lemma 3, we prove the inequality (6). Proof of Lemma 5. We can follow the same steps as in the proof of Lemma 4 and use the bound given in Corollary 10.9 [22]. A.2 Perturbation bounds for Stochastic Ho-Kalman Algorithm with SRFT Test Matrices The subsampled random Fourier transform (SRFT), which might be the simplest structured random matrix, is an n × ℓ matrix of the form Ω = n ℓ DF R where D is an n × n diagonal matrix whose entries are independent random variables uniformly distributed on the complex unit circle, F is the n × n unitary discrete Fourier transform (DFT), whose entries take the values f pq = n −1/2 e −2πi(p−1)(q−1)/n for p, q = 1, 2, . . . , n, and R is an n × ℓ matrix that samples ℓ coordinates from n uniformly at random [22]. When Ω is a SRFT matrix, we can calculate the matrix multiplication Y = AΩ using O(mn log(ℓ)) flops by applying a subsampled FFT [40]. L −L ≤ (1 + 1 + 7mT 2 l + n ) × 2 min{T 1 , T 2 } G −Ĝ (12) with failure probability at most O n −1 . Proof. We can follow the same steps as in the proof of Lemma 4 and use the bound given in Theorem 11.2 of [22] to finish the proof. A.3 Proof of Theorem 6 To prove Theorem 6, we require two auxiliary lemmas. E U Σ 1/2 −ŨΣ 1/2 S 2 F + E V Σ 1/2 −ṼΣ 1/2 S 2 F ≤ 5nE L −L .(13) Proof. : Direct application of Theorem 5.14 of [41] guarantees the existence of a unitary S such that LHS =E U Σ 1/2 −ŨΣ 1/2 S 2 F + E V Σ 1/2 −ṼΣ 1/2 S 2 F ≤ 2 √ 2 − 1 E L −L 2 F σ min (L) ,(14) where LHS refers to the left hand side of (13). To proceed, using E rank(L −L) ≤ 2n and by assumption σ min (L) ≥ 2E L −L ≥ 2/nE L −L F , we find LHS ≤ Proof. See Lemma 2.2 in [6]. Using these, we will prove the robustness of the stochastic Ho-Kalman Algorithm, which is stated in Theorem 6. The robustness will be up to a unitary transformation similar to Lemma 8. Proof. The proof is obtained by following the proof of Theorem 5.3 in [6] and substituteL for L. B Markov parameters estimation by least squares Given a sequence {z i } k−1 i=0 , the operator Toep(z) returns a k×k upper-triangular Toeplitz matrix Z, where Z i,j = Z i+1,j+1 = z i−j , if i ≤ j, and Z i,j = 0 when i > j. We will briefly introduce some existing results on learning the Markov parameter matrix G. The matrix G can be learned by solving the following OLS problem: G = argmin X∈R p×mT Y − XU 2 F = Y U T U U T −1(15) where Y = y (1) . . . y (N ) ∈ R p×N T , U = U (1) . . . U (N ) ∈ R mT ×N T . Where U (i) = Toep(u T −1 ] ∈ R p×T . Note that there are different methods from Eq 15 to formulate the least square problem, depending on how the input/output data samples are collected and utilized [6,15]. All the existing work [17,6,8,10,7,15] shows that the estimated Markov parameters converge to the true Markov parameters at a rate of O( 1 √ N ), where N is the number of trajectories, regardless of the algorithm used. Algorithm 2 2Randomized SVD: RSVD 1: Input: an m × n matrix A, a target rank k, an oversampling parameter l 2: Output: Approximate SVD s.t. A ≈ U SV T 3: Ω = randn(n, k + l) 4: P = orth(AΩ) ⊲ approx. basis for Range(A) 5: M = P T A ⊲ dim(M ) = (k + l) × n 6: [U, S, V ] = svd(M ) 7: U = P U 8: U = U (:, 1 : k), S = S(1 : k, 1 : k), V = V (:, 1 : k) Lemma 4 . 4(Average perturbation bound) Denote l ≥ 2 to be the oversampling parameter used in RSVD. Run the Stochastic Ho-Kalman Algorithm with a standard Gaussian matrix Ω ∈ R mT 2 ×(n+l) in line 3 of RSVD, where n + l ≤ min{pT 1 , mT 2 }. Then L,L satisfy the following perturbation bound: Running time of stochastic Ho-Kalman Algorithm using RSVD with oversampling parameter l. Realization error of stochastic Ho-Kalman Algorithm using RSVD with oversampling parameter l. Running time of stochastic Ho-Kalman Algorithm with varying power parameter q. The oversampling parameter l is 10. Realization error of the stochastic Ho-Kalman Algorithm with varying power parameter q. The oversampling parameter l is 10. Figure 1 : 1Oversampling and power iteration effect Lemma 7 . 7(Deviation bound) Denote l ≥ 2 to be the oversampling parameter used in RSVD algorithm. Run the Stochastic Ho-Kalman Algorithm with a SRFT matrix Ω ∈ R mT 2 ×(n+l) in computing the RSVD step, where 4[ √ n + 8 log(nmT 2 )] 2 log(n) ≤ l + n ≤ mT 2 . Then L,L satisfy the following perturbation bound: Lemma 8 . 8Suppose σ min (L) ≥ 2E L −L where σ min (L) is the smallest nonzero singular value (i.e. n-th largest singular value) of L. Let rank n matrices L,L have the singular value decomposition U ΣV * andŨΣṼ * . There exists an n × n unitary matrix S so that L −L ≤ 5nE L −L . Lemma 9. Suppose σ min (L) ≥ 2 ≥ 2E L−L . Then, E L ≤ 2 L and σ min (EL) ≥ σ min (L)/2. Table 1 : 1Comparison of SVD computation complexity. Methods below the dashed line are for computing an approximate SVD. The Moore-Penrose inverse of the matrix A denoted by A † is V Σ † U T , where Σ † is formed by transposing Σ and then taking the reciprocal of all the non-zero elements. In the following analysis, we will useŨ ,Σ,Ṽ ,L to denote the variables used in the stochastic Ho-Kalman Algorithm. There is no requirement that the systems we work with be stable. However, we are using an H∞-norm metric to judge the approximation error, so such an assumption makes things more straight forward.4 We used the publicly available python package sklearn.utils.extmath.randomized svd to compute the RSVD. AcknowledgementsJames Anderson and Han Wang acknowledge funding from the Columbia Data Science Institute. 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[ "N-States Continuous Maxwell Demon", "N-States Continuous Maxwell Demon" ]
[ "Paul Raux \nUMR 8236-LIED\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n\nIJCLab\nUniversité Paris-Saclay\nCNRS/IN2P3\n91405OrsayFrance\n", "Felix Ritort \nSmall Biosystems Lab\nCondensed Matter Physics Department\nUniversity of Barcelona\n08028BarcelonaSpain\n\nInstitut de Nanociència i Nanotecnologia (IN2UB)\nUniversitat de Barcelona\n08028BarcelonaSpain\n" ]
[ "UMR 8236-LIED\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "IJCLab\nUniversité Paris-Saclay\nCNRS/IN2P3\n91405OrsayFrance", "Small Biosystems Lab\nCondensed Matter Physics Department\nUniversity of Barcelona\n08028BarcelonaSpain", "Institut de Nanociència i Nanotecnologia (IN2UB)\nUniversitat de Barcelona\n08028BarcelonaSpain" ]
[]
Maxwell's demon is a famous thought experiment and a paradigm of the thermodynamics of information. It is related to Szilard's engine, a two-state information-to-work conversion device in which the demon performs single measurements and extracts work depending on the state measurement outcome. A variant of these models, the continuous Maxwell demon (CMD), was recently introduced by Ribezzi-Crivellari and Ritort where work was extracted after multiple repeated measurements every time that τ is in a two-state system. The CMD was able to extract unbounded amounts of work at the cost of an unbounded amount of information storage. In this work, we built a generalization of the CMD to the N-state case. We obtained generalized analytical expressions for the average work extracted and the information content. We show that the second law inequality for information-to-work conversion is fulfilled. We illustrate the results for N-states with uniform transition rates and for the N = 3 case.
10.3390/e25020321
null
256,800,404
2305.05029
6f33798aaddf54b2e1abe047b240f464f0ceaba1
N-States Continuous Maxwell Demon Published: 9 February 2023 Paul Raux UMR 8236-LIED Université Paris Cité CNRS 75013ParisFrance IJCLab Université Paris-Saclay CNRS/IN2P3 91405OrsayFrance Felix Ritort Small Biosystems Lab Condensed Matter Physics Department University of Barcelona 08028BarcelonaSpain Institut de Nanociència i Nanotecnologia (IN2UB) Universitat de Barcelona 08028BarcelonaSpain N-States Continuous Maxwell Demon Published: 9 February 202310.3390/e25020321Received: 31 December 2022 Revised: 6 February 2023 Accepted: 7 February 2023Citation: Raux, P.; Ritort, F. N-States Continuous Maxwell Demon. Entropy 2023, 25, 321. https:// Academic Editor: Jean-Noël Jaubert This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). entropy ArticleMaxwell demoninformation-to-work conversioncorrelated measurements Maxwell's demon is a famous thought experiment and a paradigm of the thermodynamics of information. It is related to Szilard's engine, a two-state information-to-work conversion device in which the demon performs single measurements and extracts work depending on the state measurement outcome. A variant of these models, the continuous Maxwell demon (CMD), was recently introduced by Ribezzi-Crivellari and Ritort where work was extracted after multiple repeated measurements every time that τ is in a two-state system. The CMD was able to extract unbounded amounts of work at the cost of an unbounded amount of information storage. In this work, we built a generalization of the CMD to the N-state case. We obtained generalized analytical expressions for the average work extracted and the information content. We show that the second law inequality for information-to-work conversion is fulfilled. We illustrate the results for N-states with uniform transition rates and for the N = 3 case. Introduction In 1867, James Clerk Maxwell proposed a thought experiment for the better understanding of the scope and limitations of the second law [1]. Known as the Maxwell demon paradox, it has spurred strong research activity for many years, setting the basis for the thermodynamics of information and information-to-work conversion [2][3][4][5][6][7][8][9][10]. In 1929, Leo Szilard introduced a simple physical model [11] in which a particle was free to move in a box of volume V with two compartments (denoted with 0 and 1) of volumes V 0 and V 1 and V = V 0 + V 1 ( Figure 1A). In Szilard's engine (SZ), the "demon" was an entity able to monitor the particle's position and store the observed compartment in a single-bit variable σ = 0, 1. Information-to-work conversion is as follows: once the particle's compartment is known, a movable wall is inserted between the two compartments, and an isothermal process is implemented to extract work. A work-extracting cycle concludes when the movable wall reaches its far end, and the measurement-work extraction process restarts. The average work extracted per cycle equals the equivalent heat transferred from the isothermal reservoir to the system: W Sz 2 = −k B T(P 0 log(P 0 ) + P 1 log(P 1 )), with P 0,1 = V 0,1 /V the occupancy probabilities of each compartment. For equal compartments P 0 = P 1 = 1/2, Szilard's engine can extract maximal work determined by the Landauer bound, W SZ ≤ k B T log(2) from the reservoir without energy consumption, meaning that heat was fully converted into work, apparently violating Kelvin's postulate. In the 1960s and 1970s, work by Landauer [12] and Bennett [13] found a solution to the paradox. The solution to this paradox considers the information content of the measurement, the work extraction, and the resetting processes of the demon [14,15]. Most importantly, to recover the initial state at the end of the thermodynamic cycle, the demon must erase the information acquired on the system [2]. The minimal erasure cost per bit of information equals k B T log(2) for equally probable outcomes. In the end, the information content stored in the demon is always larger than or equal to the extracted work, in agreement with the second law. In a recent paper, a new variant of the Maxwell demon, the continuous Maxwell demon (CMD), was introduced [16] (see also [17] for additional results), analytically solved, and experimentally tested. In the CMD, the demon performs repeated measurements of a particle's location in a two-compartment box every time τ. The first time that the demon measures that the particle changed compartments, a work extraction procedure is implemented. The main difference with the SZ engine is that, in the CMD, a work extraction cycle contains multiple measurements, whereas for the SZ, a single measurement is performed at every work cycle. Compared to the SZ, the CMD can extract a more significant amount of work W because of the larger information content of the multiple stored bits in a cycle. Interestingly, the average work per cycle in the CMD satisfies W CMD ≥ k B T log(2) being unbounded in the limits P 0 → 0 (P 1 → 1) and P 0 → 1 (P 1 → 0). A model combining the SZ and CMD work extraction protocols version showed the role of temporal correlations in optimizing information-to-energy conversion [18]. In the CMD, the time between measurements τ is arbitrary. In particular, it can be made infinitesimal, τ → 0, leading to an infinite number of measurements per cycle justifying the continuous adjective of the model. In the work extraction protocol, a pair of walls limiting the volume of the last compartment, here V 2 , are inserted. The wall between compartments 1 and 2 is fixed, whereas the wall between compartments 2 and 3 was movable and had no mass. To extract the work produced by the expansion of the particle confined in 2, the movable wall is connected to a pulley device. The average work extracted for this cycle is −k B T log V 2 V . V 2 V 0 V 1 V 2 V 3 σ σ' τ 2τ 3τ B) V 0 V 1 A) σ σ' τ 2τ V tot -V 2 M o v a b l e w a l l σ'= Ri gi d wa ll C) T σ=0,σ'=2 n=4 =0002 4-CMD W=-log(V 2 /V tot ) T σ=0,σ'=1 n=3 =0001 2-CMD Here, we generalize the CMD to the case of N-states (N-CMD). In a possible realization of the N-CMD, a particle in a box of volume V can occupy N distinct compartments of volumes V i ( Figure 1B). The demon measures in which compartment the particle is at every time τ until a change in the compartment is detected. Then, a work extraction process is implemented by inserting one fixed wall at one side and one movable wall at the other side of the compartment that can expand under the elastic collisions exerted by the particle ( Figure 1C). A pulley mechanism is attached to the movable wall to extract an average work equal to W = −k B T log P i with P i = V i /V. For N = 2, we obtain the standard CMD ( Figure 1A) (2-CMD), which corresponds to transforming the Szilard box ( Figure 1B) into a periodic torus ( Figure 1C). The outline of this work is as follows. In Section 2, we show how to generalize the mathematical formalism of the 2-CMD to N states (N-CMD). In Section 3, we analyze the performance of the N-CMD by studying the thermodynamic efficiency and power. In Section 4, we analyze several cases, and in particular case N = 3, to investigate the effect of topology on information-to-work conversion (IWC). We end with some conclusions and future directions. General Setting Let σ (= 1, . . . , N) denote the N states of a system following Markovian dynamics defined by transition rates that satisfy detailed balance, ensuring that the system relaxes to an equilibrium state. Let τ be the time between consecutive measurements. The conditional probability p τ (σ |σ) that the outcome of the measurement is σ after time τ conditioned that it starts in σ satisfies the following master equation: ∂ τ p τ (σ |σ) = N ∑ σ =1 K σ σ p τ (σ |σ)(1) with initial condition p τ→0 (σ |σ) → δ σ ,σ , where δ is the Kronecker delta function. Markov matrix K σ σ satisfies ∑ σ K σ σ = 0; ∀σ , defining transitions rates from state σ to σ : K σ σ = − ∑ σ( =σ ) k σ←σ if σ = σ k σ ←σ otherwise (2) with k σ ←σ the probability to jump from state σ to σ during time dτ. Let us denote by P σ the stationary solutions of Equation (1). The detailed balance condition reads: ∀σ, σ ; K σσ P σ = K σ σ P σ(3) The solution of Equation (1) can be written using the Perron-Frobenius theorem (see [19]) as a spectral expansion in terms of the eigenvalues and eigenvectors of K: p τ (σ |σ) = P σ ∑ α l α σ l α σ exp(λ α τ)(4) where l α is the left eigenvector of K associated with the eigenvalue λ α . The sum over the α term in Equation (4) is symmetric in σ ↔ σ . Therefore, the conditional probabilities also fulfil a detailed balance: p τ (σ|σ ) p τ (σ |σ) = P σ P σ(5) Remark 1. Detailed balance ensures that there exists a unique stationary state P σ associated with the eigenvalue λ 0 = 0 and that the other eigenvalues are real and negative, λ α =0 < 0. Equation (4) can be rewritten as follows: p τ (σ |σ) = P σ 1 + ∑ α =0 l α σ l α σ exp(λ α τ)(6) which gives p τ (σ |σ) = P σ for τ → ∞ as expected. In the CMD, a work-extraction cycle is defined by a sequence of n + 1 measurement outcomes σ i (1 ≤ i ≤ n + 1) repeatedly taken every time τ. In a cycle the first n outcomes are equal (σ) ending with σ ( = σ). We define the trajectory for a cycle as follows: T n σσ = σ, . . . , σ, n σ(7) The probability of a given trajectory T n σσ reads: P(T n σσ ) = p τ (σ|σ) n−1 p τ (σ |σ)(8) This is normalized as follows: ∑ σ ( =σ) ∞ ∑ n=1 P(T n σσ ) = 1 , ∀σ(9) Thermodynamic Work and Information-Content Like in the SZ, the work extracted by the CMD in a given cycle T n σσ equals − log(P σ ). Averaging over all the possible measurement cycles, we obtain the average extracted work: W N (τ) =< − log P σ >= − ∑ σ ∑ σ ( =σ) ∞ ∑ n=1 P σ P(T n σσ ) log P σ = − N ∑ σ=1 P σ 1 − p τ (σ|σ) ∑ σ =σ p τ (σ |σ) log P σ(10) which is positive by definition. In the limit τ → ∞ we obtain the following expression, W ∞ N = − N ∑ σ=1 P σ 1 − P σ ∑ σ =σ P σ log P σ ,(11) which can be written as follows: W ∞ N = N ∑ σ=1 P σ 1 − P σ W SZ N + N ∑ σ=1 P 2 σ log P σ 1 − P σ(12) where W SZ N is the classical statistical entropy of the system, which can also be interpreted as the average work extraction of the N-states Szilard engine, denoted as N-SZ: W SZ N =< − log(P σ ) >= − ∑ σ P σ log P σ(13) This expression can be readily minimized in the space of P σ giving the uniform solution, P σ = 1/N for which W ∞ N = log N. In contrast, W ∞ N → − log(1 − P σ ) if P σ → 1 for a given σ (and P σ → 0 ∀σ = σ) diverging in that limit. We define the average information content per cycle as the statistical entropy of the measurement-cycle probabilities [20]: I N (τ) =< − log(P σ P(T n σσ )) > = − ∑ σ ∑ σ ( =σ) ∞ ∑ n=1 P σ P(T n σσ ) log(P σ P(T n σσ )) = − N ∑ σ=1 N ∑ σ =σ P σ p τ (σ |σ) log(P σ ) + p τ (σ |σ) log(p τ (σ |σ)) ∞ ∑ n=1 p τ (σ|σ) n−1 1 1−pτ (σ|σ) − N ∑ σ=1 N ∑ σ =σ P σ p τ (σ |σ) log(p τ (σ|σ)) ∞ ∑ n=1 (n − 1)p τ (σ|σ) n−1 pτ (σ|σ) (1−pτ (σ|σ)) 2 = W SZ N − N ∑ σ P σ 1 − p τ (σ|σ) ∑ σ p τ (σ |σ) log p τ (σ |σ)(14) The positivity of I N (τ) follows from the fact that p τ (σ |σ), P σ ≤ 1. The second term in Equation (14) depends on τ and can be understood as the contribution of correlations between measurements to I N (τ). Lastly, using Equation (5), we can rearrange Equation (14) as follows: I N (τ) = W N (τ) − N ∑ σ P σ 1 − p τ (σ|σ) ∑ σ p τ (σ |σ) log p τ (σ|σ ) =∆ N >0(15) where the second term ∆ N is positive since p τ (σ|σ ) ≤ 1. Equation (15) implies the second law inequality: (16) meaning that the cost to erase the stored sequences information content is always larger than the work extracted by the demon. I N (τ) − W N (τ) > 0 ∀τ Comparison with the Szilard Engine Equating Expressions (14) and (15) for I N (τ), we obtain a relation between W N (τ) and W SZ that compares the average work extracted in the N-CMD to the N-SZ engine as follows: W N (τ) − W SZ N =< − log P σ P σ > = − ∑ σ p σ 1 − p τ (σ|σ) ∑ σ p τ (σ |σ) log p τ (σ |σ) p τ (σ|σ ) ≥ 0(17) where the first equality follows from the difference between the first right-hand side of Equations (10) and (13). This shows that the CMD's average work per cycle is always larger or equal to SZ. The equality holds for the uniform case P σ = 1/N where W N (τ) = W SZ N = log N. Thermodynamic Power and Efficiency Average Cycle Length As a preliminary, we first compute the average time of a cycle of measurement. This is similar to the mean first residence time of the system, except for the fact that (unobserved) hopping events are permitted. We define it as follows: t c N ≡ τ < n >(18) and obtain the following expression: t c N = τ 1 + ∑ σ P σ 1 − p τ (σ|σ)(19) The following equalities are shown: lim τ→0 + t c N = − ∑ i 1 ∑ α =0 (l α i ) 2 λ α > 0 (20) lim τ→∞ t c N = +∞(21) The average cycle time is the mean first passage time [21] of the discrete time random walk defined by a cycle of measurements. Thermodynamic Power We define the thermodynamic power as the average work W N extracted per cycle time t c N : Φ N (τ) = W N t c N(22) In the limit of uncorrelated measurements τ → ∞, we obtain from Equations (11) and (19): Φ ∞ N = − 1 τ ∑ N σ=1 P σ 1−P σ ∑ σ =σ P σ log P σ 1 + ∑ σ P σ 1−P σ(23) For N = 2, we recover the results in [16,17]. Information-to-Work Efficiency In the spirit of the efficiencies defined for thermal machines, we define the informationto-work conversion (IWC) efficiency of the CMD as the ratio between W N , taken to be the objective function, and I N , taken to be the cost function, for the optimization of the CMD: η N = W N I N(24) Using Equation (15), we can rewrite η N as follows: η N = 1 1 + ∆ N W N(25) From Equation (16), η N < 1. In the limit τ → ∞, we obtain: lim τ→∞ η N = 1 1 + ∑ i P i 1−P i log P i ∑ i P i 1−P i ∑ j =i P j log P j(26) In limit P σ → 1 for a given state σ, one can check that the N-CMD reaches maximal efficiency 1. Particular Cases Here, we analyze some specific examples. Case N=2 We now turn to the N = 2 case considered in [16] as an example of our formulae. The kinetic rate matrix in this case reads: K = −k 1←0 k 0←1 k 1←0 −k 0←1(27) Here, we do not need to make any particular choice of rates k σ σ to ensure detailed balance since, for two states, a detailed balance unconditionally holds. Applying the procedure sketched in Section 2, we solve the master equation: p τ = (p τ (σ|σ )) σ,σ =0,1 P 0 + P 1 exp(−Rτ) P 0 (1 − exp(−Rτ)) P 1 (1 − exp(−Rτ)) P 1 + P 0 exp(−Rτ)(28) where R = k 1←0 + k 0←1 , P 0 = k 0←1 R and P 1 = k 1←0 R such that P 0 + P 1 = 1. p τ is normalized per column: p τ (σ |σ) + p τ (1 − σ |σ) = 1 , ∀σ = 0, 1(29) First, let us consider W 2 . Since N = 2 and by normalization, there is only one term in the sum ∑ σ =σ of Equation (10). Thus, W 2 simplifies to: W 2 = −P 0 log(1 − P 0 ) − (1 − P 0 ) log(P 0 )(30) We recover the result obtained in [16] and coincidently show that the τ independence of this result is a particular feature of the N = 2 case. Moreover, since W 2 had a simple expression, we obtained a tractable expression for the comparison with the SZ average work extracted, c.f. Equation (17): W 2 − W SZ 2 = (1 − 2P 0 ) log 1 − P 0 P 0(31) This quantity is positive and vanishes only for uniform probability, P σ = 1 2 , as shown in Section 3.1. Using normalization Equation (29) again in the definition of I N Equation (14), we obtain I 2 as follows: I 2 = −P 0 log P 0 − (1 − P 0 ) log(1 − P 0 ) − P 0 p τ (0|0) p τ (1|0) log p τ (0|0) + log p τ (1|0) − (1 − P 0 ) p τ (1|1) p τ (0|1) log p τ (1|1) + log p τ (0|1)(32) which is the result obtained in [16]. The remaining results of [16] are obtained by combining Equations (28), (30), and (32). Uniform Transition Rates In this subsection, we take the following particular case for the Markov matrix K: K σ σ = R × −(N − 1) if σ = σ 1 otherwise (33) In this case, there are only two independent conditional probabilities; we can thus rewrite the master equation as follows: ∂ τ p τ (σ|σ) = R(1 − N p τ (σ|σ))(34) Via normalization, we obtain p τ (σ |σ) as follows: p τ (σ |σ) = 1 N − 1 (1 − p τ (σ|σ)) ; σ = σ(35) In the remainder of this subsection, we define the dimensionless rescaled time between two measurements asτ = Rτ. The solution of Equation (34) reads: p τ (σ|σ) = 1 N (1 + (N − 1) exp(−Nτ))(36) This particular case allows for us to obtain a glimpse of the dependence of the quantities introduced in Section 2 with N. The average work extracted is as follows: W N = log N.(37) We see that the work extracted does not depend on τ. I N reads: I N = log N − N N − 1 log 1 N (1 − exp(−Nτ)) (38) The first remark is that in the limitτ → ∞, I ∞ N = 2N−1 N−1 log N W N . One way to optimize the CMD is to maximize IWC efficiency, defined as follows: η N ≡ W N I N = log N log N − N log 1−e −Nτ N N−1(39) We find the asymptotic efficiency η ∞ N = N−1 2N−1 forτ → ∞ and η N = 1 2 for N → ∞. For the thermodynamic power, we obtain: Φ N ≡ W N t c N = log N Nτe Nτ (N−1)(e Nτ −1) +τ (40) where t c N is the average cycle time that we analyzed in Section 3.1. One can show that the maximum thermodynamic power Φ N = (N − 1) log(N) is obtained in the limitτ → 0. This shows that the maximum IWC efficiency Equation (39) and the efficiency at maximum power Equation (40) are obtained in two different limits, a general result expected for thermodynamic machines [22]. Case N = 3 The 3-CMD is the simplest case in which two different topologies of the state space are available. They are defined in Figure 2 and are denoted as triangular (Panel A) and linear (Panel B), respectively. We denote the energy of state σ(σ = 0, 1, 2) by σ . Taking β = 1, the detailed balance assumption Equation (3) then reads: K σσ K σ σ = exp(−( σ − σ )) , ∀σ = σ(41) Here we take 0 = 0. This implies that the energies of states 1, 2 read: 1 = log( P 0 P 1 ) (42) 2 = log( P 0 P 2 )(43) In the linear case, taking as a particular case k 01 = 1 and k 21 = 1, we obtain the following Markov matrix: K lin 3 =   −1 exp( 1 ) 0 1 −(exp( 1 ) + exp( 1 − 2 )) 1 0 exp( 1 − 2 ) −1   =    −1 P 0 P 1 0 1 − P 0 P 1 (1 + P 2 P 0 ) 1 0 P 2 P 1 −1   (44) where we used Equations (42) and (43) to give an expression of K lin 3 depending only on P 0 , P 1 , P 2 . In the triangular case, taking k 01 = 1 and k 21 = 1 and k 02 = 1 as a particular case, we obtain similarly the following Markov matrix: K tri 3 =   −2 exp( 1 ) exp( 2 ) 1 −(exp( 1 ) + exp( 1 − 2 )) 1 1 exp( 1 − 2 ) −(1 + exp( 2 ))   =    −2 P 0 P 1 P 0 P 2 1 − P 0 P 1 (1 + P 2 P 0 ) 1 1 P 2 P 1 −(1 + P 0 P 2 )   (45) The solution of Equation (1) with Markov matrix (44) in the linear case and (45) in the triangular case, can be written using the Perron-Frobenius theorem [19] as the following spectral expansion: P σ (τ) = Ψ 0 + c σ 1 Ψ 1 exp(λ 1 τ) + c σ 2 Ψ 2 exp(λ 2 τ)(46) where λ 1 , λ 2 < 0 and c σ 1 , c σ 2 are the coefficients determined in the limit τ → 0, which depend on the conditional state σ. These coefficients are gathered in Table 1 for both models. c σ 1 c σ 2 σ = 0 −(P 2 + P 2 2 +P 2 (P 0 +P 1 ) P 0 +P 2 ) P 1 P 2 P 0 +P 2 σ = 1 0 P 2 (P 1 −1) P 0 +P 2 σ = 2 P 0 P 0 +P 2 P 1 P 2 P 0 +P 2 Ψ 0 , Ψ 1 , Ψ 2 are the eigenvectors of both K lin 3 , K tri 3 : • Ψ 0 is the eigenvector associated to the eigenvalue 0 and it corresponds to the stationary probability. Since the detailed balance condition Equation (3) holds, the stationary probability is the Boltzmann distribution. Thus, Ψ 0 =   P 0 P 1 P 2   = 1 Z   1 exp(− 1 ) exp(− 2 )  (47) where Z = 1 + exp(− 1 ) + exp(− 2 ) • Ψ 1 is the eigenvector associated to the second eigenvalue, which reads λ lin 1 = −1 in the linear case, and λ tri 1 = −(1 + 1−P 1 P 2 ) in the triangular case. Ψ 1 reads: Ψ 1 =   −1 0 1  (48) • Ψ 2 is the eigenvector associated in both models to the eigenvalue λ 2 = − 1 P 1 . It reads: Ψ 2 =    P 0 P 2 − 1−P 1 P 2 1   (49) Uncorrelated Measurements on the 3-CMD We now turn to the limit τ → ∞. In this limit of uncorrelated measurements, the time between consecutive measurements τ is larger than the relaxation time of the system, the inverse of the lowest eigenvalue, ∼−1/λ 1 . In this limit, P σ (τ) reduces to Boltzmann distribution Equation (47) and p τ (σ |σ) = P σ . Therefore, the two models (linear and triangular) are indistinguishable. Results for work and information are shown in Figure 3. . W 2 , I 2 , W 3 , I 3 as a function of P 2 for P 1 fixed in each panel. Large work extraction is obtained in the limit of rare events P 1 → 0 and P 2 → 0, 1. First, it is clear that the second law Inequality (16) was satisfied. In the limit P 1 → 0, we recovered the 2-CMD. Our generalized expressions for work and information content reproduced well the trend observed in Figure 1c of [16]. In the limit of rare events, where P 1 → 0 and P 2 → 0, 1, we recovered the infinite average work extraction described for the 2-CMD. Large work extraction was only obtained in the 2-CMD limit. Efficiency η 3 is shown in Figure 4. For P 1 → 0 and P 2 → 0 or P 2 → 1, we recovered the limit of rare events and maximal efficiency η 3 → 1. In the 3-CMD, we have η 3 ∈ Correlated Measurements in the 3-CMD Correlated measurements are those where τ is lower than or comparable to the equilibrium relaxation time. Equation (46) shows that the dynamics of the linear and triangular topologies for the 3-CMD are very similar. Indeed, in the limit of uncorrelated measurements, the two dynamics reduce to the same Boltzmann distribution. They also collapse in the limit P 1 → 1 (with P 0 , P 2 → 0), indeed in this case λ lin 1 = λ tri 1 . In between, the topology of the network is relevant. For correlated measurements, we obtained the results shown in Figure 5. First, the average cycle time (upper-left panel in Figure 5) in the linear case was generally larger than that in the triangular case. The direct consequence, since the average work extraction was comparable in both cases, was that the thermodynamic power (upperright panel in Figure 5) extracted by the linear 3-CMD was lower than the thermodynamic power extracted by the triangular 3-CMD. Moreover, the thermodynamic power decreased logarithmically to 0 when τ increases. Thus, 3-CMD had optimal power production in limit τ → 0, i.e., in the limit of continuous measurements. The efficiency of the 3-CMD as a function of τ is plotted in the lower-left panel of Figure 5. The linear 3-CMD was generally less efficient than the triangular 3-CMD. The reason for this is in the lower-right panel of Figure 5, where W 3 and I 3 are plotted against τ for both models. For a comparable work extraction, the linear 3-CMD needs to store more information. Again, in the limit of uncorrelated measurements, the two models converge to the same result. Concluding Remarks In this work, we generalized the 2-CMD of [16,17] to N-states. We obtained generalized expressions of the average extracted work, the average information content stored in the demon's memory, and of thermodynamical quantities such as the thermodynamic power and the information-to-work efficiency of the N-CMD. We proved that the second law inequality holds for the N-CMD, thus giving bounds on the efficiency of the engine. Comparing the N-CMD to the N-SZ engine, we also showed that the N-CMD could extract more work on average than the N-SZ engine. The most efficient setting of the N-CMD was in the limit of rare events already described in [16]. In the N-CMD case, this limit was obtained by first taking the 2-CMD limit. Thus, no configuration is more efficient in the N-CMD than the 2-CMD limit. In future work on the N-CMD, it would be interesting to implement a graph theoretic procedure to obtain, for instance, a more precise explanation of the difference between the linear and triangular cases (connected graph versus fully connected graph). It would also be interesting to determine the distributions of the quantities computed here [23] and thus optimize the fluctuations of the N-CMD. Author Contributions: F.R. conceived the work, and P.R. did the calculations. All authors have read and agreed to the published version of the manuscript. Figure 1 ., 2 . 12(A) The 2-CMD is represented as a two-compartment box in which a work extraction protocol is implemented (see text). The cycle of measurement is here T for this cycle is − log V 1 V . (B) 4-CMD in circular geometry. Each compartment had volume V i . The cycle of measurement of the CMD reads: The initial state σ is 0 and the final state σ is 2, the crossing of compartment 3 remained unnoticed for measurements made at every time τ. (C) Figure 2 . 2Definition of the state spaces for the 2 topologies available for the 3-CMD: (A) Triangular 3-CMD, (B) Linear 3-CMD. Figure 3 3Figure 3. W 2 , I 2 , W 3 , I 3 as a function of P 2 for P 1 fixed in each panel. Large work extraction is obtained in the limit of rare events P 1 → 0 and P 2 → 0, 1. Figure 4 . 4W 2 , I 2 , W 3 , I 3 as a function of P 2 for P 1 fixed in each panel. Figure 5 . 5The 3-CMD for correlated measurements for P 1 = P 2 = 0.001. (upper left) Average cycle length t c 3 /τ in both models, Equation (19); (upper right) thermodynamic power; (lower left) efficiency; (lower right) average information content and work extraction in k B T units (orange and red lines collapse on top of each other). 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[]
[ "Transfer Learning for Performance Modeling of Configurable Systems: An Exploratory Analysis", "Transfer Learning for Performance Modeling of Configurable Systems: An Exploratory Analysis" ]
[ "Pooyan Jamshidi \nCarnegie Mellon University\nUSA\n", "Norbert Siegmund \nBauhaus-University\nWeimarGermany\n", "Miguel Velez \nCarnegie Mellon University\nUSA\n", "Christian Kästner \nCarnegie Mellon University\nUSA\n", "Akshay Patel \nCarnegie Mellon University\nUSA\n", "Yuvraj Agarwal \nCarnegie Mellon University\nUSA\n" ]
[ "Carnegie Mellon University\nUSA", "Bauhaus-University\nWeimarGermany", "Carnegie Mellon University\nUSA", "Carnegie Mellon University\nUSA", "Carnegie Mellon University\nUSA", "Carnegie Mellon University\nUSA" ]
[]
Modern software systems provide many configuration options which significantly influence their non-functional properties. To understand and predict the effect of configuration options, several sampling and learning strategies have been proposed, albeit often with significant cost to cover the highly dimensional configuration space. Recently, transfer learning has been applied to reduce the effort of constructing performance models by transferring knowledge about performance behavior across environments. While this line of research is promising to learn more accurate models at a lower cost, it is unclear why and when transfer learning works for performance modeling. To shed light on when it is beneficial to apply transfer learning, we conducted an empirical study on four popular software systems, varying software configurations and environmental conditions, such as hardware, workload, and software versions, to identify the key knowledge pieces that can be exploited for transfer learning. Our results show that in small environmental changes (e.g., homogeneous workload change), by applying a linear transformation to the performance model, we can understand the performance behavior of the target environment, while for severe environmental changes (e.g., drastic workload change) we can transfer only knowledge that makes sampling more efficient, e.g., by reducing the dimensionality of the configuration space.
10.1109/ase.2017.8115661
[ "https://arxiv.org/pdf/1709.02280v1.pdf" ]
24,324,165
1709.02280
24b5a87935a435182f902702676dbe8411a2ee82
Transfer Learning for Performance Modeling of Configurable Systems: An Exploratory Analysis Pooyan Jamshidi Carnegie Mellon University USA Norbert Siegmund Bauhaus-University WeimarGermany Miguel Velez Carnegie Mellon University USA Christian Kästner Carnegie Mellon University USA Akshay Patel Carnegie Mellon University USA Yuvraj Agarwal Carnegie Mellon University USA Transfer Learning for Performance Modeling of Configurable Systems: An Exploratory Analysis Index Terms-performance analysis, transfer learning Modern software systems provide many configuration options which significantly influence their non-functional properties. To understand and predict the effect of configuration options, several sampling and learning strategies have been proposed, albeit often with significant cost to cover the highly dimensional configuration space. Recently, transfer learning has been applied to reduce the effort of constructing performance models by transferring knowledge about performance behavior across environments. While this line of research is promising to learn more accurate models at a lower cost, it is unclear why and when transfer learning works for performance modeling. To shed light on when it is beneficial to apply transfer learning, we conducted an empirical study on four popular software systems, varying software configurations and environmental conditions, such as hardware, workload, and software versions, to identify the key knowledge pieces that can be exploited for transfer learning. Our results show that in small environmental changes (e.g., homogeneous workload change), by applying a linear transformation to the performance model, we can understand the performance behavior of the target environment, while for severe environmental changes (e.g., drastic workload change) we can transfer only knowledge that makes sampling more efficient, e.g., by reducing the dimensionality of the configuration space. I. INTRODUCTION Highly configurable software systems, such as mobile apps, compilers, and big data engines, are increasingly exposed to end users and developers on a daily basis for varying use cases. Users are interested not only in the fastest configuration, but also in whether the fastest configuration for their applications also remains the fastest when the environmental situation has been changed. For instance, a mobile developer might be interested to know if the software that she has configured to consume minimal energy on a testing platform will also remain energy efficient on the users' mobile platform; or, in general, whether the configuration will remain optimal when the software is used in a different environment (e.g., with a different workload, on different hardware). Performance models have been extensively used to learn and describe the performance behavior of configurable systems [15], [19], [21], [23], [33], [43]- [45], [54], [61], [63]. However, the exponentially growing configuration space, complex interactions, and unknown constraints among configuration options [56] often make it costly and difficult to learn an accurate and reliable performance model. Even worse, existing techniques usually consider only a fixed environment (e.g., fixed workload, fixed hardware, fixed versions of the dependent libraries); should that environment change, a new performance model may need to be learned from scratch. This strong assumption limits the reusability of performance models across environments. Reusing performance models or their byproducts across environments is demanded by many application scenarios, here we mention two common scenarios: • Scenario 1: Hardware change: The developers of a software system performed a performance benchmarking of the system in its staging environment and built a performance model. The model may not be able to provide accurate predictions for the performance of the system in the actual production environment though (e.g., due to the instability of measurements in its staging environment [6], [30], [38]). • Scenario 2: Workload change: The developers of a database system built a performance model using a read-heavy workload, however, the model may not be able to provide accurate predictions once the workload changes to a writeheavy one. The reason is that if the workload changes, different functions of the software might get activated (more often) and so the non-functional behavior changes, too. In such scenarios, not every user wants to repeat the costly process of building a new performance model to find a suitable configuration for the new environment. Recently, the use of transfer learning (cf. Figure 1) has been suggested to decrease the cost of learning by transferring knowledge about performance behavior across environments [7], [25], [51]. Similar to humans that learn from previous experience and transfer the learning to accomplish new tasks easier, here, knowledge about performance behavior gained in one environment can be reused effectively to learn models for changed environments with a lower cost. Despite its success, it is unclear why and when transfer learning works for performance analysis in highly configurable systems. To understand the why and when, in this paper, we con-duct an exploratory empirical study, comparing performance behavior of highly configurable systems across environmental conditions (changing workload, hardware, and software versions), to explore what forms of knowledge can be commonly exploited for performance modeling and analysis. Specifically, we explore how performance measures and models across the source and target of an environmental change are related. The notion of relatedness across environments gives us insights to consolidate common knowledge that is shared implicitly between the two environments, from knowing entire performance distributions, to knowing about the best or invalid configurations, or knowing influential configuration options, or knowing about important interactions. The various forms of shared knowledge, that we discovered in this empirical study, provide opportunities to develop novel transfer learning that are not only based on correlation concept, but also more diverse forms of similarities across environments. More specifically, we explore several hypotheses about the notion of common knowledge across environments. Our hypotheses start with very obvious relationships (e.g., correlation) that can be easily exploited, but range toward more subtle relationships (e.g., influential options or invalid regions remain stable) that can be explored with more advanced transfer learning techniques yet to be developed. We tested our hypotheses across 36 environmental changes in 4 configurable systems that have been selected purposefully covering different severities and varieties. For instance, we selected simple hardware changes (by changing computing capacity) as well as severe changes (by changing hardware from desktop to cloud). Our results indicate that some knowledge about performance behavior can be transfered even in the most severe changes we explored, and that transfer learning is actually easy for many environmental changes. We observed that, for small changes, we can frequently transfer performance models linearly across environments, while for severe environmental changes, we can still transfer partial knowledge, e.g., information about influential options or regions with invalid configurations, that can still be exploited in transfer learning, for example, to avoid certain regions when exploring a configuration space. Overall, our results are encouraging to explore transfer learning further for building performance models, showing broad possibilities of applying transfer learning beyond the relatively small changes explored in existing work (e.g., small hardware changes [51], low fidelity simulations [25], similar systems [7]). Overall, our contributions are the following: • We formulate a series of hypotheses to explore the presence and nature of common, transferable knowledge between a source and a target environment, ranging from easily exploitable relationships to more subtle ones. • We empirically investigate performance models of 4 configurable systems before and after 36 environmental changes. We performed a thorough exploratory analysis to understand why and when transfer learning works. • We discuss general implications of our results for performance modeling of configurable software systems. • We release the supplementary material including data of several months of performance measurements, and scripts for replication: https://github.com/pooyanjamshidi/ase17. II. INTUITION Understanding the performance behavior of configurable software systems can enable (i) performance debugging [14], [44], (ii) performance tuning [16], [20], [21], [32], [33], [36], [47], [51], [54], (iii) design-time evolution [2], [24], or (iv) runtime adaptation [10]- [12], [19], [25], [26]. A common strategy to build performance models is to use some form of sensitivity analysis [42] in which the system is executed repeatedly in different configurations and machine learning techniques are used to generalize a model that explains the influence of individual options or interactions [15], [44], [51]. In this paper, we are interested in how a performance model for a configurable system changes when we deploy the system in a different environment. To this end, we distinguish between configuration options -parameters that users can tweak inside the system to select functionality or make tradeoffs among performance, quality, and other attributesand environment changes -differences in how the system is deployed and used in terms of workload, hardware, and version. If a performance model remains relatively stable across environments (e.g., the top configurations remain the top configurations, the most influential options and interactions remain most influential), we can exploit this stability when learning performance models for new environments. Instead of building the model from scratch (as often exhaustively measuring the same configurations on a new environment), we can reuse knowledge gathered previously for other environments in a form of transfer learning [7], [39], [50]. That is, we can develop cheaper, faster and more accurate performance models that allow us to make predictions and optimizations of performance in changing environments [25]. For example, consider an update to faster hardware. We would often expect that the system will get faster, but will do so in a nearly uniform fashion. However, we may expect that options that cause a lot of I/O operations (e.g., a backup feature) may benefit less from a faster CPU than other options; so not all environment changes will cause uniform changes. If transfer across hardware is indeed usually easy, this encourages, for example, scenarios in which we learn performance models offline on cheap hardware and transfer it to the real system with few expensive measurements for adjustment. The question is what kind of knowledge can be exploited across environments in practice, with simple or more advanced forms of transfer learning. Specifically, we ask whether there exists common information (i.e., transferable/reusable knowledge, c.f., Figure 1) that applies to both source and target environments and, therefore, can be carried over across environments. A. Environmental changes Let us first introduce what we mean by environment, the key concept that is used throughout this paper. An environmental condition for a configurable system is determined by its hardware, workload, and software version. (i) Hardware: The deployment configuration in which the software system is running. (ii) Workload: The input of the system on which it operates on. (iii) Version: The state of the code base at a certain point in time. Of course, other environmental changes might be possible (e.g., JVM upgrade). But, we limit this study to this selection as we consider the most common changes in practice that affect performance behavior of systems. B. Preliminary concepts In this section, we provide definitions of concepts that we use throughout this study. The formal notations enable us to concisely convey concepts throughout the paper. 1) Configuration and environment space: Let C i indicate the i-th configuration option of a system A, which is either enabled or disabled (the definitions easily generalize to nonboolean options with finite domains). The configuration space is a Cartesian product of all options C = Dom(C 1 ) × · · · × Dom(C d ), where Dom(C i ) = {0, 1} and d is the number of options. A configuration is then a member of the configuration space where all the options are either enabled or disabled. We describe an environmental condition e by 3 variables e = [h, w, v] drawn from a given environment space E = H × W × V , where each member represents a set of possible values for the hardware h, workload w, and system version v. We use notation ec : [h, w 1 → w 2 , v] as shorthand for an environment change from workload w 1 to workload w 2 where hardware and version remain stable. 2) Performance model: Given a software system A with configuration space C and environment space E, a performance model is a black-box function f : C × E → R that maps each configuration c ∈ C of A in an environment e ∈ E to the performance of the system. To construct a performance model, we run A in a fixed environmental condition e ∈ E on various configurations c i ∈ C, and record the resulting performance values y i = f (c i , e) + i where i ∼ N (0, σ i ) is the measurement noise corresponding to a normal distribution with zero mean and variance σ 2 i . The training data for learning a performance model for system A in environment e is then D tr = {(c i , y i )} n i=1 , where n is the number of measurements. 3) Performance distribution: We can and will compare the performance models, but a more relax representation that allows us to assess the potentials for transfer learning is the empirical performance distribution. The performance distribution is a stochastic process, pd : E → ∆(R), that defines a probability distribution over performance measures for environmental conditions of a system. To construct a performance distribution for a system A with configuration space C, we fit a probability distribution to the set of performance values, D e = {y i }, e ∈ E, using kernel density estimation [4] (in the same way as histograms are constructed in statistics). 4) Influential option: At the level of individual configuration options, we will be interested in exploring whether options have an influence on the performance of the system in either environment; not all options will have an impact on performance in all environments. We introduce the notion of a influential option to describe a configuration option that has a statistically significant influence on performance. 5) Options interaction: The performance influence of individual configuration options may not compose linearly. For example, while encryption will slow down the system due to extra computations and compression can speed up transfer over a network, combining both options may lead to surprising effects because encrypted data is less compressible. In this work, we will look for interactions of options as nonlinear effects where the influence of two options combined is different from the sum of their individual influences [44], [45]. 6) Invalid configuration: We consider a configuration as invalid if it causes a failure or a timeout. C. Transferable knowledge As depicted in Figure 1, any sort of knowledge that can be extracted from the source environment and can contribute to the learning of a better model (i.e., faster, cheaper, more accurate, or more reliable) in the target environment is considered as transferable knowledge (or reusable knowledge [1]). There are several pieces of knowledge we can transfer, such as (i) classification or regression models, (ii) dependency graphs that represent the dependencies among configurations, and (iii) option interactions in order to prioritize certain regions in the configuration space. For transferring the extracted knowledge, we need a transfer function that transforms the source model to the target model: tf : f (·, e s ) → f (·, e t ). In its simplest form, it can be a linear mapping that transforms the source model to the target: f (·, e t ) = α × f (·, e s ) + β, where α, β are learned using observations from both environments [51]. More sophisticated transfer learning exists that reuses source data using learners such as Gaussian Processes (GP) [25]. III. RESEARCH QUESTIONS AND METHODOLOGY A. Research questions The overall question that we explore in this paper is "why and when does transfer learning work for configurable software systems?" Our hypothesis is that performance models in source and target environments are usually somehow "related." To understand the notion of relatedness that we commonly find for environmental changes in practice, we explore several research questions (each with several hypotheses), from strong notions of relatedness (e.g., linear shift) toward weaker ones (e.g., stability of influential options): RQ1: Does the performance behavior stay consistent across environments? (Section IV) If we can establish with RQ1 that linear changes across environments are common, this would be promising for transfer learning because even simple linear transformations can be applied. Even if not all environment changes may be amendable to this easy transfer learning, we explore what kind of environment changes are more amendable to transfer learning than others. RQ2: Is the influence of configuration options on performance consistent across environments? (Section V) For cases in which easy transfer learning are not possible, RQ2 concerns information that can be exploited for transfer learning at the level of individual configuration options. Specifically, we explore how commonly the influential options remain stable across environment changes. RQ3: Are the interactions among configuration options preserved across environments? (Section VI) In addition to individual options in RQ2, RQ3 concerns interactions among options, that, as described above, can often be important for explaining the effect of performance variations across configurations. Again, we explore how commonly interactions are related across environment changes. RQ4: Are the configurations that are invalid in the source environment with respect to non-functional constraints also invalid in the target environment? (Section VII) Finally, RQ4 explores an important facet of invalid configurations: How commonly can we transfer knowledge about invalid configurations across environments? Even if we cannot transfer much structure for the performance model otherwise, transferring knowledge about configurations can guide learning in the target environment on the relevant regions. B. Methodology Design: We investigate changes of performance models across environments. Therefore, we need to establish the performance of a system and how it is affected by configuration options in multiple environments. To this end, we measure the performance of each system using standard benchmarks and repeated the measurements across a large number of configurations. We then repeat this process for several changes to the environment: using different hardware, different workloads, and different versions of the system. Finally, we perform the analysis of relatedness by comparing the performance and how it is affected by options across environments. We perform comparison of a total of 36 environment changes. Analysis: For answering the research questions, we formulate different assumptions about the relatedness of the source and target environments as hypotheses -from stronger to more relaxed assumptions. For each hypothesis, we define one or more metrics and analyze 36 environment changes in four subject systems described below. For each hypothesis, we discuss how commonly we identify this kind of relatedness and whether we can identify classes of changes for which this relatedness is characteristic. If we find out that for an environmental change a stronger assumption holds, it means that a more informative knowledge is available to transfer. Severity of environment changes: We purposefully select environment changes for each subject system with the goal of exploring many different kinds of changes with different expected severity of change. With a diverse set of changes, we hope to detect patterns of environment changes that have similar characteristics with regard to relatedness of performance models. We expect that less severe changes lead to more related performance models that are easier to exploit in transfer learning than more severe ones. For transparency, we recorded the expected severity of the change when selecting environments, as listed in Table II, on a scale from small change to very large change. For example, we expect a small variation where we change the processor of the hardware to a slightly faster version, but expect a large change when we replace a local desktop computer by a virtual machine in the cloud. Since we are neither domain experts nor developers of our subject systems, recording the expected severity allows us to estimate how well intuitive judgments can (eventually) be made about suitability for transfer learning and it allows us to focus our discussion on surprising observations. C. Subject systems In this study, we selected four configurable software systems from different domains, with different functionalities, and written in different programming languages (cf. Table I). SPEAR is an industrial strength bit-vector arithmetic decision procedure and a Boolean satisfiability (SAT) solver. It is designed for proving software verification conditions and it is used for bug hunting. We considered a configuration space with 14 options that represent heuristics for solving the problems and therefore affect the solving time. We measured how long it takes to solve a SAT problem in all 16,384 configurations in multiple environments: four different SAT problems with different difficulty serve as workload, measured on three hardware system, with two versions of the solver as listed in Table II. The difficulty of the workload is characterized by the SAT problem's number of variables and clauses. x264 is a video encoder that compresses video files with a configuration space of 16 options to adjust output quality, encoder types, and encoding heuristics. Due to the size of the configuration space, we measured a subset of 4000 sampled randomly configurations. We measured the time needed to encode three different benchmark videos on two different hardware systems and for three versions as listed in Table II. Each benchmark consists of a raw video with different quality and size and we expect that options related to optimizing encoding affect the encoding time differently. We judged expected severity of environmental changes based on the difference between quality and size of benchmark videos. SQLite is a lightweight relational database management system, embedded in several browsers and operating systems, with 14 configuration options that change indexing and features for size compression useful in embedded systems, but have performance impact. We expect that some options affect certain kinds of workload (e.g., read-heavy rather than writeheavy workloads) more than others. We have measured 1000 randomly selected configurations on two hardware platforms for two versions of the database system; as workload, we have considered four variations of queries that focus on sequential reads, random reads, sequential write, and batch writes. SaC is a compiler for high-performance computing [41]. The SaC compiler implements a large number of high-level and low-level optimizations to tune programs for efficient parallel executions configurable with 50 options controlling optimizations such as function inlining, constant folding, and array elimination. We measure the execution time of a program compiled in 71,267 randomly selected configurations to assess the performance impact of SaC's options. As workloads, we select 10 different demo programs shipped with SaC, each computationally intensive, but with different characteristics. Workloads include Monte Carlo algorithms such as pfilter with multiple optimizable loops as well as programs heavily based on matrix operations like srad. To account for measurement noise, we have measured each configuration of each system and environment 3 times and used the mean for the analyses. While many performance and quality measures can be analyzed, our primary performance metric is wall-clock execution time, which is captured differently for each systems in Table I: Here, we investigate the relatedness of environments in the entire configuration space. We start by testing the strongest assumption (i.e., linear shift), which would enable an easy transfer learning (H1.1). We expect that the first hypothesis holds only for simple environmental changes. Therefore, we subsequently relax the hypothesis to test whether and when the performance distributions are similar (H1.2), whether the ranking of configurations (H1.3), and the top/bottom configurations (H1.4) stay consistent. Table II summaries the results. H1.1: The relation of the source response to the target is a constant or proportional shift. Importance. If the target response is related to the source by a constant or proportional shift, it is trivial to understand the performance behavior for the target environment using the model that has already been learned in the source environment: We need to linearly transform the source model to get the target model. We expect a linear shift if a central hardware device affecting the functionality of all configuration options homogeneously, changes such as the CPU, or homogeneous workload change. Previous studies demonstrated the existence of such cases where they trained a linear transformation to derive a target model for hardware changes [51]. Metric. We investigate whether f (c, e t ) = α × f (c, e s ) + β, ∀c ∈ C. We use metric M1: Pearson linear correlation [4] between f (c, e s ) and f (c, e t ) to evaluate the hypothesis. If the correlation is 1, we can linearly transform performance models. Due to measurement noise, we do not expect perfect correlation, but we expect, for correlations higher than 0.9, simple transfer learning can produce good predictions. Results. The result in Table II show very high correlations for about a third of all studied environmental changes. In particular, we observe high correlations for hardware changes and for many workload changes of low expected severity. Hardware change: Hardware changes often result in nearperfect correlations except for severe changes where we have used unstable hardware (e.g., Amazon cloud in ec 2 ). We investigated why using cloud hardware resulted in weak linear correlations. We analyzed the variance of the measurement noise and we observed that the proportion of the variance of the noise in the source to the target in ec 2 is σ 2 ec s 2 /σ 2 ec t 2 = 33.39, which is an order of magnitude larger than the corresponding one in ec 1 (σ 2 ec s 1 /σ 2 ec t 1 = 1.51). This suggests that we can expect a linear transformation across environments when hardware resources execute in a stable environment. For transfer learning, this means that we could reuse measurements from cheaper or testing servers in order to predict the performance behavior [6]. Moreover, it also suggests that virtualization may hinder transfer learning. Workload change: For SPEAR, we observed very strong correlations across environments where we have considered SAT problems of different sizes and difficulties. Also, when the difference among the problem size and difficulty is closer across environments (e.g., ec 3 vs. ec 4 ) the correlation is slightly higher. This observation has also been confirmed for other systems. For instance, in environmental instance ec 3 in SQLite, where the workload change is write-heavy from sequential to batch, we have observed an almost perfect correlation, 0.96, while in the read-heavy workload ec 4 (random to sequential read) the correlation is only medium at 0.5: First, the underlying hardware contains an SSD, which has different performance properties for reading and writing. Second, a database performs different internal functions when inserting or retrieving data. This implies that some environmental conditions may provide a better means for transfer learning. Version change: For SPEAR (ec 5,6,7 ) and x264 (ec 5,6,7,8 ), the correlations are extremely weak or non existence, while for SQLite (ec 5 ), the correlation is almost perfect. We speculate that the optimization features that are determined by the configuration options for SPEAR and x264 may undergo a substantial revision from version to version because algorithmic changes may significantly improve the way how the optimization features work. The implication for transfer learning is that code changes that substantially influence the internal logic controlled by configuration options may require a non-linear form of transformation or a complete set of new measurements in the target environment for those options only. Insight. For non-severe hardware changes, we can linearly transfer performance models across environments. H1.2: The performance distribution of the source is similar to the performance distribution of the target environment. Importance. In the previous hypothesis, we investigated the situation whether the response functions in the source and target are linearly correlated. In this hypothesis, we consider a relaxed version of H1.1 by investigating if the performance distributions are similar. When the performance distributions are similar, it does not imply that there exists a linear mapping between the two responses, but, there might be a more sophisticated relationship between the two environments that can be captured by a non-linear transfer function. Metric. We measure M2: Kullback-Leibler (KL) divergence [8] to compare the similarity between the performance distributions: D ec KL (pd s , pd t ) = Σ i pd s (c i ) log pds(ci) pdt(ci) , where pd s,t (·) are performance distributions of the source and target. As an example, we show the performance distributions of ec 1 and ec 13 and compare them using KL divergence in Figure 2: The lower the value of KL divergence is, the more similar are the distributions. We consider two distributions as similar if D ec KL (pd s , pd t ) < 3 [4] and dissimilar otherwise. Results. Here, we are interested to find environmental changes for which we did not observe a strong correlation, but for which there might be similarities between the performance distributions of the environment. For ec 5,6 in SPEAR, ec 3−7 in x264, ec 4,6 in SQLite, and ec 5,8 in SaC, the performance distributions are similar across environments. This implies that there exist a possibly non-linear transfer function that we can map performance models across environments. Previous studies demonstrated the feasibility of highly non-linear kernel functions for transfer learning in configurable systems [25]. Insight. Even for some severe environmental changes with no linear correlation across performance models, the performance distributions are similar, showing the potential for learning a non-linear transfer function. H1.3: The ranking of configurations stays stable. Importance. If the ranking of the configurations stays similar, the response function is then stable across environments. We can use this knowledge to prioritize certain regions in the configuration space for optimizations. Metric. Here, we use rank correlation by measuring the M3: Spearman correlation coefficient between response variables. Intuitively, the Spearman correlation will be high when observations have a similar rank. We consider rank correlations higher than 0.9 as strong and suitable for transfer learning. Results. The results in Table II show that the rank correlations are high across hardware changes and small workload changes. This metric does not provide additional insights from what we have observed in H1.1. However, in one environmental change, where, due to excessive measurement noise, the linear correlation was low, ec 2 for SPEAR, the rank correlation is high. This might hint that when unstable hardware conditions exist, the overall ranking may stay stable. Insight. The configurations retain their relative performance profile across hardware platforms. H1.4: The top/bottom performer configurations are similar. Importance. If the top configurations are similar across environments, we can extract their characteristics and use that in the transfer learning process. For instance, we can identify the top configurations from the source and inform the optimization in the target environment [23]. The bottom configurations can be used to avoid corresponding regions during sampling. Note that this is a relaxed hypothesis comparing to H1.3. Metric. We measure M4/M5: the percentage of (10th percentile) top/bottom configurations in the source that are also top/bottom performers in the target. Results. The results in Table II show that top/bottom configurations are common across hardware and small workload changes, therefore, this metric does not provide additional insights from what we have observed in H1.1. Insight. Only hardware changes preserve top configurations across environments. V. SIMILARITY OF INFLUENTIAL OPTIONS (RQ2) Here, we investigate whether the influence of individual configuration options on performance stays consistent across environments. We investigate two hypotheses about the influence strength (H2.1) and the importance of options (H2.2). H2.1: The influential options on performance stay consistent. Importance. In highly dimensional spaces, not all configuration options affect the response significantly. If we observe a high percentage of common influential options across environments, we can exploit this for learning performance models by sampling across only a subset of all configuration options, because we already know that these are the key options influencing performance. Metric. In order to investigate the option-specific effects, we use a paired t-test [4] to test if an option leads to any significant performance change and whether this change is similar across environments. That is, when comparing the pairs of configuration in which this option is enabled and disabled respectively, an influential option has a consistent effect to speed up or slow down the program, beyond random chance. If the test shows that an option make a difference, we then consider it as an influential option. We measure M6/M7: the number of influential options in source and target; We also measure M8/M9: the number of options that are influential in both/one environment. Table II show that slightly more than half of the options, for all subject systems, are influential either in the source or target environments. From the influential options, a very high percentage are common in both. This can lead to a substantial reduction for performance measurements: we can fix the non-influential options and sample only along options, which we found influential from the source. Results. The results in Insight. Only a subset of options are influential which are largely preserved across all environment changes. H2.2: The importance of options stays consistent. Importance. In machine learning, each decision variable (here option) has a relative importance to predict the response and importance of the variables play a key role for in the feature selection process [4]. Here, we use this concept to determine the relative importance of configuration options, because in configurable systems, we face many options that if prioritized properly, it can be exploited for performance predictions [25]. Metric. We use regression trees [4] for determining the relative importance of configuration options because (i) they have been used widely for performance prediction of configurable systems [15], [51] and (ii) the tree structure can provide insights into the most essential options for prediction, because a tree splits on those options first that provide the highest information gain [15]. We derive estimates of the importance of options for the trained trees on the source and target by examining how the prediction error will change as a result of options. We measure M10: correlation between importance of options for comparing the consistency across environments. Results. From Table II, the correlation coefficient between the importance of options for different environmental changes is high, and the less severe a change the higher the correlation coefficients. This confirms our intuition that small changes in the environment do not affect the influence strength of an option. Some environmental changes, where the correlation were low according to M1, show a high correlation between option importance according to M10: ec 6,7 in SPEAR, ec 3−7 in x264, ec 1,2,5,[7][8][9][10][11]14 in SaC. This observation gives further evidence that even though we did not observe a linear correlation, there might exist a non-linear relationship between performance measures. For instance, the influence of options stay the same, but interactions might change. Insight. The strength of the influence of configuration options is typically preserved across environments. VI. PRESERVATION OF OPTION INTERACTIONS (RQ3) We state two hypotheses about the preservation of option interactions (H3.1) and their importance (H3.2). H3.1: The interactions between configuration options are preserved across environments. Importance. In highly dimensional configuration spaces, the possible number of interactions among options is exponential in the number of options and it is computationally infeasible to get measurements aiming at learning an exhaustive number of interactions. Prior work has shown that a very large portion of potential interactions has no influence [29], [45]. Metric. One key objective here is to evaluate to what extent influential interactions will be preserved from source to target. Here, we learn step-wise linear regression models; a technique that has been used for creating performance influence model for configurable systems [44]. We learn all pairwise interactions, independently in the source and target environments. We then calculate the percentage of common pairwise interactions from the model by comparing the coefficients of the pairwise interaction terms of the regression models. We concentrated on pairwise interactions, as they are the most common form of interactions [29], [45]. Similar to H2.1, we measure: M11/M12: The number of interactions in the source/target; M13: The number of interactions that agree on the direction of effects in the source and the target. Table II show three important observations: (i) only a small proportion of possible interactions have an effect on performance and so are relevant (confirming prior work); (ii) for the large environmental changes, the difference in the proportion of relevant interactions across environments is not similar, while for smaller environmental changes, the proportion is almost equal; (iii) a very large proportion of interactions is common across environments. Results. The results in The mean percentage of interactions (averaged over all changes) are 25%, 28%, 10%, 6% for SPEAR, x264, SQLite, SaC respectively, where 100% would mean that all pairwise combination of options have a distinct effect on performance. Also, the percentage of common interactions across environments is high, 96%, 81%, 85%, 72% for SPEAR, x264, SQLite, SaC respectively. This result points to an important transferable knowledge: interactions often stay consistent across changes. This insight can substantially reduce measurement efforts to purposefully measure specific configurations. Insight. A low percentage of potential interactions are influential for performance model learning. H3.2: The effects of interacting options stay similar. Importance. If the effects of interacting options are similar across environments, we can prioritize regions in the configuration space based on the importance of the interactions. Metric. We measure M14: the correlation between the coefficients of the pairwise interaction terms in the linear model learned independently on the source and target environments using step-wise linear regression [18]. Results: The results in Table II reveal a very high and, in several cases, perfect correlations between interactions across environments. For several environmental changes where we previously could not find a strong evidence of transferable knowledge by previous metrics: ec 8 in x264, ec 4,6,7 in SQLite and ec 14 in SaC, we observed very strong correlations for the interactions. The implication for transfer learning is that a linear transfer function (see H1.1) may not applicable for severe changes, while a complex transfer function may exist. Insight. The importance of interactions is typically preserved across environments. VII. INVALID CONFIGURATIONS SIMILARITY (RQ4) For investigating similarity between invalid configurations across environments, we formulate two hypotheses about percentage of invalid configurations and their commonalities across environments (H4.1) and the existence of reusable knowledge that can distinguish invalid configurations (H4.2). H4.1: The percentage of invalid configurations is similar across environments and this percentage is considerable. Importance. If the percentage of invalid configurations is considerable in the source and target environments, this provides a motivation to carry any information about the invalid configurations across environments to avoid exploration of invalid regions and reduce measurement effort. Metric. We measure M15/M16: percentage of invalid configurations in the source and target, M17: percentage of invalid configurations, which are common between environments. Results. The results in Table II show that for SPEAR and x264, a considerable percentage (≈ 50%) of configurations are invalid and all of them are common across environments. For SaC, approximately 18% of the sampled configurations are invalid. For some workload changes the percentage of common invalid configuration is low (≤ 10%). The reason is that some options in SaC may have severe effects for some programs to be compiled, but have lower effects for others. Insight. A large percentage of configurations are typically invalid in both source and target environments. H4.2: A classifier for distinguishing invalid from valid configurations is reusable across environments. Importance. If there are common characteristics among the invalid configurations, we can learn a classifier in the source to identify the invalid configurations and transfer the knowledge (classifier model) to the target environment to predict invalid configurations before measuring them, thus decrease cost. Metric. We learn a classifier using multinomial logistic regression [4]. It is a model that is used to predict the probabilities of being invalid, given a set of configuration options. We measure M18: the correlation between the coefficients (i.e., the probability of the configuration being invalid) of the classification models that has been leaned independently. Results. The results in Table II show that for SPEAR and x264, the correlations between the coefficients are almost perfect. For SaC, in environmental changes where the common invalid configurations are high, the correlations between coefficients are also very high. For two cases, ec 6,7 in SPEAR, we could not find any reusable knowledge previously with other metrics. Here, we can observe that even when the influence and interactions of all options change, the region of invalid configurations may stay the same. This means that we can avoid measurements (almost half of the space) in the target. Insight. Information for identifying invalid regions can be transfered, with a high confidence, across environments. VIII. LESSONS LEARNED AND DISCUSSION Based on our analyses of 36 environment changes, we can discuss lessons learned, implications and threats to validity. A. Lessons learned Based on the empirical results presented in this paper, we have learned that there is always some similarities that relate the source and target in different forms depending on the severity of the change: • Simple changes: We observed strong correlations between response functions (interpolating performance measures) and, therefore, there is a potential for constructing simple linear transfer functions across environments (RQ1). • Large changes: We observed very similar performance distributions (e.g., version changes). In these cases, we found evidence of high correlations between either options (RQ2) or interactions (RQ3) for which a non-linear transfer may be applicable. Therefore, the key elements in a performance model that has been learned on the source will not change, but the coefficients corresponding to options and their interactions might need to be relearned for the target. • Severe changes: We have learned that a considerable part of configuration space is invalid across environmental changes that could be considered for sampling configurations in severe changes (RQ4). B. Implications for transfer learning research We provide explanations of why and when transfer learning works for performance modeling and analysis of highly configurable systems. While all research questions have positive answers for some environment changes and negative answers for others, as discussed above in Section IV-Section VII, the results align well with our expectations regarding the severity of change and their correspondence to the type of transferable knowledge: (i) For small environmental changes, the overall performance behavior was consistent across environments and a linear transformation of performance models provides a good approximation for the target performance behavior. (ii) For large environmental changes, we found evidence that individual influences of configuration options and interactions may stay consistent providing opportunities for a non-linear mapping between performance behavior across environments. (iii) Even for severe environmental changes, we found evidence of transferable knowledge in terms of reusability of detecting invalid from valid configurations providing opportunities for avoiding a large part of configuration space for sampling. The fact that we could largely predict the severity of change without deep knowledge about the configuration spaces or implementations of the subject systems is encouraging in the sense that others will likely also be able to make intuitive judgments about transferability of knowledge. For example, a user of a performance analysis approach estimating low severity of an environment change can test this hypothesis quickly with a few measurements and select the right transfer learning strategy. Transfer learning approaches for easy environment changes are readily available [7], [25], [51], [64]. For more severe environment changes, more research is needed to exploit transferable knowledge. Our results show that that even with severe environmental change, there always is some transferable knowledge that can contribute to performance understanding of configurable systems. While some learning strategies can take existing domain knowledge into account and could benefit from knowledge about influential options and interactions [44], [45], it is less obvious how to effectively incorporate such knowledge into sampling strategies and how to build more effective learners based on limited transferable knowledge. While we strongly suspect that suitable transfer learning techniques can provide significant benefits even for severe environment changes, more research is needed to design and evaluate such techniques and compare to state of the art sampling and learning strategies. Specifically, we expect research opportunities regarding: 1) Sampling strategies to exploit the relatedness of environments to select informative samples using the importance of specific regions [40] or avoiding invalid configurations. 2) Learning mechanisms to exploit the relatedness across environments and learn either a linear or non-linear associations (e.g., active learning [52], domain adaptation [31], fine tuning a pre-trained model [13], feature transfer [62], or knowledge distillation [17] in deep neural network architectures). However, efforts need to be made to make the learning less expensive. 3) Performance testing and debugging of configurable systems to benefit from our findings by transferring interesting test cases covering interactions between options [49] or detecting invalid configurations [57]- [59]. 4) Performance tuning and optimization [23] benefit from the findings by identifying the interacting options and to perform importance sampling exploiting the importance coefficients of options and their interactions. 5) Performance modeling [9] benefit from the findings by developing techniques that exploits the shared knowledge in the modeling process, e.g., tuning the parameters of a queuing network model using transfer learning. C. Threats to validity 1) External validity: We selected a diverse set of subject systems and a large number of purposefully selected environment changes, but, as usual, one has to be careful when generalizing to other subject systems and environment changes. We actually performed experiments with more environmental changes and with additional measurements on the same subject systems (e.g., for SaC we also measured the time it takes to compile the program not only its execution), but we excluded those results because they were consistent with the presented data and did not provide additional insights. 2) Internal and construct validity: Due to the size of configuration spaces, we could only measure configurations exhaustively in one subject system and had to rely on sampling (with substantial sampling size) for the others, which may miss effects in parts of the configuration space that we did not sample. We did not encounter any surprisingly different observation in our exhaustively measured SPEAR dataset. We operationalized a large number of different measures through metrics. For each measure, we considered multiple alternative metrics (e.g., different ways to establish influential options) but settled usually on the simplest and most reliable metric we could identify to keep the paper accessible and within reasonable length. In addition, we only partially used statistical tests, as needed, and often compared metrics directly using more informal comparisons and some ad-hoc threshold for detecting common patterns across environments. A different operationalization may lead to different results, but since our results are consistent across a large number of measures, we do not expect any changes to the overall big picture. For building the performance models, calculating importance of configuration options, and classifying the invalid configurations, we elected to use different machine learning models: step-wise linear regression, regression trees, and multinomial logistic regression. We chose these learner mainly because they are successful models that have been used in previous work for performance predictions of configurable systems. However, these are only few learning mechanisms out of many that may provide different accuracy and cost. Measurement noise in benchmarks can be reduced but not avoided. We performed benchmarks on dedicated systems and repeated each measurement 3 times. We repeated experiments when we encountered unusually large deviations. IX. RELATED WORK A. Performance analysis of configurable software Performance modeling and analysis is a highly researched topic [53]. Researches investigate what models are more suitable for predicting performance of the configurable systems, which sampling and optimization strategies can be used for tuning these models, and how to minimize the amount of measurement efforts for model training. Sampling strategies based on experimental design (such as Plackett-Burman) have been applied in the domain of configurable systems [15], [43], [44]. The aim of these sampling approaches is to ensure that we gain a high level of information from sparse sampling in high dimensional spaces. Optimization algorithms have also been applied to find optimal configurations for configurable systems: Recursive random sampling [60], hill climbing [55], direct search [64], optimization via guessing [37], Bayesian optimization [23], and multi-objective optimization [12]. The aim of optimization approaches is to find the optimal configuration in a highly dimensional space using only a limited sampling budget. Machine learning techniques, such as support-vector machines [61], decision trees [33], Fourier sparse functions [63], active learning [44] and search-based optimization and evolutionary algorithms [16], [54] have also been used. Our work is related to the performance analysis research mentioned above. However, we do not perform a comparison of different models, configuration optimization or sampling strategies. Instead, we concentrate on transferring performance models across hardware, workload and software version. Transfer learning, in general, is orthogonal to these approaches and can contribute to make these approaches more efficient for performance modeling and analysis. B. Performance analysis across environmental change Environmental changes have been studied before. For example, in the context of MapReduce applications [61], performance-anomaly detection [46], performance prediction based on micro-benchmark measurements on different hardware [22], consistency-analysis of parameter dependencies [64], and performance prediction of configurable systems based on hardware variants and similarity search [48]. Recently, transfer learning is used in systems and software engineering. For example, in the context of performance predictions in self-adaptive systems [25], configuration dependency transfer across software systems [7], co-design exploration for embedded systems [5], model transfer across hardware [51], and configuration optimization [3]. Although previous work has analyzed transfer learning in the context of select hardware changes [7], [25], [51], we more broadly empirically investigate why and when transfer learning works. That is, we provide evidence why and when other techniques are applicable for which environmental changes. Transfer learning has also been applied in software engineering in very different contexts, including defect predictions [28], [34], [35] and effort estimation [27]. X. CONCLUSIONS We investigated when and why transfer learning works for performance modeling and analysis of highly configurable systems. Our results suggest that performance models are frequently related across environments regarding overall performance response, performance distributions, influential configuration options and their interactions, as well as invalid configurations. While some environment changes allow simple linear forms of transfer learning, others have less obvious relationships but can still be exploited by transferring more nuanced aspects of the performance model, e.g., usable for guided sampling. Our empirical study demonstrate the existence of diverse forms of transferable knowledge across environments that can contribute to learning faster, better, reliable, and more important, less costly performance models. SaC workload descriptions: srad: random matrix generator; pfilter: particle filtering; hotspot: heat transfer differential equations; k-means: clustering; nw: optimal matching; Fig. 1 : 1Transfer learning is a form of machine learning that takes advantage of transferable knowledge from source to learn an accurate, reliable, and less costly model for the target environment. execution time, encoding time, query time, and analysis time. IV. PERFORMANCE BEHAVIOR CONSISTENCY (RQ1) Fig. 2 : 2Performance distributions of environments can be very different, D ec 1 KL = 25.02 (a,b), or very similar, D ec 13 KL = 0.32 (c,d). nbody: simulation of dynamic systems; cg: conjugate gradient; gc: garbage collector. Hardware descriptions (ID: Type/CPUs/Clock (GHz)/RAM (GiB)/Disk): h1: NUC/4/1.30/15/SSD; h2: NUC/2/2.13/7/SCSI; h3:Station/2/2.8/3/SCSI; h4: Amazon/1/2.4/1/SSD; h5: Amazon/1/2.4/0.5/SSD; h6: Azure/1/2.4/3/SCSI Metrics: M1: Pearson correlation; M2: Kullback-Leibler (KL) divergence; M3: Spearman correlation; M4/M5: Perc. of top/bottom conf.; M6/M7: Number of influential options; M8/M9: Number of options agree/disagree; M10: Correlation btw importance of options; M11/M12: Number of interactions; M13: Number of interactions agree on effects; M14: Correlation btw the coeffs; M15/M16: Perc. of invalid conf. in source/target; M17: Perc. of invalid conf. common btw environments; M18: Correlation btw coeffs TABLE I : IOverview of the real-world subject systems.d: configuration options; C: configurations; H: hardware environments; W : analyzed workload; V : analyzed versions.System Domain d |C| |H| |W | |V | SPEAR SAT solver 14 16 384 3 4 2 x264 Video encoder 16 4 000 2 3 3 SQLite Database 14 1 000 2 14 2 SaC Compiler 50 71 267 1 10 1 TABLE II : IIResults indicate that there exist several forms of knowledge that can be transfered across environments and can be used in transfer learning.SPEAR-Workload (#variables/#clauses): w 1 : 774/5934, w 2 : 1008/7728, w 3 : 1554/11914, w 4 : 978/7498; Version: v 1 : 1.2, v 2 : 2.7ec 1 : [h 2 → h 1 , w 1 , v 2 ] : [h 1 , w 1 → w 2 , v 1 → v 2 ] : [h 1 → h 2 , w 1 → w 4 , v 2 → v 1 ] VL -0.10 x264-Workload (#pictures/size): w 1 : 8/2, w 2 : 32/11, w 3 : 128/44; Version:v 1 : r2389, v 2 : r2744, v 3 : r2744 ec 1 : [h 2 → h 1 , w 3 , v 3 ] : [h 2 → h 1 , w 1 , v 3 ] : [h 1 , w 1 → w 2 , v 3 ] : [h 1 , w 1 → w 3 , v 3 ] : [h 1 , w 3 , v 1 → v 3 ] : [h 1 , w 1 → w 3 , v 2 → v 3 ] : [h 2 → h 1 , w 1 → w 3 , v 2 → v 3 ] VL SQLite-Workload: w 1 : write − seq, w 2 : write − batch, w 3 : read − rand, w 4 : read − seq; Version: v 1 : 3.7.6.3, v 2 : 3.19.0 ec 1 : [h 3 → h 2 , w 1 , v 1 ] 00 N/A N/A N/A N/A ec 2 : [h 3 → h 2 , w 2 , v 1 ] 00 N/A N/A N/A N/A ec 3 : [h 2 , w 1 → w 2 , v 1 ] : [h 2 , w 3 → w 4 , v 1 ] : [h 1 , w 1 , v 1 → v 2 ] : [h 1 , w 2 → w 1 , v 1 → v 2 ] : [h 2 → h 1 , w 2 → w 1 , v 1 → v 2 ] VLSaC-Workload: w 1 : srad, w 2 : pf ilter, w 3 : kmeans, w 4 : hotspot, w 5 : nw, w 6 : nbody100, w 7 : nbody150, w 8 : nbody750, w 9 : gc, w 10 : cgec 1 : [h 1 , w 1 → w 2 , v 1 ] : [h 1 , w 1 → w 3 , v 1 ] : [h 1 , w 1 → w 4 , v 1 ] : [h 1 , w 1 → w 5 , v 1 ] : [h 1 , w 2 → w 3 , v 1 ] : [h 1 , w 2 → w 4 , v 1 ] : [h 1 , w 2 → w 5 , v 1 ] : [h 1 , w 3 → w 4 , v 1 ] : [h 1 , w 3 → w 5 , v 1 ] ec 10 : [h 1 , w 4 → w 5 , v 1 ] ec 11 : [h 1 , w 6 → w 7 , v 1 ] N/A N/A N/A N/A ec 12 : [h 1 , w 6 → w 8 , v 1 ] ec 13 : [h 1 , w 8 → w 7 , v 1 ]ec 14 : [h 1 , w 9 → w 10 , v 1 ] ES: Expected severity of change (Sec. III-B): S: small change; SM: small medium change; M: medium change; L: large change; VL: very large change.RQ1 RQ2 RQ3 RQ4 H1.1 H1.2 H1.3 H1.4 H2.1 H2.2 H3.1 H3.2 H4.1 H4.2 Environment ES M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 S 1.00 0.22 0.97 0.92 0.92 9 7 7 0 1 25 25 25 1.00 0.47 0.45 1 1.00 ec 2 : [h 4 → h 1 , w 1 , v 2 ] L 0.59 24.88 0.91 0.76 0.86 12 7 4 2 0.51 41 27 21 0.98 0.48 0.45 1 0.98 ec 3 : [h 1 , w 1 → w 2 , v 2 ] L 0.96 1.97 0.17 0.44 0.32 9 7 4 3 1 23 23 22 0.99 0.45 0.45 1 1.00 ec 4 : [h 1 , w 1 → w 3 , v 2 ] M 0.90 3.36 -0.08 0.30 0.11 7 7 4 3 0.99 22 23 22 0.99 0.45 0.49 1 0.94 ec 5 : [h 1 , w 1 , v 2 → v 1 ] S 0.23 0.30 0.35 0.28 0.32 6 5 3 1 0.32 21 7 7 0.33 0.45 0.50 1 0.96 ec 6 L -0.10 0.72 -0.05 0.35 0.04 5 6 1 3 0.68 7 21 7 0.31 0.50 0.45 1 0.96 ec 7 6.95 0.14 0.41 0.15 6 4 2 2 0.88 21 7 7 -0.44 0.47 0.50 1 0.97 SM 0.97 1.00 0.99 0.97 0.92 9 10 8 0 0.86 21 33 18 1.00 0.49 0.49 1 1 ec 2 S 0.96 0.02 0.96 0.76 0.79 9 9 8 0 0.94 36 27 24 1.00 0.49 0.49 1 1 ec 3 M 0.65 0.06 0.63 0.53 0.58 9 11 8 1 0.89 27 33 22 0.96 0.49 0.49 1 1 ec 4 M 0.67 0.06 0.64 0.53 0.56 9 10 7 1 0.88 27 33 20 0.96 0.49 0.49 1 1 ec 5 : [h 1 , w 3 , v 2 → v 3 ] L 0.05 1.64 0.44 0.43 0.42 12 10 10 0 0.83 47 33 29 1.00 0.49 0.49 1 1 ec 6 L 0.06 1.54 0.43 0.43 0.37 11 10 9 0 0.80 46 33 27 0.99 0.49 0.49 1 1 ec 7 L 0.08 1.03 0.26 0.25 0.22 8 10 5 1 0.78 33 33 20 0.94 0.49 0.49 1 1 ec 8 0.09 14.51 0.26 0.23 0.25 8 9 5 2 0.58 33 21 18 0.94 0.49 0.49 1 1 S 0.99 0.37 0.82 0.35 0.31 5 2 2 0 1 13 9 8 1.M 0.97 1.08 0.88 0.40 0.49 5 5 4 0 1 10 11 9 1.S 0.96 1.27 0.83 0.40 0.35 2 3 1 0 1 9 9 7 0.99 N/A N/A N/A N/A ec 4 M 0.50 1.24 0.43 0.17 0.43 1 1 0 0 1 4 2 2 1.00 N/A N/A N/A N/A ec 5 M 0.95 1.00 0.79 0.24 0.29 2 4 1 0 1 12 11 7 0.99 N/A N/A N/A N/A ec 6 L 0.51 2.80 0.44 0.25 0.30 3 4 1 1 0.31 7 11 6 0.96 N/A N/A N/A N/A ec 7 0.53 4.91 0.53 0.42 0.47 3 5 2 1 0.31 7 13 6 0.97 N/A N/A N/A N/A L 0.66 25.02 0.65 0.10 0.79 13 14 8 0 0.88 82 73 52 0.27 0.18 0.17 0.88 0.73 ec 2 L 0.44 15.77 0.42 0.10 0.65 13 10 8 0 0.91 82 63 50 0.56 0.18 0.12 0.90 0.84 ec 3 S 0.93 7.88 0.93 0.36 0.90 12 10 9 0 0.96 37 64 34 0.94 0.16 0.15 0.26 0.88 ec 4 L 0.96 2.82 0.78 0.06 0.81 16 12 10 0 0.94 34 58 25 0.04 0.15 0.22 0.19 -0.29 ec 5 M 0.76 1.82 0.84 0.67 0.86 17 11 9 1 0.95 79 61 47 0.55 0.27 0.13 0.83 0.88 ec 6 S 0.91 5.54 0.80 0.00 0.91 14 11 8 0 0.85 64 65 31 -0.40 0.13 0.15 0.12 0.64 ec 7 L 0.68 25.31 0.57 0.11 0.71 14 14 8 0 0.88 67 59 29 0.05 0.21 0.22 0.09 -0.13 ec 8 L 0.68 1.70 0.56 0.00 0.91 14 13 9 1 0.88 57 67 36 0.34 0.11 0.14 0.05 0.67 ec 9 VL 0.06 3.68 0.20 0.00 0.64 16 10 9 0 0.90 51 58 35 -0.52 0.11 0.21 0.06 -0.41 L 0.70 4.85 0.76 0.00 0.75 12 12 11 0 0.95 58 57 43 0.29 0.14 0.20 0.64 -0.14 S 0.82 5.79 0.77 0.25 0.88 36 30 28 2 0.89 109 164 102 0.96 S 1.00 0.52 0.92 0.80 0.97 38 30 22 6 0.94 51 53 43 0.99 N/A N/A N/A N/A S 1.00 0.32 0.92 0.53 0.99 30 33 26 1 0.98 53 89 51 1.00 N/A N/A N/A N/A L 0.24 4.85 0.56 0.44 0.77 22 21 18 3 0.69 237 226 94 0.86 N/A N/A N/A N/A ACKNOWLEDGMENT This work has been supported by AFRL and DARPA (FA8750-16-2-0042). 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[ "https://github.com/pooyanjamshidi/ase17." ]
[ "DIVERSITREE: A NEW METHOD TO EFFICIENTLY COMPUTE DIVERSE SETS OF NEAR-OPTIMAL SOLUTIONS TO MIXED-INTEGER OPTIMIZATION PROBLEMS", "DIVERSITREE: A NEW METHOD TO EFFICIENTLY COMPUTE DIVERSE SETS OF NEAR-OPTIMAL SOLUTIONS TO MIXED-INTEGER OPTIMIZATION PROBLEMS", "DIVERSITREE: A NEW METHOD TO EFFICIENTLY COMPUTE DIVERSE SETS OF NEAR-OPTIMAL SOLUTIONS TO MIXED-INTEGER OPTIMIZATION PROBLEMS", "DIVERSITREE: A NEW METHOD TO EFFICIENTLY COMPUTE DIVERSE SETS OF NEAR-OPTIMAL SOLUTIONS TO MIXED-INTEGER OPTIMIZATION PROBLEMS" ]
[ "Izuwa Ahanor [email protected] \nDepartment of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n\n", "Hugh Medal [email protected] \nDepartment of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n\n", "Andrew C Trapp [email protected] \nDepartment of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n\n", "Izuwa Ahanor [email protected] \nDepartment of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n\n", "Hugh Medal [email protected] \nDepartment of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n\n", "Andrew C Trapp [email protected] \nDepartment of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n\n" ]
[ "Department of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n", "Department of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n", "Department of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n", "Department of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n", "Department of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n", "Department of Industrial and Systems Engineering\nData Science Program Worcester Polytechnic Institute\nUniversity of Tennessee-Knoxville\n" ]
[]
While most methods for solving mixed-integer optimization problems compute a single optimal solution, a diverse set of near-optimal solutions can often lead to improved outcomes. We present a new method for finding a set of diverse solutions by emphasizing diversity within the search for near-optimal solutions. Specifically, within a branch-and-bound framework, we investigated parameterized node selection rules that explicitly consider diversity. Our results indicate that our approach significantly increases the diversity of the final solution set. When compared with two existing methods, our method runs with similar runtime as regular node selection methods and gives a diversity improvement between 12% and 190%. In contrast, popular node selection rules, such as best-first search, in some instances performed worse than state-of-the-art methods by more than 35% and gave an improvement of no more than 130%. Further, we find that our method is most effective when diversity in node selection is continuously emphasized after reaching a minimal depth in the tree and when the solution set has grown sufficiently large. Our method can be easily incorporated into integer programming solvers and has the potential to significantly increase the diversity of solution sets.
null
[ "https://export.arxiv.org/pdf/2204.03822v3.pdf" ]
249,191,787
2204.03822
9e6bf447d42bf55d112ec2475cbe7c934fa63005
DIVERSITREE: A NEW METHOD TO EFFICIENTLY COMPUTE DIVERSE SETS OF NEAR-OPTIMAL SOLUTIONS TO MIXED-INTEGER OPTIMIZATION PROBLEMS Izuwa Ahanor [email protected] Department of Industrial and Systems Engineering Data Science Program Worcester Polytechnic Institute University of Tennessee-Knoxville Hugh Medal [email protected] Department of Industrial and Systems Engineering Data Science Program Worcester Polytechnic Institute University of Tennessee-Knoxville Andrew C Trapp [email protected] Department of Industrial and Systems Engineering Data Science Program Worcester Polytechnic Institute University of Tennessee-Knoxville DIVERSITREE: A NEW METHOD TO EFFICIENTLY COMPUTE DIVERSE SETS OF NEAR-OPTIMAL SOLUTIONS TO MIXED-INTEGER OPTIMIZATION PROBLEMS integer programming · near-optimal solutions · diversity · node-selection rules While most methods for solving mixed-integer optimization problems compute a single optimal solution, a diverse set of near-optimal solutions can often lead to improved outcomes. We present a new method for finding a set of diverse solutions by emphasizing diversity within the search for near-optimal solutions. Specifically, within a branch-and-bound framework, we investigated parameterized node selection rules that explicitly consider diversity. Our results indicate that our approach significantly increases the diversity of the final solution set. When compared with two existing methods, our method runs with similar runtime as regular node selection methods and gives a diversity improvement between 12% and 190%. In contrast, popular node selection rules, such as best-first search, in some instances performed worse than state-of-the-art methods by more than 35% and gave an improvement of no more than 130%. Further, we find that our method is most effective when diversity in node selection is continuously emphasized after reaching a minimal depth in the tree and when the solution set has grown sufficiently large. Our method can be easily incorporated into integer programming solvers and has the potential to significantly increase the diversity of solution sets. Introduction It is often important to find sets of near-optimal solutions to optimization problems rather than a single solution. In particular, for these multiple near-optimal solutions to be usable, they should be diverse, to ensure that decision makers are presented with a variety of options. There has been increasing awareness of finding not only one but multiple optima due to increased computational capabilities in the last two decades (Bertsimas et al. 2016). Some specific applications in which it is desirable to find multiple near-optimal solutions to optimization problems include the correct identification of metabolic activity of cells and tissues in metabolic networks (Rodríguez-Mier et al. 2021), aiding motif finding in computational molecular biology (Zaslavsky and Singh 2006), enabling exploration and mapping searches to broader but specific solutions in large search spaces by including near-optimal search results in search requests (Mouret andClune 2015, Zahavy et al. 2021), and providing policies that are more robust to data changes in reinforcement learning and machine learning arXiv:2204.03822v3 [cs.DM] 8 Feb 2023 (Kumar et al. 2020, Eysenbach et al. 2018, Sharifnia et al. 2021. Discovering multiple near-optimal solutions have also been applied to identifying alternative near-optimal structural designs (He et al. 2020), adding more artistic alternatives to structural topology optimization (Cai et al. 2021, He et al. 2020, providing competitive alternatives to facility location and location routing problems (Church and Baez 2020, Schittekat and Sörensen 2009), diversifying software deployment to enable stronger computer software security (Tsoupidi et al. 2020), generating multiple near-optimal group preferences in computational social choice analysis (Boehmer and Niedermeier 2021), and broadening architectural testing in processor design (Van Hentenryck et al. 2009). Many of these applications have a large number of near-optimal solutions (see Table 2 of Appendix A). For instance, both the routing problem from and the multi-period facility location problem studied in Eckstein (1996) are known to have more than 10,000 solutions with objective value within 1% of optimality. Some other problems like the image deblurring problem studied in could have an infinite number of solutions. In problems with many near-optimal solutions, there is a need to identify a small subset of near-optimal solutions that is representative of the whole. One measure of how well a subset represents the whole is the diversity of the subset. Unfortunately, methods for finding near-optimal solutions such as ONETREE find near-optimal sets consisting of solutions that are not very diverse (see example provided in §5). There are several important contexts in which it is useful to have a diverse set of near-optimal solutions. First, in some design problems decision makers seek a set of designs from which to choose, for the purpose of considering other difficult-to-model factors when selecting a single design (Joseph et al. 2015). A similar context is in statistical model selection for a specific domain. Presenting a domain expert with a diverse set of models with similar fit allows the expert to select the model that best matches intuition. Several new studies hypothesize that for many machine learning tasks a set of models exist with near-minimal loss (Semenova et al. (2019); also see §9 of Rudin et al. (2022)). Second, using optimization for decision problems can often be an iterative process in which a MIP solution is first presented to a decision maker, only to have the decision maker identify that the solution violates an important side constraint that was not included in the model. If the decision maker is only provided a single solution, the model must be re-solved after adding the side constraint. However, if a diverse set of near-optimal solutions were readily available, the decision maker may be able to find a good solution in the set that does not violate the side constraint, avoiding the need to re-solve the model. Third, in contexts in which solutions are implemented repeatedly, it can be useful to alternate between a diverse set of near-optimal solutions. For example, one may use MIP to match workers to jobs to minimize total completion time, but regularly implement different near-optimal matchings to increase cross-training. Lastly, a set of near-optimal solutions can be used to measure and explain the importance of variables in applications such as statistical model selection (see §9 of Rudin et al. (2022)). If a feature is present in a large number of diverse near-optimal solutions, this provides additional evidence that the feature is an important predictor of the response variable. One class of methods for generating diverse sets of near-optimal solutions to mixed-integer optimization (MIO) problems uses a two-phase approach (methods within this category typically also perform a precomputation to compute the optimal objective value. However, when referring to these methods as two-phase, we do not include the precomputation step as a phase). In the first solution generation phase, an oracle finds a set of near-optimal solutions without considering diversity. For example, Danna et al. (2007) developed the ONETREE oracle for this purpose. In the second diverse subset selection phase the output set from the first phase is processed by heuristics (e.g., Glover et al. (2000), Danna and Woodruff (2009)) or an optimization algorithm (e.g., Danna and Woodruff (2009)) to select a small subset of the input set with maximum diversity. While this approach works well for problems with a small number of near-optimal solutions, for MIOs with a very large set of near-optimal solutions it is not practical to find the complete set. As a result, the first phase can only compute a subset of near-optimal solutions. Unfortunately, because existing first-phase methods do not consider diversity, the near-optimal sets they generate often lack diversity (see §4). If this subset is not diverse, then the smaller subset computed by the second phase necessarily will lack diversity. The work presented in this article addresses this issue. Related Work Although there exists some work on finding near-optimal solutions to continuous optimization problems (see Lavine (2019)), most of the work has been in relation to mixed-integer optimization (MIO) problems and, in particular, with respect to (binary) integer variables. There are a number of different algorithms for generating a set of near-optimal solutions to a MIO problem. Achterberg et al. (2008) developed an approach for generating all feasible solutions to an integer programming (IP) problem called branch-and-count. This approach is based on detecting "unrestricted subtrees" in the branching tree. This method can also be used to find all solutions within a certain threshold of the optimal objective value, if known. Other methods have been proposed for finding all optimal solutions; an example is the work by Lee et al. (2000) which uses a recursive MIP algorithm to find all alternate optima. Serra (2020) used weighted decision diagrams to compactly represent all near-optimal solutions generated for an integer programming problem. The compact representation eases resolves and information retrieval from all generated solutions. Also, there is the ONETREE method (currently implemented in GAMS-CPLEX) given by Danna et al. (2007) which extends the branching tree used to solve the MIP problem to generate near-optimal solutions. In addition, approaches have been developed for specific classes of problems including using dynamic programming to generate multiple solutions to graph-based problems (Baste et al. 2022), methods to represent near-optimal solutions compactly (Serra and Hooker 2020) and algorithms specific to topology optimization problems (Wang et al. 2018). While these methods are effective at finding near-optimal solutions, they do not necessarily compute solution sets that are diverse. One of the few papers that discusses balancing diversity and optimality is Zhou et al. (2016). They developed a Dual Diverse Competitive Design (DDCD) method that formulates balancing optimality and diversity as an optimization problem that maximizes diversity, subject to constraints on the performance penalty. While their method specifically focused on generating competitive designs that would give diverse solutions, the goal of this current study is a method for general MIO problems. There are a number of different approaches for finding a diverse set of near-optimal solutions to an optimization problem. In the sequential approach (Danna et al. 2007), the optimization problem is solved multiple times, and after each solve a constraint is added that requires the next optimal solution be different from the previous. The sequential approach is also used in Trapp and Konrad (2015) to select diverse solutions to binary integer linear problems and in Petit and Trapp (2015) in the context of constraint programming. Both methods consider maximizing a metric that is the ratio of diversity of solutions to loss in objective function. Petit and Trapp (2019) extended these ideas further by introducing the notion of infusing solutions with other desirable features such as fairness, persistence and balance when generating diverse near-optimal solutions. The variable copy approach adds k copies of the variables, one copy for each near-optimal solution desired, in an optimization model and adds constraints to enforce that the solutions differ (Cameron et al. 2021). Greistorfer et al. (2008) compared the sequential and variable copy approaches for the problem of finding two diverse near-optimal solutions and found that the sequential approach usually required less computation time and yielded solutions that were nearly as good as the simultaneous approach. Further, population-based metaheuristics are a natural approach for finding diverse sets of solutions because these algorithms operate on a set of solutions. For instance, Glover et al. (2000) used a scatter-search algorithm to find a diverse set of solutions to MIPs. While metaheuristics are often effective in practice, their main drawback is their inability to guarantee that the entire set of near-optimal solutions has been found. Finally, two-phase approaches use an oracle such as the ONETREE algorithm (Danna et al. 2007) to find a (not necessarily diverse) set of near-optimal solutions. Then the second phase inputs this set of near-optimal solutions and chooses a subset that maximizes diversity. Regarding the second phase, Danna and Woodruff (2009) developed exact and heuristics algorithms to find a diverse subset. The work by Glover et al. (1998) which proposed four different heuristics algorithms for generating the most diverse solutions from a larger set does not consider how close they are to the optimal objective value. Kuo et al. (1993) proposed two different methods that use linear programming to select the most diverse solutions from a solution set. Schwind et al. (2020) developed methods for computing a small subset of solutions that represent the larger set using bi-objective optimization. In Danna et al. (2007), the authors found that the sequential and simultaneous methods do not scale as the number of solutions that are been generated increases. The approach developed in this study combines both phases of the two-phase approach, emphasizing diversity while searching for a near-optimal set. Contributions and Findings This study contributes to the existing literature on solution diversity in mixed integer programming by investigating how to obtain a diverse set of near-optimal solutions during the branch-and-bound search rather than during a post-processing step. We make the following contributions. Diversity of Solution Sets Computed by Existing Node Selection Rules. Existing branch-and-bound algorithms are not designed to find diverse sets of solutions, and it is unknown whether different algorithm configurations (e.g., node selection rules) influence the diversity of solution sets. To address this gap, we investigated the diversity of near-optimal sets generated using popular node selection rules within a branch-and-count algorithm (Achterberg et al. 2008). We found that, while the effect of node selection rules on solution set diversity is problem specific, the best-first search rule typically yields better diversity solutions overall in reasonable time. A New Node Selection Rule that Emphasizes Solution Set Diversity. The current well-established methods for finding a diverse set of near-optimal solutions do so in a two-phase approach in which a set of near-optimal solutions is first computed and then a diverse subset is identified using a post-processing algorithm. In this study we describe a new approach that computes diverse sets of near-optimal solution sets within a single branch-and-bound search 1 . To accomplish this, we introduced and subsequently investigated a new parameterized node selection rule that selects new nodes in the tree based on a weighted average of the dual bound, partial solution diversity, and tree depth. We find that each of these three metrics are needed to find diverse solution sets. Tuned Parameters for Diversity-emphasizing Node Selection Rules. Because the weighted-average metric used in our node selection rule requires parameters, we performed an experimental investigation to find the parameter combination that, on average, yields the highest solution diversity. To ensure that the identified parameter combination works well on new problems, we first tuned the parameters using a training set and then tested the resulting best parameter combinations on a test set. Results show that one particular parameter combination performs favorably well on both the training and tests sets over a variety of problem settings. Thus, our results indicate that our method generally performs well with a prescribed combination of parameters, alleviating the need for time-consuming parameter optimization and providing a set of default parameters that should work well on new problem not included in our test set. Benefits of Emphasizing Diversity in Node Selection. Using 36 problem instances randomly selected from MIPLIB (2003, 2010 and 2017), we found that using our new diversity-emphasizing node selection rule results in solution sets that are up to 190% more diverse than the ONETREE algorithm, a leading method for finding near-optimal sets currently implemented in CPLEX. Regarding runtime, we found that our method of emphasizing diversity within the node selection procedure runs within ±15% of the time used by regular node selection methods that do not emphasize diversity and results in a better tradeoff between runtime and solution diversity than competing approaches. Outline of the paper In the remainder of this article, we provide mathematical and algorithmic preliminaries in §2, followed by a description of the diversity-emphasizing node selection rules tested in this work in §3. Experimental results on the training and test set data is discussed in §4, while §5 illustrates the use of DIVERSITREE on real data. In §6, we discuss extensions to the method outlined in this study, prior to concluding in §7. Mathematical and Algorithmic Preliminaries We consider finding a diverse set of near-optimal solutions to the following problem: z * = min x∈X c T x where X = {x ∈ R d : Ax ≥ b, x i ∈ Z, ∀i ∈ I ⊆ {1, . . . , d}}. (1) Let S q = {x ∈ X : c T x ≤ (1 + q)z * } denote a set of q%-optimal solutions to (1), q ≥ 0. While mixed-integer programs are considered, diversity is only computed over binary integer variables. If |S q | is small, it may be sufficient to use a near-optimal solution generation algorithm such as ONETREE (Danna et al. 2007) to generate the entire set S q for presentation to a decision maker. In the case in which |S q | is large but not very large, not more than 1,000 elements, the following two-phase approach may be used. In the first phase, a solution generation algorithm is used to obtain the complete set of near-optimal solutions S q . Then in the second phase, use a diverse subset selection algorithm (see, e.g., Danna and Woodruff 2009) to find a small subset of S q of cardinality p, solving the following problem: max S⊆Sq,|S|=p D(S). ( 2) where D(S) is a measure of the diversity of solution set S. In many cases, however, the set S q can greatly vary in size, ranging from one or several elements, to perhaps 10,000 or more. In particular, for about half of the instances tested in Danna and Woodruff (2009), |S q | > 10, 000. For these instances, the authors limited the input to the second phase to a subset of S q consisting of the first 10,000 solutions obtained by the ONETREE algorithm Danna et al. (2007). That is, the first phase finds a subsetS ⊆ S q (|S| = 10, 000) without explicit consideration of diversity, while the second phase solves: max S⊆S,|S|=p D(S).(3) These problem instances with very large S q present two issues: • Computational. These problems could not be solved to optimality in the second-phase of the two-phase approach due to memory limitations or exceeding a 10-day time limit (Danna and Woodruff 2009). As a result, heuristics were employed. For smaller problems these heuristics appear to produce solutions that nearly maximize diversity. However, it is not known whether the heuristics produce good solutions to any of the larger problems. • Solution quality. Using the subsetS as an input to the second phase rather than the complete set S q could result in a loss of diversity if there is insufficient diversity in the input subset to either of the heuristic or exact methods. To address these issues, we seek to solve (3) directly rather than via a two-phase approach. Specifically, we examine how to modify the exploration strategy of the solution generation phase to increase the diversity of the subsetS. Computing near-optimal sets Given a mixed integer programming problem, we assume the existence of an oracle capable of enumerating a set of all or a sufficiently high number of near-optimal solutions for the problem. We use the branch-and-count method as the oracle due to its significant speed at enumerating near-optimal solutions. The branch-and-count method is implemented in SCIP (Gamrath et al. 2020), which provides access to, as well as modification privileges for, the branching tree and implementation methods of the oracle. The branch-and-count algorithm Achterberg et al. (2008), an extension of the branch-and-cut algorithm customized for detecting sets of near-optimal solutions, seeks to analyze all solution vectors contained in a subtree without complete enumeration. Infeasible subtrees, subtrees in which all of the leaf nodes are infeasible, are straightforward to detect because the descendants of an infeasible node are also infeasible. Thus, branch-and-count focuses on detecting unrestricted subtrees in which all of the leaf nodes are feasible. The authors show that a subtree rooted at a node is unrestricted if and only if all of the constraints at the node are locally redundant (i.e., satisfied by all possible variable assignments of values in the domain of that subtree). In the context of finding sets of near-optimal solutions to an optimization problem, branch-and-count first requires knowledge of the optimal objective value z * . Next the algorithm removes the objective value from the problem, adds the constraint c T x ≤ (1 + q)z * , and solves the problem using an approach much like branch-and-cut but with unrestricted subtree detection. Thus, in this context an unrestricted subtree is one in which all of the leaf nodes in the subtree represent solutions in the near-optimal set. The pseudocode for branch-and-count is given in Algorithm 1. To use this branch-and-count to generate the setS, we first solve (1) to find an optimal objective value z * . Next, in line 1 of the algorithm, we pass the MIP problem and the number of near-optimal solutions requested (p 1 ) as input to the algorithm and add the constraint c T x ≤ (1 + q)z * to the problem and solve it using the branch-and-count algorithm. We initialize a queue Q that holds nodes, each representing an MIP problem, that are available for exploration. We add the initial MIP to Q and also initialize setS = ∅. While Q is not empty and the number of solutions in the setS is less than p 1 , we select a node i from Q using a node selection rule. If the problem associated with node i is infeasible, we discard this node. If i is determined to be unrestricted, we collect solutions from i and its subtree and add them toS. Before discarding any node i, branching is performed, adding additional nodes to Q if the node is neither unrestricted nor infeasible. The entire process stops when either are no more nodes in Q to process or the cardinality of the setS equals p 1 , at which pointS is returned. Algorithm 1 Branch and Count Pseudocode During line 7 of the branch-and-count search outlined in Algorithm 1, the node selection rule decides which node from the queue of active nodes is selected as the next node. Popular node selection rules include: 1: INPUT p 1 , MIP 2: ADD constraint c T x ≤ (1 + q)z * to MIP 3: Q ← ∅ 4: ADD MIP • Best-First Search (BestFS). Selects the node with the best bound, that is, for minimization problems, select i as the next node according to i ∈ arg min i∈Q {LB i } , where LB i is the lower bound for node i. • Depth First Search (DFS). Nodes encountered as the branch and bound search tree is traversed are added to a queue and are selected in Last In First Out (LIFO) order. • Breadth First Search (BrFS). Nodes are added to a queue and processed using the First In First Out (FIFO) order. • Upper Confidence Bounds for Trees (UCT) (Gamrath et al. 2020). Selects the next node i as a node with the best UCT_score, that is: i ∈ arg min i∈Q {U CT _Score i } , U CT _score is calculated as: U CT _Score i = LB i + ρ V i v i where v i and V i are the number of times the algorithm has visited node i and its parent, respectively. ρ is a weight parameter chosen by the user. • Hybrid Estimate (HE) (Gamrath et al. 2020). Selects the next node i as a node having the best HE_Score, i.e.: i ∈ arg min i∈Q {HE_Score i } , HE_Score is calculated as: HE_Score i = (1 − ρ)LB i + ρ LB i LB i is the estimated value of the best feasible solution in subtree of node i and ρ is a user-defined weight parameter. Measuring the diversity of solutions A number of metrics exist for measuring the diversity of a set of solutions. A good metric needs to be model agnostic and ideally scaled such that diversity scores given by the metric are easy to interpret. Danna and Woodruff (2009) outlined three problem agnostic measures for the diversity of solutions: DBin, defined in more detail below and used in our tests, is scaled by the number of variables and solutions generated and considers just the binary variables; DAll which considers all variable types; and DCV which is the scaled version of DAll. The DBin metric (defined only on binary variables) is the average scaled Hamming distance between all pairs of solutions in a set S, that is, DBin(S) = 2 |S|(|S| − 1) |S| j=1 |S| k=j+1 ham(x (j) , x (k) ),(4) where x (j) is the j th solution generated by the oracle, and ham() computes the Hamming distance between a pair of solutions, that is, ham(x (j) , x (k) ) = 1 |B| i∈B |x (j) i − x (k) i |,(5) where B is the set of binary variables and |B| is the number of binary variables. An advantage of the DBin metric is that it takes values between 0 and 1 inclusive and does not depend on the size of the solution set or the number of variables. Diversity-Emphasizing Node Selection rules Within the branch-and-count algorithm (Achterberg et al. 2008), we investigated several variants of the best-first search (BestFS) node selection rule that consider solution set diversity when selecting the next node to evaluate. We focused on BestFS because, as our results in §4 show, it generated the most diverse solution sets in reasonable time when compared with other well-known node selection rules (such as DFS) when diversity was not considered in the node selection task. What follows is a description of each of the custom node selection rules tested in this work. Diverse-BFS (D-BFS(α)) The Diverse-BFS (D-BFS α) node selection rule considers both the lower bound of a node as well as the diversity of the node with respect to other solutions already in the near-optimal set. That is, this rule inputs a set of open nodes O and selects the next node i according to: i ∈ arg min i∈O {(1 − α)L i + αD i }, where α ∈ [0, 1] is a parameter that trades off the bound of node i against its diversity score D i and L i is a scaled lower bound of node i, that is, L i = LB i − min j∈O LB j max j∈O LB j − min j∈O LB j . The lower bound is scaled to [0, 1], commensurate with the diversity score D i ∈ [0, 1]. The value of D i represents the partial diversity of node i with respect to the current solution setS. We use the term partial because it is computed based only on the binary variables that have been fixed at a particular node in the branching tree. Diverse-BFS with tree depth (D-BFS(α, β)) The D-BFS(α, β) node selection rule considers the lower bound, the diversity and the depth of a node with respect to other solutions in the near-optimal set. This rule inputs a set of open nodes O and selects the next node i according to: i ∈ arg min i∈O {(1 − α − β)L i + αD i + βH i } , where (1 − α − β), α, β ∈ [0, 1] form a convex combination and are the parameters that control the weight of the scaled lower bound of node i, L i , the diversity score D i , and the scaled depth of node i, H i . Like L i , H i is scaled as: H i = Depth i − M inP lungeDepth M axP lungeDepth − M inP lungeDepth , where MinPlungeDepth and MaxPlungeDepth are set at the beginning of the computation and Depth i is the depth of node i in the tree. Diverse-BFS with solution cutoff (D-BFS(α, s)) The D-BFS(α, s) rule considers only the lower bound of a node until it has generated a small set of up to s solutions prior to incorporating the diversity of a node with respect to the solutions already in the near-optimal set. This rule selects the next node i according to: i ∈ arg min i∈O {L i } number of solutions found so far<s, arg min i∈O {(1 − α)L i + αD i } otherwise, where s is the solution cutoff parameter, that is, the number of solutions that must be accumulated before diversity is considered in node selection and α ∈ [0, 1] is a parameter that trades off the bound of node i against its diversity score D i . Diverse-BFS with depth cutoff (D-BFS(α, d)) The D-BFS(α, d) rule considers only the lower bound of a node until the depth of the active node reaches a depth d prior to also considering the diversity of the node with respect to the solutions already in the near-optimal set. This rule selects the next node i according to: i ∈ arg min i∈O {L i } depth of nodes in current iteration < d, arg min i∈O {(1 − α)L i + αD i } otherwise, where d is the depth cutoff parameter, that is, the depth that must be reached before diversity is considered in the node selection and α ∈ [0, 1] is a parameter that trades off the bound of node i against its diversity score D i . Diversity is only triggered upon reaching a depth of d or greater. DIVERSITREE -Diverse-BFS with solution cutoff and tree depth (D-BFS(α, β, s)) DIVERSITREE selects the next node i according to: i ∈ arg min i∈O {L i } number of solutions found so far < s, arg min i∈O {(1 − α − β)L i + αD i + βH i } otherwise, where α, β, s are parameters as defined in previous sections. Other Diverse-BFS methods tested We also tested several other methods for emphasizing solution set diversity. However, these additional methods were not as effective as the methods described in §3.1 - §3.5 above. They are: 1. Using the minimum of diversity and depth, that is, select next node i according to: i ∈ arg min i∈O {(1 − α)L i + α(min(D i , H i ))} . 2. Using the maximum of diversity and depth, that is, select next node i according to: i ∈ arg min i∈O {(1 − α)L i + α(max(D i , H i )} . 3. Using the product of diversity and depth, that is, select next node i according to: i ∈ arg min i∈O {(1 − α)L i + αD i H i } . These node selection rules were also tested with a nonzero solution cutoff parameter s, but were still not effective. Computational Experiments. To measure the effect of using diversity-emphasizing node selection rules on solution set diversity, we ran several sets of experiments on selected problems from MIPLIB (Koch et al. 2011, Bixby et al. 1998, Gleixner et al. 2021 using several different node selection rules, including the customized ones described in §3. We compared our approach with two state-the-art methods: ONETREE (Danna et al. 2007) and BRANCH-AND-COUNT (Achterberg et al. 2008). Experiments were run to answer the following questions: Research Question 1: Among common node selection rules (such as BestFS and DFS), do some produce a more diverse set of near-optimal solutions than others? §4.2 Research Question 2: What are the best parameters to use for the parameterized diversity-emphasizing node selection rules presented in §3? When using the best parameters, do the diversity-emphasizing node selection rules compute solution sets that are more diverse than those computed by competing approaches? §4.3 Research Question 3: What parameters should be used when using diversity-emphasizing node selection rules on a new problem? §4.4 Experimental Setup All code was implemented in C++ using SCIP Optimization suite 7.0 (Gamrath et al. 2020) and run on a server running Intel Xeon processors with sixteen cores and thirty two GB of memory. Apart from SCIP currently being one of the fastest non-commercial solvers for mixed integer programming (MIP) and mixed integer nonlinear programming, it provides a convenient way to use custom node selection rules. We evaluated our methods against the state-of-the-art ONETREE method, implemented in GAMS-CPLEX. Diversity of common node-selection rules We first examined the diversity produced by common node selection rules. For this initial set of experiments, we selected seven problems from MIPLIB that had greater than 10,000 solutions within 1% of optimality (see Table 2 in Appendix A). We used similar settings on the common node selection rules listed in §2.2 to request the generation of sets with sizes ranging from 50 to 2,000 near-optimal solutions, and computed the diversity. Figure 1 below shows the DBin diversity scores achieved by these common node selection rules for different solution set sizes (p 1 ). We labeled the different node selection rules with a "BC" prefix to indicate the use of branch-and-count to generate the solution sets. As shown in Figure 1, the best-first search (BCBFS) method finds sets with the greatest diversity in 5 out of 7 test problems. The upper confidence bounds for trees (BCUCT) node selection rule had the best performance in 2 out of the 7 test problems and achieved a similar diversity value to hybrid estimate (BCHE) in all other cases. The depth first search (BCDFS) was outperformed in terms of diversity in all of the instances. The plots also indicate that diversity (averaged for each problem instance) tends to start low and become incrementally higher as the number of requested solutions increases. We hypothesize that this may be due to a situation where as more solutions are generated, more variables are fixed or modified; increasing the Hamming distance from the first solution found. However, this trend did not hold for all of the individual problem instances. Figure 1: The final DBin diversity score achieved by common node selection rules (BestFS, DFS, UCT and HE) available on SCIP. We prefix each node selection rule name with "BC" to indicate the use of the "branch-and-count" method. The BestFS node selection rule resulted in greater diversity for a majority of problems. The superior performance of BestFS over other common rules suggested its continued use for comparison in the remainder of our experiments. Parameter optimization for diversity-emphasizing rules: training set The most general diversity-emphasizing node selection rules have up to three parameters to be tuned: α controls the emphasis on diversity for the selected node, β controls the emphasis on depth within the search tree for the selected node, and S controls the emphasis on the number of solutions generated before employing the α and β values in the node selection process. We used a grid search to tune the parameters and find the best-performing values. Toward this end, we used several problem instances from MIPLIB (Gleixner et al. 2021, Koch et al. 2011, Bixby et al. 1998) and randomly divided the set of problems into a training and testing set using a 75:25 split. We used the training set problems to find the best parameters and then tested the performance of these parameters on the testing set problems. Using a grid-search on a 20×20×20 grid over α, β, and S on Table 2 in Appendix A for different numbers (10, 50, 100, 200 and 1000) of requested near optimal solutions p 1 and values of q (% near optimal) in the range [0.01, 0.1] with increment 0.01. As a result, we obtained for each instance the best performing values of α, β, and S over every p 1 and q. It is computationally impractical to run a grid search for every new problem instance prior to applying our node selection rule. We therefore attempted to identify patterns in the mapping between problems and optimal parameter values, so as to allow the identification of a smaller number of parameter settings with good performance. Thus, our next step was to determine whether we could group problems together based on their optimal parameter settings. Toward this end, we ran a standard hierarchical clustering algorithm available in Python's Scikit-Learn package (Pedregosa et al. 2011) to cluster the problems into groups based on their best parameter settings, that is, the settings that yielded the highest DBin scores for that problem. When the number of requested solutions is small (that is, 10), the clustering algorithm found the following four general groups: 1. High α, High S, Low β (HHL): α ≥ 0.9, S ≥ 0.7, β ≤ 0.2. 2. High α, Low S, Low β (HLL): α ≥ 0.9, S ≤ 0.2, β ≤ 0.2. 3. Low α, Low S, High β (LLH): α ≤ 0.2, S ≤ 0.2, β ≥ 0.8. 4. Low α, High S, High β (LHH): α ≤ 0.2, S ≥ 0.7, β ≥ 0.8. In the HHL group, diversity is emphasized heavily (α ≥ 0.90), but only after a large number of solutions have been accumulated (S ≥ 0.70). In the HLL group, diversity is also heavily emphasized, but starting after a moderate number of solutions have been found (S ≤ 0.20). The LLH group tends to select solutions at greater depths in the tree (β ≥ 0.80) after a small number of solutions have been accumulated (S ≤ 0.20). Finally, the LHH group emphasizes diversity lightly (α ≤ 0.20) and mostly selects solutions at greater depths in the tree (β ≥ 0.80) after a large number of solutions have been accumulated (S ≥ 0.70). The structure of these four groups indicates that our method is most effective when we generate a small number of seed solutions and then emphasize diversity at greater depths in the tree. To assess the efficacy of DIVERSITREE, we then selected a single parameter setting for each of the four groups, as shown in Table 1. Using the data from the grid search, we took the weighted average of the settings in a group as G 1 and the settings that occurred the most frequently in that group as G 2 . We computed the diversity of the training set problems using G 1 and G 2 separately and selected either G 1 or G 2 as the best setting for the group, depending on which of the two produced the best diversity. We found that these parameter settings worked well for different values of p 1 and q. For each problem instance, we then ran DIVERSITREE using the parameters from its assigned group from the clustering algorithm (see Figure 6 in the Appendix). The parameter settings in Table 1 were used for all values of p 1 and q. Specifically, we used the following two-phase approach. In phase one, we generate a larger solution set p 1 with 10, 50, 100, 200, and 1000 solutions. In phase two, we use a subset selection method (similar to local search algorithm in Danna and Woodruff (2009)) to select p = 10 diverse solutions-a number that seems reasonable to present to a decision maker. We ran this process for values of q from 1% to 10% across all the problems in the training set. We compared the performance of our method with two state-of-the-art methods: with the BCBFS rule, and with the ONETREE method results reported in Danna and Woodruff (2009). These competing methods were used in phase one of the two-phase approach, with the subset selection method being employed in phase two. In Figure 2 and 3 we plot the percent improvement of the different groups of DIVERSITREE over ONETREE for values of q from 1% to 10%, and p = 10 for different p 1 values. In Figure 2, we show plots for p 1 ≤ 100 and in Figure 3, we show plots for p 1 ≥ 200. As the figure shows, DIVERSITREE achieves an improvement in diversity over ONETREE of up to 160% and at least 60%, irrespective of the four parameter settings used. Beyond outperforming ONETREE, all the parameter groups also give similar performance on average with no more than 15% gap between any group. Among the DIVERSITREE results, parameter groups HHL and HLL appear to perform best. Thus, we recommend that when using our method on a new problem either the HHL (α = 0.94, β = 0.06, s = 0.80) of HLL (α = 0.95, β = 0.06, s = 0.20) parameter settings should be used. In the next section we test the HHL parameter setting on a set of new problems that were not used for parameter tuning. Performance of tuned parameter groups on test set of problems The results of the previous section raise the question of how parameters should be tuned on a new problem. To answer this question, we tested our method using the HHL parameter setting, which was recommended in the previous section. We tested our method with the HHL setting on a test set of problems not used in parameter tuning (see Table 3 in Appendix B). We ran the same two-phase approach for diverse, near-optimal solution generation as we did in the training phase. We ran this process for p 1 ∈ {10, 50, 100, 200, 1000} and q from 1% to 10%. We kept the size of the final solution set to be p = 10. Although our method performed better than BCBFS 2 and ONETREE for all four of the parameter settings, we only present results for the recommend HHL setting. We used the HHL parameter setting on the test data and the results achieved are shown in Figure 4 and 5, which displays results similar to those in Figures 2 and 3 for the training set. DIVERSITREE generated solution sets with improvement over ONETREE between 12% and 190%. DIVERSITREE also significantly outperforms BCBFS in terms of diversity for all values of p 1 , as BCBFS generated solution sets with improvement of up to 130% and in some instances performed worse than ONETREE by more than 35%. In terms of runtime, DIVERSITREE achieves similar runtime as BCBFS, although the ONETREE method runs significantly faster than both methods, which is unsurprising given that while DIVERSITREE explicitly searches for solution, ONETREE only focuses on near-optimality. Unlike the training set data where parameter groups HHL and HLL dominated the DIVERSITREE results, the parameter groups LHH and LLH dominated the diversity results generated in the test set. In practice, a decision-maker seeking a diverse set of near-optimal solutions would need to define p (the number of near-optimal solutions), q% (a bound on how far these solutions may be from optimality), and select a setting in any of the groups we specified in §4.3 (without need for parameter tuning); p near-optimal solutions can then be directly generated within q% of the optimal, or a two-phase approach can be used to generate the solution set. The result of our tests suggest that the solution set generated would be no worse than BCBFS and would be significantly better than the state-of-the-art ONETREE method. Table 1. The plots on the right show the runtime. The runtime for DIVERSITREE were similar for the four different parameter groups, hence this figure shows the average runtime over the four groups. Example: Railway Timetabling Problem To provide a concrete example, we tested DIVERSITREE on a public transport scheduling problem within the MIPLIB2017 set (Gleixner et al. 2021). Details about the problem formulation and variables is available in Liebchen and Möhring (2003). Liebchen and Peeters (2002) discuss the general problem formulation. The problem models the cyclic railway timetabling problem where we have information about a railway network in a graph representing its infrastructure and traffic line. Each traffic line is operated every T time units and the goal of the problem is to determine periodic departure times within the interval [0, T ) at every stop of every line. The problem has 397 variables; 77 of them are binary variables, 94 integer variables and 226 continous variables. The final objective is to obtain an arrival/departure timetable with minimal passenger and vehicle waiting times. The problem is represented as a graph G = (N, A, l, u) with a set of nodes N representing a set of events, and arcs A representing set of trains. The weights on the arcs represent the time when event v i ∈ N occurs and [l, u] the allowable time interval for this event. The events are represented as triplets (train,node,arrival) or (train,node,departure) and the binary variable b ij is 1 if arc (i, j) is selected. The model is mainly based on periodic constraints, which relate the arrival and departure time variables through the time window [l, u] within which the event i must occur. For problems such as these, the solutions provided by the MIP model will be reviewed by a decision maker prior to implementation. Thus, in this context it can be useful to provide the decision maker with a set of near-optimal solutions from which to choose. This will allow the decision maker to consider not only the cost of the solution but also other factors such as variability in estimated demands, crew shortages, behavior of new equipment, or propagated delays, without having to re-solve the problem. For q = 1%, the problem had 181 near-optimal solutions, and when q increased to 3%, the number of near-optimal solutions increased to 6,264. We ran the DIVERSITREE and ONETREE algorithms on this problem for p 1 = 10, p = 10 and q = 3%. DIVERSITREE computed a solution set with a DBin value of 0.299, while ONETREE's set has a DBin value of 0.044. A further review of the generated solutions shows that of the 77 binary variables, ONETREE gave a solution set in which 68 of the binary variables had the same value in all ten solutions. In addition, ONETREE generated a set of ten solutions of which only six solutions were unique. Among the six unique solutions, five of them had at most two variables with different values indicating very close similarity of the solutions. In constrast, DIVERSITREE's gave a solution set in which only 23 of the variables were the same in all ten solutions, all ten solutions were unique and at least nine variables had different values among the solutions, indicating the diversity of the solutions. This experiment provides further evidence supporting that the DIVERSITREE approach provides a much more diverse set of train schedules from which a decision maker may select. Extensions This study focused on investigating the diversity of solutions consisting of binary variables. While a majority of integer programming problems feature binary variables, there are some problems with pure integer variables (e.g., cutting stock optimization (Gilmore and Gomory 1961)) that could also benefit from a having a diverse set of near-optimal solutions. In addition, in some contexts, a user may like to consider diversity with respect to continuous variables. While many integer programming formulations feature continuous variables that depend on binary variables in a somewhat subordinate manner (such as facility location and lot sizing), for many other problems, this may not hold. We now discuss how DIVERSITREE can be employed for problems involving variables beyond binary integer. Reformulations One way to consider diversity among general integer and continuous variables, is through reformulation. Binary Expansion of Pure Integer Variables. First, for the case of bounded pure integer variables, one approach is to replace each pure integer variable with its binary expansion. Consider a single nonnegative integer variable x ∈ Z with upper bound u. By introducing M binary variables b 0 , . . . , b M −1 we can represent x with its binary expansion: x = M −1 j=0 2 j · b j ,(6) which requires the smallest M such that u ≤ M −1 j=0 2 j so as to use as few binary variables as possible (Watters 1967). One potential drawback of this method is that the number of binary variables needed to represent a single integer variable increases logarithmically in the upper bound. Binary Discretization of Continuous Variables. It is also possible to map a continuous variable to the binary space. Consider a single continuous variable x ∈ R, and for simplicity of exposition let x ∈ [0, 1]. To represent x within accuracy = 10 −p , where p is some positive integer, then we need no more than K = log 10 log 2 p binary variables z k ∈ {0, 1} are needed, k = 1, . . . , K so that x ≈ K k=1 2 −k z k . As suggested previously, a potential drawback is that the number of binary variables needed to represent a single variable increases logarithmically with respect to p. Alternate Diversity Metrics The method presented in this paper is not limited to a specific metric for computing solution diversity. While the results in this paper are based on the DBin metric, other metrics can be used equivalent to maximizing the average elementwise variances for the binary variables. Directly generalizing DBin, Danna and Woodruff (2009) defined DAll, a metric that maximizes the variance of each variable and combines them by elementwise averaging: DAll(S) = 1 |S| |S| i=1 σ 2 i (S) R ,(7) where σ 2 i (S) = 1 |S| j∈S x i is the range of values within the bounds for variable x i . However, as Danna and Woodruff (2009) point out, a potential issue with using DAll is that the variables in the MIP need to have known bounds to scale the variances to prevent a single variable having an outsized influence. As a mitigation strategy, Danna and Woodruff (2009) propose the MIP owner provide importance weights for the different variables, which could then be a scaling factor for the calculated variances. Conclusion We introduced a novel approach to generate diverse, near-optimal solution sets by emphasizing solution diversity in the node selection step of a branch-and-bound algorithm. Our results reveal that our modified node selection rule yields up to 190% better diversity than known methods for generating diverse solutions. Our methods provide a fast way to generate diverse sets of near-optimal solutions, useful where there is utility in having a diverse set of options for decision making. We presented several methods for emphasizing diversity in node selection rules and optimal parameters that yield the most diverse set of solutions for the problems tested. We identified four groups of optimal parameters for different problems by clustering the parameter groups giving the best diversity (DBin) on the training set. Each group sets an optimal solution cutoff value and also sets how much diversity and depth to consider in determining the next solution added to the set of diverse solutions. When tested on new problems, our method (using the identified optimal parameters) ran in similar time as regular node selection rules and gave solution pools that were significantly more diverse. A positive result of our study was that the four parameter setting groups identified during the parameter tuning on the training set performed well on a previously unseen set of problems in a test set. Even so, understanding the relationship between parameter settings and different classes of problems such that we can select the best parameter for any problem with a minimal amount of tuning would be beneficial in practice. A second area of interest which is also an extension of this work is diversifying solution pools on select variables (see Voll et al. (2015)). In cases where a cluster of variables represents an attribute that is desirable in a machine learning algorithm (such as fairness or intelligence), it may be useful and informative to generate solutions pools that are diverse on only that attribute or interpret and cluster the generated solutions based the attribute. Finally, we only reported results using DBin as a diversity metric. It would be useful to understand how the new node selection method performs when using other metrics for computing diversity. [0, 1] with increment 0.01 on [0, 0.09] and 0.1 on [0.1, 1], we tested different parameters values across all training set problems shown in Figure 2 : 10 Figure 3 : 2103This figure shows plots for p1 ≤ 100. The plots on the left shows the average percent improvement on diversity (DBin) achieved by different parameter group settings of DIVERSITREE over ONETREE aggregated across all problems in the training set. The run times are shown in the plots on the right. The runtime for DIVERSITREE were similar for the four different parameter groups, hence this figure shows the average runtime over the four groups. (a) p1 = 200, p = 10 (b) p1 = 1000, p = This figure shows plots for p1 ≥ 200. The plots on the left shows the average percent improvement on diversity (DBin) achieved by different parameter group settings of DIVERSITREE over ONETREE aggregated across all problems in the training set.The run times are shown in the plots on the right. The runtime for DIVERSITREE were similar for the four different parameter groups, hence this figure shows the average runtime over the four groups. (c) p1 = 100 , p = 10 Figure 4 : 10 Figure 5 : 100104105This figure shows plots for p1 ≤ 100. Plots on the left show the percent improvement on diversity (DBin) achieved by DIVERSITREE and BCBFS on the problems in the test set using the parameter setting from group HHL shown inTable 1. The plots on the right show the runtime. The runtime for DIVERSITREE were similar for the four different parameter groups, hence this figure shows the average runtime over the four groups.(a) p1 = 200, p = 10 (b) p1 = 1000, p = This figure shows plots for p1 ≥ 200. Plots on the left show the percent improvement on diversity (DBin) achieved by DIVERSITREE and BCBFS on the problems in the test set using the parameter setting from group HHL shown in variance of variable x i in any set of solutions S. The scaling factor R = max j∈S x to queue of active nodes Q 5:S ← ∅ 6: while Q is not empty and |S| < p 1 do Use node selection rule to dequeue node i from Q if node i is unrestricted then Enumerate all child nodes of node i and add their solutions toS 10: else if node i is infeasible then Perform branching and add child nodes to queue Q7: 8: 9: 11: Discard node 12: else 13: 14: end if 15: end while 16: returnS 2.2 Node Selection Rules. Table 1 : 1The α, β and S parameter settings used in both training and testing.Number of solutions requested HHL HLL LLH LHH 10, 50, 100, 200, 1000 α :0.94, β :0.06, s :0.80 α :0.95, β :0.06, s :0.20 α :0.01, β :0.99, s :0.05 α :0.18, β :0.8, s :0.70 to be clear, our method as well as existing methods also include a precomputation step in which the optimal objective value is computed. However, we do not count this step as a phase, when referring to the number of phases of our method or other methods. In using BCBFS, we found that the default parameters on SCIP (i.e., M IN P LU N GEDEP T H = −1, M AXP LU N GEDEP T H = −1, and M AXP LU N GEQU OT = 0.25) performed poorly. Thus, we manually tuned these settings to improve the diversity obtained by BCBFS. These tuned parameter settings were used in the results shown in the remainder of the paper. AcknowledgmentsMedal gratefully acknowledges funding from the Army Research Office (grant W911NF-21-1-0079). However, the views expressed in this study do no represent those of the US Government, the US Department of Defense, or the US Army.Appendix A Problems in the training setTable 2captures the problems we solved in the training set. They are randomly selected problems from MIPLIB(Gleixner et al. 2021, Koch et al. 2011, Bixby et al. 1998. We did remove instances that did not complete computation of the objective value within 30 minutes. In total, 27 problem instances are used in this training set. We capture the characteristics of the problem below. Appendix B Problems in the testing setTable 3captures the problems we solved in the testing set. They are randomly selected problems from MIPLIB(Gleixner et al. 2021). We did remove instances that did not complete computation of the objective value within 30 minutes. In total, 9 problem instances are used in this training set. We capture the characteristics of the problem below. Appendix C Parameter groups for diversity-emphasizing rulesFigure 6shows the clustering of the problem instances into the four groups. As the figure shows, these four groups consolidate to three (HHL, HLL, and LLH) when the number of requested solutions reaches 50 and to two (HHL and HLL) for 200 requested solutions or more. The consolidation to groups HHL and HLL indicate that emphasizing diversity after generating a seed set of solutions results in a higher overall diversity of all generated solutions.One might suspect that two instances of the same problem (e.g., stein27 and stein45) would typically have the same parameter group. However, surprisingly we found that similar problems like stein27/stein45 and qnet1/qnet1_0 did not always belong to the same group. At p 1 ≥ 50, some problems took too long to complete all test cases in the grid search and thus were excluded from the parameter groups shown inFigure 6.Figure 6: The four parameter groups and the problem instances in each group as the number of requested solutions increases from 10 to 1,000. Counting solutions of integer programs using unrestricted subtree detection. 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[ "Fast as CHITA: Neural Network Pruning with Combinatorial Optimization", "Fast as CHITA: Neural Network Pruning with Combinatorial Optimization" ]
[ "Riade Benbaki ", "Wenyu Chen ", "Xiang Meng ", "Hussein Hazimeh ", "Natalia Ponomareva ", "Zhe Zhao ", "Rahul Mazumder " ]
[]
[]
The sheer size of modern neural networks makes model serving a serious computational challenge. A popular class of compression techniques overcomes this challenge by pruning or sparsifying the weights of pretrained networks. While useful, these techniques often face serious tradeoffs between computational requirements and compression quality. In this work, we propose a novel optimization-based pruning framework that considers the combined effect of pruning (and updating) multiple weights subject to a sparsity constraint. Our approach, CHITA, extends the classical Optimal Brain Surgeon framework and results in significant improvements in speed, memory, and performance over existing optimizationbased approaches for network pruning. CHITA's main workhorse performs combinatorial optimization updates on a memory-friendly representation of local quadratic approximation(s) of the loss function. On a standard benchmark of pretrained models and datasets, CHITA leads to significantly better sparsity-accuracy tradeoffs than competing methods. For example, for MLPNet with only 2% of the weights retained, our approach improves the accuracy by 63% relative to the state of the art. Furthermore, when used in conjunction with fine-tuning SGD steps, our method achieves significant accuracy gains over the state-of-the-art approaches.
10.48550/arxiv.2302.14623
[ "https://export.arxiv.org/pdf/2302.14623v1.pdf" ]
257,232,716
2302.14623
0c40850c24bf543b14ceb44124db1f4cf88211f3
Fast as CHITA: Neural Network Pruning with Combinatorial Optimization Riade Benbaki Wenyu Chen Xiang Meng Hussein Hazimeh Natalia Ponomareva Zhe Zhao Rahul Mazumder Fast as CHITA: Neural Network Pruning with Combinatorial Optimization The sheer size of modern neural networks makes model serving a serious computational challenge. A popular class of compression techniques overcomes this challenge by pruning or sparsifying the weights of pretrained networks. While useful, these techniques often face serious tradeoffs between computational requirements and compression quality. In this work, we propose a novel optimization-based pruning framework that considers the combined effect of pruning (and updating) multiple weights subject to a sparsity constraint. Our approach, CHITA, extends the classical Optimal Brain Surgeon framework and results in significant improvements in speed, memory, and performance over existing optimizationbased approaches for network pruning. CHITA's main workhorse performs combinatorial optimization updates on a memory-friendly representation of local quadratic approximation(s) of the loss function. On a standard benchmark of pretrained models and datasets, CHITA leads to significantly better sparsity-accuracy tradeoffs than competing methods. For example, for MLPNet with only 2% of the weights retained, our approach improves the accuracy by 63% relative to the state of the art. Furthermore, when used in conjunction with fine-tuning SGD steps, our method achieves significant accuracy gains over the state-of-the-art approaches. Introduction Modern neural networks tend to use a large number of parameters (Devlin et al., 2018;He et al., 2016), which leads to high computational costs during inference. A widely used 1 MIT 2 Google Research. Correspondence to: Riade Benbaki <[email protected]>, Wenyu Chen <[email protected]>, Xiang Meng <[email protected]>, Hussein Hazimeh <[email protected]>, Natalia Ponomareva <[email protected]>, Zhe Zhao <[email protected]>, Rahul Mazumder <[email protected]>. approach to mitigate inference costs is to prune or sparsify pre-trained networks by removing parameters (Blalock et al., 2020). The goal is to obtain a network with significantly fewer parameters and minimal loss in performance. This makes model storage and deployment cheaper and easier, especially in resource-constrained environments. Generally speaking, there are two main approaches for neural net pruning: (i) magnitude-based and (ii) impact-based. Magnitude-based heuristic methods (e.g., Hanson & Pratt, 1988;Mozer & Smolensky, 1989;Gordon et al., 2020) use the absolute value of weight to determine its importance and whether or not it should be pruned. Since magnitude alone may not be a perfect proxy for weight relevance, alternatives have been proposed. To this end, impact-based pruning methods (e.g. LeCun et al., 1989;Hassibi & Stork, 1992;Singh & Alistarh, 2020) remove weights based on how much their removal would impact the loss function, often using second-order information on the loss function. Both of these approaches, however, may fall short of considering the joint effect of removing (and updating) multiple weights simultaneously. The recent method CBS (Combinatorial Brain Surgeon) (Yu et al., 2022) is an optimizationbased approach that considers the joint effect of multiple weights. The authors show that CBS leads to a boost in the performance of the pruned models. However, CBS can be computationally expensive: it makes use of a local model based on the second-order (Hessian) information of the loss function, which can be prohibitively expensive in terms of runtime and/or memory (e.g., CBS takes hours to prune a network with 4.2 million parameters). In this work, we propose CHITA (Combinatorial Hessianfree Iterative Thresholding Algorithm), an efficient optimization-based framework for network pruning at scale. Our approach follows earlier works that consider a local quadratic approximation of the loss function based on the second-order Hessian information. Different from previous works, we make use of a simple yet important observation with which we can avoid computing and storing the Hessian matrix to solve the optimization problem (hence the name "Hessian-free" in CHITA)-this allows us to address large networks efficiently. Specifically, we propose an equivalent reformulation of the problem as an 0 -constrained sparse linear regression problem with a data matrix A ∈ R n×p , where p is the number of trainable parameters in the original model and n 10 3 (usually, p n) is the number of the sub-samples used in approximating the Hessian. Compared to state-of-the-art approaches that consider a dense p × p matrix approximation of the Hessian, our approach leads to a significant reduction in memory usage (up to 10 3 times for p = 10 6 ) without any approximation. Furthermore, we propose a novel approach to minimize our 0 regression reformulation, leveraging active set strategies, better stepsize selection, and various methods to accelerate convergence on the selected support. Our proposed approach leads to significant improvements over Iterative Hard Thresholding methods (IHT, Blumensath & Davies, 2009) commonly used in sparse learning literature. For instance, our framework can prune a network with 4.2M parameters to 80% sparsity (i.e., 0.84M nonzero parameters) in less than a minute and using less than 20GB of RAM 1 . Since the local quadratic model approximates the loss function only in a small neighborhood of the current solution (Singh & Alistarh, 2020), we also propose a multi-stage algorithm that updates the local quadratic model during pruning (but without retraining) and solves a more constrained problem in each stage, going from dense weights to sparse ones. Our experiments show that the resulting pruned models have a notably better accuracy compared to that of our single-stage algorithm and other pruning approaches. Furthermore, when used in the gradual pruning setting (Gale et al., 2019;Singh & Alistarh, 2020;Blalock et al., 2020) where re-training between pruning steps is performed, our pruning framework results in significant performance gains compared to state-of-the-art unstructured pruning methods. Contributions Our contributions can be summarized as follows: • We propose CHITA, an optimization framework for network pruning based on local quadratic approximation(s) of the loss function. We propose an 0 -constrained sparse regression reformulation that avoids the pitfalls of storing a large dense Hessian, resulting in a significant reduction in memory usage (we work with an n × p matrix instead of a p × p one, with often n p). • A key workhorse of CHITA is a novel IHT-based algorithm to obtain good solutions to the sparse regression formulation. Exploiting problem structure, we propose methods to accelerate convergence and improve pruning performance, such as a new and efficient stepsize selection scheme and rapidly updating weights on the support. This leads to up to 1000x runtime improvement compared to existing network pruning algorithms. • We show performance improvements across various mod-els and datasets. In particular, CHITA results in a 98% sparse (i.e., 98% of weights in dense model are set to zero) MLPNet with 90% test accuracy (3% reduction in test accuracy compared to the dense model), which is a significant improvement over the previously reported best accuracy (55%) by CBS. As an application of our framework, we use it for gradual pruning and observe notable performance gains against state-of-the-art gradual pruning approaches. Problem Setup and Related Work In this section we present the general setup with connections to related work-this lays the foundation for our proposed methods discussed in Section 3. Problem Setup Consider a neural network with an empirical loss function L(w) = 1 N N i=1 i (w), where w ∈ R p is the vector of trainable parameters, N is the number of data points (samples), and i (w) is a twice-differentiable function on the i-th sample. Given a pre-trained weight vectorw ∈ R p , our goal is to set some parameters to zero and possibly update other weights while maintaining the original model's performance (e.g., accuracy) as much as possible. In mathematical terms, given a pre-trained weightw and a target level of sparsity τ ∈ (0, 1), we aim to construct a new weight vector w ∈ R p that satisfies : • The loss function at w is as close as possible to the loss before pruning: L(w) ≈ L(w). • The number of nonzero weights at w respects the sparsity budget 2 : w 0 ≤ (1 − τ )p. Similar to LeCun et al. (1989); Hassibi & Stork (1992); Singh & Alistarh (2020), we use a local quadratic approximation of L around the pre-trained weightw: L(w) = L(w) + ∇L(w) (w −w)+ 1 2 (w −w) ∇ 2 L(w)(w −w) + O( w −w 3 ). (1) With certain choices of gradient and Hessian approximations g ≈ ∇L(w), H ≈ ∇ 2 L(w), and ignoring higher-order terms, the loss L can be locally approximated by: Q 0 (w) := L(w)+g (w−w)+ 1 2 (w−w) H(w−w). (2) Pruning the local approximation Q 0 (w) of the network can be naturally formulated as an optimization problem to minimize Q 0 (w) subject to a cardinality constraint, i.e., min w Q 0 (w) s.t. w 0 ≤ k.(3) For large networks, solving Problem (3) directly (e.g., using iterative optimization methods) is computationally challenging due to the sheer size of the p × p matrix H. In Section 3.1, we present an exact, hessian-free reformulation of Problem (3), which is key to our scalable approach. Related Work Impact-based pruning dates back to the work of LeCun et al. (1989) where the OBD (Optimal Brain Damage) framework is proposed. This approach, along with subsequent ones (Hassibi & Stork, 1992;Singh & Alistarh, 2020;Yu et al., 2022) make use of local approximation (2). It is usually assumed (but not in our work) thatw is a local optimum of the loss function, and therefore g = 0 and L(w) ≈ L(w) + 1 2 (w −w) H(w −w). Using this approximation, OBD (Optimal Brain Damage, LeCun et al. (1989)) searches for a single weight i to prune with minimal increase of the loss function, while also assuming a diagonal Hessian H. If the i-th weigth is pruned (w i = 0, w j =w j ∀j = i), then the loss function increases by δL i =w 2 i 2∇ 2 L(w) ii . This represents a score for each weight, and is used to prune weights in decreasing order of their score. OBS (Optimal Brain Surgeon, Hassibi & Stork (1992)) extends this by no longer assuming a diagonal Hessian, and using the optimality conditions to update the un-pruned weights. The authors also propose using the empirical Fisher information matrix, as an efficient approximation to the Hessian matrix. Layerwise OBS (Dong et al., 2017) proposes to overcome the computational challenge of computing the full (inverse) Hessian needed in OBS by pruning each layer independently, while Singh & Alistarh (2020) use block-diagonal approximations on the Hessian matrix, which they approximate by the empirical Fisher information matrix on a small subset of the training data (n N ): ∇ 2 L(w) ≈ H = 1 n n i=1 ∇ i (w)∇ i (w) .(4) While these approaches explore different ways to make the Hessian computationally tractable, they all rely on the OBD/OBS framework of pruning a single weight, and do not to consider the possible interactions that can arise when pruning multiple weights. To this end, Yu et al. (2022) propose CBS (Combinatorial Brain Surgeon) an algorithm to approximately solve (3). While CBS shows impressive improvements in the accuracy of the pruned model over prior work, it operates with the full dense p × p Hessian H. This limits scalability both in compute time and memory utilization, as p is often in the order of millions and more. Choices of gradient approximation g. As mentioned earlier, most previous work assumes that the pre-trained weightsw is a local optimum of the loss function L, and thus take the gradient g = 0. However, the gradient of the loss function of a pre-trained neural network may not be zero in practice due to early stopping (or approximate optimization) (Yao et al., 2007). Thus, the WoodTaylor approach (Singh & Alistarh, 2020) proposes to approximate the gradient by the stochastic gradient, using the same samples for estimating the Hessian. Namely, g = 1 n n i=1 ∇ i (w).(5) One-shot and gradual pruning. Generally speaking, one-shot pruning methods (LeCun et al., 1989;Singh & Alistarh, 2020;Yu et al., 2022) can be followed by a few finetuning and re-training steps to recover some of the accuracy lost when pruning. Furthermore, recent work has shown that pruning and re-training in a gradual fashion (hence the name, gradual pruning) can lead to big accuracy gains (Han et al., 2015;Gale et al., 2019;Zhu & Gupta, 2018). The work of Singh & Alistarh (2020) further shows that gradual pruning, when used with well-performing one-shot pruning algorithms, can outperform state-of-the-art unstructured pruning methods. In this paper, we focus on the one-shot pruning problem and show how our pruning framework outperforms other one-shot pruning methods (see Section 4.1). We then show that, if applied in the gradual pruning setting, our pruning algorithm outperforms existing approaches (see Section 4.2), establishing new state-of-the-art on Mo-bileNetV1 and ResNet50. Our Proposed Framework: CHITA In this section, we present our algorithmic framework CHITA (Combinatorial Hessian-free Iterative Thresholding Algorithm) for pruning a network to specific sparsity level. We formulate sparse network pruning by considering 0regression problem(s) and propose scalable algorithms. For example, we can address networks with size p ≈ 10 6 , n ≈ 10 3 , k ≈ 10 5 in less than one minute and using less than 20GB of memory. Our basic single-stage algorithm is an improved and scalable variant of IHT to solve (3). In Section 3.3, we propose a multi-stage framework that repeatedly refines the local quadratic approximation and optimizes it (under sparsity constraints) resulting in further performance boosts as shown in Section 4. An 0 -regression formulation Our formulation is based on a critical observation that the Hessian approximation in (4) has a low-rank structure: H = 1 n n i=1 ∇ i (w)∇ i (w) = 1 n A A ∈ R p×p ,(6) where A = [∇ 1 (w), . . . , ∇ n (w)] ∈ R n×p has rank at most n with 10 3 ≥ n p. Using observation (6) and the gradient expression (5), we note that problem (3) can be equivalently written in the following Hessian-free form: min w 1 2 b − Aw 2 s.t. w 0 ≤ k,(7) where b := Aw − e ∈ R n and e is a vector of ones. Furthermore, to improve solution quality (see discussion below), we include a ridge-like regularizer of the form w −w 2 to the objective in (7). This leads to the following problem: min w Q(w) := 1 2 b−Aw 2 + nλ 2 w−w 2 s.t. w 0 ≤ k,(8) where λ ≥ 0 is a parameter controlling the strength of the ridge regularization. Note that our algorithm actually applies to the general form (8). Importantly, the regression formulation (8) does not require us to compute or store the full Hessian matrix H ∈ R p×p . As discussed in Section 3.2, we only need to operate on the low-rank matrix A throughout our algorithm-this results in substantial gains in terms of both memory consumption and runtime. Ridge term and choices of λ. We observe empirically that the success of our final pruned model depends heavily on the accuracy of the quadratic approximation of the loss function. Since this approximation is local, it is essential to ensure that the weights w during the pruning process are sufficiently close to the initial weightsw. One way 3 to achieve this is by including a squared 2 penalty, also known as the ridge, on the difference w −w. This regularization technique does not appear to be explicitly 4 considered in previous works (Hassibi & Stork, 1992;Singh & Alistarh, 2020;Yu et al., 2022) on pruning using local quadratic approximations. The usefulness of the regularization term λ is further explored in Appendix B.2.1. We observe that a well-chosen ridge term can help improve the test accuracy on MLPNet by 3%. Relation to prior work on 0 -regression problems. There is a substantial literature on algorithms for solving 0 -regularized linear regression problems. We provide a brief overview of related work, but it is worth noting that the context of network pruning and the problem-scale we consider here makes our work different from earlier works in 0 -sparse linear regression. Blumensath & Davies (2009) developed an iterative hard thresholding method, which involves projecting the weights onto the feasible set after each gradient step. Bertsimas & Van Parys (2020); Hazimeh et al. (2022) proposed algorithms to solve sparse regression problems to optimality via branch-and-bound. Beck & Eldar (2013) explore coordinate descent-type algorithms that update one/two coordinates at a time. Hazimeh & Mazumder (2020) propose efficient algorithms based on coordinate descent and local combinatorial optimization that applies to the unconstrained 0 2 -penalized regression problem. We refer the reader to Hazimeh et al. (2022) for a more comprehensive discussion of related work. In summary, earlier methods for 0 -regularized linear regression are quite effective at discovering high-quality solutions to Problem (7) for small to medium-sized problems and require k to be sufficiently small for efficiency. However, these methods are not well-suited for tackling large network pruning problems (e.g, p ∼ 10 6 and k ∼ 10 5 ) due to slow convergence or expensive per-iteration cost. To address this issue, we propose novel approaches for scalability in Section 3.2. Additionally, we emphasize that (8) arises from a local approximation of the true loss L aroundw. Therefore, achieving a high-quality solution for (8) alone does not guarantee a pruned network with high accuracy. To this end, we study a multi-stage extension (see Section 3.3) that requires solving several problems of the form (8). (8) We present the core ideas of our proposed algorithm for Problem (8), and discuss additional details in Appendix A. Our proposed algorithm for problem Our optimization framework relies on the IHT algorithm (Blumensath & Davies, 2009;Bertsimas et al., 2016) that optimizes (8) by simultaneously updating the support and the weights. By leveraging the low-rank structure, we can avoid the computational burden of computing the full Hessian matrix, thus reducing complexity. The basic version of the IHT algorithm can be slow for problems with a million parameters. To improve the computational performance of our algorithm we propose a new line search scheme. Additionally, we use use an active set strategy and schemes to update the weights on the nonzero weights upon support stabilization. Taken together, we obtain notable improvements in computational efficiency and solution quality over traditional IHT, making it a viable option for network pruning problems at scale. STRUCTURE-AWARE IHT UPDATE The IHT algorithm operates by taking a gradient step of size τ from the current iteration, then projects it onto the set of points with a fixed number of non-zero coordinates through hard thresholding. Specifically, for any vector x, let I k (x) denote the indices of k components of x that have the largest absolute value. The hard thresholding operator P k (x) is defined as y i = x i if i ∈ I k (x), and y i = 0 if i / ∈ I k (x); where y i is the i-th coordinate of P k (x) . IHT applied to problem (8) leads to the following update: w t+1 = HT(w t , k, τ s ) := P k w t − τ s ∇Q(w t ) (9) = P k w t − τ s (A (Ab − w t ) + nλ(w t −w)) , where τ s > 0 is a suitable stepsize. The computation of HT(w t , k, τ s ) involves only matrix-vector multiplications with A (or A ) and a vector, which has a total computation cost of O(np). This is a significant reduction compared to the O(p 2 ) cost while using the full Hessian matrix as Singh & Alistarh (2020); Yu et al. (2022) do. Active set strategy. In an effort to further facilitate the efficiency of the IHT method, we propose using an active set strategy, which has been proven successful in various contexts such as (Nocedal & Wright, 1999;Friedman et al., 2010;Hazimeh & Mazumder, 2020). This strategy works by restricting the IHT updates to an active set (a relatively small subset of variables) and occasionally augmenting the active set with variables that violate certain optimality conditions. By implementing this strategy, the iteration complexity of the algorithm can be reduced to O(nk) in practice, resulting in an improvement, when k is smaller than p. The algorithm details can be found in Appendix A.3. DETERMINING A GOOD STEPSIZE Choosing an appropriate stepsize τ s is crucial for fast convergence of the IHT algorithm. To ensure convergence to a stationary solution, a common choice is to use a constant stepsize of τ s = 1/L (Bertsimas et al., 2016;Hazimeh & Mazumder, 2020), where L is the Lipschitz constant of the gradient of the objective function. This approach, while reliable, can lead to conservative updates and slow convergence-refer to Appendix A.1 for details. An alternative method for determining the stepsize is to use a backtracking line search, as proposed in Beck & Teboulle (2009). The method involves starting with a relatively large estimate of the stepsize and iteratively shrinking the step size until a sufficient decrease of the objective function is observed. However, this approach requires multiple evaluations of the objective function, which can be computationally expensive. Our novel scheme. We propose a novel line search method for determining the stepsize to improve the convergence speed of IHT. Specifically, we develop a method that (approximately) finds the stepsize that leads to the maximum decrease in the objective, i.e., we attempt to solve min τ s ≥0 g(τ s ) := Q P k w t − τ s ∇Q(w t ) .(10) For general objective functions, solving the line search problem (as in (10)) is challenging. However, in our problem, we observe and exploit an important structure: g(τ s ) is a piecewise quadratic function with respect to τ s . Thus, the optimal stepsize on each piece can be computed exactly, avoiding redundant computations (associated with finding a good stepsize) and resulting in more aggressive updates. In Appendix A.1, we present an algorithm that finds a good stepsize by exploiting this structure. Compared to standard line search, our algorithm is more efficient, as it requires fewer evaluations of the objective function and yields a stepsize that results in a steeper descent. ADDITIONAL TECHNIQUES FOR SCALABILITY While the IHT algorithm can be quite effective in identifying the appropriate support, its progress slows down considerably once the support is identified (Blumensath, 2012), resulting in slow convergence. We propose two techniques that refine the non-zero coefficients by solving subproblems to speedup the overall optimization algorithm: (i) Coordinate Descent (CD, Bertsekas, 1997;Nesterov, 2012) that updates each nonzero coordinate (with others fixed) as per a cyclic rule; (ii) Back solve based on Woodbury formula (Max, 1950) that calculates the optimal solution exactly on a restricted set of size k. We found both (i), (ii) to be important for improving the accuracy of the pruned network. Further details on the strategies (i), (ii) are in Appendix A.2 and A.4. Our single-stage algorithm CHITA glues together the different pieces discussed above into a coherent algorithm. It takes as input a low-rank matrix A, the initial weightw and the 0 -constraint k; and returns a pruned weight w that serves as a good solution to (8). A multi-stage procedure Our single-stage methods (cf Section 3.2) lead to highquality solutions for problem (8). Compared to existing methods, for a given sparsity level, our algorithms deliver a better objective value for problem (8)-for eg, see Figure 2(b). However, we note that the final performance (e.g., accuracy) of the pruned network depends heavily on the quality of the local quadratic approximation. This is particularly true when targeting high levels of sparsity (i.e., zeroing out many weights), as the objective function used in (8) may not accurately approximate the true loss function L. To this end, we propose a multi-stage procedure named CHITA++ that improves the approximation quality by iteratively updating and solving local quadratic models. We use a scheduler to gradually increase the sparsity constraint and take a small step towards higher sparsity in each stage to ensure the validity of the local quadratic approximation. Our multi-stage procedure leverages the efficiency of the single-stage approaches and can lead to pruned networks with improved accuracy by utilizing more accurate approximations of the true loss function. For example, our experiments show that the multi-stage procedure can prune ResNet20 to 90% sparsity in just a few minutes and increases test accuracy from 15% to 79% compared to the single-stage method. Algorithm 1 presents more details on CHITA++. Our proposed multi-stage method differs significantly from the gradual pruning approach described in Han et al. (2015). While both methods involve pruning steps, the gradual pruning approach also includes fine-tuning steps in which SGD is applied to further optimize the parameters for better results. However, these fine-tuning steps can be computationally expensive, usually taking days to run. In contrast, our proposed multi-stage method is a one-shot pruning method and only requires constructing and solving Problem (8) several times, resulting in an efficient and accurate solution. This solution can then be further fine-tuned using SGD or plugged into the gradual pruning framework, something we explore in Section 4.2. Algorithm 1 CHITA++: a multi-stage pruning procedure Require: Pre-trained weightsw, a target sparsity level τ , number of stages f . 1: Initialization: Construct a increasing sequence of sparsity parameters τ 1 , τ 2 , . . . , τ f = τ ; and set w 0 =w 2: for t = 1, 2, . . . , f do 3: At current solution w t−1 , calculate the gradient based on a batch of n data points and construct the matrix A given in (4). 4: Obtain a solution w t to problem (8) by invoking CHITA(A,w, k) withw = w t−1 and number of nonzeros k = (1 − τ t )p . 5: end for Experimental Results We compare our proposed framework with existing approaches, for both one-shot and gradual pruning. One shot pruning We start by comparing the performance of our methods: CHITA (single-stage) and CHITA++ (multi-stage) with several existing state-of-the-art one-shot pruning techniques on various pre-trained networks. We use the same experimental setup as in recent work (Yu et al., 2022;Singh & Alistarh, 2020). The existing pruning methods we con- Fisher sample size (n) Runtime (in seconds) Figure 1: Runtime comparison between our single-stage approaches and M-FAC (the fastest among the competitive methods) while pruning MLPNet and ResNet20 to 90% sparsity level (90% of the entries are zero). Note that Woodfisher and CBS are at least 1000 times slower than M-FAC. The error bar represents the standard error over four runs. CHITA here uses IHT to find a support and performs a back-solve on the found support. parameters) and ResNet50 (He et al., 2016, 22M parameters) trained on ImageNet (Deng et al., 2009). For further details on the choice of the Hessian approximation, we refer the reader to Appendix A.5. Detailed information on the experimental setup and reproducibility can be found in Appendix B.1.1. RUNTIME COMPARISON Recent works that use the empirical Fisher information matrix for pruning purposes (Singh & Alistarh, 2020;Yu et al., 2022) show that using more samples for Hessian and gradient approximation results in better accuracy. Our experiments also support this conclusion. However, most prior approaches become computationally prohibitive as sample size n increases. As an example, the Woodfisher and CBS algorithms require hours to prune a MobileNet when n is set to 1000, and their processing time increases at least linearly with n. In contrast, our method has been designed with efficiency in mind, and we have compared it to M-FAC, a well-optimized version of Woodfisher that is at least 1000 times faster. The results, as depicted in Figure 1, demonstrate a marked improvement in speed for our algorithm, with up to 20 times faster performance. ACCURACY OF THE PRUNED MODELS Comparison against state-of-the-art. Yu et al. (2022). We take five runs for our single-stage (CHITA) and multi-stage (CHITA++) approaches and report the mean and standard error (in the brackets). The best accuracy values (significant) are highlighted in bold. Here sparsity denotes the fraction of zero weights in convolutional and dense layers. One-shot pruning on ResNet50. We further compare our approach to competing methods on ResNet50, an even larger network where some pruning algorithms, like CBS, do not scale. In Figure 2, we evaluate the performance of our algorithm in comparison to M-FAC and Magnitude Pruning (MP) using two metrics: test accuracy and the final objective value of the 0 -constrained problem (8). As both M-FAC and our algorithm aim to minimize this objective, it can be used to judge the efficacy of our model in solving problem (8). As seen in the figure, our approach achieves a lower objective value, and in this case, it also results in a better test accuracy for the final pruned network. Sparsity schedule in multi-stage procedure. We study the effect of the sparsity schedule (i.e., choice of τ 1 ≤ τ 2 ≤ · · · ≤ τ f = τ in Algorithm 1) on the performance of CHITA++. We compare test accuracy of three different schedules: (i) exponential mesh, (ii) linear mesh, and (iii) constant mesh. For these schedules, f is set to be 15. For the first two meshes, τ 1 and τ 15 are fixed as 0.2 and 0.9, respectively. As shown in Figure 3, the exponential mesh computes τ 2 , . . . , τ 14 by drawing an exponential function, while the linear mesh adopts linear interpolation (with τ 1 and τ 15 as endpoints) to determine τ 2 , . . . , τ 14 and the constant mesh has τ 1 = τ 2 = · · · = τ 15 . Figure 4 plots the test accuracy of the three schedules over the number of stages. We observe that the linear mesh outperforms the exponential mesh in the first few iterations, but its performance drops dramatically in the last two iterations. The reason is that in high sparsity levels, even a slight increase in the sparsity rate leads to a large drop in accuracy. Taking small "stepsizes" in high sparsity levels allows the exponential mesh to fine-tune the weights in the last several stages and achieve good performance. Additional ablation studies. We perform additional ablation studies to further evaluate the performance of our method. These studies mainly focus on the effect of the ridge term (in Appendix B.2.1), and the effect of the firstorder term (in Appendix B.2.2). Performance on gradual pruning To compare our one-shot pruning algorithms against more unstructured pruning methods, we plug CHITA into a gradual pruning procedure (Gale et al., 2019), following the approach in Singh & Alistarh (2020). Specifically, we alternate between pruning steps where a sparse weight is computed and fine-tuning steps on the current support via Stochastic Gradient Descent (SGD). To obtain consistent results, we start from the same pre-trained weights used in Kusupati et al. (2020), and re-train for 100 epochs using SGD during fine-tuning steps, similarly to Kusupati et al. (2020); Singh & Alistarh (2020). We compare our approach against Incremental (Zhu & Gupta, 2018), STR (Kusupati et al., 2020), Global Magnitude (Singh & Alistarh, 2020), Wood-Fisher (Singh & Alistarh, 2020), GMP (Gale et al., 2019), Variational Dropout (Molchanov et al., 2017), RIGL (Evci et al., 2020), SNFS (Dettmers & Zettlemoyer, 2020) and DNW (Wortsman et al., 2019). Further details on training procedure can be found in Appendix B.1.2. MobileNetV1 . We start by pruning MobileNetV1 (4.2M parameters). As Table 2 demonstrates, CHITA results in significantly more accurate pruned models than previous state-of-the-art approaches at sparsities 75% and 89%, with only 6% accuracy loss compared to 11.29%, the previous best result when pruning to a sparsity of 89%. ResNet50. Similarly to MobileNetV1, CHITA improves test accuracy at sparsity levels 90%, 95%, and 98% com- pared to all other baselines, as Table 3 shows. This improvement becomes more noticeable as we increase the target sparsity, with CHITA producing a pruned model with 69.80% accuracy compared to 65.66%, the second-best performance, and previous state-of-the-art. Conclusion In this work we have presented an efficient network pruning framework CHITA , which is based on a novel, hessian-free 0 -constrained regression formulation and combinatorial optimization techniques. Our single-stage methods demonstrate comparable results to existing methods while achieving a significant improvement in runtime and reducing memory usage. Furthermore, by building upon the single-stage methods, our multi-stage approach is capable of achieving even further improvements in model accuracy. Additionally, we have demonstrated that incorporating our pruning methods into existing gradual pruning frameworks results in sparse networks with state-of-the-art accuracy. Details on the pruning step In all our gradual pruning experiments, we begin by pruning the networks to a sparsity level of 50% and proceed with six additional pruning steps to reach the target sparsity. We follow the polynomial schedule Details on the fine-tuning process We incorporate SGD with a momentum of 0.9 for 12 epochs between pruning steps. Once the networks have been pruned to the target sparsity, we continue to fine-tune the networks for an additional 28 epochs using SGD with a momentum of 0.9 (total of 100 epochs). We utilize distributed training and set the batch size to 256 per GPU during the SGD training process. We implement a cosine-based learning rate schedule similar to the one used in the STR method (Kusupati et al., 2020). Specifically, the learning rate for each epoch e between two pruning steps that occur at epochs e 1 and e 2 is defined as: In this section, we study the effect of the ridge term on the performance of our algorithm, specifically focusing on the test accuracy over the course of the algorithm. As depicted in Figure 7(a), when no ridge term is applied, the test accuracy increases initially but then experiences a sharp decline as the algorithm progresses. The underlying cause is revealed in Figure 7(b), which illustrates that without the ridge term, the distance between the original weightw and the pruned weight w keeps increasing as the algorithm progresses. As this distance increases, the local quadratic model used in (8) becomes less accurate, leading to poor test performance. One solution to this problem would be to employ early stopping to prevent the distance from growing too large. However, determining the optimal stopping point can be challenging. Practically, we instead add the ridge term nλ 2 w −w 2 to the objective function, effectively regularizing the model and maintaining its accuracy. As shown in Figure 7 Accuracy (c) Test accuracy with respect to mini-batch size, with and without the first-order term Figure 8: Effect of using a scaled first-order term on pruning MLPNet with our proposed multi-stage solver CHITA++ to a sparsity level of 0.95. All results are averaged over 5 runs. Mini-batches are used for gradient evaluations in practice instead of evaluating gradient ∇ i (w) n i=1 on n training samples. This means that each ∇ i (w) is replaced by the average gradient of a mini-batch of size m. In this scenario, the empirical Fisher matrix H is not an accurate representation of the true Hessian matrix. However, it still provides a reasonable approximation but with a scaling factor (Thomas et al., 2020;Singh & Alistarh, 2020). In scale-independent applications, e.g., minimizing L(w) ≈ L(w) + 1 2 (w −w) H(w −w) as considered in Singh & Alistarh (2020) and Yu et al. (2022), the empirical Fisher matrix H still effectively approximates the true Hessian. However, this approximation is no longer accurate in our framework, which includes a first-order term. This is supported by the results shown in Figure 8(a), where our framework with a correctly scaled term (α = 1/m) demonstrates significantly improved performance compared to one without a scaling factor (α = 1), especially when the fisher batch size m is much greater than one. To address this issue, we propose a local quadratic approximation with a scaled first-order term that reads Q(w) = L(w) + αg (w −w) + 1 2 (w −w) H(w −w).(19) Our proposed 0 -constrained framework (8) can be generalized to solving this problem by setting y = Aw − αe, where e is a vector of ones. We propose an accurate estimation of α as α = Trace(H) Trace(∇ 2 L(w)) . However, the computation cost of Trace(∇ 2 L(w)) is not negligible, even using accelerated methods as proposed in Yao et al. (2020). Through experimentation, as shown in Figure 8(b), we have discovered that the estimated value of α as given by (20) is relatively close to 1/m. Therefore, we have chosen to use 1/m as a heuristic scaling factor in our experiments, as it provides a good approximation while reducing the computational cost. In Figure 8(c), we further illustrate the benefits of using large mini-batches and a scaled first-order term. As the fisher batch size m increases, we can construct more precise local quadratic approximations through better estimation of H and g, resulting in improved test accuracy. Additionally, when m is greater than 1, using a correctly scaled first-order term provides an additional performance boost. sider include MP (Magnitude Pruning,Mozer & Smolensky, 1989), WF(WoodFisher, Singh & Alistarh, 2020), CBS (Combinatorial Brain Surgeon,Yu et al., 2022) and M-FAC (Matrix-Free Approximate Curvature,Frantar et al., 2021). The pre-trained networks we use are MLPNet (30K parameters) trained onMNIST (LeCun et al., 1998), ResNet20(He et al., 2016, 200k parameters) trained on CIFAR10(Krizhevsky et al., 2009), and MobileNet (4 The objective value in (8) for pruning ResNet50. Figure 2 : 2One-shot pruning on ResNet50 (Dense accuracy is 77.01%). The error bars are over four runs. For a fair comparison, both CHITA and M-FAC use the same hyperparameters and the same training samples for Hessian and gradient approximation. Figure 3 : 3Three different sparsity schedules: exponential, linear, and constant schedules. Figure 4 : 4Comparison of test accuracy using CHITA++ with 15 stages, pruning a ResNet20 model to a 90% sparsity rate, under different sparsity schedules. Text around the point indicates the current sparsity level of the point. Figure 6 : 6Learning rate schedule used in our gradual pruning experiments on MobileNetV1, with a target sparsity level of 0.89. Text around a point indicates the sparsity of the network at the current epoch. introduced by Zhu & Gupta (2018) as the pruning schedule and use the CHITA-BSO algorithm with block approximation (Algorithm 6) as the pruning method. The block size is set to B size = 2000 for MobileNetV1 and B size = 500 for ResNet50. Figure 6 6illustrates how such a learning rate schedule decays between pruning steps. B.2. Implementation details and ablation studies B.2.1. EFFECT OF THE RIDGE TERM Test accuracy with respect to the number of iterations in a single run of CHITA-CD (Algorithm 4). The sparsity level is 0.98. iterations (b) The distance ||w −w|| with respect to the number of iterations in a single run of CHITA-CD (Algorithm 4). The sparsity level is 0.98. Test accuracy at different sparsity levels, with best ridge term selected in range [10 −5 , 1]. The result is averaged over 4 runs. Figure 7 : 7Effect of the ridge term on the test accuracy and pruned weights of MLPNet. (c), utilizing a well-tuned ridge term results in an increase of approximately 3% on MLPNet. Table 1 1compares the test accuracy of MLPNet, ResNet20 and MobileNetV1 pruned to different sparsity levels. Our single-stage method achieves comparable results to other state-of-the-art approaches with much less time consumption. The multi-stage method (CHITA++) outperforms other methods by a large margin, especially with a high sparsity rate.Network Sparsity MP WF CBS CHITA CHITA++ MLPNet on MNIST (93.97%) 0.5 93.93 94.02 93.96 93.97 (±0.03) 95.97 (±0.05) 0.6 93.78 93.82 93.96 93.94 (±0.02) 95.93 (±0.04) 0.7 93.62 93.77 93.98 93.80 (±0.01) 95.89 (±0.06) 0.8 92.89 93.57 93.90 93.59 (±0.03) 95.80 (±0.03) 0.9 90.30 91.69 93.14 92.46 (±0.04) 95.55 (±0.03) 0.95 83.64 85.54 88.92 88.09 (±0.24) 94.70 (±0.06) 0.98 32.25 38.26 55.45 46.25 (±0.85) 90.73 (±0.11) ResNet20 on CIFAR10 (91.36%) 0.3 90.77 91.37 91.35 91.37 (±0.04) 91.25 (±0.08) 0.4 89.98 91.15 91.21 91.19 (±0.05) 91.20 (±0.05) 0.5 88.44 90.23 90.58 90.60 (±0.07) 91.04 (±0.09) 0.6 85.24 87.96 88.88 89.22 (±0.19) 90.78 (±0.12) 0.7 78.79 81.05 81.84 84.12 (±0.38) 90.38 (±0.10) 0.8 54.01 62.63 51.28 57.90 (±1.04) 88.72 (±0.17) 0.9 11.79 11.49 13.68 15.60 (±1.79) 79.32 (±1.19) MobileNetV1 on ImageNet (71.95%) 0.3 71.60 71.88 71.88 71.87 (±0.01) 71.86 (±0.02) 0.4 69.16 71.15 71.45 71.50 (±0.02) 71.61 (±0.02) 0.5 62.61 68.91 70.21 70.42 (±0.02) 70.99 (±0.04) 0.6 41.94 60.90 66.37 67.30 (±0.03) 69.54 (±0.01) 0.7 6.78 29.36 55.11 59.40 (±0.09) 66.42 (±0.03) 0.8 0.11 0.24 16.38 29.78 (±0.18) 47.45 (±0.25) Table 1 : 1Thepruning performance (model accuracy) of various methods on MLPNet, ResNet20, MobileNetV1. As to the perfor- mance of MP, WF, and CBS, we adopt the results reported in Table 2 : 2Resultsof gradually pruning MobilenetV1 in 75% and 89% sparsity regimes, comparing CHITA to other baselines (Dense accuracy: 72.00%). We also include the relative drop in accuracy to account for different methods starting from different dense weights. CHITA numbers are averaged across two runs. Numbers for other baselines are taken from Singh & Alistarh (2020). Table 3 : 3Resultsof gradually pruning a ResNet50 network in the 90%, 95%, and 98% sparsity regimes, comparing CHITA to other state-of-the-art methods (Dense accuracy: 77.01%). We also include the relative drop in accuracy to account for different methods starting from different dense weights. CHITA numbers are averaged across two runs. Numbers for other baselines are taken from Singh & Alistarh (2020). Table 4 : 4Comparisons of the Fisher sample size n and the mini-batch size m used in Hessian and gradient approximation on MLPNet, ResNet20, MobileNetV1 and ResNet50. B.1.2. GRADUAL PRUNING All experiments were carried on a computing cluster. Experiments for MobileNetV1 were run on an Intel Xeon Platinum 6248 machine with 30 CPUs and 2 GPUs; experiments for ResNet50 were run on five Intel Xeon Platinum 6248 machines with 200 CPUs and 10 GPUs. For reference, CBS and Woodfisher would run out of memory in similar circumstances. Here 0 norm w 0 denotes the number of nonzero in the vector w. Another way is to introduce a multi-stage procedure, as explained in Section 3.3. 4 It appears to be used implicitly though to obtain an invertible Fisher matrix which is achieved by adding a small diagonal λ0I to the Fisher matrix. AcknowledgementsThis research is supported in part by grants from the Office of Naval Research (N000142112841 and N000142212665), and Google. We thank Shibal Ibrahim for helpful discussions. We also thank Thiago Serra and Yu Xin for sharing with us code from their CBS paper(Yu et al., 2022).A. Algorithmic details A.1. IHT with aggressive stepsize Challenges of stepsize choice Choosing an appropriate stepsize τ s is crucial to achieving a faster convergence rate. In theory, setting τ s as the constant 1/L in (9) is a common choice in the literature to ensure the convergence to a stationary solution(Bertsimas et al., 2016;Hazimeh & Mazumder, 2020), where L is the Lipschitz constant of the gradient of Q(w). i.e., ∇Q(α) − ∇Q(β) ≤ L α − β for all α, β ∈ R p . Since Q is a quadratic objective, this quantity L is given by L = nλ + A 2 2 , where A 2 is the maximum singular value of A. This quantity could be substantial when p is large, leading to very conservative updates, sometimes negligible. Moreover, the computation of L itself may involve a few power iterations or a randomized SVD algorithm, which could be as costly as several IHT updates. An alternative method for determining the stepsize is to use a backtracking line search, as proposed inBeck & Teboulle (2009). The method involves starting with a relatively large estimate of the stepsize and iteratively shrinking the step size until a sufficient decrease of the objective function is observed. However, this method requires multiple evaluations of the quadratic objective, which can be computationally expensive.Our novel scheme We propose a new line search strategy to efficiently determine an aggressive stepsize to address the issue of slow updates in the IHT algorithm. Note that the problem of finding the best stepsize can be written as the following one-dimensional problem minSince P k is a piecewise function, g(τ s ) is a univariate piecewise quadratic function which is generally non-convex, as illustrated inFigure 5. Our key observation is that the first breaking point of g(τ s ) and the optimal stepsize on the first piece can be computed easily. More specifically, denote by τ s c the first breaking point of g(τ s ). Namely, τ s c is the largest value of τ such that the hard thresholding based on τ s ∈ [0, τ ] does not change the support, i.e.Let us denote S := supp(w). In the case where |S| = k, τ s c can be computed in closed form usingAs previously established, over the interval τ s ∈ [0, τ s c ], the function g(τ s ) = Q (w t − τ s ∇Q(w t )) is a quadratic function. Let us denote by τ s m the optimal value of τ s that minimizes g(τ s ) within the interval [0, τ s c ]. It is straightforward to see that τ s m can be computed in closed form with the same computational cost as a single evaluation of the quadratic objective function.If τ s m < τ s c , then the optimal value of τ s lies within the first quadratic piece. Practically, we have found that in this case τ s m is often also the global minimum of g(τ s ). Therefore, we can confidently take the stepsize as τ s = τ s m . Otherwise if τ s m = τ s c , then we know that g(τ s ) is monotonically decreasing on the interval [0, τ s c ]. This implies that g(τ s ) would likely continue to decrease as τ s becomes larger than τ s c . As a result, we perform a line search by incrementally increasing the value of τ s by a factor of γ > 1 starting from τ s c to approximate the stepsize that results in the steepest descent. The above procedure is summarized in Algorithm 2.Our proposed scheme offers a significant improvement in efficiency compared to standard backtracking line search by eliminating redundant steps on the quadratic piece of g(τ s ) over [0, τ s c ]. Additionally, our method directly computes the optimal stepsize on the first piece, which in many cases, results in a greater reduction in the objective function when compared to the standard backtracking line search.Finally, we note that during line search, it is always possible to find the piece of the quadratic function to which the current stepsize τ s belongs, say [τ s l , τ s u ], and calculate the optimal stepsize over that piece with small extra costs to further improve the line search. But we find it unnecessary in practice as the line search procedure usually terminates in a few steps.A.2. Cyclic coordinate descentAlthough IHT does well in identifying and updating the support, we observe that it makes slow progress in decreasing the objective in experiments. To address this issue, we use cyclic coordinate descent (CD,Bertsekas, 1997;Nesterov, 2012)Algorithm 2 A novel scheme to determine the stepsize τ s Require:as the optimal stepsize on the first piece of gτ s ← τ s m . 5: else 6: g best ← g(τ s c ), and τ s ← τ s c .7:while g best > g(γτ s ) do 8:g best ← g(γτ s ), and τ s ← γτ s . 9:end while 10: end if with full minimization in every nonzero coordinate to refine the solution on the support. CD-type methods are widely used for solving huge-scale optimization problems in statistical learning, especially those problems with sparsity structure, due to their inexpensive iteration updates and capability of exploiting problem structure, such as Lasso(Friedman et al., 2010)and L 0 L 2 -penalized regression(Hazimeh & Mazumder, 2020).Our cyclic CD updates each nonzero coordinate (with others fixed) as per a cyclic rule, and skips any coordinate with zero value to avoid violating the 0 constraint. With a feasible initialization w t and a coordinate i in the support of w t , w t+1 i is obtained by optimizing the i-th coordinate (with others fixed) through:Calculating CDUpdate(w t , i) requires the minimization of a univariate quadratic function with time cost O(n).Cyclic CD enjoys a fast convergence rate(Bertsekas, 1997;Nesterov, 2012). However, the quality of the resulting solution is limited and depends heavily on the initial solution, as CD cannot modify the support of a solution. In practice, we adopt a hybrid updating rule that combines IHT and cyclic CD for better performance in terms of both quality and efficiency. In each iteration, we perform several rounds of IHT updates and then apply cyclic CD to refine the solution on the support. This approach is summarized in Algorithm 3.Algorithm 3 IHT with CD: IHT-CD(w 0 , k, t HT , t CD )Require: w 0 , k, t HT , t CD 1: for t = 0, 1, . . . do 2:w ← w t 3:for t = 1, . . . , t HT do 4:Compute stepsize τ s using Algorithm 2. w t+1 ← w 13: end forA.3. Active set updatesThe active set strategy is a popular approach that has been shown to be effective in reducing complexity in various contexts(Nocedal & Wright, 1999;Friedman et al., 2010;Hazimeh & Mazumder, 2020). In our problem setting, the active set strategy works by starting with an initial active set (of length equal to a multiple of the required number of nonzeros k, e.g., 2k) that is selected based on the magnitude of the initial solution. In each iteration, we restrict the updates of Algorithm 3 to the current active set A. After convergence, we perform IHT updates on the full vector to find a better solution w with supp(w) ⊆ A. The algorithm terminates if such w does not exist; otherwise, we update A ← A ∪ supp(w), and the process is repeated. Algorithm 4 gives a detailed illustration of the active set method, with Algorithm 3 as the inner solver (potentially the inner solver can be replaced with any other solver, such as Algorithm 5 in the next section). In our experiments, this strategy works well on medium-sized problems (p ∼ 10 5 ) and sparse problems (k p).Algorithm 4 Active set with IHT: CHITA-CD(w 0 , k, t HT , t CD , A 0 )Require: w 0 , k, t HT , t CD , and an initial active set A 0 1: for t = 0, 1, . . . do 2:Find τ s via line search such that w t+1 ← HT(w t+1/2 , k, τ s ) satisfies (i) Q(w t+1 ) < Q(w t+1/2 ) (ii) supp(w t+1 ) ⊆ A t4:if such τ s does not exist then 5: break 6: else 7:end if 9: end forA.4. Backsolve via Woodbury formulaAs the dimensionality of the problem increases, CHITA-CD becomes increasingly computationally expensive. To address this issue, we propose a backsolve approach that further reduces the complexity while maintaining a slightly suboptimal solution. The backsolve approach calculates the optimal solution exactly on a restricted set. We first apply IHT updates a few times to obtain an initial feasible solution w, and then restrict the problem to the set S := supp(w). Under the restriction, problem (8) reduces to a quadratic problem without 0 constraint and its minimizer readswhere A S ∈ R n×k denotes a submatrix of A with columns only in S. By exploiting the low-rank structure of A S A S and utilizing Woodbury formula(Max, 1950), (15) can be computed in O(n 2 k) operations. Specifically, one can compute(15)using matrix-vector multiplications involving only A S (or its transpose) and one matrix-matrix multiplication viaThe backsolve method is stated in Algorithm 5.Algorithm 5 Backsolve: CHITA-BSO(w 0 , k, t HT )Require: w 0 , k, t HT . 1: Construct an initial active set A 0 2: w ← CHITA-CD(w 0 , k, t HT , 0,We note that prior works(Singh & Alistarh, 2020;Yu et al., 2022;Hassibi & Stork, 1992) also use the formula, but they do not exploit the problem structure to reduce the runtime and memory consumption.A.5. Stratified block-wise approximationWe describe in this subsection a block approximation strategy whereby we only consider limited-size blocks on the diagonal of the Hessian matrix and ignore off-diagonal parts. Given a disjoint partition {B i } c i=1 of {1, 2, . . . , p} and assume blocks of size B 1 × B 1 , . . . , B c × B c along the diagonal, problem (8) can then be decomposed into the following subproblemswhere b i = A BiwBi − e and c i=1 k i = k determines the sparsity in each block. The difference in the selection of {k i } c i=1 will greatly affect the quality of the solution. We observe in experiments that the best selection strategy is to first apply magnitude pruning (or other efficient heuristics) to get a feasible solution w, and then set k i = | supp(w) ∩ B i |, ∀1 ≤ i ≤ c. Algorithm 6 states the block-wise approximation algorithm, with Algorithm 5 as the inner solver for each subproblem.In our experiment, we adopt the same strategy to employ the block-wise approximation as in the prior work(Yu et al., 2022;Singh & Alistarh, 2020). We regard the set of variables that corresponds to a single layer in the network as a block and then subdivide these blocks uniformly such that the size of each block does not exceed a given parameter B size = 10 4 .We clarify that the introduction of block-wise approximation is for the sake of solution quality (accuracy of pruned network) rather than algorithmic efficiency. This differs from previous works(Singh & Alistarh, 2020;Yu et al., 2022). In fact, solving (17) for i = 1, . . . , c requires operations of the same order as solving(8)directly. On the other side, we observe in our experiments that adopting block-wise approximation will dramatically increase the network MobileNet's accuracy (from 0.2% to near 30%, given a sparsity level of 0.8).Algorithm 6 CHIAT-BSO with block approximation Require: w 0 , k, t HT , a disjoint partition {B i } c i=1 of {1, 2, . . . , p}. 1: Obtain a feasible solution via magnitude pruning w ← P k (w 0 ). Algorithmic setting We utilize the CHITA algorithm with active set strategy and coordinate descent as acceleration techniques, as outlined in Algorithm 4, to prune MLPNet and ResNet20 networks. Additionally, we use Algorithm 4 as the inner solver of our proposed multi-stage approach, CHITA++, for these networks. As to MobileNetV1 and ResNet50, we utilize CHITA-BSO with block approximation (Algorithm 6) for solving single-stage problems. We employ the exact block-wise approximation strategy as applied in previous work(Yu et al., 2022;Singh & Alistarh, 2020), see Section A.5 for details. We also use Algorithm 6 as the inner solver of our proposed multi-stage procedure CHITA++ for these networks. In our experiments, we set the number of stages in CHITA++ to 15 for MLPNet and ResNet20 and 100 for MobileNetV1. CHITA++ results on MobileNetV1 are averaged over 4 runs.Hyper-parameters For each network and each sparsity level, we run our proposed methods CHITA (single-stage) and CHITA++ (multi-stage) with ridge value λ ranging from [10 −5 , 10 3 ] and the number of IHT iterations (if Algorithm 4 is applied) ranging from[5,500]. In single-stage settings, we consider solving problem (8) with/without the first-order term. We report inTable 1the best model accuracy over all possible hyper-parameter combinations.Hyper-parameters for ResNet50 experiments To obtain consistent results, we run CHITA and M-FAC with the same set of hyperparameters (λ = 10 −5 , n = 500, B size = 10 4 ) and on the same training samples for Hessian and gradient approximation. 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[]
[ "Cached Long Short-Term Memory Neural Networks for Document-Level Sentiment Classification", "Cached Long Short-Term Memory Neural Networks for Document-Level Sentiment Classification" ]
[ "Jiacheng Xu \nSchool of Computer Science\nFudan University\n\n\n825 Zhangheng RoadShanghaiChina † ‡\n", "Danlu Chen [email protected] \nShanghai Key Laboratory of Intelligent Information Processing\nFudan University\n\n", "Xipeng Qiu [email protected] \nShanghai Key Laboratory of Intelligent Information Processing\nFudan University\n\n", "Xuanjing Huang [email protected] \nShanghai Key Laboratory of Intelligent Information Processing\nFudan University\n\n", "\nSoftware School\nFudan University\n\n" ]
[ "School of Computer Science\nFudan University\n", "825 Zhangheng RoadShanghaiChina † ‡", "Shanghai Key Laboratory of Intelligent Information Processing\nFudan University\n", "Shanghai Key Laboratory of Intelligent Information Processing\nFudan University\n", "Shanghai Key Laboratory of Intelligent Information Processing\nFudan University\n", "Software School\nFudan University\n" ]
[ "Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing" ]
Recently, neural networks have achieved great success on sentiment classification due to their ability to alleviate feature engineering. However, one of the remaining challenges is to model long texts in document-level sentiment classification under a recurrent architecture because of the deficiency of the memory unit. To address this problem, we present a Cached Long Short-Term Memory neural networks (CLSTM) to capture the overall semantic information in long texts. CLSTM introduces a cache mechanism, which divides memory into several groups with different forgetting rates and thus enables the network to keep sentiment information better within a recurrent unit. The proposed CLSTM outperforms the state-of-the-art models on three publicly available document-level sentiment analysis datasets.
10.18653/v1/d16-1172
[ "https://www.aclweb.org/anthology/D16-1172.pdf" ]
15,124,020
1610.04989
9904a69eab0792859108eec6b0578d11264b8e83
Cached Long Short-Term Memory Neural Networks for Document-Level Sentiment Classification Association for Computational LinguisticsCopyright Association for Computational LinguisticsNovember 1-5, 2016. 2016 Jiacheng Xu School of Computer Science Fudan University 825 Zhangheng RoadShanghaiChina † ‡ Danlu Chen [email protected] Shanghai Key Laboratory of Intelligent Information Processing Fudan University Xipeng Qiu [email protected] Shanghai Key Laboratory of Intelligent Information Processing Fudan University Xuanjing Huang [email protected] Shanghai Key Laboratory of Intelligent Information Processing Fudan University Software School Fudan University Cached Long Short-Term Memory Neural Networks for Document-Level Sentiment Classification Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing the 2016 Conference on Empirical Methods in Natural Language ProcessingAustin, TexasAssociation for Computational LinguisticsNovember 1-5, 2016. 2016 Recently, neural networks have achieved great success on sentiment classification due to their ability to alleviate feature engineering. However, one of the remaining challenges is to model long texts in document-level sentiment classification under a recurrent architecture because of the deficiency of the memory unit. To address this problem, we present a Cached Long Short-Term Memory neural networks (CLSTM) to capture the overall semantic information in long texts. CLSTM introduces a cache mechanism, which divides memory into several groups with different forgetting rates and thus enables the network to keep sentiment information better within a recurrent unit. The proposed CLSTM outperforms the state-of-the-art models on three publicly available document-level sentiment analysis datasets. Introduction Sentiment classification is one of the most widely used natural language processing techniques in many areas, such as E-commerce websites, online social networks, political orientation analyses (Wilson et al., 2009;O'Connor et al., 2010), etc. Recently, deep learning approaches (Socher et al., 2013;Kim, 2014;Liu et al., 2016) have gained encouraging results on sentiment classification, which frees researchers from handcrafted feature engineering. Among these methods, Recurrent Neural Networks (RNNs) are one of the most * Corresponding author. prevalent architectures because of the ability to handle variable-length texts. Sentence-or paragraph-level sentiment analysis expects the model to extract features from limited source of information, while document-level sentiment analysis demands more on selecting and storing global sentiment message from long texts with noises and redundant local pattern. Simple RNNs are not powerful enough to handle the overflow and to pick up key sentiment messages from relatively far time-steps . Efforts have been made to solve such a scalability problem on long texts by extracting semantic information hierarchically (Tang et al., 2015a;Tai et al., 2015), which first obtain sentence representations and then combine them to generate high-level document embeddings. However, some of these solutions either rely on explicit a priori structural assumptions or discard the order information within a sentence, which are vulnerable to sudden change or twists in texts especially a long-range one (Mc-Donald et al., 2007;Mikolov et al., 2013). Recurrent models match people's intuition of reading word by word and are capable to model the intrinsic relations between sentences. By keeping the word order, RNNs could extract the sentence representation implicitly and meanwhile analyze the semantic meaning of a whole document without any explicit boundary. Partially inspired by neural structure of human brain and computer system architecture, we present the Cached Long Short-Term Memory neural networks (CLSTM) to capture the long-range sentiment information. In the dual store memory model proposed by Atkinson and Shiffrin (1968), memories can reside in the short-term "buffer" for a limited time while they are simultaneously strengthening their associations in long-term memory. Accordingly, CLSTM equips a standard LSTM with a similar cache mechanism, whose internal memory is divided into several groups with different forgetting rates. A group with high forgetting rate plays a role as a cache in our model, bridging and transiting the information to groups with relatively lower forgetting rates. With different forgetting rates, CLSTM learns to capture, remember and forget semantics information through a very long distance. Our main contributions are as follows: • We introduce a cache mechanism to diversify the internal memory into several distinct groups with different memory cycles by squashing their forgetting rates. As a result, our model can capture the local and global emotional information, thereby better summarizing and analyzing sentiment on long texts in an RNN fashion. • Benefiting from long-term memory unit with a low forgetting rate, we could keep the gradient stable in the long back-propagation process. Hence, our model could converge faster than a standard LSTM. • Our model outperforms state-of-the-art methods by a large margin on three document-level datasets (Yelp 2013, Yelp 2014 and IMDB). It worth noticing that some of the previous methods have utilized extra user and product information. Related Work In this section, we briefly introduce related work in two areas: First, we discuss the existing documentlevel sentiment classification approaches; Second, we discuss some variants of LSTM which address the problem on storing the long-term information. Document-level Sentiment Classification Document-level sentiment classification is a sticky task in sentiment analysis (Pang and Lee, 2008), which is to infer the sentiment polarity or intensity of a whole document. The most challenging part is that not every part of the document is equally informative for inferring the sentiment of the whole document (Pang and Lee, 2004;Yessenalina et al., 2010). Various methods have been investigated and explored over years (Wilson et al., 2005;Pang and Lee, 2008;Pak and Paroubek, 2010;Yessenalina et al., 2010;Moraes et al., 2013). Most of these methods depend on traditional machine learning algorithms, and are in need of effective handcrafted features. Recently, neural network based methods are prevalent due to their ability of learning discriminative features from data (Socher et al., 2013;Le and Mikolov, 2014;Tang et al., 2015a). and Tai et al. (2015) integrate a tree-structured model into LSTM for better semantic composition; Bhatia et al. (2015) enhances document-level sentiment analysis by using extra discourse paring results. Most of these models work well on sentence-level or paragraph-level sentiment classification. When it comes to the document-level sentiment classification, a bottom-up hierarchical strategy is often adopted to alleviate the model complexity (Denil et al., 2014;Tang et al., 2015b;. Memory Augmented Recurrent Models Although it is widely accepted that LSTM has more long-lasting memory units than RNNs, it still suffers from "forgetting" information which is too far away from the current point (Le et al., 2015;Karpathy et al., 2015). Such a scalability problem of LSTMs is crucial to extend some previous sentence-level work to document-level sentiment analysis. Various models have been proposed to increase the ability of LSTMs to store long-range information (Le et al., 2015;Salehinejad, 2016) and two kinds of approaches gain attraction. One is to augment LSTM with an external memory (Sukhbaatar et al., 2015;Monz, 2016), but they are of poor performance on time because of the huge external memory matrix. Unlike these methods, we fully exploit the potential of internal memory of LSTM by adjusting its forgetting rates. The other one tries to use multiple time-scales to distinguish different states (El Hihi and Bengio, 1995;Koutnik et al., 2014;Liu et al., 2015). They partition the hidden states into several groups and each group is activated and updated at different frequencies (e.g. one group updates every 2 time-step, the other updates every 4 time-step). In these methods, different memory groups are not fully interconnected, and the information is transmitted from faster groups to slower ones, or vice versa. However, the memory of slower groups are not updated at every step, which may lead to sentiment information loss and semantic inconsistency. In our proposed CLSTM, we assign different forgetting rates to memory groups. This novel strategy enable each memory group to be updated at every time-step, and every bit of the long-term and shortterm memories in previous time-step to be taken into account when updating. Long Short-Term Memory Networks Long short-term memory network (LSTM) (Hochreiter and Schmidhuber, 1997) is a typical recurrent neural network, which alleviates the problem of gradient diffusion and explosion. LSTM can capture the long dependencies in a sequence by introducing a memory unit and a gate mechanism which aims to decide how to utilize and update the information kept in memory cell. Formally, the update of each LSTM component can be formalized as: i (t) = σ(W i x (t) + U i h (t−1) ), (1) f (t) = σ(W f x (t) + U f h (t−1) ),(2)o (t) = σ(W o x (t) + U o h (t−1) ),(3)c (t) = tanh(W c x (t) + U c h (t−1) ),(4)c (t) = f (t) c (t−1) + i (t) c (t) ,(5)h (t) = o (t) tanh(c (t) ),(6) where σ is the logistic sigmoid function. Operator is the element-wise multiplication operation. i (t) , f (t) , o (t) and c (t) are the input gate, forget gate, output gate, and memory cell activation vector at time-step t respectively, all of which have the same size as the hidden vector h (t) ∈ R H . W i , W f , W o ∈ R H×d and U i , U f , U o ∈ R H×H are train- able parameters. Here, H and d are the dimensionality of hidden layer and input respectively. Cached Long Short-Term Memory Neural Network LSTM is supposed to capture the long-term and short-term dependencies simultaneously, but when dealing with considerably long texts, LSTM also fails on capturing and understanding significant sentiment message (Le et al., 2015). Specifically, the error signal would nevertheless suffer from gradient vanishing in modeling long texts with hundreds of words and thus the network is difficult to train. Since the standard LSTM inevitably loses valuable features, we propose a cached long short-term memory neural networks (CLSTM) to capture information in a longer steps by introducing a cache mechanism. Moreover, in order to better control and balance the historical message and the incoming information, we adopt one particular variant of LSTM proposed by Greff et al. (2015), the Coupled Input and Forget Gate LSTM (CIFG-LSTM). Coupled Input and Forget Gate LSTM Previous studies show that the merged version gives performance comparable to a standard LSTM on language modeling and classification tasks because using the input gate and forget gate simultaneously incurs redundant information (Chung et al., 2014;Greff et al., 2015). In the CIFG-LSTM, the input gate and forget gate are coupled as one uniform gate, that is, let i (t) = 1 − f (t) . We use f (t) to denote the coupled gate. Formally, we will replace Eq. 5 as below: Figure 1 gives an illustrative comparison of a standard LSTM and the CIFG-LSTM. c (t) = f (t) c (t−1) + (1 − f (t) ) c (t) (7) Cached LSTM Cached long short-term memory neural networks (CLSTM) aims at capturing the long-range information by a cache mechanism, which divides memory into several groups, and different forgetting rates, regarded as filters, are assigned to different groups. Different groups capture different-scale dependencies by squashing the scales of forgetting rates. The groups with high forgetting rates are short-term memories, while the groups with low forgetting rates are long-term memories. Specially, we divide the memory cells into K groups {G 1 , · · · , G K }. Each group includes a internal memory c k , output gate o k and forgetting rate r k . The forgetting rate of different groups are squashed in distinct ranges. We modify the update of a LSTM as follows. r (t) k = ψ k   σ(W k r x (t) + K j=1 U j→k f h (t−1) j )   ,(8)o (t) k = σ(W k o x (t) + K j=1 U j→k o h (t−1) j ),(9)c (t) k = tanh(W k c x (t) + K j=1 U j→k c h (t−1) j ), (10) c (t) k = (1 − r (t) k ) c (t−1) k + (r (t) k ) c (t) k , (11) h (t) k = o (t) k tanh(c (t) k ),(12) where r (t) k represents forgetting rate of the k-th memory group at step t; ψ k is a squash function, which constrains the value of forgetting rate r k within a range. To better distinguish the different role of each group, its forgetting rate is squashed into a distinct area. The squash function ψ k (z) could be formalized as: r k = ψ k (z) = 1 K · z + k − 1 K ,(13) where z ∈ (0, 1) is computed by logistic sigmoid function. Therefore, r k can constrain the forgetting rate in the range of ( k−1 K , k K ). Intuitively, if a forgetting rate r k approaches to 0, the group k tends to be the long-term memory; if a r k approaches to 1, the group k tends to be the shortterm memory. Therefore, group G 1 is the slowest, while group G K is the fastest one. The faster groups are supposed to play a role as a cache, transiting information from faster groups to slower groups. Bidirectional CLSTM Graves and Schmidhuber (2005) proposed a Bidirectional LSTM (B-LSTM) model, which utilizes additional backward information and thus enhances the memory capability. We also employ the bi-directional mechanism on CLSTM and words in a text will receive information from both sides of the context. Formally, the outputs of forward LSTM for the k-th group is [ − → h (1) k , − → h (2) k , . . . , − → h (T ) k ]. The outputs of backward LSTM for the k-th group is [ ← − h (1) k , ← − h (2) k , . . . , ← − h (T ) k ] . Hence, we encode each word w t in a given text w 1:T as h (t) k : h (t) k = − → h (t) k ⊕ ← − h (t) k ,(14) where the ⊕ indicates concatenation operation. Task-specific Output Layer for Document-level Sentiment Classification With the capability of modeling long text, we can use our proposed model to analyze sentiment in a document. Figure 2 gives an overview of the architecture. Since the first group, the slowest group, is supposed to keep the long-term information and can better represent a whole document, we only utilize the Table 1: Statistics of the three datasets used in this paper. The rating scale (Class) of Yelp2013 and Yelp2014 range from 1 to 5 and that of IMDB ranges from 1 to 10. Words/Doc is the average length of a sample and Sents/Doc is the average number of sentences in a document. final state of this group to represent a document. As for the B-CLSTM, we concatenate the state of the first group in the forward LSTM at T -th time-step and the first group in the backward LSTM at first time-step. Then, a fully connected layer followed by a softmax function is used to predict the probability distribution over classes for a given input. Formally, the probability distribution p is: p = softmax(W p × z + b p ),(15) where W p and b p are model's parameters. Here z is − → h (T ) 1 in CLSTM, and z is [ − → h (T ) 1 ⊕ ← − h (1) 1 ] in B- CLSTM. Training The objective of our model is to minimize the crossentropy error of the predicted and true distributions. Besides, the objective includes an L 2 regularization term over all parameters. Formally, suppose we have m train sentence and label pairs (w (i) 1:T i , y (i) ) m i=1 , the object is to minimize the objective function J(θ): J(θ) = − 1 m m i=1 log p (i) y (i) + λ 2 ||θ|| 2 ,(16) where θ denote all the trainable parameters of our model. Experiment In this section, we study the empirical result of our model on three datasets for document-level sentiment classification. Results show that the proposed model outperforms competitor models from several aspects when modelling long texts. Datasets Most existing datasets for sentiment classification such as Stanford Sentiment Treebank (Socher et al., 2013) are composed of short paragraphs with several sentences, which cannot evaluate the effectiveness of the model under the circumstance of encoding long texts. We evaluate our model on three popular real-world datasets, Yelp 2013, Yelp 2014 and IMDB. Table 1 shows the statistical information of the three datasets. All these datasets can be publicly accessed 1 . We pre-process and split the datasets in the same way as Tang et al. (2015b) did. • Yelp 2013 and Yelp 2014 are review datasets derived from Yelp Dataset Challenge 2 of year 2013 and 2014 respectively. The sentiment polarity of each review is 1 star to 5 stars, which reveals the consumers' attitude and opinion towards the restaurants. • IMDB is a popular movie review dataset consists of 84919 movie reviews ranging from 1 to 10 (Diao et al., 2014). Average length of each review is 394.6 words, which is much larger than the length of two Yelp review datasets. Evaluation Metrics We use Accuracy (Acc.) and MSE as evaluation metrics for sentiment classification. Accuracy is a standard metric to measure the overall classification result and Mean Squared Error (MSE) is used to figure out the divergences between predicted sentiment labels and the ground truth ones. Baseline Models We compare our model, CLSTM and B-CLSTM with the following baseline methods. • CBOW sums the word vectors and applies a non-linearity followed by a softmax classification layer. • JMARS is one of the state-of-the-art recommendation algorithm (Diao et al., 2014), which leverages user and aspects of a review with collaborative filtering and topic modeling. • CNN UPNN (CNN) (Tang et al., 2015b) can be regarded as a CNN (Kim, 2014). Multiple filters are sensitive to capture different semantic features during generating a representation in a bottom-up fashion. • RNN is a basic sequential model to model texts (Elman, 1991). • LSTM is a recurrent neural network with memory cells and gating mechanism (Hochreiter and Schmidhuber, 1997). • BLSTM is the bidirectional version of LSTM, and can capture more structural information and longer distance during looking forward and back (Graves et al., 2013). • CIFG-LSTM & CIFG-BLSTM are Coupled Input Forget Gate LSTM and BLSTM, de-noted as CIFG-LSTM and CIFG-BLSTM respectively (Greff et al., 2015). They combine the input and forget gate of LSTM and require smaller number of parameters in comparison with the standard LSTM. Hyper-parameters and Initialization For parameter configuration, we choose parameters on validation set mainly according to classification accuracy for convenience because MSE always has strong correlation with accuracy. The dimension of pre-trained word vectors is 50. We use 120 as the dimension of hidden units, and choose weight decay among { 5e−4, 1e−4, 1e−5 }. We use Adagrad (Duchi et al., 2011) as optimizer and its initial learning rate is 0.01. Batch size is chosen among { 32, 64, 128 } for efficiency. For CLSTM, the number of memory groups is chosen upon each dataset, which will be discussed later. We remain the total number of the hidden units unchanged. Given 120 neurons in all for instance, there are four memory groups and each of them has 30 neurons. This makes model comparable to (B)LSTM. Table 3 shows the optimal hyper-parameter configurations for each dataset. For model initialization, we initialize all recurrent matrices with randomly sampling from uniform distribution in [-0.1, 0.1]. Besides, we use GloVe (Pennington et al., 2014) as pre-trained word vectors. The word embeddings are fine-tuned during training. Hyper-parameters achieving best results on the validation set are chosen for final evaluation on test set. Results The classification accuracy and mean square error (MSE) of our models compared with other competitive models are shown in Table 2. When comparing our models to other neural network models, we have several meaningful findings. 1. Among all unidirectional sequential models, RNN fails to capture and store semantic features while vanilla LSTM preserves sentimental messages much longer than RNN. It shows that internal memory plays a key role in text modeling. CIFG-LSTM gives performance comparable to vanilla LSTM. 2. With the help of bidirectional architecture, models could look backward and forward to capture features in long-range from global perspective. In sentiment analysis, if users show their opinion at the beginning of their review, single directional models will possibly forget these hints. 3. The proposed CLSTM beats the CIFG-LSTM and vanilla LSTM and even surpasses the bidirectional models. In Yelp 2013, CLSTM achieves 59.4% in accuracy, which is only 0.4 percent worse than B-CLSTM, which reveals that the cache mechanism has successfully and effectively stored valuable information without the support from bidirectional structure. 4. Compared with existing best methods, our model has achieved new state-of-the-art results by a large margin on all documentlevel datasets in terms of classification accuracy. Moreover, B-CLSTM even has surpassed JMARS and CNN (UPNN) methods which utilized extra user and product information. 5. In terms of time complexity and numbers of parameters, our model keeps almost the same as its counterpart models while models of hierarchically composition may require more computational resources and time. Rate of Convergence We compare the convergence rates of our models, including CIFG-LSTM, CIFG-BLSTM and B-CLSTM, and the baseline models (LSTM and BLSTM). We configure the hyper-parameter to make sure every competing model has approximately the same numbers of parameters, and various models have shown different convergence rates in Figure 3. In terms of convergence rate, B-CLSTM beats other competing models. The reason why B-CLSTM converges faster is that the splitting memory groups can be seen as a better initialization and constraints during the training process. samples. X-axis is length ranking from 0% to 100%. Effectiveness on Grouping Memory For the proposed model, the number of memory groups is a highlight. In Figure 4, we plot the best prediction accuracy (Y-axis) achieved in validation set with different number of memory groups on all datasets. From the diagram, we can find that our model outperforms the baseline method. In Yelp 2013, when we split the memory into 4 groups, it achieves the best result among all tested memory group numbers. We can observe the dropping trends when we choose more than 5 groups. For fair comparisons, we set the total amount of neurons in our model to be same with vanilla LSTM. Therefore, the more groups we split, the less the neurons belongs to each group, which leads to a worse capacity than those who have sufficient neurons for each group. Sensitivity on Document Length We also investigate the performance of our model on IMDB when it encodes documents of different lengths. Test samples are divided into 10 groups with regard to the length. From Figure 5, we can draw several thoughtful conclusions. 1. Bidirectional models have much better performance than the counterpart models. 2. The overall performance of B-CLSTM is better than CIFG-BLSTM. This means that our model is adaptive to both short texts and long documents. Besides, our model shows power in dealing with very long texts in comparison with CIFG-BLSTM. 3. CBOW is slightly better than CIFG-LSTM due to LSTM forgets a large amount of information during the unidirectional propagation. Conclusion In this paper, we address the problem of effectively analyzing the sentiment of document-level texts in an RNN architecture. Similar to the memory structure of human, memory with low forgetting rate captures the global semantic features while memory with high forgetting rate captures the local semantic features. Empirical results on three real-world document-level review datasets show that our model outperforms state-of-the-art models by a large margin. For future work, we are going to design a strategy to dynamically adjust the forgetting rates for finegrained document-level sentiment analysis. Figure 1 : 1(a) A standard LSTM unit and (b) a CIFG-LSTM unit. There are three gates in (a), the input gate, forget gate and output gates, while in (b), there are only two gates, the CIFG gate and output gate. Figure 2 : 2An overview of the proposed architecture. Different styles of arrows indicate different forgetting rates. Groups with stars are fed to a fully connected layers for softmax classification. Here is an instance of B-CLSTM with text length equal to 4 and the number of memory groups is 3. Figure 3 : 3Convergence speed experiment on Yelp 2013. X-axis is the iteration epoches and Y-axis is the classifcication accuracy(%) achieved. Figure 4 :Figure 5 : 45Classification accuracy on different number of memory group on three datasets. X-axis is the number of memory group(Study of model sensitivity on document length on IMDB. All test samples are sorted by their length and divided into 10 parts. Left most dot means classification accuracy on the shortest 10% Table 2 : 2Sentiment classification results of our model against competitor models on IMDB, Yelp 2014 and Yelp 2013. Evaluation metrics are classification accuracy (Acc.) and MSE. Models with * use user and product information as additional features. Best results in each group are in bold. Dataset IMDB Yelp13 Yelp14 Hidden layer units 120 120 120 Number of groups 3 4 4 Weight Decay 1e−4 1e−4 5e−4 Batch size 128 64 64 Table 3 : 3Optimal hyper-parameter configuration for three datasets. http://ir.hit.edu.cn/˜dytang/paper/ acl2015/dataset.7z 2 http://www.yelp.com/dataset_challenge AcknowledgmentsWe appreciate the constructive work from Xinchi Chen. Besides, we would like to thank the anonymous reviewers for their valuable comments. 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[]
[ "Density of instantaneous frequencies in the Kuramoto-Sakaguchi model", "Density of instantaneous frequencies in the Kuramoto-Sakaguchi model" ]
[ "Julio D Da Fonseca \nDepartamento de Física\nUniversidade Estadual Paulista\nBela Vista13506-900Rio ClaroSPBrazil\n", "Edson D Leonel \nDepartamento de Física\nUniversidade Estadual Paulista\nBela Vista13506-900Rio ClaroSPBrazil\n", "† ", "Rene O Medrano-T \nDepartamento de Física\nUniversidade Estadual Paulista\nBela Vista13506-900Rio ClaroSPBrazil\n\nDepartamento de Física\nUniversidade Federal de São Paulo\nUNIFESP\nCampus Diadema09913-030São PauloBrasil\n" ]
[ "Departamento de Física\nUniversidade Estadual Paulista\nBela Vista13506-900Rio ClaroSPBrazil", "Departamento de Física\nUniversidade Estadual Paulista\nBela Vista13506-900Rio ClaroSPBrazil", "Departamento de Física\nUniversidade Estadual Paulista\nBela Vista13506-900Rio ClaroSPBrazil", "Departamento de Física\nUniversidade Federal de São Paulo\nUNIFESP\nCampus Diadema09913-030São PauloBrasil" ]
[]
We obtain a formula for the statistical distribution of instantaneous frequencies in the Kuramoto-Sakaguchi model. This work is based on the Kuramoto-Sakaguchi's theory of globally coupled phase oscillators, which we review in full detail by discussing its assumptions and showing all steps behind the derivation of its main results. Our formula is a stationary probability density function with a complex mathematical structure, is consistent with numerical simulations and gives a description of the stationary collective states of the Kuramoto-Sakaguchi model. *The works of A. Winfree and Y. Kuramoto brought seminal contributions to the study of synchronization[4,6,7,8]. Inspired by Winfree's pioneering ideas[9], Kuramoto formulated a model of coupled phase oscillators today known as Kuramoto model. The Kuramoto model was introduced in Ref.[7], and its first and more detailed analysis by Kuramoto himself, published in Ref.[8]. Since then, many studies about the Kuramoto model and its variants appeared in the literature. (Reviews about the Kuramoto model can be found in Refs.[10,11]; see Refs. [12,13,14,15]for later studies related to variants of the Kuramoto model and their applications.)The Kuramoto model consists of an ensemble of oscillators with a mean-field coupling and randomly distributed natural (or intrinsic) frequencies. An oscillator is characterized by its phase, and the first-order time derivative of the oscillator's phase, which here we call instantaneous frequency, is defined by an autonomous first-order ordinary differential equation. The theoretical analysis of the Kuramoto model[8,11]evinces a transition between two stationary collective states: an incoherent state and a synchronization one. In the incoherent state, instantaneous and natural frequencies have the same statistical distribution. In the synchronization state, some oscillators have instantaneous frequencies sharing the same value. The number of synchronized oscillators depends on the model's parameter called coupling-strength, and synchronization only occurs for a coupling strength above a critical value[8,11]. In a simplified version of the Kuramoto model, identical (with the same natural frequencies) and symmetrically-coupled oscillators show multiple regular attractors[16], and the synchronization state is the most probable one[17,18].H. Sakaguchi and Y. Kuramoto created a generalization of the Kuramoto model[19]introducing into the coupling function a phase shift, also called phase-lag. The Kuramoto-Sakaguchi model and its variants appear in the study of a wide range subjects such as chimera states[20,21], chaotic transients[22], pulse-coupled oscillators[23], and Josephson-junction arrays[24]. In addition, the coupling function with a phase-lag can be interpreted as an approximate model of interactions with time-delayed phases[25]. The Kuramoto-Sakaguchi model exhibits the same stationary collective states as the original Kuramoto model[19].Collective states of Kuramoto-like models are commonly characterized by means of an order parameter, which is zero in the incoherent state and takes finite values in the synchronization state. In this work, we follow a different approach from the usual order-parameter analysis uncovering how instantaneous frequencies are statistically distributed in the stationary collective states of the Kuramoto-Sakaguchi model. Instantaneous frequencies collectively reflect the occurence of synchronized behavior[26], and they are also relevant in the study of other phenomena (e.g. frequency spirals [32]).We will show how to obtain a formula for the statistical distribution of instantaneous frequencies. The formula is defined by a stationary probability density function, which we refer to as density of instantaneous frequencies. Our goal is similar to the one pursued in Ref.[26]for the Kuramoto model, but here we will show how to obtain a more general result in a more straightforward way. A related (but still rather a different) problem was addressed by Sakaguchi
10.1016/j.chaos.2023.113553
[ "https://export.arxiv.org/pdf/2210.05011v4.pdf" ]
252,815,581
2210.05011
2338dc74495603a88c555cc56adf1ceb0e86b5d4
Density of instantaneous frequencies in the Kuramoto-Sakaguchi model December 13, 2022 Julio D Da Fonseca Departamento de Física Universidade Estadual Paulista Bela Vista13506-900Rio ClaroSPBrazil Edson D Leonel Departamento de Física Universidade Estadual Paulista Bela Vista13506-900Rio ClaroSPBrazil † Rene O Medrano-T Departamento de Física Universidade Estadual Paulista Bela Vista13506-900Rio ClaroSPBrazil Departamento de Física Universidade Federal de São Paulo UNIFESP Campus Diadema09913-030São PauloBrasil Density of instantaneous frequencies in the Kuramoto-Sakaguchi model December 13, 2022 We obtain a formula for the statistical distribution of instantaneous frequencies in the Kuramoto-Sakaguchi model. This work is based on the Kuramoto-Sakaguchi's theory of globally coupled phase oscillators, which we review in full detail by discussing its assumptions and showing all steps behind the derivation of its main results. Our formula is a stationary probability density function with a complex mathematical structure, is consistent with numerical simulations and gives a description of the stationary collective states of the Kuramoto-Sakaguchi model. *The works of A. Winfree and Y. Kuramoto brought seminal contributions to the study of synchronization[4,6,7,8]. Inspired by Winfree's pioneering ideas[9], Kuramoto formulated a model of coupled phase oscillators today known as Kuramoto model. The Kuramoto model was introduced in Ref.[7], and its first and more detailed analysis by Kuramoto himself, published in Ref.[8]. Since then, many studies about the Kuramoto model and its variants appeared in the literature. (Reviews about the Kuramoto model can be found in Refs.[10,11]; see Refs. [12,13,14,15]for later studies related to variants of the Kuramoto model and their applications.)The Kuramoto model consists of an ensemble of oscillators with a mean-field coupling and randomly distributed natural (or intrinsic) frequencies. An oscillator is characterized by its phase, and the first-order time derivative of the oscillator's phase, which here we call instantaneous frequency, is defined by an autonomous first-order ordinary differential equation. The theoretical analysis of the Kuramoto model[8,11]evinces a transition between two stationary collective states: an incoherent state and a synchronization one. In the incoherent state, instantaneous and natural frequencies have the same statistical distribution. In the synchronization state, some oscillators have instantaneous frequencies sharing the same value. The number of synchronized oscillators depends on the model's parameter called coupling-strength, and synchronization only occurs for a coupling strength above a critical value[8,11]. In a simplified version of the Kuramoto model, identical (with the same natural frequencies) and symmetrically-coupled oscillators show multiple regular attractors[16], and the synchronization state is the most probable one[17,18].H. Sakaguchi and Y. Kuramoto created a generalization of the Kuramoto model[19]introducing into the coupling function a phase shift, also called phase-lag. The Kuramoto-Sakaguchi model and its variants appear in the study of a wide range subjects such as chimera states[20,21], chaotic transients[22], pulse-coupled oscillators[23], and Josephson-junction arrays[24]. In addition, the coupling function with a phase-lag can be interpreted as an approximate model of interactions with time-delayed phases[25]. The Kuramoto-Sakaguchi model exhibits the same stationary collective states as the original Kuramoto model[19].Collective states of Kuramoto-like models are commonly characterized by means of an order parameter, which is zero in the incoherent state and takes finite values in the synchronization state. In this work, we follow a different approach from the usual order-parameter analysis uncovering how instantaneous frequencies are statistically distributed in the stationary collective states of the Kuramoto-Sakaguchi model. Instantaneous frequencies collectively reflect the occurence of synchronized behavior[26], and they are also relevant in the study of other phenomena (e.g. frequency spirals [32]).We will show how to obtain a formula for the statistical distribution of instantaneous frequencies. The formula is defined by a stationary probability density function, which we refer to as density of instantaneous frequencies. Our goal is similar to the one pursued in Ref.[26]for the Kuramoto model, but here we will show how to obtain a more general result in a more straightforward way. A related (but still rather a different) problem was addressed by Sakaguchi Introduction Synchronization is the process by which interacting oscillatory systems adjust their frequencies in order to display the same common value [1]. Power grids [2], semiconductor laser arrays [3], cardiac pacemaker cells [4], and neurosciences [5] are just a few examples in a multitude of domains where synchronization is an active research subject. and Kuramoto in Ref. [19], where they analyzed the statistical distribution of coupling-modified frequencies, namely instantaneous frequencies averaged over infinitely large time intervals (see Refs. [11,26] for further details). This work is based on the Kuramoto-Sakaguchi theory, described, as far as we know, only in Ref. [19]. We will discuss the fundamental assumptions of the Kuramoto-Sakaguchi theory and detail how its main results can be derived. Our opinion is that an explicit presentation of the Kuramoto-Sakaguchi theory is still absent. We organized this paper as follows. In Section 2, we present the Kuramoto-Sakaguchi theory and state diagrams pointing out the transition between the incoherent and synchronization states. In Section 3, we extend the Kuramoto-Sakaguchi theory by providing additional analytical results and obtaining the formula of the density of instantaneous frequencies. In Section 4, we discuss the properties of our formula in a specific application example, in which natural frequencies have a Gaussian statistical distribution. In Section 5, we check the consistency of our formula with numerical simulation data. Conclusions and an outlook on possible research directions are given in Section 6. Kuramoto-Sakaguchi theory The Kuramoto-Sakaguchi (KS) model [19] consists of an infinitely large number N of all-to-all coupled oscillators. The state of an oscillator of index i = 1...N is characterized by its phase θ i , which changes in time according tȯ θ i = ω i − K N N j=1 sin(θ i − θ j + α),(1) whereθ i is the first-order time-derivative of θ i , ω i is a random number with a prescribed density 1 , and K and α are real constant parameters. We refer toθ i as the instantaneous frequency, ω i is called natural frequency and K, the coupling strength. The oscillator of index j can be represented by the complex number exp(iθ j ). Oscillators are then points in a complex plane moving over a unit-radius circle centered at the origin. A valuable concept for the analysis of collective behavior in the KS model is that of mean field, defined by Z = 1 N N j=1 exp(iθ j ),(2) which can be interpreted as the average oscillator state. The mean field can be written as a complex number Z = R exp(iΘ)(3) where Θ denotes the mean-field phase and R, the mean-field modulus, referred to as order parameter. If the oscillators are quasi-aligned, i.e., they have approximately the same phase, then R 1. Yet, a quasi-uniform scattering of all oscillator-points over the circle results in a mean-field located near the origin, i.e. R 0. For N −→ ∞, the mean field, at a time instant t, is given by Z = +π −π exp(iθ)n(θ, t)dθ,(4) where n(θ, t) is the density of phases at the same time instant. Two simple scenarios are assumed concerning the properties of n(θ, t) in the long-time (t −→ ∞) and large-size (N −→ ∞) limits. First, n(θ, t) = 1 2π for −π < θ ≤ +π, and n(θ, t) = 0, otherwise, i.e. n(θ, t) is a time-independent and uniform density (the value 1 2π comes from the normalization condition). Second, n(θ, t) is a steadily traveling wave with velocity Ω, i.e. n(θ − Ω∆t, t) = n(θ, t + ∆t) for any time instant t and time interval ∆t. This means that the wave profile does not change in time, and the wave propagates with constant-in-time velocity Ω. We call the wave-propagation velocity, Ω, synchronization frequency. The first scenario defines the incoherent state, and the second, the synchronization state. In the incoherent state, oscillators are uniformly spread over the unit circle. In the synchronization state, a bunch of oscillators is synchronized, that is, they change collectively their phase at the same constant rate Ω. After inserting a uniform phase density in Eq. (4), we see that Z = 0. So, from Eq. (3), the order parameter (R) is zero in the incoherent state. Yet, if a KS system exhibits synchronization, then the assumption of a traveling wave with a stationary and non-uniform profile, moving with constant velocity Ω, means that R is finite and time-independent. Moreover, Z moves in the complex plane following a circular path of radius R and velocity Ω. That is, a uniform circular motion given by Z(t) = R exp [i (Ωt + Θ 0 )] ,(5) where Θ 0 is the mean-field phase at an arbitrary initial time instant. Let us consider KS oscillators in a different complex plane, with the same origin as the previous one, but with both axis rotating with angular velocity Ω. In the new rotating frame, we represent the oscillator of index j by the complex number exp(iψ j ), where ψ j is the oscillator's phase. The analogous of Eqs. (2), (3), and (4) are Z = 1 N N j=1 exp(iψ j ),(6)Z = R exp(iΨ),(7) and, for N −→ ∞, Z = +π −π exp(iψ)n(ψ)dψ.(8) The quantities Z and n(ψ) are representations of the mean field and the phase density in the rotating frame. Comparing Eq. (3) to Eq. (7), we see that Z and Z have the same length R. The mean-field length is invariant to the change of frames because the phase density profile is kept unchanged. Note also that, in Eq. (8), we removed the time dependence from the phase density, since both the rotating frame and the steadily traveling wave move together with the same velocity Ω. So, both R and Ψ are time-independent, which is the same as stating that the mean field is fixed in the rotating frame. Some conventions are useful to simplify the analysis of the KS model at a time instant t > t o = 0, with t 0 denoting the initial time instant. We choose a fixed frame such that its real axis has the same direction as the mean field at time t 0 . So, Θ 0 = 0 and Z(0) = R. Another important convention is defining a rotating frame such that, at the initial time t 0 , its real axis is dephased by α from the fixed-frame's real axis (the same parameter α of Eq. (1)). This is the same as setting Ψ = α. Thus, from Eqs. (5) and (7), at the time instant t, Z(t) = R exp (iΩt)(9) and Z = R exp iα.(10) Also, as a consequence of the above conventions, a simple geometric inquiring yields the relationṡ ψ i =θ i − Ω(11) and ψ i = θ i − Ωt + α.(12) In Eq. (11),ψ i is the instantaneous frequency of an oscillator of index i in the rotating frame. Using Eqs. (11) and (12), we can recast Eq. (1) aṡ ψ i = ω i − Ω − K N N j=1 sin(ψ i − ψ j + α).(13) Multiplying the right-hand sides of Eqs. (6) and (10) by exp [−i (ψ i + α)] and equating their imaginary parts result in N −1 N j=1 sin(ψ i − ψ j + α) = R sin ψ i .(14) Substituting the summation in Eq. (13) by the right-hand-side of Eq. (14) giveṡ ψ i = ω i − Ω − KR sin ψ i ,(15) which is a simple formulation of the KS model in the rotating frame. We emphasize that ω i , K, Ω, and R are constant-in-time numbers: ω i is a sample from a random variable with a given probability density function g; K is a given coupling strength; Ω and R are constants to be determined. The differential equation (15) gives a more detailed picture of synchronization in the KS model. For |ω i − Ω| > KR, Eq. (15) has no equilibrium point. When |ω i − Ω| = KR, a pair of stable and unstable equilibria emerges by a fold bifurcation, and they become apart as |ω i − Ω| < KR. Since the phase domain is closed (|ψ i | ≤ π), the phase of an oscillator i, for which |ω i − Ω| ≤ KR, will always converge to an attractor (stable equilibrium point) defined by ψ * i = sin −1 ω i − Ω KR ,(16) where sin −1 is an inverse of the sin function with domain [−1, +1] and image − π 2 , + π 2 . Then, − π 2 ≤ ψ * i ≤ + π 2 . The latter case, where oscillator i has a natural frequency such that −KR + Ω ≤ ω i ≤ Ω+KR, means, according to Eqs. (11), (12), (15), and (16) , that ψ i −→ ψ * i , θ i −→ ψ * i +Ωt−α, ψ i −→ 0, andθ i −→ Ω as t −→ +∞. This is the case of a synchronized oscillator, or, following Kuramoto's terminology, an S oscillator. But, for |ω i − Ω| > KR, i.e, oscillator i has a natural frequency such that ω i < −KR + Ω or Ω + KR < ω i , Eq. (15) has no equilibrium point. Then, oscillator-i's phase varies according to Eq. (15) without slowing down towards an asymptotic value. This is a desynchronized oscillator, or, simply, a D oscillator. Let n(ψ, ω) denote the joint density for a rotating-frame phase ψ and a natural frequency ω. The associated marginal phase density is given by n(ψ) = +∞ −∞ n(ψ, ω) dω,(17) where −π < ψ ≤ +π. Eq. (17) can be rewritten as n(ψ) =| ω−Ω|≤KR n(ψ, ω) dω +| ω−Ω|>KR n(ψ, ω) dω.(18) The first term in the right-hand side of Eq. (18) is the phase density for S oscillators, and the second one, the phase density for D oscillators. The two terms are functions of ψ which we denote by n S (ψ) and n D (ψ), respectively. Then, n(ψ) = n S (ψ) + n D (ψ).(19) Let n(ψ|ω) be the conditional phase density for a given natural frequency ω. If g(ω) is the natural-frequency density, replacing n(ψ, ω) with n(ψ|ω)g(ω) leads to n S (ψ) = Ω+KR Ω−KR n S (ψ|ω)g(ω) dω (20) and n D (ψ) = Ω−KR −∞ n D (ψ|ω)g(ω) dω + +∞ Ω+KR n D (ψ|ω)g(ω) dω,(21) where we use the following definitions: n S (ψ|ω) ≡ n(ψ|ω) for |ω − Ω| ≤ KR, and n D (ψ|ω) ≡ n(ψ|ω) for |ω − Ω| > KR. Thus, to find expressions for n S (ψ) and n D (ψ), we have to determine n S (ψ|ω) and n D (ψ|ω). For a generic S oscillator with natural frequency ω, Eq. (16) has the alternative form ψ * (ω) = sin −1 ω − Ω KR .(22) Since ψ * (ω) is an attractor, the phase of this oscillator is always in an arbitrarily small neighborhood of ψ * (ω) for a sufficiently long time. Then, I n S (ψ|ω) dψ = 1, ψ * (ω) ∈ I 0, ψ * (ω) / ∈ I(23) where I is an arbitrary interval contained in (−π, +π]. Eq. (23) is the same as stating that n S (ψ|ω) = δ [ψ − ψ * (ω)] .(24) We emphasize that Eq. (24) holds only for |ω − Ω| ≤ KR or, equivalently, −KR + Ω ≤ ω ≤ Ω + KR. Using (22) and (24) to solve the integral in (20), we obtain n S (ψ) = g(Ω + KR sin ψ)KR cos ψ, |ψ| ≤ π 2 0, |ψ| > π 2(25) According to Eq. (25), n S (ψ) −→ 0 as R −→ 0 + , that is, the number of S oscillators goes to zero if the order parameter becomes small by varying K and α. Moreover, if R is finite, then n S (ψ) = 0 for |ψ| > π 2 and n S (ψ) > 0 for |ψ| ≤ π 2 . This comes from the property that, for a sufficiently long time, S-oscillator phases are arbitrarily near their respective attractors, which belong, all of them, to the interval − π 2 , + π 2 (See Eq. (16)). As mentioned earlier, finding a formula for n D (ψ) requires finding a formula for n D (ψ|ω). This can be done by considering a small control interval contained in the phase domain and where the time-variation of the number of D oscillators with a given natural frequency is balanced with the flow of the same type of oscillators into and out from the interval. Defining the control interval by [ψ, ψ + δψ], we have ∂ t ψ+δψ ψ n D (ψ |ω)dψ = n D (ψ|ω)ψ(ψ) − n D (ψ + δψ|ω)ψ(ψ + δψ),(26) whereψ(ψ) andψ(ψ + δψ) are the rotating-frame instantaneous frequencies at phases ψ and ψ + δψ for D oscillators with a given natural frequency ω. The rotating-frame instantaneous frequency for a D oscillator can be defined bẏ ψ(ψ) = ω − Ω − KR sin ψ,(27) which is the same as (15) without the index notation but with the condition that |ω − Ω| > KR. In Eq. (26), on the left-hand side is the time-variation of the probability of finding D oscillators in the interval [ψ, ψ+δψ] with a given natural frequency ω. The probability flow at the endpoints of the same interval, namely ψ and ψ + δψ, is given by the right-hand side of (26). By expanding both n D (ψ |ω) and n D (ψ+δψ|ω)ψ(ψ+δψ) near ψ, taking the limit δψ −→ 0 + , and neglecting high-order terms, we obtain the continuity equation ∂ t n D (ψ|ω) + ∂ ψ [n D (ψ|ω)ψ] = 0.(28) We remind the reader that |ω − Ω| > KR for the given value of ω in Eq (28). An important assumption in KS theory is to consider that n D (ψ|ω) is a stationary density. This is consistent with the previously discussed assumption of stationarity for n(ψ). Since n(ψ) = n S (ψ) + n D (ψ) and n S (ψ) are both stationary, n D (ψ) should also be. According to Eq. (21), the simplest way to accomplish a stationary n D (ψ) is by assuming that n D (ψ|ω) is also stationary. So, from ∂ t n D (ψ|ω) = 0 and (28), n D (ψ|ω) = C(ω) ψ(ψ) ,(29) where C(ω) is a constant with respect to ψ but possibly depending on ω. Eq. (29) means that D oscillators accumulate at phases with low variation rates (low instantaneous frequencies) and are less probable to be found at phases with high variation rates (high instantaneous frequencies). Applying the normalization condition to both sides of (29) results in 1 C(ω) = π −π dψ ψ (ψ ) .(30) Using Eq. (27) and formula (2.551-3) from Ref. [27], we can solve the integral in Eq. (30) and obtain C(ω) = C + (ω), ω > Ω + KR C − (ω), ω < Ω − KR (31) where C ± (ω) = ± (ω − Ω) 2 − (KR) 2 2π .(32) By inspecting Eqs. (30) and (31), we get a simple interpretation of the quantity C(ω). For ω > Ω + KR, C(ω) = C + (ω) > 0,ψ(ψ) is positive for any ψ, and the integral in Eq. (30) corresponds to the time required for a D oscillator (with natural frequency ω) to complete a counterclockwise cycle over the unit circle, departing from the initial phase −π to the final one +π. Likewise, for ω > Ω − KR, C(ω) = C − (ω) < 0,ψ(ψ) is negative for any ψ, and the opposite of the integral in Eq. (30) is the duration time of a clockwise cycle from +π to −π . Therefore, the period of rotation over the unit circle for a D oscillator with natural frequency ω is T (ω) = 1 |C(ω)| ,(33) and the absolute value of C(ω), |C(ω)|, is the number of cycles per time unit. Now we turn our attention back to the D-oscillator phase density. We can use Eqs. (21), (29) and (31) to obtain n D (ψ) = Ω−KR −∞ C − (ω)g(ω) ψ(ψ) dω + +∞ Ω+KR C + (ω)g(ω) ψ(ψ) dω.(34) By changing the integration variable from ω to χ = ω − Ω and replacingψ(ψ) and C ± (ω) with their respective expressions (See Eqs. (27) and (32)), we can rewrite Eq. (34) as n D (ψ) = 1 2π| χ|>KR χg(Ω + χ) χ − KR sin ψ 1 − KR χ 2 dχ,(35) which is the final form of the phase density for D oscillators. Note that, according to Eq. (35), n D (ψ) −→ 1 2π as R −→ 0. Then, n(ψ) = n S (ψ) + n D (ψ) −→ 1 2π as R −→ 0, since, as discussed before, n S (ψ) −→ 0 as R −→ 0 + . This is consistent with the assumption of a uniform phase density for the incoherent state. Using the formulas of the phase densities for S and D oscillators (See Eqs. (25) and (35)), one can obtain an equation whose solution, for given K, α and g, consists of R and Ω. To obtain such an equation, we first equate the right-hand sides of Eqs. (8) and (10) with n(ψ) defined by Eq. (19). This gives Re iα = +π −π e iψ [n S (ψ) + n D (ψ)] dψ.(36) We are interested in non-trivial solutions, i.e. solutions with non-zero R, meaning that synchronization occurs for the given parameters, namely K, α, and g. So, K is also finite. Otherwise, all oscillators would be out of synchrony. By dividing both sides of Eq. (36) by the product KR and using Eqs. (25) and (35), Eq. (36) can be recast as e iα K = + π 2 − π 2 e iψ g(Ω + KR sin ψ) cos ψ dψ + iJ,(37) where J = 1 2πiKR +π −π e iψ L(ψ) dψ(38) and L(ψ) = +∞ KR g(Ω + χ) χ − KR sin ψ + g(Ω − χ) χ + KR sin ψ χ 1 − KR χ 2 dχ.(39) We aim for a simpler form for Eq. (38). By changing the integration variable in Eq. (39) from χ to γ, defined by χ = KR csc γ and 0 < γ < π 2 , we obtain L(ψ) = KR + π 2 0 h + (γ) 1 − sin γ sin ψ + h − (γ) 1 + sin γ sin ψ cot 2 γ dγ,(40) where h ± (γ) = g(Ω ± KR csc γ)(41) After substituting L(ψ) in Eq. (38) by its definition given in Eq. (40), a simple algebraic manipulation leads to J = 1 2πi + π 2 0 {[h + (γ) + h − (γ)] I 1 (γ) + sin γ [h + (γ) − h − (γ)] I 2 (γ)} cot 2 γ dγ,(42) where I 1 (γ) = +π −π e iψ f (1) γ (ψ) dψ,(43)f (1) γ (ψ) = 1 1 − (sin γ sin ψ) 2 ,(44)I 2 (γ) = +π −π e iψ f (2) γ (ψ) dψ,(45) and f (2) γ (ψ) = sin ψ 1 − (sin γ sin ψ) 2 .(46) Symmetry properties can be used to solve I 1 (γ) and I 2 (γ). Eq. (43) is the same as I 1 (γ) = +π 0 e iψ f (1) γ (ψ) + e i(ψ−π) f (1) γ (ψ − π) dψ. Since f (1) γ (ψ) = f (1) γ (ψ − π) and e iψ = −e i(ψ−π) , we conclude that I 1 (γ) = 0. Eq. (45) can be written as I 2 (γ) =´+ π 2 − π 2 e iψ f (2) γ (ψ) + e i(π−ψ) f (2) γ (π − ψ) dψ. Considering that f (2) γ (ψ) = f (2) γ (π − ψ) and e iψ + e i(π−ψ) = 2i sin ψ, we have I 2 (γ) = 2i + π 2 − π 2 sin 2 ψ 1 − (sin γ sin ψ) 2 dψ,(47) which has the alternative form I 2 (γ) = 2i sin 2 γ [Q(γ) − π] ,(48) where Q(γ) = + π 2 − π 2 1 1 − sin 2 γ sin 2 ψ dψ.(49) Solving the integral in (49) gives 2 Q(γ) = π |cos γ| . (50) 2 According to Eq. (2.562-1) in Ref. [27],´1 a+b sin 2 x dx = sign(a) √ a(a+b) arctan a+b a tan x for b a > −1. If a = 1 and b = − sin 2 γ, then, from Eq. (49), we have Q(γ) = lim →0 + 1 √ 1−sin 2 γ arctan 1 − sin 2 γ tan ψ + π 2 − − π 2 + = π |cos γ| . Also, from Eqs. (48) and (50), I 2 (γ) = 2πi sin 2 γ 1 − |cos γ| |cos γ| .(51) Back to Eq. (42), we can eliminate the term with I 1 (γ) (I 1 (γ) = 0, as mentioned above) and use Eqs. (41) and (51) to finally obtain J = + π 2 0 g Ω + KR sin ψ − g Ω − KR sin ψ cos ψ (1 − cos ψ) sin 3 ψ dψ.(52) Eq. (37), with J defined by Eq. (52), gives the values of R and Ω for K and α values such that the KS model is in the synchronization state. Eq. (37) is equivalent to the system of equations + π 2 − π 2 g(Ω + KR sin ψ) cos 2 ψ dψ = 1 K cos α (53) + π 2 − π 2 g(Ω + KR sin ψ) cos ψ sin ψ dψ + J = 1 K sin α(54) In Figs. 1(a-d), we show numerical approximations to solutions of the system (53-54) for different pairs of K and α values. All numerical solutions are computed by assuming that g(ω) = 1 √ 2π exp − ω 2 2 ,(55) which is the standard normal density, and using the software library MINPACK [28]. If convergence is not achieved, we assume that the system (53-54) has no solution, set R = 0, and attribute no specific value to Ω. This means that the KS model is in the incoherent state. Figure1(a) shows a 100×100 resolution grid of points (K, α). Each point has a color defined according to the value of R, e.g. dark blue for R = 0. A well-defined boundary separates two regions, one with R = 0 and another where R > 0. Figure 1(a) can be seen as a phase diagram pointing out the transition from a fully desynchronized state to a hybrid state consisting of both D and S oscillators. Figure1(b) shows a similar grid to that of Fig.1(a). The grid has the same set of points (K, α), but the color of each point is defined by the corresponding value of Ω. If Ω is not defined for a specific point, we associate this point with the white color. In Figs. 1(c) and (d), we show slices of the three-dimensional graphs of Figs. 1(a) and (b). For each two-dimensional profile, K is kept fixed, and α varies between −1 and +1. The graphs . We assume that g is a standard normal density. of Figure 1(c) suggest that, for constant K, R is an even function of α with a maximum at α = 0. In Figure 1(d), the graphs indicate that Ω varies monotonically as an odd function of α. Moreover, at least for the set of K values considered, Ω is more sensitive to variations in α than in K. Density of instantaneous frequencies In the previous section, we showed how to obtain known results from the KS theory relevant to this work. We are now able to proceed towards our core result: the density of instantaneous frequencies in the KS model, which we represent by the probability density function G. The quantity G(ν) dν is then the probability that a KS oscillator i has its fixed-frame instantaneous frequency,θ i , in the interval [ν, ν + dν). In the process of obtaining G, we are concerned about the case R > 0 (synchronization state). According to Eqs. (11) and (15), for R = 0,θ i = ω i . Then, in the incoherent state, the densities of instantaneous and natural frequencies are identical, i.e. G(ν) = g(ν). Before we show how to obtain the density G, we introduce some basic facts and definitions to simplify the notation. First, the product KR is defined by a ≡ KR,(56) and x ≡ x − Ω a(57) for a generic variable x. We can then rewrite Eqs. (22), (27), and (31) as ψ * (ω) = sin −1 ( ω) ,(58)ψ(ψ) = a ( ω − sin ψ) ,(59) and C (ω) = a 2π sgn ω √ ω 2 − 1.(60) In Eq. (60), sgn ω is the sign of ω and | ω| > 1. Another set of useful definitions iṡ ψ ≡ ν − Ω,(61)∆ν ≡ ν − ω,(62) and the function f ν (ψ) ≡ ∆ν + sin(ψ). The symbolψ is used to denote the rotating-frame instantaneous frequency of an oscillator with a fixed-frame instantaneous frequency ν. Definitions (61) and (62) imply ν =ψ a(64) and ∆ν =ψ a − ω. From the formulas for n S (ψ|ω) and n D (ψ|ω), given by Eqs. (24) and (29), the phase density for an oscillator with known natural frequency ω is n(ψ|ω) =      δ [ψ − ψ * (ω)] , | ω| ≤ 1 C(ω) ψ(ψ) , | ω| > 1(66) where ψ * (ω) and C (ω) is defined by Eqs. (58) and (60). We now have enough mathematical tools to determine G. As a first step, we define p(ψ|ω) as the probability density that an oscillator has a rotating-frame instantaneous frequencyψ given that the oscillator's natural frequency is ω. From the random variable transformation theorem [29], we have p(ψ|ω) =ˆ+ π −π δ[ψ −ψ(ψ)]n(ψ|ω) dψ(67) In Eq. (67),ψ is the argument of function p(ψ|ω),ψ(ψ) is the function of ψ defined by Eq. (59), and n(ψ|ω) is the conditional density (66). From Eqs. (66) and (67), p(ψ|ω) =      p S (ψ|ω), | ω| ≤ 1 p D (ψ|ω), | ω| > 1 (68) where p S (ψ|ω) = 1 aˆ+ π −π δ [ψ − ψ * (ω)] δ [f ν (ψ)] dψ(69) and p D (ψ|ω) = 1 2πa sgn ω √ ω 2 − 1ˆ+ π −π δ [f ν (ψ)] ω − sin ψ dψ.(70) which expresses the certainty that S oscillators have rotating-frame instantaneous frequencies equal to zero, i.e., their fixed-frame instantaneous frequencies are equal to Ω. To solve the integral in Eq. (70), we write δ [f ν (ψ)] as δ [f ν (ψ)] = β∈O[fν ] δ(ψ − β) |f ν (β)| ,(72) where f ν is the derivative of f ν , and β runs through O [f ν ], defined as the set of the simple zeros of f ν . If O [f ν ] is an empty set, which is the case for |∆ν| ≥ 1, δ [f ν (ψ)] = 0. For |∆ν| < 1, f ν has two simple zeros: β 1 = sin −1 (−∆ν) and β 2 = π−β 1 . Since |f ν (β 1,2 )| = cos sin −1 (∆ν) = 1 − (∆ν) 2 , Eq. (72) gives δ [f ν (ψ)] =        1 √ 1−(∆ν) 2 [δ (ψ − β 1 ) + δ (ψ − β 2 )] , |∆ν| < 1 0, |∆ν| ≥ 1.(73) From Eqs. (70) and (73), we have p D (ψ|ω) =        sgn ω πa ν ω 2 −1 1−(∆ν) 2 , |∆ν| < 1 0, |∆ν| ≥ 1,(74) whose explicit dependence onψ can be obtained using Eqs. (64) and (65). With Eqs. (71) and (74), we complete the definition of the conditional density p(ψ|ω), given by (68). In a similar way to what we have done to obtain the density of phases (See Eqs. (19), (20) and (21)), we can define the densities of instantaneous frequencies as p(ψ) = p S (ψ) + p D (ψ),(75) where p S (ψ) =| ω|≤1 p S (ψ|ω)g(ω) dω(76) and p D (ψ) =| ω|>1 p D (ψ|ω)g(ω) dω.(77) In Eqs. (76) and (77), p S (ψ) and p D (ψ) are the densities of instantaneous frequencies in the rotating frame for S and D oscillators, respectively. Since the integration domain in (76) is the interval [Ω − a, Ω + a], substituting (71) in (76) results in p S (ψ) = S(K, α)δ(ψ),(78) where S(K, α) =ˆΩ +a Ω−a g(ω)dω.(79) ν ≤ −2 −2 <ν < 0 0 <ν < +2 +2 ≤ν (80). For ν = 0, D(ν) is the empty set. ω + νν + 1 −1ν + 1ν + 1 ω − νν − 1ν − 1 +1ν − 1 The quantity S(K, α) has an important meaning: S(K, α) is the probability that an oscillator is synchronized. So, S(K, α) quantifies the fraction of S oscillators. Note that S(K, α) has implicit dependencies on K and α through Ω and a. As already mentioned, a = KR, and the pair {Ω, R} is the solution of the system of Eqs. (53-54). In order to obtain a formula for p D (ψ) from Eq. (77), it is useful to define D(ν) as the set of real numbers ω such that | ω| > 1 and |∆ν| < 1. That is, D(ν) ≡ { ω ∈ IR − [−1, +1] | ν − 1 < ω < ν + 1} .(80) Note that, for ν = 0, D(ν) is the empty set. For finite ν, D(ν) is an open interval ( ω − ν , ω + ν ), whose endpoints ω − ν and ω + ν change according to the value of ν. The endpoints of D(ν) are given in Table 1 for different intervals ofν. A more compact way of defining ω − ν and ω + ν is ω ± ν = (ν ± 2) Θ [ν (ν ± 2)] ∓ 1,(81) where Θ denotes the Heaviside step function with the standard definition Θ(0) = 1 2 . Let D(ν) be the set of real numbers ω such that | ω| > 1 and |∆ν| ≥ 1. So, from (74) and (77), p D (ψ) =ˆD (ν) p D (ψ|ω)g(ω) dω +ˆD (ν) p D (ψ|ω)g(ω) dω.(82) Since p D (ψ|ω) = 0 for |∆ν| ≥ 1 (See Eq. (74)), the second integral in (82) is zero. The first integral is also zero only if D(ν) is empty, that is, if ν = 0 (ψ = 0). If ν = 0 (ψ = 0), the first integral can be determined using the endpoints of D(ν) as integration limits and the expression of p D (ψ|ω) for |∆ν| < 1 defined in Eq. (74). Thus, Eq. (82) can be rewritten as p D (ψ) = a πψˆ ω + ψ+Ω ω − ψ+Ω sgn ω ω 2 − 1 1 − ψ a − ω 2 g(a ω + Ω) d ω(83) for non-zero values ofψ, and p D (ψ) = 0 forψ = 0. From (81), the limits of integration in (83) are ω ± ψ+Ω = ψ a ± 2 Θ ψ a ψ a ± 2 ∓ 1.(84) In order to show the explicit dependence of p D (ψ) onψ, we used (61), (64), and (65). We also changed the previous integration variable, ω, to ω. Equations (75), (78) and (83) give a full description of the density of instantaneous frequencies in the rotating frame. Our goal now is to obtain the density of fixed-frame instantaneous frequencies, which we defined, at the beginning of this section, as a probability density function G(ν). Instantaneous frequencies in the rotating and fixed frames are related through Eq. (61). Therefore, G(ν) = p(ν − Ω), which is the same as G(ν) = p S (ν − Ω) + p D (ν − Ω).(85) Let G S (ν) = p S (ν − Ω) and G D (ν) = p D (ν − Ω). From (78), (83), and (85), we have G(ν) = G S (ν) + G D (ν),(86) where G S (ν) = S(K, α)δ(ν − Ω),(87)G D (ν) = 1 π | ν|ˆ ω + ν ω − ν ω 2 − 1 1 − ( ν − ω) 2 g(a ω + Ω) d ω,(88) for |ν − Ω| > 0, and G D (Ω) = 0. Note that we replaced sgn ω ν with 1 | ν| to obtain Eq. (88). This is possible due to the integral-limits signs, which can be determined from Table 1. If ν is positive, both ω − ν and ω + ν are positive. If ν is negative, both ω − ν and ω + ν are negative. So, the integration variable, ω, which is in the interval ω − ν < ω < ω + ν , has the same sign as ν. This implies ν = sgn ω | ν|. An alternative formula for Eq. (88) can be obtained by changing the integration variable from ω to ψ = sin −1 ( ω − ν). This change results in: ω 2 − 1 = (sin ψ + ν) 2 − 1, 1 − ( ν − ω) 2 = cos ψ, and g(a ω + Ω) = g(a sin ψ + ν). In addition, if ψ − ν and ψ + ν denote the new integral limits, then, from ψ ± ν = sin −1 ( ω ± ν − ν) and Θ(x) + Θ(−x) = 1, ψ ± ν = sin −1 {− (ν ± 2) Θ [−ν (ν ± 2)] ± 1} .(89) The formulas above allow us to rewrite Eq. (88) as G D (ν) = 1 π | ν|ˆψ + ν ψ − ν (sin ψ + ν) 2 − 1g(a sin ψ + ν) dψ.(90) Equations (86), (87), and (90) are identical to those which define the density of instantaneous frequencies in the Kuramoto model [26]. However, although the formulas in both models depend on R and Ω in the same manner, R and Ω depend implicitly on α in the KS model, while the parameter α is not defined in the Kuramoto model. We conclude this section in a summary fashion presenting again our main result with the original notation restored and together with all auxiliary equations. From Eqs. (86), (87) and (88), G(ν) = S(K, α)δ(ν − Ω) + G D (ν),(91) where S(K, α) =ˆΩ +KR Ω−KR g(ω)dω,(92) and G D (ν) = KR π [|ν − Ω| + Θ(− |ν − Ω|)]ˆ ω + ν ω − ν ω 2 − 1 1 − ν−Ω KR − ω 2 g(Ω + KR ω) d ω,(93) which have the integral limits ω ± ν = ν − Ω KR ± 2 Θ ν − Ω KR ν − Ω KR ± 2 ∓ 1.(94)e iα K = + π 2 − π 2 e iψ g(Ω + KR sin ψ) cos ψ dψ + iJ,(95) where J = + π 2 0 g Ω + KR sin ψ − g Ω − KR sin ψ cos ψ (1 − cos ψ) sin 3 ψ dψ.(96) Interestingly, from Eqs. (91) to (96), Eq. (95) is the only one in which the parameter α is present. Application: Gaussian density of natural frequencies In this section, we illustrate our analytical result assuming that natural frequencies are distributed according to the standard normal density (See Eq. (55)). Since a complete plot of G cannot be drawn due to the delta term, we show in Figs. 2 and 3 only graphs of S(K, α) and G D . The graphs come from Eqs. (92) and (93). Each graph consists of 250 points. 2(a) shows graphs of G D with α = 0. This is the case for which the KS model reduces to the Kuramoto model. Each graph is plotted with K fixed. The dashed vertical line points out the delta-term position, given by Ω = 0. The area below the curve connecting neighboring points of a graph of G D corresponds to the fraction of D oscillators. Fig. 2(a) indicates that the fraction of D oscillators diminishes as K increases. In Fig. 2(b), we show how S(K, 0), the fraction of S oscillators for α = 0, varies with K. For values of K less than a critical value K c 1.5957 3 , S(K, 0) = 0, i.e. oscillators are all of D type. This is consistent with Fig. 1(a) and Eq. (92): for K < K c and α = 0, we have R = 0, and then the integration range of Eq. (92) collapses, leading to S(K, 0) = 0. Note also that, in Fig. 2(a), for K = 1.596, which is slightly greater than K c , the graph of G D resembles, except near ν = 0, the profile of the standard normal density. As K increases beyond K c , Fig. 2(b) shows an increase of S(K, 0). The plots in Figs. 3(a) show the effects of the parameter α on S(K, α). Each graph is plotted with K fixed and α varying in the interval [−1.0, +1.0]. For higher values of K, S(K, α) has non-zero values over wider ranges of α, and the maximum points of S(K, α) are positioned at higher levels. The graphs of S(K, α) are similar to those of R, shown in Fig. 1(c): if R = 0, then S(K, α) = 0, as expected from Eq.(92); for R > 0, the variation of S(K, α) follows closely that of R, suggesting a monotonic dependence of S(K, α) on R. (b-e) Graphs of g (in blue) and G D . As in Fig. 2(a), the dashed vertical lines represent the delta term of G. R = 0 for a given pair (K, α), a graph of g, instead of a G D graph, is shown (with blue dots) to illustrate the profile of the instantaneous-frequency distribution in the incoherent state. The set of K values used is the same as the one of Fig. 3(a). Figures 3(b-e) show that, for α = 0, G D has non-symmetric profiles. Yet, for α = 0, G D profiles are symmetric (See Fig. 2(a)). As discussed in Ref. [26], if g has a profile with a symmetry axis (e.g. the standard normal density), G D profiles have the same symmetry axis in the Kuramoto model (KS model with α = 0). Another property of the graphs in Figs. 3(b-e) is that, for α = 0, the dashed-line position also differs from zero. The dashed-line position, given by Ω, has a sign opposite to the sign of α, as shown in Fig. 1(d). A remarkable feature of the graphs in Figs. 3(b-e) is the discontinuity at ν = Ω. For finite α, the left and right non-symmetric branches of the graphs are visibly disconnected. For α = 0, the symmetric branches seem connected at ν = Ω = 0, but, from Eq. (93), we know that G D (0) = 0, and the graphs indicate that G D (0 + ) = G D (0 − ) > 0. Thus, a discontinuity seems to occur even in the symmetric profiles. Numerical analysis In Figs. 4 and 5, we show graphs of G D (in orange) and normalized histograms (in blue) of instantaneous frequencies obtained in numerical simulations of the KS model. By numerical simulation we mean numerical integration of the N differential equations of the KS model from the initial time instant 0 to a final one T > 0. All simulations were performed with the numerical library ODEPACK [30]. The ODEPACK's solver used is LSODA, a hybrid implementation of Adams and BDF methods [31]. Again, we assume a standard normal density of natural frequencies. Before initiating a simulation, two random samples are generated: a sample {ω i } N i=1 of random natural frequencies and another one of random initial phases, {θ i (0)} N i=1 . A standardnormal generator of random number s is used to create the sample of natural frequencies. The random initial phases are generated according to a uniform distribution in the interval −π < θ i (0) < π. After a simulation is concluded, a normalized histogram (a histogram with the unit area) with bars of equal width is built from the instantaneous frequencies θ i (T ) We avoided showing the peaks entirely because their height are much higher than the other histogram bars. The smaller the width of the peak, the higher its height. The peaks emerge due to the large accumulation of instantaneous frequencies, most of them from S oscillators. We draw the reader's attention to the fact that the area of the peak is not exactly equal to S(K, α): a small part of the peak area contributes to the fraction of D oscillators, given by 1 − S(K, α). Yet, the peak area can be a good approximation to S(K, α), depending on how small is the peak width. A noteworthy property is the seeming reflection symmetry around the peaks. The reflection symmetry is related to sign inversions in the parameter α: if the signal of α is inverted, keeping fixed its absolute value, the right (or left) side of the graphs and histograms are reflected on the left (or right) side. The same type of symmetry is observed in Figs. 5(a-d), where K = 1.60, and the set of α values is the same as in Figs. 4(a-d). In Figs. 5(a) and 5(d) we plot g (in red) instead of G D , since R = 0 (incoherent state) for the values of K and α considered (as indicated in Fig. 1(c)). The curves of g fit histograms showing no clearly visible peaks. A synchronization state is represented in Figs. 5(b) and 5(c). The curves of G D (in orange) also fit the histograms, but small deviations occur due to time fluctuations in the histogram bars. Such fluctuations and deviations are also present in the histograms of Figs. 6(a-c). The deviations occur near the synchronization peak. The sequence of figures 6(a), 6(b) and 6(c) depicts the evolution of the instantaneous-frequency distribution in histograms from three different time instants of the same simulation: T 1 = 500, T 2 = 1000, and T 3 = 1500. The simulation time interval is [0, T 3 ], and we define K = 1.64, α = 0.25, and N = 5 × 10 5 . Again, initial phases and natural frequencies are random numbers generated according to the uniform and standard normal distributions. The histograms of Figs .6(d), 6(e), and 6(f) comes from a simulation similar to the previous one. The difference is that the number of oscillators is three times higher, requiring a new random sample of initial phases and natural frequencies. The histograms show a more stationary profile in the sequence of time instants T 1 , T 2 , and T 3 . In addition, the graphs of G D are in better agreement with the histogram profiles. Conclusions In this work, we showed how to obtain the density of instantaneous frequencies in the Kuramoto-Sakaguchi model. The density of instantaneous frequencies is a stationary probability density function with a complex formula given by the sum of two terms: a Dirac-delta-type function and a discontinuous one. The Dirac-delta term is located in the synchronization frequency and carries information about the number of synchronized oscillators. The other term is discontinuous at a point defined also by the synchronization-frequency value. The discontinuous term have profiles with varied and unexpected shapes, and the area below them gives the fraction of out-of-synchrony oscillators. Our formula is a generalization of the one obtained in Ref. [26] for the Kuramoto model (i.e. Kuramoto-Sakaguchi model with a phase-lag equal to zero). The formulas are mathematically quite similar, particularly concerning the property that the generalization has no explicit dependence on the phase-lag parameter. Indeed, the dependence on the phase-lag is implicit and takes place through the order parameter and the synchronization frequency. Contrary to what was shown for the Kuramoto model in Ref. [26], natural-frequency densities with a symmetry axis in the synchronization frequency does not imply the same type of symmetry in the density of instantaneous frequencies for the Kuramoto-Sakaguchi model. However, this density exhibits a reflection symmetry, characterized by a flip of the density profile induced by sign inversions in the phase-lag parameter. Our result is in accordance with numerical simulations of the Kuramoto-Sakaguchi model, provided simulations are performed with a large enough number of oscillators. If the number of oscillators is such that the instantaneous-frequency distribution from simulations (normalized histograms) shows significant non-stationary behavior, one observes a less robust fit to simulation data. But increasing the number of oscillators suppresses the non-stationarity and improves the quality of the theoretical prediction. These finite-size effects, also analyzed in the context of the Kuramoto model [26], are expected to occur in the Kuramoto-Sakaguchi model. Our result is based on the Kuramoto-Sakaguchi theory, which includes equilibrium assumptions devised under the requirement of an infinite number of oscillators and infinitely long times. New research directions can be taken with this work as a starting point. A mathematicallyoriented subject would be finding asymptotic analytical properties of the density function near the synchronization frequency as well as in its tails. Extending the study presented here, considering other types of natural-frequency densities, including non-symmetric and non-unimodal ones, is also an interesting topic. In addition, our formula can be used to determine the instantaneous-frequency statistical moments, which can then be compared to the moments of the natural frequencies and used to compute relevant quantities such as expected values and variances. More importantly, we envision a non-stationary model capturing the collective dynamics of the instantaneous frequencies. We conjecture that the model solution would be a timedependent density of instantaneous frequencies, and our formula would reflect the solution behavior in the long-time limit. Figure 1 : 1(a) Order-parameter diagram. (b) Synchronization-frequency diagram. (c) Graphs of R for constant K and varying α. (d) Graphs of Ω for constant K and varying α. The values of R and Ω are numerical approximations to solutions of the system of Eqs. (53-54) By solving the integral in Eq. (69), we get p S (ψ|ω) = δ {af ν [ψ * (ω)]}. So, from (58), (61), (62), and (63), p S (ψ|ω) = δ(ψ), We introduced the function Θ(− |ν − Ω|) in (93) so that G D (Ω) = 0, as stated earlier. The quantities R and Ω, which appear in Eqs. (91), (92), (93), and (94), form the solution of Eq. (37), namely Figure 2 : 2(a) Graphs of G D for α = 0. The dashed vertical line represents the delta term in (91). (b) Fraction of synchronized oscillators, S(K, α), for α = 0. Fig. Fig. 2(a) shows graphs of G D with α = 0. This is the case for which the KS model reduces to the Kuramoto model. Each graph is plotted with K fixed. The dashed vertical line points out the delta-term position, given by Ω = 0. The area below the curve connecting neighboring points of a graph of G D corresponds to the fraction of D oscillators. Fig. 2(a) indicates that the fraction of D oscillators diminishes as K increases. In Fig. 2(b), we show how S(K, 0), the fraction of S oscillators for α = 0, varies with K. For values of K less than a critical value K c 1.5957 3 , S(K, 0) = 0, i.e. oscillators are all of D type. This is consistent with Fig. 1(a) and Eq. (92): for K < K c and α = 0, we have R = 0, and then the integration range of Eq. (92) collapses, leading to S(K, 0) = 0. Note also that, in Fig. 2(a), for K = 1.596, which is slightly greater than K c , the graph of G D resembles, except near ν = 0, the profile of the standard normal density. As K increases beyond K c , Fig. 2(b) shows an increase of S(K, 0). The plots in Figs. 3(a) show the effects of the parameter α on S(K, α). Each graph is plotted with K fixed and α varying in the interval [−1.0, +1.0]. For higher values of K, S(K, α) has non-zero values over wider ranges of α, and the maximum points of S(K, α) are positioned at higher levels. The graphs of S(K, α) are similar to those of R, shown in Fig. 1(c): if R = 0, then S(K, α) = 0, as expected from Eq.(92); for R > 0, the variation of S(K, α) follows closely that of R, suggesting a monotonic dependence of S(K, α) on R. Graphs of G D and g are shown in Figs. 3(b-e) for different values of K and α. Given K and α, if R > 0, we show the corresponding G D graph and dashed line with the same color. If Figure 3 : 3Graphs of G D and g are shown in Figs. 3(b-e) for different values of K and α. Given K and α, if R > 0, we show the corresponding G D graph and dashed line with the same color. (a) Fraction of synchronized oscillators, S(K, α). and the numerically-obtained set of phases {θ i (T )} N i=1 . For the histograms of Figs. 4 and 5, the simulation parameters are N = 5 × 10 5 and T = 5 × 10 2 . The occurrence of both S and D oscillators is depicted in the histograms of Figs. 4(a-d), where K = 1.80, and α takes values in the set {−0.5, −0.25, +0.25, +0.5}.Figs. 4(a-d)show that the graphs of G D are in agreement with the histogram profiles. The histograms exhibit, at ν = Ω, tall and thin bars, which we refer to as peaks. The peaks are related to the delta term of the function G (See Eq. (91)). Figure 4 : 4In blue: normalized histograms of instantaneous frequencies obtained in numerical simulations of the Kuramoto-Sakaguchi model. In orange: graphs of G D . For all simulations and graphs, K = 1.80. Model size: N = 5 × 10 5 . Simulation time: T = 5 × 10 2 . Figure 5 : 5In blue: normalized histograms of instantaneous frequencies obtained numerically. In red: graphs of g (a,d). In orange: graphs ofG D (b,c). The values of α are the same as inFig. 4. In all simulations and graphs, K = 1.67. Model size: N = 5 × 10 5 . Simulation time: T = 5 × 10 2 . Figure 6 : 6In blue: normalized histograms of instantaneous frequencies at the time instants T 1 , T 2 , and T 3 obtained in two simulations of the KS model performed in the time interval [0, T 3 ] and with different numbers of oscillators: one with N = 5 × 10 5 and the other with N = 1.5 × 10 6 . The histograms in (a), (b) and (c) result from the simulation with the smallest N , while those in (d), (e), and (f), from the one with the highest N . For all simulations and graphs of G D (in orange), K = 1.64 and α = 0.25. Table 1 : 1Endpoints of the open interval D(ν) = ( ω − ν , ω + ν ), defined by For the sake of simplicity, hereafter we always use the term "density" to refer to a probability density function. We calculated the critical value, K c , using the formula K c = 2 πg(Ω) (See Ref.[11] for details). An approximate value for K c can also be determined from Eq. (95) with α = 0. 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[ "Spinor-helicity formalism for massive and massless amplitudes in five dimensions", "Spinor-helicity formalism for massive and massless amplitudes in five dimensions", "Spinor-helicity formalism for massive and massless amplitudes in five dimensions", "Spinor-helicity formalism for massive and massless amplitudes in five dimensions" ]
[ "Marco Chiodaroli [email protected] \nDepartment of Physics and Astronomy\nUppsala University\n75108UppsalaSweden\n", "Murat Günaydin [email protected] \nInstitute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA\n", "Henrik Johansson [email protected] \nDepartment of Physics and Astronomy\nUppsala University\n75108UppsalaSweden\n\nStockholm University\nNordita\n\nKTH Royal Institute of Technology\nHannes Alfvéns väg 1210691StockholmSweden\n", "Radu Roiban \nInstitute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA\n", "Marco Chiodaroli [email protected] \nDepartment of Physics and Astronomy\nUppsala University\n75108UppsalaSweden\n", "Murat Günaydin [email protected] \nInstitute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA\n", "Henrik Johansson [email protected] \nDepartment of Physics and Astronomy\nUppsala University\n75108UppsalaSweden\n\nStockholm University\nNordita\n\nKTH Royal Institute of Technology\nHannes Alfvéns väg 1210691StockholmSweden\n", "Radu Roiban \nInstitute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA\n" ]
[ "Department of Physics and Astronomy\nUppsala University\n75108UppsalaSweden", "Institute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA", "Department of Physics and Astronomy\nUppsala University\n75108UppsalaSweden", "Stockholm University\nNordita", "KTH Royal Institute of Technology\nHannes Alfvéns väg 1210691StockholmSweden", "Institute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA", "Department of Physics and Astronomy\nUppsala University\n75108UppsalaSweden", "Institute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA", "Department of Physics and Astronomy\nUppsala University\n75108UppsalaSweden", "Stockholm University\nNordita", "KTH Royal Institute of Technology\nHannes Alfvéns väg 1210691StockholmSweden", "Institute for Gravitation and the Cosmos\nThe Pennsylvania State University\n16802University ParkPAUSA" ]
[]
Five-dimensional gauge and gravity theories are known to exhibit striking properties. D = 5 is the lowest dimension where massive tensor states appear naturally, providing a testing ground for perturbative insights into six-dimensional tensor theories. Five-dimensional supergravities are highly constrained and admit elegant geometric and algebraic formulations, with global symmetries manifest at the Lagrangian level.In this paper, we take a step towards the systematic investigation of amplitudes in five dimensions, and present a five-dimensional version of the spinor-helicity formalism, applicable to massless, massive and supersymmetric states. We give explicit representations for on-shell spinor and polarization variables such that the littlegroup symmetry and gauge redundancy are manifest. Massive self-dual tensor states are discussed in some detail, as well as all the on-shell supermultiplets that can appear in matter-coupled gauge and supergravity theories. As a byproduct of considering supersymmetry in the presence of central charge, we obtain massless ten-dimensional Majorana-Weyl spinors as products of five-dimensional massive spinors.We present compact expressions for superamplitudes at multiplicity three and four, including several novel superamplitudes that either do not straightforwardly uplift to six dimensions, or have not appeared in the six-dimensional literature. We discuss several examples of five-dimensional double-copy constructions in the context of gravitational theories with massive vectors and tensors, illustrating that the formalism we construct can also be used to considerably streamline the double-copy construction of N = 2 Maxwell-Einstein supergravities.Conclusion 43A Gamma matrices and reality properties for on-shell spinors 46
10.1007/jhep02(2023)040
[ "https://export.arxiv.org/pdf/2202.08257v1.pdf" ]
246,867,390
2202.08257
be3eba8babca3724d28f869030f66c3f2b8f58fd
Spinor-helicity formalism for massive and massless amplitudes in five dimensions 16 Feb 2022 Marco Chiodaroli [email protected] Department of Physics and Astronomy Uppsala University 75108UppsalaSweden Murat Günaydin [email protected] Institute for Gravitation and the Cosmos The Pennsylvania State University 16802University ParkPAUSA Henrik Johansson [email protected] Department of Physics and Astronomy Uppsala University 75108UppsalaSweden Stockholm University Nordita KTH Royal Institute of Technology Hannes Alfvéns väg 1210691StockholmSweden Radu Roiban Institute for Gravitation and the Cosmos The Pennsylvania State University 16802University ParkPAUSA Spinor-helicity formalism for massive and massless amplitudes in five dimensions 16 Feb 2022Prepared for submission to JHEP Five-dimensional gauge and gravity theories are known to exhibit striking properties. D = 5 is the lowest dimension where massive tensor states appear naturally, providing a testing ground for perturbative insights into six-dimensional tensor theories. Five-dimensional supergravities are highly constrained and admit elegant geometric and algebraic formulations, with global symmetries manifest at the Lagrangian level.In this paper, we take a step towards the systematic investigation of amplitudes in five dimensions, and present a five-dimensional version of the spinor-helicity formalism, applicable to massless, massive and supersymmetric states. We give explicit representations for on-shell spinor and polarization variables such that the littlegroup symmetry and gauge redundancy are manifest. Massive self-dual tensor states are discussed in some detail, as well as all the on-shell supermultiplets that can appear in matter-coupled gauge and supergravity theories. As a byproduct of considering supersymmetry in the presence of central charge, we obtain massless ten-dimensional Majorana-Weyl spinors as products of five-dimensional massive spinors.We present compact expressions for superamplitudes at multiplicity three and four, including several novel superamplitudes that either do not straightforwardly uplift to six dimensions, or have not appeared in the six-dimensional literature. We discuss several examples of five-dimensional double-copy constructions in the context of gravitational theories with massive vectors and tensors, illustrating that the formalism we construct can also be used to considerably streamline the double-copy construction of N = 2 Maxwell-Einstein supergravities.Conclusion 43A Gamma matrices and reality properties for on-shell spinors 46 Introduction Over the past decades, four-dimensional spinor-helicity variables have been crucial for streamlining scattering amplitude calculations, uncovering novel structures in gauge theories and gravity. They are a fundamental ingredient of the Parke-Taylor formula for tree-level MHV amplitudes [1], are closely related to the twistor-space description of amplitudes [2][3][4][5], and provide a simple presentation for modern on-shell recursion relations [4,6]. The built-in advantage of the spinor-helicity formalism consists in providing a covariant and dimension-specific solution for the on-shell conditions, leading to simple expressions for momenta and physical polarization vectors and tensors. In turn, this leads to the observation that scattering amplitudes become simple once they are written in terms of physical (i.e. on-shell) quantities. Spinor-helicity methods have been used most successfully for amplitudes involving four-dimensional massless states. Detailed extensions of the formalism have been introduced in three, six and ten dimensions [7][8][9][10][11], as well as for massive fields in four dimensions [12] (see also Refs. [13][14][15][16][17][18]). While helicity cannot be defined in higher dimensions and for massive fields, the term higher-dimensional spinor-helicity is commonly used (with a slight abuse of notation) to refer to the higher-dimensional extension of the 4D spinor-helicity formalism. The invariant content of the higherdimensional approach is to properly classify all variables by both their little-group and Lorentz group representations, and as such spinor/little-group formalism can be used as a synonym. Five-dimensional theories are interesting because they populate a special corner of the parameter space of known theories. On the one hand, all theories in five dimensions can be dimensionally reduced to four dimensions. This reduction, however, obscures properties that are indigenous to five dimensions, such as supergravity U-duality symmetries being symmetries of the action rather than of the equations of motion. On the other hand, not all five-dimensional theories can be lifted to six dimensions, either because of their field content, or because of their interactions. Thus, studying effects that depend strongly on the dimensional regulator, such as rational terms in the amplitudes, may be subtle. It may be more appropriate to make all variables explicitly five dimensional. In this work, we tackle the problem of defining natural variables for scattering amplitudes in five spacetime dimensions, and initiate the study of the 5D spinor-helicity formalism. Large classes of N = 2 Maxwell-Einstein and Yang-Mills-Einstein supergravities are naturally formulated in five dimensions [19][20][21][22][23][24][25][26]. Their Lagrangians are considerably simpler than those of their four-dimensional relatives, and yet present a far richer structure than theories that uplift to higher dimension. While the study of scattering amplitudes in these theories has been yielding interesting results [27][28][29][30][31][32][33][34][35][36] (see also Ref. [37] for a comprehensive review), until now the analysis relied on using four-dimensional spinor-helicity methods after dimensional reduction. To streamline and advance the study of amplitudes in five-dimensional theories, it is therefore desirable to develop tools that are designed to work directly in five dimensions, bypassing the need for dimensional reduction. Additionally, various gaugings of maximal supergravity have been explicitly studied in five dimensions [38][39][40][41], including some very recent results [42,43]. While the application of amplitude methods to the study of gauged maximal supergravity is still in its infancy [44], the five-dimensional spinor-helicity methods developed here should provide a valuable tool for advanced calculations [45]. A central aspect of D = 5 is that it is the lowest dimension where massive anti-symmetric tensor states and fields can arise, which cannot be dualized to other types of interacting fields [22][23][24][25]. Studying theories with massive tensors in five dimensions may prove to be crucial for understanding six-dimensional theories with tensors, possibly including the famed (2, 0) theory. The study of amplitudes in the (2, 0) theory has so far presented notable difficulties, for example, due to the lack of suitable three-point amplitudes that can be used as building blocks [46] (see also Refs. [47,48]). 1 In many cases, the study of amplitudes in five dimensions has required adapting tools and methods developed in six dimensions. This includes dimensional reduction of the spinor-helicity formalism from six dimensions [46], techniques based on scattering equations [58,59], as well as ambitwistor strings [60,61]. Venturing away from Minkowski backgrounds, five-dimensional theories are of great interest from the perspective of the AdS/CFT correspondence [62][63][64]. In its simplest formulation it relates correlation functions in N = 4 super-Yang-Mills theory to boundary correlation functions in 5D gauged supergravity given by the reduction of type IIB supergravity on a five-sphere [65,66]. Supergravity fields are then naturally organized with a split notation in which spacetime indices along the compact directions are related to R-symmetry indices. The massive 5D spinors with central charge that we will study have a similar organization, and it is likely that they can be suitably adapted to describe states in AdS 5 × S 5 background. This paper is structured as follows. In Section 2, we provide the basis of the 5D formalism we are using and introduce the relevant notation. In Section 3, we discuss supersymmetry and the organization of the on-shell states into 5D superfields. We also briefly outline the relation between our 5D superfields and 4D and 6D superfields. In Section 4, we discuss superamplitudes at three and four points for interacting vector multiplets. Finally, in Section 5, we discuss theories of gravitationally interacting fields, including massive tensors, from the point of view of the double-copy construction [67][68][69]. Among other things, we obtain compact expressions for amplitudes for Maxwell-Einstein and Yang-Mills-Einstein theories, recovering previous results in a considerably simpler form. We then conclude with a discussion of open problems and direction for future investigation. Five-dimensional spinor-helicity formalism To make the 5D spinor-helicity notation accessible, in this section we spell out the necessary details, including the explicit parametrization for the frequently-encountered spinors and polarization vectors and tensors. Five-dimensional on-shell spinors: massless and massive Working in mostly-minus signature (+−−−−), we consider the SO(1, 4) → SO (1,3) decomposition of a 5D momentum and gamma matrices, p µ = (pμ, p 4 ), γ µ = (γμ, γ 4 ), respectively. Here 4D indices are barred for clarity,μ = 0, . . ., 3, and what we call γ 4 is the usual γ 5 matrix in 4D. Thus, in the Weyl basis, we have / p ≡ p µ γ µ = −ip 4 p · σ p ·σ ip 4 ,(2.1) where σμ = (1, σ i ) andσμ = (1, −σ i ) are the 4D sigma matrices. Because of the isomorphism SO(1, 4) ∼ = USp(2, 2), we can lower the upper index on / p and write the 5D momentum as an antisymmetric 4 × 4 matrix, p AB = p µ γ µ AB ≡ −p µ (γ µ ) C A Ω CB = −(/ p Ω) AB . (2.2) We choose the symplectic metric Ω of USp(2, 2) to be block diagonal, Ω AB = ǫ αβ 0 0 −ǫαβ = Ω BA ,(2.3) where A, B are fundamental USp(2, 2) indices decomposed as A = α ⊕α in terms of the SL(2, C) indices of the Levi-Civita symbols and sigma matrices σμ αβ ,σβ ᾱ µ . We use Ω to lower and raise indices, according to the left-multiplication convention, e.g. X A = Ω AB X B , X A = Ω AB X B . (2.4) The two-dimensional Levi-Civita symbol is normalized as ǫ 12 = ǫ 21 = 1, implying that Ω AC Ω CB = δ B A . The gamma matrices with lowered indices, γ µ AB = Ω BC (γ µ ) C A , are antisymmetric and Ω-traceless, γ µ AB Ω BA = 0. Furthermore, in addition to the Clifford algebra {γ µ , γ ν } = 2η µν , they obey the special 5D identity (γ µ ) B A (γ µ ) D C = −2Ω AC Ω BD + 2δ D A δ B C − δ B A δ D C . (2.5) See Appendix A for further details on the gamma matrices. The determinant of p AB evaluates to Det(p AB ) = ( 1 4 p AB p AB ) 2 = (p 2 ) 2 . Thus, for massless 5D momentum, p 2 = 0, the matrix p AB has rank two. It can therefore be factorized over the SU(2) little group using on-shell USp(2, 2) spinors, p AB = |p a A |p a B or / p A B ≡ p A B = |p a A p a | B ,(2.6) where a, b, . . . are SU(2) little-group indices and the "bra" is defined as p a | B = Ω BA |p a A . Little-group indices can be lowered and raised through left-multiplication, similarly to Eq. (2.4), by the Levi-Civita symbol ǫ ab = ǫ ba . An explicit parametrization of the massless USp(2, 2) spinor is |p a A =      p 0 + p 3 0 p 1 + ip 2 − ip 4 p 0 +p 3 ip 4 − p 1 −ip 2 p 0 +p 3 0 1      , (2.7) where we have scaled the little-group components to make the spinor entries free of square roots. Using spinors that are rational functions of the momentum components will greatly simplify practical calculations. The parametrization (2.7) is such that the contraction of two USp(2, 2) spinors of the same massless momentum gives a vanishing result, p a |p b ≡ Ω BA |p a A |p b B = 0 . (2.8) Eq. (2.8) implies that the spinor satisfies the massless Dirac equation, p AB |p a B = 0 , (2.9) and hence it is an on-shell spinor. 2 Here p AB is obtained by raising the indices in Eqs. (2.1)-(2.2). We proceed to extend the construction above to massive 5D momenta obeying p 2 = m 2 . The matrix p AB now has rank 4, so it can be factorized over the massive little group SO(4) ∼ SU(2) × SU (2). To construct this factorization, we note that we can always split p into two massless momenta k and q, p µ = k µ + m 2 q µ , (2.10) where q µ is a reference null-vector that satisfies 2p · q = 2k · q = 1, and k µ is defined by the above relation. Since both k and q are massless, we can reuse for each of them the parametrization from Eq. (2.7). After contracting Eq. (2.10) with the gamma matrices, we get an expression that can be factorized over the SU(2) × SU(2) little group, 11) or, alternatively, p AB = |k a A |k a B + m 2 |q a A |q a B = 1 2 |p a A |p a B + 1 2 |pȧ] A |p˙a] B ,(2.p B A = 1 2 |p a A p a | B + 1 2 |pȧ] A [p˙a| B , (2.12) 2 Here we treat the on-shell spinor as a function of the momentum following the parametrization in Eq. (2.7). However, if the spinor is used to define the null momentum, then Eq. (2.8) becomes a one-parameter constraint. The presence of this constraint differentiates the 5D spinor-helicity formalism from its 4D and 6D relatives. where we have defined new massive spinors |p a A = |k a A + m|q a A , |p˙a] A = |k˙a A − m|q˙a A . (2.13) While the little-group indices of the massless spinors run over the diagonal SU(2) subgroup, here we assign the indices a andȧ to run over the left and right factors of the massive little group SU(2) × SU(2), respectively. Let us further spell out the properties of the reference spinor |q a A . We will demand that it is normalized relative to the |k a A spinor, to satisfy k a |q b = ǫ ab . (2.14) Indeed, this relation can be solved as |q a = |q b q|k −1 bc ǫ ac , (2.15) where |q b is an arbitrary massless USp(2, 2) spinor, and q|k −1 bc is the matrix inverse of k c |q b . It then follows that q µ = 1 4 q a |γ µ |q a =q µ /(2q · p). In particular, it implies that 2p · q = 2k · q = 1 2 k a |q b q b |k a = 1 as desired. The constraint (2.14) is stronger than the momentum constraint 2k · q = 1, as it also imposes an alignment of the little groups of the k, q vectors. We will assume that this stronger constraint holds throughout the paper. The constraint in Eq. (2.14) implies that the contractions of the two spinors corresponding to a massive momentum are p a |p b = 2mǫ ab , [p˙a|p˙b] = −2mǫ˙a˙b , p a |p˙b] = 0 . (2.16) It therefore follows that the massive spinors obey the massive Dirac equation, p AB |p a B = −m p a | A , p AB |p˙a] B = m [p˙a| A . (2.17) While it is tempting to identify the above spinors with the standard particle u and antiparticle v solutions to the Dirac equation, this is not quite accurate. As shown in the Appendix A, the reality properties of the square and angle spinors are compatible with them being symplectic-Majorana spinors. In terms of the pair of massless spinors that satisfy Eq. (2.14), the USp(2, 2) identity operator can be written as |q a A k a | B + |k a A q a | B = δ B A ,(2.18) from which the completeness relations of the massive spinors follow, |p a A p a | B = p B A + m δ B A ≡ 2m(P + ) B A , |pȧ] A [p˙a| B = p B A − m δ B A ≡ 2m(P − ) B A . (2.19) The projectors satisfy P ± P ± = ±P ± and P ± P ∓ = 0. The above relations are analogous to the completeness relations of the standard u and v spinors. Before ending this section, we note that there exists a reference spinor that is particularly natural to work with. For massless momenta k µ , a convenient parametrization of the reference spinor is |q a A =      0 0 0 1 k 0 +k 3 1 0 0 0      . (2.20) which corresponds to the null vector q µ = 1 2(k 0 +k 3 ) (1, 0, 0, −1, 0), and the global null vectorq µ = (1, 0, 0, −1, 0). However, this spinor cannot be obtained by plugging the momentum q µ into the parametrization in Eq. (2.7), since this gives a singular expression. Indeed, the above reference spinor is located at infinity in the parametrization (2.7). This choice has the benefit that inner products are simple, q a i |q b j = 0 , k a i |q b j = 0 1 − k 0i +k 3i k 0j +k 3j 0 (2.21) where i, j are the particle labels. Furthermore, with the choice (2.20), the massive-spinor parametrizations simplify considerably. From Eq. (2.13) and the identity k 0 + k 3 = p 0 + p 3 , we obtain the massive spinor |p a A =      p 0 + p 3 0 p 1 + ip 2 m−ip 4 p 0 +p 3 m + ip 4 − p 1 −ip 2 p 0 +p 3 0 1      . (2.22) The spinor |p˙a] A can be obtained by flipping the sign of the mass as |p] = |p m→−m . We will use these massive spinors in the remainder of the paper. When displaying amplitudes in the following sections, we will find convenient to dress the free little-group indices with bosonic auxiliary variables z a andzȧ. In formulae in which we do not wish to explicitly display these variables, we will adopt the following short-hand notation, | i ≡ |k a i z ia , | i ≡ |p a i z ia , | i ] ≡ |p˙a i ]z iȧ ,(2.23) where the index i is the particle label. However, for the reader's convenience, we will keep the little-group indices and momentum explicit for the remainder of this section. Massless and massive polarization vectors An advantage of spinor-helicity variables is that they provide convenient parametrizations for general asymptotic states. Here we construct 5D polarization vectors, using the previously-introduced spinor variables. We start with the polarizations of massless vectors. The only natural choice is 3 ε µ ab (k, q) = k (a |γ µ |q b) √ 2 = − q (a |γ µ |k b) √ 2 . (2.24) where a, b are the SU(2) little-group indices and the symmetrization is normalized as A (a B b) = 1 2 A a B b + 1 2 A b B a . The reference spinor |q b must satisfy Eq. (2.14) or, equivalently, Eq. (2.15). The massless polarization vector constructed in this way is transverse, k · ε ab (k, q) = q · ε ab (k, q) = 0, because |k and |q obey (by construction) the massless Dirac equation. It also satisfies the following completeness relations: ε µ ab (k, q)ε µ,cd (k, q) = − 1 2 (ǫ ac ǫ bd + ǫ ad ǫ bc ) , ε µ ab (k, q)ε ν,ab (k, q) = −η µν + 2k (µ q ν) . (2.25) Any massless polarization can be mapped to one written in terms of the reference spinor given in Eq. (2.20) with a gauge transformation. 4 Using the massive on-shell spinors we have constructed in the previous section, we can also construct polarization vectors for massive vectors (W bosons), ε µ aȧ (p) = − p a |γ µ |pȧ] 2 √ 2m . (2.26) By construction, these are transverse as before, p · ε aȧ (p) = 0, and span the SO(4) little group, ε µ aȧ (p)ε µ,bḃ (p) = −ǫ ab ǫ˙a˙b . (2.27) Moreover, they obey the Minkowski-space completeness relation appropriate to vectors transverse to a massive on-shell momentum, − ε µ aȧ (p)ε ν,aȧ (p) = η µν − p µ p ν m 2 ≡η µν ,(2.28) which is also the massive-vector physical-state projector. We may also construct a linearized field strength for the massive vector using the spinor. It takes the form F µν aȧ = − p a |γ µν |pȧ] 2 √ 2 = 2 p [µ ε ν] aȧ ,(2. 29) 3 Upon dimensional reduction, these expressions reproduce the ones from Ref. [70] with the identifications ε µ 11 → ε µ − , ε µ 22 → ε µ + . 4 Note that enacting different gauge transformations for the different polarizations does not change amplitudes and hence does not break the SU (2) little-group symmetry for the asymptotic states. where γ µν = 1 2 (γ µ γ ν − γ ν γ µ ) are the rank-2 elements of the Clifford algebra as usual, and the antisymmetrization includes a factor of 1/2. In the massless limit, the four on-shell vector states in Eq. (2.26) split into three symmetric and one antisymmetric one with respect to the diagonal little group. The latter mode is divergent in the massless limit, m 2 (ε µ aḃ (p) − ε µ bȧ (p)) m→0,ȧ→a,ḃ→b = − 1 √ 2 p µ ǫ ab . (2.30) The symmetric states reduce to the massless polarization vector in Eq. (2.24), 1 2 ε µ aḃ (p) + ε µ bȧ (p) m→0,ȧ→a,ḃ→b = ε µ ab (k, q) . (2.31) The linearized field strength for the massive vector has no mass factor in the denominator, so it has a smooth massless limit. In giving explicit representations for amplitudes in 5D, it will be convenient to dress the little-group indices with (Grassmann-even) auxiliary variables z a ,z˙a, so that external states carry dressed polarization vectors in the massless and massive cases, ε µ i = ε µ ab (k i , q i )z a i z b i , ε µ i = ε µ aḃ (p i )z a izḃ i ,(2.32) where we also introduced particle labels i. An added benefit of this notation is that the auxiliary variables take care of the symmetrization of the little-group indices for the massless polarization. Tensor polarizations Five dimensions is the lowest dimension in which theories can exhibit asymptotic states corresponding to massive (anti)self-dual tensor fields [71], B µν = ± i 3!m ǫ µνρσλ H ρσλ ,(2.33) where H µνρ = ∂ [µ B νρ] is the field strength of the anti-symmetric tensor B µν and the 5D Levi-Civita symbol is normalized as ǫ 01234 = ǫ 01234 = 1. For an on-shell tensor with p 2 = m 2 and ∂ µ B µν = 0, the massive (anti)self-duality relation can also be written as ∂ ρ H ρµν = ∓ im 3! ǫ µνρσλ H ρσλ . (2.34) The variables introduced in previous sections allow us to construct polarization tensors obeying all the required physical constraints. For a massive self-dual and antiself-dual tensors they are ε µν ab (p) = p a |γ µν |p b 4 √ 2m , ε µν aḃ (p) = [pȧ|γ µν |p˙b] 4 √ 2m ,(2.35) respectively. These polarization tensors are transverse, p µ ε µν ab (p) = 0 , p µ ε µν aḃ (p) = 0 . (2.36) They also satisfy the little-group completeness relation for each SU(2) factor of the little group, ε µν aḃ (p)ε µν,ċḋ (p) = 1 2 (ǫȧ˙cǫ˙bḋ + ǫ˙aḋǫ˙b˙c) , ε µν ab (p)ε µν,cd (p) = 1 2 (ǫ ac ǫ bd + ǫ ad ǫ bc ) ,(2.37) as well as the Minkowski-space completeness relation in the space of transverse twotensors, ε µν ab (p)ε ρσ,ab (p) = 1 4 (η µρησν −η µσηρν ) − ǫ µνρσλ p λ 4m ≡ P SD , ε µν aḃ (p)ε ρσ,ȧḃ (p) = 1 4 (η µρησν −η µσηρν ) + ǫ µνρσλ p λ 4m ≡ P ASD . (2.38) Hereη µν is the massive-vector physical-state projector defined in Eq. (2.28), and P SD and P ASD are the (anti-)self-dual tensor projectors. The latter satisfy (P SD ) 2 = P SD , (P ASD ) 2 = P ASD and P SD P ASD = 0. The polarization tensors (2.35) satisfy the momentum-space form of the (anti)selfduality relations in Eq. (2.33), p ρ ǫ µνρσλ ε µν ab (p) = −2m ε σλ,ab (p) , p ρ ǫ µνρσλ ε µν aḃ (p) = 2m ε σλ,ȧḃ (p) . (2.39) The polarization tensors in Eq. (2.35) can also be obtained from the massive polarization vectors we have constructed through the relations ε µν ab (p) = − 1 √ 2 ε [µ aȧ (p)ε ν] bḃ (p)ǫ˙a˙b , ε µν aḃ (p) = 1 √ 2 ε [µ aȧ (p)ε ν] bḃ (p)ǫ ab . (2.40) This provides one way of realizing massive tensors in a gravitational theory using the double copy of massive vectors from a gauge theory. The massless limit of the polarization tensors (2.35) is singular because of the manifest factors of m −1 ; one can however dualize them to polarization vectors which are finite in the massless limit. For example, the self-dual tensor becomes ε µ ′ ab (k, q) = 2ε µν ab (p)k ρ q σ ǫ µνρσλ η λµ ′ m→0 ,(2.41) where p µ = k µ + m 2 q µ as before, and the polarization vector is identical to the one introduced in Eq. (2.24). For the anti-self-dual tensor ε µν aḃ (p), the same limit holds up to an overall minus sign. In fact, the self-dual and anti-self-dual tensor can be combined into a tensor that is finite in the massless limit, ε µν ab (k, q) = ε µν ab (p) − ε µν aḃ (p) ȧ,ḃ→a,b; m→0 . (2.42) This corresponds to the physical state of a massless vector, albeit written as tensor. Combining the above relations we have the following (non-linear) relation for the massless polarization vector, 43) which is equivalent to the statement that, in 5D, the double copy of two massless vectors contains a massless vector in its antisymmetric part. ε µ ′ ab (k, q) = − √ 2ε µ ac (k, q)ε ν bd (k, q)ǫ cd k ρ q σ ǫ µνρσλ η λµ ′ ,(2. Conformal generators In this subsection, we briefly comment on conformal symmetry in five and six dimensions, and their possible supersymmetric extensions. It is convenient to start in 6D, and infer details of the 5D case via dimensional reduction. The unitary representations of the 6D conformal group SO(6, 2) = SO * (8) and their extensions to superconformal algebras OSp(8 * |N ), with even N , using twistorial oscillators were studied in Refs. [72][73][74]. The generators of SO(6, 2) were realized as bilinears of twistorial oscillators and one finds that the group admits infinitely many representations, referred to as doubletons, that describe massless conformal fields of ever increasing spins. These massless conformal fields correspond to symmetric tensors of the spinor representation of the 6D Lorentz group SU * (4). For the doubletons, the Poincaré mass operator vanishes identically. Similarly, the conformal superalgebras OSp(8 * |N ), with even subalgebras SO * (8) ⊕ USp(N ), exist for any even N and admit infinitely-many conformally-massless unitary supermultiplets of ever increasing spins. In six dimensions, coordinates and momenta can be described by anti-symmetric tensors of twistorial variables in coordinate-space or momentum-space pictures. The twistorial oscillators formulated in Ref. [74] satisfy the commutation relations [ξ a , λB b ] = 1 2 δB A δ b a ,(2.44) whereÂ,B = 1, 2, 3, 4 are the spinor representation indices of SU * (4) and a, b = 1, 2. Then, the translation Pμ and special-conformal generators Kμ in six dimensions can be represented as PÂB = − 1 2 (Σμ)ÂBPμ = λ a λB b ǫ ba , KÂB = − 1 2 (Σμ)ÂBKμ = ξ a ξB b ǫ ba , (2.45) where Σμ are the anti-symmetric 6D sigma matrices andμ,ν, . . . = 0, 1, . . . , 5. Under commutation, they close into the Lorentz group SU * (4) generators MÂB and dilatation generator D. Under the 5D Lorentz group USp(2, 2), the 6D translation and special conformal generator decompose as 5 + 1 which corresponds to taking the symplectic traces in the antisymmetric tensor representation of SU * (4), PÂB = P AB + 1 4 Ω AB P , KÂB = K AB + 1 4 Ω AB K ,(2.46) where as before Ω AB is the USp(2, 2) invariant symplectic metric, and P AB = PÂB − 1 4 Ω AB P , P = 1 4 Ω AB PÂB = P 5 , (2.47) K AB = KÂB − 1 4 Ω AB K , K = 1 4 Ω AB KÂB . (2.48) The generators P AB and K AB close into the USp(2, 2) generators M A B and dilatation generator. Together, they generate the 5D conformal group SO(5, 2). The conformally-massless representations of SO(6, 2) lead to massive representations of SO(5, 2) under the above reduction since PμPμ = 0 =⇒ P µ P µ = (P 5 ) 2 = M 2 , (2.49) where M is the mass. Restricting to the massless SO(5, 2) representations is not possible without modifying the oscillator construction. It follows from general results that conformal groups SO(D, 2) in even dimensions D admit infinitely many unitary representations describing massless conformal fields of ever increasing spins, while, in odd dimensions, one finds only two unitary representations that describe massless conformal fields, namely scalar and spinor fields [75]. At a deeper level, this follows from the following fact. Massless conformal fields in any dimension [76] are described by the so-called minimal unitary representations of the conformal group SO(D, 2) and their deformations. A true minimal unitary representation of SO(D, 2) describes a massless conformal scalar in D dimensions. This minimal unitary representation admits infinitely many deformations labeled by the quadratic Casimir of the little group SO(D − 2) of massless particles in even dimensions. They describe massless conformal fields of ever increasing spins. On the other hand, in odd dimensions, the minimal unitary representation of the conformal group admits a single deformation that describes a massless spinor field [76]. Therefore, in five dimensions, only the scalar and spinor fields can be conformally massless [77]. The so-called two remarkable representations of the 3D conformal group Sp(4, R) = SO(3, 2) that were first studied by Dirac and labeled as singletons correspond simply to the minimal unitary representation and its spinorial deformation. The minimal unitary representation of the 5D conformal group SO(5, 2) and its deformation as well as their supersymmetric extension were studied in Refs. [77,78]. There exists a unique simple superconformal algebra in five dimensions, namely the exceptional superalgebra F (4) with the even subalgebra SO(5, 2) ⊕ SU(2) [79]. The minimal unitary supermultiplet of F (4) consists of two complex scalar fields in the doublet of R-symmetry group SU(2) R and a symplectic-Majorana spinor, which are both conformally massless [77]. The superalgebra F (4) is not a subalgebra of any of the 6D superconformal algebras OSp(8 * |N ). In this paper we will not study the F (4) superalgebra, however we will study the massive representations that come from 6D superconformal algebra. In particular, the Poincaré subalgebra of 6D superconformal algebra OSp(8 * |N ) descends directly to N extended Poincaré superalgebra in 5D with the momentum generator in the fifth spatial dimension acting as a singlet central charge under the R-symmetry group USp(N ). Supersymmetry We now consider the Poincaré superalgebras in five dimensions. N -extended Poincaré superalgebras with central charges have the general form [80] {Q I A , Q J B } = Ω IJ (γ µ ) AB P µ + Ω AB (Z IJ + Z Ω IJ ) , (3.1) where the supercharge Q I A carries a lower USp(2, 2) Lorentz index and an upper USp(N ) R-symmetry index. The Ω IJ is the symplectic metric of USp(N ), and the central charge is here decomposed into a singlet Z and non-singlet Z IJ of this group. The non-singlet central charge Z IJ is antisymmetric and Ω-traceless. Massless representations of the Poincaré superalgebras have vanishing central charges. In this section, we discuss in detail the realization of both the massless and massive cases for (minimal) N = 2 and N = 4 supersymmetry. 5 This is sufficient for describing super-Yang-Mills amplitudes and theories that have tensor multiplets. For gravitational theories, which may have up to N = 8 supersymmetry, we will infer the details via the double copy. All relevant 1/2-BPS on-shell superfields will be given, and their double-copy relations will be exhibited. N = 2 supersymmetry without central charge The N = 2 Poincaré superalgebra in five dimensions without central charges is given by {Q α A , Q β B } = ǫ αβ γ µ AB P µ ,(3.2) where α, β are SU(2) R-symmetry indices and A and B are as before the fundamental USp(2, 2) indices. The supercharge for a one-particle state of null momentum p µ can be written as a product of an on-shell spinor, |p a A , and Grassmann-odd variables θ α a , Q α A 1-pt = |p a A θ α a ≡ |q α A , (3.3) where we defined the Grassmann-odd symplectic-Majorana spinor |q α A . In term of this spinor, it follows that θ α a = q a |q α , where q a | is the usual reference spinor satisfying q a |p b = ǫ ab . As a consequence of (3.2), θ α a must satisfy the supersymmetry algebra projected on the little group, {θ α a , θ β b } = −ǫ αβ ǫ ab . (3.4) The possible solutions of Eq. (3.4) are related by SU(2) × SU(2) transformations, 6 θ ′α a = E b a θ β b E α β ,(3.5) where Det E = DetẼ = 1. For our purpose, it is convenient to choose a solution that maintains little-group covariance, at the expense of manifest R-symmetry. An explicit parametrization of θ α a that satisfies this algebra is given by the oscillator representation θ α a = (θ a ) α = η a − ∂ ∂η a , (3.6) where η a are two unconstrained Grassmann-odd auxiliary parameters, which together transform as a spinor in the little group and a scalar in the Lorentz group. We may also assign a U(1) charge of 1/2 to η a , and charge −1/2 to ∂ ∂η a , which corresponds to the Cartan generator of the broken SU(2). Thus, the η a are complex variables. Having defined the one-particle supercharge, it follows that the multi-particle supercharge is the sum Q α A = n i=1 |q α i A , (3.7) where i is the particle label. It is easy to check the full supersymmetry algebra, {Q α A , Q β B } = n i,j=1 {|q α i A , |q β j B } = n i,j=1 |p a i A |p b j B {θ α ai , θ β bj } = ǫ αβ γ µ AB P µ ,(3.8) where we used that {θ α ai , θ β aj } = −δ ij ǫ αβ ǫ ab , since the little group of each particle is independent, and the total momentum is defined as P µ = i p µ i . N = 2 supersymmetry with central charge The N = 2 supersymmetry algebra in 5D admits a singlet central charge, {Q α A , Q β B } = ǫ αβ γ µ AB P µ + Zǫ αβ Ω AB . (3.9) On dimensional grounds, it is possible to identify the central charge with mass, hence we rewrite it as Z = M . (3.10) We can repeat the construction of the single-particle supercharge as in the massless case above, while replacing the massless spinor with a massive one. For a one-particle state of momentum p µ and mass m, we now use a massive spinor to extract the littlegroup dependence, Q α A 1-pt = |p a A θ α a ≡ |q α A . (3.11) This implies that the supercharges can now be chiral in the sense that they only involve one of the SU(2) factors of the little group SO(4) ∼ SU(2) × SU(2). The fermionic oscillator algebra is now chiral in the same sense, {θ α a , θ β b } = −ǫ αβ ǫ ab . (3.12) As before, this relation has a family of solutions parameterized by SU(2) × SU (2); a representative, written in terms of an unconstrained Grassmann variable and the corresponding derivative, is θ α a = (θ a ) α = η a − ∂ ∂η a ,(3.13) where, as in the massless case, we have chosen to break the SU(2) R-symmetry down to a manifest U(1), while maintaining the SU(2) little-group symmetry. The multi-particle supersymmetry algebra, whose generators are sums of single-particle supersymmetry generators, follows straightforwardly, {Q α A , Q β B } = n i,j=1 {|q α i A , |q β j B } = n i,j=1 |p a i A |p b j B {θ α ai , θ β bj } = ǫ αβ (γ µ AB P µ + Ω AB M) . (3.14) The last equality follows from Eq. (2.19), P µ = i p µ i and M = i m i . N = 4 supersymmetry without central charge The massless N = 4 supersymmetry algebra in 5D is {QȦ A , QḂ B } = ΩȦḂ γ µ AB P µ ,(3.15) whereȦ,Ḃ are USp(4) R-symmetry indices. 7 As before, the supercharge for a oneparticle state of null momentum p µ can be written by factoring out an on-shell spinor carrying the Lorentz index, QȦ A 1-pt = |p a A θȦ a ≡ |qȦ A ,(3.16) where θ α a are Grassmann-odd and satisfy the supersymmetry algebra projected on the little group, {θȦ a , θḂ b } = −ΩȦḂǫ ab . (3.17) An explicit parametrization of θ α a that satisfies the algebra is given by the oscillator representation θȦ a = (θ a )Ȧ = θ 1α a θ 2α a with θî α a = ηî a − ∂ ∂η â ı , (3.18) where we used the SO(2) × SU(2) subgroup of SO(5) by writing the four-component USp(4) index as a product of two fundamental indices,Ȧ =î ⊗ α. Alternatively, we can write a solution that does not break the diagonal SU(2) factor, 8 θȦ a = (θ a )Ȧ = i √ 2 η α a + ∂ ∂η a α η α a − ∂ ∂η a α . (3.19) This solution gives the maximal R symmetry that can be realized in amplitudes without breaking little-group symmetry. The unconstrained Grassmann-odd auxiliary parameters η α a have four complex components, making the SU(2) little group and U(1) × SU(2) ⊂ USp(4) R-symmetry subgroup manifest. Similarly to previous cases, the multi-particle supersymmetry algebra follows by summing over particle labels i, j, {QȦ A , QḂ B } = n i,j=1 {|qȦ i A , |qḂ j B } = n i,j=1 |p a i A |p b j B {θȦ ai , θḂ bj } = ΩȦḂγ µ AB P µ . (3.20) N = 4 supersymmetry with central charge The massive N = 4 supersymmetry algebra in 5D includes an antisymmetric central charge ZȦḂ, {QȦ A , QḂ B } = ΩȦḂ γ µ AB P µ + ZȦḂ Ω AB . (3.21) We will first consider the case in which the central charge is a singlet of R-symmetry group USp(4). Thus, it is of the form ZȦḂ = M ΩȦḂ , (3.22) which is a direct extension of the N = 2 case in Section 3.2. We will construct a supercharge which only depend on the massive angle spinor, QȦ A 1-pt = |p a A θȦ a ≡ |qȦ A . (3.23) Thus, in this case, the supersymmetry is chiral, which, in 6D language, corresponds to (2, 0) supersymmetry. The Grassmann-odd parameters satisfy the same little-group superalgebra as in the massless case (3.17), {θȦ a , θḂ b } = −ΩȦḂǫ ab . (3.24) The explicit solution for this algebra is then identical to Eq. (3.19). This is not surprising, since the little-group and R symmetries are identical to the massless case. The full algebra is then Next, consider the case in which ZȦḂ is not a singlet and, thus, breaks USp(4). We start with the simplifying assumption that the central charge is proportional to a SO(5) gamma matrix. For example, taking iΓ 9 gives the block-diagonal form {QȦ A , QḂ B } = n i,j=1 {|qȦ i A , |qḂ j B } = n i,j=1 |p a i A |p b j B {θȦ ai , θḂ bj } = ΩȦḂ(γ µ AB P µ + MΩ AB ) ,ZȦḂ = iM (Γ 9 )ȦḂ = M ǫ αβ 0 0 −ǫαβ . (3.26) Here we take the SO(5) index to run from 5 to 9, and the internal SO(5) gamma matrices are simply related to the spacetime ones by a Wick rotation, Γ nȦḂ = −iγ 0 AB , γ µ>0 AB A→Ȧ,B→Ḃ,n=µ+5 . (3.27) With the central charge pointing in a fixed direction, the R-symmetry is broken to SO(4) ∼ SU(2) × SU (2), and the supercharges separate into those that transform in each copy of SU (2), QȦ A → Q α à Qα A , (3.28) where, as before, the Greek indices are R-symmetry and the dot distinguishes the left and right SU (2) factors. For a one-particle state of momentum p µ and mass m, we use the massive spinors to extract the little-group dependence, Q α A 1-pt = |p a A θ α a ≡ |q α A , Qα A 1-pt = |p˙a] Aθα a ≡ |qα] A . (3.29) We are led to two independent oscillator algebras {θ α a , θ β b } = −ǫ αβ ǫ ab , {θα a ,θβ b } = −ǫαβǫ˙a˙b , {θ α a ,θβ b } = 0 ,(3.30) which are solved by θ α a = (θ a ) α = η a − ∂ ∂η a ,θα a = (θȧ)α = ηȧ − ∂ ∂ηȧ , (3.31) where now we have two sets of complex unconstrained auxiliary Grassmann parameters η a ,η˙a. They make the little group SU(2) × SU(2) manifest and break the R-symmetry down to U(1) × U(1) ⊂ SU(2) × SU (2). Note that this is isomorphic to the solution we wrote down in Eq. (3.18). Now consider the more general case where the central charge is a generic SO(5) vector, 9 ZȦḂ = iΓ nȦḂ M n . (3.32) On general grounds, we expect that we need 10D Majorana-Weyl spinors to construct single-particle supercharges, and that we should think of a ten-dimensional momentum p [10] as decomposed as p 4) momentum, and µ n = (µ 5 , µ 6 , µ 7 , µ 8 , µ 9 ) is SO(5) momentum. For p [10] to be massless, the 5-vectors must satisfy the constraint p 2 − m 2 = 0 and µ 2 + m 2 = 0 (with negative SO(5) signature such that µ 2 < 0). For a generic multi-particle state the mass vector appearing in the central charge is given as M n = j µ n j , where j labels the particles. [10] N = (p ν , µ n ), where p ν = (p 0 , p 1 , p 2 , p 3 , p 4 ) is SO(1, We can then obtain new spinors by tensoring the 5D massive spinors as . The resulting 10D Majorana-Weyl spinor is the direct sum of the above spinors, and it has 4 + 4 = 8 degrees of freedom, as is manifest from inspecting the little-group indices. The one-particle supercharge can now be written more covariantly as λ aα AȦ ≡ 1 √ 2m |p a A |µ α Ȧ andλ˙aα AȦ ≡ 1 √ 2m |p˙a] A |µα]Ȧ ,(3.QȦ A 1-pt = 1 √ 2m |p a A θ α a µ α |Ȧ + |p˙a] Aθα a [µα|Ȧ ≡ |qȦ A ,(3.34) where we may now think of |qȦ A as either a 5D symplectic-Majorana spinor or a 9 Antisymmetry rules out higher-rank elements of the Clifford algebra in the decomposition of the central charge. Also, for 1/2-BPS multiplets in super-Yang-Mills theory, the singlet and SO(5)vector central charge are mutually exclusive [80]. 10 The insertion of i's are chosen such that the completeness relations for the spinors are |µ a Ȧ µ a |Ḃ = iµḂ A + m δḂ A and |µȧ]Ȧ[µȧ|Ḃ = −iµḂ A + m δḂ A . 10D Majorana-Weyl spinor. The supersymmetry algebra now becomes {QȦ A , QḂ B } = i,j=1 {|qȦ i A , |qḂ j B } = i,j=1 1 2m i |p a i A |p b j B {θ α ai , θ β bj } µ αi |Ȧ µ βj |Ḃ + |p˙a i ] A |p˙b j ] B {θα ai ,θβ bj }[µα i |Ȧ[µβ j |Ḃ = ΩȦḂγ µ AB P µ + iΩ AB (Γ n )ȦḂM n ,(3.35) where θ α ai andθα ai obey the little-group supersymmetry algebras (3.30) with the same solution (3.31). Indeed, the two last cases we considered, namely the central charge proportional to iΓ 9 or to a general SO(5) vector, are related to each other by a SO(5) rotation, so it is not surprising that the little-group details are the same. Let us briefly explain the appearance of the imaginary unit in front of the mass in the central charge. If follows from constructing the 10D gamma matrices, they are given by the following 32-by-32 matrices, (Γ N ) B A = 0 (Σ N ) BḂ AȦ (Σ N ) BḂ AȦ 0 , (3.36) where the 16-by-16 sigma matrices are (Σ N ) BḂ AȦ = (γ ν ) B A δḂ A , iδ B A (Γ n )Ḃ A and (Σ N ) BḂ AȦ = (γ ν ) B A δḂ A , − iδ B A (Γ n )Ḃ A . From this we see that iΓ n is the natural expression appearing in the sigma matrices, and it is Σ N that appears on the final line of the anti-commutator Eq. (3.35). Hence, the supersymmetry algebra can be written more compactly as, {Q Ḃ A , QḂ A } = (Σ N ) BḂ AȦ P [10] N ,(3.37) where P [10] N = (P ν , M n ) is the 10D momentum generator. On-shell superfields for massless 5D multiplets The unconstrained Grassmann variables η are the building blocks of the 5D onshell multiplets. We list here the on-shell superfields describing massless 1/2-BPS multiplets; the massive ones will be described in the next subsection. All displayed indices are either SU(2) little-group indices (Latin), or SU(2) R-symmetry indices (Greek). The on-shell vector supermultiplet with N = 2 supersymmetry is V N =2 a = χ 1 a + A ab + ǫ ab √ 2 φ η b + χ 2 a (η) 2 ,(3.38) where we have defined the square of the Grassmann variables as (η) 2 = 1 2 η a η a = η 1 η 2 . This superfield contains four bosonic 11 (A ab =A ba , φ) and four fermionic χ α a degrees of freedom, the latter are symplectic-Majorana spinors. 11 Imposing proper normalization for the scalar gives the √ 2 factor, since ǫ ab / √ 2 has unit norm. The superfield transforms as a covariant SU(2) little-group spinor, while the SU(2) R-symmetry of the gaugino χ α a is not manifest. This is a consequence of the chosen oscillator representation of the supersymmetry algebra, which breaks manifest R-symmetry, and thus realizes linearly only half of the supercharges. 12 Only a U(1) remnant of the R-symmetry remains manifest in the charge carried by the η and the fermions. The hypermultiplet is a little-group singlet and an R-symmetry doublet, Φ α N =2 = φ α + χ α a η a + φ α (η) 2 ,(3.39) and contains four bosonic and four fermionic degrees of freedom. The vector and hypermultiplets are related through the double copy involving a fermion χ α a , V a N =2 = Φ α N =2 ⊗ χ a α ,(3.40) where the index contraction is needed to obtain the same number of degrees of freedom as in Eq. (3.38). The graviton multiplet with N = 2 supersymmetry transforms as a spin-3/2 representation of the little group, H N =2 abc = ψ 1 abc + h abcd + 1 √ 2 A (ab ǫ c)d η d + ψ 2 abc (η) 2 ,(3.41) where the tensors with little-group indices a, b, c, .. are symmetric. The parenthesis denote total symmetrization of the little-group indices, which includes a 1/n! factor. The multiplet has eight bosonic and eight fermionic degrees of freedom. ψ α abc are the gravitini and the vector A ab represents the graviphoton. The above graviton multiplet can be obtained as a double copy of the N = 2 vector multiplet and a non-supersymmetric vector field, H N =2 abc = V N =2 (a ⊗ A bc) ,(3.42) where the little-group indices are symmetrized. If instead we antisymmetrize a pair of indices (contracting with ǫ ab ), we get the axidilaton-vector multiplet V N =2 c = 1 2 ǫ ab V N =2 a ⊗ A bc = χ 1 c + A axi cb + ǫ cb √ 2 ϕ dil η b + χ 2 c (η) 2 .(3.43) Next considering N = 4, we find the maximally-supersymmetric vector multiplet V N =4 = φ + χ α a η a α + A ab η a α η bα + φ αβ η aα η a β +χ α a η a α (η) 2 +φ(η) 4 ,(3.44) where (η) 2 = 1 4 η a α η α a . It contains eight bosonic and eight fermionic degrees of freedom, and it has manifest SU(2) × U(1) ⊂ USp(4) R-symmetry. The vector multiplet can also be obtained from two different double copies: V N =4 = 1 2 Φ α N =2 ⊗ Φ β N =2 ǫ αβ = 1 2 V N =2 a ⊗ V N =2 b ǫ˙a˙b . (3.45) 12 The choice here is analogous in spirit with the non-chiral 4D superspace of Ref. [81]. The N = 4 graviton multiplet transforms as a little-group vector, H N =4 ab = A ab + ψ α abc + 1 √ 2 ψ α (a ǫ b)c η c α + h abcd + 1 √ 2à (a(c ǫ b)d) + 1 2 ǫ a(c ǫ d)b φ η c α η dα + η cα η c β A αβ ab + ψ α abc + 1 √ 2ψ α (a ǫ b)c η c α (η) 2 +Ā ab (η) 4 ,(3.46) where the six graviphotons are distributed as three singlets and a triplet of the manifest SU(2) R symmetry. The multiplet has 24 bosonic and 24 fermionic degrees of freedom. It can be obtained as the double copy of an N = 4 vector and nonsupersymmetric vector, 47) or, alternatively, as the symmetric double copy of two N = 2 vector multiplets: H N =4 ab = V N =4 ⊗ A ab ,(3.H N =4 ab = V N =2 (a ⊗ V N =2 b) . (3.48) The little-group indices are symmetrized, and the single-copy Grassmann variables η α a = (η a ,η a ) form a doublet of the expected manifest SU(2) R symmetry of the double-copy multiplet. Finally, the N = 6 and maximal N = 8 supersymmetry graviton multiplets can be obtained as the double copies H N =6 a = V N =4 ⊗ V N =2 a , H N =8 = V N =4 ⊗ V N =4 ,(3.49) respectively. It is not difficult to carry out the multiplication and separate the various monomials in Grassmann variables to identify the component states. We will not write them out explicitly, but instead note that these multiplets contain 64 + 64 and 128 + 128 bosonic and fermionic states, respectively. On-shell superfields for massive 5D multiplets From the perspective of the little group, 5D massive multiplets and massless 6D multiplets are the same. We will therefore use the established notation for the latter to also denote the former. The 5D massive vector multiplet can also be seen as a 6D chiral (1, 0) multiplet, 13 V (1,0) a = χ 1 a + W aȧ η a + χ 2 a (η) 2 ,(3.50) where the fermions χ˙a α are symplectic-Majorana spinors. Similarly, the hypermultiplet is Φα (1,0) = ϕ 1α + χα a η a + ϕ 2α (η) 2 . (3.51) The above fermions χ α a and scalars ϕ αα are transforming in the SO(4) R-symmetry, but the realization of the supersymmetry algebra using η a partially breaks the Rsymmetry to U(1) × SU (2). An (1, 0) tensor multiplet can be written as T (1,0) a = χ 1 a + B ab + ǫ ab √ 2 ϕ η b + χ 2 a (η) 2 ,(3.52) where B ab is a self-dual tensor, and the fermions χ α a are symplectic-Majorana spinors of opposite SO(4) chirality compared to the spinors in the (1, 0) vector multiplet. A 6D (1, 0) graviton multiplet, and the corresponding massive 5D multiplet, is H (1,0) aȧḃ = ψ 1 aȧḃ + h abȧḃ + ǫ ab √ 2 B˙a˙b η b + ψ 2 aȧḃ (η) 2 . (3.53) It has 12 bosonic and 12 fermionic states and it is symmetric in the dotted little-group indices. B˙a˙b is the anti-self-dual gravitensor and ψ α aȧḃ are the gravitini. Through the double copy, we can also write the (1, 0) graviton multiplet as H (1,0) aȧḃ = V (1,0) a ⊗ W aḃ − ǫ˙a˙b T (1,0) a ,(3.54) where we "subtract" the tensor multiplet following the ghost prescription outlined in Ref. [82]. The subtracted tensor multiplet has its own double-copy decomposition, T (1,0) a = 1 2 ǫαβΦα (1,0) ⊗ χβ a . (3.55) The construction (3.54) corresponds to the double-copy realization of pure 6D (1, 0) supergravity via tensor ghosts from Ref. [82]. It should be noted that we can also identify T B˙a˙b = 1 2 W a(ȧ ⊗W bḃ) ǫ ab ,(3.56) while the "axitensor" contained in the first product and subtracted by T (1,0) a are B axi ab = − 1 2 W (aȧ ⊗W b)ḃ ǫ˙a˙b = 1 2 χ α a ⊗ χ bα . (3.57) The second identification is necessary for the subtraction in Eq. (3.54) to remove the "axitensor" together with its partners and yield the (1,0) graviton multiplet. We turn next to maximal supersymmetry. The (1, 1) vector multiplet (or, equivalently, the massive 5D N = 4 vector multiplet) is V (1,1) = ϕ 11 + χ a1 η a + χ 1ȧηȧ + ϕ 12 (η) 2 + ϕ 21 (η) 2 + W aȧ η aηȧ + χ a2 η a (η) 2 + χ 2ȧηȧ (η) 2 + ϕ 22 (η) 2 (η) 2 . (3.58) Introducing the variables ζ α = 1, (η) 2 andζα = 1, (η) 2 , it can also be written more compactly as V (1,1) = ϕ αα ζ αζα + χ aα η aζα + χ αȧηȧ ζ α + W aȧ η aηȧ . (3.59) If we assign SO(4) ∼ SU(2)×SU(2) transformations to the ζ,ζ variables, the massive N = 4 vector multiplet exhibits the complete SO(4) R-symmetry unbroken by the central charge. The (2, 0) tensor multiplet can be written as T (2,0) = φ + χ α a η a α + B ab η a α η bα + φ αβ η aα η a β +χ α a η a α (η) 2 +φ(η) 4 ,(3.60) where states exhibit a manifest SU(2) × U(1) ⊂ SO(5) R-symmetry. This multiplet has exactly the same form as the massless N = 4 vector multiplet, except that the massless vector A ab is here replaced by a massive self-dual tensor B ab . The (2, 0) tensor multiplet can be obtained as a double copy in two different ways, T (2,0) = 1 2 Φα (1,0) ⊗ Φβ (1,0) ǫαβ = 1 2 V (1,0) a ⊗ V (1,0) b ǫ˙a˙b ,(3.61) in terms of two (1, 0) hypermultiplets or two (1, 0) vector multiplets, respectively. The (1, 1) graviton multiplet can be written as a double copy, H (1,1) bḃ = V (1,1) ⊗ W bḃ = W ααbḃ ζ αζα +ψ aαbḃ η aζα +ψ αȧbḃηȧ ζ α +h aȧbḃ η aηȧ , (3.62) where W ααbḃ are four massive vectors,ψ aαbḃ = ψ aαbḃ + ǫ ab ψα˙b/ √ 2 consist of a massive gravitino and a fermion (similarly forψ αȧbḃ ), andh aȧbḃ = h aȧbḃ + ǫ ab B˙a˙b/ √ 2 + ǫ˙a˙bB ab / √ 2 + ǫ ab ǫ˙a˙bφ/2 consists of the graviton together with a tensor and a scalar. The (2, 0) graviton multiplet can also be obtained as a double copy, H (2,0) aḃ = V (1,0) a ⊗ V (1,0) b − ǫ˙a˙b T (2,0) ,(3.63) as the little-group-traceless part of the product of two (1, 0) vector multiplets. In a similar spirit, (2, 0) non-metric graviton multiplet is a double copy of a (2, 0) tensor multiplet and another tensor, H (2,0) ab = T (2,0) ⊗ B ab . (3.64) The "graviton" h abcd is non-metric in the sense that it only has chiral little-group indices and descends from the field strength of a mixed tensor gauge field and not that of a symmetric metric tensor in six dimensions. It contains five degrees of freedom (i.e. the same as the 5D massless graviton). For 3/4-maximal and maximal supersymmetry, we can obtain massive graviton multiplets as various double copies. The most interesting are the following: We will not spell out here the details of the component fields, as they are simply a straightforward but tedious exercise of distributing the multiplets over the tensor product and identifying fields. H (2,1) a = V (1,1) ⊗ V In the massive case, we can also consider gravitino multiplets of Poincaré supergravity. They appear when supersymmetry is spontaneously broken, and can be obtained from a double-copy construction. When maximal supersymmetry is partially broken down to 3/4-maximal supersymmetry, the massive 5D gravitino multiplet is of (2, 1) type, Ψ (2,1) α = V (1,1) ⊗ Φ (1,0) α . (3.66) For 3/4-maximal supersymmetry partially broken down to 1/2-maximal supersymmetry, the gravitino multiplet is of (1, 1) type Ψ (1,1) aα = V (1,1) ⊗ χȧ α = V (1,0) a ⊗ Φ (0,1) α , (3.67) or (2, 0) type Ψ (2,0) aα = V (1,0) a ⊗ Φ (1,0) α . (3.68) Finally, for partial breakings that preserve 1/4-maximal supersymmetry, the gravitino multiplets are of (1, 0) type Ψ (1,0) aḃα = V (1,0) (ȧ ⊗ χ˙b )α . (3.69) Five-dimensional amplitudes with vectors and tensors In our conventions, m-point color-dressed gauge-theory amplitudes are written as A m = g m−2 σ∈S m−2 A m (1, σ(2), · · · , σ(m − 1), m)Tr(T 1 T σ(2) · · · T σ(m−1) T m ) , (4.1) where A m (1, . . . , m) are color-ordered partial amplitudes and representation matrices are chosen to obey Tr(T a T b ) = δ ab and [T a , T b ] =f abc T c . We use A m (1, . . . , m) to denote the corresponding partial superamplitudes. Similarly, we use M m to denote component gravitational amplitudes and M m to denote gravitational superamplitudes. Three-point amplitudes and superamplitudes As illustrated in Section 2, we dress the SU(2) little-group indices with auxiliary bosonic variables z a ,zȧ. To avoid explicitly displaying little-group indices, we use the short-hand notation introduced in Eq. (2.23), | i ≡ |k a i z ia , |q i ≡ |q a i z ia , | i ≡ |p a i z ia , | i ] ≡ |p˙a i ]z iȧ .(4.2) For example, in this notation, the massless polarization vector corresponding to the i-th leg is written as ε µ i = i |γ µ |q i √ 2 , (4.3) where q i is a reference momentum obeying the condition (2.14), as well as q i q j = 0. The three-gluon partial amplitude in 5D Yang-Mills theory is given by A 3 (1A, 2A, 3A) = i z a 2 2 a q 3 b z b 3 z c 3 3 c q 2 d z d 2 z e 1 1 e 2 f 2 f q 1 g z g 1 + cyclic(1, 2, 3) ,(4. 4) where we have chosen to display the little-group indices and to collect little-group singlets in parenthesis. Using the short-hand notation, this expression can be rewritten in the more compact form Three-point partial amplitudes in 5D Yang-Mills theory with massless matter are A 3 (1φ, 2φ, 3A) = i 31 a 1 a q 3 = i 3|k 1 |q 3 , (4.6) A 3 (1χ 2 , 2χ 1 , 3A) = i 13 q 3 2 − (3 ↔ q 3 ) , (4.7) A 3 (1χ 2 , 2χ 1 , 3φ) = − i √ 2 12 . (4.8) These amplitudes can be obtained from the superamplitude A N =2 3 (1V, 2V, 3V) = −iη a 1 1 a |p 2 |q 1 3 i=1 η b i i b q 2 3 j=1 η c j j c q 3 + cyclic(1, 2, 3) , (4.9) by acting with the appropriate Grassmann derivatives corresponding to the desired external states. More explicitly, the component amplitudes in Eqs. (4.5)-(4.8) are A 3 (1A, 2A, 3A) = − z a 1 ∂ ∂η a 1 z b 2 ∂ ∂η b 2 z c 3 ∂ ∂η c 3 A N =2 3 (1V, 2V, 3V) , (4.10) A 3 (1φ, 2φ, 3A) = − ǫ ab √ 2 ∂ ∂η a 1 ∂ ∂z b 1 ǫ cd √ 2 ∂ ∂η c 2 ∂ ∂z d 2 z e 3 ∂ ∂η e 3 A N =2 3 (1V, 2V, 3V), (4.11) A 3 (1χ 2 , 2χ 1 , 3A) = ∂ ∂η 2 1 ∂ ∂η 1 1 z c 3 ∂ ∂η c 3 A N =2 3 (1V, 2V, 3V) ,(4. 12) A 3 (1χ 2 , 2χ 1 , 3φ) = ∂ ∂η 2 1 ∂ ∂η 1 1 ǫ cd √ 2 ∂ ∂η c 3 ∂ ∂z d 3 A N =2 3 (1V, 2V, 3V) ,(4.13) where the operators corresponding to each leg have been constructed so that they extract the components of the corresponding on-shell superfields with the correct normalization. The superamplitude (4.9) can also be rewritten in terms of the supercharges |Q = i |Q i = i |i a η ia as (1, 2, 3) . A N =2 3 (1V, 2V, 3V) = −i Q 1 |p 2 |q 1 q 2 |Q q 3 |Q + cyclic (4.14) Note that this object is totally symmetric under permutations. This is appropriate given the fermionic nature of the superfields since the color factor (which has been stripped off) is totally antisymmetric. In a similar way, we can write down the superamplitude between two massless hypermultiplets and one vector multiplet in 5D N = 2 super-Yang-Mills theory. It has an even simpler expression, A N =2 3 (1Φα, 2Φβ, 3V) = i 2 ǫαβ q 3 |Q Q|Q . (4.15) Moving to the massive case, it is instructive to first consider three-point amplitudes between two massive spinors and a massless vector. With a chiral spinor χ and an anti-chiral spinorχ, we have the following candidate amplitudes: A(1χ, 2χ, 3A) = −i √ 2 1|ε 3 |2 = i 13 q 3 2 − (3 ↔ q 3 ) , (4.16) A(1χ, 2χ, 3A) = −i √ 2[1|ε 3 |2] = i[13 q 3 2] − (3 ↔ q 3 ) ,(4. 17) A(1χ, 2χ, 3A) ? = −i √ 2 1|ε 3 |2] = i 13 q 3 2] − (3 ↔ q 3 ) . (4.18) Taking the two masses to have opposite sign, i.e. m 1 = m = −m 2 , one can check that the third candidate amplitude A(1χ, 2χ, 3A) is not gauge invariant. For ε 3 → p 3 this amplitude does not vanish, unlike the other two, and hence it must vanish identically. This implies that χ andχ are not related by CPT symmetry, and hence massive 5D theories have a notion of chirality that is preserved by the interactions, similar to 6D massless theories. We also give the component amplitude between two massive and one massless vectors, which has the following expression: A 3 (1W, 2W, 3A) = − i 4m 2 [23 q 3 2 − (3 ↔ q 3 ) [1|k 3 |1 (4.19) + 1 2 12 [12] − 12][12 ) 3|p 1 |q 3 − (1 ↔ 2) . It corresponds to the partial amplitude in Yang-Mills theory with spontaneouslybroken gauge symmetry (hence the W -boson label) or, alternatively, to a Kaluza-Klein Yang-Mills theory. If we have three massive vector bosons, the spontaneously-broken 5D Yang-Mills amplitude can be cast in the following form, 14 where the permutations run over the dihedral group S 3 with negative sign for odd permutations. The amplitude we give here is only supported on the mass conservation relation, m 1 + m 2 + m 3 = 0 . Q|Q = 3 i,j=1 η a i i a j b η b j , [Q|Q] = 3 i,j=1ηȧ i [iȧj˙b]η˙b j . (4.23) The (1,0) superamplitude for two massive hypermultiplets and a massive Coulombbranch vector multiplet in 5D super-Yang-Mills theory is A (1,0) 3 (1Φα, 2Φβ, 3V) = i ǫαβ 4m 3 [3Q Q|Q ,(4.24) where we have again assumed mass conservation. Moving to the case of massless N = 4 supersymmetry, we find no straightforward generalization of the superamplitude (4.9). To explore other possible forms, let us turn to the known 6D construction of three-point amplitudes [7,8], and adapt it to 5D. We begin by noticing that, with massless three-point kinematics, all the 2by-2 matrices i a j b have rank one, and hence have no inverses. Instead one can decompose them into SU(2) little-group spinors u a i as 1 a 2 b = u a 1 u b 2 , 2 a 1 b = −u b 1 u a 2 ,(4.25)2 a 3 b = u a 2 u b 3 , 3 a 2 b = −u b 2 u a 3 ,(4.26)3 a 1 b = u a 3 u b 1 , 1 a 3 b = −u b 3 u a 1 . (4.27) All u a i variables are uniquely determined, up to an overall sign, by this system. There exist corresponding reference spinors w a i that satisfy u a i w b i − u b i w a i = ǫ ab (no sum over i) (4.28) and that can be further constrained by 3 i=1 |i a w a i = 0 . With these constraints, w a i are not unique, but the remaining two degrees of freedom will cancel out once the amplitude is assembled [8]. Note that the u a i and w a i variables only exists for on-shell three-point kinematics, which has degenerate and complex momenta. With the above variables, the massless three-point superamplitude in 5D N = 4 super-Yang-Mills theory can be written as A N =4 3 (1V, 2V, 3V) = i 4 δ 2 i η α ia w a i α=1,2 Q α |Q α , (4.30) where |Q α = i |Q α i = i |i a η α ia is the supercharge. Note that one cannot write down a delta function of all the eight supercharges as this object vanishes for threepoint kinematics. 15 Nevertheless, is possible to verify that the above superamplitude is invariant under all supersymmetry generators. So far, we have only given implicit definitions for the u a i and w a i spinors, but we can do a bit better. Given that our spinor parametrization |i a is linear in momentum in the first little-group component a = 1, we have via momentum conservation the convenient relation 3 i=1 |i 1 = 0 . (4.31) This implies that 1 1 2 1 = 2 1 3 1 = 3 1 1 1 ≡ r 2 , which then gives the unique solution for the u a i spinors, u a 1 = 1 r 1 a 2 1 , u a 2 = 1 r 2 a 3 1 , u a 3 = 1 r 3 a 1 1 . (4.32) One can easily confirm that u a i u b j gives by construction three correct entries of the corresponding 2-by-2 matrix ± i a j b , and by the reduced rank it then follows that the forth entry (a = b = 2) is also correct. Our choice of parametrization also gives a simple solution for the w a i spinors, the fact that all little-group spinors have the same first entry u 1 i = r. We may further simplify the construction by noticing that, for any choice of the w a i variables, one can find a corresponding global reference spinor ρ| such that w a i = ρ|i a . (4.34) 15 The spinors |Q α i , with i = 1, 2, 3, span a three-dimensional subspace of U Sp(2, 2) and δ 8 (Q) has the interpretation as the square of the corresponding four-volume, hence it vanishes. w a i = w a = 1 r (0, 1) . Thus, we can write also the last delta function in Eq. (4.30) in terms of the supercharge, A N =4 3 (1V, 2V, 3V) = i 4 α=1,2 ρ|Q α Q α |Q α = i 36 α=1,2 det |Q α , |Q α , |Q α , |ρ . (4.35) Because the last formula is a 4-by-4 determinant, there exists a three-fold family of |ρ spinors that gives the same amplitude. For the choice of w a i = 1 r (0, 1), we find that the global reference spinor can be chosen simply as ρ| = 1 r (0, 0, 0, 1) . (4.36) Generalization to the massive case follows even more closely the 6D case [8]. Three point kinematics implies that det i a j b = m i m j = det[i˙aj˙b] , det i a j˙a] = 0 . (4.37) One can define the little-group spinors With our parametrization of the massive spinors, an explicit solution is 16 Recall the vector amplitude from Ref. [8], (u a 1 u b 2 w c 3 + cyclic(1, 2, 3))(ũȧ 1ũḃ 2wċ 3 + cyclic (1, 2, 3)), may be obtained from Eq. (4.45) using the identities 1 a 2˙b] = u a 1ũḃ 2 , 2 a 1˙b] = −u a 2ũḃ 1 , (4.38) 2 a 3˙b] = u a 2ũḃ 3 , 3 a 2˙b] = −u a 3ũḃ 2 , (4.39) 3 a 1˙b] = u a 3ũḃ 1 , 1 a 3˙b] = −u a 1ũḃ 3 ,u a i = 1 r i a j 1 ] ,ũ˙a i = 1 r [i˙aj 1 , w a i = 1 r (0, 1) =w˙a i ,1 a 2 b = u a 1 u b 2 − 2m 2 w a 1 u b 2 + 2m 1 u a 1 w b 2 and 1 a 2 b w c 3 − u a 1 u b 2 w c 3 + cyclic(1, 2, 3) = 0. where the global reference spinors satisfy ρ|i a = w a i and [ρ|i˙a] =w˙a i , which for our simple choice can be obtained by ρ| = [ρ| = 1 r (0, 0, 0, 1). Again, we have imposed mass conservation. It is interesting to note that, from the simple factorized form of the massive (1, 1) superamplitude, one may attempt to write down a massive (2, 0) superamplitude for non-abelian self-dual tensor multiplets (interpreted as a Kaluza-Klein reduction of a 6D (2, 0) tensor theory). The naive guess is A (2,0) 3 (1T , 2T , 3T ) ? = i 4 α=1,2 ρ|Q α Q α |Q α = i 36 α=1,2 det |Q α , |Q α , |Q α , |ρ , (4.46) which superficially looks indistinguishable from the massless N = 4 superamplitude. However, we do not find any covariant expressions (in terms of momenta and tensor polarizations) that match the corresponding candidate non-abelian three-tensor component amplitude A 3 (1B ab , 2B cd , 3B ef ) ? = idet |1 (a , |2 (c , |3 (e , |ρ det |1 b) , |2 d) , |3 f ) , |ρ , (4.47) nor for the corresponding candidate scalar-tensor component amplitude A 3 (1φ, 2φ, 3B ab ) ? = iu a 3 u b 3 = i √ 2m 3 ε ab 3,µν [1 1 |γ µν |2 1 ] [1 1 2 1 ] . (4.48) The fact that we do not find covariant formulas is consistent with the three-pointamplitude analysis of Ref. [46], and indicate that these are not well-behaved amplitudes. In the next subsection the corresponding four-tensor amplitude will be analyzed with the same conclusion. The three-graviton superamplitude in massive 5D Kaluza-Klein (2,0) supergravity is given by the double copy Note the same tensor amplitude can be obtained as the double copy of (1, 0) super-Yang-Mills amplitudes with massive half-hyper multiplets and a Coulombbranch vector, 2Φ, 3V˙b) , (4.51) and the equivalence of the two double copies follow from the kinematic identities where, as before, mass conservation is assumed. The above equivalence agrees with the fact that the different tensors can be embedded into the (2, 2) gravitational theory where they are related by R-symmetry. The three-graviton superamplitude in massive Kaluza-Klein (2,2) supergravity is where we assumed that both massive amplitudes originate from massless 6D kinematics via Kaluza-Klein compactification. All the massive amplitudes given above can equivalently describe 6D massless amplitudes. M (2,0) 3 (1T , 2T , 3H˙a˙b) = −iA (1,0) 3 (1Φ, 2Φ, 3V˙a)A (1,0) 3 (1Φ,u˙a 3ũḃ 3 = − 1 2 det |1˙c],M (2,2) 3 (1H, 2H, 3H) = −iA (1,1) 3 (1V, 2V, 3V)A (1,1) 3 (1V, 2V, 3V) (4.53) = i 36 2 α=1,2 det |Q α , |Q α , |Q α , |ρ Superamplitudes at four points The most convenient expressions for superamplitudes are relevant for gauge theories with maximal N = 4 supersymmetry. As it is to be expected, the basic object appearing at four points is the Grassmann delta function, which takes the following expression with the 5D spinor-helicity notation, δ 8 (Q) = 1 4 2 α=1 i<j η αa i i a j b η αb j 2 = 1 64 2 α=1 Q α |Q α 2 . (4.55) The color-ordered superamplitude between four vector multiplets is then readily written as A N =4 4 (1V, 2V, 3V, 4V) = −4i δ 8 (Q) st . (4.56) We can directly verify that this superamplitude yields the appropriate component amplitudes. For example, the amplitude between four massless vectors is The N = 2 vector superfields can then be embedded in the N = 4 superfields, which leads to the identification A N =2 4 (1V, 2V, 3V, 4V) = −4i z a 1 ∂ ∂η a2 1 · · · z d 4 ∂ ∂η d2 4 δ 8 (Q) st = −2i δ 4 (Q) st 12 34 + cyclic(1, 2, 3) . (4.58) Since the complement of an N = 2 vector superfield in an N = 4 vector superfield is a hypermultiplet, from the amplitude above one can extrapolate that the amplitudes with hypermultiplets take a similar form, 59) and A N =2 4 (1Φα, 2Φβ, 3V, 4V) = −2iǫαβ δ 4 (Q) st 3|p 1 |4 ,(4.A N =2 4 (1Φα, 2Φβ, 3Φ˙γ, 4Φδ) = −2iδ 4 (Q) ǫαβǫ˙γδ s + ǫβγǫδα t . (4.60) It is easy to promote the massless four-point amplitudes to massive amplitudes by replacing the expressions for the massless supercharges with the supercharges in the massive case. As before, we have several distinct representations of the supersymmetry algebra that differ based on the chirality of the supercharges under the SO(4) little group. Let us start by introducing the needed Grassmann delta functions for massive supercharges. For standard non-chiral (1, 1) supersymmetry, we have δ 4 (Q)δ 4 (Q) = 1 4 i<j η a i i a j b η b j 2 i<jηȧ i [iȧj˙b]η˙b j 2 = 1 64 Q|Q [Q|Q] 2 , (4.61) whereas for chiral (2,0) and (0,2) supersymmetry one has the following Grassmann delta functions: δ 8 (Q) = 1 4 2 α=1 i<j η αa i i a j b η αb j 2 = 1 64 2 α=1 Q α |Q α 2 , (4.62) δ 8 (Q) = 1 4 2 α=1 i,jη αȧ i [iȧj˙b]η αḃ j 2 = 1 64 2 α=1 [Q α |Q α ] 2 . Based on these expressions, we can obtain massive superamplitudes between four massive vector multiplets, A (1,1) 4 (1V, 2V, 3V, 4V) = −4i δ 4 (Q)δ 4 (Q) (s − m 2 s )(t − m 2 t ) , (4.63) where m s , m t denote the masses for the s-and t-channel poles. They are given by the mass-conservation condition: m s = m 1 + m 2 , m t = m 2 + m 3 . It is useful to explicitly write out the component amplitude between four massive vector fields, it is Having obtained the (1, 1) vector amplitude, one may guess that a naive candidate for the color-ordered superamplitude between four massive tensor (2, 0) multiplets is A 4 (1W, 2W, 3W, 4W ) = −i 12 34 + cyclic(1, 2, 3) [12][34] + cyclic(1, 2, 3) (s − m 2 s )(t − m 2 t ) = i (s − m 2 s )(t − m 2 t ) 12A (2,0) 4 (1T , 2T , 3T , 4T ) ? = −4i δ 8 (Q) (s − m 2 s )(t − m 2 t ) . (4.65) To check if this is a well-behaved amplitude we want to check if it factorizes properly. In terms of the already guessed three-point amplitudes, we can work out the needed s-channel factorization, d 4 η 5 A (2,0) 3 (1T , 2T , 5T )A (2,0) 3 (−5T , 3T , 4T ) = −4iδ 8 (Q) τ 2 (t − m 2 t ) 2 ,(4.66) where τ = 1 r 34 [1 1 2 1 ][3 1 ρ] + [2 1 3 1 ][1 1 ρ] + [3 1 1 1 ][2 1 ρ] and ρ = 1 r 12 (0, 0, 0, 1), with r 2 ij = i 1 j 1 ]. In order for the factorization to work, we need the double pole to cancel out. However, the relation between τ and the pole is ττ = t − m 2 t , whereτ is given by swapping square and angle spinors in τ . Hence, the double pole only cancels out in the massless limit or, alternatively, if we had done the same calculation for (1, 1) super-Yang-Mills theory. One may wonder if the three-point amplitudes can be modified such that they absorb the unwanted τ 2 /(t−m 2 t ) factor. However, given that it depends non-trivially on momenta belonging to different three-point amplitudes, this is unlikely to work out. Our conclusion agree with the analysis of Ref. [58], where the non-abelian (2,0) candidate amplitude was shown to have irreconcilable factorization properties. In contrast to the problematic non-abelian tensor case, there is interesting and well-behaved gravitational amplitude that involves abelian tensor multiplets. The four-tensor superamplitude in (2,0) supergravity is given by the double copy M (2,0) 4 (1T , 2T , 3T , 4T ) = −i(s−m 2 s )A (1,0) 4 (1V, 2V, 3V, 4V)A (1,0) 4 (1V, 2V, 4V, 3V) = iδ 8 (Q) 1 s − m 2 s + 1 t − m 2 t + 1 u − m 2 u , (4.67) where the poles corresponds to (massive) graviton exchange. Comparison with superamplitudes in other dimensions Massive multiplets in five dimensions are closely related with massless multiplets in six dimensions since the little group is the same. This relation was used in [46] to dimensionally-reduce the 6D spinor-helicity formalism to 5D and discuss certain three-point amplitudes potentially related to those of the (2,0) 6D theory. Following the standard 6D notation, a massless momentum is written as [7,8,73] p AB [6] = λ Aa λ B a , p [6] AB =λ Aȧλȧ B , (4.68) where λ A a are 6D chiral spinors, A, B = 1, . . . , 4, and a, b are SU(2) indices, and similarly for the anti-chiral spinorsλ Ȧ a . In our massive formalism, we have the natural identification λ A a ≡ p a | A ,λ˙a A ≡ |p˙a] A . (4.69) The sixth component of the 6D momentum is interpreted as the mass in five dimensions (leading to a natural interpretation for the mass-conservation condition i m i = 0). Formulas for the polarization vectors and supercharges follow straightforwardly from their 6D analogues. It should also be noted that the USp(2, 2)invariant matrix Ω can be seen as the sixth entry to the Σ matrices in 6D, where the remaining five entries are given by the 5D gamma matrices. Moving to the massless case, it can be obtained from the 6D spinor-helicity formalism through the identification of the chiral and anti-chiral spinors λ andλ. In a sense, massless superfields and amplitudes in five dimensions present a closer analogy with 4D superamplitudes in a non-chiral representation [81]. Starting from the conventional chiral representation of four-point amplitudes, one can act with a Grassmann Fourier transform with respect to half of the Grassmann coordinates, Φ = dη 3 dη 4 e η 3η3 +η 4η4 Φ chiral . (4.70) For example, the maximally supersymmetric on-shell vector superfield in four dimensions now have the expansion V N =4 4D = ϕ 34 +χ α34 η α + χαηα + A − η 2 + A +η2 + ϕ αα η αηα + χ α η α (η) 2 +χ 12αηα (η) 2 + ϕ 12 (η) 2 (η) 2 ,(4.71) with α = 1, 2 andα = 3, 4. This closely reflects our 5D superfields. The corresponding superamplitude at four points in 4D spinor-helicity notation is given by A N =4 4D (1, 2, 3, 4) = −i 2 α=1 i<j η α i ij η α j ) 4 α=3 i<jηα i [ij]ηα j ) st ,(4.72) The relation between 4D and 5D variables is then simply η α ia i a j b η bα j 4D kin = η α i ij η α j +ηα i [ij]ηα j . (4.73) It is also immediate to verify that, with this relation, Eq. (4.57) reproduces the familiar expression for the 4D amplitudes upon dimensional reduction with the correct normalization. Amplitude decomposition in terms of total SU(2) weights We now turn to the massless amplitudes in five dimensions, and ask what is the 5D equivalent of the 4D decomposition in helicity sectors. Since the 4D helicity follows the weight under little group rephrasing and the 4D little group is a subgroup of the 5D one, U(1) ⊂ SU (2), it is natural to expect that the desired decomposition is that in irreducible representations of the 5D little group. With the particular choice of reference momenta from Eq. (2.20), we have the relations (ε 11 i · ε 11 j ) = 0 , (ε 11 i · ε 12 j ) = 0 , (ε 22 i · ε 12 j ) = 0 , (ε 22 i · ε 22 j ) = 0 , (4.74) for all i, j. A three-point massless amplitude can be decomposed in terms of the total SU(2) little-group representation carried by the external states. This is done by tensoring three three-dimensional representations of SU(2), 3 ⊗ 3 ⊗ 3 = 7 ⊕ 2 × 5 ⊕ 3 × 3 ⊕ 1 . (4.75) To generate all amplitudes within a given SU(2) sector, we can start from the lowestweight state in which there is a maximal number of 1 in the little-group labels for the polarization, and act with raising operators. Eq. (4.74) has as a consequence that we need at least two 1 and two 2 labels to have a nonzero results, hence the 7 and 5 sectors are automatically zero for YM theory in 5D, since the corresponding lowestweight states do not have this property. This therefore constrains the structure of the allowed polynomials in auxiliary variables that are dressing the amplitudes with free SU(2) indices: where we have defined z a 1 1 z b 1 1 z a 2 2 z b 2 2 z a 3 3 z b 3 3 3 and z a 1 1 z b 1 1 z a 2 2 z b 2 2 z a 3 3 z b 3 3 1 as the degree-six polynomials in the auxiliary variables that transform in the three-dimensional SU (2) representation and in the singlet representation. At four points, the decomposition is carried out in a similar way except that now the starting point is the product of four three-dimensional representations of SU(2): 3 ⊗ 3 ⊗ 3 ⊗ 3 = 9 ⊕ 3 × 7 ⊕ 6 × 5 ⊕ 6 × 3 ⊕ 3 × 1 . (4.77) As before, the amplitudes in the 9 and 7 sectors are equal to zero, mirroring the vanishing of all plus and one-minus amplitudes in four dimensions. As in four dimensions, these sectors become nonzero once a higher-dimensional operator is added to the Lagrangian of the theory. More on double-copy amplitudes In this section, we revisit the double-copy construction of Maxwell-Einstein and Yang-Mills-Einstein theories with the new formalism. Maxwell-Einstein and Yang-Mills-Einstein supergravities revisited The main advantage of the 5D formalism is that it does not require reducing the supergravity Lagrangian to four dimensions to match amplitudes from the double copy, as it was done in the earlier work [28,30]. Being able to formulate the construction directly in five dimensions streamlines and simplifies the derivation. One of the remarkable properties of Maxwell-Einstein theories with N = 2 supersymmetry in five dimensions is that they exhibit the simple cubic vector couplings [19,83] 1 6 √ 6 d 5 x C IJK ǫ µνρστ F I µν F J ρσ A K τ ,(5.1) where the C IJK tensor is symmetric in the I, J, K indices running over the number of vector fields present in the theory. The full Lagrangian of the theory is fixed once the C IJK tensor in determined, as done in Refs. [19][20][21]. In turn, this tensor can be read off the three-vector amplitude, which has the following simple expression, M 3 (1A, 2A, 3A) = −i 8 3 C IJK ǫ k 1 , ε 1 , k 2 , ε 2 , ε 3 = −i 2C IJK √ 3 12 13 q 3 2 −(3 ↔ q 3 ) , (5.2) where ǫ(a, b, c, d, f ) = a µ b ν c ρ d σ f λ ǫ µνρσλ . In short, these theories are completely specified by their three-point amplitudes. The simplest double-copy realization of a family of such theories involves, as one of the gauge theory factors, N = 2 super Yang-Mills theory and, as the other factor, a Yang-Mills theory with additional adjoint scalars φ i . At three points, all N = 2 amplitudes can be packaged in a single superamplitude. The two non-zero amplitudes on the non-supersymmetric side are A 3 (1A, 2A, 3A) and A 3 (1A, 2φ i , 3φ j ). With the intent of identifying the relevant C IJK tensors, we focus on amplitudes between vectors in the supergravity theory. In five dimensions, we have three possible double-copy origins for vector states: A i ab = A ab N =2 ⊗ φ i N =0 ,(5. 3) A 1 ab = φ N =2 ⊗ A ab N =0 , (5.4) A 0 ab = A c(a N =2 ⊗ A c b) N =0 ,(5.5) where the gauge-theory scalars carry an index i = 2, . . . , n V and we label the vector states with their little-group indices. It is not difficult to see that there are only two non-zero amplitude between three vectors: M 3 (1A 0 , 2A i , 3A j ) and M 3 (1A 0 , 2A 1 , 3A 1 ) . The former is given by 17 M 3 (1A 0 , 2A i , 3A j ) = − i 4 ǫ ab ∂ ∂z a 1 A 3 (1A, 2A, 3A) ∂ ∂z b 1 A 3 (1A, 2φ i , 3φ j ) ,(5.6) where the differential operators in the antisymmetric indices extract the correct littlegroup representation. After some additional work, it is possible to cast the doublecopy amplitude in the form M 3 (1A 0 , 2A i , 3A j ) = iδ ij 12 13 q 3 2 − (3 ↔ q 3 ) . (5.7) The second nonzero amplitude is M 3 (1A 0 , 2A 1 , 3A 1 ) = −i ǫ ab 4 ∂ ∂z a 1 A 3 (1A, 2φ, 3φ) ∂ ∂z b 1 A 3 (1A, 2A, 3A) = −i 12 13 q 3 2 − (3 ↔ q 3 ) ,(5.8) which despite its appearance is invariant under the full S 3 permutation symmetry. The difference in sign compared to Eq. (5.7) can be traced to the fact that in the two amplitudes the scalars entering the double-copy construction for the supergravity vectors come from different gauge-theory factors. We can now read off the C IJK for this theory directly by comparing the doublecopy amplitudes in Eqs. (5.8) and (5.7) with the supergravity expression Eq. (5.2): C 011 = √ 3 2 , C 0ij = − √ 3 2 δ ij . (5.9) This is the well-known generic Jordan family of Maxwell-Einstein supergravities in five dimensions [19][20][21]. It should be noted that analyzing amplitudes in five dimensions gives us the option to avoid explicitly expanding around a base point. 18 The earlier formulation of the double-copy construction for these theories requires matching of amplitudes in four dimensions, so the 5D spinor-helicity formalism considerably streamlines the derivation. We can extend the supersymmetric gauge theory by including hypermultiplets and the non-supersymmetric theory by including fermions. This yields extra vectors in the double-copy theory as A α ab = χ (a N =2 ⊗ χ α b) N =0 ,(5.10) where α is an extra flavor index carried by the gauge-theory fermions. Introducing these fields turns on additional three-vector amplitudes, which are of the form M 3 (1A i , 2A α , 3A β ) = −iA 3 (1A, 2χ, 3χ)A 3 (1φ i , 2χ α , 3χ β ) . (5.11) The amplitude between two fermions and one scalar in the non-supersymmetric theory is taken to be proportional to a matrix in the global indices that appears in the Yukawa couplings, A 3 (1φ i , 2χ α , 3χ β ) = − i √ 2 Γ i αβ 23 . (5.12) The resulting supergravity amplitude is M 3 (1A i , 2A α , 3A β ) = − i √ 2 Γ i αβ 23 21 q 1 3 − (3 ↔ q 3 ) ,(5.13) which leads to the additional non-zero entries of the C IJK tensor, C iαβ = 1 2 3 2 Γ i αβ . (5.14) We have recovered the construction for homogeneous supergravities first given in Ref. [30]. Here we do not discuss the constraints imposed by color/kinematics duality on the matrix Γ i αβ , i.e. we just assume that we have gauge theories obeying color/kinematics duality from which we can take the amplitudes entering the double copy. Color/kinematics duality can be conveniently studied at the level of the gauge theories, and in this particular case requires that the Γ i αβ matrices obey Clifford-algebra relations, as shown in Ref. [30]. The net result is that we recover the classification of homogeneous Maxwell-Einstein supergravities from the supergravity literature [26]. For completeness, we also give the double-copy map for the other bosonic super-gravity fields, 19 h abcd = A (ab N =2 ⊗ A cd) N =0 , (5.15) ϕ 1 = A ab N =2 ⊗ A ab N =0 , (5.16) ϕ i = φ N =2 ⊗ φ i N =0 , (5.17) ϕ α = 1 √ 2 χ a N =2 ⊗ χ aα N =0 . (5.18) Finally, it is completely straightforward to turn on trilinear scalar couplings in the non-supersymmetric gauge theory entering the construction while preserving the duality between color and kinematics. This yields non-Abelian gauge interactions in the gravitational double-copy theory, which becomes a Yang-Mills-Einstein theory with gauge symmetry given by the flavor symmetry of the trilinear couplings in the non-supersymmetric gauge theory Lagrangian, as shown in Ref. [28]. The component amplitudes discussed in this section can be supersymmetrized using the superamplitudes introduced in Section 4. The amplitude between three supergravity massless vector multiplets can be written as 20 M 3 (1V i , 2V j , 3V k ) = −i 2 √ 3 C ijk 12 q 3 |Q Q|Q ,(5.19) where we used Eq. (4.15) and this expression is invariant under cyclic symmetry due to nontrivial properties of three-point kinematics. Amplitudes with massive vectors and tensors In this section, we study double-copy amplitudes with massive external fields. Their simplest realization is in a gravitational theory containing massive vectors. We begin by studying amplitudes with gravitons and massive vector fields, and then proceed to include other massless and massive fields. Similarly to the massless vectors discussed in the previous section, massive vectors can originate either as the double copies of a massive vector and a massive scalar, or as the double copies of two massive fermions. We start by considering amplitudes between two massive vector fields and a graviton. The first type of massive vectors leads to the double-copy amplitude 19 Note that the field combinations appearing in the double-copy map are canonically normalized, with the exception of ϕ 1 , which has a 3 2 (∂ µ ϕ 1 ) 2 kinetic term in the supergravity Lagrangian expanded at the appropriate base-point. 20 Here, the indices i, j, k run over the matter vector multiplets in the theory. This expression requires the C IJK tensors to be given in the canonical basis with C 000 = 1, C 0ij = − 1 2 δ ij , C 00i = 0 and C ijk arbitrary. M 3 (1W, 2W, 3h) = −iA 3 (1φ, 2φ, 3A)A 3 (1W, 2W, 3A) = −i(ε 3 · p 1 ) 2(ε 1 · p 2 )(ε 2 · ε 3 ) + (ε 1 · ε 2 )(ε 3 · p 1 ) − (1 ↔ 2) . (5.20) The second type of double copy leads to M 3 (1W, 2W, 3h) = −iA 3 (1χ, 2χ, 3A)A 3 (1χ, 2χ, 3A) = i 13 q 3 2] − (3 ↔ q 3 ) [13 q 3 2 − (3 ↔ q 3 ) . (5.21) Upon explicit evaluation, one can check that the two expressions are equivalent. This is a consequence of the graviton interacting universally with matter, and an important check that we obtained the correct amplitudes. A change of normalization in the double-copy map would result in these two amplitudes becoming different, i.e. this is a check of the correctness of the normalization of the double-copy fields. At the level of amplitudes, we can write six gauge-invariant structures with the right little-group indices, 21 12] 21] (p 1 · ε 3 ), 12 [21] (p 1 · ε 3 ), 1|p 2 |1] 2|ε 3 |2] − (1 ↔ 2), m 1 12 [2|ε 3 |1], m 1 [12] 2|ε 3 |1 , m 1 [12 [2|ε 3 |1 − (1 ↔ 2) . (5.22) These structures are however not independent, and it is possible to verify the relations We will use these relations in the remainder of this section and organize the threepoint matrix elements from the supergravity Lagrangian and from the double copy in a basis of gauge-invariant structures drawn from Eq. (5.22). 2 12 [21] − 12] 21] (p 1 · ε 3 ) + 1 2 1|p 2 |1] 2|ε 3 |2] − (1 ↔ 2) +m 1 12 [2|ε 3 |1] − In the supergravity Lagrangian we can write three terms that are gauge invariant with respect to the massless vector. The first is a bilinear in the covariant derivative of the massive vector fields, O 1 = D [µ W ν] D µ W ν . (5.25) The corresponding three-point matrix element is g s t a W W A |DW | 2 = ig s t a (ε 1 · ε 2 (p 1 − p 2 ) · ε 3 + ε 2 · ε 3 p 2 · ε 1 − ε 1 · ε 3 p 1 · ε 2 ) = i g s t a 2m 2 12 [21] − 21] 12] (p 1 · ε 3 ) + 1 4 1|p 2 |1] 2|ε 3 |2] − (1 ↔ 2) , (5.26) 21 As in the previous section, we take the two masses to obey m 1 = −m 2 = m. where t a is the representation matrix of the massive vectors with respect to the unbroken gauge group and g s is the coupling constant of the supergravity gauge interactions. Representation indices for the massive vectors are not explicitly displayed. The second gauge-invariant term in the supergravity Lagrangian has the form O 2 = F a µν W µ t a W ν . (5.27) Its contribution to the three-point amplitude is Lastly, as we have seen in our analysis of the massless theories, an additional operator appears in the supergravity Lagrangian, 22 g s t a W W A W ·F ·W = ig s t a (ε 1 · ε 3 p 3 · ε 2 − ε 2 · ε 3 p 3 · ε 1 ) = i g s t a 8m 2 1|p 2 |1] 2|ε 3 |2] − (1 ↔ 2O 3 = DW ∧ A ∧ DW . (5.30) The corresponding contribution to the three-vector amplitude is t a W W A DW ∧A∧DW = 4it a ǫ(ε 1 , ε 2 , ε 3 , p 1 , p 2 ) = it a 12 [1|ε 3 |2] + [12] 1|ε 3 |2 . (5.31) We now consider amplitudes obtained through the double-copy construction. Specifically, massive fields can have two distinct origins: they can arise as double copies of two massive fermions, or as double copies of one massive vector and one massive scalar. We will take the massive vector either as the vector arising from the double copy of two gluons or as a gluon times an adjoint scalar. Altogether, we have five distinct classes of three-point amplitudes that we need to discuss. We start from the bosonic double copies. The simplest one is 23 22 A fourth possible term in the supergravity Lagrangian has the form D (µ W ν) D µ W ν and will not be considered in this section. Together with the three we discuss they exhaust the basis of independent three-point gauge invariants. 23 For amplitudes between fields transforming in matter non-adjoint representations, instead of three-point color-ordered amplitudes, we use three-point amplitudes in which the color factor T a ij has been stripped. M 3 (1W , 2W, 3A) = −iA 3 (1W , 2W, 3A)A 3 (1ϕ, 2ϕ, 3φ) = λ 2 W W A |DW | 2 + W W A W ·F ·W (5.32) where λ is the constant appearing in the three-scalar coupling, which is normalized as λ ϕφϕ. This leads to the supergravity gauge coupling constant being given by g s = κλ 4 , (5.33) which is consistent with the result in Ref. [30] (we have restored κ in the above equation). A 3 (1W , 2W, 3A) must be an amplitude from a spontaneously-broken gauge-theory amplitude in order to satisfy color/kinematics duality. Another bosonic double copy is M 3 (1W , 2W, 3A) = −iA 3 (1W , 2W, 3φ)A 3 (1φ, 2φ, 3A) = im(ε 1 · ε 2 )(ε 3 · p 1 − ε 3 · p 2 ) = m W W A |DW | 2 − m W W A W ·F ·W . (5.34) In order to have an amplitude of this form, one needs to take φ to be the Higgs field on the left gauge theory. For this amplitude, the supergravity coupling constant is g s = κm 2 . (5.35) There is one last amplitude obtained from a bosonic double copy, M 3 (1W , 2W, 3A) = −i ǫ ab 4 ∂ ∂z a 3 A 3 (1W , 2W, 3A) ∂ ∂z b 3 A 3 (1ϕ, 2ϕ, 3A) = − W W A DW ∧A∧DW . (5.36) Interestingly, the double copy does not give any term proportional to the mass, and therefore it is the same as in the massless case given in the previous section. We now inspect amplitudes in which the massive vectors come from fermionic double copies. There are two such amplitude. The first one is M 3 (1W , 2W, 3A) = −i ǫ ab 4 ∂ ∂z a 3 A 3 (1χ, 2χ, 3A) ∂ ∂z b 3 A 3 (1χ, 2χ, 3A) = √ 2im (ε 1 · ε 3 p 3 · ε 2 − ε 2 · ε 3 p 3 · ε 1 ) = √ 2im W W A W ·F ·W . (5.37) The second amplitude is M 3 (1W , 2W, 3A) = −iA(1χ, 2χ, 3A)A(1χ, 2χ, 3φ) = i 2 [12] 1|ε 3 |2 = m W W A |DW | 2 + 1 4 W W A DW ∧A∧DW . (5.38) The result gives both a term proportional to the C IJK tensor and a term that vanishes in the massless limit. In formulating the above double-copy constructions, we have paired massive fermions with opposite little-group chiralities. However, we can also choose to have the double-copy map pair fermions with equal chiralities to give supergravity tensors, M 3 (1B, 2B, 3A) = −iA(1χ, 2χ, 3A)A(1χ, 2χ, 3φ) = i 2 12 1|ε 3 |2 . (5.39) Several comments on these results are now in order. First, we have seen that, in contrast to amplitudes between two vectors and a graviton, which are universal, massive and massless supergravity vectors admit several different kinds of couplings. In particular, vectors with different double-copy origin give rise to different operators in the supergravity Lagrangian. Some of these couplings disappear in the massless limit. This is an indication that the double-copy theory does not admit an unbroken gauge phase. In fact, these classes of amplitudes arise naturally in the doublecopy construction for Yang-Mills-Einstein theories with non-compact gauge groups, which always need to have the gauge group be spontaneously broken to a compact subgroup to preserve unitarity. In some cases, one has the choice on how to pair gauge-theory states in the double-copy map. Different choices can result in massive vectors or massive tensors in the output of the double copy. However, the pairing of gauge-theory states depends on their respective gauge-group representations, so that a supergravity state is associated to a gauge-invariant bilinear with states from the two gauge-theory factors. In this section, we have used three-point gauge amplitudes for the input of the double copy without studying in detail the theories from which they are originating. In particular, we have not analyzed the constraints coming from color/kinematics duality, which become important for ensuring the consistency of amplitudes at higherpoints. Color/kinematics duality may require a delicate balance between the various gauge-group representations and impose constraints on the parameters of the theory. For example, fermions carrying both little-group chiralities can be present in a given representation, which implies that both tensors and vectors will be generated through the double copy. The analysis of the double-copy construction for theories with noncompact gauge groups is beyond the scope of this article and will be carried out in a separate publication [84]. Conclusion In this paper, we have introduced a 5D version of the spinor-helicity formalism, which provides convenient variables valid for 5D massive and massless kinematics, as well as extended on-shell supersymmetry. We have discussed two main applications: the study of amplitudes involving massless and massive Yang-Mills fields and the formulation of the previously-known double-copy construction for N = 2 Maxwell-Einstein and Yang-Mills-Einstein theories in a purely 5D language. Furthermore, we have elaborated on massive self-dual tensor fields. These appear naturally in some supergravity theories in five dimensions and, as such, these theories can be regarded as an important stepping stone towards understanding the mysterious 6D theories of non-abelian tensor fields. The 5D N = 2 Maxwell-Einstein and Yang-Mills-Einstein supergravities, which are uniquely determined by their trilinear couplings, are a prime testing ground for understanding whether all supergravity theories exhibit a double-copy structure [37,69]. To systematically explore these research directions, efficient methods that make use of the special properties of 5D kinematics are necessary. Earlier discussions appeared in Ref. [85] and especially Ref. [46], in which a 5D formalism is obtained by dimensionally-reducing the 6D spinor-helicity formalism of Refs. [7,8]. See also Refs. [60,61] for a discussion of five-dimensional amplitudes in the context of ambitwistor strings and Refs. [58,59] for methods based on scattering equations. In this paper, we constructed manifestly-supersymmetric three-point superamplitudes whose components reproduce the amplitudes determined by the standard cubic terms of 5D N = 2 supergravity couplings around Minkowski vacua. The form of the superamplitudes and the structure of the superfields, with origins in 6D massless theories, suggest a possible generalization to amplitudes of non-abelian (2, 0) self-dual tensor multiplets. However, our analysis, suggesting the absence of covariant expressions corresponding to the candidate spinor-helicity expressions, are consistent with the negative results obtained in Ref. [46]. Furthermore, while the candidate four-point formulas for the non-abelian tensor amplitudes are suggestive, they do not pass standard factorization checks, in agreement with the analysis of Ref. [58]. The superamplitudes for maximally supersymmetric gauge and gravity theories are similar to the ones obtained through dimensional reduction from 6D. This is a consequence of the fact that theories with N = 4 supersymmetry in five dimensions always uplift to higher dimensions. However, superamplitudes with only halfmaximal supersymmetry do not always follow a similar pattern. The expressions constructed in Section 4.1 are distinct from the truncations of the corresponding maximally-supersymmetric superamplitudes, and appear to be native to five dimensions. Indeed, the explicit mass dependence in the denominator implies that they cannot be straightforwardly uplifted to six dimensions. This is consistent with the fact that not all N = 2 theories in five dimensions possess a higher-dimensional uplift, in close analogy with e.g. 4D N = 2 theories that do not have a 5D uplift. Concretely, whether the superamplitude in Eq. (5.19) uplifts to six dimensions depends on the particular form of the C IJK tensors. The simplest example of non-upliftable theories discussed in the literature are those in the so-called generic non-Jordan family (see Ref. [19]). In Section 3.4, we analyzed the 5D N = 4 supersymmetry algebra with a nonsinglet central charge and interpreted the central charge as components of the momentum in 10 dimensions. We can expose the full 10D Lorentz symmetry by combining the two products of massive 5D spinors in Eq. where I = i ⊕ j is an SO(8) little-group vector index, i ⊕ j is its decomposition over the subgroup SO(4) × SO(4) ⊂ SO (8), and σ i aα , σ j aα are the respective SO(4) sigma matrices. Finally, the index A = A ⊗Ȧ is a 16-component Weyl spinor index. We thus obtain a spinor-helicity-like formalism in 10D, with manifest SO(1, 4) × SO(5) symmetry, which is different from that of Ref. [11]. It would be desirable to explore this formalism further and explicitly construct the low multiplicity superamplitudes of 10D super-Yang-Mills and type IIA/B supergravity. Finally, one of the reasons for developing a systematic 5D formalism, and perhaps its main application, is to streamline the study of 5D supergravities in the double-copy realization of their amplitudes. In five dimensions, N = 2 Maxwell-Einstein and Yang-Mills-Einstein theories are uniquely specified [19,21] by the C IJK tensor which enters the three-vector terms in the Lagrangian. We have shown how to extract these C IJK tensors from the double-copy form of the three-vector amplitudes and recovered it for all the homogeneous supergravities, classified in Ref. [26] and first given in a double-copy form in Ref. [30]. The formalism developed here can be further applied to the study of gauged supergravities and Yang-Mills-Einstein theories with non-compact gauge groups. We will return on the latter in forthcoming work [84]. A Gamma matrices and reality properties for on-shell spinors The 5D gamma matrices with lowered USp(2, 2) indices are antisymmetric, Ω-traceless and satisfy the following quadratic relations where E ab is a complex unit-determinant matrix (SL(2, C) matrix) that depends on the momentum, the reference vector and the precise choice of spinors in Eq. (2.7). If we consider the spinor parametrization in Eq. (2.7), and pick a reference vector q µ = (1, 0, 0, 1, 0), then the matrix E ab takes the simple form E ab = 0 − x −1 x 0 , (A.5) where x = p 0 + p 3 − m 2 p 0 −p 3 . Since it has unit determinant, the general E ab matrix satisfies E ac ǫ cd E bd = −ǫ ab , (A.6) and hence it acts as a complex conjugation on the little-group metric. Indeed, we have that (ǫ ab ) * = −ǫ ab . (A.7) Note that the γ 0,AB matrix satisfies the analogous relation This can be interpreted as the statement that the complex conjugated spinor is equal to the original spinor, up to a similarity transform and a sign flip. This is consistent with the spinors being symplectic-Majorana spinors. decomposition in terms of total SU(2) weights 35 5 More on double-copy amplitudes 36 5.1 Maxwell-Einstein and Yang-Mills-Einstein supergravities revisited 36 5.2 Amplitudes with massive vectors and tensors 39 mass term M = i m i comes from the completeness relation in Eq. (2.19). ⊗ W aḃ ǫ˙a˙b. It is interesting to contrast the double-copy form of some of the component fields in Eq.(3.54). The gravitensor is , H (2,2) = V (1,1) ⊗ V (1,1) ,H (3,0) a = T (2,0) ⊗ T (1,0) a ,H (4,0) = T (2,0) ⊗ T(2,0) .(3.65) A 3 ( 31A, 2A, 3A) = i 2q 3 3q 2 1|k 2 |q 1 + cyclic(1,2,3) . A 3 ( 31W, 2W, 3W ) = i 4m 2 m 3 [12] 23 [31 ± perms(1, 2, 3) , (4.20) 14 Note that the identity (m 1 −m 3 )[12] 23 [31 ±perms = 0 can be used to rewrite the amplitude. be derived from the superamplitude of 5D half-maximal super-Yang-Mills theory on the Coulomb branch, invariant under chiral (1,0) supersymmetry, provided that the mass conservation condition in Eq. (4.21) is obeyed. The bilinears in the chiral and anti-chiral supercharges can be written more explicitly as choice, Eq. (4.29) follows from Eq. (4.31), while u [a i w b] = ǫ ab follows from reference spinors w a i andw˙b i that satisfy u a i w b i − u b i w a i = ǫ ab (no sum over i) , (4.41)u˙a iwḃ i −ũ˙b iwȧ i = ǫ˙a˙b (no sum over i) , = i + 1 Mod 3, and r 2 = 1 1 2 1 ] = 2 1 3 1 ] = 3 1 1 1 ]. The three-point massive (1, 1) superamplitude in maximal 5D super-Yang-Mills on the Coulomb branch is then the direct generalization of Eq. (4.30), |Q , |Q , |Q , |ρ det |Q], |Q], |Q], |ρ] , (4.45) |1 , |2 , |3 , |ρ det |Q , |Q , |Q , |ρ (4.54) × det |1], |2], |3], |ρ] det |Q], |Q], |Q], |ρ] , 2η µν Ω AD , γ µ AB Ω BC γ ν CD Ω DA = 4η µν , γ µ AB (γ µ ) CD = 2(Ω AC Ω BD − Ω AD Ω BC ) − Ω AB Ω CD , (A.1)where the first identity is the Clifford algebra. Under complex conjugation the gamma matrices transform into themselves up to a similarity transform through time-like matrix γ 0,AB ,(γ µ AB ) * ≡ γ µ,AB = γ 0,AC γ µ CD γ 0,BD . (A.2)Hence it is the time-like gamma matrix γ 0,AB = γ 0 AB = −(γ 0 BA ) lowers and raises USp(2, 2) indices under complex conjugation. By inspection we see that the massive spinors obey the following reality property (|p a A ) * ≡ | p a A = γ 0,AB |p b B E ab , (A.4) γ 0 , 0AC Ω CD γ 0,BD = −Ω AB , (A.8) which similarly implies that (Ω AB ) * = −Ω AB .We can invert the conjugation in Eq. 33 ) 33where the SO(5) spinors can be obtained from Eq. (2.22) by an appropriate Wick rotation: 10 |µ = |p p j>0 →iµ j+5and |µ] = i|p]p j>0 →iµ j+5 det |Q α , |Q α , |Q α , |ρ .i 36 det |1], |2], |3], |ρ] 2 α=1,2 Likewise, there exist a well-behaved massive abelian tensor-graviton superamplitude in the Kaluza-Klein (2,0) supergravity theory M (2,0) 3 (1T , 2T , 3H˙a˙b) = −iA (1,0) 3 (1V˙c, 2Vḋ, 3V˙a)A (1,0) 3 (1Vċ, 2Vḋ, 3V˙b) = i 36 det |1˙c], |2ḋ], |3˙a], |ρ] det |1ċ], |2ḋ], |3˙b], |ρ] α=1,2 det |Q α , |Q α , |Q α , |ρ = − i 18ũ˙a 3ũḃ 3 α=1,2 det |Q α , |Q α , |Q α , |ρ . (4.50) m 1 [12] 2|ε 3 |1 = 0 (5.23) and 12 [2|ε 3 |1] + [12] 2|ε 3 |1 − [12 [2|ε 3 |1 − 12] 2|ε 3 |1] = 0 .(5.24) Note that in the Yang-Mills Lagrangian with spontaneously-broken gauge symmetry both O 1 and O 2 are present with the same normalization, and) . (5.28) W W A YM = W W A |DW | 2 + W W A W ·F ·W . (5.29) For a different approach for understanding perturbative six-dimensional self-dual tensor theories see Refs.[49][50][51][52][53][54][55][56][57]. In our convention, the number of supercharges is 4N . For massive 1/2-BPS multiplets we use a label (r + , r − ) to denote the number of chiral and anti-chiral little-group components of the supercharges, where N = 2(r + + r − )[80]. Naively, the right-hand side of this relation is invariant under two independent SL(2, C) transformations, one acting on the indices a, b, ... and one acting on the R-symmetry group indices α, β, ... However R-symmetry group is always compact and the indices a, b, .. refer to the little-group SU (2) indices inside SL(2, C). Since the R-symmetry group U Sp(4) ∼ = SO(5) only differs from the Lorentz group U Sp(2, 2) ∼ = SO(1, 4) by a signature change, it is convenient recycle the notation by putting dots on all indices. Specifically, we take ΩȦḂ to be given by Eq.(2.3). Note that a simpler SU (2)-preserving parametrization of θȦ a can be obtained after doing a similarity transform on ΩȦḂ, but here we work with the block-diagonal symplectic metric (2.3). 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[ "DETECTING ADVERSARIAL ATTACKS ON AUDIO-VISUAL SPEECH RECOGNITION", "DETECTING ADVERSARIAL ATTACKS ON AUDIO-VISUAL SPEECH RECOGNITION" ]
[ "Pingchuan Ma [email protected] \nDept. of Computing\nImperial College London\nUK\n", "Stavros Petridis [email protected] \nDept. of Computing\nImperial College London\nUK\n\nSamsung AI Center\nCambridgeUK\n", "Maja Pantic \nDept. of Computing\nImperial College London\nUK\n\nSamsung AI Center\nCambridgeUK\n" ]
[ "Dept. of Computing\nImperial College London\nUK", "Dept. of Computing\nImperial College London\nUK", "Samsung AI Center\nCambridgeUK", "Dept. of Computing\nImperial College London\nUK", "Samsung AI Center\nCambridgeUK" ]
[]
Adversarial attacks pose a threat to deep learning models. However, research on adversarial detection methods, especially in the multi-modal domain, is very limited. In this work, we propose an efficient and straightforward detection method based on the temporal correlation between audio and video streams. The main idea is that the correlation between audio and video in adversarial examples will be lower than benign examples due to added adversarial noise. We use the synchronisation confidence score as a proxy for audio-visual correlation and based on it we can detect adversarial attacks. To the best of our knowledge, this is the first work on detection of adversarial attacks on audio-visual speech recognition models. We apply recent adversarial attacks on two audiovisual speech recognition models trained on the GRID and LRW datasets. The experimental results demonstrated that the proposed approach is an effective way for detecting such attacks.
10.1109/icassp39728.2021.9413661
[ "https://arxiv.org/pdf/1912.08639v1.pdf" ]
209,404,979
1912.08639
741258d52f3c5c3030de43aa09c7e55a45ff1a84
DETECTING ADVERSARIAL ATTACKS ON AUDIO-VISUAL SPEECH RECOGNITION 18 Dec 2019 Pingchuan Ma [email protected] Dept. of Computing Imperial College London UK Stavros Petridis [email protected] Dept. of Computing Imperial College London UK Samsung AI Center CambridgeUK Maja Pantic Dept. of Computing Imperial College London UK Samsung AI Center CambridgeUK DETECTING ADVERSARIAL ATTACKS ON AUDIO-VISUAL SPEECH RECOGNITION 18 Dec 2019Index Terms-Audiovisual Speech RecognitionAdver- sarial Attack DetectionAudiovisusal Synchronisation Adversarial attacks pose a threat to deep learning models. However, research on adversarial detection methods, especially in the multi-modal domain, is very limited. In this work, we propose an efficient and straightforward detection method based on the temporal correlation between audio and video streams. The main idea is that the correlation between audio and video in adversarial examples will be lower than benign examples due to added adversarial noise. We use the synchronisation confidence score as a proxy for audio-visual correlation and based on it we can detect adversarial attacks. To the best of our knowledge, this is the first work on detection of adversarial attacks on audio-visual speech recognition models. We apply recent adversarial attacks on two audiovisual speech recognition models trained on the GRID and LRW datasets. The experimental results demonstrated that the proposed approach is an effective way for detecting such attacks. INTRODUCTION Deep networks achieve state-of-the-art performance on several tasks such as image classification, image segmentation and face recognition. However, recent studies [1,2] show that such networks are susceptible to adversarial attacks. Given any input x and a classifier f (·), an adversary tries to carefully construct a sample x adv that is similar to x but f (x) = f (x adv ). The adversarial examples are indistinguishable from the original ones but can easily degrade the performance of deep classifiers. Existing studies on adversarial attacks have mainly focused in the image domain [2,3,4,5]. Recently, adversarial attacks in the audio domain have also been presented [6,7]. One of the most prominent studies is the iterative optimisation-based attack [7], which directly operates on an audio clip and enables it to be transcribed to any phrase when a perturbation is added. Works on defense approaches against adversarial attacks can be divided into three categories: adversarial training [2], gradient masking [8] and input transforma-tion [9]. The first one adds adversarial examples in the training set whereas the second one builds a model which does not have useful gradients. Both of them require the model to be retrained, which can be computationally expensive. In contrast, the latter one attempts to defend adversarial attacks by transforming the input. On the other hand, work on how to detect adversarial attacks is very limited. To the best of our knowledge, the only work in the audio domain was proposed by Yang et al. [10] and exploits the inherent temporal dependency in audio samples to detect adversarial examples. The main idea is that the transcribed results from an audio sequence and segments extracted from it are consistent in benign examples but not in adversarial ones. In other words, the temporal dependency is not preserved in adversarial sequences. Inspired by the idea of using temporal dependency to detect audio adversarial examples, we propose a simple and efficient detection method against audio-visual adversarial attacks. To the best of our knowledge, this is the first work which presents a detection method of adversarial attacks on audio-visual speech recognition. The key idea is that the audio stream is highly correlated with the video of the face (and especially the mouth region). In case of an adversarial example, the added noise on the audio and video streams is expected to weaken the audio-visual correlation. Hence, we propose the use of audio-visual synchronisation as a proxy to correlation. In other words, we expect higher synchronisation scores for benign examples and lower scores for adversarial examples. The proposed detection method is tested on speech recognition attacks on models trained on the Lip Reading in the Wild (LRW) [11] and GRID datasets [12]. Our results show that we can detect audio-visual adversarial attacks with high accuracy. DATABASES For the purposes of this study, we use two audiovisual datasets, the LRW [11] and GRID [12] datasets. The LRW dataset is a large-scale audio-visual dataset consisting of clips from BBC programs. The dataset has 500 isolated words from more than 1000 speakers and contains 488766, 25000, and 25000 examples in the training, validation and test sets, respectively. Each utterance is a short segment with a length of 29 frames (1.16 seconds), where target words are centred in the segment of utterances. The GRID dataset consists of 33 speakers and 33000 utterances (1000 per speaker). Each utterance is composed of six words taken from the combination of the following components: <command: 4><colour: 4><preposition: 4><letter: 25><digit: 10><adverb: 4>, where the number of choices for each component is indicated in the angle brackets. In this work, we follow the evaluation protocol from [13] where 16, 7 and 10 subjects are used for training, validation and testing, respectively. BACKGROUND Attacks In this study, we consider two attack methods, Fast Gradient Sign Method (FGSM) [2] and the iterative optimisation-based attack [7]. FGSM, which is suitable for attacks on classification models, computes the gradient with respect to the benign input and each pixel can be updated to maximise the loss. Basic Iterative Method (BIM) [14] is an extended version of FGSM by applying it multiple times with a small step size. Specifically, given a loss function J(·, ·) for training the classification model f (·), the adversarial noise x adv is generated as follows: x adv 0 = x x adv N +1 = Clip x,ǫ {x adv N + αsign(∇ x J(f (x adv N ), y true )}(1) where α is the step size, x adv N is the adversarial example after N -steps of the iterative attack and y true is the true label. After each step, pixel values in the adversarial images x adv are clamped to the range [x − ǫ, x + ǫ], where ǫ is the maximum change in each pixel value. This method was proposed for adversarial attacks on images but can also be applied to audio clips by crafting perturbation to the audio input. The second type of attack [7] has been recently proposed and is suitable for attacks on continuous speech recognition models. Audio adversarial examples can be generated, which can be transcribed to any phrase but sound similar to the benign one. Specifically, the goal of this targeted attack is to seek an adversary input x adv , which is very close to the benign input x, but the model decodes it to the target phrase z target . The objective of the attack is the following: minimize J(f (x + δ), z target ) such that δ < ǫ(2) where ǫ is introduced to limit the maximum change for each audio sample or pixel and δ is the amount of adversarial noise. Audio-visual Speech Recognition Threat Model The architecture is shown in Fig. 1a. We use the end-toend audiovisual model that was proposed in [15]. The video stream consists of spatiotemporal convolution, a ResNet18 network and a 2-layer BGRU network whereas the audio stream consists of a 5-layer CNN and a 2-layer BGRU network. These two streams are used for feature extraction from raw modalities. The top two-layer BGRU network further models the temporal dynamics of the concatenated feature. According to the problem type, two different loss functions are applied for training. The multi-class cross entropy loss, where each input sequence is assigned a single class, is suitable for word-level speech recognition. The CTC loss is used for sentence-level classification. This loss transcribes directly from sequence to sequence when the alignment between inputs and target outputs is unknown. Given an input sequence x = (x 1 , ..., x T ), CTC sums over the probability of all possible alignments to obtain the posterior of the target sequence. SYNCHRONISATION-BASED DETECTION METHOD Chung et al. [16,17] introduced the SyncNet model, which is able to predict the synchronisation error when raw audio and video streams are given. This error is quantified by the synchronisation offset and confidence score. A sliding window approach is used to determine the audio-visual offset. For each 5-frame video window, the offset is found when the distance between the visual features and all audio features in a ± 1 second range is minimised. The confidence score for a particular offset is defined as the difference between the the minimum and the median of the Euclidean distances (computed over all windows). Audio and video are considered perfectly matched if the offset approaches to zero with a high level of confidence score. In this work, we aim to explore if such synchronisation is affected by adversarial noise. The detection method is shown in Fig. 1b and 1c. In the detection model, we measure the temporal consistency between the audio and video streams via a model trained for audio-visual synchronisation. For benign audio and video streams, the confidence score should be relatively high since audio and video are aligned and therefore highly synchronised. However, for adversarial audio and video examples, the confidence score is expected to be lower. The added perturbation, which aims to alter the model toward the target transcription, reduces the correlation between the two streams, hence they are less synchronous. Fig. 1d. shows the confidence distribution of 300 benign and adversarial examples from the GRID dataset. EXPERIMENTAL SETUP Attacks We evaluate our proposed method using two adversarial attacks on both modalities. We assume a white-box scenario, where the parameters of models are known to the attacker. Attacks against Word-level Classification: Attacks such as FGSM and BIM are suitable for word recognition models trained on the LRW dataset. For FGSM, ǫ A for the audio stream and ǫ V for the video stream, were chosen heuristically. In our case, we set ǫ A to 1024 and ǫ V to 16 1 step size α V was set to 1 in the image domain, which means the value of each pixel is changed by 1 at each iteration. The step size α A in the audio domain is set to 64. We follow the number of iterations setting suggested by [14], which is selected to be min(ǫ V + 4, 1.25ǫ V ). Attacks against Continuous Speech Recognition: For attacking a speech recognition model trained on GRID we use a recently proposed targeted attack [7]. The maximum change allowed as defined by ǫ (see Eq. 2) is initialised to 2048 and 32 for audio and video, respectively, and is reduced during iterative optimisation. We implement the attack with 1000 iterations. In our studies, 10 random utterances are selected as target utterances. 300 adversarial examples are randomly selected for each target utterance. Evaluation Metrics We use the Euclidean distance (L 2 ) for measuring the similarity between two images. We also use the L ∞ norm to measure the maximum change per pixel. For audio samples we follow [7] and convert the L ∞ norm to the scale of Decibels (dB): dB(x) = max i 20 · log 10 (x i ), where x i is an arbitrary audio sample point from the audio clip x. The audio distortion is specified as the relative loudness to the benign audio, which can be defined as dB x (δ) = dB(δ) − dB(x). The Area Under the Curve (AUC) score is used for evaluating the detection approach. We compute the synchronisation confidence score in benign and adversarial examples and Table 1. Results for the proposed adversarial attack detection approach on word recognition models trained on the LRW dataset. L V ∞ is 4 and 8 pixels when ǫ V is 16 and 32, respectively. L A ∞ is -19dB when ǫ A is 1024. Finally, in order to compare how this approach would work in a real scenario, we select the threshold (from Fig. 1c) which maximises the average F 1 score of adversarial and benign classes on the validation set. Then we use this threshold to compute the average F 1 score on the test set. Attacks CR(%) L V 2 AUC F 1 FGSM (ǫ A =1024, ǫ V =16 RESULTS Word-level Speech Recognition Detection results for attacks on word-level speech recognition are shown in Table 1. In the presence of adversarial noise, the classification rate drops from 97.20% 2 to 12.40% using FGSM whereas it drops to 3.27% after running 36 iterations using BIM. On the other hand, this detection method achieves an AUC score as high as 0.99 using FGSM and 0.93 using BIM. The corresponding F 1 scores are 0.96 and 0.85, respectively. We also notice that when the attack is stronger, e.g., BIM is used instead of FSGM, the classification rate goes down, i.e., the attack is more successful, and at the same time the distortion (L V 2 ) becomes smaller. Consequently, detection becomes more difficult and this is reflected to the lower AUC and F 1 scores. We also investigate the detection performance when the ǫ V decreases to 16 from 32, i.e., the pixel values change less. It is clear from Table 1 that for both types of attacks the distortion is smaller and as a consequence detection becomes harder, both AUC and F 1 scores go down. However, such attacks are less successful since the classification rate goes up. Sentence-level Speech Recognition Detection results for fully targeted attacks on sentence-level speech recognition, i.e., the goal of the attack is that the transcribed result is the same as the desired target phrase, are shown in Table 2. It is clear that the attack is almost always successful no matter what the target sentence is, more than 92% of the examples are transcribed as the target phrase. At the same time the detection rates are quite high with an Table 2. Results of the proposed audio-visual synchronisation detection on fully targeted adversarial attacks, i.e., the goal of the attack is to make the WER between transcribed and target phrases 0, on continuous speech recognition models trained on GRID. The success rate is the proportion of adversarial examples with WER equal to 0. (ǫ A = 1024, AUC between 0.93 and 0.97 and an F 1 score between 0.81 and 0.86. We also observe that the maximum distortions applied to the audio and video signals are similar in most cases. We also consider another scenario where the WER between the transcribed results and target phrases is up to 50%. Results are shown in Table 3. In this case the attack is always successful. In addition the generated audio and video adversarial examples are less distorted than the ones generated by the fully targeted attacks. In turn, this leads to smaller AUC scores, between 0.88 and 0.92, and F 1 scores, between 0.76 and 0.80. ǫ V = 16) Target Phrases Success Rate L V 2 L V ∞ L A ∞ (dB) AUC F 1 bin CONCLUSION In this work, we have investigated the use of audio-visual synchronisation as a detection method of adversarial attacks. We hypothesised that the synchronisation confidence score will be lower in adversarial than benign examples and demonstrated that this can be used for detecting adversarial attacks. In future work, we would like to investigate more sophisticated approaches for measuring the correlation between audio and visual streams. Fig. 1 . 1An overview of our proposed detection method. (a) A video and an audio clip are fed to the end-to-end audio-visual speech recognition model. They are also fed to the synchronisation network (b) which estimates a synchronisation confidence score which is used for determining if the audio-visual model has been attacked or not (c). The confidence distribution of 300 adversarial and benign examples from the GRID dataset is shown in (d). . For BIM, the 1 Pixel values are in the range of [0, 255]. Audio samples are in the range of [-32768, 32767]. Fig. 2 . 2One example using iterative optimisation-based attack on the GRID dataset. (a): benign example; (b): adversarial noise; (c): adversarial example; Raw audio waveforms, audio log-spectrum and raw images are presented from top to bottom. the threshold we compute the Receiver Operating Characteristic (ROC) curve. Table 3 . 3Results of the proposed audio-visual synchronisation detection on targeted adversarial attacks on continuous speech recognition models trained on GRID. The WER between transcribed and target phrases is up to 50%. The success rate is the proportion of adversarial examples with WER less than 50%. (ǫ A = 1024, ǫ V = 16)Target Phrases Success Rate L V 2 L V ∞ L A ∞ (dB) AUC F 1 bin blue at a zero please 1.00 5.10 7.27 -51.39 0.91 0.79 bin white by o nine now 1.00 5.25 7.47 -48.86 0.91 0.79 lay green with y seven again 1.00 5.24 7.45 -49.28 0.91 0.79 lay red at c eight soon 1.00 5.01 7.13 -49.54 0.91 0.79 place blue at p one again 1.00 4.83 6.90 -51.71 0.88 0.76 place red by a one soon 1.00 5.02 7.14 -49.93 0.89 0.79 place red by z two soon 1.00 5.14 7.31 -48.33 0.91 0.80 set green in f one again 1.00 5.14 7.32 -47.43 0.92 0.80 set red in x four now 1.00 5.15 7.33 -46.19 0.92 0.80 set white in p five now 1.00 5.19 7.40 -46.39 0.91 0.78 This is the performance of the model trained on the LRW dataset when benign examples are fed to it. . Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, Rob Fergus, Intriguing properties of neural networks," in ICLRChristian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus, "Intriguing properties of neural networks," in ICLR, 2014. Explaining and harnessing adversarial examples. J Ian, Jonathon Goodfellow, Christian Shlens, Szegedy, ICLR. Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy, "Explaining and harnessing adversarial exam- ples," in ICLR, 2015. One pixel attack for fooling deep neural networks. Jiawei Su, Danilo Vasconcellos Vargas, Kouichi Sakurai, IEEE Transactions on Evolutionary Computation. Jiawei Su, Danilo Vasconcellos Vargas, and Kouichi Sakurai, "One pixel attack for fooling deep neural net- works," IEEE Transactions on Evolutionary Computa- tion, 2019. Synthesizing robust adversarial examples. Anish Athalye, Logan Engstrom, Andrew Ilyas, Kevin Kwok, ICML. Anish Athalye, Logan Engstrom, Andrew Ilyas, and Kevin Kwok, "Synthesizing robust adversarial exam- ples," ICML, 2018. Decision-based adversarial attacks: Reliable attacks against black-box machine learning models. Wieland Brendel, Jonas Rauber, Matthias Bethge, in ICLR. Wieland Brendel, Jonas Rauber, and Matthias Bethge, "Decision-based adversarial attacks: Reliable attacks against black-box machine learning models," in ICLR, 2018. Did you hear that? adversarial examples against automatic speech recognition. Moustafa Alzantot, Bharathan Balaji, Mani Srivastava, NIPS Machine Deception workshop. Moustafa Alzantot, Bharathan Balaji, and Mani Srivas- tava, "Did you hear that? adversarial examples against automatic speech recognition," in NIPS Machine De- ception workshop, 2017. Audio adversarial examples: Targeted attacks on speech-to-text. Nicholas Carlini, David Wagner, IEEE Security and Privacy Workshops. Nicholas Carlini and David Wagner, "Audio adversarial examples: Targeted attacks on speech-to-text," in IEEE Security and Privacy Workshops, 2018. Distillation as a defense to adversarial perturbations against deep neural networks. Nicolas Papernot, Patrick Mcdaniel, Xi Wu, Somesh Jha, Ananthram Swami, IEEE Symposium on Security and Privacy. Nicolas Papernot, Patrick McDaniel, Xi Wu, Somesh Jha, and Ananthram Swami, "Distillation as a defense to adversarial perturbations against deep neural networks," in IEEE Symposium on Security and Privacy, 2016. Feature squeezing: Detecting adversarial examples in deep neural networks. Weilin Xu, David Evans, Yanjun Qi, NDSS. Weilin Xu, David Evans, and Yanjun Qi, "Feature squeezing: Detecting adversarial examples in deep neu- ral networks," in NDSS, 2018. Characterizing audio adversarial examples using temporal dependency. Zhuolin Yang, Bo Li, Pin-Yu Chen, Dawn Song, ICLR. Zhuolin Yang, Bo Li, Pin-Yu Chen, and Dawn Song, "Characterizing audio adversarial examples using tem- poral dependency," in ICLR, 2019. Lip reading in the wild. Son Joon, Andrew Chung, Zisserman, ACCV. Joon Son Chung and Andrew Zisserman, "Lip reading in the wild," in ACCV, 2016. An audio-visual corpus for speech perception and automatic speech recognition. Martin Cooke, Jon Barker, Stuart Cunningham, Xu Shao, The Journal of the Acoustical Society of America. Martin Cooke, Jon Barker, Stuart Cunningham, and Xu Shao, "An audio-visual corpus for speech percep- tion and automatic speech recognition," in The Journal of the Acoustical Society of America, 2006. End-to-end speech-driven facial animation with temporal gans. Konstantinos Vougioukas, Stavros Petridis, Maja Pantic, BMVC. Konstantinos Vougioukas, Stavros Petridis, and Maja Pantic, "End-to-end speech-driven facial animation with temporal gans," BMVC, 2018. Adversarial examples in the physical world. Alexey Kurakin, Ian Goodfellow, Samy Bengio, ICLR workshop. Alexey Kurakin, Ian Goodfellow, and Samy Bengio, "Adversarial examples in the physical world," in ICLR workshop, 2017. Investigating the lombard effect influence on end-to-end audio-visual speech recognition. Pingchuan Ma, Stavros Petridis, Maja Pantic, INTERSPEECH. Pingchuan Ma, Stavros Petridis, and Maja Pantic, "In- vestigating the lombard effect influence on end-to-end audio-visual speech recognition," in INTERSPEECH, 2019. Out of time: automated lip sync in the wild. Son Joon, Andrew Chung, Zisserman, ACCVJoon Son Chung and Andrew Zisserman, "Out of time: automated lip sync in the wild," in ACCV, 2016. Perfect match: Improved cross-modal embeddings for audio-visual synchronisation. Soo-Whan Chung, Joon Son Chung, Hong-Goo Kang, IEEE ICASSP. Soo-Whan Chung, Joon Son Chung, and Hong-Goo Kang, "Perfect match: Improved cross-modal em- beddings for audio-visual synchronisation," in IEEE ICASSP, 2019.
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[ "Signatures of Self-Interacting Dark Matter in the Matter Power Spectrum and the CMB", "Signatures of Self-Interacting Dark Matter in the Matter Power Spectrum and the CMB" ]
[ "Ran Huo \nDepartment of Physics and Astronomy\nUniversity of California\n92521RiversideCaliforniaUSA\n", "Manoj Kaplinghat \nDepartment of Physics and Astronomy\nUniversity of California\n92697IrvineCaliforniaUSA\n", "Zhen Pan \nDepartment of Physics\nUniversity of California\n95616DavisCaliforniaUSA\n", "Hai-Bo Yu \nDepartment of Physics and Astronomy\nUniversity of California\n92521RiversideCaliforniaUSA\n" ]
[ "Department of Physics and Astronomy\nUniversity of California\n92521RiversideCaliforniaUSA", "Department of Physics and Astronomy\nUniversity of California\n92697IrvineCaliforniaUSA", "Department of Physics\nUniversity of California\n95616DavisCaliforniaUSA", "Department of Physics and Astronomy\nUniversity of California\n92521RiversideCaliforniaUSA" ]
[]
We consider a self-interacting dark matter model in which the massive dark photon mediating the selfinteraction decays to light dark fermions to avoid over-closing the universe. We find that if the model is constrained to explain the dark matter halos inferred for spiral galaxies and galaxy clusters simultaneously, there is a strong indication that dark matter is produced asymmetrically in the early universe. It also implies the presence of dark radiation, late kinetic decoupling for dark matter, and a suppressed linear power spectrum due to dark acoustic damping. The Lyman-α forest power spectrum measurements put a strong upper limit on the damping scale and the model has little room to reduce the abundances of satellite galaxies. Future observations in the matter power spectrum and the CMB, in tandem with the impact of self-interactions in galactic halos, makes it possible to measure the gauge coupling and masses of the dark sector particles even when signals in conventional dark matter searches are absent.
10.1016/j.physletb.2018.06.024
[ "https://web.archive.org/web/20200623035400/https:/escholarship.org/content/qt5k58r329/qt5k58r329_noSplash_b7902b8f844768c6f10a236cdc298763.pdf?t=pwuuv1" ]
59,364,361
1709.09717
c3a96b7654b40682a4b2feb1dc41de2e5982753d
Signatures of Self-Interacting Dark Matter in the Matter Power Spectrum and the CMB Ran Huo Department of Physics and Astronomy University of California 92521RiversideCaliforniaUSA Manoj Kaplinghat Department of Physics and Astronomy University of California 92697IrvineCaliforniaUSA Zhen Pan Department of Physics University of California 95616DavisCaliforniaUSA Hai-Bo Yu Department of Physics and Astronomy University of California 92521RiversideCaliforniaUSA Signatures of Self-Interacting Dark Matter in the Matter Power Spectrum and the CMB (Dated: July 3, 2018) We consider a self-interacting dark matter model in which the massive dark photon mediating the selfinteraction decays to light dark fermions to avoid over-closing the universe. We find that if the model is constrained to explain the dark matter halos inferred for spiral galaxies and galaxy clusters simultaneously, there is a strong indication that dark matter is produced asymmetrically in the early universe. It also implies the presence of dark radiation, late kinetic decoupling for dark matter, and a suppressed linear power spectrum due to dark acoustic damping. The Lyman-α forest power spectrum measurements put a strong upper limit on the damping scale and the model has little room to reduce the abundances of satellite galaxies. Future observations in the matter power spectrum and the CMB, in tandem with the impact of self-interactions in galactic halos, makes it possible to measure the gauge coupling and masses of the dark sector particles even when signals in conventional dark matter searches are absent. I. Introduction. The existence of dark matter (DM) in the universe is inferred from its gravitational influence on normal matter. Null results in terrestrial DM searches have put strong constraints on the DM interaction with the standard model particles, e.g., [1][2][3]. However, it does not preclude the possibility that DM may interact strongly with itself [4,5]. Strong DM self-interactions can change the inner halo structure, leading to a better agreement with small-scale observations than the cold DM (CDM) model (see [6] for a review and reference therein). In particular, kinetic thermalization due to the DM self-collisions ties the baryonic and DM distributions in galaxies together [7,8]. It has been shown [9,10] that this can lead to the observed diversity in rotation curves of low and high surface brightness galaxies [11,12]. A generic feature of self-interacting DM models is the existence of a light force carrier for mediating strong DM self-interactions in galactic halos. A mass hierarchy between the mediator and the DM particle is required to get a self-scattering cross section that decreases for velocities of O(1000 km/s) (cluster scale) [5]. This mediator must decay to avoid the over-closing the universe [13,14], unless it is (almost) massless (see, e.g., [15,16]). The minimal models where the mediator decays to standard model particles have been strongly constrained by DM direct detection experiments [14,17,18], since the DM-nucleus scattering cross section is enhanced due to the smallness of the mediator mass. In addition, the s-wave DM annihilation can be boosted, resulting in strong constraints from indirect detection experiments [19][20][21]. A simple solution is to introduce a massless particle species (f ) in the dark sector, in addition to the DM particle (χ) and the mediator (φ). In the early universe, φ can be in thermal equilibrium with f , so that its number density becomes Boltzmann suppressed when the temperature is below its mass, avoiding the over-closure problem. Since φ is not necessary to couple to the standard model in this case, conventional DM signals can be absent. Aside from usual χ-χ self-scattering, φ also mediates χ-f collisions in the early thermal bath. A tight coupling between matter and radiation in the early universe will lead to a cutoff in the linear matter power spectrum [15,[22][23][24][25][26][27][28][29][30][31][32][33]. This model was invoked previously to simultaneously flatten the density profiles of dwarfs and reduce their abundances [27,34,35]. In this Letter, we use this model to explicitly demonstrate how astrophysical observations can pin down the particle physics parameter space. After constraining it to explain the dark matter halos inferred for dwarf galaxies and galaxy clusters, we explore the presence of the damping scale and the dark radiation using the Lyman-α forest, satellite counts, and CMB. In particular, we highlight two major findings. • Astrophysical data favor an asymmetric production mechanism for SIDM. When we require DM selfinteractions to explain the diversity of inferred dark matter halo profiles in dwarf galaxies to clusters of galaxies, there is a minimal annihilation cross section for the inevitable process, χχ → φφ. For symmetric DM (both DM and anti-DM particles are equally populated), the allowed DM mass is in the narrow range of ∼ 9-240 MeV. The corresponding coupling constants have to be unnaturally small to give rise to a relic density consistent with the observed value. • The Lyman-α forest power spectrum measurements mute the impact of the damping scale (induced by the χ-f interaction). We explicitly demonstrate that the kinetic decoupling temperature dictates the deviation of the SIDM matter power spectrum from the standard CDM case and map it to the warm DM (WDM) mass space. After taking into account the most recent Lyman-α constraints, we show that the model is unlikely to solve the missing satellites problem as suggested in [27,34,35]. II. A Constrained Simplified SIDM Model We consider a simplified SIDM model with the following interaction La-arXiv:1709.09717v3 [hep-ph] 2 Jul 2018 1: SIDM parameter space (2σ CL) favored by a wide range of astrophysical data from dwarf galaxies to galaxy clusters, for both asymmetric (gray) and symmetric DM (magenta). Inset: the DM self-scattering cross section vs. velocity for a best fit case and the data points with error bars are from [5]. grangian [25,27], L int = −ig χχ γ µ χφ µ + m χχ χ + 1 2 m 2 φ φ µ φ µ − ig ff γ µ f φ µ , (1) where we assume that the SIDM particle (χ) and the massless fermion f couple to a gauge boson (φ) with coupling constants g χ and g f , respectively. We assume g χ = g f , since they are expected to be similar from the model building perspective. The dark sector could evolve independently from the visible sector in the early universe, and we use ξ to parameterize the ratio of dark-to-visible temperatures, T f /T γ [26,36]. This model, with four parameters in total (g χ , m χ , m φ , ξ), can be regarded as a simplified version of more general and complex hidden charged DM models [26,36]. In the early universe, DM particles can annihilate to the mediator. For symmetric DM, the required annihilation cross section is ξ × 6 × 10 −26 cm 3 /s, which fixes g χ ≈ 0.02(m χ /GeV) 1 2 ξ 1 4 (we take ξ = 0.48), as denoted in Fig. 1 (lower, magenta). We further determine m φ for given m χ by fitting to the preferred σ χχ /m χ values in [5], which are extracted from SIDM fits to galactic rotation curves and lensing and kinematic measurements in clusters of galaxies. We find the allowed DM mass range is very limited, 9-240 MeV, after simultaneously imposing σ χχ /m χ 1 cm 2 /g (galaxies) and 0.1 cm 2 /g (clusters), as shown in Fig. 1 (upper, magenta). Without the cluster constraint, it is possible to have strong DM self-interactions in galaxies for larger m χ , due to the non-perturbative enhancement effects in the quantum and classical regions [37,38], where σ χχ /m χ has a strong velocity dependence and becomes negligible in clusters. For asymmetric DM, the abundance is set up by a primordial DM asymmetry [39,40]. We do not impose a prior constrain on g χ from the abundance consideration and determine both m φ and g χ from the cross section measurements, as shown in Fig. 1 (gray). Compared to the symmetric case, it is clear that asymmetric DM has a much larger mass range to be consistent observations from dwarfs to clusters. When m χ 200 GeV, the model becomes non-perturbative. For m χ 40 MeV, χχ → φφ itself can deplete the symmetric component [39]. While for a smaller mass, additional annihilation channels are required. In the rest of the paper, we will focus on asymmetric DM with m χ 1 GeV and see these constraints provide concrete predictions for the matter power spectrum and the CMB. The presence of light fermions f contributes to the relativistic degrees of freedom parameterized as N eff = 3.046 + ∆N eff , with ∆N eff = (11/4) 4/3 ξ 4 . The temperature ratio, ξ, remains constant through kinetic decoupling and later because there is no entropy transfer in the two sectors. The analysis of Planck data indicates that N eff = 3.15 ± 0.23 [41], which can be recast as an upper bound on the temperature ratio at the recombination epoch, ξ 0.62, at 2σ CL. After fitting to the Planck 2015 polarization and temperature data, [42] found a stronger bound ξ 0.48 for imperfect fluid at 2σ CL, with some dependence on the χ-f interaction that we neglect. III. Dark Radiation, Acoustic Damping and Kinetic Decoupling. In the early universe, the elastic scattering process χf → χf can damp the linear power spectrum of SIDM. When the momentum transfer rate goes below the Hubble rate, kinetic decoupling occurs. We calculate the visible sector temperature when kinetic decoupling of dark matter happens as [27,32] T kd ≈ 1.38 keV √ g χ g f m χ 100 GeV 1 4 m φ 10 MeV g * 3.38 1 8 0.5 ξ 3 2 (2) where g * is the number of massless degrees of freedom at decoupling. Fig. 2 shows the T kd contours for the SIDM model. There is clear degeneracy between m χ and ξ, i.e., a stronger momentum transfer rate in the χ-f collision can compensate a colder hidden sector thermal bath in determining T kd . In the high mass regime, m χ 9 GeV, m χ increases with decreasing ξ to keep a constant decoupling temperature. As shown in Fig. 1, when m χ increases from 20 to 200 GeV, m φ decreases from 20 to 7 MeV and g χ increases from 0.5 to 3.5. The net result is a larger momentum transfer rate for larger m χ and hence a colder hidden sector to maintain the same T kd . For m χ 9 GeV, this behavior changes because the required m φ increases sharply with m χ , suppressing momentum transfer. For given ξ, T kd reaches its maximum, 5 keV(0.5/ξ) 3/2 , when m χ ≈ 9 GeV. If the two sectors were thermalized after inflation, e.g., through the collision process mediated by the inflaton [44], then the temperature ratio is ξ ≈ 0.5 (∆N eff ≈ 0.24) and we predict a maximal decoupling temperature of 5 keV, which we use to set a lower limit on the minimum halo mass as we discuss in Sec. V. If the visible sector has additional massive new particles, ξ could be lower. For example, with the minimal supersymmetric standard model, we get ξ ≈ 0.43 (∆N eff ≈ 0.13). This is within the reach of the CMB-S4 experiment, with a projected sensitivity of ∆N eff ≈ 0.02 (ξ ≈ 0.27) [45]. In Fig. 3, we compare the matter power spectra for three cases (denoted by the black dots in Fig. 2) with fixed T kd , generated using the modified version of the Boltzmann code CAMB [46] developed for the ETHOS simulations [32]. For the model parameters, T kd m χ , which implies that freestreaming effects are not relevant. The presence of dark acoustic oscillations [15,26,27,47,48] for k a(T kd )H(T kd ) ≈ 10(T kd /keV)Mpc −1 is clearly evident in Fig. 3. The resulting suppression of the power spectrum is only dependent on T kd to a good approximation, until dark Silk damping becomes important on smaller scales. The dependence on ξ through the expansion rate and sound horizon is weak, which we ex- We also show the corresponding thermal WDM mass that gives rise to the same δA. The strong (weak) Lyman-α constraints exclude thermal WDM with a mass below 5.3 keV (3.5 keV) at 95% CL [43], which correspond to the lower limits on T kd in the SIDM model, T kd ≈ 1 keV (0.6 keV). Lower: The number of subhalos with masses larger than 10 8 M /h in a MW-sized halo predicted in the SIDM model. A conservative lower limit of 63 using counts of satellites in the MW is shown by the short horizontal line. Given the strong (weak) Lyman-α constraints from Ref. [43], the number of subhalos in the SIDM model can only be suppressed by 10% (30%) compared to the CDM prediction, leaving little room for the model to impact the abundances of satellite galaxies. plicitly verify in Fig. 3 for the parameters we take. IV. Lyman-α Constraints. The Lyman-α forest absorption spectrum measures the neutral hydrogen density fluctuation on very large scales. Since the density of H atoms tracks that of the DM distribution, the Lyman-α forest can be used to constrain DM properties [49][50][51][52][53][54][55][56][57]. In particular, it has put strong constrains on WDM models, where the free-streaming effect damps the DM linear power spectrum. A recent combined analysis of XQ-100 and HIRES/MIKE samples put a lower limit on the thermal WDM mass, 5.3 keV at 2σ CL, when the temperature evolution of the inter galactic medium is modeled as a power law in redshift [43]. The limit is relaxed to 3.5 keV, if one allows a non-smooth evolution of the temperature with sudden temperature changes up to 5000 K [43]. To recast the Lyman-α constraints on thermal WDM as constraints on the SIDM damping scale, we use the estimator introduced in [43 ], δA = (A CDM − A)/A CDM , where A = kmax kmin dkP 1D (k)/P CDM 1D (k) with P 1D (k) = (1/2π) ∞ k dk k P (k ) . δA measures the power suppression relative to CDM. In calculating δA for the model, we compute the 3D linear matter power spectrum, P (k ), at redshift z = 0, and take k max = 20h/Mpc and k min = 0.5h/Mpc for the range of scales probed [43], with h = 0.67. In Fig. 4 (upper), we map T kd to δA for ξ = 0.48. In comparison, we also compute δA for WDM and find 5.3 keV (3.5 keV) WDM and SIDM with T kd ≈ 1 keV (0.6 keV) have the same δA. In Fig. 2, we also show the same constraints in the ξ − m χ plane. A non-zero ∆N eff delays matter-radiation equality and suppresses growth, which is reflected in the matter power spectrum. However, the redshift of equality is measured to roughly 1% [41], which can be obtained by changing other cosmological parameters [58]. For simplicity, we have fixed ΛCDM cosmological parameters to the Planck best-fit values [41] in calculating the matter power spectrum. This is a good approximation because ∆N eff is small for ξ = 0.48. Changing to ξ = 0.27 (CMB-S4 predicted sensitivity) in our approximate analysis only weakens the constraints on T kd by 10%. IV. The halo mass function. A damped DM matter power spectrum will lead to a decrease in the number of low mass field halos and subhalos. The onset of this suppression in the field HMF is controlled by M cut ≈ 0.7 × 10 8 ( keV/T kd ) 3 M [47,48,59]. We find that the ansatz dn SIDM /dM = (1 + M cut /M ) −1.34 dn CDM /dM reproduces the field HMFs in the ETHOS simulations well, see Fig. 5 for comparison. The predicted kinetic decoupling temperature of the SIDM model constrained to fit galaxy and cluster-scale halos (Fig. 1) and allowed by Lyman-α constraints (Fig. 4) is in the range of 0.6-5 keV. The upper limit is obtained if the two sectors were thermalized in the early universe. The corresponding minimum halo mass is M cut ∼ 10 8 -10 5 M . In Fig. 6, we show the power spectra for four SIDM benchmark models and two WDM models, together with their HMFs for field halos. Satellite counts provide an important constraint on the HMF [64,65]. We follow the procedure in [66] (see Eq. 17 therein) to calculate the subhalo mass function for the model. Assuming a MW halo mass 1.7 × 10 12 M /h, we obtain the number of subhalos with masses larger than 10 8 M /h, N subhalo , for given T kd , as shown in Fig. 4 (lower). We de-mand that this number should be greater than 63, which is the sum of 11 classical satellites and 15 SDSS satellites × 3.5 for incomplete sky coverage [64,67]. This constraint is weaker than the Lyman-α limits but we expect it to be a comparable constraint as more satellites are discovered. With the new satellites in the DES footprint [68][69][70][71], the total number of satellites (corrected for selection biases) may be consistent with ΛCDM expectations [72][73][74]. This is still an open issue. The potential mismatch between the number of predicted subhalos and the observed satellites ("missing satellites problem" [75,76]) has been used to motivate the presence of dark acoustic damping [27]. Our analysis shows that the constraints from the Lyman-α forest power spectrum, assuming no significant unmodeled systematic effects, leave little room to modify the abundance of satellites. For the strong (weak) Lyman-α constraint, T kd = 1 keV (0.6 keV), the corresponding number of subhalos for masses larger than 10 8 M /h is N subhalo ≈ 142 (113), as shown in Fig. 4 (lower), which is only 10% (30%) less than that predicted in the CDM model. Since T kd determines the damped power spectrum, modifying the particle masses or couplings will not change this result. On the other hand, self-interactions (leading to core formation) could change the distribution of satellites in the inner region of MW and Andromeda due to tidal effects [77], while the distribution in the outer parts remains similar to the collisionless case [78,79]. It is also possible that the early star formation feedback effects are different in cored SIDM halos and this may impact the faint-end luminosity function. These effects are clearly relevant for the "missing satellites problem," and remained to be quantified. Observationally, we expect LSST to provide a definite statement in terms of the census of the ultra-faint satellites [73]. In addition, the dark subhalos (or ultra-faint galaxies) could be discovered gravitationally through lensing [80][81][82][83][84][85][86][87], and tidal streams in the MW [88][89][90][91][92][93][94][95]. Does the model solve the too-big-to-fail problem at the bright end of the satellite luminosity function [96,97]? The ETHOS-4 model has T kd = 0.5 keV and it agrees broadly with observations of the dwarf spheroidals in the MW [59]. This is mainly due to damping (see also [98]) because σ χχ /m χ ∼ 0.2 cm 2 /g in this model, which cannot change the inner halo of dwarf galaxies significantly [99]. Given the Lyman-α constraint, the small-scale power in our case is less suppressed (T kd 1 keV (0.6 keV)) compared to ETHOS-4 model. However, in our case, the self-interaction cross section is large in dwarf galaxies (σ χχ /m χ ∼ 2 cm 2 /s), which would lower the subhalo densities due to core formation to roughly the right values [78,100]. It is of interest to perform a more detailed assessment of the too-big-to-fail problem in our case. In the future, if a cut-off scale in the HMF is observed, we can derive its corresponding T kd and fix the relation in the ξ-m χ plane (see Fig. 2). The particle parameters, e.g., m χ , m φ , and g χ , can be further determined (up to the two-fold degeneracy) if the presence of dark radiation is detected or even a stronger upper bound on ξ is obtained. Similar conclu- We also show 5.3 keV (3.5 keV) thermal WDM, corresponding to the strong (weak) lower bounds from the Lyman-α constraints derived in Ref. [43], and CDM. Right: Field halo mass functions, for our SIDM benchmark cases, together with thermal warm DM and CDM. We use the extended Press-Schechter theory [60,61] to generate the CDM halo mass function with the fitting parameters given in [62], and multiply it with the scaling relation for the SIDM cases (see Fig. 5). The warm DM ones are estimated with the fitting formula in [63]. sions seem to also apply to the case of the atomic dark matter model [15,16,29,101], when it is constrained to solve the small-scale puzzles [102]. V. Conclusions. SIDM is a compelling alternative to CDM. It keeps all the success of CDM on large scales, while modifying the inner halo structure in accord with observations. Using a simplified particle physics realization, we have shown that SIDM generically prefers asymmetric DM, and predicts the existence of dark radiation and a damped linear DM power spectrum, with the damping scale set by the selfscattering cross section and the temperature ratio between the two sectors. We have mapped out the favored model parameters, combining observations of stellar kinematics of spiral galaxies and galaxy clusters, the CMB, and Lyman-α forest, to narrow down the SIDM model parameter space. The predictions from this viable region of parameter space are the presence of dark radiation and the cut-off in the mass function of halos, both potentially observable in the future. FIG. 1: SIDM parameter space (2σ CL) favored by a wide range of astrophysical data from dwarf galaxies to galaxy clusters, for both asymmetric (gray) and symmetric DM (magenta). Inset: the DM self-scattering cross section vs. velocity for a best fit case and the data points with error bars are from [5]. FIG. 2 :FIG. 3 : 23SIDM kinetic decoupling temperature contours on the mχ vs ξ plane, T kd = 0.4-5 keV. For given mχ, m φ and gχ are fixed as their best fit values shown inFig. 1. The regions above the arrows are constrained by the Planck[42] and Lyman-α (corresponding to the 5.3 keV and 3.5 keV WDM limits[43]) observations, respectively. The horizontal line denotes the projected sensitivity of CMB Stage-IV experiments. Black dots denote the cases with their matter power spectra presented inFig. 3.CDM m Χ 175 GeV, Ξ 0.27 m Χ 91 GeV, Ξ 0.48 m Χ 0.15 GeV, The linear matter power spectra are similar at fixed T kd , here 1 keV. The models shown have parameters, mχ = 175 GeV (ξ = 0.27), mχ = 91 and 0.15 GeV (ξ = 0.48). FIG. 4 : 4Upper: Average deviation of one-dimensional SIDM power spectrum normalized to CDM vs. the kinetic decoupling temperature for ξ = 0.48. ��� �� [�/(� -� � ⊙ )] �� ���� /�� ��� FIG. 5: Normalized halo mass functions for ETHOS-1 (red), 2 (blue), and 3 (green), from the simulations [59] (solid) and our analytical scaling relation (dashed), dnSIDM/dnCDM = (1 + Mcut/M ) −1.34 . 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[]
[ "Cosmic censor of shock-wave singularities", "Cosmic censor of shock-wave singularities" ]
[ "Uwe R Fischer \nDepartment of Physics and Astronomy\nCenter for Theoretical Physics\nSeoul National University\n08826SeoulKorea\n", "Satadal Datta \nDepartment of Physics and Astronomy\nCenter for Theoretical Physics\nSeoul National University\n08826SeoulKorea\n" ]
[ "Department of Physics and Astronomy\nCenter for Theoretical Physics\nSeoul National University\n08826SeoulKorea", "Department of Physics and Astronomy\nCenter for Theoretical Physics\nSeoul National University\n08826SeoulKorea" ]
[]
A dispersionless shock wave in a fluid without friction develops a sonic spacetime singularity which is naked (not hidden by a horizon). We construct a fluid-dynamical analogue of the cosmic censorship hypothesis of Penrose, wherein this naked nondispersive shock-wave singularity is censored (prohibited) due to the microscopic structure of the underlying aether and the resulting effective trans-Planckian dispersion. We find that including quantum pressure in Bose-Einstein condensates provides such a censor: Approaching the instant of shock t shock , rapid spatial oscillations of density and velocity develop, which begin to emerge already slightly before t shock . These oscillations render the spacetime structure completely regular, and therefore lead to a removal (censoring) of the spacetime singularity.
null
[ "https://export.arxiv.org/pdf/2209.02763v2.pdf" ]
252,111,060
2209.02763
bba0e309c6c08d15ce066f0fad8cd23c9544aeb2
Cosmic censor of shock-wave singularities Uwe R Fischer Department of Physics and Astronomy Center for Theoretical Physics Seoul National University 08826SeoulKorea Satadal Datta Department of Physics and Astronomy Center for Theoretical Physics Seoul National University 08826SeoulKorea Cosmic censor of shock-wave singularities (Dated: November 28, 2022), A dispersionless shock wave in a fluid without friction develops a sonic spacetime singularity which is naked (not hidden by a horizon). We construct a fluid-dynamical analogue of the cosmic censorship hypothesis of Penrose, wherein this naked nondispersive shock-wave singularity is censored (prohibited) due to the microscopic structure of the underlying aether and the resulting effective trans-Planckian dispersion. We find that including quantum pressure in Bose-Einstein condensates provides such a censor: Approaching the instant of shock t shock , rapid spatial oscillations of density and velocity develop, which begin to emerge already slightly before t shock . These oscillations render the spacetime structure completely regular, and therefore lead to a removal (censoring) of the spacetime singularity. I. INTRODUCTION In Einsteinian gravity, singularities are ubiquitous [1][2][3]. However, the physical spacetime nature of these singularities is still under debate. The singularity theorems by Stephen Hawking and Roger Penrose state that if there either exists a trapped surface due to gravitational collapse or the Universe is assumed to be spatially closed, spacetime singularities are formed with the following conditions being satisfied: We have Einstein gravity at zero or negative cosmological constant, the weak energy condition is maintained, closed timelike curves are absent, and every timelike or null geodesic enters a region where the curvature is not specially alined with the geodesic [4][5][6][7]. As these theorems guarantee that if there exists a trapped surface in spacetime, a singularity must form, one may ask the question if the reverse holds true, and whether a singularity may form without a horizon enclosing it (naked singularity). The cosmic censorship hypothesis (CCH), then, in its weak form, states that generic gravitational collapse, starting from a nonsingular initial state, can not create a naked singularity in spacetime [2,8,9]. However, explicit counterexamples to the CCH, for physically viable processes, have been found cf., e.g., [10][11][12][13][14]. On the other hand, mechanisms arguing that naked singularities are indeed hidden were developed, among which backreaction is a prominent example [15][16][17][18]. It is thus fair to say that the CCH is still widely debated, as regards the possible mechanisms for either violating or preserving it, and whether these mechanisms are of quantum or classical origin, also cf. Ref. [19]. This is largely due to the fact that there is no applicable quantum theory of gravity, in particular complete in the ultraviolet, with which to ascertain whether a given argument for (or against) the CCH is true. The seminal paper of Unruh [20] triggered, especially recently, with a substantial improvement of experimental capabilities, on a broad front a field which was coined analogue gravity [21]. Its essence is that it models the propagation of classical and quantum fields on curved spacetime backgrounds, exploring various phenomena inaccessible at present in the realm of gravity proper, see, e.g., Refs. [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. A particularly promising arena are Bose-Einstein condensates (BECs) due to the atomic precision control and accurate correlation function resolution they offer . Acoustic black holes ("dumb" holes [62]) or cosmological horizons are thus well established and experimentally realized within the analogue gravity realm. On the other hand, distinct from Einstein gravity, where singularities are ubiquitous, singularities in quantum fluids, and with particular regard to their acoustic spacetime properties, have not been experimentally investigated yet, to the best of our knowledge. It is important here to pause, and to clearly state at the outset the most important differences of analogue gravity and Einstein gravity: In analogue gravity, the acoustic spacetime metric is governed by nonlinear fluid dynamics and not by a solution of the Einstein equations. In Einstein gravity, black holes (and, as a result, also singularities in spacetime due to the theorems by Hawking and Penrose) are formed from gravitational collapse of matter. In fluids, it is the transition of subsonic to supersonic flow which creates an effective dumb hole horizon for linear sound in the medium. Distinct from Einstein gravity, this analogue gravitational field, providing a background effective spacetime for linear perturbations on top of it, is governed by a velocity scalar [63], in a comparable way to a nonlinear self-interacting scalar field theory of gravity [64]. In the present work, we establish a highly nonlinear process creating a naked singularity in the acoustic spacetime metric, physically represented by a shock wave in a BEC without dispersion included (that is in the so-called Thomas-Fermi limit). For this nondispersive shock, the nonlinearity causes a stepwise discontinuity in the acoustic metric components, and as a result a naked timelike Ricci curvature singularity of the effective spacetime emerges. In the real BEC quantum fluid, dispersive effects can however not be neglected, due to the quantum pressure, which occurs because of the stiffness of the condensate order parameter (scalar field) against spatial variations of its modulus. We reveal a cosmic censor in action, due to the existence of quantum pressure, when a nondispersive shock-wave [65] would develop a singularity of the effective spacetime at its front. Due to the dynamical differences of Einstein and analogue gravity, here the singularity is censored (prohibited to form), instead of being dressed by a spacetime horizon. II. FLUID DYNAMICS OF DILUTE BOSE-EINSTEIN CONDENSATES A. Fluid perturbations Dilute BECs represent inviscid, barotropic, and irrotational fluids, where, importantly, the quantum pressure term is added to the Euler equation. Setting the atomic mass m = 1, we have to solve the following set [66]: ∂ t ρ + ∇ · (ρv) = 0,(1)∂ t v + v · ∇v = − ∇p ρ + 2 2 ∇ ∇ 2 √ ρ √ ρ − ∇V ext , (2) p = p(ρ) = 1 2 gρ 2 ,(3)∇ × v = 0 ⇒ v = ∇Φ.(4) These equations are the only field equations occurring in our problem for condensate density ρ(r, t) and condensate velocity v(r, t), and the spacetime metric for sound is then a derived and not fundamental (also see below). In the above relation (4), Φ is a velocity potential due to the irrotationality of the superfluid (excluding quantized vortex lines). The scalar potential V ext is employed by the cold quantum gas experimentalist to create certain classes of effective spacetimes (see for an overview [21]), while the condensate pressure p arises from the two-body repulsive contact interaction between atoms, where the coefficient g is proportional to the s-wave scattering length in the dilute gas [66]. Finally, the term 2 2 ∇ ∇ 2 √ ρ √ ρ in the Euler equation (2), is the so-called quantum pressure term [66]. From the barotropic equation of state (3), the sound speed c s = dp dρ = √ gρ; stability implies that g > 0. We linearize the fluid equations over the background of a dispersive shock wave in a BEC [67]. The quantum pressure term is negligible until the shock is closely approached. Due to the quantum pressure term, the discontinuity in the flow, which were expected to be present in the nondispersive post-shock phase [65], is regularized. One observes instead an oscillation pattern in the density profile upon approaching the shock (Fig. 5 in Appendix B). To physically distinguish classical sound wave from the background, one works with a linear perturbation with different space and time scale than the background flow, as discussed in the literature for linear sound propagation over background [21], and for nonlinear sound as well [63]. We denote background quantities with subscript (0) and the linear perturbations with subscript (1). We write v = v (0) + ∇Φ (1) by following the conventions of Ref. [63]. For example, with a dispersive nonlinear wave as the background, initially, when t is much less than the shock time t shock , the wave is linear and nondispersive. For t t shock , such a linear wave satisfies the massless Klein-Gordon (KG) field equation over the analogue Minkowski spacetime of a uniform static medium as background. We call this the initial background, and denote it with subscript 0. According to the Riemann wave equation for travelling one-dimensional (1D) waves, see Eq. (12) below, the intrinsic nonlinearity of the fluiddynamical equations becomes significant in the course of time as the wave approaches the shock [65]. The KG analogy then does not hold anymore. In Ref. [63], we have described the classical backreaction of the nonlinear perturbation onto the accoustic metric, and defined a new background by absorbing these nonlinear perturbations into it. Here, we go near and beyond the shock time, with now in addition the quantum pressure, which originates from the spatial stiffness of the macroscopic BEC wavefunction against deformations, becoming significant. Linearizing (1) gives ∂ρ (1) ∂t + ∇ · (ρ (0) ∇Φ (1) + ρ (1) v (0) ) = 0.(5) The linearized Euler equation follows from the Eq. (2): Φ (1) + c 2 s(0) ρ (0) ρ (1) + v (0) · ∇Φ (1) + 2 ρ (1) 4ρ 2 (0) ∇ 2 ρ (0) − 1 ρ (0) (∇ρ (0) ) 2 + 2 4ρ (0) 1 ρ (0) ∇ρ (0) · ∇ρ (1) − ∇ 2 ρ (1) = 0. (6) Incorporating only the gradient terms from the background, thus neglecting ∇ρ (1) , and ∇ 2 ρ (1) , we get ρ (1) (1 + 2 α)c 2 s(0) ρ (0) = −Φ (1) − v (0) · ∇Φ (1) .(7) Here, we introduced a parameter α via α = 1 4c 2 s(0) ∇ · ∇ρ (0) ρ (0) .(8) We can then define a new length scale l = l(x, t) via l −2 := 2 |α|/ξ 2 which characterizes the background spatial variation, and where the spatiotemporally local healing length is given by ξ(x, t) = ξ(ρ (0) ) = √ gρ (0) . The competition of the "microscopic" structure dictated by ξ and the "background" scale l is expressed by α(x, t) which thus appears in the metric q µν in Eq. (9) below. B. Spacetime metric in the dispersive fluid Now, we substitute ρ (1) from Eq. (7) into Eq. (5), dropping the terms in the last closed bracket of Eq. (6). This is the limit where the linear perturbation of all physical quantities such as ρ (1) , p (1) can be written in terms of partial derivatives in Φ (1) , and the full solution can be obtained when Φ (1) over a known background has been solved for. Going beyond this limit requires to solve for ρ (1) also, and the equation of motion for Φ (1) becomes an integro-differential equation [41]. As a result, the acoustic spacetime metric is not local in space and time anymore. Here, we restrict ourselves to small wave number k excitations, i.e, perturbations with wavelength larger than the coherence length ξ(ρ (0) ). In this limit, we can construct an acoustic metric local in spacetime. Linearizing in the perturbation amplitude now proceeds still as conventionally carried out in the analogue gravity literature [20,21]. The difference is found in the dispersive nature of the background. The latter is controlled by well-posed initial (and/or boundary) conditions by the experimentalist. Over such an externally fixed, albeit nonlinear and dispersive background, any excitation to linear order is called a perturbation. In our particular case, the highly nonlinear and dispersive background flow is clearly distinct from the linear nondispersive perturbations which experience the effective spacetime produced from such a background medium. We then compare the equation of the scalar field Φ (1) to that of a minimally coupled massless KG field equation, and find the following effective spacetime metric in 3+1D, q µν := ρ (0) c (0)     −(c 2 (0) − v 2 (0) ) . . . −v T (0) · · · · · · · · · · · · −v (0) . . . I 3×3     ,(9) with a modified local sound speed c (0) = c s(0) 1 + 2 α(10) due to the dispersive nature of the background. Evidently, the 2 small length scale correction term is present for a general background flow. Note that for stability, we have to impose the lower bound α > −1/ 2 . The q µν are no longer simple algebraic functions of background density and velocity, and interpolate between the fully nonlinear metric without dispersion g µν introduced in [63] and the linear perturbations metric without dispersion g µν . See Table I for an overview of the various concepts and the classification of spacetime metrics in the presence of nonlinearity and/or dispersion due to quantum pressure. We note that the effective spacetime metric for linear perturbations of wavelength larger than the healing length, q µν , does not represent a so-called rainbow spacetime [68,69]. Distinct from such a rainbow spacetime, the metric q µν does not depend on the wave vector k of the excitations. III. DISPERSIVE SHOCK WAVES We consider the propagation of a wave, initially created as a Gaussian distribution, in the condensate. We consider a realistic situation, with the effect of quantum pressure included, i.e., a highly nonlinear dispersive wave [67]. The acoustic metric of such nonlinear dispersive pulse wave in our quasi-1D BEC set up, is given by the Eq. (9) with v (0) having only one component along x axis, v (0) (x, t). We choose the initial wave profile [67] as the Gaussian ρ (0) (x, t = 0) = ρ ∞ 1 + 2η exp − x 2 2σ 2 , v (0) (x, t = 0) = 0,(11) where σ(ρ) ξ(ρ). Here, at the center of our quasi-1D BEC set up, we produce a source of gravitational wave (GW) with density being almost uniform towards the boundary, mimicking asymptotically flat effective spacetime with a GW source. This longitudinal GW is different from its counterpart in Einstein gravity, in that the spacetime lacks general covariance, and the GW can not be represented in its usual transverse and traceless form, cf. the discussion in [61]. The Thomas-Fermi profile in Eq. (11) (neglecting the impact of quantum pressure on the initial state) can be created by focusing a laser detuned from atomic resonance onto the center of the one dimensional condensate, with a size σ [67]. Switching off the laser creates a nonlinear dispersive propagating wave with high frequency oscillations when the shock occurs, as previously described in [67], see for a detailed description Appendix B. Shock waves in quasi-1D BECs have been experimentally observed [70], and also in nonlinear photon fluids (8). For Background (i), ρ0, v0 represent a solution of the nondispersive fluid equations without quantum pressure, and are initially chosen as the background before the shock develops, with perturbations treated to linear order. This initial Background (i) corresponds to the conventional analogue gravity metric and may or may not derive from nonlinear fluid equations; for example a uniform static medium does not represent a nonlinear background. For the Background (ii), ρ (0) , v (0) , are found from the fully nonlinear, coupled fluid equations for both background and perturbations, however without quantum pressure included. cf. Ref. [63]. Finally, for the Background (iii), ρ (0) , v (0) are found from the nonlinear fluid equations applied to the background motion alone, but now with quantum pressure included. . We use here units in terms of a length L, chosen appropriately for the purpose of our numerical calculation. For example, in the experiment [70] the size of the condensate is roughly 500 µm, whereas the full x-axis range of our numerical simulation is 500. Therefore L would be approximately 1 µm with the parameters of Ref. [70]. Time t is then measured in units of L 2 , when setting = 1. In these units, the parameters we choose are g = 7500, ρ∞ = 0.002, η = 0.2, σ = 8.838. We then obtain t shock 13.43, using the method described in Ref. [67]. Only after and slightly before the instant of shock, the dispersive nature of the background flow becomes important in the oscillatory region, and hence in qµν . Top left (at t = 27): We plot qtt in the post-shock phase; the behavior of other metric components is similar. (Bottom row): In the post-shock phase, the amplitude of the parameter α defined in Eq. (8) in the nonoscillatory region is essentially negligible compared to its amplitude in the oscillatory region; it however increases rapidly as t approaches t shock . Background (i) Background (ii) Background (iii) ρ0, v0 ρ (0) , v (0) ρ (0) , v (0) l ξ l ξ l ∼ ξ gµν gµν qµν [71]. In particular, Ref. [70] captures density modulations which may be compared to the high-frequency post-shock oscillations predicted by Damski [67]. We numerically solve the fluid equations (in a box potential with ∇V ext = 0), that is Eqs. (1) to (4), employing a 4th order Runge-Kutta method to perform the time integration, and expanding the spatial derivatives within a central difference method scheme up to the same 4th order accuracy [72]. We now consider only nonlinearity taken into account for the fluid motion, i.e., Eq. (1)-(4) without quantum pressure, and with the initial profile of Eq. (11). After a certain time, the initial Gaussian density wave profile separates completely into two identical smaller pieces (while respecting mass conservation), and moving in opposite directions. The right-moving travelling wave in the polytropic gas with pressure p ∝ ρ γ (for BECs γ = 2) can be described in terms of single variable v (0) (x, t) by the Riemann wave equation [73]: ∂v (0) ∂t + c s0 + γ + 1 2 v (0) ∂v (0) ∂x = 0,(12)ρ (0) = ρ 0 1 + γ − 1 2 v (0) c so 2 γ−1 .(13) The second identity directly relating density to flow speed perturbations is valid for a simple wave [65]. The leftmoving travelling wave comes with a '−' sign in front of c s0 in the above equations; ρ (0) = ρ ∞ for v = 0, ρ ∞ ρ 0 of Eq. (11) since σ l. This first-order quasi linear partial differential equation leads to multivalued valued solution by the method of characteristics [74]. By obeying momentum and mass conservation across the discontinuity, one is led to the equal area rule (x − x s )dv = 0 where x s is the shock location (location of discontinuity), to avoid such a multivalued solution from the shock time (= t shock ) onward [65]. We discuss this issue further in Appendix A. In the presence of quantum pressure, the solution (density, velocity etc) becomes oscillatory around the discontinuity, in comparison in the Fig. 2. Therefore, the solution becomes a well behaved function of x and t [67], see Appendix B. Numerical solution of Eq. (1) together with Eq. (4) produces q tt in Fig. 1. As expected, α is practically zero in the nonoscillatory region. The α-correction term in the metric q µν , which is usually hidden in a slowly varying background, is amplified in a region where quantum pressure is important: It is a significant contribution relative to the other forces in the Euler-type evolution of momentum (2) in the oscillatory region (cf. Fig. 7 in the Appendix B). Remarkably, the oscillations in the solution starts just slightly before the shock time t shock (see Fig. 6 Appendix B), whereas t shock is computed in the zero quantum pressure limit. Therefore, t shock maintains its importance as a time scale even with quantum pressure, signifying the time of initiation of oscillation. A linear travelling 1D wave can not stay linear forever, after a certain time nonlinearity makes the v profile steeper, with negative ∂v ∂x . This renders, in turn, the quantum pressure significant. Thus nonlinearity invites dispersion due to quantum pressure to play a significant role, also see the Appendices A and B. IV. CENSORING THE NAKED SINGULARITY We now aim to find what a discontinuity in the solution means for the effective spacetime. We denote the acoustic metric for the nondispersive metric as g µν , cf. Table I. We stress that, while the metric is derived nondispersively, it is still taking the nonlinearity of the fluid into account [63]. It reads [75] ds 2 = g µν dx µ dx ν = ρ (0) c s(0) −(c 2 s(0) − v 2 (0) )dt 2 −2v (0) dtdx + i=1,2,3 (dx i ) 2   . (14) In the above metric, ρ (0) , v (0) are found from the solution of the fluid equations without quantum pressure. This is Background (ii) in Table I. For nonlinear dispersive shock wave, Background (ii) and Background (iii) coincide very well in every region except in the oscillatory region, i.e., the region around shock location x s . In the asymptotic region, i.e., near the condensate wall, Background (ii) and Background (iii) coincide with the Background (i) which is uniform and static, i.e., an acoustic analogue of Minkowski spacetime. Evidently the acoustic metric is discontinuous at x = x s after the shock has occurred. We compute the Ricci scalar, R [76] for g µν for the right moving travelling wave satisfying Eq. (12). We perform the calculations in Mathematica, replacing ∂ t by ∂ x derivatives, employing the Riemann wave equation (12). This procedure leads to the surprisingly simple relation R = (1 + γ) ρ (0) ∂ 2 v (0) (t, x) ∂x 2 .(15) expressing the curvature scalar solely by the second spatial derivative of the background flow field. At x = x s , v (0) = v 1 and ρ (0) = ρ 1 which are the pre-shock values of velocity and density respectively, related to each other by the Eq. (13). Since in this case, the wave is propagating from left to right, at x = x s v (0) first has v 1 then it jumps to post-shock value v 2 (< v 1 ), thus unrealistic multivalued v (0) is avoided. At x = x s , x limit of v (0) doesn't exist, but it has a definite value which is v 1 , and as a consequence; this discontinuity can be written mathematically in terms of a Heaviside step function, see Appendix A. ∂v (0) (t,x) ∂x = −∞ at x = x s , and ∂ 2 v (0) (t,x) ∂x 2 at x = x s can be expressed as a summation of δ(0) and δ (0) (with definite coefficients) type of infinities (in Appendix A); where denotes a x derivative. We discuss the visualization of Dirac delta distributions through a delta-sequence function in Fig. 4 of Appendix A. We plot in Fig. 3 the Ricci scalar of the nondispersive wave as it approaches the curvature singularity in the pre-shock phase t < t shock . As can be seen, the expression (15) implies the existence of a (strong) curvature singularity at x = x s , where x s is the position of discontinuity at t ≥ t shock . Since the velocity at any x remains always very much less than the minimum value of sound speed c s0 (= gρ 0 ), there is no event horizon present in the acoustic metric. Since at x = x s , v (0) = v 1 ; sound speed c s(0) = c s1 = c s0 + γ+1 2 v 1 , and the travel speed of the discontinuity is u = c s0 + γ+1 4 (v 1 +v 2 ), v 2 (< v 1 ) is the post-shock value of v (0) [65]. Hence c s1 > u. In Eq. (14), by putting dx = udt, dy = dz = 0, we find ds 2 = ρ1 cs1 −c 2 s1 + (u − v 1 ) 2 dt 2 , from the above discussion, we notice that c s1 > |u − v 1 |. Therefore, at x = x s , the discontinuity follows a timelike trajectory, representing a naked singularity. When we, on the other hand, solve the fluid equations with quantum pressure, the solution oscillates instead of discontinuity, we render the curvature for the metric q µν ∀ x and t finite, thus removing the singularity, cf. Fig. 2. However, for nondispersive waves, the discontinuity does not persist for t → ∞, and (v 1 − v 2 ) then falls to zero [65]. V. CONCLUSION We demonstrated that the quantum pressure term leads to a regular oscillatory numerical solution for travelling waves in a quasi-1D BEC, thus prohibiting the otherwise naked singularity. Analogue gravity is effectively an aether theory, for which we have shown, using a particular initial condition, that the occurrence of a naked singularity is forbidden. Whether singularities in the dispersive aether of the BEC arise for any given nonsingular initial condition is an open question. We have thus provided, for a BEC laboratory analogue simulating curved spacetimes, a censor prototype operating in the trans-Planckian sector of the dispersion relation, which is based on microscopic quantum many-body physics of the system, and is thus naturally complete in the ultraviolet. To ultimately resolve the question of whether the CCH holds true, this latter property is crucial also for any proper quantum gravity. In a cosmological context, the Trans-Planckian Censorship Conjecture (TCC) can be viewed as a momentum space version of the CCH, replacing the timelike singularity by the set of trans-Planckian wavelengths and the event horizon by the Hubble horizon [77], to avoid (any variant of) the so-called trans-Planckian problem [78]. An observer with access to infrared (sub-Hubble) modes) must, for the TCC to hold, be shielded by the Hubble horizon from the trans-Planckian modes. Extending the line of argument presented, we expect that a cold atom analogue of the TCC can also be established, cf. [47]. In this Appendix, first we briefly introduce the equal area principle introduced in [65] for nondispersive shock waves, and then we proceed to calculating the Ricci scalar curvature for such a nondispersive shock wave. The Riemann wave Eq. (12) can be solved by the analytical techniques for partial differential equations, i.e., the method of characteristics. This analytical solution [61] gives rise to multivalued solution after a certain time, t shock . At t = t shock , ∂v ∂x reaches infinity [65]. If we follow the method of characteristics [63,65] to solve Eq. (12) for the case without quantum pressure to avoid multivalued solution of density and velocity after t shock , the solution has to become discontinuous. This jump in velocity (and density) approximately satisfies the equal area rule [65]: v2 v1 (x − x s )dv = 0,(A1) where v 1 and v 2 (v 1 > v 2 ) are the pre-shock and postshock values of discontinuous velocity v (0) across the position of discontinuity (shock) at x = x s . As a result, ρ 1 and ρ 2 are pre-shock and post-shock values of density ρ (0) related to v 1 and v 2 by ρ 1,2 = ρ o 1 + γ − 1 2 v 1,2 c so 2 γ−1 . (A2) With this discontinuity, velocity and density profiles are not multivalued anymore, which is discussed in detail by the classic textbook [65] . The expression of Ricci scalar (Eq. (15)) in the nondispersive limit is proportional to the second derivative in v (0) , here we discuss an analytical way to calculate the second derivative of v (0) with a discontinuity at x = x s . This discontinuous velocity profile v(x, t) at fixed time t > t shock can written in a compact approximate way, v (0) (x, t) = (1 − Θ(x − x s )) f 1 (x) + Θ(x − x s )f 2 (x),(A3) where Θ is the Heaviside step function, defined by Θ(x − x s ) = 1 for x > x s and Θ(x − x s ) = 0 for x ≤ x s . Furthermore, f 1 (x), f 2 (x) are Newton interpolation polynomials [72], constructed from a finite number of points on the pre-shock curve segment and on the post-shock curve segment of v(x, t) respectively, at a fixed time t > t shock , e.g., from the left subfigure of the Fig. 2. Thus we approximately describe v (0) (x, t) at fixed t > t shock by these two polynomials with finite coefficients in a compact way. Therefore, f 1 (x) and f 2 (x), for a reasonably accurate fitting, should satisfy (a) f 1 (x s ) ∼ v 1 > f 2 (x s ) ∼ v 2 , and (b) the slopes of f 1 (x) and f 2 (x), at x = x s smoothly fits into the pre-shock curve segment and post-shock curve segment, respectively. We find ∂v (0) ∂x = (1 − Θ(x − x s )) df 1 dx + Θ(x − x s ) df 2 dx + δ(x − x s ) (f 2 (x) − f 1 (x)) , (A4) where δ(x − x s ) is the Dirac delta distribution. The first two finite terms of the equation has a similar pattern to the Eq. (A3) for obvious reasons. Therefore, ∂v (0) ∂x | x=xs = δ(0)(v 2 − v 1 ) + df 1 dx | x=xs .(A5) Evidently, the first term on the right hand side dominates over the second term, rendering ∂v ∂x | x=xs to be −∞, since v 2 < v 1 . ∂ 2 v (0) ∂x 2 = (1 − Θ(x − x s )) d 2 f 1 dx 2 + Θ(x − x s ) d 2 f 2 dx 2 + 2δ(x − x s ) df 2 dx − df 1 dx + δ (x − x s ) (f 2 (x) − f 1 (x)) (A6) Therefore, at x = x s , ignoring the finite term d 2 f1 dx 2 | x=xs , we write down the infinite terms as follows, ∂ 2 v (0) ∂x 2 | x=xs = 2δ(0) df 2 dx | x=xs − df 1 dx | x=xs + δ (0) (v 2 − v 1 ) .(A7) According to Fig. 2, df2 dx | x=xs is negative; it always stays negative in the post-shock phase, and df1 dx | x=xs is positive. Numerics in fact shows that, initially after t shock , df1 dx | x=xs is negative, but eventually it becomes positive over time. The quantity ∂ 2 v (0) ∂x 2 | x=xs above consists of two tive (right). lim n→∞ ∞ −∞ f (x)δn(x)dx = ∞ −∞ f (x)δ(x)dx, and lim n→∞ ∞ −∞ f (x)δ n (x)dx = ∞ −∞ f (x)δ (x)dx (for arbitrary f (x)) are used as defining relation between Dirac delta distribution and its derivative with their sequence functions respectively [79] [80]. different kinds of infinity. One can represent them by δsequence functions [79]. We choose here a particular one to describe these infinities (see also Fig. 4), δ n (x) = n √ π exp ( −n 2 x 2 ) (A8) δ n (x) = − 2n 3 x √ π exp ( −n 2 x 2 ),(A9) where n is a positive integer. Using relation xδ ( x) = −δ(x), x 2 δ (x) = −xδ(x) = 0 [80], we observe from Eq. (A6), (x − x s ) ∂ 2 v (0) ∂x 2 = (x − x s ) (1 − Θ(x − x s )) d 2 f 1 dx 2 + (x − x s )Θ(x − x s ) d 2 f 2 dx 2 − δ(x − x s ) (f 2 (x) − f 1 (x)) (A10) And it follows that (x − x s ) n ∂ 2 v (0) ∂x 2 | x=xs = −δ n,1 δ(0) (v 2 − v 1 ) . (A11) This is how "strange" the second derivative ∂ 2 v (0) ∂x 2 | x=xs in fact behaves. Appendix B: Initiation of oscillations in dispersive shock waves In this Appendix, we collect our numerical findings on dispersive shock waves with initial conditions (11), as described in the main text. Some of these results have been presented already in Ref. [67], but for the convenience of the reader we reproduce here these results together with a few additional observations, where our overall aim is to inspect closely the initiation of the oscillation of the dispersive shock waves, which is due to the quantum pressure term. Specifically, in Fig. 5, we observe how the oscillation region is slowly spreading with progressing time. In Fig. 6, we display how the shock wave enters the oscillation phase, just prior to the shock time t shock . Finally, in Fig. 7, we display in some detail the onset of oscillations due to the quantum pressure becoming significant. (2) at t = 27. We see that the quantum pressure term becomes significant only in the oscillatory region. (Bottom) The quantum pressure is very much smaller, by several orders of magnitudes, in the nonoscillatory region. The parameters are identical to those in Fig. 1. FIG. 1 . 1Behavior of qµν for the dispersive shock wave as background, found by numerically solving Eqs. (1)-(4) FIG. 2 . 2(Left) Multivalued solution of postshock Riemann wave by the method of characteristics (dotted), physical solution with discontinuity (nondispersive shock) by equal area rule[65] (solid line). (Right) Discontinuity in the flow is avoided as the wave approaches the shock time when we take quantum pressure into account. Inclusion of quantum pressure in the equation creates oscillation and thus continuous solution of v removes singularity in the acoustic metric, i.e., censorship of singularity. The wave profile with quantum pressure is a good match with the nondispersive nonlinear wave profile in the region except the rapidly oscillatory region. Parameters as inFig. 1. FIG. 3 . 3The Ricci scalar of gµν approaching the shock singularity of the nondispersive Riemann wave. Parameters are identical to those ofFig. 1. ACKNOWLEDGMENTS We thank B.Damski and F. Marino for helpful discussions on dispersive shock waves. This work has been supported by the National Research Foundation of Korea under Grants No. 2017R1A2A2A05001422 and No. 2020R1A2C2008103. Appendix A: Nondispersive shock waves and the curvature singularity FIG. 4 . 4Sequences up to n = 4 are shown to represent the delta distribution function (left) and its first deriva- FIG. 5 .FIG. 6 .FIG. 7 . 567(Top) Evolution of density profile with time. At t = 0, the laser at the center of the condensate is switched off. 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[ "The pathfinder Dragonfly Spectral Line Mapper: Pushing the limits for ultra-low surface brightness spectroscopy", "The pathfinder Dragonfly Spectral Line Mapper: Pushing the limits for ultra-low surface brightness spectroscopy" ]
[ "Deborah M Lokhorst \nDepartment of Astronomy & Astrophysics\nNRC Herzberg Astronomy & Astrophysics Research Centre\n5071 West Saanich RoadV9E2E7VictoriaBCCanada\n\nUniversity of Toronto\n50 St. George StreetM5S3H4TorontoONCanada\n", "Seery Chen \nDunlap Institute\nUniversity of Toronto\n50 St. George StreetM5S3H4TorontoONCanada\n", "Imad Pasha \nDepartment of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA\n", "Jeff Shen \nDunlap Institute\nUniversity of Toronto\n50 St. George StreetM5S3H4TorontoONCanada\n", "Evgeni I Malakhov \nNew Mexico Skies, Inc\n\n", "Roberto G Abraham \nDunlap Institute\nUniversity of Toronto\n50 St. George StreetM5S3H4TorontoONCanada\n", "Pieter Van Dokkum \nDepartment of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA\n", "David A Dunlap ", "\nContentment Crest, Mayhill\n88339NMUSA\n" ]
[ "Department of Astronomy & Astrophysics\nNRC Herzberg Astronomy & Astrophysics Research Centre\n5071 West Saanich RoadV9E2E7VictoriaBCCanada", "University of Toronto\n50 St. George StreetM5S3H4TorontoONCanada", "Dunlap Institute\nUniversity of Toronto\n50 St. George StreetM5S3H4TorontoONCanada", "Department of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA", "Dunlap Institute\nUniversity of Toronto\n50 St. George StreetM5S3H4TorontoONCanada", "New Mexico Skies, Inc\n", "Dunlap Institute\nUniversity of Toronto\n50 St. George StreetM5S3H4TorontoONCanada", "Department of Astronomy\nYale University\n52 Hillhouse Ave06511New HavenCTUSA", "Contentment Crest, Mayhill\n88339NMUSA" ]
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The pathfinder Dragonfly Spectral Line Mapper is a distributed aperture telescope based off of the Dragonfly Telephoto Array with additional instrumentation (the Dragonfly "Filter-Tilter") to enable ultranarrow bandpass imaging. The pathfinder is composed of three redundant optical tube assemblies (OTAs) which are mounted together to form a single field of view imaging telescope (where the effective aperture diameter increases as the square-root of the number of OTAs). The pathfinder has been on sky from March 2020 to October 2021 equipped with narrowband filters to provide proof-of-concept imaging, surface brightness limit measurements, on sky testing, and observing software development. Here we describe the pathfinder telescope and the sensitivity limits reached along with observing methods. We outline the current limiting factors for reaching ultra-low surface brightnesses and present a comprehensive comparison of instrument sensitivities to low surface brightness line emission and other methods of observing the ultra-faint line emission from diffuse gas. Finally, we touch on plans for the upcoming 120-OTA Dragonfly Spectral Line Mapper, which is currently under construction.
10.1117/12.2629396
[ "https://export.arxiv.org/pdf/2209.07487v1.pdf" ]
251,876,905
2209.07487
93c8b792d73afd2fe4a1b1e742da975b200f69e3
The pathfinder Dragonfly Spectral Line Mapper: Pushing the limits for ultra-low surface brightness spectroscopy Deborah M Lokhorst Department of Astronomy & Astrophysics NRC Herzberg Astronomy & Astrophysics Research Centre 5071 West Saanich RoadV9E2E7VictoriaBCCanada University of Toronto 50 St. George StreetM5S3H4TorontoONCanada Seery Chen Dunlap Institute University of Toronto 50 St. George StreetM5S3H4TorontoONCanada Imad Pasha Department of Astronomy Yale University 52 Hillhouse Ave06511New HavenCTUSA Jeff Shen Dunlap Institute University of Toronto 50 St. George StreetM5S3H4TorontoONCanada Evgeni I Malakhov New Mexico Skies, Inc Roberto G Abraham Dunlap Institute University of Toronto 50 St. George StreetM5S3H4TorontoONCanada Pieter Van Dokkum Department of Astronomy Yale University 52 Hillhouse Ave06511New HavenCTUSA David A Dunlap Contentment Crest, Mayhill 88339NMUSA The pathfinder Dragonfly Spectral Line Mapper: Pushing the limits for ultra-low surface brightness spectroscopy low surface brightnessnarrowband imagingwide-field imagingcircumgalactic mediumground- based telescopes The pathfinder Dragonfly Spectral Line Mapper is a distributed aperture telescope based off of the Dragonfly Telephoto Array with additional instrumentation (the Dragonfly "Filter-Tilter") to enable ultranarrow bandpass imaging. The pathfinder is composed of three redundant optical tube assemblies (OTAs) which are mounted together to form a single field of view imaging telescope (where the effective aperture diameter increases as the square-root of the number of OTAs). The pathfinder has been on sky from March 2020 to October 2021 equipped with narrowband filters to provide proof-of-concept imaging, surface brightness limit measurements, on sky testing, and observing software development. Here we describe the pathfinder telescope and the sensitivity limits reached along with observing methods. We outline the current limiting factors for reaching ultra-low surface brightnesses and present a comprehensive comparison of instrument sensitivities to low surface brightness line emission and other methods of observing the ultra-faint line emission from diffuse gas. Finally, we touch on plans for the upcoming 120-OTA Dragonfly Spectral Line Mapper, which is currently under construction. INTRODUCTION Directly imaging low surface brightness structures presents a unique challenge as the features of interest are potentially millions of times fainter than the brightest objects in a typical image. This huge dynamic range coupled with systematic errors that are typically negligible or correctable (e.g., scattering in telescope optics, detector amplification) greatly impacts our ability to detect low surface brightness features. The archetype of low surface brightness phenomena is the largest and faintest structure in the universe: the 'cosmic web'. Cosmological simulations predict that on large scales dark matter collapses to form the cosmic web, taking on a foamlike structure that permeates the universe with galaxies forming at the nodes of this web. Gas flows along the filaments of the cosmic web (the intergalactic medium) into giant reservoirs of gas surrounding galaxies (the circumgalactic medium), then transitions into galaxies to fuel star formation. The gas in the cosmic web is nearly invisible, emitting mainly through fluorescence as the gas slowly cools. The prospect of directly detecting the ultra-low surface brightness line emission from cosmic web has spurred the development of technological upgrades in astronomical instrumentation. In particular, there have been several recent and ongoing projects to build instruments that aim to directly observe line emission from the circumgalactic and intergalactic medium, such as the Cosmic Web Imager (CWI) at Palomar Observatory 1 and the Keck Cosmic Web Imager (KCWI) at the W. M. Keck Observatory. 2,3 Imaging the circumgalactic and intergalactic media remains very difficult due to the faintness of emission from the diffuse gas, but there has been recent success at high redshifts where rest-frame ultraviolet (UV) emission lines are targeted by integral field spectrometers such as CWI and KCWI, as well as the Multi-Unit Spectroscopic Explorer (MUSE) on the Very Large Telescope (VLT). 4 In the past decade, integral field units with spectroscopic capability have discovered an abundance of extragalactic gaseous structures, including gaseous bridges between galaxies and quasars, 5, 6 enormous hundreds-of-kpc-sized gas clouds, 7,8 and extensive gaseous haloes detected in a variety of emission lines including Lyα, [Oii], and Mg ii. [9][10][11][12] These techniques work well for medium to high redshift observations, where bright UV lines such as the Lyα λ1216 and Mg ii λ2796, 2803 emission lines are redshifted into the visible wavelength regime and where the 1 arcmin 2 fields of view of these instruments can cover an appreciable area around the targeted galaxies. In the local universe, though, the bright UV emission lines are not accessible from the ground and while visible wavelength emission lines such as Hα and [Oiii] are accessible, they are an order of magnitude fainter, requiring sensitivity to surface brightnesses down to ∼ 10 −20 erg cm −2 s −1 arcsec −2 to detect. 13 As we have shown in Ref. 13, and discuss in detail below, it should be possible for an upgraded Dragonfly Telephoto Array 14 (Dragonfly) with ultranarrowband imaging capability to reach this low surface brightness limit and directly detect visible wavelength line emission from gas in the circumgalactic medium. 13,15 Dragonfly is a telescope specially designed for low surface brightness imaging. Its design is based on the innovative concept of building an extremely fast refracting telescope using lenses instead of mirrors to reduce scattering of light in the optics and multiplexing together an array of high-end commercial lenses to synthesize a larger effective aperture. Over the past few years, Dragonfly has made a series of groundbreaking discoveries, helping to rekindle the longneglected study of the low surface brightness universe. Dragonfly's success in detecting low surface brightness stellar structures prompted the question of whether Dragonfly, with its unprecedented sensitivity to the diffuse stellar continuum, could be modified to detect the extremely faint line emission from the circumgalactic and intergalactic media. The pathfinder Dragonfly Spectral Line Mapper was built to test this concept. Its basic design and components were modeled off of the Dragonfly Telephoto Array with the addition of instrumentation to incorporate ultranarrowband imaging capability. The pathfinder Dragonfly Spectral Line Mapper uses the Dragonfly Filter-Tilter instrumentation described in Ref. 15 to implement ultranarrowband filters on the telescope at the front of the optics. This is important for two main reasons: 1) to prevent degradation of the filter transmission profile that inevitably occurs when interference filters are placed in converging beams and 2) to incorporate a rotational mechanism to tilt the filter, which smoothly shifts the bandpass in wavelength space. Rotating an ultranarrowband filter by ∼ 20 • enables it to target the same cosmological volume as a typical narrowband filter of bandwidth ∼ 10 nm. The pathfinder Dragonfly Spectral Line Mapper was used to carry out an imaging survey on the M81 group of galaxies utilizing ultranarrowband filters selected to target Hα and [Nii] emission. These observations functioned as both a proof-of-concept and a scientific survey of the group. The Hα and [Nii] emission from the group was imaged separately with the same filters by rotating the ultranarrowband filters to two different angles with respect to the optical axis. The resulting data revealed many new gaseous features in the group, including a giant cloud of gas in the outskirts of the M82 galaxy. 16 Here we describe the pathfinder Dragonfly Spectral Line Mapper and the observing methods used to carry out the imaging survey. This includes several upgrades and additions to the observing methods and software used by the Dragonfly Telephoto Array, such as calibration of the Dragonfly Filter-Tilter and additional hardware for flatfield imaging. Finally, we compare the sensitivity of the Dragonfly Spectral Line Mapper with other observatories and instruments that are designed to detect similar features, and extrapolate on the final sensitivities expected to be reached by the full Dragonfly Spectral Line Mapper, which is under construction. The Dragonfly Spectral Line Mapper will have 120 lenses, 40× the collected area of the pathfinder, including a specialized suite of filters to enable sub-percent calibration of the science frames. This paper is one of three in a series in these proceedings on imaging the low surface brightness universe with distributed aperture telescopes and the Dragonfly Telephoto Array. Table 1 summarizes the topics covered by each paper. THE DRAGONFLY SPECTRAL LINE MAPPER PATHFINDER The pathfinder Dragonfly Spectral Line Mapper is a 3-lens version of the Dragonfly Telephoto Array with Dragonfly Filter-Tilter instrumentation that implements ultranarrow-bandpass imaging capability on the telescope (as described in Ref. 15). The pathfinder Dragonfly Spectral Line Mapper was built at New Mexico Skies Observatories in Mayhill, New Mexico, in the spring of 2019. Commissioning and science data were collected from 2019 through 2021. The pathfinder is composed of three separate and identical optical tube assemblies (OTAs; pictured in the left panel of Fig. 1). A schematic of the pathfinder is shown in the right panel of Fig. 1 Table 3 Mount Paramount ME II 15 for further details on the theory and instrument design of the Dragonfly Filter-Tilter. A custom designed baffle was built for each OTA to block stray light contamination from entering the optical path. The pathfinder instrument also includs electroluminescent flat field panels (commercially available Aniltak Flip-Flats) that were mounted at the front of each OTA to collect flat field images. The panels were used to collect flat field images during the night in between science exposures at the same pointing and filter rotation as used for the science frame collection. The elements of the pathfinder Dragonfly Spectral Line Mapper are summarized in Table 2 for quick reference. Observing Software The pathfinder Dragonfly Spectral Line Mapper used the observing software package written for the Dragonfly Telephoto Array 14 with several additions to integrate the required ultranarrowband instrumentation. The Dragonfly observing software is designed to allow completely autonomous observing, through all manner of weather conditions. The observing software is adaptive; upon starting observations for the night, it checks before each science exposure whether it should continue or not (e.g., it will stop taking science images if the weather has reported cloudy or dome is closed). Depending on dome and sky conditions, it carries out the optimal task, such as a science exposure, dark frame exposure (due to cloudy weather conditions), or flat-field exposure (due to the rising of the sun). Dragonfly uses an Internet-of-Things framework to control each OTA of the telescope. In this framework, API commands are sent from a control PC to the OTAs, each of which has its own Intel Compute Stick. The observing commands are sent over web servers that run on each Intel Stick. Each Intel Stick is connected through USB or serial ports to the components of the OTA, allowing it to control the camera, focuser, Filter-Tilter and Flip-Flat for that specific OTA. The connections between the computers and devices are shown diagrammatically in the bottom-middle panel of Fig. 1. The Dragonfly observing software worked well for controlling the pathfinder Dragonfly Spectral Line Mapper, but it was not a direct drop-in replacement. In particular, the Dragonfly Filter-Tilter control and flat-frame collection needed to be developed and integrated with the observing software. In addition, with the change from broadband to narrowband filters, several smaller changes, such as the adjustment of exposure times for focusing frames, were necessary. We detail these modifications and additions for the interested reader here. 1. Filter-Tilter Control Software. In order to carry out observations on the pathfinder Dragonfly Spectral Line Mapper, the observing software needed to be updated to also include an API to control the Filter-Tilter rotation. The software to control the Filter-Tilters was integrated into the observing software to allow the filters to be automatically tilted to the correct rotation before each integration. This required adding RESTful API calls to the Dragonfly observing software. Tilting the filters now requires just a single extra parameter when starting the observations and is fully automated during the night without needing any intervention from the observer. Additional software for carrying out automated Filter-Tilter calibration was also developed and is described in Section 2.2. 2. Flat field frame collection. The observing software was updated to include gathering flat field frames using illumination panels ("Flip-Flats") which are commercially available. Each lens on the pathfinder has its own Flip-Flat. During the night, flat frames at each observing tilt of the filter can be taken to allow an accurate calibration in the case of any flexure in the equipment. Observing software was added to carry out the flat frame collection automatically after each set of science exposures. 3. Lens focusing. The script for focusing the lenses was modified to allow the exposure time for the focusing images to be adjusted. Longer exposure times were necessary because of the significantly narrower bandwidth of the pathfinder Dragonfly Spectral Line Mapper filters compared to the Dragonfly broadband filters. Filter-Tilter Calibration Method The operation of the pathfinder Dragonfly Spectral Line Mapper relies on the ability to set the desired central wavelength of the filter by tilting the filters. In order to tilt to a specific wavelength on command, a calibration of the zeropoint of the filter tilt is required. In this Section, we describe the current calibration method. The shift in the central wavelength of a filter is a smoothly varying monotonic function of the tilt of the filter and the angle of incidence of the light across the field-of-view can be easily modelled. This means that the calibration of the tilt of the filter is a straightforward process using a target radiating line emission (or at least quasi-monochromatic emission) at a wavelength within the range swept by the filter bandpass during tilting (i.e., within the 653.9 nm to 659.9 nm range scanned by the pathfinder Dragonfly Spectral Line Mapper filters during tilting, which have a central wavelength of 659.9 nm). We used two different light sources for this calibration: 1) a laser with a central wavelength of 657.4 nm and bandwidth 0.61 nm (measured at room temperature) and 2) line emission from a planetary nebula. The first method of calibration consists of taking images of the laser shining on a target at a distance of about 20 m from the telescope. The chosen laser is stable over periods of about 30 minutes, which is sufficient for laser calibration, which takes 5 to 10 minutes. However, the wavelength of the inexpensive laser used has a strong dependence on the temperature and, as the laser is mounted inside the telescope dome in the open air, there is a temperature difference of ∆T ≈ 20 • C between winter and summer nights. This results in a variation in the lasing wavelength of up to 5 nm for our observations. To mitigate this uncertainty in our calibration, we either use corrected laser specifications for the temperature during calibration or make the setup independent of the lasing wavelength by tilting to an angle of 20 • in both directions and using the midpoint distance between the two peaks as the filter bandpass shifts through the lasing wavelength to calibrate the tilt. A series of images of the laser-illuminated target were taken at different tilts, after which the images were analysed to determine the brightness of the reflected laser through the filter in each image. These results were compared to a model of the throughput of the laser light as a function of filter tilt, which is shown in the left panel of Fig. 2. The laser flux was measured from a tilt of −20 • to +20 • to create the two peaks described above and by fitting the model to the data, the zeropoint of the filter tilt was determined. The second method of calibration is identical to the first method, except that instead of using a target illuminated by a laser, an astronomical source of line emission is used. In the second panel of Fig. 2 the results for imaging the planetary nebula NGC 6543 (the "Cat's Eye" Nebula) at a series of filter tilts from 0 − 20 • are shown. In this case, the known wavelength of Hα emission from the planetary nebula is used to model the transmission through the filter bandpass and fit to the calibration observations to determine the zeropoint of the filter. For this set of calibration observations, there is an excess of emission redward of the Hα emission, which is attributed to [Nii] emission from NGC 6543. A combined model of Hα and [Nii] emission at the redshift of NGC 6543 was created to determine a better fit to the data. A relative fraction of 0.12 between the Hα and the [Nii] emission yielded the best fit to the data. This method can be used to properly fit the observed emission curve for a variety of astronomical sources during calibration analysis. ULTRA-LOW SURFACE BRIGHTNESS OBSERVATIONS An imaging campaign on the M81 group of galaxies was undertaken in the spring of 2020 with the pathfinder version of the Dragonfly Spectral Line Mapper. Narrowband imaging of Hα and [Nii] emission from the galaxy group was collected from March 2020 to October 2021. These observations served as proof-of-concept imaging to determine the surface brightness limits of the instrument, as well as scientific imaging, the results of which are presented in Ref. 16. The observations followed the Dragonfly automated observing model, where the telescope is set up every night for observing at the beginning of the night and the telescope carries out observations autonomously, adapting to changing weather conditions and pausing observations when necessary. In total, this resulted in 73 nights of data collection over the months of February to June 2020 and a total of 652 on-target science frames collected with individual exposure times of 1800 seconds. After the removal of frames with low image quality (e.g., due to poor focusing, poor guiding, poor seeing, etc.), the total exposure time with the 3-lens pathfinder that was included in the final science frames was 31.7 hours and 15.3 hours for the Hα and Overlaid in red is Hα data obtained with the 3-unit Dragonfly Spectral Line Mapper pathfinder. The Hα imaging revealed a multitude of star formation complexes in M81 and the expansive M82 outflow that connects M82 to the "Hα cap" above the disk. 18,19 In this imaging not only is the Hα cap clearly visible but also a variety of new features including a tidal dwarf galaxy candidate at the edge of the M82 disk (Ref. 20; within the dashed gray box), Hα emission from the tidal streamer appearing to connect the edge of the disk to the Hα cap, and the most striking discovery of a giant cloud of gas seeming to hover above the M82 galaxy, the Hα shell. 16 [Nii] science images, respectively (see Ref. 15 for further details). The data were taken with the filters at two different tilts: 12.5 • to target the Hα λ6563 emission line and 7 • to target the [Nii] λ6583 emission line. Tilting the filters smoothly shifts the filter central wavelength, and these two tilts shifted the filter central wavelengths to 656.3 nm and 683.5 nm, respectively. In addition, data were collected with the Dragonfly Telephoto Array (in the configuration most recently described in Ref. 17) yielding r and g band continuum images with exposure times of 12.5 minutes and 10 minutes, respectively. The final science images had a total field of view of ∼2 • × 3 • and reached an Hα surface brightness limit of ∼ 5 × 10 −19 erg cm −2 s −1 arcsec −2 to the 3σ level on a 4 spatial scale. A composite r and g band image along with the high signal-to-noise regions of the pathfinder Dragonfly Spectral Line Mapper Hα data is shown in Fig. 3. The deep, wide-field data confirmed previously discovered low surface brightness features in the field such as the M82 Hα cap, 18,19 as well as several new features and extensions to already known features in the group. These included a secondary ridge of Hα emission between the Hα cap and the M82 galaxy, as well as emission filling the region between the M82 galaxy and the Hα cap. In addition, Hα emission from a tidal dwarf candidate and emission corresponding to the HI tidal streamer on the northeast side of the M82 disk were discovered (indicated by dashed box in Fig. 3; Ref. 20). The most striking result of this imaging campaign was the discovery of a colossal cloud of gas in the outskirts of the M82 galaxy with an extent of 0.8 deg (labelled in Fig. 3), which is analyzed further in Ref. 16. The main goal with this instrument was to reach the surface brightnesses required to directly image the ultra-faint line emission from gas in the surroundings of galaxies, and determine what improvements upon the instrument would be necessary before undertaking the construction of the full-scale Dragonfly Spectral Line Mapper. In order to reach these low surface brightness limits, the calibration of our data needed to be undertaken with care. To ensure that our observations are sky noise limited, dark noise and flat-field corrections needed to be carried out to below 0.1%. The data collection followed the Dragonfly model of collecting data regardless of sky conditions and then removing frames that are classified as "bad" during the data reduction pipeline. This automatically removed a significant fraction of the frames due to poor image quality. In addition to this data cut, we also inspected the frames by eye to search for and remove frames with contamination that were not caught by the automated pipeline. Together, this resulted in using only 279 out of the 652 science frames collected to create the final science images. In the following Sections, we delve into the analysis of the main sources of noise in the narrowband data and the methods for their removal. We identify several improvements in hardware and calibration frame collection that will be incorporated in the full-scale Dragonfly Spectral Line Mapper design and operation. Dark Frame Stability The Aluma CCD694 cameras used with the Dragonfly Spectral Line Mapper pathfinder were selected for their low dark current (0.025 electron s −1 pixel −1 ) and readout noise (4 electron RMS). In half an hour dark frame exposures, it was observed that there were additional significant sources of noise in the images above the dark current and readout noise: glow from amplifier and bias circuitry (e.g. , Figs 4 and 5). While the dark current and readout noise met the requirements for the low surface brightness limits in the data, the noise from the amplifier and bias circuitry proved difficult to remove due to their fluctuation over time. To quantify this variation, we collected a series of half an hour dark frames over each night for one week and a set of dark frames spread out over the time frame of a month. While the background noise level in the regions of the frame unaffected by glow from the amplifier and bias circuitry stayed consistent with the Poisson noise of the dark current (RMS∼30 counts in the raw images), the amplifier and bias circuitry noise varied considerably. The amplifier glow and bias circuitry glow is visible in the dark images displayed in Fig. 4, which shows a master dark formed out of 10 dark frames with exposure times of 1800s on the left panel, a single dark with an exposure time of 1800s in the middle panel, and the residual from the subtraction of the master dark from the single dark in the right panel. This residual is a replica of the dark noise that would remain after dark frame subtraction in a science exposure of 1800s, representing the combination of noise from the dark current and readout. An increase in noise is visible in the areas affected by the amplifier and bias circuitry. Fig. 5 shows averages over rows in 10 dark frames taken during one night, as well as the master dark. It is apparent that the Figure 5. Comparison of the counts in rows 100, 500, and 1500 (averaged over the surrounding 10 rows, and averaged over 25 columns to reduce noise) in 10 1800s dark exposures taken on the same night along and the master dark made from the combination of these 10 dark frames. There is a shift in the bias level between darks, which is nonlinear in the regions affected by bias circuitry and amplifier glow (e.g., row 100). Figure 6. Comparison of Hα science exposure stack with and without removing frames with poor dark frame subtraction. Including the frames with poor dark subtraction (left panel) results in large variation with a cutoff along a specific declination. Removing those frames and redoing the science exposure stack (right panel) removes that linear feature, but also results in lower signal-to-noise throughout the frame. This effect is partly due to repeated dither positions used during exposures on different nights which ended up aligning the regions with dark noise in the science image. underlying bias level shifts through the night, so the master dark is not representative of the complete sample of dark frames. The shift in the bias level adds a noise floor to the data which is removed during the sky modeling and subtraction step of the Dragonfly pipeline. Any nonlinearity in the noise floor such as the amplifier glow and bias circuitry glow remains in the science frame, though, as they are not fit and subtracted during the sky modeling step. This added a systematic noise source into the science frames, creating an artificial low surface brightness feature within the final science frames. This feature was apparent in the final science image despite the ∼ 15 dithering of the telescope between science exposures and ∼ 15 offset in pointing between the three OTAs. In order to remove this noise source, the data reduction pipeline was modified with an additional step of cropping the raw images to remove the bottom physical section of the image where the amplifier and bias circuitry glow showed up. This successfully removed the false feature caused by the systematic noise in the science frame. This is shown in Fig. 6, which displays Hα science image stacks with and without removing frames with poor dark frame subtraction. Including the frames with poor dark subtraction (left panel) results in large variation with a cutoff along a specific declination. Removing those frames and redoing the science exposure stack (right panel) removes that linear feature, but also results in lower signal-to-noise throughout the frame. This effect is partly due to repeated dither positions used during exposures on different nights which ended up aligning the regions with dark noise perfectly in the science image. We will be using randomized dither patterns with the full Dragonfly Spectral Line Mapper to avoid this effect, which is magnified as the exposure times increase in length. Newer Aluma CCD694 cameras use modified electronics and firmware, and the amplifier and bias circuitry glow have been significantly reduced since this data was collected (see our companion paper, Chen, S. et al. 2022, for further details). Flat Fielding Stability Flat-fields for the Dragonfly Spectral Line Mapper pathfinder were collected using electroluminescent flat-fielding panels (Alnitak Flip-Flats). This allowed us to collect flats at a series of different tilts. To investigate the effects of different filter tilt angles on the flat fielding, flats were taken at tilt angles from 0 to 10 degrees, in increments of 1 degree. Five flats were taken at each angle, and combined to make a master flat at each angle. We modeled the bright center and dimmer edges as cos 4 vignetting for the master flat at each angle; an example is shown in Fig. 7 for a filter rotated to an angle of 0 deg. The cos 4 vignetting is given by the following function: z = A cos 4   x − x c a 2 + y − y c b 2   + B(1) where z is the pixel value, x c is the center x-pixel, y c is the center y-pixel, a and b are the semi minor/major axes of the ellipse, and A is the scaling amplitude, and B is the offset. For simplicity, a = b, as after fitting the data with both a and b variable we find a ≈ b, differing by about 1-3 pixels. The center point of the vignetting moved ∼ 10 − 100 pixels depending on the tilt of the filter. The right panel of Fig. 7 shows the variation of the center pixel location in the fitted model as a function of the tilt of the filter. At larger angles the center pixel (x c , y c ) moves "left and up". The vignetting depends on the tilt of the filter, making it necessary to collect flats at the same tilt angles as those used during Dragonfly science observations to carry out a robust calibration from the flat field images. The nonlinearity of the shift in the position of the center pixel with filter angle may be due to the shift in the shape of the spectral response across the field of view which is symmetric at a filter tilt of 0 • then becomes a linear gradient across the field of view at a filter of tilt of ≈ 3 • (see Figure 6 of Ref. 15). In addition, we were careful to collect flats while the telescope was at the same pointing as the science frames were collected. By utilizing the Flip-Flats, we were able to collect flats throughout the night, before and after each science exposure. This is necessary due to flexure in the optical tube assembly, in particular caused by shifting of the image stabilisation lens in the Canon telephoto lenses and flexure in the connection between the lens and focuser-camera assembly. Similar to that from varying tilts, this flexure resulted in a shift in the centroid of the flat-field image. While the centroid shift was very small (on order of a few pixels), the resulting additional source of noise was significant. Stray Light Contamination In this narrowband setup, we are extremely sensitive to any sources of scattered light or light leakage through the optical tube assembly onto the detector. During the observations, we noticed that two strong types of contaminating light structures appeared in a subset of the data frames on different nights. Two example images showing data contaminated by these two light structures are shown in the left column of Fig. 8. Models of the background light structure are shown in the right column of Fig. 8. All three of the OTAs in the pathfinder Dragonfly Spectral Line Mapper were affected by light contamination to a low level that usually showed up as a gradient in the images. The structures shown in Fig. 8 are unique in that they are extremely prominent (at least an order of magnitude larger than typical light contamination in the images) and appeared solely in data taken with the OTA that had the latest generation of the telephoto lens, the 400mm f/2.8 IS III version. After further investigation, we determined that at least part of this light contamination was due to faulty baffling of the Filter-Tilter that allowed light to leak into the front of the optics. Indeed, the situation for one of these prominent light structures (shown in the bottom panels of Fig. 8) resolved itself without any intervention from the science team, so perhaps it was a hardware issue that was fixed as part of routine servicing of the telescope by the observatory staff. To address the other large source of light contamination (shown in the top panels of Fig 8), we carried out an extensive investigation to attempt to pinpoint the location of the light leak inside the optics. This included switching out every single component of our optical array. We finally concluded that it was the telephoto lens itself that was the culprit for causing the light contamination and we hypothesize that this light leakage is due to removal of shielding material in the telephoto lens allowing infrared emission to penetrate the lens and enter the optics, where it was then incident upon the detector. To test this theory, we inserted a UV-IR blocking filter into the available slot on the telephoto lens, which is at the back of the lens in front of the camera. This greatly reduced the light contamination in the optics and we were able to recover a clean signal in the image. COSMIC WEB DETECTORS In this Section, we return to the earlier discussion of the telescopes and instruments being used to investigate the intergalactic and circumgalactic media of galaxies. There are now a large number of experiments targeting these media in the universe, ranging from local observations of galaxies to high redshift observations of quasars. In this Section, we provide a comparison of these observational techniques and the surface brightness limits they have reached. Each technique has relative strengths and weaknesses, so a side-by-side comparison allows one to choose the optimal instrument for specific observations. In Table 4 we list the fields of view of several instruments that have been used to investigate the circumgalactic and/or intergalactic media of galaxies at different ranges of redshift, along with the surface brightness limits reached by these instruments when undertaking these studies. In addition to instruments with measured surface brightness limits, we have included an estimate of the surface brightness limits that will be observed with the full Dragonfly Spectral Line Mapper. This estimate was calculated by scaling the pathfinder Dragonfly Spectral Line Mapper surface brightness limit by both increasing the aperture of the telescope (40× the size of the pathfinder) and reducing the pathfinder's bandpass (0.8 nm compared to the 3 nm bandpass of the pathfinder instrument). We predict that the Dragonfly Spectral Line Mapper will reach a limiting surface brightness of 1 × 10 −19 erg cm −2 s −1 arcsec −2 for a 3σ detection over a spatial scale of 4 in 7 hours. The fields of view of the instruments listed in Table 4 are displayed in Fig. 9, projected on an image of the One tool to measure the survey efficiency of an instrument is to calculate the sky survey rate, Γ, which is defined as the ratio of the observed sky area to the time needed to reach the desired sensitivity. 29 We evaluate the instruments using this scale in Fig. 10, where we have plotted the fields of view of the instruments against the time required to reach a sensitivity of 1 × 10 −19 erg cm −2 s −1 arcsec −2 with a 3σ detection. The surface brightness limits for the values in Fig. 10 are calculated over a spatial scale of 10 arcsec 2 and scaled from the values in Table 4 where necessary. Scaling the detection limits from Nσ to 3σ is straightforward as the surface brightness limit is calculated from the RMS fluctuations in the sky background, so a simple scaling of 3/N can be implemented. The required exposure times were scaled from the values in Table 4 by assuming that the signalto-noise ratio scales with the square root of the integration time. Creating an equivalent comparison between the spatial scale is less straightforward, and in this case, the values were scaled empirically using relative amplitudes from instruments where surface brightness limits calculated at different scales exists (e.g., Refs. 28 and 21 provide two different estimates of the sky background observed by the MUSE instrument). This introduces error into this estimate, which is encompassed by the error bars in the plot. Table 4. We scaled these values empirically using relative numbers from instruments where the surface brightness limits have been calculated at different scales (e.g., Refs. 28 and 21 provide two different estimates of the sky background observed by the MUSE instrument). This introduces error into this estimate, particularly for the CFHT/MegaCam and Keck/KCWI estimates. Lines of constant survey speed are shown for comparison, where a higher survey speed value corresponds to a more efficient survey instrument. Lines of constant survey rate are plotted in Fig. 10 to aid the comparison of the survey efficiency of the instruments. As one may expect, the wide-field instruments have the highest survey efficiency due to their large fields of view. It is necessary that this large field of view is accompanied with sensitivity, though, which separates CFHT/MegaCam and the Dragonfly Spectral Line Mapper from the Kitt Peak/Burrell Schmidt instrument. This plot also shows that when large fields of view are not important, the IFUs on the larger telescopes are more efficient for observations, due to the lower exposure times required. There are limits to this comparison, which include that it does not take into account the spectral range of the IFUs, which adds the third spatial dimension into the survey, changing it from a survey area into a survey volume. Thus, this specific comparison is useful for considering single targets or objects within some fixed cosmological volume. Including the spectral range of the instruments would boost the survey efficiency of the IFUS. Another limitation of this comparison is the simplicity in calculating the limiting surface brightnesses. Not only is it approximate due to empirical scaling of sensitivities, but it also does not take into account systematic errors in surface brightness measurements. As was discussed earlier, Dragonfly's strength is the removal of systematic sources of noise in low surface brightness imaging, which isn't considered in this simple surface brightness limit estimate, which is based on the random noise in the images. SUMMARY We have developed an upgrade to the Dragonfly Telephoto Array to implement ultranarrow bandpass imaging capability on the telescope with the goal of targeting the line emission from the circumgalactic medium around nearby galaxies, and the brightest pockets of the cosmic web in the local Universe. The Dragonfly Filter-Tilter 15 was developed to incorporate ultranarrowband filters on Dragonfly, mounting the filters at the front of the optics to avoid the degradation of the filter transmission profile and to enable the rotation of the filter with respect to the optical axis. We built the Dragonfly Spectral Line Mapper pathfinder, which is a 3-unit version of Dragonfly with the Filter-Tilter instrumentation, to test this concept. We carried out an imaging on the M81 group of galaxies with the pathfinder reaching surface brightness limits comparable to those reached by state-of-the-art instruments on large optical telescopes. The low surface brightness levels reached by the pathfinder Dragonfly Spectral Line Mapper forecast what the full Dragonfly Spectral Line Mapper will reach. With narrower filter bandpasses and 40× more collecting area, we predict that the Dragonfly Spectral Line Mapper will reach surface brightness limits of 1 × Figure 1 . 1The pathfinder Dragonfly Spectral Line Mapper, on sky from the spring of 2019 to the fall of 2021 is pictured in the left panel. A schematic representation of the pathfinder is also shown, with the components and connections between computers and devices labeled. Figure 2 . 2Left panel: The laser flux through the narrowband filters measured for tilts between -20 and +20 degrees; the orange, purple and gray data points are measured from the units of the narrowband Dragonfly pathfinder independently. A model of the laser throughput through the filters is shown as the green line. The model and data are well-matched, providing a straight-forward calibration method by fitting the model to our data to find the zeropoint of the filter tilt. Right panel: The measured brightness of the planetary nebula NGC 6543 at a series of filter tilts from 0 to 20 deg (shown for each unit of the Dragonfly narrowband set up as the orange, purple, and gray datapoints). The measurements are compared to models of the planetary nebula throughput, taking into account Hα and [Nii] emission from the planetary nebula. Figure 3 . 3False colour image of M81 and M82 obtained with the Dragonfly Telephoto Array in g and r band imaging. Figure 4 . 4A master dark formed from the combination of 10 1800s dark frames is shown in the left panel, a single dark frame of 1800s is shown in the middle panel, and the residual from the subtraction of the master dark from the single dark frame is shown in the right panel. There is significant amplifier glow and bias circuitry glow visible in the dark frames (locations most affected are indicated on the frames). Figure 7 . 7The left plot shows a calibrated and stacked flat taken by one of the pathfinder Dragonfly Spectral Line Mapper OTAs at a filter tilt of 0 degrees using the electroluminescent flat-fielding panel. The middle plot shows the cos 4 vignetting model (Equation 1) fitted to the data. The center x-pixel, xc, and center y-pixel, yc, for each flat as a function of filter angle are shown in the right panel, where the center position is calculated from the cos 4 vignetting model. Figure 8 . 8Many of the frames collected by the pathfinder Dragonfly Spectral Line Mapper where contaminated by light structures. The left column of images shows two prominent examples that together affected ∼ 100 data frames. In the right column is models of the background sky which shows two distinct types of light features that appeared for some time in the data. Figure 9 . 9The fields of view of several instruments used to map the cosmic web as projected onto a false colour image of NGC 4565 (the Needle Galaxy) taken by the Dragonfly Telephoto Array.27 The moon is shown for scale.NGC 4565 galaxy (NGC 4565 has a radial distance of D ≈ 13 Mpc; Ref.26). This image demonstrates the complementary nature of the various instruments when targeting structures in the nearby Universe (D < 50 Mpc). Integral field units (IFUs) on 8-m class telescopes carry out high angular resolution studies, which are suited to smaller scale structures, while the wide-field imagers such as Dragonfly and CFHT MegaCam are suited to studying larger scale structures such as the galaxy environment. Figure 10 . 10A comparison of the fields of view of select instruments against the exposure time required to reach a sensitivity of 10 −19 erg cm −2 s −1 arcsec −2 with a 3σ detection. The surface brightness limits are calculated over a spatial scale of 10 arcsec 2 for the exposure time estimates. The surface brightness estimates were either taken directly from the literature or scaled from values in 10 −19 erg cm −2 s −1 arcsec −2 on scales of 4 in 7 hours. The Dragonfly Spectral Line Mapper is under construction in a phased timeline, deploying 10 to 30 units at a time in stages. There are currently 10 units of the Dragonfly Spectral Line Mapper on sky, which are being used to carry out commissioning and filter testing (see Chen, S., et al. 2022 for further details). This number will increase to 60 units in the fall of 2022, and to the final number of 120 units in the summer of 2023. Table 1 . 1Content summary of the three papers in this series, with the topics covered by this paper in bold.Distributed aperture telescope general concepts Abraham et al. Low surface brightness imaging challenges Abraham et al. Lessons learned from Dragonfly Abraham et al. Narrowband imaging concepts and methods This article Narrowband imaging survey speed This article Dragonfly Spectral Line Mapper pathfinder results & lessons learned This article Dragonfly Spectral Line Mapper design Chen et al. Dragonfly Spectral Line Mapper laboratory tests Chen et al. Dragonfly Spectral Line Mapper roadmap Chen et al. Table 2 . 2Specifications of the pathfinder Dragonfly Spectral Line Mapper.Parameter Value Effective aperture 24.6 cm (14.3 cm × 3) Effective focal length 40 cm (f /1.6) Field of view 1 • .4 × 1 • .9 Camera SBIG Aluma CCD694 Detector Sony ICX 694, 2750 × 2200 CCD Pixel size 4.54 µm (2.45 ) Optics Canon Telephoto 400mm f/2.8 IS II, Canon Telephoto 400mm f/2.8 IS II, Canon Telephoto 400mm f/2.8 IS III Filters See Table 3 . 3Specifications of the Iridian Spectral Technologies Inc. Dragonfly Filter.Parameter Value Central Wavelength at 0 • angle of incidence 659.9 nm Bandwidth 3 nm Out-of-band blocking <OD4 200 -1100 nm Clear Aperture Diameter 152 mm Operating temperature -10 • C to +20 • C Transmittance Peak Tx >95% with Tx >90% for bandwidth >2.0 nm of the narrowband interference filters, this shifts the central wavelength of the filter from its 0 • angle of incidence value by up to ≈ 8 nm. The filters used for these observations have a central wavelength of 659.9 nm in vacuum and FWHM of 3 nm. See Ref. Table 4 . 4Instrument surface brightness limit and field of view comparisons for the Multi-Unit Spectroscopic Explorer (MUSE) on ESO's Very Large Telescope (VLT), the Cosmic Web Imager (CWI) at the Palomar Observatory, the Keck Cosmic Web Imager (KCWI) at the Keck Observatory, MegaCam imager at the Canada-France-Hawaii Telescope (CFHT), the Burrell Schmidt Telescope at Kitt Peak National Observatory, and the pathfinder and full-scale Dragonfly Spectral Line Mappers (DSLM).Instrument Surface Brightness Limit Spatial Scale Exposure FOV Refs (erg cm −2 s −1 arcsec −2 ) Time (s) VLT/MUSE 2.4 × 10 −19 at 3σ per 3.75Å 10 1 hr 60 × 60 21 Palomar/CWI 1.3 × 10 −19 at 1σ per 4Å 10 − 15 11 hr 60 × 40 22, 23 Keck/KCWI 4.8 × 10 −19 at 1σ per 5Å 1 3.5 hr 33.1 × 20.4 10 CFHT/MegaCam 2 × 10 −18 at 1σ 3 2 hr 40 × 30 24 Kitt Peak/ 3.09 × 10 −18 at 1σ 2 900s 1 • × 1 • 25 Burrell Schmidt Pathfinder DSLM 5 × 10 −19 at 3σ 4 32 hr 1 • .4 × 1 • .9 16 DSLM 1 × 10 −19 at 3σ 4 7 hr 1 • .4 × 1 • .9 16, this work ACKNOWLEDGMENTSWe are very grateful to the staff at New Mexico Skies Observatories, without whom this work couldn't have been carried out. This work is supported by a Canadian Foundation for Innovation (CFI) grant. We are thankful for contributions from the Dunlap Institute (funded through an endowment established by the David Dunlap family and the University of Toronto), the Natural Sciences and Engineering Research Council of Canada (NSERC), and the National Science Foundation (NSF), without which this research would not have been possible. The Cosmic Web Imager: an integral field spectrograph for the Hale Telescope at Palomar Observatory: instrument design and first results," in [Ground-based and Airborne Instrumentation for. M Matuszewski, D Chang, R M Crabill, D C Martin, A M Moore, P Morrissey, S Rahman, Proceedings of the SPIE. 773577350PMatuszewski, M., Chang, D., Crabill, R. M., Martin, D. C., Moore, A. 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[]
[ "Robust and Provably Monotonic Networks", "Robust and Provably Monotonic Networks" ]
[ "Ouail Kitouni [email protected] \nNSF AI Institute for Artificial Intelligence and Fundamental Interactions Laboratory for Nuclear Science\nMIT\nCambridgeMAUSA\n", "Niklas Nolte [email protected] \nNSF AI Institute for Artificial Intelligence and Fundamental Interactions Laboratory for Nuclear Science\nMIT\nCambridgeMAUSA\n", "Mike Williams \nNSF AI Institute for Artificial Intelligence and Fundamental Interactions Laboratory for Nuclear Science\nMIT\nCambridgeMAUSA\n" ]
[ "NSF AI Institute for Artificial Intelligence and Fundamental Interactions Laboratory for Nuclear Science\nMIT\nCambridgeMAUSA", "NSF AI Institute for Artificial Intelligence and Fundamental Interactions Laboratory for Nuclear Science\nMIT\nCambridgeMAUSA", "NSF AI Institute for Artificial Intelligence and Fundamental Interactions Laboratory for Nuclear Science\nMIT\nCambridgeMAUSA" ]
[]
The Lipschitz constant of the map between the input and output space represented by a neural network is a natural metric for assessing the robustness of the model. We present a new method to constrain the Lipschitz constant of dense deep learning models that can also be generalized to other architectures. The method relies on a simple weight normalization scheme during training that ensures the Lipschitz constant of every layer is below an upper limit specified by the analyst. A simple monotonic residual connection can then be used to make the model monotonic in any subset of its inputs, which is useful in scenarios where domain knowledge dictates such dependence. Examples can be found in algorithmic fairness requirements or, as presented here, in the classification of the decays of subatomic particles produced at the CERN Large Hadron Collider. Our normalization is minimally constraining and allows the underlying architecture to maintain higher expressiveness compared to other techniques which aim to either control the Lipschitz constant of the model or ensure its monotonicity. We show how the algorithm was used to train a powerful, robust, and interpretable discriminator for heavy-flavor-quark decays, which has been adopted for use as the primary dataselection algorithm in the LHCb real-time data-processing system in the current LHC data-taking period known as Run 3. In addition, our algorithm has also achieved state-of-the-art performance on benchmarks in medicine, finance, and other applications.
null
[ "https://export.arxiv.org/pdf/2112.00038v2.pdf" ]
244,772,981
2112.00038
1cccd9c84a40dd8466e27f20d76cca533625ece6
Robust and Provably Monotonic Networks Ouail Kitouni [email protected] NSF AI Institute for Artificial Intelligence and Fundamental Interactions Laboratory for Nuclear Science MIT CambridgeMAUSA Niklas Nolte [email protected] NSF AI Institute for Artificial Intelligence and Fundamental Interactions Laboratory for Nuclear Science MIT CambridgeMAUSA Mike Williams NSF AI Institute for Artificial Intelligence and Fundamental Interactions Laboratory for Nuclear Science MIT CambridgeMAUSA Robust and Provably Monotonic Networks The Lipschitz constant of the map between the input and output space represented by a neural network is a natural metric for assessing the robustness of the model. We present a new method to constrain the Lipschitz constant of dense deep learning models that can also be generalized to other architectures. The method relies on a simple weight normalization scheme during training that ensures the Lipschitz constant of every layer is below an upper limit specified by the analyst. A simple monotonic residual connection can then be used to make the model monotonic in any subset of its inputs, which is useful in scenarios where domain knowledge dictates such dependence. Examples can be found in algorithmic fairness requirements or, as presented here, in the classification of the decays of subatomic particles produced at the CERN Large Hadron Collider. Our normalization is minimally constraining and allows the underlying architecture to maintain higher expressiveness compared to other techniques which aim to either control the Lipschitz constant of the model or ensure its monotonicity. We show how the algorithm was used to train a powerful, robust, and interpretable discriminator for heavy-flavor-quark decays, which has been adopted for use as the primary dataselection algorithm in the LHCb real-time data-processing system in the current LHC data-taking period known as Run 3. In addition, our algorithm has also achieved state-of-the-art performance on benchmarks in medicine, finance, and other applications. Introduction The sensor arrays of the LHC experiments produce over 100 TB/s of data, more than a zettabyte per year. After drastic data-reduction performed by custom-built read-out electronics, the annual data volumes are still O(100) exabytes, which cannot be stored indefinitely. Therefore, each experiment processes the data in real-time and decides whether each proton-proton collision event should remain persistent or be discarded permanently, referred to as triggering in particle physics. Trigger classification algorithms must be designed to minimize the impact of effects like experimental instabilities that occur during data taking-and deficiencies in simulated training samples. (If we knew all of the physics required to produce perfect training samples, there would be no point in performing the experiment.) The need for increasingly complex discriminators for the LHCb trigger system [1,2] calls for the use of expressive models which are both robust and interpretable. Here we present an architecture based on a novel weight normalization technique that achieves both of these requirements. Robustness A natural way of ensuring the robustness of a model is to constrain the Lipschitz constant of the function it represents, defined such that for every pair of points on the graph of the function, the absolute value of the slope of the line connecting them is not greater than the Lipschitz constant. To this end, we developed a new architecture whose Lipschitz constant is constrained by design using a novel layer-wise normalization which allows the architecture to be more expressive than the current state-of-the-art with more stable and faster training. Interpretability An important inductive bias in particle detection at the LHC is the idea that particular collision events are more interesting if they are outliers, e.g., possible evidence of a particle produced with a longer-than-expected (given known physics) lifetime would definitely warrant further detailed study. The problem is that outliers are often caused by experimental artifacts or imperfections, which are included and labeled as background in training; whereas the set of all possible interesting outliers is not possible to construct a priori, thus not included in the training process. This problem is immediately solved if outliers are better is implemented directly using an expressive monotonic architecture. Some work was done in this regard [3][4][5] but most implementations are either not expressive enough or provide no guarantees. We present Monotonic Lipschitz Networks which overcome both of these problems by building an architecture that is monotonic in any subset of the inputs by design, while keeping the constraints minimal such that it still offers significantly better expressiveness compared to current methods. Monotonic Lipschitz Networks The goal is to develop a neural network architecture representing a scalar-valued function f (x) : R n → R (2.1) that is provably monotonic in any subset of inputs and whose gradient (with respect to its inputs) has a constrained magnitude in any particular direction. In an experimental setting, this latter property is a measure of robustness to small changes in experimental conditions or to small deficiencies in the training samples. Constraints with respect to a particular L p metric will be denoted as Lip p . We start with a model g(x) that is Lip 1 with Lipschitz constant λ if ∀ x, y ∈ R n (we show below how to train such a model) |g(x) − g(y)| ≤ λ x − y 1 . (2. 2) The choice of 1-norm is crucial because it allows a well defined maximum directional derivative for each input regardless of the gradient direction. This has the convenient side effect that we can tune the robustness requirement for each input individually. Note that rescaling the inputs x i allows for λ directional dependence. Enforcing Monotonicity Assuming we have trained a model that satisfies (2.2), we can make an architecture with built-in monotonicity by adding a term that is linear (or has gradient λ) in each direction in which we want to be monotonic: f (x) = g(x) + λ i∈I x i ,(2.3) where I denotes the set of indices of the input features for which we would like to be monotonic. This residual connection enforces monotonicity: ∂f ∂x i = ∂g ∂x i + λ ≥ 0 ∀ i ∈ I . (2.4) Note that the construction presented here only works with Lip 1 constraints as Lip p =1 functions introduce dependencies between the partial derivatives. In addition, we stress that monotonicity is defined via partial derivatives. The value of f is guaranteed to increase when x i is increased while keeping all x =i constant. It is therefore advisable to look out for ill defined edge cases. For instance, let x 2 ≡ −x 1 in the training data and define I = {1, 2}. This is incompatible with the architecture and produces unwanted results unless λ = 0 for both x 1 and x 2 (otherwise the problem is ill posed). To the best of our knowledge, the only use of residual connections in the literature when trying to learn monotonic functions is in the context of invertible ResNets [6]. Instead, the state-of-the-art approach for learning monotonic functions involves penalizing negative gradients in the loss, then certifying the final model is monotonic, rather than enforcing it in the architecture (e.g. in [3]). Enforcing Lipschitz Constraints Ideally, the construction g(x) should be a universal approximator of Lip 1 functions. Here, we discuss possible architectures for this task. Lip 1 constrained models Fully connected networks can be Lipschitz bounded by constraining the matrix norm of all weights [7,8]. Given the fully connected network with activation σ g(x) = W m σ(W m−1 σ(...σ(W 1 x + b 1 )...) + b m−1 ) + b m , (2.5) where W m is the weight matrix of layer m, g(x) satisfies Eq. (2.2) if m i=0 W i 1 ≤ λ (2.6) and σ has a Lipschitz constant less than or equal to 1. There are multiple ways to enforce Eq. (2.6). Two possibilities that involve scaling by the operator norm of the weight matrix [7] are W i → W i = λ 1/m W i max(1, W i 1 ) or W i → W i = W i max(1, λ −1/m · W i 1 ) . (2.7) In our studies thus far, the latter variant seems to train slightly better. However, in some cases it might be useful to use the former to avoid the scale imbalance between the neural network's output and the residual connection used to induce monotonicity. In order to satisfy Eq. (2.6), it is not necessary to divide the entire matrix by its 1-norm. It is sufficient to ensure that the absolute sum over each column is constrained: W i → W i = W i diag   1 max 1, λ −1/m j |W i jk |   . (2.8) This novel normalization scheme tends to give even better training results in practice. While Eq. (2.8) is not suitable as a general-purpose scheme, e.g. it would not work in convolutional networks, its performance in training in our analysis motivates further study of this approach in future work. The constraints in Eqs. (2.7) and (2.8) can be applied in different ways. For example, one could normalize the weights directly before each call such that the induced gradients are propagated through the network like in [8]. While one could come up with toy examples for which propagating the gradients in this way hurts training, it appears that this approach is what usually is implemented for spectral norm [8] in PyTorch and TensorFlow. Alternatively, the constraint could be applied by projecting any infeasible parameter values back into the set of feasible matrices after each gradient update as in Algorithm 2 of [7]. Algorithm 1 summarizes our approach. Algorithm 1 Training with enforced Lipschitz constraint using weight-norming Require: {D i } n i=1 , a collection of n training batches. Require: w, the non-normalized weight parameter at some layer. These are the optimized leaf parameters Require: Norm, the function used to normalize the weights, e.g. as given by Eq. (2.8). Require: Cost, the loss computed using a neural network with weight parametersŵ on a given batch. w ← Norm(w) This is the weight used in the neural network matrix multiplication while not converged do for i from 1 to n do L ← Cost(D i ,ŵ) w ← w − ∇ wŵ · ∇ŵL w ← Norm(w) end for end while At inference time, onlyŵ is used. Preserving expressive power Some Lipschitz network architectures (e.g. [8]) tend to overconstrain the model in the sense that these architectures cannot fit all functions λ-Lip 1 due to gradient attenuation. For many problems this is a rather theoretical issue. However, it becomes a practical problem for the monotonic architecture since it often works on the edges of its constraints, for instance when partial derivatives close to zero are required. The authors of [9] showed that ReLU networks are unable to fit the function f (x) = |x| if the layers are norm-constrained with λ = 1. The reason lies in fact that ReLU, and most other commonly used activations, do not have unit gradient with respect to the inputs over their entire domain. While element-wise activations like ReLU cannot have unit gradient over the whole domain without being exactly linear, the authors of [10] explore activations that introduce nonlinearities by reordering elements of the input vector. They propose the following activation function: σ = GroupSort, (2.9) which sorts its inputs in chunks (groups) of a fixed size. This operation has gradient 1 with respect to every input and gives architectures constrained with Eq. (2.6) increased expressive power. In addition, we have found that using this activation function also results in achieving sufficient expressiveness with a small number of weights, making the networks ideal for use in resource-constrained applications. Example Applications to Simple Models Before applying our new architecture to real-time data-processing at the LHC, we first demonstrate that it behaves as expected on some simple toy problems. Robustness to Outliers We will demonstrate the robustness that arises from the Lipschitz constraint by making a simple toy regression model to fit to data sampled from a 1-dimensional function with one particularly noisy data point. The underlying model that we sample from here has the form Each model is trained using 10 random initialization seeds. The dark lines are averages over the seeds, which are each shown as light lines. The unconstrained models exhibit overfitting of the noisy outlier, whereas the Lipschitz networks are robust. In addition, the Lipschitz constraint produces much smoother models as expected. N.b., here we set the Lipschitz constant to be λ = 1, whereas the slope of the true model is cos x. This allows for more variation in the fit model than the true model. In this exercise we assumed that all we know is that the slope is bounded by unity. If we did have more precise a priori information about the slope, we could easily employ this by rescaling x as discussed in Sec. 2. y = sin(x) + (x),(3. where (x) is Gaussian noise with unit variance for one data point and 0.01 otherwise. While this toy problem will explicitly show that the Lipschitz network is more robust against outliers than an unconstrained network due to its bounded gradient, it also serves as a proxy for any scenario with deficiencies in the training data. N.b., due to its bounded gradient a Lipschitz network is also more robust against adversarial attacks and data corruption than an unconstrained model. Figure 1 shows that the unconstrained model overfits the data as expected, whereas applying our approach from Sec. 2 does not. The Lipschitz model effectively ignores the outlier, since there is no way to accommodate that data point while respecting its built-in gradient bound. In addition, we see that the Lipschitz constraint enforces much smoother functions over the full range-the degree of this smoothness determined by us via the chosen Lipschitz constant. Monotonic Dependence To demonstrate monotonicity, we will make a simple toy regression model to fit to data sampled from the following 1-dimensional function: f (x) = log(x) + (x), (3.2) where is a Gaussian noise term whose variance is linearly increasing in x. In this toy model, we will assume that our prior knowledge tells us that the function we are trying to fit must be monotonic, despite the non-monotonic behavior observed due to the noise. This situation is ubiquitous in realworld applications of AI/ML, but is especially prevalent in the sciences (see, e.g., Sec. 4). First, we train standard (unconstrained) neural networks on several different samples drawn from Eq. (3.2). Here, we also consider two generic situations where the training data are missing: one that requires extrapolation beyond the region covered by the training data, and another that requires interpolation between two occupied regions. Figure 2 shows that the unconstrained models overfit the data as expected, resulting in non-monotonic behavior. Furthermore, when extrapolating or interpolating into regions where training data were absent, the unconstrained models exhibit highly undesirable and in some cases unpredictable behavior. (This problem is exacerbated in higher dimensions and sparser data.) In the case of extrapolation, the behavior of the unconstrained model is largely driven by the noise in the last one or two data points. The interpolation scenario is less predictable. While the overfitting observed here could be reduced by employing some form of strong regularization, such an approach would not (in general) lead to monotonic behavior, nor would it formally bound the gradient. Applying our approach from Sec. 2 does both. Figure 2 demonstrates that our method always produces a monotonic function, even in the extrapolation scenario where the slope of the noise terms in the last few data points is strongly suggestive of non-monotonic behavior. In addition, the Lipschitz constraint produces much smoother models than in the unconstrained case. Therefore, we conclude that the monotonicity and Lipschitz constraints do act as strong regularization against fitting random non-monotonic noise as expected. Expressiveness GroupSort weight-constrained neural networks can describe arbitrarily complex decision boundaries in classification problems provided the proper objective function is used in training (the usual cross entropy and MSE losses may be sub-optimal for Lipschitz models in some scenarios [11], see Sec. 5). Here we will directly regress on a synthetic boundary to emulate a classification problem. The boundary is the perimeter of circle with oscillating radius and is given by ∂ = {r + α[cos ωθ, sin ωθ]) | θ ∈ [0, 2π]} ,(3.3) where r and α are chosen to be 1.5 and 0.18, respectively. Figure 3 shows an example where this complicated decision boundary is learned by a Lipschitz network (as defined in Sec. 2) trained on the boundary while achieving zero loss, demonstrating the expressiveness that is possible to obtain in these models. Example Application: The LHCb inclusive heavy-flavor Run 3 trigger The architecture presented in Sec. 2 has been developed with a specific purpose in mind: The classification of the decays of heavy-flavor particles produced at the Large Hadron Collider, which are bound states that contain a beauty or charm quark that live long enough to travel an observable distance O(1 cm) before decaying. The dataset used here is built from simulated proton-proton (pp) collisions in the LHCb detector. Charged particles that survive long enough to traverse the entire detector before decaying are reconstructed and combined pairwise into decay-vertex (DV) candidates. The task concerns discriminating between DV candidates corresponding to the decays of heavyflavor particles versus all other sources of DVs. The signatures of a heavy-flavor DV are substantial separation from the pp collision point, due to the relatively long heavy-flavor particle lifetimes, and sizable transverse momenta, p T , of the component particles, due to the large heavy-flavor particle Each model is trained using 10 random initialization seeds. The dark lines are averages over the seeds, which are each shown as light lines. The unconstrained models exhibit overfitting of the noise and non-monotonic behavior, and when extrapolating or interpolating into regions where training data were absent, these models exhibit highly undesirable and unpredictable behavior. Conversely, the monotonic Lipschitz models always produce a monotonic function, even in scenarios where the noise is strongly suggestive of non-monotonic behavior. In addition, the Lipschitz constraint produces much smoother models as expected. N.b., here we set the Lipschitz constant to be λ = 1, whereas the slope of the true model is 1/x. This allows for more variation in the fit model than the true model. In this exercise we assumed that all we know is that the slope is bounded by unity. If we did have more precise a priori information about the slope, we could easily employ this by rescaling x as discussed in Sec. 2. masses. There are three main sources of background DVs. The first involves DVs formed from particles that originated directly from the pp collision, but where the location of the DV is measured to have non-zero displacement due to resolution effects. These DVs will typically have small displacements and small p T . The second source of background DVs arises due to particles produced in the pp collision interacting with the LHCb detector material, creating new particles at a point in space far from the pp collision point. Such DVs will have even larger displacement than the signal, but again have smaller p T . The third source involves at least one fake particle, i.e. a particle inferred from detector information that did not actually exist in the event. Since the simplest path through the detector (a straight line) In the first decision-making stage of the LHCb trigger, a pre-selection is applied to reject most background DVs, followed by a classifier based on the following four DV features: p T , the scalar sum of the p T of the two particles that formed the DV; min[χ 2 IP ], the smaller of the two increases observed when attempting to instead include each component particle into the pp-collision vertex fit, which is large when the DV is far from the pp collision point; the quality of the DV vertex fit; and the spatial distance between the DV and pp-collision locations, relative to their resolutions. N.b., the threshold required on the classifier response when run in real time during data taking is fixed by the maximum output bandwidth allowed from the first trigger stage. Unfortunately, extremely large values of both displacement and momentum are more common for backgrounds than for heavy-flavor signals. For the former, this is easily visualized by considering a simplified problem using only the two most-powerful inputs, p T and χ 2 IP . Figure 4 (left) shows that an unconstrained neural network learns to reject DVs with increasing larger displacements, corresponding to the lower right corner in the figure. Figure 5 (left) shows that this leads to a dependence of the signal efficiency on the lifetime of the decaying heavy-flavor particle. Larger lifetimes are disfavored since few heavy-flavor particles live more than O(10 ps Fig. 4 at the expected Run 3 working point versus the proper lifetime of the decaying heavy-flavor particle selected. The monotonic models produce a nearly uniform efficiency above a few ps at the expense of a few percent lifetime-integrated efficiency. Such a trade off is desirable as explained in the text. undesirable because in many cases studying the longest-lived heavy-flavor particles is of more interest than simply collecting the largest decay sample integrated over lifetime (see, e.g., [12]). Furthermore, many proposed explanations of dark matter and other types of new physics predict the existence of new particles with similar properties to heavy-flavor particles, but with longer lifetimes [13,14]. This classifier would reject these particles because it is unaware of our inductive bias that highly displaced DVs are worth selecting in the trigger and studying in more detail later. Since the LHCb community is generally interested in studying highly displaced DVs for many physics reasons, we want to ensure that a larger displacement corresponds to a more signal-like response. The same goes for DVs with higher p T . Enforcing a monotonic response in both features is thus a desirable property, especially because it also ensures the desired behaviour for data points that are outside the boundaries of the training data. Multiple methods to enforce monotonic behavior in BDTs already exist [15], and Figs. 4 (middle) and 5 (middle) show that this approach works here. However, the jagged decision boundary can cause problems, e.g., when measuring the heavy-flavor p T spectrum. Specifically, the jagged BDT decision boundary can lead to sharp changes in the selection efficiency. If there is not perfect alignment of where these changes occur with where the interval boundaries of the spectrum are defined, then correcting for the efficiency can be challenging. Figure 4 (right) shows that our novel approach, outlined in Sec. 2, successfully produces a smooth and monotonic response, and Fig. 5 (right) shows that this provides the monotonic lifetime dependence we wanted in the efficiency. Not only does our architecture guarantee a monotonic response in whatever features the analyst wants, it is guaranteed to be robust with respect to small changes to the inputs as governed by the constrained Lipschitz constant. Because calibration and resolution effects play a role in obtaining the features during detector operation, robustness is a necessary requirement for any classification performed online. Downstream analyses of these data depend on their stability. Figure 6 shows that the cost in terms of signal efficiency loss of enforcing monotonicity and robustness is small, even under the unrealistic assumption that the training data were, in fact, perfect. Therefore, the actual cost is likely negligible, while the benefits of the guarantees provided is hard to quantify but immediately obvious to the LHCb collaboration. Our algorithm runs in the LHCb trigger software stack and has been chosen to replace Refs. [16,17] as the primary trigger-selection algorithm used by LHCb in Run 3. Due to its guaranteed robustness-and excellent expressiveness even for small networksthis architecture is being explored for other uses within the LHCb trigger system for Run 3, since robustness and monotonicity are ubiquitous inductive biases in experimental particle physics. Experiment details The default LHCb model shown here is a 4-input, 3-layer (width 20) network with GroupSort activation (here, all outputs are sorted), λ = 2, constrained using Eq. (2.8). Inference times in the fully GPU-based LHCb trigger application [18] are 4 times faster than the Run 3 trigger BDT that was the baseline algorithm before ours was chosen to replace it (the BDT baseline was based on the model used during data taking in Run 2 [16,17]). We performed O(1000) runs with different seeds but the differences were negligible, at the level of O(0.1%). For the unconstrained network, we use the same architecture but without the linear term and without the weight constraints during training. The depth and width are the same as used for the monotonic Lipschitz network. The BDT is a LightGBM [19] gradient boosted classifier with 1000 base trees and a maximum 25 leaves per tree. Monotonicity is enforced there via the built-in monotone_constraints keyword. Code for the monotonic network implementation of the architecture developed here can be found at https://github.com/niklasnolte/MonotoneNorm. Limitations and Potential Improvements We are working on improving the architecture as follows. First, common initialization techniques are not optimal for weight-normed networks [20]. Simple modifications to weight initialization could likely improve convergence significantly. Second, it appears from empirical investigation that the networks described in Eq. (2.6) with GroupSort activation could be universal approximators, as we have yet to find a function that could not be approximated well enough with a deep enough network. A proof for universality is still required and could be developed in the future. Note that universal approximation is indeed proven for a similar architecture that only differs slightly in the normalization scheme, see [10]. Neither of these limitations has any practical impact on the example applications discussed in the previous sections. In many scenarios, Lipschitz-constrained architectures are considered inferior to unconstrained architectures because of their inability to offer competitive performance on standard benchmarks. This low performance is partly due to the fact that standard losses (such as cross-entropy) are not an adequate proxy of the metric of interest (accuracy) for the Lipschitz-constrained models. At a fundamental level, for any maximally accurate unconstrained classifier f (x) with Lipschitz constant λ, there exists a Lipschitz 1 classifier that replicates the former's decision boundary, namely, f (x)/λ. In the following, we will demonstrate a basic toy setting in which a maximally accurate Lipschitz classifier exists but cannot be obtained using standard losses. To understand the effect of the choice of objective function, we train a Lipschitz-constrained model to separate the two-moons dataset as shown in Fig. 7. This example is special in that the two samples do not overlap and can be completely separated by a Lipschitz-bounded function; however, that function cannot return the true label values for any data points due to the Lipschitz bound. Therefore, a loss function that penalizes any difference of the model output to the true label now faces a misalignment of the optimization target and the actual goal: While the classification goal is to have high accuracy, i.e. correct output sign, the optimization target is to minimize deviations of the output from the true label. This misalignment becomes irrelevant for a function with unbounded Lipschitz constant. We will show below that for examples such as this there is an important dependence on the objective used and its hyperparameters. First, we note that losses with exponential tails (in the sense that they require large weights to reach zero) are in general not suitable for maximizing accuracy. In practice, this can be remedied in cross-entropy by increasing the temperature. Note that cross-entropy with temperature τ is defined as L BCE τ (y,ŷ) = L BCE (y, τŷ),(5.1) where L BCE (y,ŷ) is the usual binary cross-entropy loss on targets y and predictionsŷ. Following Py-Torch conventions,ŷ are logits which will be normalized before computing the negative log-likelihood. An accurate classification boundary comes at the expense of reduced margins when classes have small separation. A maximally robust accurate classifier will, however, have optimal margins if trained using the appropriate objective. In the case of separable data (i.e. when classes have disjoint support), the maximally robust accurate Lipschitz classifier is the signed-distance function (SDF) [11] defined, in the binary case as Figure 7: A Lipschitz network trained to classify the Two-Moons dataset using different objects. Ordered from left to right and from top to bottom: Mean Squared Error, Binary Cross Entropy, Binary Cross Entropy with high temperature (τ = 16), Hinge loss with margin 1, Hinge loss with margin 0.1, and Hinge with dynamic margin. The network is evaluated on a uniform grid and its output is shown as a heatmap. The average absolute prediction (mean margin) on the validation set is also shown. SDF(x) = sign d(C +1 , x) − d(C −1 , x) · d(B, x),(5. where C +1 and C −1 are the sets of points for which x has label +1 and −1, respectively, and B is the boundary between classes defined as B ≡ x d(C +1 , x) = d(C −1 , x) . For a closed set S, the distance to x is defined as d(S, x) = min y∈S d(y, x). A naive objective minimized by the SDF is the hinge loss with margin given by d(B, x). Because we do not have access to the true decision boundary a priori, as a proxy, we use the following objective: While this objective produces the highest margins for an accurate classifier, as depicted in Fig. 7, it may encounter scalability issues when applied to higher-dimensional problems due to the unavoidable spareness of the training data. There are many possible alternative approaches that could resolve this issue, though this remains an open problem. For lower-dimensional problems with overlapping datasets-as studied in the various examples above and the most common scenario in scientific applications-this non-optimal loss issue does not appear to be relevant. Another factor that restricts the perceived expressiveness of Lipschitz architectures is the lack of access to standard techniques that improve convergence in unconstrained networks. For example, batch norm cannot be directly used with Lipschitz architectures. If the variance is too small, it may exceed the Lipschitz bound, and if it is too large, it can reduce the effective Lipschitz constant substantially. Summary & Discussion The Lipschitz constant of the map between the input and output space represented by a neural network is a natural metric for assessing the robustness of the model. We developed a new method to constrain the Lipschitz constant of dense deep learning models that can also be generalized to other architectures. Our method relies on a simple weight normalization scheme during training that ensures the Lipschitz constant of every layer is below an upper limit specified by the analyst. A simple monotonic residual connection can then be used to make the model monotonic in any subset of its inputs, which is useful in scenarios where domain knowledge dictates such dependence. Our implementation of Lipschitz constrained networks is minimally constraining compared to other weight-normed models. This allows the underlying architecture to be more expressive and easier to train while maintaining explicit robustness guarantees. We showed how the algorithm was used to train a powerful, robust, and interpretable discriminator for heavy-flavor decays in the LHCb trigger system. Furthermore, thanks to the expressive capacity of the architecture, we were able to shrink the number of model parameters to meet the memory and latency requirements of the LHCb trigger, which allows for faster event selection. This translates to higher sensitivity to the elusive physics phenomena we aim to observe. Our algorithm has been adopted for use as the primary data-selection algorithm in the LHCb trigger in the current LHC data-taking period known as Run 3. Given that the desire for robustness and interpretability and benevolent out-of-distribution behavior is ubiquitous when performing experiments, we expect that our architecture could have wideranging applications in science. In addition, our architecture could also be used in various applications in which robustness is required such as safety-critical environments and those which need protection against adversarial attacks. Monotonicity is a desirable property in various applications where fairness and safety are a concern. There are many scenarios in which models which are not monotonic are unacceptable. For example, in Ref. [21] we showed that our algorithm achieves state-of-the-art performance on benchmarks in medicine, finance, and other applications with monotonic inductive biases. In addition, in Ref. [22] we presented a new and interesting direction for the architecture developed here: Estimation of the Wasserstein metric (Earth Mover's Distance) in optimal transport by employing the Kantorovich-Rubinstein duality to enable its use in geometric fitting applications. Therefore, we expect that our algorithm will have far-reaching impact well beyond experimental physics. Figure 1 : 1Training robust and unconstrained models using a realization of the toy model in Eq. (3.1). Figure 2 : 2Training monotonic and unconstrained models using four realizations (purple data points) of the toy model in Eq. (3.2). The shaded regions represent the (top) extrapolation or (bottom) interpolation regions of interest, where training data are absent. Each panel presents a different realization of the Gaussian noise. Figure 3 : 3Regression example to emulate a complex decision boundary in two dimensions. The training data points are shown in blue (the inner radius is labeled 0, the middle is labeled 0.5, and the outer radius is labeled 1), while the output of the network is shown in color. The contour lines of the network output are shown in purple and red for the values of 1.0 and 0.5, respectively, which properly trace out the curves populated by the outer and middle sets of data points.corresponds to the highest possible momentum, DVs involving fake particles can have large p T . Figure 4 :Figure 5 : 45Simplified version of the LHCb inclusive heavy-flavor trigger problem using only 2 inputs, which permits displaying the response everywhere in the feature space; shown here as a heat map with more signal-like (background-like) regions colored blue (red). The dark solid line shows the decision boundary predicted to give the required output bandwidth in Run 3. Efficiency of each model shown in Figure 6 : 6Performance as quantified by the difference in signal efficiency (true positive rate) relative to the unconstrained NN at the expected Run 3 working point for the (left) 24 beauty and (right) 17 charm decays currently being used to benchmark this trigger. Each colored data point shows the change in efficiency for a given decay, while the shaded bands represent the local density of points. The white points show the median values for each set of points. L DynamicHinge (y,ŷ, x) = L Hinge δ(x|y) (y,ŷ), (5.3) where δ(x|y) = d(C −y ,x) 2 and L Hinge is defined as L Hinge m (y,ŷ) = max (0, m − yŷ) . (5.4) ). While rejecting DVs with the largest displacements does maximize the integrated signal efficiency in the training sample, this islog (min[χ 2 IP ]) p T [GeV] unconstrained NN AUC=0.94 monotonic BDT AUC=0.93 monotonic Lipschitz NN AUC=0.93 AcknowledgementThe authors would like to thank the LHCb computing and simulation teams for their support and for producing the simulated LHCb samples used to benchmark the performance of RTA software. This work was supported by NSF grants PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi.org/) and OAC-2004645. The LHCb trigger and its performance in 2011. R Aaij, 10.1088/1748-0221/8/04/P04022P04022 LHCb-DP-2012-004JINST. 81211.3055R. Aaij et al., The LHCb trigger and its performance in 2011, JINST 8 (2013) P04022 LHCb-DP-2012-004, [1211.3055]. Performance of the LHCb trigger and full real-time reconstruction in Run 2 of the LHC. R Aaij, 10.1088/1748-0221/14/04/P040131812.10790JINST. 144013R. 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Ma et al., Lightgbm: A highly efficient gradient boosting decision tree, in Advances in Neural Information Processing Systems 30 (NIP 2017), December, 2017, https://www.microsoft.com/en-us/research/publication/lightgbm-a-highly-efficient-gradient- boosting-decision-tree/. How to initialize your network? robust initialization for weightnorm & resnets. D Arpit, V Campos, Y Bengio, D. Arpit, V. Campos and Y. Bengio, How to initialize your network? robust initialization for weightnorm & resnets, 2019. Expressive Monotonic Neural Networks. O Kitouni, N Nolte, M Williams, International Conference on Learning Representations. 2023ICLR 2023O. Kitouni, N. Nolte and M. Williams, Expressive Monotonic Neural Networks, in International Conference on Learning Representations (ICLR 2023), 2023. Geometric Fitting using Neural Estimation of the Energy Mover's Distance. O Kitouni, N Nolte, M Williams, Finding Neemo, Advances in Neural Information Processing Systems. Machine Learning and the Physical Sciences. 2022 [2209.15624O. Kitouni, N. Nolte and M. Williams, Finding NEEMo: Geometric Fitting using Neural Estimation of the Energy Mover's Distance, in Advances in Neural Information Processing Systems (NeurIPS 2022), Machine Learning and the Physical Sciences, 2022 [2209.15624].
[ "https://github.com/niklasnolte/MonotoneNorm." ]
[ "Fighting Fire with Fire: Contrastive Debiasing without Bias-free Data via Generative Bias-transformation", "Fighting Fire with Fire: Contrastive Debiasing without Bias-free Data via Generative Bias-transformation" ]
[ "Yeonsung Jung [email protected] ", "Hajin Shim ", "June Yong Yang ", "Eunho Yang [email protected] ", "Kaist ", "Aitrics ", "South Korea " ]
[]
[]
Despite their remarkable ability to generalize with overcapacity networks, deep neural networks often learn to abuse spurious biases in the data instead of using the actual task-related information. Since such shortcuts are only effective within the collected dataset, the resulting biased model underperforms on real-world inputs, or cause unintended social repercussions such as gender discrimination. To counteract the influence of bias, existing methods either exploit auxiliary information which is rarely obtainable in practice, or sift for bias-free samples in the training data, hoping for the sufficient existence of clean samples. However, such presumptions about the data are not always guaranteed. In this paper, we propose Contrastive Debiasing via Generative Bias-transformation (CDvG) which is capable of operating in more general environments where existing methods break down due to unmet presumptions such as insufficient bias-free samples. Motivated by our observation that not only discriminative models, as previously known, but also generative models tend to focus on the bias when possible, CDvG uses a translation model to transform the bias in the sample to another mode of bias while preserving task-relevant information. Through contrastive learning, we set transformed biased views against another, learning biasinvariant representations. Experimental results on synthetic and real-world datasets demonstrate that our framework outperforms the current state-of-the-arts, and effectively prevents the models from being biased even when bias-free samples are extremely scarce.
null
[ "https://arxiv.org/pdf/2112.01021v1.pdf" ]
244,798,580
2112.01021
ba92bdebc9488cadf03d66ea66c7032a8b7473d8
Fighting Fire with Fire: Contrastive Debiasing without Bias-free Data via Generative Bias-transformation Yeonsung Jung [email protected] Hajin Shim June Yong Yang Eunho Yang [email protected] Kaist Aitrics South Korea Fighting Fire with Fire: Contrastive Debiasing without Bias-free Data via Generative Bias-transformation Despite their remarkable ability to generalize with overcapacity networks, deep neural networks often learn to abuse spurious biases in the data instead of using the actual task-related information. Since such shortcuts are only effective within the collected dataset, the resulting biased model underperforms on real-world inputs, or cause unintended social repercussions such as gender discrimination. To counteract the influence of bias, existing methods either exploit auxiliary information which is rarely obtainable in practice, or sift for bias-free samples in the training data, hoping for the sufficient existence of clean samples. However, such presumptions about the data are not always guaranteed. In this paper, we propose Contrastive Debiasing via Generative Bias-transformation (CDvG) which is capable of operating in more general environments where existing methods break down due to unmet presumptions such as insufficient bias-free samples. Motivated by our observation that not only discriminative models, as previously known, but also generative models tend to focus on the bias when possible, CDvG uses a translation model to transform the bias in the sample to another mode of bias while preserving task-relevant information. Through contrastive learning, we set transformed biased views against another, learning biasinvariant representations. Experimental results on synthetic and real-world datasets demonstrate that our framework outperforms the current state-of-the-arts, and effectively prevents the models from being biased even when bias-free samples are extremely scarce. Introduction Recent advances in deep learning have showcased that deep neural networks (DNN) are capable of reaching stateof-the-art performance in various fields of machine learning, such as computer vision [12], natural language processing [4], reinforcement learning [28] and more. However, it * Equal contribution. is also known that the over-parametrized nature of DNNs not only exposes them to general overfitting, but also renders them susceptible to spurious biases present in collected datasets [31] which are detrimental to the generalization performance. In supervised learning, neural networks tend to prefer learning shortcut solutions based on biases rather than learning to classify with real signal [25,37]. Since spurious biases do not provide task-related information, DNNs that learned to predict using these biases will ultimately fail on future data. For instance, a classifier trained to identify car racing images using a dataset dominated by track racing images will exploit the track road information for classification. However, the classifier will fail to exhibit the same performance on images of off-road rallies. Unfortunately, this form of model overfitting cannot be screened effectively by standard validation processes since such biases are introduced in the dataset curation phase and are most likely shared across the training and validation sets. To this end, debiased learning -effective learning in the presence of bias -is imperative in utilizing DNNs for real-world applications outside the laboratory. A tautological solution to the bias problem is to construct a bias-free dataset from the start. However, curating a dataset devoid of all bias is extremely costly at best, and generally infeasible. A more practical attempt at neutralizing dataset bias is to fortify a dataset with explicit supervision signals with regards to the bias [18,30]. However, additional expenditure of human labor in procuring such information cannot be avoided, which renders the option less appealing. In most cases where such explicit supervision for bias is absent, the following two lines of works are recently proposed. One line of works has focused on mitigating the influence of bias by leveraging the bias type (e.g. texture) [2,10,15,33] to design bias-oriented auxiliary models or to augment texture-perturbed samples. However, the acquisition of such prior knowledge on bias is by no means guaranteed, and even when presented with such information, designing bias-oriented architectures is not always straightforward. Acknowledging the shortcomings of this direction, another line of works leverages an empirical prior knowledge that malignant biases are usually learned faster than task-relevant features [20,21,26,29] to capture bias attributes. However, these methods tend to break down in regimes where bias-free samples are extremely scarce or nonexistent. Thus, they are restricted to operate under the presumption that a sufficient number of bias-free samples exist in the dataset and can be recognized. Since dataset bias may come in many shapes and sizes, the success of these methods cannot be ensured. To tackle these shortcomings, we develop a general debiased learning method capable of operating even in regimes devoid of bias-free data, thus not reliant on any presumptions regarding the data. In this paper, we propose Contrastive Debiasing via Generative Biastransformation (CDvG), which contrasts the bias modes within the dataset against each other to attenuate the influence of bias while effectively learning the task-relevant information contained in all samples. Through preliminary experiments, we find that generative models -specifically image-to-image translation models -favor learning nonessential biases over task-relevant signals, as discriminative models are known to do so (See Section 4.1). Motivated by this observation, we train a biased generative model in an unsupervised manner that learns the bias distribution over the signal, obtaining capabilities to transform the bias feature of a given input to another bias. Using the learned generator, we synthesize novel views with altered bias features. Fighting fire with fire, we pit one bias type against another via contrastive learning. By maximizing agreement between the views with different biases, the model is encouraged to learn bias-invariant representations. Unlike existing methods, CDvG does not require explicit supervision, domain knowledge, or other meta-knowledge -the existence of biasfree samples. Our contributions are three-fold: • We experimentally discover that certain generative models are also prone to consider conspicuous but incidental information such as background or texture first rather than task-related information, as discriminative models do. • We propose Contrastive Debiasing via Generative Bias-transformation (CDvG) with generative view transformation that transforms bias factors of input images. CDvG does not presume the existence of bias labels, bias type information or even the existence of bias-free samples. • With experiments on both synthetic and real-world biased datasets, we demonstrate that our method outperforms others in a variety of bias situations. We show that our method is particularly effective com-pared to existing methods when bias-free samples are extremely scarce. Related Works In real-world datasets across diverse domains, there exist various kinds of incidental biases strongly correlated with the task-related information but essentially irrelevant. However, when the bias factors are more noticeable and easier to learn than the task-related signals, DNNs tend to lean on such biases [29], causing failures in generalization. To counter this effect, several lines of works were developed. First, explicit bias supervision can be used to screen or mitigate the influence of bias. Kim et al. [18] uses known explicit bias supervision to train an auxiliary network that helps in reducing the influence of bias. Sagawa et al. [30] use the bias supervision to group data samples for grouped distributionally robust optimization (DRO). Kim et al. [19] address the problem by minimizing the mutual information not to learn biased instances by using bias supervision. Recently, Hong et al. [15] propose bias-contrastive loss and bias-balanced regression that encourages the model to pull together the samples in the same class with different bias features, with balancing target-bias distribution. When acquiring bias supervision is impractical, we can leverage domain-specific knowledge about bias. For example, it was shown that ImageNet-trained classifiers exploit texture information in the image rather than information contained in the object of interest [3,9]. Utilizing this fact, Geirhos et al. [10] construct an augmented dataset by applying various textures to the original images for texture debiasing. On the other hand, the following works exploit a separate auxiliary model in which bias is injected using prior knowledge such as the type of the bias. Wang et al. [33] trains a texture-debiased classifier by projecting classifier representations to a subspace orthogonal to the previously learned bias space. Bahng et al. [2] lead the model representation to be statistically independent to the biased one produced from the biased auxiliary model using Hilbert-Schmidt independence criterion. Hong et al. [15] further propose a soft version of bias-contrastive loss for the case where the type of bias rather than the label of the bias is available. However, these methods are grounded on obtaining such knowledge -which is often costly, or even impossible. In addition, designing a bias-oriented auxiliary model may not be so intuitive depending on the type of bias. Without the domain knowledge, Li et al. [26] introduced an alternating minimization scheme between bias identification and sample reweighting to combat bias without any side information. Using the fact that biases are usually learned faster than salient task-relevant features, Nam et al. [29] train an auxiliary model slightly ahead in terms of training iterations using generalized cross entropy [35] to absorb fast-learned bias, and assign more weight to the biasfree samples for the debiased follower. Kim et al. [21] proposed feature-level data augmentation that disentangles bias features by using a biased auxiliary model obtained by following [29] and swaps latent bias features within the minibatch. Another augmentation-based debiasing method, Bi-aSwap [20], employs task-related features in abundant biasaligned samples by synthesizing a new image that takes the bias-irrelevant core features from the biased sample and the bias attribute from the bias-free sample. However, all these methods presume that bias-free samples do exist in sufficient quantities and can be distinguished, which cannot always be guaranteed. Learning without Bias-free Samples In this section, we first define our setup: learning without bias-free samples, then analyze the behavior of existing methods under such environments as preliminary. To formally define our target task, we introduce the following random variables: input image X, target label Y , signal S, and bias B comprising input X = {S, B}. Here, B is a feature unrelated to Y , that is, Y and B are independent given S. We further define the random variable for bias label Y B which is hidden. Ideally, the image classification model parameterized by θ predicts the label based on signal as P θ (Y |X) = P θ (Y |B, S) = P θ (Y |S). However, when the training data consists of highly correlated bias and target, i.e., H(Y tr |B tr ) ≈ 0, predicting Y based on B is also one of the possible solutions that can be deemed effective in the training phase. When B is easier to be perceived than Table 1. Comparison of debiasing methods on three biased datasets devoid of bias-free samples (ratio 0%). We report averaged accuracy on the last epoch and the standard deviation over 3 runs. Method ColoredMNIST CorruptedCIFAR10 BFFHQ (ratio (%)) 0 S, the model preferentially takes B as a clue to predict Y . However, this is obviously an unintended consequence and impairs generalizability due to the discrepancy of the biastarget correlation Y |B between the training and test phases. Although recent state-of-the-art debiasing methods [2,15,20,22,29] work as intended when they can appropriately uncover the existing bias-free samples, they neglect the case where such bias-free samples are absent (or extremely scarce) from the beginning. In other words, This implies that these methods function under a presumption that enough number of bias-free samples exist within the dataset. Thus, in regimes where the mentioned presumption does not hold true, their behaviors are practically unknown, and are likely to break down. More specifically, in Table 1, we empirically demonstrate that previous methods fail to effectively de-bias the model (showing low classification scores, almost comparable to Vanilla) when a perfectly biased training set is given, i.e. the proportion of bias-free samples is 0%. We further analyse the reweighting scheme [22,29] on CorruptedCI- . Changes in the ratio of bias-free weight (blue bar) and accuracy (red line) of DisEnt [22] as the ratio of bias-free samples decreases. The green and black horizontal line are the accuracy of ours (GDvG) and Vanilla respectively on perfectly biased data. FAR10 in Figure 3. We evaluate whether the method assigns high weight to the bias-free samples, as the available bias-free samples are reduced. To this end, we measure the ratio of the sum of bias-free weights to the total sum of weights for all samples (Ratio of Bias-free Weight in Figure 3) which means how much the bias-free samples are focused. As the bias-free samples are more scarce, the accuracy decreases as the ratio of bias-free weight decreases. The ratio is controlled by gradually removing bias-free samples of the datasets with 0.5% ratio. Meanwhile, [15,20] require bias-free samples necessarily to construct pairs of the bias-aligned and bias-free samples in the same class. Based on this observation, we consider a more general task of learning-under-bias that handles the situations where there are no to only a few bias-free samples as well as the cases where such samples are sufficiently provided. Method In this section, we propose CDvG to solve the bias problem defined in Section 3 where there is no assumption of the existence of bias-free samples nor any other prior knowledge about the bias. Section 4.1 presents our empirical findings that are key for this new approach, and Section 4.2 describes CDvG in detail. Bias Is Generated First An intuitive solution to address the bias problem is to obtain sufficient bias-free samples. One possible approach toward this goal is to exploit a generative model to generate a sufficient amount of bias-free samples. Since discriminative models are known to be susceptible to bias, it is not farfetched to first enquire whether generative models also carry the same frailty. For instance, one might intuitively suspect that the family of Generative Adversarial Networks (GAN) are susceptible to bias, as their composition by nature involves a discriminator, which is bias-pregnable. To this end, we investigate the case of a particular member of the GAN family, namely, image translation models. Image-to-image translation methods [8,16,36] render an image x from a source domain y to a target domain y . Ideally, they find out the representative characteristics s of the target domain y and combine them with the input image. However, when the dataset is biased, the generator of the translation model interacting with the biased discriminator is also prone to learn to translate biases rather than core domain features. For example, CycleGAN [36], a representative milestone of image-to-image translation, exhibit a number of typical failure modes that when the source domain is an apple and the target domain is an orange, a transformed image is not the orange counterpart of the input, but an apple with the color and texture of an orange. This implies that despite not using a biased dataset, the model perceives color and texture rather than shape as the representative traits for the target domain. With this in consideration, it is plausible to speculate that this phenomenon would be exacerbated when handling highly biased datasets. To verify this, we examine the behavior of StarGAN On the biased dataset, the domain classifier D cls , which has absorbed bias attributes from the data, transfers its knowledge to G. As a result, G captures biases as representative traits. To quantitatively evaluate whether the translation model truly favor learning biases over task-relevant signals, we measure the classification loss of generated images x = G(x, y ) with the bias classifier C B and the signal classifier C Y while training StarGAN on the biased dataset Colored MNIST (Figure 1a) and BFFHQ ( Figure 1d) which have color biases and gender biases, respectively. As alternatives of oracles, the classifiers C B and C Y are trained with the bias label y B (color and gender) and the true class label y (digits and age) of the given image x respectively, on the unbiased Colored MNIST and FFHQ which means there are no correlation of colordigits and gender-age respectively. The bias and signal loss are defined as E (x,y)∼D,y ∼Cat(|Y|) [L CE (C B (x ), y )] and E (x,y)∼D,y ∼Cat(|Y|) [L CE (C Y (x ), y )] respectively. As expected, Figure 4 shows that the bias is more favorable than the signal to translation model in both synthetic and real-world datasets. For Colored MNIST, the signal loss starts quite low due to the simplicity of the dataset resulting in easily generating the average shape of each digit class y without considering x. However, as x is recognized, the signal loss increases rapidly since G concentrates on the color biases rather than digits. For BFFHQ, although both signal and bias classification losses decrease, we speculate that this is because gender bias is not particularly easier to learn than the signal. Nevertheless, G still focuses more on the bias than the signal. Next, we also present the qualitative results in Figure 1. The leftmost images of each subfigure are the original x for each dataset and each column is translated results x = G(x, y ) of each class y . Interestingly, we found that the translated results retain their contents to some extent and color, texture, background (e.g. rock wall for climbing, water for fishing), and gender characteristics (e.g. makeup for female, beard for male), which are the respective bias features of the datasets, are altered. BFFHQ is curated to have Algorithm 1: Contrastive Debiasing via Generative Bias-transformation (CDvG) Input: Encoder E, Projection head H, Classifier C, Biased generator G, Augmentation family T Data: Training set D = {(x, y)} ⊂ X × Y 1 for minibatch {(x k , y k )} N k=1 do 2 # Generate bias-transformed view 3 y k ∼ Categorical(|Y|) 4 x k = G(x k , y k ) 5 6 # Augmentation operator 7 t 1 ∼ T , t 2 ∼ T 8x k ,x k = t 1 (x k ), t 2 (x k ) 9 10 Update E, C, H to minimize k (L CE (C(E(x k )), y k ) 11 +L CE (C(E(x k )), y k )) 12 +λ * L CL (E, H) 13 end for age-gender as highly correlated signal-bias, i.e, young female and old male person images. Although generated images of BFFHQ in Figure 1d look quite different from the real images, the core contents are still discernible while the gender-specific features emerge as pink lips and eye makeup in the second column and thick eyebrow and beard for the third column. Contrastive Debiasing without Bias-free Data via Bias-transformed Views Motivated by the findings of Section 4.1, we propose to use the biased translation model to transform an image to have different biases of other classes and integrate it with the contrastive learning framework. By comparing the diverse biased views generated with the transformation function, the contrastive loss encourages the encoder to learn representations invariant to bias transformations without distinguishing any bias-free data. Contrastive learning is a self-supervised learning method proven to learn representations substantially beneficial to numerous downstream tasks, achieving state-of-the-art performance [5][6][7]11]. A representative work [5] defines the contrastive prediction task in two steps: a) augmenting two views from the same image by strong data augmentations [1,13,24] and b) maximizing agreement between the augmented views on the latent space by employing the contrastive loss [32] to capture core features of the image. In the process of maximizing the agreement between the augmented views, the encoder discards the deviating features between the views and learns transformation-invariant representations. For example, using color distortion methods such as color jitter and gray scale in the augmentation step encourages the encoder to learn color-invariant representations. From the observation in Section 4.1 that generator G learns biases as representative characteristics of the domain rather than task-related features, we adopt StarGAN, the most widespread image-to-image translation model, to learn the bias transformation. Note that other multi-class domain translation models can also be adopted. First, we train StarGAN on a biased dataset to acquire the bias-transformation generator G. It is worth noting that we do not need bias-free samples at hand in this process. With, G, we translate an input image x with a target label y to obtain a bias-transformed view x = G(x, y ), where y is uniformly sampled for every iteration to generate views with diverse biases. After the bias-transformation step, we additionally apply the random augmentation operators t 1 and t 2 to the original and the bias-transformed images respectively asx = t 1 (x), x = t 2 (x ), where t 1 and t 2 are sampled from the same augmentation family T . According to Chen et al. [5], strong augmentations such as color distortion and Gaussian blur are essential to the performance of the contrastive learning framework as they prevent the encoder from easily finding a solution that two views come from the same image. By following [5], T is composed of the following sequential augmentations: random resized cropping, random horizontal flipping and random color distortion. The resulting learning objective is given by the combination of the cross entropy loss and the contrastive loss. First, the encoder E and the following classifier C are optimized to minimize the cross entropy loss L CE which is applied to bothx andx : min E,C L CE (C(E(x)), y) + L CE (C(E(x )), y). Also, we train the encoder E and the projection head H to minimize the contrastive loss L CL to tie the original view x and the bias-transformed viewx as follows: L CL (E, H) = N k−1 (2k − 1, 2k) + (2k, 2k − 1). The loss (i, j) for a positive pair (i, j) is defined as (i, j) = − log exp(sim i,j /τ ) 2N k=1 1 k =i exp(sim i,k /τ ) , Where sim i,j = z i z j /(||z i ||||z j ||) is the cosine similarity, and z i = H(E(x i )) is a projected representation ofx i with the base encoder E followed by the projection head H. By tying the original viewx and the bias-transformed viewx as a positive pair, the encoder attempts to attenuate biases while emphasizing the true signals shared by the views via maximizing the mutual information between their latent representations. Experiments To validate the effectiveness of our method compared to recent debiased learning methods, we conduct image classification experiments on standard benchmark datasets for debiased learning. We first present the settings for our experiment including datasets and baselines (Section 5.1) then report the comparisons of our method and baselines regarding their debiasing performance (Section 5.2). In addition, we perform ablation studies to demonstrate that each component of our method contributes to respective performance improvements (Section 5.3). Experimental settings Dataset We experiment on two types of datasets: Synthetic datasets injected with synthetic biases and real-world datasets with natural biases. Colored MNIST is a biased version of MNIST with colors as biases. Corrupted CIFAR-10 is an artificially corrupted version of CIFAR-10 [23] to carry biases as proposed in [14]. Specifically, the dataset has been corrupted by the following types of method: {Snow, Frost, Fog, Brightness, Contrast, Spatter, Elastic transform, JPEG, Pixelate and Sat-urate}. By setting the proportion of bias-free samples to {0%, 0.5%, 1%, 2%, 5%}, we evaluate the performance considering the highly biased setting and, moreover, the hardest case where bias-free samples are absent. The datasets with 0% bias-free samples are constructed by excluding bias-free samples from the datasets with the ratio of 0.5%. Biased Action Recognition (BAR) is a real-world action dataset proposed by [29]. There are six action labels that are biased to background places. Another real-world data, BFFHQ, is proposed by [20] which is curated from Flickr-Faces-HQ [17]. It consists of face images where an age (young/old) as a task label and a gender (male/female) as bias attribute. Setting For the generator G which transforms biased input image to have an another bias, we use the StarGAN [8] basic architecture which is composed of total 3 blocks and 6 blocks for Colored MNIST and the others, respectively. Each block consists of 2 convolutional layers and skip connection. For the encoder E, we use MLP with three hidden layers for Colored MNIST, randomly initialized ResNet-18 [12] for Corrupted CIFAR-10 and BFFHQ and use pretrained ResNet-18 1 for BAR datasets following the convention. Also, We use the Adam optimizer with default parameters. We train the model for 90, 200 and 500 epochs for BAR, {Colored MNIST, BFFHQ} and Corrupted CIFAR-10, respectively. We defer the further details to Supplementary. Baselines To benchmark the performance of our method, we compare our method with diverse categories of debiased learning methods. We compared with HEX [33] and Rebias [2] which leaveraging the domain knowledge about bias and LfF [29] which deliberately trains a auxiliary biased classifier to distinguish bias-free samples and utilize. Also, DisEnt [22] and BiaSwap [20], which are based on the bias-free sample selection scheme of [29], try to combine the other biases with the instance on feature space and image space, respectively. For HEX and BiaSwap, we borrow the available performance directly from the papers and remain as blank for the others because the official code of BiaSwap is unavailable and HEX is a replaceable with Rebias which is similar but more advanced. Evaluation Due to the nature of debiasing without bias labels, unbiased validation set cannot be obtained. Therefore, recent works tend to choose the best test accuracy for evaluation. However, when the model fluctuates, a high value is likely to achieve which is not the true performance. Therefore, we not only report the best test accuracy but also compare the last epoch test accuracy with state-of-the-art works for the accurate evaluation. We report the best test and last epoch averaged accuracy and the standard deviation over three runs in Table 2 and Table 3, respectively. In addition, we evaluate the performance when bias-free samples do not exist (ratio 0%) which is likely to occur in real-world settings but not considered in recent studies. Main results We report the results of baselines and our method on the standard benchmark datasets including synthetic and realworld datasets to validate the effectiveness of our method for debiased learning when bias label is not available. Synthetic dataset In Table 2, we report the best test performances on Colored MNIST which has color biases, and Corrupted CIFAR-10 which is artificially corrupted for evaluating debiasing procedures. We observe that the proposed method significantly outperforms the baselines on both datasets' overall ratio, especially with a large margin when the bias-free samples are absent. When the bias-free samples are extremely scarce (0.5%), our method significantly improves the performance against the state-of-the-art by 11.63% and 14.74% in Colored MNIST and Corrupted CIFAR10, respectively. Especially, without bias-free samples, our method outperforms the baseline by a significant margin of 74.14% and 11.27% in each dataset, respectively. In relatively high ratio settings (5.0%, 2.0%), where recent state-of-the-art methods have aimed, our method also shows better performance. In Table 3, we report last test accuracy. Our method also shows better performance in all cases. Importantly, [22] shows remarkable performance degradation of 4.6%, 4.2% on average than test accuracy in each dataset, respectively. However, our method shows much smaller gap of 0.1%, 0.8% on average with the stable standard deviation. To sum up, our model can handle various types of synthetic bias and works well on a wide range of bias ratios, especially when bias-free samples are scarce and even absent, compared with state-of-the-art methods in both best test accuracy and last test accuracy. Real-world dataset To verify the effectiveness of our method in a realistic biased setting, we additionally conduct the experiments on BAR and BFFHQ dataset which has a background bias and a gender bias, respectively. In BFFHQ, the proposed method show improved performance of 64.80% and 50.73% with small standard deviations when bias-free samples are scarce and absent, respectively. Note that, since the bias of BFFHQ not particularly more noticeable than the signal as mentioned in Section 4.1, total debiasing without bias label is difficult but we address it to some extent. In BAR, our method improves the performance by 3.26% with the small standard deviation. we report BAR performance by last test accuracy only because baselines use pretrained ResNet-18 which show high performance in the early stage of training and slowly decreasing. In Table 3, we observe that, in real-world datasets, our method also show 2.7% of performance degradation which is smaller compared to 10.4% of state-of-the-art. Overall consistent results empirically demonstrate that our method debias a place bias and a gender bias which are difficult real-world biases better than previous approaches. Ablation study We study the effectiveness of each component of our method. In Table 4, our method without bias-transformation generator (CDvG w/o G) shows improvement. It mainly comes from some advantages of using contrastive learning in debiased learning. Firstly, according to Wang et al. [34], contrastive loss optimizes uniformity which makes feature vectors uniformly distributed to keep the distance from each other on the unit hypersphere. Therefore, in the case of the image classification on a biased dataset, it prevents instances of the same class from quickly gathering together on the latent space. Also, stochastic view augmentation contributes to the encoder being less susceptible to bias by its regularization effect. However, despite these advantages, contrastive learning would be still vulnerable to various biases as it leaves room for selecting biases as shortcut solutions when they remain in both views. Therefore, con-trastive learning by itself cannot be an answer for the biased problem. Also, by adopting our bias-transform generator only for augmentation (CDvG w/o CL), there is a further improvement. It shows that our generator successfully augments bias-transformed images which work well for the debiasing problem. Our whole framework shows an obvious improvement which means that since bias-transformed images are not perfect, it is essential to induce bias-invariance and capture the true signal shared by the bias-transformed view and original view by contrastive loss. Based on the ablation results, we confirm that our whole framework effectively works to address the bias problem not a single component as intended. Conclusion In this paper, we have proposed Contrastive Debiasing via Generative bias-transformation, a general debiased learning method that does not require presumptions about the data such as the existence of bias-free samples or any domain knowledge. Motivated by the observation that not only discriminative models but also generative models tend to focus on bias, we utilize a translation model as a biastransformation function to generate diverse biased views. With contrastive learning, we compare the biased views to obtain bias-invariant representations. Our experimental results show that our method outperforms the current state-ofthe-arts in both synthetic and real-world biases, especially when bias-free samples are scarce or even absent. Limitations Since the image translation model is a key component of our proposed method, general failure of this component will result in model malfunction. In this paper, the results and analyses are conducted on the regime where biased samples are the dominant majority. Thus, it is unclear how these methods will behave in the regime where bias-free data are the majority, i.e. when the data is undeceiving. However, such situations would not require debiased learning techniques from the start. B. Bias-transformed images by CycleGAN [36] As an example that other translation models also can be used in our framework, we show bias-transformed images generated by CycleGAN, which translates between two domains, trained on BAR dataset. The leftmost column of the figure contains the original images and each column is the transformed images of each target domain. The resulting images show the translated background bias attributes. Figure 1 . 1Bias-transformed images by our generative bias-transformation on synthetic and real-world biased datasets. The leftmost column of each subfigure contains the original images and each column is the transformed images of each target domain. The resulting images show the translated bias attributes such as color, texture, background, and gender characteristics respectively. Figure 2 . 2Illustration of our Contrastive Debiasing via Generative Bias-transformation (CDvG). Figure 3 3Figure 3. Changes in the ratio of bias-free weight (blue bar) and accuracy (red line) of DisEnt [22] as the ratio of bias-free samples decreases. The green and black horizontal line are the accuracy of ours (GDvG) and Vanilla respectively on perfectly biased data. [8] on biased datasets. StarGAN introduces a domain classifier D cls to enable translation between multiple domains. D cls , trained on the original images with their domain labels, induces the generator G to translate the image x to the target domain y as x = G(x, y ) by optimizing the domain clas-sification loss L cls : L cls (G, D cls ) = E (x,y)∼D,y ∼Y [− log D cls (y |G(x, y ))]. Figure 4 . 4Capturing a bias shortcut in StarGAN. Figure 5 . 5Bias-transformed images by CycleGAN on BAR dataset. Table 2 . 2The average and the standard deviation of best accuracy over 3 runs. Values with †, ‡ are directly from[22] and[20], respectively.Dataset Ratio(%) Vanilla HEX [33] ReBias [2] LfF [29] DisEnt [21] BiaSwap [20] CDvG Colored MNIST 0.0 18.12±1.18 - 14.64±0.50 22.06±3.82 18.43±1.19 - 96.20±0.09 0.5 39.80±1.44 30.33 † ±0.76 70.47 † ±1.84 69.04±2.88 68.26±4.75 85.76 ‡ 97.39±0.12 1.0 56.43±1.39 43.73 † ±5.50 87.40 † ±0.78 81.72±1.14 79.10±0.81 83.74 ‡ 97.38±0.03 2.0 69.57±1.99 56.85 † ±2.58 92.91 † ±0.15 84.97±0.68 83.18±0.79 85.29 ‡ 97.56±0.06 5.0 84.05±0.80 74.62 † ±3.20 96.96 † ±0.04 88.43±0.82 88.65±0.30 90.85 ‡ 97.59±0.10 Corrupted CIFAR-10 0.0 21.77±0.26 - 21.93±0.37 21.86±0.89 24.44±1.80 - 35.71±0.59 0.5 23.08 † ±1.25 13.87 † ±0.06 22.27 † ±0.41 28.57 † ±1.30 29.95 † ±0.71 29.11 ‡ 44.69±0.31 1.0 25.82 † ±0.33 14.81 † ±0.42 25.72 † ±0.20 33.07 † ±0.77 36.49 † ±1.79 32.54 ‡ 45.63±0.27 2.0 30.06 † ±0.71 15.20 † ±0.54 31.66 † ±0.43 39.91 † ±0.30 41.78 † ±2.29 35.25 ‡ 49.23±0.40 5.0 39.42 † ±0.64 16.04 † ±0.63 43.43 † ±0.41 50.27 † ±1.56 51.13 † ±1.28 41.62 ‡ 54.11±0.47 BFFHQ 0.0 46.27±0.74 - 46.47±0.77 49.47±1.59 49.34±1.20 - 50.73±1.18 0.5 56.87 † ±2.69 - 59.46 † ±0.64 62.20 † ±1.00 63.87 † ±0.31 - 64.80±0.33 Table 3 . 3Averaged accuracy and standard deviation on the last epoch, over 3 runs. 12.53±0.92 13.16±1.87 11.65±0.61 95.97±0.18 0.5 39.12±0.91 66.54±3.80 62.13±3.96 97.26±0.12 1.0 56.02±1.81 79.83±2.23 75.49±0.21 97.25±0.10 2.0 69.32±0.22 82.66±0.39 80.08±0.45 97.47±0.05 5.0 83.93±0.89 83.30±1.23 85.00±1.17 97.49±0.11Dataset Ratio (%) Vanilla LfF [29] DisEnt [21] CDvG Colored MNIST 0.0 Corrupted CIFAR-10 0.0 16.05±0.13 15.88±0.45 18.76±0.88 31.24±0.31 0.5 20.87±0.34 25.58±0.23 28.62±1.74 43.41±0.31 1.0 24.05±0.61 30.68±0.50 32.31±0.03 44.50±0.21 2.0 29.47±0.20 37.96±1.09 36.51±2.34 48.20±0.24 5.0 41.12±0.16 48.49±0.16 46.41±0.62 53.49±0.48 BFFHQ 0.0 37.93±0.96 39.67±1.00 38.13±2.13 42.80±0.33 0.5 52.40±1.88 58.07±0.82 54.33±0.92 59.40±1.34 BAR -66.39±0.70 63.68±0.49 64.84±1.98 68.10±0.42 Table 4 . 4Ablation study for bias-transformed view generation (w/o G) and for contrastive loss (w/o CL). 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In 32nd Conference on Neural Information Processing Systems (NeurIPS), 2018. 2 Unpaired image-to-image translation using cycleconsistent adversarial networks. Jun-Yan Zhu, Taesung Park, Phillip Isola, Alexei A Efros, abs/1703.10593CoRR413Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle- consistent adversarial networks. CoRR, abs/1703.10593, 2017. 4, 13 Object recognition with and without objects. Zhuotun Zhu, Lingxi Xie, Alan Yuille, Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI-17. the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI-17Zhuotun Zhu, Lingxi Xie, and Alan Yuille. Object recog- nition with and without objects. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial In- telligence, IJCAI-17, pages 3609-3615, 2017. 1 Implementation details Basically, we follow the same experimental settings in baselines. A , 22A. Implementation details Basically, we follow the same experimental settings in baselines [22, 29]. Datasets All the datasets except BAR can be obtained from the official Github repository of DisEnt. BAR is available in the BAR Github repository. 22Datasets All the datasets except BAR can be obtained from the official Github repository of DisEnt [22] (https://github.com/kakaoenterprise/Learning-Debiased-Disentangled). BAR is available in the BAR Github repository (https://github.com/alinlab/BAR) provided by [29]. stargan) across all datasets. For discriminator, we set the option 'd repeat num' (the number of strided convolutional layers in Discriminator) to 4 and 5 for {Colored MNIST, BAR, BFFHQ} and Corrupted CIFAR10 respectively. For Colored MNIST, we reduce the number of residual blocks of the generator to 3 and set the reconstruction weight to 500. StarGAN for Colored MNIST is trained for 5000 iterations without random horizontal flipping. Training StarGAN First we specify the details of training biased StarGAN. We basically follow the default settings for architectures, optimizers weights for loss terms and other training configurations in the Github repository of StarGAN. Except for BAR, we use the original image size. For BAR, we resize them to 224 with following random cropTraining StarGAN First we specify the details of training biased StarGAN. We basically follow the default settings for architectures, optimizers weights for loss terms and other training configurations in the Github repository of StarGAN (https://github.com/yunjey/stargan) across all datasets. For discriminator, we set the option 'd repeat num' (the number of strided convolutional layers in Discriminator) to 4 and 5 for {Colored MNIST, BAR, BFFHQ} and Corrupted CIFAR10 respectively. For Colored MNIST, we reduce the number of residual blocks of the generator to 3 and set the reconstruction weight to 500. StarGAN for Colored MNIST is trained for 5000 iterations without random horizontal flipping. Except for BAR, we use the original image size. For BAR, we resize them to 224 with following random crop. We use learning rates of 0.001, 0.0001 for {Colored MNIST, Corrupted CIFAR-10, BFFHQ} and BAR, respectively. Also, we use cosine annealing from initial learning rates lr to lr * 0.1 3 for learning rate scheduling [27] for all datasets. Note that, we do not use random horizontal flipping for ColoredMNIST. Additionally, the original image size of BFFHQ. Mnist {colored, Cifar-10} Corrupted, {bffhq Bar}, 256Training Configuration For training. is 128 but we resize them to 224 by following the previous work [22Training Configuration For training, we use batch sizes of 256, 64 for {Colored MNIST, Corrupted CIFAR-10} and {BFFHQ, BAR}, respectively. We use learning rates of 0.001, 0.0001 for {Colored MNIST, Corrupted CIFAR-10, BFFHQ} and BAR, respectively. Also, we use cosine annealing from initial learning rates lr to lr * 0.1 3 for learning rate scheduling [27] for all datasets. Note that, we do not use random horizontal flipping for ColoredMNIST. Additionally, the original image size of BFFHQ is 128 but we resize them to 224 by following the previous work [22]. Encoder For Colored MNIST, we use a 3-layer MLP encoder with a hidden dimension of 100 across all layers. Encoder For Colored MNIST, we use a 3-layer MLP encoder with a hidden dimension of 100 across all layers. Evaluation Note here that, by following [22], the performances on BFFHQ whose task is a binary classification are evaluated only on bias-free test samples that consist of young male and old female samples. Evaluation Note here that, by following [22], the performances on BFFHQ whose task is a binary classification are evalu- ated only on bias-free test samples that consist of young male and old female samples.
[ "https://github.com/kakaoenterprise/Learning-Debiased-Disentangled).", "https://github.com/alinlab/BAR)", "https://github.com/yunjey/stargan)" ]
[ "PatchRD: Detail-Preserving Shape Completion by Learning Patch Retrieval and Deformation", "PatchRD: Detail-Preserving Shape Completion by Learning Patch Retrieval and Deformation" ]
[ "Bo Sun \nAustin\n", "Vladimir G Kim \nAdobe Research\n\n", "Noam Aigerman \nAdobe Research\n\n", "Qixing Huang \nAustin\n", "Siddhartha Chaudhuri \nAdobe Research\n\n\nIIT Bombay\n\n" ]
[ "Austin", "Adobe Research\n", "Adobe Research\n", "Austin", "Adobe Research\n", "IIT Bombay\n" ]
[]
This paper introduces a data-driven shape completion approach that focuses on completing geometric details of missing regions of 3D shapes. We observe that existing generative methods lack the training data and representation capacity to synthesize plausible, fine-grained details with complex geometry and topology. Our key insight is to copy and deform patches from the partial input to complete missing regions. This enables us to preserve the style of local geometric features, even if it drastically differs from the training data. Our fully automatic approach proceeds in two stages. First, we learn to retrieve candidate patches from the input shape. Second, we select and deform some of the retrieved candidates to seamlessly blend them into the complete shape. This method combines the advantages of the two most common completion methods: similarity-based single-instance completion, and completion by learning a shape space. We leverage repeating patterns by retrieving patches from the partial input, and learn global structural priors by using a neural network to guide the retrieval and deformation steps. Experimental results show our approach considerably outperforms baselines across multiple datasets and shape categories. Code and data are available at
10.48550/arxiv.2207.11790
[ "https://export.arxiv.org/pdf/2207.11790v1.pdf" ]
251,041,034
2207.11790
2d8e79ca4c090bd3ba4030d06fe499199af6b630
PatchRD: Detail-Preserving Shape Completion by Learning Patch Retrieval and Deformation Bo Sun Austin Vladimir G Kim Adobe Research Noam Aigerman Adobe Research Qixing Huang Austin Siddhartha Chaudhuri Adobe Research IIT Bombay PatchRD: Detail-Preserving Shape Completion by Learning Patch Retrieval and Deformation This paper introduces a data-driven shape completion approach that focuses on completing geometric details of missing regions of 3D shapes. We observe that existing generative methods lack the training data and representation capacity to synthesize plausible, fine-grained details with complex geometry and topology. Our key insight is to copy and deform patches from the partial input to complete missing regions. This enables us to preserve the style of local geometric features, even if it drastically differs from the training data. Our fully automatic approach proceeds in two stages. First, we learn to retrieve candidate patches from the input shape. Second, we select and deform some of the retrieved candidates to seamlessly blend them into the complete shape. This method combines the advantages of the two most common completion methods: similarity-based single-instance completion, and completion by learning a shape space. We leverage repeating patterns by retrieving patches from the partial input, and learn global structural priors by using a neural network to guide the retrieval and deformation steps. Experimental results show our approach considerably outperforms baselines across multiple datasets and shape categories. Code and data are available at Introduction Completing geometric objects is a fundamental problem in visual computing with a wide range of applications. For example, when scanning complex geometric objects, it is always difficult to scan every point of the underlying object [33]. The scanned geometry usually contains various levels of holes and missing geometries, making it critical to develop high-quality geometry completion techniques [61,10,68,13,1,24,18]. Geometry completion is also used in interactive shape modeling [7], as a way to suggest additional content to add to a partial 3D object/scene. Geometry completion is challenging, particularly when the missing regions contain non-trivial geometric content. Early geometry completion techniques focus on hole filling [61,10,68,13,1,24,18]. These techniques rely on the assumption that the missing regions are simple surface patches and can be filled by smoothly extending hole regions. Input PatchRD (Ours) Generative Model [71] Input PatchRD (Ours) Generative Model [71] Fig. 1: We propose PatchRD, a non-parametric shape completion method based on patch retrieval and deformation. Compared with the parametric generation methods, our method is able to recover complex geometric details as well as keeping the global shape smoothness. Filling regions with complex shapes rely on data priors. Existing approaches fall into two categories. The first category extracts similar regions from the input shape. The hypothesis is that a physical 3D object naturally exhibits repeating content due to symmetries and texture. While early works use user-specified rules to retrieve and fuse similar patches, recent works have studied using a deep network to automatically complete a single image or shape [62,20,21]. The goal of these approaches is to use different layers of the neural network (e.g., a convolutional neural network) to automatically extract repeating patterns. However, these approaches are most suitable when the repeating patterns are prevalent within the partial input. They cannot infer correlations between the missing surface and the observed surface. Another category [78,59,12,73,43,46,17,71] consists of data-driven techniques, which implicitly learn a parametric shape space model. Given an incomplete shape, they find the best reconstruction using the underlying generative model to generate the complete shape. This methodology has enjoyed success for specific categories of models such as faces [4,80,74,48] and human body shapes [1,38,45,30,26], but they generally cannot recover shape details due to limited training data and difficulty in synthesizing geometric styles that exhibit large topological and geometrical variations. This paper introduces a shape completion approach that combines the strengths of the two categories of approaches described above. Although it remains difficult to capture the space of geometric details, existing approaches can learn high-level compositional rules such as spatial correlations of geometric primitives and parts among both the observed and missing regions. We propose to leverage this property to guide similar region retrieval and fusion on a given shape for geometry completion. Specifically, given an input incomplete shape, the proposed approach first predicts a coarse completion using an off-the-shelf method. The coarse completion does not necessarily capture the shape details but it provides guidance on locations of the missing patches. For each coarse voxel patch, we learn a shape distance function to retrieve top-k detailed shape patches in the input shape. The final stage of our approach learns a deformation for each retrieved patch Fig. 2: Approach pipeline. Given an incomplete shape S, we first predict a coarse shape C with the rough structure and no details. For each patch on C, K detailed patch candidates are retrieved from the input shape. Then we predict deformations and blending weights for all retrieved candidates. Finally, the output shapeŜ is computed by summing up the deformed patches and their blending weights. and a blending function to integrate the retrieved patches into a continuous surface. The deformation prioritizes the compatibility scores between adjacent patches. The blending functions optimize the contribution of each patch and ensure surface smoothness. Experimental results on the ShapeNet dataset [6] show that our approach outperforms existing shape completion techniques for reconstructing shape details both qualitatively and quantitatively. In summary, our contributions are: -We propose a non-parametric shape completion method based on patch retrieval and deformation. -Our method preserves local shape details while enforcing global consistency. -Our method achieves state-of-the-art shape completion results compared with various baselines. Related Work Shape Completion. Shape completion is a crucial and long-studied task in geometry processing. Non-data-driven works [54,40,27,28] address hole-filling in a purely geometric manner. Without any high-level prior on the resulting shape, they target filling holes in a "smooth-as-possible" manner with membranes. To complete more complex shapes, several works [55,29,36,44,34,50] rely on data-driven methods to get the structure priors or part references. Similarly to our method, [44,34,50] retrieve some candidate models from a database, then perform a non-rigid surface alignment to deform the retrieved shape to fit the input. However, our approach operates at the patch level and can reconstruct shapes that are topologically different from those in the training data. With the development of deep learning, neural networks can be used to learn a shape prior from a large dataset and then complete shapes. Voxel-based methods [69,12,70] are good at capturing rough structures, but are limited to low resolution by the cubic scaling of voxel counts. Our framework is especially designed to circumvent these resolution limitations. Alternatively, point cloud completion [78,59,73,42,77,72,67,71] has become a popular venue as well. [73,25,66] use coarse-to-fine structures to densify the output point cloud and refine local regions. NSFA [79] and HRSC [19] used a two stage method to infer global structures and refine local geometries. SnowflakeNet [71] modeled the progressive generation hierarchically and arranged the points in locally structured patterns. As point clouds are sparse and unstructured, it is difficult to recover fine-grained shape details. 3D-EPN [12] and our method both use coarse-to-fine and patch-based pipelines. However, their method only retrieves shapes from the training set and directly copies the nearest patches based on low-level concatenation of distance fields. Our method retrieves patch-level details from the input and jointly learns deformation and blending, which enables our method to handle complex details as well as maintain global coherence. Patch-based Image In-painting. In the 2D domain, many works utilize detailed patches to get high-resolution image inpainting results. Traditional methods [14,2,32,22] often synthesize textures for missing areas or expanding the current images. PatchMatch [2] proposed an image editing method by efficiently searching and replacing local patches. SceneComp [22] patched up holes in images by finding similar image regions in a large database. Recently, with the power of neural networks, more methods [60,37,47,76,49,75] use patch-guided generation to get finer details. [60,37,47] modeled images to scene graphs or semantic layouts and retrieve image patches for each graph/layout component. [76,49,75] add transformers [65], pixel flow and patch blending to get better generation results respectively. Our method leverages many insights from the 2D domain, however these cannot be directly transferred to 3D, for two reasons: i) the signals are inherently different, as 2D pixels are spatially-dense and continuous, while voxels are sparse and effectively binary; ii) the number of voxels in a domain scales cubically with resolution, as opposed to the quadratic scaling of pixels. This significantly limits the performance of various algorithms. The novel pipeline proposed herein is tailor-made to address these challenges. 3D Shape Detailization. Adding or preserving details on 3D shapes is an important yet challenging problem in 3D synthesis. Details can be added to a given surface via a reference 3D texture [56,23,81]. More relevant to use various geometric representations to synthesize geometric details [9,5,16,8,35]. DLS [5] and LDIF [16] divide a shape to different local regions and reconstruct local implicit surfaces. D2IM-Net [35] disentangles shape structure and surface details. DECOR-GAN [9] trained a patch-GAN to transfer details from one shape to another. In our case, we focus on the task of partial-to-full reconstruction, and use detailization as a submodule during the process. 3D Generation by Retrieval. Instead of synthesizing shapes from scratch with a statistical model, it is often effective to simply retrieve the nearest shape from a database [58,31,15,57]. This produces high-quality results at the cost of generalization. Deformation-aware retrieval techniques [63,64,51,39] improve the representation power from a limited database. Our method also combines deformation with retrieval, but our retrieval is at the level of local patches from the input shape itself. RetrievalFuse [52] retrieves patches from a database for scene reconstruction. An attention-based mechanism is then used to regenerate the scene, guided by the patches. In contrast, we directly copy and deform retrieved patches to fit the output, preserving their original details and fidelity. Overview Our framework receives an incomplete or a partial shape S as input and completes it into a full detailed shapeŜ. Our main observation is that local shape details often repeat and are consistent across different regions of the shape, up to an approximately rigid deformation. Thus, our approach extracts local regions, which we call patches, from the given incomplete shape S, and uses them to complete and output a full complete shape. In order to analyze and synthesize topologically diverse data using convolutional architectures, we represent shapes and patches as voxel grids with occupancy values, at a resolution of s shape cells. The key challenges facing us are choosing patches from the partial input, and devising a method to deform and blend them into a seamless, complete detailed output. This naturally leads to a three-stage pipeline: (i) complete the partial input to get a coarse complete structure C to guide detail completion; (ii) for each completed coarse patch in C, retrieve candidate detailed patches from the input shape S; (iii) deform and blend the retrieved detailed patches to output the complete detailed shapeŜ (see Figure 2). Following is an overview of the process; we elaborate on each step in the following sections. Coarse Completion. We generate a full coarse shape C from the partial input S using a simple 3D-CNN architecture. Our goal is to leverage advances in 3D shape completion, which can provide coarse approximations of the underlying ground truth, but does not accurately reconstruct local geometric details. Patch Retrieval (Section 4). We train another neural network to retrieve k candidate detailed patches from S for each coarse patch in C. Namely, we learn geometric similarity, defined by a rigid-transformation-invariant distance d, between the coarse and detailed patches. Deformation and Blending of Patches (Section 5). Given k candidate patches, we use a third neural network to predict rigid transformations and blending weights for each candidate patch, which together define a deformation and blending of patches for globally-consistent, plausible shape completion. Patch Retrieval The input to this stage is the partial detailed shape S, and a coarse and completed version of the shape, C. The goal of this step is to retrieve a set of patch candidates that can be deformed and stitched to get a fully detailed shape. A ! " # " !" ! " ! ! !(# ! , %) (small) ! " ! ! !(# " , %) (large) ! " Pairwise Metric Learning: # ! # " Retrieval Components: ' # ! # " ' Source Query Target Fig. 3: Retrieval learning. We learn a feature mapping to predict geometric distances between the query coarse patches and the sampled detailed patches. We use the geometric distances between the GT detailed patches and the sampled patches as the supervision. Distances for patches that are close up to a rigid transformation are small. Otherwise, distances are large. patch is a cube-shaped sub-region extracted from a shape, composed of s 3 patch voxels. Our patch sampling process P takes a shape as input and outputs a collection of patches, where coarse patches P(C) serve as queries and detailed patches from the partial input P(S) as sources. In order to decide whether a retrieved detailed patch could be an apt substitution for a true detailed patch, we propose a geometric distance metric invariant to rigid deformations (Section 4.1). This geometric distance will be used to supervise the neural network used during testing time, which learns similarities between coarse patches P(C) and their detailed counterparts P(S) (Section 4.2). Finally, we describe how to use this network at inference time to retrieve candidate patches for the full shape (Section 4.3). Geometric Distance We define a measure of geometric distance, d, between two detailed shape patches (p 1 , p 2 ). This metric should be agnostic to their poses, since the pose can be fixed at the deformation stage, hence we define the distance as the minimum over all possible rigid transformations of the patch: d(p 1 , p 2 ) = min T ∥T (p 1 ) − p 2 ∥ 1 ∥T (p 1 )∥ 1 + ∥p 2 ∥ 1(1) where T is a rigid motion composed with a possible reflection, i.e., T = (R, t, f ), R ∈ SO(3) is the rotation, t ∈ R 3 is the translation, f ∈ {0, 1} denotes if a reflection is applied, and || · || 1 denotes the L1 norm of positional vectors to patch centers. To practically compute this distance, we run ICP [3], initialized from two transformations (with and without reflection enabled) that align patch centers. While this geometric distance can be computed for detailed patches, at inference time we only have coarse patches. Therefore, we train a network to embed coarse patches into a latent space in which Euclidean distances match the geometric distances of the detailed patches they represent. Metric embedding We train two neural networks to act as encoders, one for coarse patches and one for detailed patches, E c and E d , respectively. We aim to have the Euclidean distances between their generated codes reflect the distances between the true detailed patches observed during training. Given a coarse patch c ∈ P(C) with its true corresponding detailed patch q ∈ P(Ŝ gt ), as well as a some other detailed patch p ∈ P(S), we define a metric embedding loss: L r = (c,p,q)∈T ∥∥E c (c) − E d (p)∥ 2 − d(p, q))∥ 2 .(2) where d(p, q) is the geometric distance defined in Equation (1). Our training triplets are composed of true matches and random patches: T = T true ∪ T rnd . Where in both sets c is a random coarse patch, q is the corresponding true detailed patch. We either set p = q for T true or randomly sample p ∈ P(S) for T rnd . See Figure 3 for an illustration. Retrieval on a Full Shape We can now use trained encoder networks at inference time to retrieve detailed patches for each coarse patch. First, we encode all the detailed patches in P(S) via E d . Similarly, for each non-empty coarse patch c ∈ P(C) with lowest corner at location l, we encode it with E c and find the K-nearest-neighbor detailed codes. We store the list of retrieved patches for each location, denoted as R l . We sample the coarse patches using a fixed-size (s 3 patch ) sliding window with a stride γ patch . Note that in the retrieval stage we do not assume that we know which parts of the detailed shape need to be completed. Since our feature learning step observed a lot of positive coarse/fine pairs with the detailed input, we found that the input is naturally reconstructed from the retrieved detailed patches. Deformation and Blending of Patches The input to this stage is the coarse shape C, partial input S, and the retrieval candidates. The output is the full detailed shapeŜ, produced by deforming and blending the retrieved patches. As illustrated by Figure 2 we first apply a rigid transformation to each retrieved patch and then blend these transformed patches into the final shape. Our guiding principle is the notion of partition-ofunity [41], which blends candidate patches with optimized transformations into a smooth completion. Unlike using fixed weighting functions, we propose to learn the blending weights. These weights serve the role of selecting candidate patches and stitching them smoothly. We observe that learning the blending weights requires some context (our method needs to be aware of at least a few neighboring patches), but does not require understanding the whole shape (coarse shape and retrieved patches Average Chair Plane Car Table 1: Shape completion results on the random-crop dataset on 8 ShapeNet categories. We show the L 2 Chamfer distance (CD) (×10 3 ) between the output shape and the ground truth 16384 points from PCN dataset [78] (lower is better). Our method reduces the CD drastically compared with the baselines. already constrain the global structure). Thus, to maximize efficiency and generalizability, we opt to perform deformation and blending at the meso-scale of subvolumes V ⊂ S with size s subv . Next, we provide more details on our blending operator (Section 5.1) and how to learn it from the data (Section 5.2). The Deformation and Blending Operator Given a subvolume V , we first identify [r m , m = 1...M ] an ordered list of M best patches to be considered for blending. These patches are from the retrieved candidates R l such that l ∈ V , and sorted according to two criteria: (i) retrieval index, (ii) x, y, z ordering. If more than M such patches exist, we simply take the first M . Each patch r m is transformed with a rigid motion and possible reflection: T m , and we have a blending weight for each patch at every point x in our volume: ω m [x]. The output at voxel x is the weighted sum of the deformed blending candidates: V [x] = 1 ξ[x] m=1...M ω m [x] · T m (r m )[x](3) where ω m [x] is the blending weight for patch m at voxel x, and T m (r m ) is the transformed patch (placed in the volume V , and padded with 0), and ξ[x] = m=1..M ω m [x] is the normalization factor. At inference time, when we need to reconstruct the entire shape, we sample V over the entire domainŜ (with stride γ subv ), and average values in the region of overlap. Learning Deformation and Blending Directly optimizing deformation and blending is prone to being stuck in local optimum. To address this we develop a neural network to predict deformations and blending weights and train it with reconstruction and smoothness losses. Prediction network. We train a neural network g to predict deformation and blending weights. The network consists of three convolutional encoders, one for each voxel grid: the coarse shape (with a binary mask for the cropped subvolume V ), the partial input, and the tensor of retrieved patches (M channels at resolution of V ). We use fully-connected layers to mix the output of convolutional encoders into a bottleneck, which is than decoded into deformation T and blending ω parameters. Reconstruction loss. The first loss L rec aims to recover the target shapeŜ gt : L rec = ∥V gt − V ∥ 2 ,(4) where V gt ⊂Ŝ gt and V ⊂Ŝ are corresponding true and predicted subvolumes (we sample V randomly for training). Blending smoothness loss. The second loss L sm regularizes patch pairs. Specifically, if two patches have large blending weights for a voxel, then their transformations are forced to be compatible on that voxel: L sm = x∈V m,n ∥ω m [x] · ω n [x] · (T m (r m )[x] − T n (r n )[x])∥ Where x iterates over the volume and m, n over all retrieved patches. Note that r m and r n are only defined on a small region based on where the patch is placed, so this sum only matters in regions where transformed patches T m (r m ) and T n (r n ) map to nearby points x accordingly. Final Loss The final loss term is L = L rec + αL sm .(5) Experiments We primarily evaluate our method on the detail-preserving shape completion benchmark (Section 6.1), and demonstrate that our method outperforms stateof-the-art baselines (Section 6.2). We further demonstrate that our method can generalize beyond the benchmark setup, handling real scans, data with large missing areas, and novel categories of shapes (Section 6.3). Finally, we run an ablation study (Section 6.4) and evaluate sensitivity to the size of the missing area (Section 6.5). Experimental Setup Implementation Details. We use the following parameters for all experiments. The sizes of various voxel grids are: s shape = 128, s patch = 18, s subv = 40 with strides γ patch = 4, γ subv = 32. We sample |T rnd | = 800 and |T true | = 400 triplets to train our patch similarity (Section 4.2). Our blending operator uses M = 400 best retrieved patches (Section 5.1). We set α = 10 for Equation 5. To improve performance we also define our blending weights ω m at a coarser level than V . Table 2: FID comparison on the chair class (note that we can only apply this metric to volumetric baselines). Our method produces more plausible shapes. Input Conv-ONet SnowFlakeNet Ours GT 3D-GAN 3D-GAN VRCNet Input 3D-GAN Conv-ONet VRCNet Snowflake PatchRD (Ours) GT In particular, we use windows of size s 3 blend = 8 3 to have constant weight, and compute the blending smoothness term at the boundary of these windows. Dataset. We use shapes from ShapeNet [6], a public large-scale repository of 3D meshes to create the completion benchmark. We pick eight shape categories selected in prior work PCN [78]. For each category, we use the same subset of training and testing shapes with 80% / 20% split as in DECOR-GAN work [9]. For voxel-based methods, we convert each mesh to a 128 3 voxel grid, and for point-based baselines, we use the existing point clouds with 16384 points per mesh [78]. We create synthetic incomplete shapes by cropping (deleting) a random cuboid with 10%−30% volume with respect to the full shape. This randomly cropped dataset is generated to simulate smaller-scale data corruption. We also show results on planar cutting and point scans in Section 6.3. Metrics. To evaluate the quality of the completion, we use the L 2 Chamfer Distance (CD) with respect to the ground truth detailed shape. Since CD does not really evaluate the quality of finer details, we also use Frechet Inception Distance (FID), to evaluate plausibility. FID metric computes the distance of the layer activations from a pre-trained shape classifier. We use 3D VGG16 [53]) trained on ShapeNet and activations of the first fully connected layer. Baseline Approaches To the best of our knowledge, we are the first to do the 3D shape completion task on the random-cropped dataset. Considering the task similarity, we compare our method with the other shape completion and reconstruction baselines. Our baselines span different shape representations: PCN [78], TopNet [59], GRNet [73], VRCNet [42], and SnowFlakeNet [71] are point-based scan completion baselines, 3D-GAN [69] is a voxel-based shape generation method, Conv-ONet [46] is an implicit surfaces-based shape reconstruction methods, and AtlasNet [17] is an atlas-based shape reconstruction method. We show our method outperforms these baselines both quantitatively and qualitatively. Table 1 and Table 2 show quantitative comparisons between PatchRD and baselines, demonstrating that our method significantly outperforms all baselines. PatchRD achieved superior performance on all categories except airplanes, which shows that it generalizes well across different classes of shapes. Shape Completion Results Specifically, the voxel-based baseline [69] produces coarse shapes where fine details are missing. Point-based baselines [42,71] often have noisy patterns around on fine structures while our method has clean geometry details. The implicit surface based method [46] could capture the details but the geometry is not smooth and the topology is not preserved. Our method keeps the smooth connection between geometry patches. More results can be found in the supplemental materials. Figure 4 shows qualitative comparisons. We pick four representative baselines for this visualization including point-based methods that performed the best on the benchmark [42,71] as well as voxel-based [69] and implicit-based methods [46]. Our results show better shape quality by recovering local details as well as preserving global shape smoothness. Other Applications Real-World Application: Scan Completion. We test our method on realworld shape completion from scans. We use shapes from ScanNet[11], 3D indoor scene dataset as input to our method. Objects in ScanNet often have some missing parts, especially thin structures, due to occlusion and incompleteness of scan viewpoints. We convert these point clouds to voxel grids and apply our completion technique trained on ShapeNet, see results in Figure 5a. Note how our method completes the undersampled areas, while preserving the details of the input and smoothly blending new details to the existing content. Shapes with Large Missing Areas. We also demonstrate that our method can handle large missing areas (Figure 5b). In this experiment we cut the shape with a random plane, where in some cases more than half of the shape might be missing. Our method recovers the shape structure and extends the local shape details to the whole shape when only given a small region of reference details. Completion on Novel Categories We further evaluate the ability of our method to generalize to novel categories. Note that only the prediction of the complete coarse shape relies on any global categorical priors. Unlike other generative techniques that decode the entire shape, our method does not need to learn how to synthesize category-specific details and thus succeeds in detailpreservation as long as the coarse shape is somewhat reasonable. In Figure 5c we demonstrate the output of different methods when tested on a novel category. Note how our method is most successful in conveying overall shape as well as matching the details of the input. Ablation Study We evaluate the significance of deformation learning, patch blending, and blending smoothness term via an ablation study (see Table 3). Table 3: Ablation study. In the left figure, we visualize the effect of different components in our experiment. Patch alignment can't get good patch transformation. Results with no blending are subjective to bad retrievals. Results with no smoothing show discontinuity between neighboring patches. Results with all components contain geometric details as well as smoothness. In the right table, We show the reconstruction error CD and shape plausibility score FID on ShapeNet chair class. Results with all components get both better CD and FID. No Deformation Learning We simply use ICP to align the retrieved patch to the query. Table 3 (Patch Alignment) illustrates that this leads to zigzagging artifacts due to patch misalignments. No Patch Blending Instead of blending several retrieved patches, we simply place the best retrieved patch at the query location. Table.3 show that our method with all components performs the best with respect to reconstruction and plausibility metrics. Sensitivity to the Size of Missing Regions The completion task is often ill-posed, and becomes especially ambiguous as the missing region increases. We evaluate the sensitivity of our method to the size of the region by trying to increase the crop size from 10% to 50% of the volume. Table 4 demonstrates that our method can produce plausible completions even under severe crops. However, in some cases it is impossible to reconstruct details that are completely removed. We report quantitative results in Table 4. While both reconstruction and plausibility error increases for larger crops, we observe that plausibility (FID) score does not deteriorate at the same rate. Conclusions, Limitations and Future Work Conclusions. This paper proposed a novel non-parametric shape completion method that preserves local geometric details and global shape smoothness. Our method recovers details by copying detailed patches from the incomplete shape, and achieves smoothness by a novel patch blending term. Our method obtained state-of-the-art completion results compared with various baselines with different Table 4: Sensitivity to the size of missing regions. The left figure shows results with different crop ratios. Input geometries and shape contours influence the output shapes. The right table shows the reconstruction error and the shape plausibility with the increase of crop ratios. As the ratio increases, the reconstruction error keeps growing although the output shapes remain fairly plausible. [78]. On the left, we show our method is able to produce cleaner and more plausible results than the structurebased baseline. On the right, we show some failure cases where shape details are missing in the input shape. 3D representations. It also achieved high-quality results in real-world 3D scans and shapes with large missing areas. Limitations. Our method has two limitations: (1) It builds on the assumption that the shape details are present in the partial input shape, which might not hold if large regions are missing in the scan. For completeness, we still evaluate our method in this scenario using the PCN benchmark, which focuses on largescale structure recovery from very sparse input. In Figure 6 we show that our method succeeds when there are enough detail references in the input, and fails if the input is too sparse. We also provide quantitative evaluations in supplemental material. These results suggest that our method is better suited for completing denser inputs (e.g., multi-view scans). In the future, we plan to address this issue by incorporating patches retrieved from other shapes. (2) Our method cannot guarantee to recover the global structure because the retrieval stage is performed at the local patch level. To address this issue, we need to enforce suitable structural priors and develop structureaware representations. We leave both for future research. Future work. Recovering geometric details is a hard but important problem. Our method shows that reusing the detailed patches from an incomplete shape is a promising direction. In the direction of the patch-based shape completion, potential future work includes: (1) Applying patch retrieval and deformation on other 3D representations such as point cloud and implicit surfaces. This can handle the resolution limitation and computation burden caused by the volumetric representation. (2) Unifying parametric synthesis and patch-based non-parametric synthesis to augment geometric details that are not present in the partial input shape. the GT shape. However, our method can still produce plausible output, i.e. semantically correct and smoothly connected shapes. If stage 1 fails, the overall structure will be different from the GT shape. If stage 2 fails, the local details will be inaccurate. If stage 3 fails, the connection between patches will not be smooth, causing irregular or noisy shapes. Some examples of failure cases from each step are shown in Fig. ??. Network Architectures We show the detailed network architectures for coarse completion, retrieval metric learning, and deformation and blending weight prediction in Figure 8, Figure 9, and Figure 10 respectively. 9: Architecture of the feature encoder in the retrieval learning part. The input is a patch with size (18,18,18). The output is a feature vector with size 128. In the figure, blue boxes are tensors and white boxes are layers between two tensors. The array after Conv3d means (input channel, output channel, kernel size, kernel size,kernel size). s means stride. p means padding. Coarse Shape Encoder Partial Shape Encoder Sub-Volume Encoder Sub-Volume Decoder (symmetric to Sub-Volume Encoder) Deformation: Blending Confidence: (1,400,40,40,40) Deformation Prediction Head Blending Confidence Prediction Head Fig. 10: Architecture of the patch deformation and blending weight prediction network. There are 3 branches to encode the coarse shape, partial shape and the sub-volume to one-dimensional feature vectors. Then two heads decode the concatenated feature vectors to deformation and blending weights. In the figure, blue boxes are tensors and white boxes are layers between two tensors. The array after Conv3d means (input channel, output channel, kernel size, kernel size,kernel size). s means stride. Fig. 4 : 4Qualitative shape completion results on the Random-Crop Dataset. Our results recover more geometric details and keep the shape smooth while other baselines often produce coarse, noisy, or discontinuous results. Fig. 5 : 5More applications on real scans, shapes with large missing areas and novel categories. Fig. 7 : 7Failure cases caused by different steps. Fig. Fig. 9: Architecture of the feature encoder in the retrieval learning part. The input is a patch with size (18, 18, 18). The output is a feature vector with size 128. In the figure, blue boxes are tensors and white boxes are layers between two tensors. The array after Conv3d means (input channel, output channel, kernel size, kernel size,kernel size). s means stride. p means padding. Table Cabinet Lamp CabinetBoat Couch AtlasNet[17] 7.03 6.08 2.32 5.32 5.38 8.46 14.20 6.01 8.47 Conv-ONet[46] 6.42 2.91 2.29 8.60 7.94 12.6 5.82 4.03 7.21 TopNet[59] 6.30 5.94 2.18 4.85 5.63 5.13 15.32 5.60 5.73 3D-GAN[69] 6.00 6.02 1.77 3.46 5.08 7.29 12.23 7.20 4.92 PCN[78] 4.47 3.75 1.45 3.58 3.32 4.82 10.56 4.22 4.03 GRNet[73] 2.69 3.27 1.47 3.15 2.43 3.35 2.54 2.50 2.84 VRCNet[42] 2.63 2.96 1.30 3.25 2.35 2.98 2.86 2.23 3.13 SnowflakeNet[71] 2.06 2.45 0.72 2.55 2.15 2.76 2.17 1.33 2.35 PatchRD (Ours) 1.22 1.08 0.98 1.01 1.32 1.45 1.23 0.99 1.67 Table 3 ( 3No Blending) shows 10% 20% 40% 60% L2-CD 0.88 1.22 2.35 6.64 FID 9.74 10.32 13.34 15.63Input Output Input Output Input Output Input Output Qualitative Comparison Input Snow[71]Ours Input Snow[71]Ours Failure Cases Input Output GT Input Output GTFig. 6: Qualitative results on the PCN Dataset TopNet[?] GRNet[?] SnowFlakeNet[?] PatchRD(Ours)CD-L1 13.43 9.37 7.78 8.79 Table 1 : 1Quantitative results on chair class of the PCN Dataset[?]. All methods are trained on chair class only. We report the L 1 chamfer distance ×10 − 3. 3D-GAN Conv-ONet VRCNet Snowflake PatchRD (Ours) GTFig. 1: More qualitative shape completion results on the Random-Crop Dataset. 3D-GAN Conv-ONet VRCNet Snowflake PatchRD (Ours) GT Fig. 2: More qualitative shape completion results on the Random-Crop Dataset.Fig. 3: More shape completion results on real scans for ScanNet[?] objects.Fig. 4: More shape completion results on shapes with large missing areas. Input Conv-ONet[?] Snow[?] PatchRD Input Conv-ONet[?] Snow[?] PatchRD Fig. 5: More testing results on novel categories. For each row, we note the training categories and testing categories on the left top corners. Lamp→Chair means training on lamp and testing on chair shapes.Fig. 6: More qualitative comparison on PCN Dataset[?].Input arXiv:2207.11790v1 [cs.CV] 24 Jul 2022 Failure Cases AnalysisOur pipeline has 3 stages: (1) coarse completion, (2) patch retrieval, and (3) patch deformation and blending. If one stage fails, the result might be different fromMore ResultsWe show more qualitative results of shape completion results on random-crop dataset (Figure 1andMore Results on PCN BenchmarkWe show the quantitative results and more qualitative results on the PCN dataset [?] inTable 1andFigure 6respectively. Our method is quantitatively a little worse than the best-performing SnowFlakeNet because our method might fail when there's no reference details in the input shape, and CD-L1 is more sensitive to structure than details. Importantly, visual results inFigure 6indicate that our method produces more clean and plausible shapes, especially in the missing areas.More Training DetailsFor the coarse completion, the input shape is the partial detailed shape and the ground truth is the coarse version (4× downsampled) of the detailed full shape. The loss function is the cross-entropy loss between the GT and the output. We use Kaiming Uniform method for weight initialization and the Adam optimizer to train 100 epochs for each shape category on a single Titan X. Training takes ∼3 hrs for the 3D CNN, ∼12 hrs for retrieval, and ∼2 hrs for deformation and blending. Inference for a shape with 128 3 voxels takes ∼20s on a single 12GB Titan X.Fig. 8: Architecture of the coarse completion network. The input is a partial shape with size (128, 128, 128) and the output is a coarse shape with the same size. We only show the encoder in detail here. The decoder is symmetric to the encoder. 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[ "FairDistillation: Mitigating Stereotyping in Language Models", "FairDistillation: Mitigating Stereotyping in Language Models" ]
[ "Pieter Delobelle \nDepartment of Computer Science\nKU Leuven\nBelgium Leuven.AI institute\nBelgium\n", "Bettina Berendt \nDepartment of Computer Science\nKU Leuven\nBelgium Leuven.AI institute\nBelgium\n\nFaculty of Electrical Engineering and Computer Science\nWeizenbaum Institute\nBerlinTUGermany, Germany\n" ]
[ "Department of Computer Science\nKU Leuven\nBelgium Leuven.AI institute\nBelgium", "Department of Computer Science\nKU Leuven\nBelgium Leuven.AI institute\nBelgium", "Faculty of Electrical Engineering and Computer Science\nWeizenbaum Institute\nBerlinTUGermany, Germany" ]
[]
Large pre-trained language models are successfully being used in a variety of tasks, across many languages. With this ever-increasing usage, the risk of harmful side effects also rises, for example by reproducing and reinforcing stereotypes. However, detecting and mitigating these harms is difficult to do in general and becomes computationally expensive when tackling multiple languages or when considering different biases.To address this, we present FairDistillation: a cross-lingual method based on knowledge distillation to construct smaller language models while controlling for specific biases. We found that our distillation method does not negatively affect the downstream performance on most tasks and successfully mitigates stereotyping and representational harms. We demonstrate that FairDistillation can create fairer language models at a considerably lower cost than alternative approaches.
10.48550/arxiv.2207.04546
[ "https://export.arxiv.org/pdf/2207.04546v2.pdf" ]
250,426,284
2207.04546
91642d9c31bf9021cd02e50fa6049236de750702
FairDistillation: Mitigating Stereotyping in Language Models Pieter Delobelle Department of Computer Science KU Leuven Belgium Leuven.AI institute Belgium Bettina Berendt Department of Computer Science KU Leuven Belgium Leuven.AI institute Belgium Faculty of Electrical Engineering and Computer Science Weizenbaum Institute BerlinTUGermany, Germany FairDistillation: Mitigating Stereotyping in Language Models Knowledge distillationFairnessBERTLanguage models Large pre-trained language models are successfully being used in a variety of tasks, across many languages. With this ever-increasing usage, the risk of harmful side effects also rises, for example by reproducing and reinforcing stereotypes. However, detecting and mitigating these harms is difficult to do in general and becomes computationally expensive when tackling multiple languages or when considering different biases.To address this, we present FairDistillation: a cross-lingual method based on knowledge distillation to construct smaller language models while controlling for specific biases. We found that our distillation method does not negatively affect the downstream performance on most tasks and successfully mitigates stereotyping and representational harms. We demonstrate that FairDistillation can create fairer language models at a considerably lower cost than alternative approaches. Introduction Pre-trained transformer-based Language Models (LMs), like BERT [14], are not only pushing the state-of-the-art across many languages, they are also being deployed in various services, ranging from machine translation to internet search [14,22,34]. However, these deployed language models have been shown to exhibit problematic behaviour. For instance, BERT and other models (i) replicate gender stereotypes [1,12,32], (ii) exhibit dubious racial correlations [32] and (iii) reproduce racial stereotypes [26]. These behaviours are all present in pre-trained models that are used in a wide range of applications, which are referred to as downstream tasks. Without precautions, downstream tasks could use such problematic behaviour to make biased predictions. LMs are generally finetuned for such tasks, where allocation harms (i.e. allocating or withholding a resource) might occur [3]. These can originate from the fine-tuning dataset or the pre-trained model or a combination of both. We focus on the pre-training, where representation harms (i.e. encoding stereotypes) can occur in the pre-trained LMs [3,36]. Multiple methods have been proposed to reduce representational harms in language models [1,36], These methods are based on pre-processing of the data, for example Counterfactual Data Augmentation (CDA) [23] or Counterfactual Data Substitution (CDS) [16]. In both cases, gendered words in input sequences are replaced by a predefined counterfactual, e.g. "He is a doctor " → "She is a doctor ". CDA can significantly increase the training dataset, with longer training times as a consequence, so CDS-based methods replace input sequences instead. Nevertheless, both techniques require retraining the model with an augmented dataset, instead of leveraging the efforts done to train the original model. We propose a framework for mitigating representational harms based on knowledge distillation [17], which we demonstrate on gender stereotypes. Our approach uses existing language models as a teacher, which provides a richer training signal and does not require retraining from scratch. To prevent the transfer of learnt correlations to new LMs, our framework replaces CDA's augmentation strategy with probabilistic rules between tokens. Since our approach can be performed at a fraction of the original training cost and also creates smaller models, it becomes more feasible to create domain-specific bias-controlled LMs. In this paper, we start in Section 2 with an overview of language models and fairness interventions ( § 2.1). In Section 3, we present our method to create debiased language models, which we call FairDistillation. Section 4 describes the evaluation set-up and Section 5 presents the results. Section 6 gives an overview of future work and ethical considerations and we conclude in Section 7. Background BERT [14] is a language model that is trained in two phases: (i) selfsupervised pre-training with a Masked Language Modeling (MLM) objective and afterwards (ii) supervised finetuning for downstream tasks. The intuition behind the first learning task is that learning to reconstruct missing words in a sentence helps with capturing interesting semanticsand because this relies on co-occurrences it also captures stereotypes. Formally, a token xm in the input sequence x1, . . . , xN is replaced by a masked token (<mask>) and the MLM objective is to predict the original token xm based on the context x = x1, . . . , xm−1, xm+1, . . . , xN , following max θ N i=1 1x i =xm log (P (xi | x; θ)) with 1x i =xm as an indicator function whether the token is correctly predicted. This training setup results in a good estimator of the contextualized probability of a word P (xi | x; θ). Aside from the MLM objective, the original BERT model also incorporated a Next Sentence Prediction (NSP) objective. Liu et al. [22] later concluded that the NSP objective did not improve training and removed it when constructing RoBERTa. Because of this, we do not further consider this objective during distillation or evaluation. After pre-training, the newly obtained model can be reused and finetuned for different classification and regression tasks, like sentiment analysis. Finetuning requires different datasets, that can also introduce biases that are referred to as extrinsic biases [11]. Mitigating extrinsic biases in downstream tasks is out of scope for this work. Nevertheless, since LMs are used both for downstream tasks and for generating contextualized embeddings, mitigating intrinsic biases is still crucial. Mitigating intrinsic biases Bolukbasi et al. [5] presented two intrinsic debiasing methods based on removing the observed gender axis in static word embeddings. Mitigating problematic correlations is more challenging for LMs because of the contextualization that models like BERT incorporate. This means that word representations from LMs cannot be considered in isolation, so mitigation strategies for word embeddings cannot be applied. Models like BERT can only generate meaningful representations for a given sequence, so for this reason, mitigation strategies have mostly been based on Counterfactual Data Augmentation (CDA) [1,16,23,36]. This strategy augments the pre-training dataset with sequences where certain words, like pronouns or names, are swapped. Unfortunately, this requires re-training the model from scratch, which can be extremely costly and with many negative side effects [2]. One of few mitigation strategies that does not alter the training data was also presented by Webster et al. [36], namely using dropout as a regularisation method against problematic correlations. Regularisation as a means to mitigate problematic correlations thus seems a feasible option, but albeit effective, the method still requires retraining the model from scratch. It should also be noted that these efforts are mostly focused on English. Results of performing CDA on a German model were less successful, likely due to gender marking [1]. All previous methods require retaining a language model. Lauscher et al. [21] presents a unique approach, ADELE, that addresses this issue by using adapters [18,29,30]. These adapters are inserted after each attention layer and are the only trainable parameters, so the majority of parameters of a language model are shared over different tasks. ADELE trains these adapters on a subset (1/3th) of the original BERT corpus 3 with the MLM objective and CDA to mitigate biases. Although ADELE works very different from our distillation method, both methods aim to reduce the computational requirements and associated costs, by reusing existing models. Knowledge distillation Knowledge distillation is a method to transfer learnt knowledge from one model-originally proposed as an ensemble of models-to another, usually smaller model [7,17]. Buciluǎ et al. [7] introduced this technique as model compression with an ensemble of models that are used to label a dataset. This was later adapted for neural networks [17]. The teacher outputs a label probability distribution zi where some labels have higher probabilities, for example names or pronouns are more likely than verbs in the sentence "<mask> is a doctor.". To incorporate this information, a variation of the softmax function (Equation 1) can be used with a temperature T to tune the importance of these labels. pi = exp z i T j exp z j T .(1) Sanh et al. [31] focus on the distillation of the MLM task from pre-trained LMs. Their models, DistilBERT and DistilRoBERTa, are trained on a linear combination of a distillation loss Lce with the softening function from Equation 1, the original MLM loss L mlm , and additionally a cosine loss Lcos for the last hidden states, following L = αceLce + α mlm L mlm + αcosLcos. TinyBERT [19] takes the same approach but also proposes a set of loss functions that perform distillation on (i) the embeddings layer, (ii) each of the transformer layers, and (iii) the prediction layer for specific tasks. These different loss functions make TinyBERT perform slightly better than DistilBERT, but these functions require additional transformations to be learnt. In addition, if the student and teacher have a different number of layers, a mapping function is also required to transfer the knowledge between both. FairDistillation In this section, we introduce FairDistillation, a method to mitigate problematic correlations in pre-trained language models. We first present the distillation architecture ( § 3.1) and afterwards, we will discuss the probabilistic rules that our method relies on ( § 3.2). Architecture Our method trains a newly initialized model (the student) from an already trained model (the teacher). Often, the teacher model has already been evaluated for biases, for example stereotypical gender norms for professions, which can lead to representational harms [3]. In this example, models like BERT-base [14] predict that the input sentence "<mask> is a doctor." should be filled with 'He' instead of 'She'. The LM encoded that the token 'He' is more frequent in the training dataset, both in isolation and in combination with words like 'doctor '. To prevent representational harms from being encoded in the final model, we apply a set of user-specified rules to the predictions of the original model. By doing so, we can train a new model with these predictions. Predictions of a teacher model provide a richer training signal and thus require less training time compared to CDA and CDS [17,19,31]. Moreover, we can simultaneously reduce the student's model size to improve both training and inference times, which boils down to knowledge distillation as is done for DistilBERT [31]. Figure 1 illustrates our method, which consists of 5 steps. First, an input sequence x is passed to both the teacher and the student model, both with an MLM prediction head. Second, the MLM predictions of the teacher model are passed to the rule engine. Third, the predictions for certain tokens, like 'He', are modified based on the provided rules. Figure 1 demonstrates how a rule where we assume equal probabilities P ('He' | x) = P ('She' | x) alters the MLM prediction, which we discuss more in-depth later in this section. Fourth, both MLM outputs, after applying possible rules to the teacher outputs, are used to calculate the distillation loss Lce between the teacher and student outputs. Finally, the MLM outputs of the student model can also be used to calculate an additional loss term L mlm to train the student model in the same manner as the original model. Student architecture. The student models use the same base architecture as the teacher models, but with 6 attention layers instead of the typical 12 layers, following Sanh et al. [31]. The weights are initialized at random, which we prefer over smarter initialization strategies [31] to prevent an accidental transfer of problematic correlations. We also reuse the teacher's tokenizer for the student, since these are already specifically constructed for the targeted language and no complex token translation is needed. Applying probabilistic rules. The MLM head outputs a vector for each position in the input sequence, so for BERT-base this means at most 512 vectors. Each value in this vector represents the probability that a token fits in this position. Consequently, there will be 30,522 values for BERTbase-uncased. We assume that some probabilities should be equal, like P ('He' | "<mask> is a doctor") = P ('She' | "<mask> is a doctor"), so our method can enforce these kind of equality rules. During distillation, our method applies these equality rules to all the MLM outputs of the teacher. For efficiency reasons, the tokens of interest are translated into a small lookup table at the start of the distillation loop so that applying each rule only requires a few lookup operations. The corresponding values of the tokens are set to the mean of both values. Consequently, the outcome is also normalized and each prediction still sums up to 1. Currently, our method only supports equalization between two or more tokens. We did experiment with implementing these and more complex rules in ProbLog, a probabilistic logic programming language [10], but this proved to be unfeasible because of inference times that frequently exceeded 0.5 seconds per training example. Nevertheless, future work could focus on adding more complex rules that also depend for example on context or on part-of-speech tags to distinguish between adjectives ('His car' → 'Her car') or pronouns ('. . . is his' → '. . . is hers'). Knowledge distillation. We follow the DistilBERT [31] distillation method, as discussed in Section 2. FairDistillation applies a set of rules to affect the distillation loss, but the student not only learns from the distillation task, but also from the MLM task. It is possible to concurrently train on this MLM objective for little additional cost. Although this can be another source of problematic correlations, we opted to use this loss without correcting any associations. We reason that the contextual probability for a single input sequence can also be a useful signal. Obtaining probabilistic rules Until now, we used a running example of a probabilistic rule where the contextualized probability, as generated by the teacher LM, has to be equal for two tokens, namely 'He' and 'She'. CDA achieves something similar by augmenting the dataset based on word mappings [15,23]. These mappings are very similar to our probabilistic rules; in fact, AugLy, a popular data augmentation framework [28], has the same mapping 4 that we use for our running example in the context of gender bias. Depending on which biases one wants to mitigate, different sets of rules are required. We focus in this work on gender bias, so we rely on the same kind of rules as CDA. Simple rules to balance predictions highlight the robustness of our method and do not require lists of professions, which come with their own issues and biases [4]. However, creating more fine-grained, domain-specific rules might improve our results. Such rules could aim at balancing, for example, profession titles or proper names. Experimental setup We evaluate our method in two Indo-European languages: (i) English and (ii) Dutch, of which the results are discussed further in Section 5. Both languages have their own set of models, pre-training corpora and evaluation datasets, which we briefly cover in this section. The evaluation of gender biases is also highly language-dependent and to illustrate generalization of our method beyond English, we also used a monolingual model for Dutch [12] with an architecture similar to RoBERTa [22]. We opted for this language since it has some interesting, challenging characteristics, namely it is one of only two languages with cross-serial dependencies that make it non-context free, with the other one being Swiss-German [6]. It also has gendered suffixes for some, yet not all, nouns. This affects such evaluations since these rely on implicit associations between nouns (e.g. for professions). However, grammatical gender can also be an opportunity to evaluate how e.g. gendered professions align with the workforce [1] or with equal opportunity policies. We compare our method based on three popular metrics that we discuss in this section, Delobelle et al. [11] provides a more comprehensive overview of intrinsic fairness measures. SEAT. The Word Embedding Association Test (WEAT) [8] measures associations between target words ('He', 'She', . . . ) and attribute words ('doctor', 'nurse', . . . ). Between the embeddings of each target and attribute word, a similarity measure like cosine similarity can be used to quantify the association between word pairs. To add context, SEAT uses some 'semantically bleached' template sentences [26]. LPBS. Kurita et al. [20] observe that using SEAT for the learned BERT embeddings fails to find many statistically significant biases, which is addressed in the presented log probability bias score (LPBS). This score computes a probability ptgt for a target token t (e.g. 'He' or 'She') from the distribution of the masked position Xm following ptgt = P (Xm = t | x; θ) , for a template sentence, e.g. "<mask> is a doctor". Since the prior likelihood P (Xm = t) can skew the results, the authors correct for this by calculating a template prior pprior by additionally masking the token(s) with a profession or another attribute xp, following pprior = P (Xm = t | x\{xp}; θ) . Both probabilities are combined in a measure of association log p tgt p prior and the bias score is the difference between the association measures for two targets, like 'He' and 'She'. Kurita et al. [20] applied their method to the original English BERT model [14] and found statistically significant differences for all categories of the WEAT templates. DisCo. Webster et al. [36] also utilize templates to evaluate possible biases which their approach also mitigates (see § 2.1). As an intrinsic measure, the authors present discovery of correlations (DisCo). Compared to previously discussed metrics, this metric measures the difference in predictions for the attribute token xp when varying gendered tokens (i.e. '<P> is a <mask>' for different pronouns or names instead of '<mask> is a <P>' with different professions). We experimented with the original DisCo metric, which performs statistical tests between predicted tokens, but we found that it didn't produce any statistically significant tokens. So, we simplified the metric to measure the differences in probabilities for the predicted tokens. The resulting score of our DisCo implementation can therefore also be negative, while the original version has a lower bound of 0 as it counts the number of statistically significant fills. In the remainder of this section, we discuss our evaluations of English and Dutch in their respective subsections, where we define the used datasets, models and language-specific evaluation aspects. English setup The first model we use as a teacher is the original uncased BERT model (BERT-base-uncased) as released by Devlin et al. [14], which is also the most-studied LM with regard to gender stereotypes. This model was trained on the Toronto Bookcorpus and Wikipedia, but the Toronto Bookcorpus is no longer publicly available anymore and thus hinders reproduction. For this reason, Jiao et al. [19] use only Wikipedia. We used a portion of the English section of the OSCAR corpus [27] to keep the training dataset size similar. More specifically, we used the first two shards of the unshuffled version. We recognise that there is a mismatch between the domains of the Bookcorpus and OSCAR, but we believe this is acceptable to increase reproducibility. As introduced in § 3.2, we use a set of gendered pronouns and define which ones should have the same probability. Since we use the uncased variant, we only need to define one set of rules, since 'She' and 'she' result in the same token. The tokenization method used by BERT, WordPiece, splits words and adds a merge symbol (e.g. 'word' + '##piece'), so no special care is required. For RoBERTa [22], which uses Byte Pair Encoding (BPE), a word boundary symbol is used. Consequently, a word can have different tokens and representations depending if a space, punctuation mark, mask token or sequence start token are in front of the target token. Since the Dutch model uses BPE, we will revisit this issue in § 4.2. Evaluation. To evaluate our method's performance trade-off, we finetune the obtained model on the Internet Movie Database (IMDB) sentiment analysis task [24], which was also done by BERT [14] and DistilBERT [31]. The dataset contains 25k training examples, from which we used 5k as a separate validation set, and another 25k test sequences. This is a high-level task where no gendered correlations should be used for predictions. Predicting entailment is a high-level task covered multiple times in GLUE [35], on which we also evaluated our method with the pre-trained model. For a description of this benchmark and all datasets, we refer to Wang et al. [35]. Bias evaluation. To evaluate possible problematic correlations with regard to gender stereotypes, we compute DisCo and LPBS, which we introduce at the beginning of this section. We use the Employee Salary dataset 5 [20]. Following Kurita et al. [20], we filter on the top 1000 highestearning instances as a proxy for prestigious jobs and test this for the same two templates ('<mask> is a <P>' and '<mask> can do <P>'). However, we additionally filter digits from the job titles and remove duplicate titles, to not skew the results towards more popular professions. Dutch setup We use a Dutch RoBERTa-based model called RobBERT [12] as a teacher, more specifically robbert-v2-dutch-base. This model was pre-trained on the Dutch section of the shuffled version of OSCAR 6 . Similar to the distilled version of RobBERT [13], we select a 1GB portion of the OSCAR corpus (using head, 2.5%) to illustrate the ability to perform successful knowledge distillation with only a small fraction of the data required in comparison to the pre-trained model. To create our model, we used a defined a set of rules based on the gendered pronouns 'Hij' and 'Zij' ('He' and 'She'). The tokens corresponding to these pronouns were grouped based on capitalization and included spaces, since the BPE tokenizer includes a word boundary character at the beginning of some tokens. Our method then used these rules to equalise the distributions predicted by the teacher during distillation, which we performed for 3 epochs. This took approximately 40 hours per epoch on a Nvidia 1080 Ti and a traditionally distilled model required the same time, indicating our method has very limited effect on training time. Evaluation. We compare the model created with FairDistillation to RobBERT and RobBERTje [12,13] on the same set of benchmark tasks: (i) sentiment analysis on book reviews (DBRD) [33], (ii) NER, (iii) POS tagging, and (iv) natural language inference with SICK-NL [37].These tasks are fairly high-level sequence-labelling tasks that can exhibit allocational harms, such as the predictive difference for sentiment analysis that was illustrated by Delobelle et al. [12]. Bias evaluation. We also evaluate numerically using the LPBS and DisCo metrics, but the RobBERT LM has also been evaluated by the authors on gender stereotyping using a different technique. This evaluation technique is based on a set of templates and a translated set of professions 7 from Bolukbasi et al. [5]. These professions have a perceived gender (e.g. 'actress' is a female profession and 'surveyor' is neutral), which can be correlated with the predictions by the model. The authors rank the tokens based on the predicted probability instead of using this probability directly. Interestingly, a correlation was not considered problematic, but male pronouns were predicted higher on average, even for by definition female professions (e.g. 'nun'). To compare these results, we recreate the same plot and report the Mean Ranking Difference (MRD). We focus on the gendered pronouns 'zij ' ('she') and 'hij ' ('he') for our evaluation. Table 1. English results on IMDB (sentiment analysis), GLUE [35], and two bias measures. Following Devlin et al. [14], we report F1 scores for QQP and MRPC, Spearman correlations for STS-B, and accuracy for all other tasks. Results reported by Devlin et al. [14] on the GLUE dashboard are indicated with an obelisk ( † ), while the results from [31] are also on the GLUE dev set, indicated with an asterisk ( * ). For LPBS, positive values represent more stereotypical associations, and for DisCo, lower values are more favorable. Results In this section, we present the results of the experiments (Section 4). We discuss English ( § 5.1) and Dutch ( § 5.2) results separately. We also performed experiments on French using the CamemBERT model [25], but we chose to ommit those results due to our limited understanding of the language, which we address further in Section 6. To eliminate any possible effect from hyperparameter assignments on the results, we ran each finetuning training 10 times with random hyperparameter assignments. We varied the (i) learning rate, (ii) weight decay, and (iii) the number of gradient accumulation steps to effectively scale the batch size while still fully utilizing the GPU. The full set of hyperparameters is listed in Table 3 in the supplementary materials. For the Dutch benchmarks and for the English IMDB, we select the bestperforming model based on the validation set and present the results on the held-out test set. The results from the GLUE benchmark are from the dev set, which were also the results reported by [31]. Unless indicated otherwise, all training runs are done on a single Nvidia 1080 Ti with 11 GB VRAM. All models are also comparably sized, with 66M trainable parameters each. This is 50% of the model size of the teachers. English results We observe that problematic correlations are reduced on all three metrics, as is shown in Table 1. One interesting observation-which also holds for Dutch-is that distillation in itself is already successful in mitigating these correlations. This might be related to regularization as a method to control correlations [36], but we leave this for a future study. On the IMDB task, our model suffers a 10% accuracy drop, which is significant. However, as noted in Section 4, we used a smaller training set for finetuning than BERT and DistilBERT, because we created a separate validation set from the original training set. For GLUE, the results are in line with distilBERT. We do observe some diminished scores, notably CoLA, but the overall trade-off is limited. Table 2. Dutch results on several benchmarks, namely Dutch Book Reviews (DBRD, sentiment analysis), named entity recognition (NER), part-of-speech (POS), tagging, and language inference (SICK-NL). We report bias as measured with LPBS and DisCo and additionally the mean ranking difference (MRD), which measures the preference of a language model to fill in male tokens (negative score) or female tokens (positive score). Benchmarks are reported with accuracy with 95% CI, except for the NER task, where we report the F1 score. Results indicated with † were reported by Delobelle et al. [12]. For MRD, smaller ranking differences are more favorable, for LPBS, positive values represent more stereotypical associations, and for DisCo, lower values are more favorable. Unlike the other models, we performed our FairDistillation method on 4 Nvidia V100's for 3 epochs, which took 70h per epoch. Finetuning was done on an Nvidia 1080 Ti for 4 epochs for IMDB, which took approximately 1h per run and was replicated 10 times. For GLUE, we report the dev results and did not do any hyperparameter search. We used the same hyperparameters as distilBERT [31], who also report the development set results. Benchmark scores Bias Dutch results Both the distilled RobBERT model and the model obtained with FairDistillation perform only slightly worse (within 97.5% of the original model) for both downstream tasks (see Table 2). Both models have only half the parameters compared to the original RobBERT model and are thus faster to train and deploy, making this a decent trade-off between model size and predictive performance. With no significant differences in performance between the distilled model and our FairDistillation model, this highlights the potential of our method. With regards to the bias evaluation, we observe a reduction between the original model and ours (Table 2): correlations are significantly reduced as measured by DisCo and the mean ranking of female associated tokens improved by 3.5 tokens. The only exception is LPBS [20], which incorporates a correction based on the prior probability of a token. Our method effectively corrects this prior, while still allowing the context to affect individual results with the MLM objective. Since Dutch has gendered nouns for some professions, a correlation is not necessarily undesirable, but the prior is (e.g. assuming all physicians are men). Further graphical analysis of the predicted rankings for the third person singular pronouns confirms this, as shown in Figure 2. These charts reveal that most professions are now less associated with the masculine pronoun. When considering which pronoun is ranked higher (i.e. above or below y = 0), this result is even more pronounced. RobBERT only predicted a feminine pronoun for a single profession [12], while with our method this increased to 15 professions. Limitations and ethical considerations Despite the promising results, there are several potential improvements possible to our methods, as well as some ethical considerations. First, we rely on facts that express probabilities for a single token at a time. For gender stereotyping, this is sufficient as the vocabulary usually contains the tokens of interest. However, this is not the case for many other problematic correlations, especially those affecting minority groups. Tokens that are interesting here, like names, are not in the tokenizer's vocabulary because this is created based on occurrence counts. Addressing this limitation would require extending our method to support facts that span multiple tokens. Second, the effects of our method on 'low-level' grammatical tasks require further study in English, as we focused on GLUE and sentiment analysis. The used Dutch benchmarks do cover more tasks and seem to indicate favourable reductions. Third, our work focuses on binary gender stereotypes. This leaves out a wide range of people who do not identify as such and although our method supports equalization over more than two tokens, this might be challenging if the intended words span multiple tokens. This also poses a challenge for generalizing our method beyond gender bias, since this frequently involves words that are not a single token in BERT's vocabulary. Fourth, since none of the authors is a native speaker of a language like French, we only performed a limited, exploratory evaluation of our methods with CamemBERT [25]. Our method appears successful, with improved scores of 0.04 (DisCo) and -0.85 (LPBS) compared to -1.15 (DisCo) and 1.99 (LPBS) for the original CamemBERT model [25]. The performance of the LM was also still high, with XNLI [9] scores 75.6 compared to 82.5 for CamemBERT. However, constructing correct probabilistic rules and evaluating them is tricky for non-native speakers. For example, the female variant of a profession can refer to a woman practising said profession, but also to the spouse of a man with this profession. When discussing these results with native French speakers from Wallonia, Belgium and from northern France, we realised that we are not well-suited to address this. We thus leave a more comprehensive evaluation across languages as future work. Finally, by presenting a method to remove correlations with gender stereotypes in pre-trained language models, we risk it being used as a 'rubber stamp' to absolve model creators from their responsibilities. Therefore, we urge creators to critically analyse LMs within the social context that these models will be deployed in, both with respect to both the pre-trained and the finetuned model. Conclusion We introduced a method called FairDistillation that allows to use probabilistic rules during knowledge distillation. We showed that this can effectively mitigate gender stereotypes in language models. Our method demonstrates that knowledge distillation of language models with probabilistic rules is a possible alternative to re-training in order to reduce representational harms. Even though comes at a slight cost for some downstream tasks, but we find that the overall cost is limited and can mostly be attributed to the distillation process. Fig. 1 . 1input x e.g. P('he')=P('she') Overview of the training procedure with FairDistillation for a single input sequence in English. Fig. 2 . 2Differences in predictions for the Dutch template '<mask> is een <P>' for our and the original RobBERT model. 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B Van Der Burgh, S Verberne, arXiv:1910.00896van der Burgh, B., Verberne, S.: The merits of Universal Language Model Fine-tuning for Small Datasets -a case with Dutch book reviews. arXiv:1910.00896 [cs] (Oct 2019), URL http://arxiv.org/ abs/1910.00896 Attention is all you need. A Vaswani, N Shazeer, N Parmar, J Uszkoreit, L Jones, A N Gomez, L Kaiser, I Polosukhin, I Guyon, U V Luxburg, S Bengio, H Wallach, R Fergus, S Vishwanathan, Advances in Neural Information Processing Systems. Garnett, R.Curran Associates, Inc30Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, L., Polosukhin, I.: Attention is all you need. In: Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 30, pp. 5998-6008, Curran Associates, Inc. (2017) A Wang, A Singh, J Michael, F Hill, O Levy, S R Bowman, GLUE: A multi-task benchmark and analysis platform for natural language understanding. the Proceedings of ICLRWang, A., Singh, A., Michael, J., Hill, F., Levy, O., Bowman, S.R.: GLUE: A multi-task benchmark and analysis platform for natural language understanding (2019), in the Proceedings of ICLR. K Webster, X Wang, I Tenney, A Beutel, E Pitler, E Pavlick, J Chen, S Petrov, arXiv:2010.06032Measuring and Reducing Gendered Correlations in Pre-trained Models. Webster, K., Wang, X., Tenney, I., Beutel, A., Pitler, E., Pavlick, E., Chen, J., Petrov, S.: Measuring and Reducing Gendered Correlations in Pre-trained Models. arXiv:2010.06032 [cs] (Oct 2020) G Wijnholds, M Moortgat, arXiv:2101.05716SICKNL: A dataset for Dutch natural language inference. arXiv preprintWijnholds, G., Moortgat, M.: SICKNL: A dataset for Dutch natural language inference. arXiv preprint arXiv:2101.05716 (2021), URL https://arxiv.org/abs/2101.05716
[ "https://github.com/keitakurita/contextual" ]
[ "PACS 68.35.Ja, 78.55.Cr, 78", "PACS 68.35.Ja, 78.55.Cr, 78" ]
[ "N M Litovchenko \nInstitute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine\n", "D V Korbutyak \nInstitute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine\n", "O M Strilchuk [email protected] \nInstitute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine\n", "N M Litovchenko \nInstitute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine\n", "D V Korbutyak \nInstitute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine\n", "O M Strilchuk \nInstitute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine\n", "V E Lashkaryov \nInstitute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine\n" ]
[ "Institute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine", "Institute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine", "Institute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine", "Institute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine", "Institute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine", "Institute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine", "Institute of Semiconductor Physics\nNat. Acad. of Sci\nUkraine" ]
[]
Characteristics of GaAs/InxGa1−xAs/GaAs heterostructures with a single quantum well, which were obtained at various growth parameters, are evaluated according to the results of measurements of low-temperature photoluminescence (PL) spectra and their corresponding theoretical analysis. The experimentally obtained temperature dependences of the energy position of the PL band maximum, hνmax, band half-width, W0, and intensity, I, are examined. The values of energy of local phonons, E ph , exciton binding energy, Eex, and the Huang-Rhys factor, N , are determined. A comparison between the values obtained for those quantities and the growth parameters of considered specimens allowed us to assert that the highest-quality specimens are those that are characterized by low N values and one-mode phonon spectra.K e y w o r d s: photoluminescence, quantum well, exciton, phonon
10.15407/ujpe58.03.0260
[ "https://arxiv.org/pdf/1303.3884v1.pdf" ]
117,910,830
1303.3884
e495e232bb84bee4d321d54557e21ff63e25f57c
PACS 68.35.Ja, 78.55.Cr, 78 DeCopyright De15 Mar 2013 N M Litovchenko Institute of Semiconductor Physics Nat. Acad. of Sci Ukraine D V Korbutyak Institute of Semiconductor Physics Nat. Acad. of Sci Ukraine O M Strilchuk [email protected] Institute of Semiconductor Physics Nat. Acad. of Sci Ukraine N M Litovchenko Institute of Semiconductor Physics Nat. Acad. of Sci Ukraine D V Korbutyak Institute of Semiconductor Physics Nat. Acad. of Sci Ukraine O M Strilchuk Institute of Semiconductor Physics Nat. Acad. of Sci Ukraine V E Lashkaryov Institute of Semiconductor Physics Nat. Acad. of Sci Ukraine PACS 68.35.Ja, 78.55.Cr, 78 De6715 Mar 2013EXCITONIC PARAMETERS OF In x Ga 1−x As-GaAs HETEROSTRUCTURES WITH QUANTUM WELLS AT LOW TEMPERATURES Characteristics of GaAs/InxGa1−xAs/GaAs heterostructures with a single quantum well, which were obtained at various growth parameters, are evaluated according to the results of measurements of low-temperature photoluminescence (PL) spectra and their corresponding theoretical analysis. The experimentally obtained temperature dependences of the energy position of the PL band maximum, hνmax, band half-width, W0, and intensity, I, are examined. The values of energy of local phonons, E ph , exciton binding energy, Eex, and the Huang-Rhys factor, N , are determined. A comparison between the values obtained for those quantities and the growth parameters of considered specimens allowed us to assert that the highest-quality specimens are those that are characterized by low N values and one-mode phonon spectra.K e y w o r d s: photoluminescence, quantum well, exciton, phonon Introduction In x Ga 1−x As-GaAs heterostructures are widely used in modern optoelectronics as structures that are capable of being adapted for the convenient reception, transmission, and transformation of radiation in various spectral ranges. Special attention is attracted to heterostructures with quantized layers owing to their enhanced sensitivity and a possibility to additionally vary the optical spectrum [1][2][3][4][5][6][7]. The energy of an emitted quantum in such heterostructures is governed by the distance between the sizequantization levels of electrons and holes, E e1-hh1 , which, in turn, depend on the quantum well (QW) width d and the composition of a substitutional solid solution (x is the indium content). For instance, for the typical values x = 0.2 and d = = 80Å, the changes of x by 1% and the well width by the width of a monolayer (approximately 3Å) give rise to the variations by 9 and 4 meV, respectively, in the transition energy [4]. Such a high sensitivity to the parameters imposes strict requirements on both the width and the element composition of QW. Moreover, a shortcoming of those heterostructures consists in a considerable mismatch between the lattice constants in the epitaxial layer and the substrate, which results in the emergence of substantial mechanical stresses and the generation of numerous dislocations. Buffer layers or the formation of quaternary alloys with the phosphorus additive usually reduce the influence of those undesirable factors, but not completely. Therefore, there arises the requirement in a non-destructive quantitative control over the deformation and defect factors. In this report, we pay attention to a possibility of using the interaction between phonons and excited electrons (the Huang-Rhys factor N ) for the characterization of the imperfection degree. This parameter is determined on the basis of the temperature behavior experimentally found for the exciton photoluminescence band half-widths and subjected to the corresponding theoretical analysis. Experimental Specimens and Technique Low-temperature (5-200 K) photoluminescence (PL) researches are carried out with the use of heterostructures with a single quantum well, GaAs/In x Ga 1−x As/GaAs. The specimens were grown up following the MOCVD technology. They were characterized by various contents of indium, various widths of In x Ga 1−x As quantum wells, and various thicknesses of protective GaAs layers (Table 1). Luminescence was excited by a He-Ne laser (a quantum energy of 1.96 eV, and the radiation intensity L = (3 × 10 17 ÷ 10 19 ) quantum/(cm 2 ·s)). To analyze the PL spectra, we used an MDR-23 monochromator with the spectral resolution not worse than 0.2 meV. The signal was registered with the use of a cooled FEP-62 photoelectronic multiplier. Experimental Part In the photoluminescence spectra of researched specimens, we observed the intensive bands, which correspond to the exciton recombination e1-hh1 in the quantum well in the interval T = (5 ÷ 40) K and to the recombination of free charge carriers in the quantum well in the interval T = (50 ÷ 200) K. This fact is confirmed by the dependences of the radiation intensity on the excitation one, I(L). The main growth parameters of studied specimens and the corresponding photoluminescence bands are quoted in Table 1. In Fig. 1, a, the normalized PL spectra of the specimens under investigation obtained at T = 5 K are depicted. Attention should be paid to some spectral features. First of all, it is the spread in the energy positions of PL bands, which stems from different values of quantum well widths d and indium contents x in the In x Ga 1−x As quantum well. However, even provided that the corresponding values of d and x are identical, the energy positions of PL bands can differ substantially. For instance, for specimens 4-1 and 4-2 with the identical x = 0.35 and d = 73Å, the energy positions of PL maxima are different (see Table 1); namely, hν max = 1.396 eV for specimen 4-2 and 1.253 eV for specimen 4-1. In our opinion, the origin of such a dis-crepancy consists in fluctuations of the In concentration in the QWs of specimens 4-1 and 4-2. Really, if the energy difference between the radiation maxima, ∆hν max = 0.143 eV, was caused by the difference between the quantum well widths, the latter would be amount to about 107Å (this value can be obtained with regard for the fact mentioned above that a change of the QW width by about 3Å gives rise to a variation of about 4 meV in the transition energy), but such a value is unreal. This reasoning agrees with the results of work [4], where it was noticed that a variation in the spectral positions of exciton peaks for the In x Ga 1−x As quantum well is mainly associated with the variations in the In concentration x. Different half-widths W ( Table 1) and shapes of examined PL spectra exhibited in Fig. 1, a compose another feature of those spectra. The PL band halfwidth depends on both the degree of exciton localization in the QW and the character of exciton scattering by phonons, defects, inhomogeneities at heterointerfaces, and so forth. The radiation emission spectra of some specimens demonstrate a characteristic tail of the PL band in the low-energy interval, which may be caused by the participation of phonons in the radiative recombination of excitons in the QW. As an example, we decomposed the PL band of specimen 3-2 into two components: the zero-phonon one and the phonon replica ( Fig. 1, b), with the use of the procedure proposed in work [8]. In so doing, we used an approximation that phonons of only one type participate in PL. This approach enabled us to obtain the value a b Fig. 1. PL spectra of the studied GaAs/InxGa 1−x As/GAs quantum heterostructures (a). Decomposition of PL band 3 in panel a in two components (b): the zero-phonon one and the phonon replica. Emax = 1.372 eV, E ph = 9.8 meV, the Huang-Rhys factor N = 0.3 E ph = 9.8 meV for the energy of interacting local phonons and N = 0.3 for the Huang-Rhys factor, which characterizes the strength of exciton-phonon interaction. As to the half-width of PL bands (Table 1), it changes from 7.4 meV for specimen 3-1 to 15.2 meV for specimen 4-1. In our case, the Bohr radius of an exciton in the QW is comparable with the QW width. Therefore, with a high probability, the exciton is localized at inhomogeneities of heterointerfaces. The smaller width of the PL band corresponds to a larger localization degree, as it takes place for PL by bound excitons in a bulk semiconductor. The temper-ature dependences of the PL intensity typical of the specimens under consideration are shown in Fig. 2. At low temperatures (T = 5 ÷ 40 K), the PL intensity varies weakly. As the temperature grows, the PL intensity decreases for specimens 1 and 2 and, for specimens 3 and 4, first slightly increases and then falls down, which is connected with the temperatureinduced ejection of charge carriers from the quantum well into the barrier. In Fig. 3, the temperature dependences of the radiation maximum position typical of examined specimens are depicted. The dashed curve demonstrates the results of calculation obtained in the framework of the Varshni model. For all specimens, the temperature dependences of the PL maximum position have an S-like form. At low temperatures, a considerable deviation of the calculated values from experimental ones is observed: first, the position of the PL maximum shifts toward low energies (red shift); then, up to a certain temperature, the maximum shifts backward toward high energies. At T > (60 ÷ 89) K, the maximum position shifts toward low energies in accordance with the Varshni model. In Fig. 4, the typical dependences of the PL band half-width on the temperature are shown. For specimens 1 and 2, a monotonous increase of the half-width with the temperature (the dependence of type I) is observed. Specimens 4 are characterized by a sharp initial (to a temperature of 40-80 K) growth of the half-width, then by an insignificant reduction of this parameter followed by its subsequent growth, as the temperature grows further (the dependence of type II). For specimens 3, the dependences of both types are observed. Discussion of Experimental Results The temperature dependences of the PL intensity obtained for specimens 2 and 4-2 (Fig. 2) were analyzed with the use of the Arrhenius formula I(T ) = C/[1+a 1 exp(−E a1 /kT )+a 2 exp(−E a2 /kT )]. This enabled us to determine two temperature intervals with different slopes: low-and high-temperature ones with the activation energies E a1 and E a2 , respectively (Table 1). In the low-temperature interval, E a1 = (2.5 ÷ 5) meV. Such a small value of E a1 testifies that this quantity corresponds to the delocalization energy of excitons bound at inhomogeneities of heterointerfaces in the QW at low temperatures. It is significant that the higher the delocalization energy (and, accordingly, the deeper is the potential well, which is associated with the corrugation of heterointerfaces), the narrower is the PL band, which corresponds to a more localized state of excitons (see, e.g., the PL bands for specimens 1 and 3-1). At temperatures T > 40 K, the activation energy for the temperature-induced quenching of the PL band owing to the e1-hh1 transitions equals E a2 = (50÷85) meV, and, as was indicated above, this is connected with the temperature-induced ejection of charge carriers into the barrier. The temperature dependences of the PL maximum depicted in Fig. 3 and their comparison with the results of calculations following the Varshni formula allow the binding energy of excitons in the QW to be evaluated. As one can see from Fig. 3, the energy of a quantum emitted at low temperatures (T = (5 ÷ 40) K) is lower than the energy of interband transitions calculated by the Varshni formula. The corresponding difference E ex ≈ 10 meV is just the binding energy of excitons in the QW to within the accuracy of the energy of exciton localization at heterointerfaces. The theoretical analysis of the temperature dependences obtained for the scattering parameter (the PL band half-width W ) is based on the fact that this quantity comprises the probability of the momentum scattering as a result of several independent processes (by impurities, phonons, and others), W ∼ τ t , 1 τ t ∼ W t (T ) + W ph.opt (T ) + W ph.local (T ) + ...,(1) where τ t is the lifetime of nonequilibrium charge carriers. The temperature dependences of the scattering probability are different for different mechanisms. Hence, the temperature dependence can be used to distinguish between their contributions. In particular, the Coulomb scattering by local centers depends on T , which is the most pronounced at low enough temperatures. At the same time, the role of the phonon mechanism grows with T [9, 10], according W = ΣW oi cth ω ph 2kT 1/2 , W oi = 2(2 ln 2) 1/2 N −1/2 phi ω phi ∼ N −1/2 phi ,(2) where E ph = hω ph is the energy of a phonon localized at a radiative-recombination center, and N ph is the phonon emission probability at the recombination (the Huang-Rhys factor). In the case N < 1, the latter is given by the following relation: 4. Dependences of the photoluminescence band halfwidth W on the temperature N ∼ 5 8 (E ex /E ph ) (ε 0 /ε ∞ − 1) ∼ Fig.∼ e 2 /E ph 1 3a B 1 ε ∞ − 1 ε 0 ,(3) where E ex is the exciton binding energy, a B the Bohr radius of an exciton, and ε 0 and ε ∞ are the static and high-frequency dielectric permittivities, respectively. From this formula and knowing the phonon energy E ph , it is easy to find the binding energy of an exciton [11], E ex ≈ N E ph 3 2 ε ∞ ε 0 − ε ∞ .(4) Hence, the temperature dependence W (T ) allows a number of parameters that characterize the heterostructure state -such as W 0 , E ph , and N [12] -to be obtained, as well as the Stokes shift, ∆ω st = 2N ph ω ph . It also enables one to evaluate the position of the phonon-free line by the formula ω 0 = ω max +n ω ph , where n is the number of phonon replica. Special attention should be paid to the quantity W . Its magnitude is reciprocal to the charge carrier mobility and is predicted to be much less for a perfect quantum well than that for the corresponding bulk material. However, mechanical stresses and defects can compensate this useful effect. The proposed analysis allows the contributions of different mechanisms to be estimated separately. The features in the temperature dependences of the PL intensity, maximum position, and half-width obtained in this work can be explained by the presence of localized (defect) states in the studied specimens [6,7], which are induced by fluctuations of QW dimensions, and/or by a variation of the QW composition. At low temperatures, photo-induced charge carriers (excitons) are captured by the localized potential. As the temperature is elevated to a value that corresponds to the localization energy maximum, a shift of the PL maximum position toward lower energies (the red shift) is observed, because excitons obtain a sufficient thermal energy to overcome the potential barrier and become relatively free. Some of those excitons relax into lower states, which capture them, and recombine there. In this temperature interval, we observe a drastic increase in the half-width W of the PL band, in accordance with the growth in the population of states owing to the capture of released charge carriers onto them. As the temperature grows further, the PL maximum shifts into the range of high energies, and the band half-width becomes somewhat narrower due to the thermally equilibrium distribution of excitons. This occurs until the temperature corresponding to the complete delocalization of charge carriers is attained. At higher temperatures, the e1-hh1 transitions dominate in the PL spectrum, and, according to the Varshni formula, the maximum position changes with the temperature as the energy gap width. With the use of Eqs. (2)-(4) and the experimentally obtained temperature dependences for the PL band half-width, we determined the parameters E ph , N , E ex , and the Bohr exciton radius a B (see Table 2). Let us consider this dependence of type I (it is inherent to specimens 1, 2, and 3-1). It has a monotonous character and can be described well by Eq. (2), in which the scattering processes with phonons of energies 6 to 11 meV are taken into consideration (Fig. 5, a). For structures 4, the character of the band halfwidth dependence on the temperature is of the other type (type II). For those specimens, we determined two values for the energy of local phonons (Fig. 5): (i) in the interval from 5 to 20 K, where a drastic increase of the band half-width is observed, the energy of phonons is 3.5-4 meV; (ii) in the interval from 30 to 200 K, an insignificant narrowing of the PL band is observed, which is followed by the increase of its half-width with the temperature; here, the energy of phonons equals 18-19 meV. Structures 3 revealed the dependences of both types. Specimen 3-1 demonstrated the dependence of monotonous type I, and the corresponding energy of phonons was 11 meV (Fig. 5,c, curve 1 ). Specimen 3-2 was characterized by the dependence of type II: the energy of phonons was 8 meV in the interval 5-60 K and 12 meV in the interval 60-200 K (Fig. 5, c, curve 2 ). The magnitudes of exciton binding energy obtained with the help of Eq. (4) for various specimens (E ex > 6.7 meV) considerably exceed the corresponding energy for bulk excitons in In x Ga 1−x As (E ex ≈ 3 meV) [1], which testifies to the quantization of excitons in the In x Ga 1−x As-GaAs quantum well. Let us compare the values of E ph obtained from the temperature dependences of the line width with the theoretical relations (1)-(4) (see Table 2). For every specimen, it turned out several times less than the characteristic values for bulk or surface (confinement) phonons. An evident reason is the fact that the studied specimens with heterojunctions had rather a large number of defects, probably localized at interfaces. It is known that one of the mechanisms of defect emergence in InGaAs structures consists in the segregation of clusters of the In phase if the optimum epitaxy and temperature regimes were not followed at the stage of heterostructure formation [3]. Hence, the typical defects have to include precipitates of the redundant element, i.e. indium. Now let us estimate the energy of local phonons that correspond to vibrations in vicinities of defects at the interfaces between InGaAs and In precipitates. It is known that the maximum frequency of harmonic vibrations is determined by the a b c Fig. 5. Temperature dependences of the PL band half-width, W (T 0.5 ) for (a) specimens 1 and 2, (b) specimen 4, and (c) specimens 3-1 and 3-2. Symbols demonstrate experimental results, and dashed curves correspond to their approximation by formula (2) relation [13] ω ph = 2π d E m ρ 1/2 = 2 d v,(5) where E m is Young's modulus, v is the thermal velocity, ρ = m/V i is the substance density, m is the atomic (molecular) mass, and V i is the atomic (molecular) volume. In vicinities of defect centers, E m becomes several times smaller, and ρ increases as the ratio between the densities of defect components. The lattice constant d increases as the ratio between the atomic sizes r of film components, r(GaAs)/r(InGaAs). Hence, according to the estimations made for the InGaAs heterostructure, (d V /d D ) ∼ 1/2 and ρ v /ρ D ∼ m V /m D ∼ (30/45). Whence, ω D ω V = d V d D E D E V S V S D 1/2 = 1 2 1 2 · 1 2 1/2 ∼ 1 4 .(6) Therefore, we may expect that the energy of local phonons in InGaAs is several times lower than that in the bulk, i.e. E ph ∼ 1 4 E phV ∼ 8 ÷ 10 meV. When comparing the values obtained for E ph , E ex , and N with the growth parameters of specimens, we may assert that the specimens with small values of N have the highest quality, i.e. the specimens with the highest mobility and the one-mode phonon spectrum. Conclusions Heterostructures GaAs/In x Ga 1−x As/GaAs that have a single quantum well and are characterized by various growth parameters were studied with the use of the method of low-temperature photoluminescence. The following facts were revealed. (i) The photoluminescence spectra of examined specimens demonstrate intense radiation bands. These bands are induced by the recombination of excitons in the quantum well in a temperature interval of 5-40 K and by the recombination of free charge carriers in the quantum well in the interval T = 50 ÷ 200 K. (ii) For all specimens, the temperature dependence of the photoluminescence intensity maximum position has an S-like shape. In the low-temperature interval, the values calculated within the Varshni model considerably deviate from the experimental data. (iii) The researched specimens revealed both monotonous and nonmonotonous dependences of the photoluminescence band half-width on the temperature. The features observed in the temperature dependences of the maximum position and the half-width of the PL band testify that all examined specimens contain defect states, in one quantity or another, induced by fluctuations in the QW composition; in particular, the inhomogeneities may occur owing to the segregation of In-phase clusters (in the form of 3D islands). The values of E ph determined from the temperature dependences of the PL band width turned out several times lower than the corresponding characteristic values for bulk and surface phonons for all studied specimens. Specimens with a high intensity of radiation emission and a narrow radiation band were found to be characterized by a small value of the Huang-Rhys factor and a one-mode phonon spectrum. Local phonons of two types -with energies of 3.5-4 and 18-19 meV, respectively -take part in the process of exciton scattering in specimens with low intensities of radiation and wide radiation bands (i.e. with a worse structural quality). c N.M. LITOVCHENKO, D.V. KORBUTYAK, O.M. STRILCHUK, 2013 Fig. 2 . 2Temperature dependences of the PL intensity: symbols demonstrate experimental results, dashed curves correspond to the approximation by the Arrhenius formula Fig. 3 . 3Temperature dependences of the photoluminescence maximum position, hvm: symbols demonstrate the experimental results, dashed curves show the results of calculations by the Varshni formula to the law Table 1 1Specimen is the relative content of indium, d is the quantum well width, cap is the thickness of the protective GaAs layer, hνmax is the maximum position in the PL spectrum, W is the line half-width, I is the PL intensity, E a1 and E a2 are the activation energies.x In d,Å cap,Å T = 5 E a1 , meV E a2 , meV hνmax, eV W , meV Imax, rel. units No. 1 0.16 84 220 1.3568 7.4 1093 4.5 52 No. 2 0.21 88 230 1.33 9.9 582 2.8 55 No. 3-1 0.20 92 600 1.355 7.3 4876 5 70 No. 3-2 0.20 92 600 1.3721 10.9 949 2.5 46 No. 4-1 0.35 73 600 1.253 15.2 302 2.5 85 No. 4-2 0.35 73 600 1.396 13.2 988 2.5 60 N o t a t i o n: x Table 2 2SpecimenE ph , meV N Eex, meV a B ,Å No. 1 8 0.15 9.1 122 No. 2 6.5 0.42 20.8 54 No. 3-1 11 0.08 6.7 168 No. 3-2 8 0.27 16.5 8 12 0.149 13.6 82 No. 4-1 3.2 3.1 78.7 14 18 0.13 19.0 58.8 No. 4-2 3.5 3.4 95.4 12 19 0.22 33.5 33 ISSN 2071-0186. Ukr. J.Phys. 2013. Vol. 58, No. 3 The authors express their sincere gratitude to Corresponding Member of the NAS of Ukraine V.G. Litovchenko for the discussion of the results of this work and useful advices. . I A Avrutskii, V A Sychugov, B A Usievich, Fiz. Tekh. Poluprovodn. 251787I.A. Avrutskii, V.A. Sychugov, and B.A. Usievich, Fiz. Tekh. Poluprovodn. 25, 1787 (1991). . I A Avrutskii, V G Litovchenko, Fiz. Tekh. Poluprovodn. 31875I.A. Avrutskii and V.G. Litovchenko, Fiz. Tekh. Poluprovodn. 31, 875 (1997). . M M Grigoriev, E G Gule, A I Klimovska, Yu A Korus, V G Litovchenko, Ukr. Fiz. Zh. 45853M.M. Grigoriev, E.G. Gule, A.I. Klimovska, Yu.A. Ko- rus, and V.G. Litovchenko, Ukr. Fiz. Zh. 45, 853 (2000). . I A Avrutskii, O P Osaulenko, V G Plotnichenko, Yu N Pyrkov, Fiz. Tekh. Poluprovodn. 261907I.A. Avrutskii, O.P. Osaulenko, V.G. Plotnichenko, and Yu.N. Pyrkov, Fiz. Tekh. Poluprovodn. 26, 1907 (1992). . H D Sun, R Macaluso, S Calvez, M D Dawson, J. Appl. Phys. 947581H.D. Sun, R. Macaluso, S. Calvez, and M.D. Dawson, J. Appl. Phys. 94, 7581 (2003). . N V Kryzhanovskaya, A Yu, V V Egorov, N K Mamutin, A F Polyakov, A R Tsatsulnikov, N N Kovsh, V M Ledentsov, D Ustinov, Bimberg, Fiz. Tekh. Poluprovodn. 39735N.V. Kryzhanovskaya, A.Yu. Egorov, V.V. Mamutin, N.K. Polyakov, A.F. Tsatsulnikov, A.R. Kovsh, N.N. Ledentsov, V.M. Ustinov, and D. Bimberg, Fiz. Tekh. Poluprovodn. 39, 735 (2005). . F.-I Lai, S Y Kuo, J S Wang, R S Hsiao, H C Kuo, J Chi, S C Wang, H S Wang, C T Liang, Y F Chen, J. Cryst. Growth. 29127F.-I. Lai, S.Y. Kuo, J.S. Wang, R.S. Hsiao, H.C. Kuo, J. Chi, S.C. Wang, H.S. Wang, C.T. Liang, and Y.F. Chen, J. Cryst. Growth 291, 27 (2006). . M Soltani, M Certier, R Evrard, E Kartheusev, J. Appl. Phys. 78M. Soltani, M. Certier, R. Evrard, and E. Kartheusev, J. Appl. Phys. 78, 5626 (1995). . S I Pekar, Zh. Eksp. Teor. Fiz. 20510S.I. Pekar, Zh. Eksp. Teor. Fiz. 20, 510 (1950). . C J Hwang, Phys. Rev. 180827C.J. Hwang, Phys. Rev. 180, 827 (1969). . V G Litovchenko, N L Dmitruk, D V Korbutyak, A V Sarikov, Fiz. Tekh. Poluprovodn. 36447V.G. Litovchenko, N.L. Dmitruk, D.V. Korbutyak, and A.V. Sarikov, Fiz. Tekh. Poluprovodn. 36, 447 (2002). . V A Zuev, D V Korbutyak, V G Litovchenko, A V Drazhan, Fiz. Tverd. Tela. 173300V.A. Zuev, D.V. Korbutyak, V.G. Litovchenko, and A.V. Drazhan, Fiz. Tverd. Tela 17, 3300 (1975). . I Bolesta, Solid State Physics (Lviv, Lviv. Nats. Univ.in UkrainianI. Bolesta, Solid State Physics (Lviv, Lviv. Nats. Univ., 2003) (in Ukrainian). з вiдповiдним теоретичним аналiзом. Проаналiзованi експериментально отриманi температурнi залежностi енергiї максимуму смуги ФЛ (hνmax), пiвширини (W 0 ) та iнтенсивностi I. Визначено параметри E ph (енергiя локальних фононiв), Eex (енергiя зв'язку екситонiв) та N (фактор Хуанга-Рiс). Проведена оцiнка характеристик гетероструктур з одиночною квантовою ямою GaAs/InxGa 1−x As/GaAs з рiзними ростовими параметрами за результатами вимiрювань низькотемпературних спектрiв фотолюмiнесценцiї (ФЛ). Проведене зiставлення отриманих значень E ph , Eex та N з ростовими параметрами зразкiв дає пiдставу стверджувати, що найбiльш якiсними є зразки з малим значенням N i одномодовим фононним спектромПроведена оцiнка характеристик гетероструктур з одино- чною квантовою ямою GaAs/InxGa 1−x As/GaAs з рiзними ростовими параметрами за результатами вимiрювань низь- котемпературних спектрiв фотолюмiнесценцiї (ФЛ), з вiд- повiдним теоретичним аналiзом. Проаналiзованi експери- ментально отриманi температурнi залежностi енергiї ма- ксимуму смуги ФЛ (hνmax), пiвширини (W 0 ) та iнтенсивно- стi I. Визначено параметри E ph (енергiя локальних фоно- нiв), Eex (енергiя зв'язку екситонiв) та N (фактор Хуанга- Рiс). Проведене зiставлення отриманих значень E ph , Eex та N з ростовими параметрами зразкiв дає пiдставу стверджу- вати, що найбiльш якiсними є зразки з малим значенням N i одномодовим фононним спектром.
[]
[ "Cosmic Background of Gravitational Waves from Rotating Neutron Stars", "Cosmic Background of Gravitational Waves from Rotating Neutron Stars" ]
[ "T Regimbau \nObservatoire de la Côte d'Azur\nB.P. 422906304, Cedex 4NiceFrance\n", "J A De \nObservatoire de la Côte d'Azur\nB.P. 422906304, Cedex 4NiceFrance\n", "Freitas Pacheco \nObservatoire de la Côte d'Azur\nB.P. 422906304, Cedex 4NiceFrance\n" ]
[ "Observatoire de la Côte d'Azur\nB.P. 422906304, Cedex 4NiceFrance", "Observatoire de la Côte d'Azur\nB.P. 422906304, Cedex 4NiceFrance", "Observatoire de la Côte d'Azur\nB.P. 422906304, Cedex 4NiceFrance" ]
[]
The extragalactic background of gravitational waves produced by tri-axial rotating neutron stars was calculated, under the assumption that the properties of the underlying pulsar population are the same of those of the galactic population, recently derived byRegimbau & de Freitas Pacheco (2000). For an equatorial ellipticity of ε = 10 −6 , the equivalent density parameter due to gravitational waves has a maximum amplitude in the range 2×10 −11 -3×10 −9 , around 0.9-1.5 kHz. The main reasons affecting the theoretical predictions are discussed. This background is comparable to that produced by the "ring-down" emission from distorted black holes. The detection possibility of this background by a future generation of gravitational antennas is also examined.
10.1051/0004-6361:20011005
[ "https://export.arxiv.org/pdf/astro-ph/0105260v2.pdf" ]
18,498,930
astro-ph/0105260
2e1da104e7119add264aabac6ac6f8739c89dd03
Cosmic Background of Gravitational Waves from Rotating Neutron Stars May 2001 T Regimbau Observatoire de la Côte d'Azur B.P. 422906304, Cedex 4NiceFrance J A De Observatoire de la Côte d'Azur B.P. 422906304, Cedex 4NiceFrance Freitas Pacheco Observatoire de la Côte d'Azur B.P. 422906304, Cedex 4NiceFrance Cosmic Background of Gravitational Waves from Rotating Neutron Stars May 2001Received date; accepted datearXiv:astro-ph/0105260v2 16 A&A manuscript no. (will be inserted by hand later) Your thesaurus codes are:Pulsars, Gravitational Waves The extragalactic background of gravitational waves produced by tri-axial rotating neutron stars was calculated, under the assumption that the properties of the underlying pulsar population are the same of those of the galactic population, recently derived byRegimbau & de Freitas Pacheco (2000). For an equatorial ellipticity of ε = 10 −6 , the equivalent density parameter due to gravitational waves has a maximum amplitude in the range 2×10 −11 -3×10 −9 , around 0.9-1.5 kHz. The main reasons affecting the theoretical predictions are discussed. This background is comparable to that produced by the "ring-down" emission from distorted black holes. The detection possibility of this background by a future generation of gravitational antennas is also examined. Introduction In the past years, a large number of papers devoted to stochastic backgrounds of gravitational waves appeared in the literature (see Maggiore 2000 for a recent review). Besides processes that took place very shortly after the big-bang, the emission from a large number of unresolved sources can produce a stochastic background. Supernovas (Blair et al. 1997) and distorted black holes (Ferrari et al. 1999a;de Araújo et al. 2000) are examples of sources able to generate a shot noise, while a truly continuous background could be produced, for instance, by the " r-mode" emission from young and hot neutron stars (Owen et al. 1998;Ferrari et al. 1999b). Detection of such backgrounds may probe the cosmic star formation rate up to redshifts of z ∼ 4-5, the mass range of the progenitors of neutron stars and black holes as well as the initial angular momentum of these objects. The contribution of the entire population of rotating neutron stars to the continuous galactic background of gravitational waves was considered by different authors (Schutz 1991;Giazotto et al. 1997;de Freitas Pacheco & Horvath 1997) and, more recently, this subject was revisited by Regimbau & de Freitas Pacheco (2000, hereafter RP00). In the latter, the "true" population of rotating neutron stars was synthesized by Monte Carlo techniques and its contribution to the galactic background of gravitational waves was estimated. If the planned sensitivity of the first generation of laser beam interferometers is taken into account (in particular that of the French-Italian project VIRGO), then the simulations by RP00 suggest that only few objects will contribute to the signal, if the mean equatorial ellipticity of neutron stars is of the order of ε = 10 −6 . Upper limits on ε have been obtained by assuming that the observed spin-down of pulsars is essentially due to the emission of gravitational waves. In this case, one obtains ε ≤ 10 −3 for "normal" pulsars whereas recycled or rejuvenated pulsars seem to have equatorial deformations less than 10 −8 . Although the galactic population will not produce a truly background, it remains to be investigated the integrated contribution of these objects in a fairly large volume of the universe. Estimates of this emission are important because it may rival with a possible background of cosmological origin. Since there is an upper limit to the wave frequency of the pulsar gravitational radiation, a putative cosmic background will dominate at low frequencies and, the knowledge of the spectral energy distribution of the background produced by discrete sources may help in the choice of the best frequency domain to search for a relic emission. In the present paper, the integrated gravitational emission of rotating neutron stars to the background is calculated under the assumption that the distributions of the rotation period and magnetic field derived by RP00 can conveniently be scaled to other galaxies. The plan of this paper in the following: in section 2 the model computations are described; in section 3 the results are discussed and finally, in section 4 the conclusions are given. The Model The main working hypothesis of our computations concerns the true rotation period and magnetic field distributions of pulsars. For galactic objects, these distributions were derived by RP00 using Monte Carlo simulations to reproduce the different observed distributions of physical parameters, like the period and its first derivative as well as distances, when selection effects are taken into account. These simulations permitted to establish the parameters of the initial distribution of period and magnetic field. "Faute de mieux", we assumed here that pulsars are born everywhere with rotation periods and fields obeying the same distribution laws. For a single pulsar, the frequency distribution of the total emitted gravitational energy in the source's frame is dE dν = dE dt dt dν(1) It is worth mentioning that in spite of (dE/dt) in the above equation be the pulsar gravitational wave emission rate, the time variation of the frequency is fixed by the magnetic dipole emission, responsible for the deceleration of the star. This means that angular momentum losses by gravitational waves will never overcome those produced by magnetic torques, which is equivalent to say that the average equatorial deformation is always less than 10 −3 . Under these conditions, the energy frequency distribution is (the fact that the gravitational wave frequency is twice the rotational frequency was already taken into account) dE dν = 256Gπ 6 5c 5 ε 2 I 2 ( τ m P 2 o )ν 3 = Kν 3(2) where I is the moment of inertia of the star, τ m is the magnetic braking timescale (see RP00), P o is the initial period of the pulsar and the other symbols have their usual meaning. In order to estimate the average ratio (τ m /P 2 o ), we adopted the following procedure. We have performed Monte Carlo simulations in which the distribution probabilities of the variables τ m and P o are the same as RP00. The resulting distribution of the quantity log(τ m /P 2 o ) is given in fig.1, and it can be fitted by a Gaussian with a mean equal to <log(τ m /P 2 o ) > = 12.544. Adopting this value as representative of the whole population, the constant K in eq. (2) is K = 9.27×10 35 (ε −6 ) 2 erg.Hz −4 , where we have introduced the notation ε −6 = (ε/10 −6 ). The gravitational wave flux at frequency ν o (observer's frame) due to sources localized in the redshift shell z,z+dz is dF νo = 1 4πd 2 L dE dν dν dν o dR(z)(3) where d L =(1+z)r is the distance-luminosity, r is the proper distance and the observer's frequency ν o is related to the frequency ν at the source by ν = (1 + z)ν o . The event rate inside the shell z,z+dz is dR(z) = λ p R c (z) dV dz dz(4) In the above equation, R c (z) is the "cosmic" star formation rate, λ p = 40 10 ξ(m)dm is the mass fraction of formed stars in the range 10-40 M ⊙ , supposed to be the mass range of the pulsar progenitors, with ξ(m) being the initial mass function. For a Salpeter's law (ξ(m) ∝ m −2.35 ), λ p = 4.84×10 −3 M −1 ⊙ . The element of the comoving volume is dV = 4πr 2 c H o dz E(Ω i , z)(5) where H o is the Hubble parameter and the function E(Ω i , z) is defined by the equation E(Ω i , z) = (1 + z) 2 (1 + zΩ m ) − z(2 + z]Ω v 1/2 (6) where Ω m and Ω v are respectively the density parameters due to matter (baryonic and non-baryonic) and the vacuum, corresponding to a non-zero cosmological constant. The equivalent density parameter due to the spatial curvature satisfies Ω k = 1 − Ω m − Ω v . Combining these equations, the expected gravitational wave flux at the frequency ν o is F νo = Kν 3 o λ p ( c H o )(ε −6 ) 2 zmax 0 (1 + z) 2 R c (z) E(Ω i , z) dz (7) In the literature is often used an equivalent density parameter due to gravitational waves to measure the strength of the background at a given frequency, defined by the equation Ω GW = ( 8πG 3H 2 o ) ν o F νo c 3 (8) Numerical calculations In order to evaluate numerically eqs. (7) (Perlmutter et al. 1999;Schmidt et al. 1998), which suggest that the expansion of the Universe is accelerating, support a spatially flat geometry and a non-zero cosmological constant. Both set of data are consistent with Ω m = 0.30 (including baryonic and non-baryonic matter) and Ω v = 0.70, which will be adopted in our computations. However, no significant differences in our results were observed if a model defined by Ω m = 1 and Ω v = 0 is adopted. The Hubble parameter H o was taken to be equal to 68 km/s/Mpc (Krauss 2001). RP00 assume in their simulations that the maximum rotation frequency of a newly born pulsar is 2000 Hz, which corresponds to a gravitational wave frequency of 4000 Hz. If the upper limit of the integral in eq. (7) is z max = 5, then the maximum frequency seen by the observer is ≈ 660 Hz. For higher frequencies, only near objects will contribute to the integral and the upper limit should be replaced by z max = (4000/ν o ) − 1, with ν o in Hz. This parameter affects the resulting spectrum as we shall see below. Thus, calculations with a different cutoff were also performed. Figure 2 shows the density parameter Ω GW as a function of the frequency. Labels M1 Results and H1 correspond to star formation rates given respectively by eqs. (9) and (10) and a maximum gravitational wave frequency equal to 4000 Hz. The labels M2 and H2 have the same meaning but here the maximum gravitational wave frequency cutoff is at 2000 Hz, corresponding to a minimum rotation period of 1ms. All these curves were calculated for an equatorial deformation ε=10 −6 , and we recall that the results scale as ε 2 . The numerical calculations indicate a broad maximum around 1.5 kHz, if the maximum possible pulsar rotation frequency is 2 kHz. Decreasing this limit by a half, the spectrum narrows and the maximum shifts toward lower frequencies (∼ 0.9 kHz). The amplitude is also affected, being reduced by almost one order of magnitude. This happens because according to eq.(8), the amplitude grows as ν 4 o but a lower frequency cutoff implies that only nearby objects will contribute to the integrated signal and thus reducing the amplitude at maximum. If the equatorial ellipticity may reach values of the order of 10 −6 , then the energy density of the background generated by pulsars may be comparable and even higher than that expected from newly born black holes (Ferrari et al. 1999a;de Araújo et al. 2000), originated from the collapse of massive stars (M ≥ 40 M ⊙ ). For a comparison, the spectrum corresponding to the ring-down emission from distorted black holes calculated by Ferrari et al. (1999a) is also plotted in fig.(2), appropriately scaled to the Hubble parameter here adopted. In the case of distorted black holes, the uncertainties on the estimates of the background energy density rest on the conversion efficiency of the mass energy into gravitational waves as well as on the minimum mass of the progenitor. Hot and fast rotating newly formed neutron stars may be unstable against the r-mode instability. Ferrari et al. (1999b) estimated that if all newly born neutron stars cross the "instability window" (see, for instance, Andersson et al. 2000), then the resulting density parameter has a maximum amplitude of Ω GW ≈ 2 × 10 −8 in the frequency range 0.5-1.7 kHz. This signal by far would be the dominant component of the background at these frequencies. However, according to the simulations by RP00, only few pulsars are born within the instability window, reducing the amplitude of the background due to such a mechanism by orders of magnitude. Unless the equatorial ellipticity of pulsars be substantially higher than the present expectations, the background generated by rotating neutron stars will hardly be detected by the present generation of laser beam interferometers and/or resonant detectors, but this could be a possibility for future projects presently under consideration, as the Large Scale Cryogenic Gravitational Wave Telescope (LCGT), sponsored by the University of Tokyo and the European antenna EURO (W. Winkler, private communication). The former, with a baseline of 3 km, is expected to have a 100W laser and cooled sapphire mirrors among other technological improvements. Therefore, one may expect that advanced laser beam interferometers may attain in a near future a sensitivity around 1 kHz, corresponding to a strain noise S n (ν) of about 10 −25 Hz −1/2 . On the other hand, the best strategy to detect the signal, when the detector output is dominated by the noise, which is the present case, is to correlate data from two different gravitational antennas and to assume that they have independent noise. One interesting possibility would be to correlate the output of such an advanced detector with a resonant mass detector located at the same site, having a spherical or truncated icosahedron geometry. The advantages of this geometry with respect to a longitudinal bar is that a free elastic sphere has five degenerate quadrupole modes, each of which is sensitive to a different polarization and wave direction. Moreover, for a given material and resonant frequency, a spherical detector has a cross section larger than a cylindrical one. The sensitivity of resonant spheres is limited by Brownian motion noise associated with dissipation in the antenna and transducer, as well as by the electronic noise from amplifiers. In this case, the strain noise at resonance is approximately (Coccia & Fafone 1997) S n (ν) = ( 4kT F n Q n M s v 2 s ν n ) 1/2(11) where k is the Boltzmann constant, T is the sphere temperature, F n is a dimensionless coefficient depending on each quadrupole mode (F 1 = 2.98, F 2 = 1.14, F 3 = 0.107), Q n is the quality factor of the mode, M s is the mass of the sphere, v s is the velocity of the sound and ν n is the mode frequency. For practical purposes, let us consider a sphere constituted of the aluminium alloy Al-5056. This material has a sound velocity of 5440 m.s −1 and Coccia et al. (1996) have reported Q values as high as 10 8 for temperatures below 100 mK. A sphere with a diameter of 3.5 m (mass of 60.3 tons) has the two main frequencies of the quadrupole modes at 0.8 kHz and 1.5 kHz, covering quite well the predicted interval where the maximum amplitude of the pulsar background should occur. Assuming a typical temperature of 20 mK, the expected strain noise derived from eq.(11) is S n (ν 1 ) ≈ 1.5×10 −24 Hz −1/2 . If ∆ν ≈ 20 Hz is the bandwidth of the resonant mass detector and τ is the integration time, then the expected optimized signal-to-noise S/N of the correlated outputs is (Allen 1997) S N = 3H 2 o √ 50π 2 √ ∆ντ Ω GW ν 3 S 1 (ν)S 2 (ν)(12) For one year integration, one obtains from the equation above S/N ≈ 0.2, indicating that new technology detectors may reach in the future the required sensitivity to detect such a signal. Conclusions The contribution of rotating neutron stars to the extragalactic background of gravitational waves was calculated, under the assumption that the parameters characterizing the galactic population of pulsars derived by RP00 are the same everywhere. The amplitude of the equivalent density parameter attains a maximum in the frequency interval 0.9-1.5 kHz and is in the range 10 −11 up to 3×10 −9 . The amplitude scales as ε 2 and, for a given equatorial ellipticity, the main uncertainties in the amplitude are essentially due to the cosmic star formation rate and to the rotation frequency limit at the pulsar birth, which depends on the equation of state of the nuclear matter. For "realistic" equations of state, these limits are in the rotation period range 0.5 -1.0 ms, values adopted in our calculations. The present estimates indicate that this background, having a duty cycle (measured by the product between the typical duration of the signal and the mean birth frequency of pulsars) greater than one, may have an energy density comparable to that produced by "ring-down" black holes. This emission is unlikely to be detected by the present generation of detectors. Correlated advanced detectors may reach a limit of about Ω GW ≈ 10 −10 for a flat spectrum (Maggiore 2000), which is not the present case. However, new technology detectors, which are presently under consideration, may attain the required sensitivity. In particular, taking into account the low cost of a resonant mass detector when compared with that of a laser interferometer, the installation in the same site of a "sphere" operating near the maximum predict frequency (∼ 0.9-1.5 kHz), could be the adequate strategy to detect such a signal in the future. -(8), it is necessary to specify the cosmic star formation rate R c (z) and the parameters of the world model, namely, the values of H o , Ω m and Ω v .Madau & Pozzetti (1999) have reviewed the constraints imposed by the observed extragalactic background light on the cosmic star formation rate (CSFR). They concluded that after an extinction correction of A 1500 = 1.2 mag (A 2800 = 0.55 mag), a star formation rate given by the relationR c (z) = 0.23e 3.4z (44.7 + e 3.8z ) M ⊙ .yr −1 .M pc −3(9)fits well all measurements of the UV-continuum and Hα luminosity densities from the present epoch up to z = 4. However, according toHopkins et al. (2001), even when reddening corrections are taken into account, significant discrepancies still remain between the CSFR derived from UV-Hα measurements and those derived from far-infrared and radio luminosities, which are not affected by dust extinction.Hopkins et al. (2001) assumed a reddening correction dependent on the star formation rate and obtained a good agreement between the CSFRs derived from different set of measurements. We have fitted their results by a function similar to eq.(9), namely, R c (z) = 1.207e 3.836z (39.970 + e 4.163z ) M ⊙ .yr −1 .M pc −3 (10) Taking into account the uncertainties still present in the derivation of the CSFR, we have performed calculations using both rates. Recent BOOMERANG and MAXIMA results (de Bernardis et al. 2000; Hanany et al. 2000) on the power spectra of the cosmic microwave background and observations of distant type Ia supernovas . B Allen, J.-A. Marck, J.-PAllen B., 1997, in Relativistic Gravitation and Gravitational Radiation, eds. J.-A. Marck, J.-P. . Lasota, Cambridge University Press373Lasota, Cambridge University Press, p.373 . N Andersson, D I Jones, K D Kokkotas, N Stergioulas, ApJ. 53475Andersson N, Jones D.I., Kokkotas K.D., Stergioulas N. 2000, ApJ 534, 75 . D G Blair, R Burman, L J S Woodings, M Mulder, M G Zadnik, W.F. Velloso, O.D. Aguiar, N.S. MagalhãesWorld Scientific251Blair D.G., Burman R., Woodings L.J.S., Mulder M., Zadnik M.G., 1997, in Omnidirectional Gravitational Radiation Observatory, eds. W.F. Velloso, O.D. Aguiar, N.S. Magalhães, World Scientific, p.251 . E Coccia, V Fafone, G Frossati, E Ter Haar, M W Meisel, PRLA. 219263Coccia E., Fafone V., Frossati G., ter Haar E., Meisel M.W., 1996, PRLA 219, 263 . E Coccia, V Fafone, W.F. Velloso, O.D. Aguiar, N.S. MagalhãesWorld Scientific113Coccia E., Fafone V., 1997, in Omnidirectional Gravitational Radiation Observatory, eds. W.F. Velloso, O.D. Aguiar, N.S. Magalhães, World Scientific, p.113 . J C N De Araújo, O D Miranda, O D Aguiar, PRD. 61124015de Araújo J.C.N., Miranda O.D., Aguiar O.D. 2000, PRD 61, 124015 . P De Bernardis, Nature. 404995de Bernardis P. et al, 2000, Nature 404, 995 . J A De Freitas Pacheco, J E Horvath, 56859de Freitas Pacheco J.A., Horvath J.E., 1997, PRD 56, 859 . V Ferrari, S Matarrese, R Schneider, MNRAS. 303247Ferrari V., Matarrese S., Schneider R., 1999a, MNRAS 303, 247 . V Ferrari, S Matarrese, R Schneider, MNRAS. 303258Ferrari V., Matarrese S., Schneider R., 1999b, MNRAS 303, 258 . A Giazotto, S Bonazzola, E Gourgoulhon, 55Giazotto A., Bonazzola S., Gourgoulhon E., 1997, PRD 55, 2015 . S Hanany, ApJL. 5455Hanany S. et al, 2000, ApJL 545, 5 . A M Hopkins, A J Connoly, D B Haarsma, L E Cram, astro-ph/0103253Hopkins A.M., Connoly A.J., Haarsma D.B., Cram L.E., 2001, astro-ph/0103253 . L M Krauss, astro-ph/0102305Krauss L.M., 2001, astro-ph/0102305 . P Madau, L Pozzetti, MNRAS. 3129Madau P., Pozzetti L., 1999, MNRAS 312, 9 . M Maggiore, Phys.Rep. in press (gr-qc/9909001Maggiore M., 2000, Phys.Rep. in press (gr-qc/9909001) . B J Owen, L Lindblom, C Cutler, B F Schutz, A Vechio, N Andersson, 5884020Owen B.J., Lindblom L., Cutler C., Schutz B.F., Vechio A., Andersson N., 1998, PRD 58, 084020 . 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[ "Mapping Dark Matter in the Milky Way using Normalizing Flows and Gaia DR3", "Mapping Dark Matter in the Milky Way using Normalizing Flows and Gaia DR3" ]
[ "Sung Hak Lim \nDept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA\n", "Eric Putney \nDept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA\n", "Matthew R Buckley \nDept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA\n", "David Shih \nDept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA\n" ]
[ "Dept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA", "Dept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA", "Dept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA", "Dept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA" ]
[]
We present a novel, data-driven analysis of Galactic dynamics, using unsupervised machine learning -in the form of density estimation with normalizing flows -to learn the underlying phase space distribution of 6 million nearby stars from the Gaia DR3 catalog. Solving the collisionless Boltzmann equation with the assumption of approximate equilibrium, we calculate -for the first time ever -a model-free, unbinned, fully 3D map of the local acceleration and mass density fields within a 3 kpc sphere around the Sun. As our approach makes no assumptions about symmetries, we can test for signs of disequilibrium in our results. We find our results are consistent with equilibrium at the 10% level, limited by the current precision of the normalizing flows. After subtracting the known contribution of stars and gas from the calculated mass density, we find clear evidence for dark matter throughout the analyzed volume. Assuming spherical symmetry and averaging mass density measurements, we find a local dark matter density of 0.47 ± 0.05 GeV/cm 3 . We fit our results to a generalized NFW, and find a profile broadly consistent with other recent analyses.
null
[ "https://export.arxiv.org/pdf/2305.13358v1.pdf" ]
258,840,880
2305.13358
b11f23f7f8490349ca00c7bcf44899f6b345b045
Mapping Dark Matter in the Milky Way using Normalizing Flows and Gaia DR3 Sung Hak Lim Dept. of Physics and Astronomy NHETC 08854Rutgers, PiscatawayNJUSA Eric Putney Dept. of Physics and Astronomy NHETC 08854Rutgers, PiscatawayNJUSA Matthew R Buckley Dept. of Physics and Astronomy NHETC 08854Rutgers, PiscatawayNJUSA David Shih Dept. of Physics and Astronomy NHETC 08854Rutgers, PiscatawayNJUSA Mapping Dark Matter in the Milky Way using Normalizing Flows and Gaia DR3 We present a novel, data-driven analysis of Galactic dynamics, using unsupervised machine learning -in the form of density estimation with normalizing flows -to learn the underlying phase space distribution of 6 million nearby stars from the Gaia DR3 catalog. Solving the collisionless Boltzmann equation with the assumption of approximate equilibrium, we calculate -for the first time ever -a model-free, unbinned, fully 3D map of the local acceleration and mass density fields within a 3 kpc sphere around the Sun. As our approach makes no assumptions about symmetries, we can test for signs of disequilibrium in our results. We find our results are consistent with equilibrium at the 10% level, limited by the current precision of the normalizing flows. After subtracting the known contribution of stars and gas from the calculated mass density, we find clear evidence for dark matter throughout the analyzed volume. Assuming spherical symmetry and averaging mass density measurements, we find a local dark matter density of 0.47 ± 0.05 GeV/cm 3 . We fit our results to a generalized NFW, and find a profile broadly consistent with other recent analyses. I. INTRODUCTION Multiple lines of evidence indicate that the majority of matter in the Universe is dark -that is, it does not interact with the known particles through electromagnetic or strong nuclear interactions. Measurements of galaxy rotation curves [1][2][3], galaxy clusters [4,5], the early Universe [6], and gravitationally-lensed systems [7] cannot be explained without the addition of new particles beyond the Standard Model. Despite a robust experimental program, dark matter has resisted attempts to measure its particle physics interactions, and astrophysical probes remain a vital window into its properties. In this work, we employ a novel method to determine the dark matter density around the Solar location within the Milky Way, one that has never before been applied to data. Our method is fully data-driven, uses the measurements of stellar position ⃗ x and velocities ⃗ v made possible by the Gaia Space Telescope [8,9], and is powered by modern, unsupervised machine learning methods. The phase space density f (⃗ x, ⃗ v) of the population of stars within the Milky Way obeys the collisionless Boltzmann Equation: ∂f ∂t + v i ∂f ∂x i = ∂Φ ∂x i ∂f ∂v i .(1) Here, Φ is the total gravitational potential, which can be related to the total mass density ρ using the Poisson Equation 4πGρ = ∇ 2 Φ.(2) Assuming that the phase space density of the stars is in equilibrium ∂f /∂t = 0, the 3D acceleration field −∂Φ/∂x i can be derived from knowledge of the stellar phase space density f today. A further derivative of Φ then gives the total mass density ρ; the dark matter density can then be calculated assuming knowledge of the baryonic components. Measuring the stellar phase space density and its derivatives has traditionally been difficult, given the relatively high dimensionality (six) of the data. Instead, measurements of the local Galactic potential have used moments of the Boltzmann Equation -the Jeans Equation -along with simplifying assumptions (axisymmetry, specific functional forms, and/or small mixed radial and altitude "tilt" terms) which allows for relatively stable calculation of numeric derivatives from stellar data binned in the position coordinates. We refer to Refs. [10][11][12][13][14][15] for recent examples of these techniques. The modern machine learning method known as normalizing flows provides a new approach to this problem that allows direct, unbinned access to the phase space density, independent of symmetry assumptions. Normalizing flows (reviewed in Ref. [16]) are a class of unsupervised deep learning algorithms that are sufficiently expressive to allow accurate modeling of the phase space density of high-dimensional data. Using normalizing flows, Refs. [17][18][19] directly solved the Boltzmann Equation for synthetic mock stellar data drawn from smooth analytic simulations of a galaxy [20,21]. In Ref. [22], we demonstrated this approach on a fully-cosmological N -body simulation of a Milky Way-like galaxy, including realistic Gaia-like measurement errors and the impact of departures from symmetry and lack of equilibrium. Here, we apply the algorithm developed in Ref. [22] to Gaia Data Release 3 (DR3) [23] itself. Using a population of stars within 4 kpc of the Sun for which full kinematic solutions are available, we measure the gravitational acceleration and total density everywhere within the sphere except in regions near the disk where dust extinction is significant. We also estimate the total uncertainty on our acceleration and density measurements throughout this region. At each location, these uncertainties include statistical uncertainty, Gaia measurement uncertainty, and an estimate of fit uncertainty from the normalizing flows. As the flow is extremely expressive, our errors should encompass a fuller range of possible shape variations of the density profile consistent with data, compared to many other approaches that fit the profile to a (perhaps overly-restrictive) functional form. Using existing measurements of the baryon density, we find clear evidence of a non-baryonic component to the mass density throughout the Solar neighborhood. Though these measurements do not rely on any assumptions of symmetry within the data, imposing spherical symmetry on the dark matter density allows us to average measurements at different locations and reduce errors. Under this additional assumption, we find a dark matter density of 0.47 ± 0.05 GeV/cm 3 at the Sun's distance from the Galactic Center. We also fit our density measurements to a generalized Navarro-Frenk-White (NFW) [24,25] profile, though with considerable uncertainties on the best-fit parameters. Future data releases from Gaia will increase the number of stars with full kinematic information by a factor of ∼ 5, as well as decreasing the proper motion measurement errors by ∼ 2. Combined with anticipated improvements in understanding the error model and quantifying dust extinction, the accuracy of the dark matter density measurements obtained using this method can be greatly increased in the near future. In Section II, we introduce the Gaia DR3 dataset used to train our normalizing flows. Section III contains the core results from our analysis: here we show our estimates of the phase space density using normalizing flows, followed by the calculations of accelerations and mass density using the collisionless Boltzmann Equation. Additionally, we investigate evidence for departures from equilibrium in the data and perform self-consistency checks. In Section IV we discuss our results and future directions for flow-based modelling of Galactic dynamics. II. GAIA DR3 The Gaia space telescope [23,26] has revolutionized precision astrometry. As of its third data release (DR3), Gaia has measured the full 6D kinematics of nearly 33 million stars [27]. This unprecedented volume of data, combined with state-of-the-art density estimation techniques, allows for robust mapping of the Milky Way's phase space density. For those stars with full kinematic information, the angular positions (α, δ), proper motions on the sky (µ * α , µ δ ), and parallax ϖ are measured by the Gaia photometer while the radial velocity V rad and apparent magnitude G RVS are measured by the Gaia spectrometer (RVS). Nearly 100% of stars with apparent magnitude brighter than G RVS = 14 are expected to have 6D kinematics in Gaia DR3 [27]. In addition to a maximum apparent magnitude, the Gaia spectrometer also has a minimum apparent magnitude due to saturation [28]. Stars brighter than G RVS ≈ 3 are not included in the dataset used for this analysis. In terms of G RVS and the parallax-derived distance (d/kpc) = (1 mas/ϖ), the absolute magnitude M G is M G = G RVS − 5 log 10 (d/kpc) − 10.(3) In this analysis, we do not use other spectral information (such as BP − RP color). The three coordinate systems used in this work are shown: the Galactocentric Cartesian coordinates (x, y, z), the spherical coordinates (r, θ, ϕ), and the cylindrical coordinates (R, ϕ, z). The lines through the observational volume with low dust extinction along which we measure accelerations and mass densities are shown in color. In orange, we show two lines at z = +1.5 kpc, one varying r and another ϕ. The two corresponding lines at z = −1.5 kpc are shown in purple. The line parameterized by polar arclength s = r⊙ × (π/2 − θ) passing through the Solar location is shown in green. Given the approximate axisymmetry of the baryonic disk of the Galaxy and the approximate spherical symmetry of the dark matter distribution, it is useful to consider the data in both spherical and cylindrical coordinates, as well as Galactocentric Cartesian coordinates. The three coordinate systems we use are shown in Figure 1 and are defined as follows: 1. Our Cartesian coordinate system places the x − y plane in the Galactic disk, with x = 0 at the Galactic center and the Sun along the +x axis. The +z axis (perpendicular to the disk) is oriented so that the net rotation of the disk stars in the −y direction. From the fundamental kinematic properties (parallax, angular position, etc.) measured by Gaia, the positions and velocities in these three coordinate systems can be obtained, using the parallax to calculate distance and assuming a Galactocentric Solar position and velocity of (8.122, 0.0, 0.0208) kpc [29,30] and (−12.9, −245.6, 7.78) km/s [29,31,32], respectively. A. Tracer Population Selection To extract the accelerations from the derivatives of stellar phase space density using the Boltzmann Equation, the population of stars in question must be complete, unbiased, and in dynamic equilibrium. These conditions are not satisfied by the full Gaia 6D dataset, which is complete in observed (but not absolute) magnitude. Stars which are intrinsically dim but nearby are included, while intrinsically dim but more distant stars were not observed and thus are absent. We first remove from our analysis dataset stars without spectroscopic or photometric magnitudes, as well as stars with large relative parallax errors (3σ ϖ > ϖ). Of the 31,532,490 stars in Gaia DR3 with full 6D kinematic solutions available within 10 kpc, 29,855,114 stars remain after these cuts. We further remove 13 abnormally fast stars, with speeds higher than 1000 km/s relative to the Galactic rest frame. Within 4 kpc of the Sun, 24,789,061 stars remain, their position-space distributions are shown in the left column of Figure 2, and their color-magnitude diagram (BP − RP versus M G ) is shown in Figure 3. In order to correct for the bias in the dataset towards stars with low observed magnitude, we require that every star in our restricted sample is bright enough to have been observed by Gaia regardless of its position within a sphere around the Sun. We set this sphere to be 4 kpc in radius, both to limit fractional parallax errors (which grow with distance), as well as to allow a sufficiently large number of stars to pass the completeness criteria. That is, we require that the absolute magnitude M G of every star in our sample is bright enough that, if the star was located at 4 kpc, it would have an observed magnitude G RVS < 14 and thus be above the completeness limit of the Gaia spectrometer: M G < 14 − (5 log 10 4 + 10).(4) This selection criterion is shown as the white dashed line in Figure 3. Applying this selection removes 63.6% of the stars in the sample with distance < 10 kpc, leaving 10, 876, 430. Note that we place this selection criterion on the magnitude without extinction-correction. We find that the dust extinction corrections available from Gaia are not uniform across the 4 kpc sphere centered on the Sun. As a result, correcting for extinction induces position-dependent suppression factors in the derived phase space density. In the future, these corrections may be more uniformly available, which would likely improve this analysis. Due to the saturation limit G RVS ≈ 3, the closest star to the Sun in our final dataset is 30.4 pc away, and there are somewhat fewer stars within 50 pc in the analysis than would otherwise be expected. As this is a very small volume and total number of missing stars (relative to the entire dataset), we do not find that the normalizing flows greatly suppress the phase space density of stars near the Solar location. For our final analysis, we require the measured distance of stars from the Sun to be less than 4 kpc, leaving 5,811,956 stars. The larger dataset (including stars out to 10 kpc) will be used when quantifying the impact of measurement errors on our final results, as we will discuss below. In the middle column of Figure 2, we show the stellar number density as a function of Galactocentric position after selecting on bright stars Eq. (4). As can be seen, the many large density variations present in the number density prior to the magnitude selection are suppressed after requiring nearby stars to be bright enough to be seen if they were at the edge of the dataset. Additionally, the observational bias causing the stellar densities to be higher near the Sun is suppressed after the selection and the number density of stars is (correctly) seen to be rising towards the Galactic center. The sight-lines to stars are less affected by dust off of the Galactic disk. Within the dust-filled disk, significant density variations remain after the application of Eq. (4). This is most obviously seen as a triangular wedge of apparent low stellar density towards the Galactic center at |z| ∼ 0, but striations can also be seen in other directions within the disk. These features are primarily in the disk (|z| ≲ 1 kpc) and largely trace known dust features, as can be seen in the left-hand column Figure 4, where we overlay extinction maps of dust [33] with the observed number density of stars after the magnitude selection criterion has been applied. For this comparison, we stitch together two three-dimensional maps: bayestar19 [34] covering declination > 30 • , and marshall [35] covering b ∈ [−10 • , 10 • ] and ℓ ∈ [−100 • , 100 • ]. For the position not covered by either 3D maps, we use a 2D dust map SFD [36] for distances d > 1.0 kpc. The regions of low stellar density match to the known dust clouds, including the Serpens-Aquila rift, the Rho Ophiuchi cloud complex, Lupus, the Dark River, Pipe Nebula, the Northern Coalsack, and the Vela molecular ridge. In the right column of Figure 2, we show the stars after the magnitude selection Eq. (4) with the additional requirement of |z| > 1 kpc. No dusty features are visible outside the disk. Although the effects of dust occlusion are greatest inside the disk (|z| ≲ 1 kpc), one notable exception is in the neighborhood closest to the Sun: for the stars nearest to us, dust does not have enough opportunity to accumulate along the sight-lines, and so extinction remains relatively low. As a result, measurements of phase space density along lines perpendicular to the disk and passing through the Solar location should be reliable and mostly unaffected by dust. This is seen in Figure 5, where we show the binned stellar densities as a function of s and azimuthal angle ϕ, along with contours of dust extinction. The magnitude selection results in a sample of tracer stars that is unbiased by apparent magnitude within 4 kpc, and thus complete in absolute magnitude (modulo the effects of dust). Roughly 69% of these stars belong to the red clump, a sizable sub-population of the red giant branch (RGB) that is tightly clustered in the color-magnitude diagram (indicated by a white rectangle in Figure 3). Red clump stars are typically between 1-4 Gyrs old [37]. Older stellar populations are preferred for kinematic studies of the Milky Way, as they have had sufficient time to equilibrate over the Galaxy's dynamic timescales [38]. Stars from the red clump meet this criteria, and have been used in a recent precision Jeans analysis of the Solar neighborhood [11]. In principle, the percentage of the sample composed of red clump stars could be increased by selecting based on extinction-corrected color in addition to M G . However, similar to extinctions for the Gaia spectrometer, extinction-corrected colors are not uniformly available across the sky. Using corrected colors would lead to selection effects in the phase space density and errors in the solution to the Boltzmann equation. While we expect this to be less of an issue in future data releases from Gaia, for this analysis we do not apply color selection criteria. We will apply our normalizing flow algorithm to the complete, unbiased dataset within 4 kpc, but with the knowledge that gravitational accelerations or mass densities within the disk far from the Sun are not reliable due to dust extinction. In Figure 1, we show the dustavoiding lines along which we measure accelerations and mass densities. B. Measurement Errors Gaia DR3 provides measurement errors in the form of Gaussian standard deviations for the measured quantities of angular position, parallax, proper motion, RVS, and G RVS . In the left and center panels of Figure 6, we show histograms of the kinematic errors for all stars within 4 kpc which pass the selection criteria described in Section II A. Propagating the errors from the angular positions, proper motions, parallax, and RVS measurements results in a covariance matrix for the measurement errors in the Cartesian coordinates used for training the normalizing flows. The median standard deviation in dis- tance and speed of stars perturbed by the Cartesian error model is shown in the right panel of Figure 6. The parallax and RVS measurements contribute the dominant sources of error to the kinematic solutions. In particular, parallax errors in some cases can be larger than the measured parallax; this often results in negative parallaxes when varying within errors. Considerable literature exists on the conversion of parallax to distance, including the impact of measurement errors [39][40][41][42][43]. We note that Gaia also provides a secondary distance estimate distance gspphot, which is inferred from a Markov Chain Monte Carlo algorithm that uses spectra from the BP and RP bands, apparent G magnitude, and parallax ( [44,45]). Unfortunately, the availability of distance gspphot (as well as other data products from this fit) is not uniform across the sky. Given this limitation, we find the most uniform reliable distance estimate to be the inverse parallax after stars with large relative parallax errors are removed. Our error propagation procedure follows the outline of Ref. [22]: we generate multiple variations of the original dataset after varying every star's kinematic features within their Gaussian errors, using the correlation matrix given the errors provided by Gaia DR3 converted into Cartesian coordinates. For each varied iteration of the data, we calculate first the phase space density, then the gravitational acceleration and mass density (using the algorithms described in Section III). The variations of these quantities over multiple error-smeared datasets provides an estimate of the impact of measurement errors on the derived quantities. In order to avoid the edge effect of stars migrating out of the 4 kpc-radius sphere when errors are applied without a corresponding inward migration of more distant stars, for each variation of the dataset we apply the error-smearing to stars in the 10 kpc-radius sphere. After applying the errors to the larger dataset, we select those stars whose error-smeared distances place them within 4 kpc of the Sun. III. ACCELERATIONS AND MASS DENSITIES FROM NORMALIZING FLOWS The techniques used in this work to calculate accelerations and mass density from the collisionless Boltzmann equation using normalizing flow-derived phase space density are described fully in Ref. [22]. We describe each component of our analysis briefly here, and present results using Gaia DR3. A. Phase Space Densities To determine the gravitational accelerations − ⃗ ∇Φ from the collisionless Boltzmann Equation applied to a stellar population, we must first determine the phase space density f (⃗ x, ⃗ v) for that population. We accomplish this using normalizing flows, an unsupervised machinelearning technique for density estimation. Normalizing flows are based on invertible transformations of a simple base distribution (such as standard normal distribution) into a more complicated distribution. As long as the transformation is expressive enough, normalizing flows are able to model a variety of distributions so that this model can be used as a free-form density estimator. This expressivity is generally achieved using neural networks with bijective constraints. We use two normalizing flows to model the stellar number density n(⃗ x) and the conditional velocity distribution p(⃗ v|⃗ x) separately: f (⃗ x, ⃗ v) = n(⃗ x)p(⃗ v|⃗ x).(5) Note that the estimation of p(⃗ v|⃗ x) requires conditional density estimation; this is easily implemented in the normalizing flows architecture by simply making the transformation conditioned on the position vector ⃗ x. In addition to increasing the accuracy of the overall phase space density, this decomposition is also helpful for solving the Boltzmann equation. In both cases, the loss functions L x and L v for training the density models n(⃗ x) and p(⃗ v|⃗ x) are the negative log-likelihoods of the data: L x = − 1 N N i=1 log n(⃗ x i ) (6) L v = − 1 N N i=1 log p(⃗ v i |⃗ x i ),(7) where N is the size of an input dataset. In this work, we implement the normalizing flows using the Masked Autoregressive Flow (MAF) [46] architecture as the transformation model. GELU activation [47] is used in order to model a smooth differentiable transformation. We use the ADAM optimizer [48] to train the flows. We construct a validation dataset using 20% of the stars, randomly selected. Early stopping with a patience of 50 epochs is used, and we select the model with the lowest validation loss. For the central values, we use Monte Carlo cross-validation and ensemble averaging in order to fully utilize the dataset and reduce noise in our density estimation. That is, we prepare 100 different random splits of training and validation datasets and train a MAF for each. The density is estimated by ensemble averaging the probabilities given by each of the MAFs. All the neural networks are implemented using PyTorch [49] and nflows [50]. The details of our neural network architectures and data preprocessing prior to training are identical to those described in Ref. [22]. In Figure 7, we plot one-dimensional histograms of the selected data compared to the density estimated by the normalizing flows. Black markers are the histogram of selected stars randomly downsampled to 20% of the original size (this matches the size of the randomly-selected validation dataset). Red lines are the histograms of synthetic stars generated by sampling from the MAFs. For this figure, we upsample from the MAF to 100 times the size of the Gaia dataset. The small deviations near the Solar location in x and y histograms are due to dust extinction in the disk (as seen in Figures 2 and 4). Along lines of sight with significant dust, there are sharp falloffs in the apparent stellar density which do not reflect the true density of stars. The MAFs are constrained to be smooth functions, and as a result, they have difficulties modeling such discontinuities, and the quality of density estimation may degrade. The bump in the x histogram is mainly due to dust clouds in Cygnus (ℓ ∼ 75 • ) and Vela (ℓ ∼ −95 • ), Figure 4), we can see that the MAF is correctly learning the overall density scales and substructures visible in the stellar counts within the disk, though detailed comparison suggests that the MAF may have difficulty replicating the small-scale sharp features of the dust. MAFs are known to generate spurious "wrinkles" around sharp edges or topologically non-trivial structures within data [51]. Though the wrinkles in density themselves are small, numerical artifacts will be amplified in density derivative estimations, resulting in biases in the acceleration and mass density estimations. As previously mentioned, we avoid low-|z| regions which are far from the Solar location to minimize the effects of dust on our measurements of the acceleration and mass density fields. Density estimators that improve over the MAF results are available [52], but are computationally expensive for the density derivative estimations; we leave studies of such architectures for future work. σ v rad (km/s) 10 −2 10 −1 10 0 σ µ α i (mas/yr) σ α σ δ σ µ α σ µ δ B. Accelerations Using the learned phase space density, we estimate the acceleration field ⃗ a = − ⃗ ∇Φ by solving the collisionless Boltzmann equation. As observed in Refs. [18,22], since Φ is a function of position only, we can approximately solve the Boltzmann Equation for the acceleration by the least square method using generated velocity samples at a given ⃗ x and minimizing the residual ∂f /∂t. That is, at a given position ⃗ x, we calculate the accelerations from the trained MAFs by finding the value of ⃗ a(x) that minimizes the following mean square error (MSE): L a = d 3 ⃗ v p(⃗ v|⃗ x) v i ∂f ∂x i + a i (⃗ x) ∂f ∂v i 2 .(8) We evaluate this integral by quasi-Monte Carlo integration [22] and minimize it analytically to determine the best-fit acceleration value. For each position, we sample 10,000 velocities with |⃗ v| < 600 km/s in order to obtain a stable acceleration solution with small statistical errors. Given that Eq. (8) is highly overconstrained, the residual ∂f /∂t is not guaranteed to be zero. However, our minimization procedure follows (in spirit) the approximation typically made in density measurements based on the Boltzmann Equation, which assume the phase space density f is in equilibrium (∂f /∂t = 0). Within the Gaia EDR3 [56] This work ax (10 −10 m/s 2 ) −2.32 ± 0.16 −1.94 ± 0.22 ay (10 −10 m/s 2 ) 0.04 ± 0.16 0.08 ± 0.08 az (10 −10 m/s 2 ) −0.14 ± 0.19 −0.06 ± 0.08 |⃗ a| (10 −10 m/s 2 ) 2.32 ± 0.16 1.94 ± 0.22 Milky Way this assumption would imply an axial and z-symmetric potential (and thus mass density and acceleration fields which respect these symmetries). There is evidence that the Milky Way is not in dynamic equilibrium [53][54][55]. As we do not enforce these symmetries in our acceleration calculation and since our MSE minimization can result in residual non-zero ∂f /∂t, we can perform closure tests to estimate the amount of deviation from equilibrium and validate whether the accelerations are derived in a self-consistent manner. We discuss this in detail in Section III C. In Table I, we show the estimated acceleration at the Solar location, averaged over a ball centered at the Sun and with a radius 100 pc. 1 The measured radial acceleration at the Solar location is (1.94±0.22)×10 −10 m/s 2 ; the other components are negligible in comparison. The error includes measurement and statistical errors estimated by resampling and bootstrapping, the variation from multiple independent trainings of the MAFs, and the standard deviation of the estimated acceleration within the 100 pc ball. Our result agrees with the acceleration measurement of Gaia EDR3 [56] within 2σ. This latter measurement is obtained from the proper motion of quasars caused by secular aberration due to the orbital motion of the Solar system in the Milky Way. Differences between these two measurement techniques may indicate local disequilibrium in the Solar neighborhood, as we discuss in Section III C. In the future, the dataset available to Gaia DR4 or DR5 may be sufficient to resolve any differences at a statistically significant level. In Figure 8, we show the estimated accelerations in spherical coordinates (a r , a θ , a ϕ ), measured along curves of r, ϕ, and s = r ⊙ · (π/2 − θ) through the volume around the Sun (avoiding regions of significant dust where the MAF results may be inaccurate). The orientations of these measurement curves within the Galaxy are sum-marized in Figure 1, and we describe them further here: • Accelerations as a function of r are measured along lines passing through (x, y, z) = (8.122, 0, ±1.5) kpc -that is, through points 1.5 kpc above (towards Galactic North) and below (towards Galactic South) the Galactic disk at the Solar location. • Accelerations as a function of ϕ are calculated along curves located 1.5 kpc North and South of the Solar location passing through (x, y, z) = (7.982, 0, ±1.5), constrained so that r = r ⊙ along the curve. These 1.5 kpc offsets above and below the disk are chosen to avoid the dominant sources of dust extinction and resulting wrinkles in the MAF number densities, as seen in Figures 2 and 4. • We measure accelerations as a function of the polar arc s passing through the Solar location. These are only acceleration measurements we make closer to the disk than |z| = 1.5 kpc. Sight-lines to this arc pass through the disk only near to the Earth, where extinction is less of a concern (see Figure 5). For accelerations within 150 pc of the Sun, we average over a sphere (as in Table I) to smooth over dust within the disk. No averaging is applied to points further away. C. Testing Equilibrium The acceleration field calculated using the MAF comes from minimizing ∂f /∂t, but makes no assumptions about symmetry. In the self-gravitating, rotationally supported Milky Way, departures from axisymmetry would presumably imply disequilibrium, and hence ∂f /∂t ̸ = 0. Our method's sensitivity to the residual ∂f /∂t and freedom from enforced symmetry allows us to perform self-consistency checks in a manner that has not typically been possible for previous measures of acceleration and mass density built on the Boltzmann Equation or its moments. We consider two such checks here. First, in the absence of measurement errors or disequilibrium and assuming maximally expressive MAFs, the northern (z > 0) and southern (z < 0) measurements of a r should be equal, and the a θ values should be equal up to a negative sign. If the system respects axisymmetry, a ϕ should be zero. In the subpanels of Figure 8, we show the deviations from North-South symmetry for a r and a θ over the magnitude of the acceleration, as well as a ϕ /|⃗ a|. As can be seen, these measures of disequilibrium and/or departure from axisymmetry are at most 10%, suggesting that the system is in equilibrium at least to this level. Note that several of the largest deviations occur for s ∼ 0 and at the edge of the observational volume. The former location is in the disk where residual dust extinction may be a concern, and the latter is where measurement errors are large. In these regions then, these measures may , and a ϕ (bottom row) as a function of spherical radius r (left column), polar arclength s = r⊙ × (π/2 − θ) (middle column), and azimuthal arclength (r 2 ⊙ − z 2 ) 1/2 × ϕ (right column). Radial and azimuthal measurements are taken along lines off-set from the Galactic midplane, at z = 1.5 kpc (yellow lines, labeled "North") and z = −1.5 kpc (purple lines, labeled "South"). See Figure 1 for the orientation of these measurement lines within the Galaxy. The inner dark bands and outer light bands denote 1σ and 2σ uncertainties respectively. Subpanels in the top and middle rows show the fractional difference between the North and South measurements (appropriately mirrored for a θ ). Subpanels in the bottom row show the ratio of a ϕ to the magnitude of ⃗ a. overestimate the amount of disequilibrium in the tracer stars. A second test of equilibrium can be obtained if a sepa-rate measure of the acceleration at specific point is available. In such circumstances, we can directly calculate ∂f /∂t, or the equivalent ∂ ln f /∂t (which can be inter- preted as an inverse timescale). The acceleration measured at the Solar location by referencing distant quasars [56] is such an independent measurement. At the Solar location, we sample 10 6 velocities from the MAF. For each velocity we solve Eq. (1) for ∂ ln f /∂t assuming the acceleration −∇Φ is given by the acceleration found by Ref. [56] (Table I, left column). The resulting distribution of ∂ ln f /∂t is shown in Figure 9. As can be seen, this quantity is roughly peaked at zero (suggesting equilibrium), but it is unclear whether the spread in the distribution is due to non-equilibrium structures, or is equivalent to the expected residual one would obtain given the over-constrained system of equations and imperfections in the MAF modelling of the data. To probe this, we also show in Figure 9 the residual ∂ ln f /∂t which results when the averaged MAF-derived acceleration (Table I, right column) at the Solar location is used as −∇Φ over the sample of 10 6 velocities. The mean of this distribution is consistent with zero, indicating that the dark matter accelerations obtained from minimizing Eq. (8) have themselves no preference for non-equilibrium. Furthermore, the distributions of ∂ ln f /∂t assuming either acceleration are very similar. The quasar-derived acceleration results in a ∂ ln f /∂t distribution which is slightly broader, with some trace of multi-modality. However, there is no statistically significant difference between the two. We therefore conclude that the MAF results are consistent with local equilibrium given the current data, as might be expected given the similarity of the accelerations obtained from our method and Ref. [56]. To summarize, under these closure tests, we find that our assumption of equilibrium in the analysis of the Boltzmann Equation is good to within the ∼ 10% level, as quantified by deviations from axisymmetry and northsouth reflection symmetry in the measured accelerations. Using an independently determined acceleration at the Solar location [56], we also find closure (self-consistency) of our equilibrium assumption to within the level of precision of the MAF density estimation. In the future, larger datasets, better control of errors, and more expressive normalizing flow architectures may allow tighter distributions of ∂ ln f /∂t, allowing statistically significant measures of equilibrium and axisymmetry. D. Mass Densities Given an acceleration calculated at every point in the volume around the Sun, the total mass density ρ can be calculated using the Poisson Equation Eq. (2). This requires taking an additional numeric derivative of Φ. Again following the algorithm of Ref. [22], we calculate the second derivative of Φ at position ⃗ x by convolving the accelerations over a truncated Gaussian kernel K centered at ⃗ x: 4πG ρ * K = (∇ 2 Φ) * K.(9) Here * indicates convolution with the kernel. The Gaussian kernel is truncated at |⃗ x/ ⃗ h| = 2 where ⃗ h = (0.5, 0.5, 0.2) kpc. This ellipsoidal kernel averages the mass density at scales below ⃗ h, and thus we are not sensitive to density fluctuations at scales smaller than this. We draw 3,200 points to calculate the mass density at each ⃗ x, again using quasi-Monte Carlo sampling. With the total mass density calculated at a point, we can then extract the dark matter mass density using a model for the baryonic components of the Milky Way. The details of our baryonic model are discussed in Appendix A. Briefly, we model the baryonic components of the Galaxy at the Solar cylindrical radius with 15 components (ten stellar and five gas components) as per Ref. [57], with refinements from Refs. [58,59]. Each component has an exponential or Gaussian suppression as a function of height |z| off of the disk -note that all these baryonic components assume axial symmetry. The parameters of each model can be found in Table A. Baryons dominate the mass density within |z| ∼ 0.5 kpc of the disk. In Table II, we report the total mass density, the baryonic density, and the inferred dark matter density calculated at the Solar location, using a single averaging kernel centered at the Sun. In Figure 10, we show the total mass density ρ, the modeled baryonic mass, and the dark matter mass density as a function of arc length s above and below the disk from the Solar location at ϕ = 0. Due to the finite kernel size, we cannot calculate densities at the edge of the 4 kpc sphere centered on the Sun, and (as errors increase at larger distances) we instead show densities only up to s = 3 kpc. 1.18 ± 0.14 0.47 ± 0.05 1. 38 TABLE II: MAF-estimated densities at the Solar location or averaged at the Solar radius r⊙. The dark matter density is the difference between the total mass density ρ⊙ and the baryonic mass density (obtained from the model described in Appendix A). The averaged mass density at r = r⊙ is the weighted average of the dark matter mass density evaluated at 15 independent points at r = r⊙ (as in Figure 10) and subtracting the baryonic mass density in that region. The curve in Figure 10 is created from mass densities sampled with kernel centers more densely packed than the kernel size. As a result, neighboring density values are correlated. This smooths the resulting curve as a function of s, and statistical fluctuations in individual uncorrelated measurements appear as extended bumps. This effect is also the likely source of the slight off-set in the peak in the total mass density from s = 0. The mass densities of 15 points with non-overlapping kernels are indicated in Figure 10. Considering these independent measures 2 of the total mass density, the dark matter density at r = r ⊙ is statistically consistent with a constant value, as expected for a spherically symmetric dark matter profile. Assuming this symmetry, we can use these measurements of the total mass density at independent locations within the dataset to obtain an averaged dark matter density at fixed radius r after subtracting the baryonic density at each location (note that this is an imposition of a symmetry assumption that has not been used in this analysis otherwise). The resulting averaged dark matter mass density is reported in Table II. In Figure 11, we compare our result for ρ DM (r = r ⊙ ) with a set of recent measurements (made using a variety of methods). As can be seen, our result is consistent with the previous literature, with competitive and comprehensive error bars. Our errors contain both statistical and systematic errors in the calculation of the total (dark matter plus baryonic) mass density. If we again assume spherical symmetry, we can further investigate the dependence of the dark matter mass density on spherical radius r. We show in Figure 12 the dark matter mass densities evaluated at various Galactocentric radii r, along z = ±1.5 kpc off-sets from the disk (and at different values of ϕ). Considering independent measurements over a range of r values within the 4 kpc observational sphere, we fit the measured ρ DM as a function of r to the generalized NFW profile ρ DM (r) = ρ 0 r rs β 1 + r rs 3−β ,(10) with free parameters ρ 0 , r s , and β (in the standard NFW, β = 1). In performing our fit, we adopted a truncated Gaussian prior for β centered at 2 with a width of 2. We adopt This is broadly in agreement with other recent fits to the dark matter density profile (e.g., Refs. [59,86]). We plot our best-fit profile in Figure 12. We note that -given the range of complete data available from Gaia, measurement errors, and dust extinction -our dataset does not extend to the low-r regime, and so does not yet provide significant discriminating power between different models of dark matter density profiles. IV. DISCUSSION Using normalizing flows to model the phase space density of bright stars within Gaia DR3, we have -for the first time -measured the gravitational acceleration and mass density within the local volume of the Milky Way without assumption of functional form or symmetry. The resulting acceleration and mass density maps across a three dimensional volume around the Sun provide a unique window into Galactic structure and dynamics. We find the acceleration at the Sun's location to be nearly entirely radial, with a r = (1.94 ± 0.22) × 10 −10 m/s 2 . This is within 2σ of recent measurements of the acceleration using quasars [56]. If the differences between the two measurements were taken at face value, this would suggest ∼ 10% disequilibrium. Similar deviations from the assumptions of equilibrium and axisymmetry are found when comparing the measurements in the Galactic North and South. Though at this time the statistical and systematic errors on our MAF-derived accelerations are too large to draw confident conclusions about departures from equilibrium, our method allows us to test assumptions of equilibrium and symmetry in a way that has not been previously possible. Without assuming any symmetries, our measurement of dark matter at the Solar location is ρ DM,⊙ = 0.32 ± 0.18 GeV/cm 3 . This large uncertainty can be significantly reduced with the further imposition of spherical symmetry -allowing the averaging of measurements at different locations. Under this assumption, we find a dark matter density at the Solar radius of 0.47 ± 0.05 GeV/cm 3 , in agreement with previous measurements using a variety of other techniques. Recall that our density measurements encompass a wide range of possible dark matter density profiles due to the expressivity of the MAF; this variation is fully incorporated into our error budget. The dark matter density profile is consistent with a generalized NFW. We find a preference for a small scale radius r s , in keeping with recent measurements from rotation curves [59]. However, our current range of reliable data does not yet extend deep into the central region of the Galaxy, and so our statistical preference for this value of r s is not high. These machine-learning assisted measurements of Galactic acceleration and mass density are expected to improve significantly in the near future. Gaia DR4 and DR5 are expected to expand the number of stars with full six-dimensional kinematics by a factor of five, with an associated decrease in statistical errors in our determination of the phase space density. Measurement errors are likewise expected to improve by a factor of two, and improvements in our analysis technique to correct for the bias introduced by these errors are possible. Dust extinction is a major limitation in applying our algorithm to regions close to the disk or toward the Galactic center. An improved understanding of the effects of dust on the measured Gaia stellar features which are uniform across the sky will allow greater accuracy in our measurements of phase space, acceleration, and mass density. Indeed, normalizing flows may play a role in data-driven modelling of dust extinction, which we will Overall, we can expect improvements in architecture, analysis, data quantity, and data quality to allow great advances over these first results. In addition to greatly improved precision and better constraints on the density profile, future analyses based on these techniques may be able to directly probe the departures from equilibrium within the Milky Way, especially when combined with other measurements of local acceleration, such as those based on quasars [56], pulsars [87], or binary systems [88]. Orange points were evaluated at z = +1.5 kpc, purple points were evaluated at z = −1.5 kpc. The average dark matter density at the Solar radius r⊙ is shown as a red star. The maximum likelihood fit to a generalized NFW profile is shown in a solid black line, with 1σ and 2σ variance across the posterior distribution of models explored in Figure 13 shown as green and yellow error bands, respectively. A recent fit [86] to a standard NFW profile to the rotation curve of the Milky Way is shown as a dashed black line. Table A. The McKee model draws on a collection of pre-Gaia star counts and gas surveys. Recent updates using Gaia data [59,92] have not substantially altered the model. The stellar bulge and halo do not significantly contribute to the mass density in the Solar neighborhood, and so they are not independently modelled. However, this model does include halo stars within the disk. The McKee model characterizes the surface densities Σ 0,i and effective scale heights h z,i , as well as the functional form for the number density as a function of z, for each component. Assuming direct proportionality between number and mass density -i.e., assuming no chemical evolution of each component as a function of z -we can model each mass density in the same way as the number density. All but 12 components were fit to an exponential mass density profile ρ exp,i (⃗ x) = Σ 0,i h i e −|z|/hi e −(R−r⊙)/h R .(A1) The remaining three components (H 2 , HI CNM , and HI WNM,1 ) were fit to the following Gaussian mass density profile ρ gauss,i (⃗ x) = Σ 0,i √ πh i e −|z| 2 /h 2 i e −(R−r⊙)/h R . We supplement each component of the McKee model with a radial scale length h R,i , informed by the baryonic model used in Ref. [59]. Stellar populations were assigned a scale radius of 2.35 kpc, all HI gasses were assigned a large scale radius of 18.24 kpc, and H 2 gas was assigned h R = 2.57 kpc. HII gas was assigned an arbitrary scale radius of 2.5 kpc. Fue to its overall small contribution to the surface density, uncertainty in this scale length does not have a significant effect on the mass density. It should also be emphasized that the precise details of the baryonic radial profile are insignificant for |z| > 500 pc and for all R within our observational window. Values and uncertainties (when available) for Σ 0,i , h z,i , h R,i , and the corresponding ρ 0,i ≡ ρ i (z = 0) for all 15 components are given in Table A. We follow Ref. [58] in assigning 10% uncertainties to any unreported surface density errors in the McKee model. In total, we expect approximately 8% uncertainty in the baryonic mass density at the Solar location ρ b,⊙ . We do not follow Ref. [57] in inflating this uncertainty to 15%, although we agree that the systematic uncertainties in the original error model are likely underestimated. As discussed in Ref. [85], de-projecting the McKee model out of the plane into a volume density ρ b (z) comes with systematic uncertainties. Based on comparisons to the MWPotential2014 Milky Way mass model implemented in the galpy library, our de-projection of the McKee model does not deviate significantly from other standard baryonic mass distributions. Additionally, these systematic uncertainties become subdominant to our other measurement and statistical uncertainties of ρ(z) for |z| > 500 pc, where ρ b is greatly sub-dominant to ρ DM . Only the estimate for the dark matter density at the Solar location ρ DM,⊙ is significantly affected by our choice and interpretation of the McKee model, as the Solar System is located near the midplane. Finally, when evaluating the baryonic mass density at a particular point, we convolve our estimate for ρ b (⃗ x) over the same quasi-random Gaussian kernel used to estimate ρ(⃗ x). The convolved baryonic mass density profile ρ b (⃗ x) * K is comparable to ρ b (⃗ x) everywhere except for the disk, where the peak at z = 0 is widened and shortened due to this convolution. As a result, if the vertical profile falls off too quickly with z, ρ b (z = 0) * K will be underestimated. This introduces systematic uncertainty in the estimate of ρ b * K for |z| ≲ 500 pc in the disk, compared to |z| ≳ 500 pc in the more robust halo region. [57], some values of hz,i are a weighted average of two scale heights, representing an "effective" scale height. ρ0,i = ρ(z = 0)i, where ρ(z = 0)i is the volume mass density of each component in the midplane. ρ(z = 0)i is computed from Σ0,i and hz,i via ρ(z)i = (1/2)∂Σ(z)i/∂z and by assuming a form for Σ(z). For an exponential mass profile, Σ(z)i = Σ0,i(1 − exp(−|z|/hz,i)). hR,i is the exponential scale radius of each component, capturing the first-order radial behavior of baryonic mass density (ignoring detailed features such as spiral arms or clouds). FIG. 1 : 1Schematic representation of the Solar location (red dot) relative to the Galactic Center (black dot). The 4 kpc observation volume is shown as a transparent grey sphere. FIG. 2 : 2Density plots of the stars with full 6-dimensional kinematic information available from Gaia within 4 kpc of the Solar location in the x − y (top row) and x − z (bottom row) planes. The left column shows all 24,789,061 fully-characterized stars. The middle column shows the 5,811,956 remaining stars after applying the selection criteria described in the text. The right column applies the additional requirement of |z| > 1 kpc, resulting in 470,702 stars. FIG. 3 : 3Absolute magnitude MG and color BP − RP of all 24,789,061 stars within 4 kpc of the Sun with 6-dimensional kinematic information measured by Gaia. The horizontal dashed white line denotes the magnitude completeness criteria Eq. (4). All 5,811,956 stars above the dashed white line are bright enough to be observable for Gaia regardless of position in the 4 kpc sphere. The visible peak in the white box is the red clump. All features in the space of uncorrected MG and BP − RP will appear to be smeared towards the bottomright of this figure, as dust extinction both dims and reddens stars. FIG. 4 : 4Stellar number density from Gaia data (left) and MAF-learned number density (right) n(⃗ r) on Galactic longitude (ℓ) and latitude (b) planes at distance (top) 1 kpc, (center) 2 kpc, and (bottom) 3 kpc. The stellar number densities are obtained by directly counting the number of stars within 0.1 kpc from the center of the pixel. The MAF-learned number densities are described in Section III. The Ks-band extinction maps at a given distance (obtained from the dustmaps package[33]) are shown as contours. The maps are Gaussian-kernel smoothed with a bandwidth of 8 • . We show four extinction value contours from white to blue: 0.15, 0.3, 0.45, and 0.60. FIG. 5 : 5Stellar number density from Gaia data (left) and MAF-learned number density (right) as a function of s and r⊙ · ϕ, at r = r⊙ = 8.122 kpc. The stellar number densities are obtained by directly counting the number of stars within 0.1 kpc from the center of the pixel. The left-hand plot contains 206,852 stars. The MAF-learned number densities are described in Section III. We overlay the contour plot of three dimensional Ks band extinction map obtained from the dustmaps package[33]. The map is Gaussian-smoothed with a bandwidth 0.2 kpc. The dashed black contour is an extinction of 0.05, and the four contours from white to blue are extinction values of 0.15, 0.3, 0.45, and 0.60. FIG. 6 : 6Left and Center: 1σ standard deviations (as reported by Gaia DR3) of the measured kinematic parameters for stars passing the distance, magnitude, and parallax error selections of Section II A. Errors are shown for parallax ϖ (left plot, lower axis), radial velocity v rad (left plot, upper axis), angular ICRS position (α, δ) (center plot, lower axis), and ICRS proper motion (µα, µ δ ) (center plot, upper axis). We discard stars with relative parallax errors larger than 1/3. Stars with radial velocity uncertainties larger than 40 km/s are not included in Gaia DR3[27].Right: Median standard deviation in Cartesian position (blue line, left vertical axis) and velocity (yellow line, right vertical axis) as a function of stellar distance d. The 16 th and 84 th percentiles of these standard deviations are shown as asymmetric bands around the central median value. while the bump in the y histogram is due to dust clouds closer to the Galactic center. These variations within the learned probability densities are not mismeasurement of the data; rather the MAFs are correctly capturing the dust-filled substructures of the Milky Way. The right-hand column of Figure 4 shows the estimated number density n(⃗ r) on the Galactic longitude and latitude (ℓ, b) planes at different distances d from the Sun, again with the dust map overlaid. Comparing to the observed binned number densi-ties (left column of FIG. 7 : 7Normalized histograms of (top) position components and (bottom) velocity components for selected stars in Gaia DR3 (downsampled to 20% of the original size). The red lines are the histograms for synthetic stars sampled from the normalizing flows. The error bars are the 1σ statistical uncertainty. Below the main plots, we show the pull distributions, (i.e., the difference between Gaia and MAF histograms divided by the 1σ statistical uncertainty). FIG. 8 : 8Accelerations ar (top row), a θ (middle row) FIG. 9 : 9Distribution of ∂ ln f /∂t at the Solar location using the MAF-derived Solar acceleration a⊙ (blue) and the Gaia EDR3 quasar-derived measurement of a⊙ (orange). a flat prior for r s within [0, 20], and a truncated Gaussian prior centered at 40 × 10 −2 M ⊙ /pc 3 with a width of 80 × 10 −2 M ⊙ /pc 3 for ρ 0 in the range [0, 200] × 10 −2 M ⊙ /pc 3 . This choice of priors restricts the model space to a physically realistic domain: the Gaussian prior for ρ 0 prevents arbitrarily large density parameters, leading to extremely small values of the scale radius r s . We show the posterior distribution for ρ 0 , r s , and β in Figure 13 with median values and 16-th and 84-th percentile uncertainties of ρ 0 = 30.1 +64.4 −25.1 × 10 −2 M ⊙ /pc 3 , r s = 3.5 +5.4 −1.4 kpc, and β = 1.0 +1.2 −0.7 . The best fit model is ρ 0 = 23.5 × 10 −2 M ⊙ /pc 3 , r s = 3.6 kpc, and β = 1.1. FIG. 10 : 10Top: Independent measurements of total mass density as a function of polar arclength s, for three values of ϕ: ϕ = 0 (black circles), ϕ = +0.209 (blue triangles), and ϕ = −0.209 (red squares). Points at nonzero ϕ have been offset for visibility. Horizontal error bars indicate the 1σ width of the kernel in the s-direction. The mass density at ϕ = 0 for s values with overlapping kernels is shown in the black curve (note that due to kernel overlap these measurements are correlated at length scales of ∼ 0.4 kpc). The baryonic mass density model is shown with the blue curve. Dark and light bands represent 1σ and 2σ uncertainties, respectively. Bottom: Independent measurements of dark matter mass density as a function of s and ϕ, obtained by subtracting the baryonic density in the top panel from the corresponding total mass density. The best-fit value of ρDM(r = r⊙) = 0.47 ± 0.05 GeV/cm 3 is shown as a horizontal black line enclosed by green and yellow bands denoting the 1σ and 2σ uncertainties, respectively. FIG. 12 : 12Dark matter mass densities measured at 12 independent points as a function of Galactic spherical radius r. for a fit to a generalized NFW profile Eq. (10). The median values of each parameter are shown above the marginalized 1D histograms, with the 16 th and 84 th percentile values shown as error bounds. The maximum likelihood model is shown in black. The McKee mass model is broken into 15 components: five types of gas, seven stellar populations, and three populations of compact objects. Each component is described in TABLE I : IGalactic acceleration at the Solar location ⃗ a⊙ in Cartesian coordinates, calculated by averaging the solution to the Boltzmann equation within a 100 pc sphere centered on the Sun. We list for comparison the acceleration at the Solar location obtained from Gaia DR3 quasar measurements[56]. TABLE III : IIIAll parameters of the baryonic mass model used in this work, as well as their respective references. Σ0,i is the surface density of each baryonic component at the Solar radius R = r⊙ = 8.122 kpc. hz,i is the scale height, indicating how far above and below the midplane each component extends. As in Ref. . The cylindrical coordinate system (R, ϕ, z) uses the same z axis as the Galactocentric Cartesian coordinates. The Sun is located at ϕ = 0, with positive ϕ increasing towards the +y axis.3. The spherical coordinate system (r, θ, ϕ) has the same ϕ angle as the cylindrical coordinates, and measures +θ relative to +z axis, with the Galactic disk at θ = π/2. We define the polar arc length above or below the Galactic disk at the Solar radius (along the θ direction) as s ≡ r ⊙ × (π/2 − θ), with r ⊙ = 8.122 kpc. Recall that our data has a small void at the Solar location with radius ∼ 50 pc, due to the Gaia lower limit on the apparent magnitude. Although our normalizing flows have smoothed out this region, we apply this averaging in order to suppress any artifacts from the interpolation. As MAFs are highly expressive models with a very large number of parameters, it is reasonable to assume that points in space separated at sufficiently large length scales are described by unique model parameters and are not correlated. For all practical purposes, if the averaging kernels of two mass density estimates do not overlap, the two estimates are independent measurements. AcknowledgementsWe thank Adrian Price-Whelan and Mitchell Weikert for helpful discussions. This work was supported by the DOE under Award Number DOE-SC0010008. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos. esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www. cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. We thank the Gaia Project Scientist Support Team and DPAC for their work in development and maintenance of the PyGaia code. The authors acknowledge the Office of Advanced Research Computing (OARC) at Rutgers, The State Uni-versity of New Jersey for providing access to the Amarel cluster and associated research computing resources that have contributed to the results reported here. URL: https://oarc.rutgers.eduAppendix A: Baryon Mass ModelIn order to estimate the local density field of dark matter ρ DM (⃗ x) given the total density field ρ(⃗ x), we must estimate the local distribution of baryonic mass density ρ b (⃗ x) in the Milky Way. We follow Refs.[15,58,79,83,87,[89][90][91]and base our model for ρ b (⃗ x) in the Solar neighborhood on the work of Ref.[57](hereafter referred to as the McKee model), an extensive compilation of estimates of the surface mass densities of gas, stars, and compact objects in the Solar neighborhood. . Casagrande, 60Casagrande, (2020) [60] . Pato, Pato, et al., (2015) . Huang, Huang, et al., (2016) . Benito, Benito, et al., (2019) . Karukes, Karukes, et al., (2019) . Lin, 65Lin, et al., (2019) [65] . De Salas, de Salas, et al., (2019) . Ablimit, Ablimit, et al., (2020) . Benito, 68Benito, et al., (2020) [68] . Sofue, 69Sofue, (2020) [69] . Zhou, Zhou, et al., (2022) . Ou, 59Ou, et al., (2023) [59] . 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[ "Generating functions of multiple t-star values of general level *", "Generating functions of multiple t-star values of general level *" ]
[ "Zhonghua Li \nSchool of Mathematical Sciences\nTongji University No\n1239 Siping Road200092ShanghaiChina\n", "† \nSchool of Mathematical Sciences\nTongji University No\n1239 Siping Road200092ShanghaiChina\n", "Lu Yan \nSchool of Mathematical Sciences\nTongji University No\n1239 Siping Road200092ShanghaiChina\n" ]
[ "School of Mathematical Sciences\nTongji University No\n1239 Siping Road200092ShanghaiChina", "School of Mathematical Sciences\nTongji University No\n1239 Siping Road200092ShanghaiChina", "School of Mathematical Sciences\nTongji University No\n1239 Siping Road200092ShanghaiChina" ]
[]
In this paper, we study the explicit expressions of multiple t-star values of general level. We represent the generating functions of multiple t-star values of level N with an arbitrary number of blocks of twos, which generalize the results for multiple zeta-star values and multiple t-star values. These generating functions can provide a formula of multiple t-star values of level N . As applications, some evaluations of multiple t-star values of level N with one-two-three or more general indices are given.
null
[ "https://export.arxiv.org/pdf/2305.14031v1.pdf" ]
258,841,005
2305.14031
fb839cfdc31d307aa32f8c7a50db27c794f85f1e
Generating functions of multiple t-star values of general level * May 2023 Zhonghua Li School of Mathematical Sciences Tongji University No 1239 Siping Road200092ShanghaiChina † School of Mathematical Sciences Tongji University No 1239 Siping Road200092ShanghaiChina Lu Yan School of Mathematical Sciences Tongji University No 1239 Siping Road200092ShanghaiChina Generating functions of multiple t-star values of general level * May 2023multiple t-star valuegenerating function 2010 Mathematics Subject Classification 11M3205A15 In this paper, we study the explicit expressions of multiple t-star values of general level. We represent the generating functions of multiple t-star values of level N with an arbitrary number of blocks of twos, which generalize the results for multiple zeta-star values and multiple t-star values. These generating functions can provide a formula of multiple t-star values of level N . As applications, some evaluations of multiple t-star values of level N with one-two-three or more general indices are given. Introduction Let C, R, Z and N be the set of complex numbers, real numbers, integers and positive integers, respectively. And let N 0 = N ∪ {0}. A finite sequence s = (s 1 , . . . , s r ) of positive integers is called an index. If s 1 > 1,the index s is called admissible. The quantities s 1 + · · · + s r and r are called the weight and depth of the index s. In [26], H. Yuan and J. Zhao introduced the multiple zeta values of level N, where N is a fixed positive integer. For an admissible index s = (s 1 , . . . , s r ) and any a = (a 1 , . . . , a r ) ∈ R N r , R N = Z/NZ, the multiple zeta values of level N is defined by ζ N (s; a) = k 1 >···>kr ≥1 k i ≡a i mod N 1 k s 1 1 · · · k sr r . Recently, the multiple t-(star) values of level N were studied by Z. Li and Z. Wang in [17]. For an admissible index s = (s 1 , . . . , s r ) and any a ∈ {1, 2, . . . , N}, the multiple t-value of level N and the multiple t-star value of level N are defined by t N,a (s) = t N,a (s 1 , . . . , s r ) := m 1 >···>mr >0 m i ≡a mod N 1 m s 1 1 · · · m sr r = k 1 >···>kr≥0 1 (Nk 1 + a) s 1 · · · (Nk r + a) sr , t ⋆ N,a (s) = t ⋆ N,a (s 1 , . . . , s r ) := m 1 ≥···≥mr >0 m i ≡a mod N 1 m s 1 1 · · · m sr r = k 1 ≥···≥kr≥0 1 (Nk 1 + a) s 1 · · · (Nk r + a) sr , respectively. Here if the index is empty, we treat the values t N,a (∅) = t ⋆ N,a (∅) = 1. Setting N = a = 1, we get the multiple zeta value and the multiple zeta-star value(cf. [12,27]) which are the multiple t-value and the multiple t-star value [13] respectively. In [28], D. Zagier gave explicit evaluation formulas for ζ({2} a , 3, {2} b ) and ζ ⋆ ({2} a , 3, {2} b ) by establishing the generating functions, where a, b ∈ N 0 and the {s} a denotes the sequence of s repeated a times.. After that, other proofs of Zagier's evaluation formulas have appeared, see for example [11,[14][15][16]. Especially in 2017, Kh. Hessami Pilehrood and T. Hessami Pilehrood [8] gave another proof of the theorem of Zagier by using a new representation of the generating function of ζ ⋆ ({2} a , 3, {2} b ). Later in [9], Kh. Hessami Pilehrood and T. Hessami Pilehrood studied the generating function of multiple zeta-star values with an arbitrary number of blocks of twos. As applications, they [10] provided another way to derive some known evaluations of multiple zeta-star values, and gave new explicit evaluations of ζ ⋆ ({{2} a , 3, {2} a , 1} d ) and ζ ⋆ ({{2} a , 3, {2} a , 1} d , {2} a+1 ), where a ∈ N 0 and d ∈ N. Note that there have been numerous contributions on evaluations of multiple zeta values and multiple zeta-star values, for example, see [1,2,11,20,22,27,29] and the references therein. On the other hand, by the method of establishing the generating functions, T. Murakami [21] obtained the evaluation of t({2} a , 3, {2} b ) and S. Charlton [3] gave the evaluation of t({2} a , 1, {2} b ). In a recent paper [16], L. Lai, C. Lupu and D. Orr provided another proof of Murakami's result of t({2} a , 3, {2} b ). In [17], Z. Li and Z. Wang found evaluations of t ⋆ ({2} a , 3, {2} b ) and t ⋆ ({2} a , 1, {2} b ) . Similarly as in [9], we [18] studied the generating functions of multiple t-star values with an arbitrary number of blocks of twos. And as a result, we obtained similar evaluation formulas as in [17] for t ⋆ ({2} a , 3, {2} b ) and t ⋆ ({2} a , 1, {2} b ), and gave some evaluations of multiple t-star values with more general indices. In this paper, we consider the generating functions of multiple t-star values of level N with an arbitrary number of blocks of twos, which generalize the results for multiple zeta-star values and multiple t-star values in [9,18]. In Section 2, we study the generating functions of the finite form of multiple t-star values of level N. Then we obtain explicit expressions of the generating functions of multiple t-star values of level N, and get a formula of multiple t-star values of level N in Section 3. Finally, we give some evaluations of multiple t-star values of level N in Section 4. The following notations will be used throughout the paper. For any index s = (s 1 , . . . , s r ) and k, m ∈ N 0 , we define that W # k,m (s) =      k≥l 1 ≥···≥lr≥m 2 △(k,l 1 )+△(l 1 ,l 2 )+···+△(lr ,m) (N l 1 +a) s 1 ···(N lr+a) sr if s = ∅ and k ≥ m, 2 △(k,m) otherwise, (1.1) where △(k, m) = 0 if k = m, 1 if k = m. For any c ∈ N 0 , define δ(c) =      2 if c = 0, 1 if c = 1, 0 if c ≥ 3.1 (Nk 1 + a) s 1 · · · (Nk r + a) sr , where N is a fixed positive integer and a ∈ {1, 2, . . . , N}. Here we treat n t ⋆ N,a (∅) = 1. For d ∈ N 0 , c = (c 1 , . . . , c d ) ∈ N d and z = (z 0 , z 1 , . . . , z d ) ∈ C d+1 , we denote the generating function of the finite form of multiple t-star values of level N with an arbitrary number of blocks of twos as D n (c; z) = b 0 ,b 1 ,...,b d ≥0 n t ⋆ N,a ({2} b 0 , c 1 , {2} b 1 , . . . , c d , {2} b d )z 2b 0 0 z 2b 1 1 · · · z 2b d d , Recall that the {s} a denotes the sequence of s repeated a times. Here we give an explicit expression of the generating function D n (c; z). Theorem 2.1. Let a, N be integers with 1 ≤ a ≤ N. For any n, d ∈ N 0 , c = (c 1 , . . . , c d ) ∈ (N \ {2}) d and z = (z 0 , z 1 , . . . , z d ) ∈ C d+1 with |z j | < 1, j = 0, 1, . . . , d, we have D n (c; z) = n≥k 0 ≥k 1 ≥···≥k d ≥0 N a N n+1 2 (n − k 0 )! 2a N n+k 0 +1 · 2a N k d k d !(Nk d + a) × d i=0 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ), (2.1) where k −1 = −1, δ i = δ(c i ) + δ(c i+1 ) with c 0 = 1 and c d+1 = 0, the Pochhammer symbol (a) n is defined by (a) n = Γ(a + n) Γ(a) = 1 if n = 0, a(a + 1) · · · (a + n − 1) if n > 0. Applying power series expansion 1 (2k−1) 2 −z 2 = ∞ b=0 z 2b (2k−1) 2b+2 and comparing the coefficients of z 2b 0 0 z 2b 1 1 · · · z 2b d d in (2.1), we get the following corollary. Corollary 2.2. Let a, N be integers with 1 ≤ a ≤ N. For any n, d ∈ N 0 , c 1 , . . . , c d ∈ Z + \ {2} and b 0 , b 1 , . . . , b d ∈ N 0 , we have n t ⋆ N,a ({2} b 0 , c 1 , {2} b 1 , . . . , c d , {2} b d ) = n≥k 0 ≥k 1 ≥···≥k d ≥0 N a N n+1 2 (n − k 0 )! 2a N n+k 0 +1 · 2a N k d k d !(Nk d + a) × d i=0 (−1) k i δ i (Nk i + a) 2b i +3−δ i W # k i−1 ,k i ({1} c i −3 ), where k −1 = −1, δ i = δ(c i ) + δ(c i+1 ) with c 0 = 1 and c d+1 = 0, and (a) n is the Pochhammer symbol. Preliminary Lemmas We prove several finite sum formulas that will be needed in the proof of Theorem 2.1. Lemma 2.3. For integers a, N with 1 ≤ a ≤ N and n, l ∈ N 0 , we have n k=l+1 2 k + a N (n − k)! 2a N n+k+1 = n − l (n − l)! 2a N n+l+1 (2.2) and n k=l+1 2(−1) k (n − k)! 2a N n+k+1 = (−1) l+1 (n − l) n + a N (n − l)! 2a N n+l+1 . (2.3) Proof. We may assume that n ≥ l + 1. To prove (2.2), we observe that 2 k + a N (n − k)! 2a N n+k+1 = I(n; k + 1) − I(n; k) (2.4) for any integer k with l + 1 ≤ k ≤ n, and I(n; j) = −1 (n − k)! 2a N n+k for l + 1 ≤ j ≤ n and I(n; n + 1) = 0. Then summing both sides of (2.4) over k from l + 1 to n, (2.2) follows easily. Similarly, we set J(n; j) = (−1) j+1 n + a N (n − j)! 2a N n+j for l + 1 ≤ j ≤ n and J(n; n + 1) = 0. It is easy to see that for any integer k with l + 1 ≤ k ≤ n, 2(−1) k (n − k)! 2a N n+k+1 = J(n; k + 1) − J(n; k). Then summing both sides of the above equation over k from l + 1 to n, we obtain (2.3). Lemma 2.4. For integers a, N with 1 ≤ a ≤ N and n, l, c ∈ N 0 , we have n k=l (−1) k (n − k)! 2a N n+k+1 W # k,l ({1} c ) = (−1) l (Nl + a) (Nn + a) c+1 (n − l)! 2a N n+l+1 . (2.5) Proof. We may assume that n ≥ l. The proof is by induction on c. For c = 0, we need to show n k=l (−1) k (n − k)! 2a N n+k+1 · 2 △(k,l) = (−1) l (Nl + a) (Nn + a)(n − l)! 2a N n+l+1 . (2.6) By the definition of △(k, l), the left-hand side of (2.6) is (−1) l (n − l)! 2a N n+l+1 + n k=l+1 2(−1) k (n − k)! 2a N n+k+1 . Applying (2.3), we find the left-hand side of (2.6) becomes (−1) l (n − l)! 2a N n+l+1 + (−1) l+1 (n − l) n + a N (n − l)! 2a N n+l+1 , which exactly is the right-hand side of (2.6). If c ≥ 1, the left-hand side of (2.5) is n k=l (−1) k (n − k)! 2a N n+k+1 k≥l 1 ≥···≥lc≥l 2 △(k,l 1 )+△(l 1 ,l 2 )+···+△(lc,l) (Nl 1 + a) · · · (Nl c + a) = n≥l 1 ≥···≥lc≥l n k=l 1 (−1) k 2 △(k,l 1 ) (n − k)! 2a N n+k+1 2 △(l 1 ,l 2 )+···+△(lc,l) (Nl 1 + a)(Nl 2 + a) · · · (Nl c + a) . Using (2.6) to deal with the part in the curly brace, we find the left-hand side of (2.5) is n≥l 1 ≥···≥lc≥l (−1) l 1 (Nl 1 + a) (Nn + a)(n − l 1 )! 2a N n+l 1 +1 2 △(l 1 ,l 2 )+···+△(lc,l) (Nl 1 + a)(Nl 2 + a) · · · (Nl c + a) , which is 1 Nn + a n l 1 =l (−1) l 1 (n − l 1 )! 2a N n+l 1 +1 W # l 1 ,l ({1} c−1 ). Then formula (2.5) easily follows from the inductive hypothesis for c − 1. Proof of Theorem 2.1 We apply induction on n + d. For n = 0, we have D 0 (c; z) = b 0 ,b 1 ,...,b d ≥0 z 2b 0 0 · · · z 2b d d a 2b 0 +···+2b d +c 1 +···+c d = 1 a c 1 +···+c d d i=0 a 2 a 2 − z 2 i , and the right-hand side of (2.1) is 1 2 d i=0 a δ i −1 a 2 − z 2 i · W # −1,0 (∅)W # 0,0 ({1} c 1 −3 ) · · · W # 0,0 ({1} c d −3 ) = d i=0 a 2 a 2 − z 2 i · a 2δ(c 1 )+···+2δ(c d )−3d a c 1 −3+2δ(c 1 ) · · · a c d −3+2δ(c d ) = 1 a c 1 +···+c d d i=0 a 2 a 2 − z 2 i . So (2.1) is true for n = 0. For d = 0, we observe that for n > 0, n t ⋆ N,a ({2} b 0 ) = b 0 k=0 n−1 t ⋆ N,a ({2} k ) (Nn + a) 2(b 0 −k) . Hence we have a recursive formula D n (; z 0 ) = (Nn + a) 2 (Nn + a) 2 − z 2 0 D n−1 (; z 0 ). Using the above formula repeatedly, we get D n (; z 0 ) = n j=1 (Nj + a) 2 (Nj + a) 2 − z 2 0 D 0 (; z 0 ) = n j=0 (Nj + a) 2 (Nj + a) 2 − z 2 0 . By the method of partial fractional decomposition, we have D n (; z 0 ) = n k=0 2(−1) k N a N n+1 2 2a N k k!(n − k)! 2a N n+k+1 (Nk + a) · (Nk + a) 2 (Nk + a) 2 − z 2 0 = n k=0 N a N n+1 2 (n − k)! 2a N n+k+1 · 2a N k k!(Nk + a) · 2(−1) k (Nk + a) 2 (Nk + a) 2 − z 2 0 So (2.1) is proved for d = 0. Now assume that n > 0 and d > 0. Since n t ⋆ N,a ({2} b 0 , c 1 , {2} b 1 , . . . , c d , {2} b d ) = b 0 k=0 1 (Nn + a) 2b 0 −2k n−1 t ⋆ N,a ({2} k , c 1 , {2} b 1 , . . . , c d , {2} b d ) + 1 (Nn + a) 2b 0 +c 1 n t ⋆ N,a ({2} b 1 , c 2 , {2} b 2 , . . . , c d , {2} b d ), we find that D n (c; z) = b 0 ,...,b d ≥0 b 0 k=0 1 (Nn + a) 2b 0 −2k n−1 t ⋆ N,a ({2} k , c 1 , {2} b 1 , . . . , c d , {2} b d )z 2b 0 0 · · · z 2b d d , + b 0 ≥0 z 2b 0 0 (Nn + a) 2b 0 +c 1 D n (c − ; z − ), where c − = (c 2 , . . . , c d ) and z − = (z 1 , z 2 , . . . , z d ). Then it is easy to get the following recursive formula D n (c; z) = (Nn + a) 2 (Nn + a) 2 − z 2 0 D n−1 (c; z) + (Nn + a) 2−c 1 (Nn + a) 2 − z 2 0 D n (c − ; z − ). (2.7) Let D n (c; z) denote the right-hand side of (2.1). Therefore according to (2.7), we need to prove (Nn + a) 2 (Nn + a) 2 − z 2 0 D n−1 (c; z) − D n (c; z) = − (Nn + a) 2−c 1 (Nn + a) 2 − z 2 0 D n (c − ; z − ). (2.8) By the induction hypothesis for D n−1 (c; z), the left hand-side of (2.8) is (Nn + a) 2 (Nn + a) 2 − z 2 0 n−1≥k 0 ≥k 1 ≥···≥k d ≥0 N a N n 2 (n − 1 − k 0 )! 2a N n+k 0 · 2a N k d k d !(Nk d + a) × d i=0 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ) − n≥k 0 ≥k 1 ≥···≥k d ≥0 N a N n+1 2 (n − k 0 )! 2a N n+k 0 +1 · 2a N k d k d !(Nk d + a) × d i=0 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ), which is n≥k 0 ≥···≥k d ≥0 d i=0 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ) · N 2a N k d k d !(Nk d + a) ×      (Nn + a) 2 (Nn + a) 2 − z 2 0 · a N n 2 (n − 1 − k 0 )! 2a N n+k 0 − a N n+1 2 (n − k 0 )! 2a N n+k 0 +1      . As the part in braces can be simplified as − a N n+1 2 (n − k 0 )! 2a N n+k 0 +1 · (Nk 0 + a) 2 − z 2 0 (Nn + a) 2 − z 2 0 , the left-hand side of (2.8) becomes −2N a N n+1 2 (Nn + a) 2 − z 2 0 n≥k 0 ≥k 1 ≥···≥k d ≥0 (−1) k 0 (1+δ(c 1 )) (Nk 0 + a) δ(c 1 ) (n − k 0 )! 2a N n+k 0 +1 · 2a N k d k d !(Nk d + a) × d i=1 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ). (2.9) Let k 0 denote the inner sum over k 0 in (2.9), that is k 0 = n k 0 =k 1 (−1) k 0 (1+δ(c 1 )) (Nk 0 + a) δ(c 1 ) (n − k 0 )! 2a N n+k 0 +1 W # k 0 ,k 1 ({1} c 1 −3 ). If c 1 = 1, δ(c 1 ) = 1, we obtain k 0 = n k 0 =k 1 Nk 0 + a (n − k 0 )! 2a N n+k 0 +1 2 △(k 0 ,k 1 ) . = Nk 1 + a (n − k 1 )! 2a N n+k 1 +1 + 2 n k 0 =k 1 +1 Nk 0 + a (n − k 0 )! 2a N n+k 0 +1 . Then by (2.2), we have k 0 = Nk 1 + a (n − k 1 )! 2a N n+k 1 +1 + Nn − Nk 1 (n − k 1 )! 2a N n+k 1 +1 = Nn + a (n − k 1 )! 2a N n+k 1 +1 . Using the above result together with (2.9), we find the left-hand side of (2.8) equals −(Nn + a) (Nn + a) 2 − z 2 0 n≥k 1 ≥···≥k d ≥0 N a N n+1 2 (n − k 1 )! 2a N n+k 1 +1 · 2a N k d k d !(Nk d + a) × 2(−1) k 1 (1+δ(c 2 )) (Nk 1 + a) δ(c 2 ) (Nk 1 + a) 2 − z 2 1 d i=2 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ), which by the inductive hypothesis for D n (c − ; z − ) is − Nn + a (Nn + a) 2 − z 2 0 D n (c − ; z − ). Therefore, the theorem is proved in this case. If c 1 ≥ 3, δ(c 1 ) = 0, by Lemma 2.4, we get k 0 = n k 0 =k 1 (−1) k 0 (n − k 0 )! 2a N n+k 0 +1 W # k 0 ,k 1 ({1} c 1 −3 ) = (−1) k 1 (Nn + a) 2−c 1 (Nk 1 + a) (n − k 1 )! 2a N n+k 1 +1 Combining this formula with (2.9), we obtain the left-hand side of (2.8) is −(Nn + a) 2−c 1 (Nn + a) 2 − z 2 0 n≥k 1 ≥···≥k d ≥0 N a N n+1 2 (n − k 1 )! 2a N n+k 1 +1 · 2a N k d k d !(Nk d + a) × 2(−1) k 1 (1+δ(c 2 )) (Nk 1 + a) δ(c 2 ) (Nk 1 + a) 2 − z 2 1 d i=2 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ), which by the inductive hypothesis for D n (c − ; z − ) is − (Nn + a) 2−c 1 (Nn + a) 2 − z 2 0 D n (c − ; z − ). Therefore, we conclude that D n (c; z) = D n (c; z). The proof is complete. Multiple t-star values of level N and their generating functions In this section, we prove the correctness of limit transfer from the generating functions of the finite form of multiple t-star values of level N to the generating functions of multiple t-star values of level N. For d ∈ N 0 , c = (c 1 , . . . , c d ) ∈ N d with c 1 > 1 and z = (z 0 , z 1 , . . . , z d ) ∈ C d+1 , we define the generating function D(c; z) = b 0 ,b 1 ,...,b d ≥0 t ⋆ N,a ({2} b 0 , c 1 , {2} b 1 , . . . , c d , {2} b d )z 2b 0 0 z 2b 1 1 · · · z 2b d d . The following theorem generalize the results of [ Pochhammer symbol, B(a, b) is the Beta function. = N B a N , a N k 0 ≥k 1 ≥···≥k d ≥0 2a N k d k d !(Nk d + a) d i=0 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ), (3.1) where k −1 = −1, δ i = δ(c i ) + δ(c i+1 ) with c 0 = 1 and c d+1 = 0, (a) n is the The following theorem gives explicit expressions of multiple t-star values of level N with an arbitrary number of blocks of twos. Pochhammer symbol, B(a, b) is the Beta function. and the convergence is uniform in any closed region E: Theorem 3.2. Let a, N be integers with 1 ≤ a ≤ N. For any d, b 0 , b 1 , . . . , b d ∈ N 0 , c 1 , . . . , c d ∈ N \ {2} with c 1 ≥ 3 if b 0 = 0 and d ≥ 1, we have t ⋆ N,a ({2} b 0 , c 1 , {2} b 1 , . . . , c d , {2} b d ) = N B a N , a N k 0 ≥k 1 ≥···≥k d ≥0 2a N k d k d !(Nk d + a) d i=0 (−1) k i δ i (Nk i + a) 2b i +3−δ i W # k i−1 ,k i ({1} c i −3 ), (3.2) where k −1 = −1, δ i = δ(c i ) + δ(c i+1 ) with c 0 = 1 and c d+1 = 0, (a) n is the Lemmas |z 0 | ≤ u 0 < 1, |z 1 | ≤ u 1 < 1, . . . , |z d | ≤ u d < 1. Proof. For |z| < 1, we find that n k=m 1 − z 2 (Nk + a) 2 −1 = ∞ l=0 n,m t ⋆ N,a ({2} l )z 2l . (3.4) Here we define that, for integers n ≥ m > 0 and s = (s 1 , . . . , s r ) ∈ N r , n,m t ⋆ N,a (s) = n,m t ⋆ N,a (s 1 , . . . , s r ) = n≥k 1 ≥···≥kr≥m 1 (Nk 1 + a) s 1 · · · (Nk r + a) sr . We also set n,m t ⋆ N,a (∅) = 1. If s 1 > 1, we may let n tend to infinity and define Hence the lemma is true for d = 0. Now assume that d ≥ 1. Using (3.4), we get ∞,m t ⋆ N,a (s) = k 1 ≥···≥kr≥m 1 (Nk 1 + a) s 1 · · · (Nk r + a) sr .D n (c; z) = n≥k 1 ≥···≥k d ≥0 n k=k 1 1 − z 2 0 (N k+a) 2 −1 k 1 k=k 2 1 − z 2 1 (N k+a) 2 −1 · · · k d k=0 1 − z 2 d (N k+a) 2 −1 (Nk 1 + a) c 1 · · · (Nk d + a) c d . Setting D ∞ n (c; z) = n≥k 1 ≥···≥k d ≥0 ∞ k=k 1 1 − z 2 0 (N k+a) 2 −1 k 1 k=k 2 1 − z 2 1 (N k+a) 2 −1 · · · k d k=0 1 − z 2 d (N k+a) 2 −1 (Nk 1 + a) c 1 · · · (Nk d + a) c d , then |D ∞ n (c; z) − D n (c; z)| ≤ n≥k 1 ≥···≥k d ≥0 n k=k 1 1 − z 2 0 (N k+a) 2 −1 k 1 k=k 2 1 − z 2 1 (N k+a) 2 −1 · · · k d k=0 1 − z 2 d (N k+a) 2 −1 (Nk 1 + a) c 1 · · · (Nk d + a) c d × ∞ k=n+1 1 − z 2 0 (Nk + a) 2 −1 − 1 . (3.5) Assume that |z 0 | ≤ u 0 < 1, |z 1 | ≤ u 1 < 1, . . . , |z d | ≤ u d < 1. Using the infinite product formula for the sine function sin πz πz = ∞ k=1 1 − z 2 k 2 < ∞ k=0 1 − z 2 (Nk + a) 2 , we obtain n≥k 1 ≥···≥k d ≥0 n k=k 1 1 − z 2 0 (N k+a) 2 −1 k 1 k=k 2 1 − z 2 1 (N k+a) 2 −1 · · · k d k=0 1 − z 2 d (N k+a) 2 −1 (Nk 1 + a) c 1 · · · (Nk d + a) c d < d i=0 π|z i | sin(π|z i |) n t ⋆ N,a (c) < d i=0 πu i sin(πu i ) t ⋆ N,a (c). (3.6) Using (3.4), we get ∞ k=n+1 1 − z 2 0 (Nk + a) 2 −1 − 1 = ∞ l=1 ∞,n+1 t ⋆ N,a ({2} l )z 2l 0 < ∞ l=1 ∞,n+1 t ⋆ N,a ({2} l ). Note that ∞,n+1 t ⋆ N,a ({2} l ) = k 1 ≥···≥k l ≥n+1 1 (Nk 1 + a) 2 · · · (Nk l + a) 2 < ∞ k=n+1 1 (Nk + a) 2 l < ∞ n dx (Nx + a) 2 l = 1 N(Nn + a) l , then we have ∞ k=n+1 1 − z 2 0 (Nk + a) 2 −1 − 1 < ∞ l=1 1 N(Nn + a) l = 1 N(Nn + a) − 1 . (3.7) Using (3.5), (3.6) and (3.7), we get |D ∞ n (c; z) − D n (c; z)| < 1 N(Nn + a) − 1 d i=0 πu i sin(πu i ) t ⋆ N,a (c) → 0, as n → ∞ on the closed region E. Similarly, we have |D(c; z) − D ∞ n (c; z)| ≤ k 1 ≥···≥k d ≥0 k 1 >n ∞ k=k 1 1 − |z 0 | 2 (N k+a) 2 −1 k 1 k=k 2 1 − |z 1 | 2 (N k+a) 2 −1 k d k=0 1 − |z d | 2 (N k+a) 2 −1 (Nk 1 + a) c 1 · · · (Nk d + a) c d < d i=0 πu i sin(πu i ) t ⋆ N,a (c) − n t ⋆ N,a (c) → 0, as n → ∞ on the closed region E. Therefore, the proof is finished. Lemma 3.4. Let N ∈ N, a ∈ {1, 2, . . . , N}, ε, p, q, C ∈ R with ε > 0, p > 1, C > 0, and let M k be real numbers satisfying |M k | < C log q (Nk + N + a) k p + ε for any k ∈ N 0 . Then we have lim n→∞ n k=0 |M k | N a N n+1 2 (n − k)! 2a N n+k+1 = ∞ k=0 |M k | N B a N , a N , where B(a, b) is the Beta function. Proof. Let U k,n = N a N n+1 2 (n − k)! 2a N n+k+1 = NΓ 2a N Γ a N 2 · Γ n + 1 + a N 2 Γ(n + 1 − k)Γ n + 1 + k + 2a N , where Γ(a) is the Gamma function. For any k ∈ N 0 , applying the well-known Stirling's formula (cf. [5,7]) Γ(x) ∼ √ 2π(x − 1) x− 1 2 e −(x−1) , x → ∞, and the limit formula 1 + 1 x x ∼ e, x → ∞, we obtain that Γ n + 1 + a N 2 Γ(n + 1 − k)Γ n + 1 + k + 2a N ∼ n + a N (n − k) n + k + 2a N 1 + k + a N n − k n−k 1 + −k − a N n + k + 2a N n+k+ 2a N ∼1, n → ∞. Therefore, we have lim n→∞ U k,n = NΓ 2a N Γ a N 2 = N B a N , a N , which leads to the conclusion that U k,n is a bounded sequence of numbers. It is easy to verify that U k,n is decreasing with respect to k, so we have 0 ≤ U k,n ≤ U 0,n . Note that U k,n+1 U k,n = n + 1 + a N 2 (n − k + 1) n + k + 1 + 2a N = n 2 + 2 + 2a N n + 1 + a N 2 n 2 + 2 + 2a N n + (1 − k) k + 1 + 2a N > 1, so U k,n is increasing with respect to n. Since |M k | < C log q (N k+N +a) k p +ε with p > 1, ∞ k=0 |M k |U k,n − n k=0 |M k |U k,n = ∞ k=n+1 |M k |U k,n → 0 (3.8) as n → ∞. By Monotone Convergence Theorem, we have lim n→∞ ∞ k=0 |M k |U k,n = ∞ k=0 |M k | lim n→∞ U k,n = ∞ k=0 |M k | N B a N , a N . (3.9) Hence, (3.8) and (3.9) can deduce the desired result. D k 0 (c; z) = k 0 ≥k 1 ≥···≥k d ≥0 2a N k d k d !(Nk d + a) d i=0 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ), (3.10) where k −1 = −1, δ i = δ(c i ) + δ(c i+1 ) with c 0 = 1 and c d+1 = 0. Then there exist ε, p, q, C ∈ R with p > 1 and ε, C > 0 such that for all k 0 ∈ N 0 , D k 0 (c; z) < C log q (Nk 0 + N + a) k p 0 + ε . Proof. For i = 1, 2, . . . , d, if c i ∈ {1, 3}, we have W # k i−1 ,k i ({1} c i −3 ) = 2 △(k i−1 ,k i ) ≤ 2, and if c i > 3, we have W # k i−1 ,k i ({1} c i −3 ) = k i−1 ≥l 1 ≥···≥l c i −3 ≥k i 2 △(k i−1 ,l 1 )+△(l 1 ,l 2 )+···+△(l c i −3 ,k i ) (Nl 1 + a) · · · (Nl c i −3 + a) < C 1 log c i −3 (Nk i−1 + N + a), where C 1 > 2 is some positive constant. Then we can express the above in a unified form W # k i−1 ,k i ({1} c i −3 ) < C 2 log c i −1 (Nk 0 + N + a), where C 2 > 2(log 2) −2 is a positive constant. Notice that δ 0 = 1, δ d ∈ {2, 3}, δ i ∈ {0, 1, 2} for i = 1, 2, . . . , d − 1, and 2a N k d k d !(Nk d + a) < k d + 1 Nk d + a ≤ 1. So we find D k 0 (c; z) = 2 |(Nk 0 + a) 2 − z 2 0 | k 0 ≥k 1 ≥···≥k d ≥0 2a N k d k d !(Nk d + a) d i=1 (−1) k i δ i (Nk i + a) δ i −1 (Nk i + a) 2 − z 2 i W # k i−1 ,k i ({1} c i −3 ) <C 3 · log c 1 +···+c d −d (Nk 0 + N + a) (Nk 0 + a) 2 − |z 0 | 2 k 0 ≥k 1 ≥···≥k d ≥0 d i=1 Nk i + a (Nk i + a) 2 − |z i | 2 , where C 3 is a positive constant. Let |z max | = max{|z 1 |, . . . , |z d |}, then D k 0 (c; z) < C 4 · log c 1 +···+c d −d (Nk 0 + N + a) (Nk 0 + a) 2 − |z 0 | 2 k 0 k=1 1 Nk + a − |z max | 2 d < C 5 · log c 1 +···+c d (Nk 0 + N + a) (Nk 0 + a) 2 − |z 0 | 2 < C 6 · log c 1 +···+c d (Nk 0 + N + a) k 2 0 + a 2 − |z 0 | 2 , where C 4 , C 5 and C 6 are positive constants. Therefore by setting ε = a 2 − |z 0 | 2 , q = c 1 + · · · + c d , p = 2 and C = C 6 , we conclude the result. Proofs of the theorems D(c; z) = lim n→∞ D n (c; z) = lim n→∞ n k 0 =0 N a N n+1 2 (n − k 0 )! 2a N n+k 0 +1 D k 0 (c; z), where D k 0 (c; z) is defined in (3.10). Then using Lemmas 3.4 and 3.5, we have D(c; z) = N B a N , a N ∞ k 0 =0 D k 0 (c; z),D k 0 (c; b) = k 0 ≥k 1 ≥···≥k d ≥0 2a N k d k d !(Nk d + a) d i=0 (−1) k i δ i (Nk i + a) 2b i −δ i +3 W # k i−1 ,k i ({1} c i −3 ). By the proof of Lemma 3.5, it is easy to deduce that there exist ε, p, q, C ∈ R with p > 1 and ε, C > 0 such that for all k 0 ∈ N 0 , Setting d = 0 in Theorem 3.2, we obtain the following formula. For d = 1, setting c 1 = 1, 3 and c + 3 with c > 3 respectively in Theorem 3.2, we have the following three theorems, t N,a (2b 0 + 1 + s 1 , s 2 , . . . , s r−1 , 2b 1 + 1 + s r ). D k 0 (c; b) < C log q (Nk 0 + N + a) k p 0 + ε . Here for r = 1, t N,a (2b 0 + 1 + s 1 , s 2 , . . . , s r−1 , 2b 1 +1+s r ) is treated as t N,a (2b 0 + 2b 1 + 4 + c). Setting c 1 = · · · = c d = 1, b 0 = b d = 1 and b 1 = · · · = b d−1 = 0 in Theorem 3.2, we deduce the following result. Here for r = 1, t N,a (s 1 + 2, s 2 , . . . , s r−1 , s r + 2) becomes t N,a (d + 5). 2 The finite form of multiple t-star values of level N In this section, we prove a finite version of the generating function of multiple t-star values of level N. Let n be a fixed nonnegative integer. For any index s = (s 1 , . . . , s r ), we define the finite form of multiple t-star value of level N by n t ⋆ N,a (s) := n≥k 1 ≥···≥kr≥0 Theorem 3. 1 . 1Let a, N be integers with 1 ≤ a ≤ N. For any d ∈ N 0 , c = (c 1 , . . . , c d ) ∈ (N \ {2}) d with c 1 ≥ 3, and z = (z 0 , z 1 , . . . , z d ) ∈ C d+1 with |z j | < 1, j = 0, 1, . . . , d, we have D(c; z) Lemma 3. 3 . 3Let d ∈ N 0 , c = (c 1 , . . . , c d ) ∈ (N \ {2}) d with c 1 ≥ 3, and let z = (z 0 , z 1 , . . . , z d ) ∈ C d+1 with |z j | < 1, j = 0, 1, . . . , d. Then lim n→∞ D n (c; z) = D(c; z), (3.3) Lemma 3. 5 . 5For N ∈ N, a ∈ {1, 2, . . . , N}, k 0 , d ∈ N 0 , c = (c 1 , . . . , c d ) ∈ (N \ {2}) d with c 1 ≥ 3, z = (z 0 , z 1 , . . . , z d ) ∈ C d+1 with |z j | < 1, j = 0, 1, . . . , d, let Proof of Theorem 3.2 Set b = (b 0 , b 1 ,. . . , b d ) and ( the limit n → ∞ in Corollary 2.2 and applying (3.11) and Lemma 3.4, we get the desired result. For an admissible index s = (s 1 , . . . , s r ) and any a ∈ {1, 2, . . . , N}, the multiple t-value of level N and the multiple t-star value of level N are defined by 4 Evaluations for multiple t-star values of level N In this section, we give some evaluations of multiple t-star values of level N. For an index s = (s 1 , . . . , s r ), σ 1 , . . . , σ r ∈ {1, −1} with s 1 > 1 if σ 1 = 1, we define the alternating multiple t-value of level N containing factorial and pochhammer symbols by t N,a (s 1 , . . . , s r ; σ 1 , . . . , σ r ) = Nk 1 + a) s 1 · · · (Nk r + a) sr · 2a N kr k r ! . where (a) n is the Pochhammer symbol. To save space, we may put a bar on the top of s i if σ i = −1. For example, t N,a (s 1 , s 2 ) = t N,a (s 1 , s 2 ; −1, 1) = k 1 >k 2 ≥0 (−1) k 1 (Nk 1 + a) s 1 (Nk 2 + a) s 2 · 2a N k 2 k 2 ! . B(a, b)is the Beta function. Theorem 4 . 2 . 42For any b 0 , b 1 ∈ N 0 , we have t ⋆ N,a ({2} b 0 , 3, {2} b 1 Theorem 4. 3 .. 3 , 33For any b 0 ∈ N, b 1 ∈ N 0 , we have t ⋆ N,a ({2} b 0 , 1, {2} b 1 For any b 0 , b 1 ∈ N 0 , c ∈ N, we have t ⋆ N,a ({2} b 0 , c + {2} s 1 ,...,sr ≥1 s 1 ,...,sr ≥1 t N,a (s 1 + 2, s 2 , . . . , s r−1 , s r + 2). We now prove Theorems 3.1 and 3.2. Proof of Theorem 3.1 By Lemma 3.3 and Theorem 2.1, we deduce that Combinatorial aspects of multiple zeta values. J M Borwein, D M Bradley, D J Broadhurst, P Lisoněk, Electron. J. Comb. 538J. M. Borwein, D. M. Bradley, D. J. Broadhurst and P. Lisoněk, Combinatorial aspects of multiple zeta values, Electron. J. Comb. 5 (1998), R38. J M Borwein, D M Bradley, D J Broadhurst, P Lisoněk, Special values of multiple polylogarithms. 353J. M. Borwein, D. M. Bradley, D. J. Broadhurst and P. Lisoněk, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001), no. 3, pp. 907-941. On motivic multiple t values, Saha's basis conjecture, and generators of alternating MZV's, preprint. S Charlton, arXiv:2112.14613S. Charlton, On motivic multiple t values, Saha's basis conjecture, and generators of alternating MZV's, preprint, arXiv: 2112.14613. On the sum relation of multiple Hurwitz zeta functions. C Chung, Quaest. Math. 423C. Chung, On the sum relation of multiple Hurwitz zeta functions, Quaest. Math. 42 (2019), no. 3, pp. 297-305. An elementary proof of Stirling's formula. P Diaconis, D Freedman, Amer. Math. Monthly. 93P. Diaconis and D. Freedman, An elementary proof of Stirling's formula, Amer. Math. Monthly 93 (1986), pp. 123-125. Institutiones calculi differentialis. L Euler, Opera Omnia Ser. G. Kowalewski101; opera matL. Euler, Institutiones calculi differentialis, G. Kowalewski (ed.), Opera Omnia Ser. 1; opera mat., 10, Teubner (1980). A direct proof of Stirling's formula. W Feller, Amer. Math. Monthly. 74W. Feller, A direct proof of Stirling's formula, Amer. Math. Monthly 74 (1967), pp.1223-1225. An alternative proof of a theorem of Zagier. Kh, T. Hessami Hessami Pilehrood, Pilehrood, J. Math. Anal. Appl. 4491Kh. Hessami Pilehrood and T. Hessami Pilehrood, An alternative proof of a the- orem of Zagier, J. Math. Anal. Appl. 449 (2017), no. 1, pp. 168-175. Generating functions for multiple zeta star values. Kh, T. Hessami Hessami Pilehrood, Pilehrood, J. Théor. Nombres Bordeaux. 312Kh. Hessami Pilehrood and T. Hessami Pilehrood, Generating functions for mul- tiple zeta star values, J. Théor. Nombres Bordeaux 31 (2019), no. 2, pp. 343-360. Multiple zeta star values on 3 − 2 − 1 indices. Kh, T. Hessami Hessami Pilehrood, Pilehrood, Ramanujan J. Kh. Hessami Pilehrood and T. Hessami Pilehrood, Multiple zeta star values on 3 − 2 − 1 indices, Ramanujan J (2022). New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner¡ s series. Kh, T Hessami Pilehrood, R Pilehrood, Tauraso, Trans. Amer. Math. Soc. 3666Kh. Hessami Pilehrood, T. Hessami Pilehrood and R. Tauraso, New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner¡ s series, Trans. Amer. Math. Soc. 366 (2014), no. 6, pp. 3131-3159. Multiple harmonic series. M E Hoffman, Pacific J. Math. 1522M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), no. 2, pp. 275-290. An odd variant of multiple zeta values. M E Hoffman, Comm. Number Theory Phys. 133M. E. Hoffman, An odd variant of multiple zeta values, Comm. Number Theory Phys. 13 (2019), no. 3, pp. 529-567. Alternating double Euler sums, hypergeometric identities and a theorem of Zagier. T Lee-Peng, J. Math. Anal. Appl. 4621T. Lee-Peng, Alternating double Euler sums, hypergeometric identities and a the- orem of Zagier, J. Math. Anal. Appl. 462 (2018), no. 1, pp. 777-800. Another proof of Zagier's evaluation formula of the multiple zeta values ζ. Z Li, Math. Res. Lett. 22Z. Li, Another proof of Zagier's evaluation formula of the multiple zeta values ζ(2, . . . , 2, 3, 2, . . . , 2), Math. Res. Lett. 20 (2013), no. 5, pp. 947-950. Elementary proofs of Zagier's formula for multiple zeta values and its odd variant. L Lai, C Lupu, D Orr, arXiv:2201.09262preprintL. Lai, C. Lupu and D. Orr, Elementary proofs of Zagier's formula for multiple zeta values and its odd variant, preprint, arXiv: 2201.09262. Relations of multiple t-values of general level. Z Li, Z Wang, arXiv:2210.16854preprintZ. Li and Z. Wang, Relations of multiple t-values of general level, preprint, arXiv: 2210.16854. Zhonghua Li, Lu Yan, arXiv:2212.09070Generating functions of multiple t-star values. Zhonghua Li and Lu Yan, Generating functions of multiple t-star values, arXiv: 2212.09070. A family of multiple harmonic sum and multiple zeta star value identities. E Linebarger, J Zhao, Mathematika. 611E. Linebarger and J. Zhao, A family of multiple harmonic sum and multiple zeta star value identities, Mathematika 61 (2015), no. 1, pp. 63-71. On some explicit evaluations of multiple zeta-star values. S Muneta, J. Number Theory. 1289S. Muneta, On some explicit evaluations of multiple zeta-star values, J. Number Theory 128 (2008), no. 9, pp. 2538-2548. On Hoffman's t-values of maximal height and generators of multiple zeta values. T Murakami, Math. Ann. 3821-2T. Murakami, On Hoffman's t-values of maximal height and generators of multiple zeta values, Math. Ann. 382 (2022), no. 1-2, pp. 421-458. Y Ohno, W Zudilin, Zeta stars. 2Y. Ohno and W. Zudilin, Zeta stars, Commun. Number Theory Phys. 2 (2008), no. 2, pp. 325-347. Alternating double t-values and T -values. J Quan, Adv. Differ. Equ. 2020450J. Quan, Alternating double t-values and T -values, Adv. Differ. Equ. 2020 450 (2020). . J Wallis, Arithmetica Infinitorum. 1656J. Wallis, Arithmetica Infinitorum, Oxford, England, 1656. Parametric Euler T -sums of odd harmonic numbers. C Xu, L Yan, arXiv:2203.13996preprintC. Xu and L. Yan, Parametric Euler T -sums of odd harmonic numbers, preprint, arXiv: 2203.13996. Double shuffle relations of double zeta values and double Eisenstein series of level N. H Yuan, J Zhao, J. London Math. Soc. 922H. Yuan and J. Zhao, Double shuffle relations of double zeta values and double Eisenstein series of level N, J. London Math. Soc. 92 (2) (2015), pp. 520-546. Values of zeta functions and their applications. D Zagier, First European Congress of Mathematics. IIBirkhäuserProgr. Math.D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, vol. II (Paris, 1992), pp. 497-512, Progr. Math., vol. 120, Birkhäuser, Basel, 1994. D Zagier, Evaluation of the multiple zeta values ζ. 2D. Zagier, Evaluation of the multiple zeta values ζ(2, . . . , 2, 3, 2, . . . , 2), Ann. Math. 175 (2012), no. 2, pp. 977-1000. Identity families of multiple harmonic sums and multiple zeta star values. J Zhao, J. Math. Soc. Japan. 684J. Zhao, Identity families of multiple harmonic sums and multiple zeta star values, J. Math. Soc. Japan 68 (2016), no. 4, pp. 1669-1694.
[]
[ "ε-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves", "ε-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves" ]
[ "Xuhang Jiang \nSchool of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n", "Xing Wang \nPhysik Department\nTechnische Universität München\nTUM School of Natural Sciences\n85748GarchingGermany\n", "Lin Li ", "Yang \nZhejiang Institute of Modern Physics\nSchool of Physics\nZhejiang University\n310027HangzhouChina\n", "Jing-Bang Zhao \nSchool of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n" ]
[ "School of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina", "Physik Department\nTechnische Universität München\nTUM School of Natural Sciences\n85748GarchingGermany", "Zhejiang Institute of Modern Physics\nSchool of Physics\nZhejiang University\n310027HangzhouChina", "School of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina" ]
[]
In this paper, we investigate two-loop non-planar triangle Feynman integrals involving elliptic curves. In contrast to the Sunrise and Banana integral families, the triangle families involve non-trivial sub-sectors. We show that the methodology developed in the context of Banana integrals can also be extended to these cases and obtain ε-factorized differential equations for all sectors. The letters are combinations of modular forms on the corresponding elliptic curves and algebraic functions arising from the sub-sectors. With uniform transcendental boundary conditions, we express our results in terms of iterated integrals order-by-order in the dimensional regulator, which can be evaluated efficiently. Our method can be straightforwardly generalized to other elliptic integral families and have important applications to precision physics at current and future high-energy colliders.
null
[ "https://export.arxiv.org/pdf/2305.13951v1.pdf" ]
258,841,052
2305.13951
d159c315ecb3c8ed87129ec57ca21783e358a701
ε-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves 23 May 2023 Xuhang Jiang School of Physics and State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina Xing Wang Physik Department Technische Universität München TUM School of Natural Sciences 85748GarchingGermany Lin Li Yang Zhejiang Institute of Modern Physics School of Physics Zhejiang University 310027HangzhouChina Jing-Bang Zhao School of Physics and State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina ε-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves 23 May 2023 In this paper, we investigate two-loop non-planar triangle Feynman integrals involving elliptic curves. In contrast to the Sunrise and Banana integral families, the triangle families involve non-trivial sub-sectors. We show that the methodology developed in the context of Banana integrals can also be extended to these cases and obtain ε-factorized differential equations for all sectors. The letters are combinations of modular forms on the corresponding elliptic curves and algebraic functions arising from the sub-sectors. With uniform transcendental boundary conditions, we express our results in terms of iterated integrals order-by-order in the dimensional regulator, which can be evaluated efficiently. Our method can be straightforwardly generalized to other elliptic integral families and have important applications to precision physics at current and future high-energy colliders. Introduction Precision predictions for observables in quantum field theories are crucial for extracting information from experimental data at facilities such as the CERN Large Hadron Collider (LHC). To achieve this, one needs to calculate scattering amplitudes at the multi-loop level, which boils down to calculating corresponding Feynman integrals. The first step in this process is to reduce all relevant Feynman integrals to linear combinations of a smaller set of master integrals (MIs). There are several ways to perform this reduction, with the most commonly used being the integration-by-parts (IBP) technique with the Laporta algorithm [1]. Several public packages implementing this algorithm are available, e.g., Litered [2,3], FIRE [4], Reduze [5] and Kira [6]. However, for cutting-edge applications, the IBP systems can become prohibitively expensive to solve. In recent years, novel approaches have emerged to bypass these cumbersome linear systems, e.g., the GKZ system [7][8][9] and the intersection theory [10,11], just to name a few. After reduction, a system of differential equations can be written down for the master integrals with respect to kinematic variables or some auxiliary variables [12][13][14][15][16]. With suitable boundary conditions, this system can always be solved numerically by direct integration or by series expansions [17][18][19]. It is conceptually more interesting to solve the differential equations analytically, which is, however not an easy task in general. In dimensional regularization, with the spacetime dimension D = 4 − 2ε, the differential equations get simpler if the dependence on ε is entirely factorized out [20]. In this case, the solutions can be obtained order-by-order in ε as iterated integrals, which often lead to analytic expressions in terms of multiple polylogarithms (MPLs) [21,22]. Hence, the problem of analytically calculating Feynman integrals often reduces to finding appropriate transformations to turn the differential equations into this kind of ε-form. For an integral family involving only logarithmic singularities, it is generally believed that a system of differential equations in the ε-form always exists. Various methods to find it have been proposed in the literature [23][24][25][26][27][28][29][30][31][32][33][34]. For more generic integrals, it is unclear whether the factorization of ε-dependence can always be achieved. For integrals involving elliptic curves or even beyond, there is growing evidence that this statement may be true. In particular, this has been shown for equal-mass Banana integrals at arbitrary loop order [35][36][37]. In this paper, we provide more examples to support this statement using the geometry-inspired method for Banana integrals [35][36][37][38]. Our understanding of Feynman integrals and scattering amplitudes benefits greatly from the geometric point of view. The simplest class of functions, the MPLs, are related to Abelian differentials of the first kind on a Riemann sphere (a genus-0 object). Their integrands are rational functions, and can usually be cast into the d log-form [33,[39][40][41][42]. The next-to-simplest class of functions are related to elliptic curves, which have received much interest in recent decades [43][44][45][46][47][48][49][50][51][52][53][54][55]. These elliptic-curve related functions appear in many cutting-edge calculations and are of crucial importance for precision predictions at future colliders. An elliptic Feynman integral family may be associated with one or more elliptic curves (an example with two curves contributes to top quark pair production at the LHC and was studied in [56]). In this paper, we focus on integral families involving only one elliptic curve. Given an integral family, the associated elliptic curve can be obtained from the maximal cut in the Figure 1: Two non-planar triangle integral families at two loops. Family (a) (the left diagram) involves a closed loop with the same mass and was studied in [60]. Family (b) (the right diagram) involves two massive propagators. Baikov representation [57,58] or from the variety of the second graph polynomial (see [59] for a comprehensive review). The curve is generically parameterized by external variables such as momentum scalar products and masses. In this paper, we consider elliptic curves depending on only one dimensionless parameter. This can happen if the integrals involve only two physical scales or if one constrains to a two-dimensional surface in a higher-dimensional parameter space. In these single-parameter cases, evidences show that the best variable to use is the modular variable τ defined by the ratio of the two periods of the underlying elliptic curve, which has been employed to cast the differential equations into the ε-form for Sunrise and Banana families [35][36][37][38]. In this paper, we revisit two non-planar triangle integral families from the above perspective. The diagrams for the two families are shown in Fig. 1. Compared to the Sunrise and Banana families, these triangle families involve non-trivial sub-sectors. It is hence interesting to investigate whether the method developed in [35][36][37][38] can be extended to find ε-factorized differential equations in these cases, and to express the solutions as well-behaved iterated integrals with nice analytic and algebraic properties, that can be easily evaluated numerically. Besides being theoretically interesting, these integrals are also of phenomenological importance. The family (a) appears, e.g., in bottom-quark loop contributions to the H → gg decay, in top-quark loop contributions to the di-photon and di-jet productions at the LHC, as well as in the pair production and associated production of the Higgs boson under suitable limits. The family (b) appears, e.g., in the next-to-next-to-leading order (NNLO) electroweak (EW) corrections to the H → bb decay and to the di-jet production. The investigation in this paper, therefore, serves as a step towards a fully analytic understanding of these important scattering amplitudes. The paper is organized as follows. In Section 2 we set up our notations and present the results for the sub-sector integrals in both family (a) and (b). For the elliptic top-sector integrals, we first use the simpler case of family (a) in Section 3 to introduce our method and then apply it to the more complicated case of family (b) in Section 4. We conclude in Section 5. All the results are provided in auxiliary files with the preprint. Setup and canonical sub-sector integrals We consider the integral families plotted in Fig. 1, where thick internal lines have the same mass m, and all the other internal lines are massless. The thick external leg has the offshellness s = (p 1 + p 2 ) 2 , and the other two external legs are massless p 2 1 = p 2 2 = 0. In momentum space, the two integral families involve 6 propagators and an irreducible scalar product (ISP) D 7 = l 2 1 . The integrals in the two families can be written as I ν 1 ν 2 ···ν 7 = e 2εγ E (m 2 ) ν−D d D l 1 iπ D/2 d D l 2 iπ D/2 D −ν 7 7 D ν 1 1 D ν 2 2 D ν 3 3 D ν 4 4 D ν 5 5 D ν 6 6 ,(1) with D = 4 − 2ε, ν = ν 1 + · · · + ν 6 + ν 7 . The propagator denominators are D 1 = (l 1 − p 1 ) 2 , D 2 = (l 2 − p 1 ) 2 − m 2 , D 3 = (l 1 + p 2 ) 2 , D 4 = (l 1 − l 2 + p 2 ) 2 − m 2 , D 5 = (l 1 − l 2 ) 2 − κ m 2 , D 6 = l 2 2 − κ m 2 ,(2) where κ = 1 for family (a) and κ = 0 for family (b). We have made the Feynman i0-prescription in the propagators implicit. With the pre-factor (m 2 ) v−D , the integrals depend on only one dimensionless variable that we take as y = − m 2 s ∈ R + i0 ,(3) where the infinitesimal imaginary part is determined by the Feynman i0 prescription. We use Litered [3] and Kira [6] to perform the IBP reductions. For family (a), there are 11 master integrals in total, 2 of which are in the top sector, while the others are in sub-sectors. In this section, we first construct ε-factorized differential equations for the sub-sectors. Those for the top sectors will be discussed in the latter sections. Canonical basis for sub-sectors The sub-sectors in both families do not involve elliptic curves. We use the method of [33,34] to construct integrals with uniform transcendentality (UT) in the Baikov representations. Briefly speaking, we construct integrands in the generalized d log forms: [P(z z z)] ε d log α 1 (z z z) ∧ d log α 2 (z z z) ∧ · · · ∧ d log α n (z z z) ,(6) where z z z denotes the set of Baikov variables, which is simply a subset of the propagator denominators, i.e., z i ∈ {D i }. P(z z z) is a rational function that defines the representation for the particular sector, and α i (z z z) are algebraic functions of z z z. As an example, in the sub-sector {1, 1, 1, 0, 1, 1, 0} of family (b), we can have the following construction [P(z z z)] ε d log z 1 ∧ d log z 2 ∧ d log z 3 ∧ d log z 5 ∧ d log z 6 ∧ d log(z 7 − z 1 ) .(7) The above integrand can be converted to a linear combination of Feynman integrals following the method presented in [33,34]. With suitable normalization factors, it corresponds to the UT integral ε 4 x I (b) 1110110 . Proceeding with each sub-sector, we arrive at the following 15 UT integrals in the sub-sectors of family (b): M (b) 4 = ε 4 x I (b) 1111100 , M (b) 5 = ε 3 x 2 I (b) 1120110 , M (b) 6 = ε 4 x I (b) 1110110 , M (b) 7 = ε 3 x I (b) 1102110 , M (b) 8 = ε 4 x I (b) 1101110 , M (b) 9 = ε 2 x(4 + x) −x I (b) 1122000 + ε 2(1 + 2ε) I (b) 0202000 , M (b) 10 = ε 3 x I (b) 1112000 , M (b) 11 = ε 2 x 2 (1 + x)I (b) 0211200 − (1 + x)I (b) 0121200 − 2I (b) 0120200 + 2 − x x I (b) 0202000 , M (b) 12 = ε 2 x 4 (3 + x)I (b) 0121200 + (1 − x)I (b) 0211200 + 2I (b) 0220100 + 4I (b) 0120200 + I (b) 0202000 , M (b) 13 = ε 3 x I (b) 1010120 , M (b) 14 = ε 2 x I (b) 1220000 , M (b) 15 = ε(ε − 1) 2 I (b) 1200200 , M (b) 16 = −ε 2 x 2 I (b) 0220100 , M (b) 17 = −ε 2 1 − x 4 I (b) 0120200 + 2I (b) 0220100 , M (b) 18 = ε 2 I (b) 0202000 ,(8) where x ≡ −1/y = s/m 2 . The differential equations for this UT basis take the ε-form (hereafter, we will suppress the superscript (a) or (b) when it is clear from the context): (10) 1 We thank Christoph Nega and Lorenzo Tancredi for sharing preliminary results for related topics. d dy M 4 = ε y −2M 10 + 3 2 M 11 + M 12 + M 16 − 3 2 M 18 , d dy M 5 = ε y(1 + y) (2M 5 − 2M 6 − 2M 13 + M 14 − M 15 − 2M 16 − 2M 17 ) , The ε-form differential equations for the sub-sectors are given by: d dy               M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 M 11               = ε y                 0 0 0 0 −1 0 0 0 0 0 1 0 −1 1 2 0 0 0 0 0 0 4y−3 4y−1 3 √ 1−4y 0 0 0 −1 √ 1−4y −1 √ 1−4y 0 0 −1 √ 1−4y −1 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 −2 √ 4y+1 0 0 0 0 0 0 0 3 √ 4y+1 4(y+1) 4y+1 0 −1 √ 4y+1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0                               M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 M 11               .(11) Iterated integrals The solutions to the ε-factorized differential equations such as Eqs. (9) and (11) can be naturally represented by iterated integrals [62] combined with suitable boundary conditions. Here we summarize the definitions that we will need in our calculations and refer the readers to [59] for a systematic review. Let f i (z)'s be functions of z, called letters. We define the n-fold iterated integral of these letters by: I( f 1 , f 2 , . . . , f n ; z 0 , z) = z z 0 dz 1 f 1 (z 1 ) z 1 z 0 dz 2 f 2 (z 2 ) . . . z n−1 z 0 dz n f n (z n ) ,(12) where z 0 is the base point. By convention, one sets I(; z 0 , z) = 1. When the integration develops logarithmic singularity for z i → z 0 , the standard "trailing zero" or "tangential base point" regularization [43,63] is adopted. Acting on an iterated integral, the symbol map [64] simply picks up the sequence of its letters: S[I( f 1 , f 2 , . . . , f n ; z 0 , z)] = f n (z) ⊗ · · · ⊗ f 2 (z) ⊗ f 1 (z) .(13) For the sub-sector integrals of family (a) and (b), the asymptotic boundary conditions at y = 0 can be easily obtained and are collected in the Appendix. Combining the boundary conditions and the iterated integrals from the differential equations, we obtain the solutions for Eqs. (9) and (11) and give them in the electronic files attached to this paper (where the iterated integrals are represented by their symbols). When the letters are rational functions, the iterated integrals can be cast into MPLs. Supposing that f i (z) = 1/(z − z i ), the MPLs are defined as G(z 1 , z 2 , . . . , z n ; z) ≡ I( f 1 , f 2 , . . . , f n ; 0, z) . In the MPL cases, the "trailing zero" or "tangential base point" regularization implies that G(0, . . . , 0 r-times ; z) = 1 r! ln r (z) .(15) In our cases, the symbols of the solutions involve algebraic letters (square roots). They can be transformed into rational letters via variable changes. This is usually called "rationalization". Family (b) involves only one square root √ 1 − 4y, which can be rationalized by y = t (1 + t) 2 , t = 1 − √ 1 − 4y 1 + √ 1 − 4y .(16) Family (a) involves two square roots √ 1 ± 4y. However, they never appear simultaneously in a symbol. Hence, we can rationalize them individually. For √ 1 − 4y, we use the same transformation as family (b), while for √ 1 + 4y, we use y = u (1 − u) 2 , u = √ 1 + 4y − 1 √ 1 + 4y + 1 .(17) After rationalization, the solutions can be easily written as MPLs of the new variables t and u. We solve the sub-sector integrals in these two families up to transcendental weight 6 using the expressions of harmonic polylogarithms (HPL) [65,66], which consist of a subset of MPL. The analytic continuation beyond the vicinity of y = 0 in sub-sectors can be performed just by the analytic continuation of HPL, which can be performed by packages like [67]. The results of sub-sectors are given in the auxiliary files attached to this preprint. In later sections, we will encounter iterated integrals with letters related to modular forms. In these cases, the base point is conventionally chosen as τ = i∞ where τ is the modular variable, and there is a factor of 2πi in each integration measure: I(η 1 , η 2 , · · · , η n ; i∞, τ) = (2πi) n τ i∞ dτ 1 η 1 (τ 1 ) τ 1 i∞ dτ 2 η 2 (τ 2 ) · · · τ n−1 i∞ dτ n η n (τ n ) .(18) In practice, one introduces the q variable as q = exp(2πiτ), and the iterated integrals become I(η 1 , η 2 , · · · , η n ; 0, q) = q 0 dq 1 q 1 η 1 (q 1 ) q 1 0 dq 2 q 2 η 2 (q 2 ) · · · q n−1 0 dq n q n η n (q n ) .(19) These iterated integrals can be evaluated via the q-expansion within the convergence region. In particular, the results for the top-sector integrals in terms of q-expansions are given in the auxiliary files attached to this paper. It is worthwhile to remark on the concepts of "weight" that will be used throughout the paper. The first concept is "transcendental weight". As usual, we assign weight −1 to ε and weight n to the Riemann zeta value ζ n . For MPLs, the transcendental weight is equal to the depth of the iterated integral. For example, Eq. (14) has depth n and weight n. It is a different story for iterated integrals involving elliptic kernels, as transcendental weights may not agree with the depth of the iterated integral [68]. Here we adopt the definition in [69]. As we will see, the boundary conditions in the limit y → 0 are UT functions in the MPL sense. Combined with the εfactorized differential equations for the top-sector integrals, our results are UT or pure functions in the elliptic sense according to [69]. Here, the "elliptic transcendental weight" is again equal to the depth of the iterated integral. Finally, we will also mention the concept of "modular weight", which is a property of modular forms under modular transformations. The top-sector integrals in family (a) We now turn to investigate the top sectors that involve elliptic curves. We begin with the simpler case of family (a), shown in the left panel of Fig. 1. The differential equations of the two topsector integrals I 1 and I 2 depend on each other and also depend on the sub-sector integrals. Our basic strategy is as follows: Firstly, we impose the maximal cut to eliminate the sub-sector integrals from the differential equations. We then use the Picard-Fuchs operator to obtain the εfactorization within the top sector. Finally, we recover the sub-sector dependence and bring them to ε by suitable subtractions (this step turns out to be unnecessary for family (a)). This strategy will also be applicable to family (b). With the ε-factorized differential equations at hand, we study the solutions for the top-sector integrals as functions of the modular variable and present numeric results. The Picard-Fuchs operator and ε-factorization The two top-sector integrals I 1 and I 2 in family (a) satisfy a system of two first-order differential equations, with dependence on sub-sector integrals M 3 , . . . , M 11 . This system is equivalent to the following second-order differential equation for I 1 : L (ε) 2 (y) I 1 ≡ 2 ∑ n=0 r n (y, ε) d dy n I 1 = 1 ε 2 (16y − 1) f sub (y) −8M 4 + 12 √ 1 − 4y M 5 + 28M 6 − 16M 7 − 10M 8 ,(20) with the coefficients r 2 (y, ε) = 1 , r 1 (y, ε) = −3 + 32y + 2ε(−1 + 8y) y(1 − 16y) , r 2 (y, ε) = 4(1 − 9y + ε(1 − 6y)) y 2 (1 − 16y) .(21) The superscript "(ε)" in the Picard-Fuchs operator L (ε) 2 (y) reminds us that it depends on ε. The maximal cut for an integral of the form (1) amounts to taking the residue around zero for each of the propagator denominators D 1 , . . . , D 6 . Under the maximal cut, the sub-sector integrals automatically vanish, and the above second-order differential equation for I 1 becomes homogeneous. The information about the top sector is then entirely contained in the operator L (ε) 2 (y) that we are going to study. The basic idea to obtain the ε form in the top sector is to modulo ε independence as much as one can, which is controlled by the operator at ε = 0, namely L (0) 2 (y). Later we will see that this operator is closely related to the underlying geometric object: the elliptic curve. The two independent kernels ψ 0 (y) and ψ 1 (y) of this operator, satisfying L (0) 2 (y) ψ k (y) = 0, correspond to the two periods of the elliptic curve. We resort to the Frobenius method [70,71] to solve for them in a suitable region of y. The operator L (0) 2 has three regular singular points: {0, 1/16, ∞}. By studying the indicial equations around these points, we find that around y = 0, its two solutions have the same asymptotic behavior, so is the case for y = ∞. Here we focus on the point y = 0, where the multiplicity of the indicial equation is maximized. Around this point, there exists a holomorphic solution as a power series in y, and the other solution exhibits logarithmic behavior. We write the two solutions in the following form 2 : ψ k (y) = 1 (2πi) k k ∑ j=0 ln j y j! ∞ ∑ n=0 a k− j,n y n+2 , (k = 0, 1) .(22) The relative normalization between the two solutions is fixed by the above choice, and the overall normalization is chosen as a 0,0 = 1 for convenience. We solve the coefficients a i, j iteratively and obtain: ψ 0 (y) = y 2 1 + 4y + 36y 2 + 400y 3 + 4900y 4 + O(y 7 ) , ψ 1 (y) = 1 2πi ψ 0 ln y + y 2 8y + 84y 2 + 2960 3 y 3 + 37310 3 y 4 + O(y 7 ) .(23) The modular variable is defined to be the ratio of the two periods: τ = ψ 1 /ψ 0 . A function defined on the related elliptic curve can be Fourier-expanded in terms of the q variable: q ≡ exp(2πiτ). In particular, the y variable can be treated as a function of τ or q and vice versa. In the vicinity of y = 0, the relations read y(q) = q − 8q 2 + 44q 3 − 192q 4 + 718q 5 − 2400q 6 + O(q 7 ) , q(y) = y + 8y 2 + 84y 3 + 992y 4 + 12514y 5 + 164688y 6 + O(y 7 ) . Note that the first coefficients in the above series are fixed to unity by the relative normalization between ψ 0 and ψ 1 chosen in Eq. (22). The q-expansion of y in Eq. (24) can be identified as the function y(τ) = η 8 (τ)η 16 (4τ) η 24 (2τ) = 1 16 λ(2τ) ,(25) using the Online Encyclopedia of Integer Sequences (OEIS) [72], where η(τ) is the Dedekind eta function and λ(τ) is the modular λ function. This is a well-defined extension of the seriesexpansion relation in Eq. 24, i.e., given a τ, one obtains a y. However, it is not straightforward to write down the inverse map from y to τ or q. We will show in the next subsection how to perform the analytic continuation for the inverse map. After that, given a y ∈ R + i0, we obtain a τ via τ(y) and return the same value of y via y τ(y) without ambiguity. The rest of this subsection, though, does not rely on this continuation. We only require the modular variable to be defined as the ratio of two periods controlled by the Picard-Fuchs operator and formulate the following as generally as possible. Any function or operator f of y can be treated as that of τ or q. We will abuse the notation a bit and write f (y), f (τ), or f (q) depending on the context. For differential operators, the Jacobian is given by J(y) = 1 2πi dy dτ = Θ q y , with Θ q ≡ q d dq = J(y) d dy .(26) It can be expressed in terms of the Wronskian W (y) ≡ 2πi ψ 0 d dy ψ 1 − ψ 1 d dy ψ 0 = y 3 1 − 16y ,(27) as J(y) = ψ 2 0 W (y) .(28) The above two identities can be derived from the differential equation satisfied by ψ k and the definition for τ = ψ 1 /ψ 0 . Note that the initial Picard-Fuchs operator L 0 2 (y) is an irreducible second-order operator. However, with the variable change, one can verify that it factorizes in q-space as: L (0) 2 (y) = ψ 0 J 2 Θ 2 q 1 ψ 0 .(29) This is a special case of the generalized factorization of Calabi-Yau operators [37,73,74]. After the analytic continuation of the modular map, Eq. (29) holds for all values of y. The factorization pattern is the starting point of the canonical basis in the top sector below. We now proceed to choose the ansatz for the top-sector canonical MIs as M 1 = ε 4 I 1 ψ 0 , M 2 = J(y) ε d dy M 1 − F 11 (y) M 1 ,(30) We require the two MIs to satisfy the following ε-factorized differential equations under the maximal cut: J(y) d dy M 1 M 2 mc = ε A mc (y) M 1 M 2 mc .(31) These lead to constraints on the coefficient function F 11 , which can be easily solved as F 11 (y) = 1 − 8y y 4 ψ 2 0 .(32) The connection matrix for the top sector takes the following form A mc = F 11 1 F 2 11 F 11 .(33) We now bring back the sub-sector dependence with the inhomogeneous term in (20). Note that the function f sub (y) consists of UT MIs and the ε-dependence is already factorized out in Eq. (20). These tell us that the differential equations for M 1 and M 2 are automatically ε-factorized without the maximal cut: J(y) d dy M 1 M 2 = ε A mc (y) M 1 M 2 + ε 16y − 1 y 6 ψ 3 0 (y) f sub (y) 0 1 .(34) Hence, we have obtained the ε-form differential equations for family (a), with the canonical MIs in Eqs. (30) and (10). Elliptic curve and analytic continuation of the modular map The map between y and q in Eqs. (23) and (24) only works in the vicinity of y = 0. In this subsection, we analytically continue this map to be valid for the whole kinematic space. This is achieved by studying the associated elliptic curve. We formulate the ε form in the top sector in terms of y. Hence all the formulae in the previous section carry on in terms of τ with the continued map, valid for all kinematical values (except the threshold point). Taking the maximal cut of I 1 in the Baikov representation, we obtain the associated elliptic curve: E : v 2 = (u − u 1 ) (u − u 2 ) (u − u 3 ) (u − u 4 ) ,(35) where the four roots are u 1 = 0 , u 2 = 1 − √ 1 − 16y 2 , u 3 = 1 + √ 1 − 16y 2 , u 4 = 1 .(36) In the region 0 < y < 1/16, the four roots are ordered as u 1 < u 2 < u 3 < u 4 . We define the elliptic modulus as k 2 = (u 2 − u 1 )(u 4 − u 3 ) (u 3 − u 1 )(u 4 − u 2 ) = 1 − √ 1 − 16y 1 + √ 1 − 16y 2 .(37) The periods of the elliptic curve reside in the solution space of the Picard-Fuchs operator L (0) 2 , which is spanned by the two complete elliptic integrals of the first kind, {K(k 2 ), iK(1 − k 2 )}. It attempts to relate the modular variable τ to the ratio of these two elliptic integrals. However, as k 2 changes with y, the complete elliptic integrals will develop discontinuity when across branch cuts. As a result, the periods naively defined by the complete elliptic integrals as functions of τ are not smooth. This can be fixed by compensating for the discontinuity when crossing the branch cuts. In this context, we define the two periodsψ 0 (y) andψ 1 (y) as linear combinations of the two elliptic integrals: ψ 1 (y) ψ 0 (y) = 2 π y 2 (u 3 − u 1 )(u 4 − u 2 ) γ(y) i K(1 − k 2 ) K(k 2 ) ,(38) where the monodromy matrix γ(y) is given by γ(y) =            1 2 0 1 , y < 0 or y ≥ 1 8 , 1 0 0 1 , 0 ≤ y < 1 8 .(39) The matrix is determined by the discontinuity [75] of the complete elliptic integral K(k 2 ), whose branch cut in terms of k 2 is [1, ∞). At y = 1/8 + i0, 1 − k 2 goes across the branch cut of K(1 − k 2 ), and one needs to compensate the change of the imaginary part to make the function continuous. Performing the series expansion around y = 0 and comparing with Eq. (23), we may identify ψ 0 =ψ 0 and ψ 1 =ψ 1 /4, and τ =ψ 1 (y) 4ψ 0 (y) . It is easy to verify that the above function is indeed the inverse map of Eq. (25) in the entire phase space. Thus Eq. 25 and Eq. 40 give one-to-one maps between y and τ. Interestingly, we find that the three kinematic singular points are exactly mapped to the three cusps of the congruence subgroup Γ 1 (4): τ(y = 0) = i∞ , τ(y = ∞) = 1 2 , τ(y = 1/16) = 0 .(41) We show the behaviors of τ and q as functions of y in Fig. 2, and the two periods are shown in Fig. 3. Letters and modular forms We may now consider letters in the ε-factorized differential equations as functions of the modular variable τ. It is instructive to study the properties of these functions under the modular transformation with a, b, c, d ∈ Z and ad − bc = 1. As mentioned earlier, the relevant congruence subgroup is Γ 1 (4), meaning that a, d ≡ 1 (mod 4) and c ≡ 0 (mod 4). The functions should have specific scaling behaviors under this group transformation. First of all, the kinematic variable y as a function of τ should not change under the transformation, i.e., τ → aτ + b cτ + d ,(42)y aτ + b cτ + d = y(τ) .(43) In other words, y(τ) is a modular function under the congruence subgroup. It is a meromorphic function in the complex upper half-plane H and at the cusps. In Eq. (25), we have given the analytic expression of y(τ) in terms of an eta quotient, which is valid in the whole domain. The periods and the coefficients in the differential equations are, in general, modular forms, transforming with certain modular weight k: f aτ + b cτ + d = (cτ + d) k f (τ) ,(44) which are holomorphic in H ∪ {cusps}. The building block at hand is ψ 0 , a modular form with weight 1. Note that the dimension of the modular weight-1 Eisenstein subspace of Γ 1 (4) is 1, and that of the modular weight-2 Eisenstein subspace is 2. They are spanned by {e 1,1 } and {e 1,2 , e 2,2 } respectively: ψ 0 = λ(2τ) 2 64 e 1,1 .(45) We refer the readers to [59] for their explicit definitions. The Jacobian J and the coefficient F 11 manifest in modular weight 2, while F 2 11 is of modular weight 4. The mixing with sub-sectors involves a modular form of weight 3. It is convenient to introduce another basis for the weight 1 and weight 2 modular forms: b 1,1 = ψ 0 y 2 , b 1,2 = 1 − 8y y 4 ψ 2 0 , b 2,2 = 1 − 16y y 4 ψ 2 0 ,(46) which is related to the Eisenstein basis by e 1,1 = 1 4 b 1,1 , e 1,2 = 1 6 b 1,2 − 1 8 b 2,2 , e 2,2 = 1 4 b 1,2 − 1 8 b 2,2 .(47) Four letters in the differential equation are modular forms. They are η 1,2 = b 1,2 , η 2,2 = b 2,2 , η 3 = b 1,1 b 2,2 , η 4 = b 2 1,2 .(48) Five extra letters are not modular forms, given by ρ = η 3 √ 1 − 4y , ϑ = η 2,2 √ 1 − 4y , ϕ = η 2,2 √ 1 + 4y , ϖ 1 = η 2,2 1 − 4y , ϖ 2 = η 2,2 1 + 4y .(49) The differential equation for the whole family (a) can then be reformulated as 1 2πi d ⃗ M dτ = ε                                    ⃗ M,(50) with ⃗ M = (M 1 , M 2 , M 3 , · · · , M 11 ) T . Comparing the above equation with Eq. (11), we see that the letters with modular weight 2 actually correspond to algebraic letters in terms of y. Here we note that η 1,2 and η 2,2 are weight-2 modular forms, and y is a weight-0 modular function of τ. The derivative with respect to τ increases the modular weight by 2. An example is the Jacobian J(y(τ)), which has weight 2. The relations between the τ-representation and the y-representation of these weight-2 letters are given by: η 1,2 · 2πi dτ = d log y − 1 2 d log(1 − 16y) , η 2,2 · 2πi dτ = d log y , ϑ · 2πi dτ = d log 1 − √ 1 − 4y 1 + √ 1 − 4y = d logt , ϕ · 2πi dτ = d log √ 1 + 4y − 1 √ 1 + 4y + 1 = d log u , ϖ 1 · 2πi dτ = d log y − d log(1 − 4y) , ϖ 2 · 2πi dτ = d log y − d log(1 + 4y) .(51) The weight-3 and weight-4 letters (η 4 , η 3 and ρ), on the other hand, are genuinely elliptic and do not have d log representations. It is worth mentioning that modular weight 2 does not guarantee d log representations, either. For example, for three-loop Banana integrals, whose geometric object is a K3 surface, logarithmic differential one-forms are insufficient for modular weight-2 letters. See [37] for an extensive discussion. Boundary conditions We need the boundary conditions to solve the differential equations, which we choose as y = 0. Since this is a singular point, we need to calculate the asymptotic behaviors of the canonical MIs in the limit y → 0. The sub-sector integrals are simple, and we collect the boundary conditions in the Appendix. For the top sector, it is enough to calculate I 1 . From the indicial equation of L (0) 2 around y = 0, we know that I 1 scales as y 2 for small y, with some logarithmic corrections. Then the boundary terms are given by these logarithmic corrections. They can be calculated by Mellin-Barnes techniques. The Mellin-Barnes representation of I 1 reads: I 1 = e 2ε γ E 3 ∏ i=1 dz i 2πi y 2+2ε+z 1 Γ(−z 1 )Γ(−z 2 )Γ(−z 3 ) Γ(1 + z 2 ) 2 Γ(−ε − z 1 ) 2 Γ(1 + z 2 + z 3 ) Γ(−2ε − 2z 1 )Γ(−3ε − z 1 ) × Γ(−1 − 2ε − z 1 − z 2 )Γ(1 + 2ε + z 1 − z 3 )Γ(−2ε − z 1 + z 3 ) 2 Γ(1 − ε + z 2 + z 3 ) Γ(−2ε − z 1 + z 2 + z 3 + 1) 2 ,(52) and then we perform the asymptotic expansion for y → 0. With the help of MBTools [76] and XSummer [77], we obtain the first three orders in ε: where L y = log y. I 1 y 2 y→0 = 1 3 L 4 y − 6ζ 2 L 2 y − 40ζ 3 L y − 49ζ 4 + ε 1 5 L 5 y − 20ζ 2 3 L 3 y − 42ζ 3 L 2 y − 29ζ 4 L y + 64ζ 2 ζ 3 + 32ζ 5 + ε 2 7 90 L 6 y − 11ζ 2 3 L 4 y − 32ζ 3 L 3 y − 69ζ 4 L 2 y + (124ζ 2 ζ 3 − 236ζ 5 ) L y − 280ζ 6 + 246ζ 2 3 + O(ε 3 ),(53) It is straightforward to derive the boundary conditions for M 1 and M 2 from that of I 1 . Using ψ 0 = y 2 + O(y 3 ), F 11 = 1 + O(y), J(y) = y + O(y 2 ) and y = q + O(q 2 ), we have: M 1 y→0 = ε 4 1 3 L 4 q − 6ζ 2 L 2 q − 40ζ 3 L q − 49ζ 4 + ε 5 1 5 L 5 q − 20 3 ζ 2 L 3 q − 42ζ 3 L 2 q − 29ζ 4 L q + 64ζ 2 ζ 3 + 32ζ 5 + O(ε 6 ), M 2 y→0 = ε 3 4 3 L 3 q − 12ζ 2 L q − 40ζ 3 + ε 4 2 3 L 4 q − 14ζ 2 L 2 q − 44ζ 3 L q + 20ζ 4 + O(ε 5 ),(54) where L q = log q. Results The solutions to the differential equations can be formally written as iterated integrals over the letters, with the boundary conditions obtained above. Armed with all the above, it is straightforward to write down the leading terms of M 1 and M 2 in ε expansion here: M 2 = ε 3 − Within the radius of convergence, the iterated integrals can be efficiently computed via q-expansion. 3 The convergence radius of the expansion is determined by the nearest singularity in relevant letters. We emphasize that the modular form letters are holomorphic in the upper half plane H. Thus the singularities are present only in the algebraic letters from sub-sector dependence. Three singular points 4 affect the top-sector integrals: y = 0, −1/4, ∞. To cover the whole phase space, we can perform the expansion around each of these points. Here we only consider the expansion around y = 0 (i.e., q = 0). We show the radius of convergence for this expansion in Fig. 4 2 . The q-expansion converges rather fast in this region. With the first 8 orders in the expansion, our results perfectly agree with those computed using AMFlow [16,18,19]. For regions of y outside (−1/4, 1/20), we note that in the case of MPLs, one may employ various transformations [81] to bring them back to the regions where series expansions converge. In the elliptic cases, we will need to employ the modular transformations in addition. We leave this to future investigations. The top sector integrals in family (b) We now turn to the top sector of family (b). Our treatment is essentially similar to family (a), with some new ingredients. These ingredients are generic and appear in other cases as well. The first issue is that there are more than two MIs in the top sector, and we need to introduce new objects in the ansatz of canonical MIs. Secondly, the dependence on the sub-sector integrals is not automatically ε-factorized as the inhomogeneous term of the Picard-Fuchs equation. We introduce a subtraction scheme, which can be applied to other cases, to achieve factorization. The Picard-Fuchs operator and ε-factorization The maximal cut There are three MIs in the top sector. The first-order differential equations for them are equivalent to the following third-order differential equation for I 1 : where the inhomogeneous terms on the right-hand side are sub-sector integrals and will be discussed later. The coefficients in the Picard-Fuchs operator are L (ε) 3 (y) I 1 = 3 ∑ i=0 r i (y, ε) d i dy i I 1 = R(y, ε) ,(56)r 3 (y, ε) = 1 , r 2 (y, ε) = 6y (1 + 4y)ε − 1 + 3 y(y + 1)(8y − 1) , r 1 (y, ε) = 16y 3 + 12y 2 + 4 ε 2 + −24y 3 − 18y 2 − 12y ε + 8y 3 + 30y 2 + 6y − 7 y 2 (y + 1) 2 (8y − 1) , r 0 (y, ε) = 1 y 3 (y + 1) 2 (8y − 1) −(8y 2 + 8y)ε 3 − (16y 3 + 4y 2 + 8y + 8)ε 2 +(24y 3 + 20y 2 + 14y)ε − 8y 3 − 32y 2 − 4y + 8 .(57) Although this operator is of order 3, when ε = 0, it factorizes into the composition of a secondorder operator L (0) 2 (y) and a first-order operator L (0) 1 (y), found via DFactor [82] in Maple [83]: L (0) 3 (y) = d dy + 8 8y − 1 L (0) 1 (y) d 2 dy 2 + 1 y + 1 + 8 8y − 1 − 3 y d dy + 8y(y + 2) − 4 y 2 (y + 1)(8y − 1) L (0) 2 (y) .(58) The irreducible second-order operator L (0) 2 (y) is associated with an elliptic curve, similar to that in family (a), while the first-order one acts as a "decoration". This factorized structure appears very often in elliptic integral families. In general, the complexity of the underlying geometric object increases with the order of the Picard-Fuchs operator when it is irreducible. For example, there are cases associated with hyperelliptic curves [84] or Calabi-Yau n-folds [73,[85][86][87][88][89][90]. However, when the operator is factorable, like in Eq. (58), the geometry becomes simpler. In this case, although the operator is a third-order one, the essence is still an elliptic curve instead of more complicated objects. One may generalize the above observation to higher-order operators. Suppose that an (n + 1)-order operator L (0) n+1 (y) factorizes into L (0) n+1 (y) = d dy + 1 y + a L (0) n (y) .(59) We can expect that the geometry is determined by the n-order operator L (0) n (y). In fact, we can say that these two operators are equivalent up to an inhomogeneous term. The solution space of L where c is a constant. The operators L ψ k (y) = 1 (2πi) k k ∑ j=0 ln j y j! ∞ ∑ n=0 a k− j,n y n+2 , (k = 0, 1, 2) .(61) The overall normalization is fixed by a 0,0 = 1. The first solution ψ 0 is holomorphic, while the other two have logarithmic behaviors. The expansion coefficients can be obtained iteratively, and the results read ψ 0 (y) = y 2 1 + 2y + 10y 2 + 56y 3 + 346y 4 + 2252y 5 + O(y 8 ), ψ 1 (y) = 1 2πi ψ 0 ln y + y 3 20 60 + 330y + 2000y 2 + 12805y 3 + 85262y 4 + O(y 8 ) , ψ 2 (y) = 1 (2πi) 2 −ψ 0 ln 2 y 2 + 2πi ψ 1 ln y + 3y 4 16 12 + 120y + 931y 2 + 6910y 3 + O(y 8 ) .(62) One can check that ψ 0 and ψ 1 are annihilated by L (0) 2 , i.e., they are associated with the two independent periods of the underlying elliptic curve. We again define the modular variable τ ≡ ψ 1 /ψ 0 and q ≡ exp(2πiτ). The relation between y and q (around y = 0) is given by y(q) = q − 3q 2 + 3q 3 + 5q 4 − 18q 5 + 15q 6 + 24q 7 + O(q 8 ) , q(y) = y + 3y 2 + 15y 3 + 85y 4 + 522y 5 + 3366y 6 + 22450y 7 + O(y 8 ) . (63) The q-expansion of y coincides with the sequence A123633 in the OEIS [91], whose closed form is given by y(τ) = η(τ) 3 η(6τ) 9 η(2τ) 3 η(3τ) 9 .(64) Like family (a), this is the well-defined extension of the first line of the series expansion above. We will extend the second line by studying the associated elliptic curve later on, such that given a y ∈ R + i0, we obtain a τ via τ(y) and then return the same y via y τ(y) without ambiguity. The rest of this subsection does not rely on these but on the requirement that ψ i are periods of the Picard-Fuchs operator. As a result, the construction is as general as possible. The Jacobian J(y) for the variable change from y to τ or q is defined similar as Eq. (26), and can be expressed as J(y) = ψ 2 0 (y) W (y) ,(65) where the Wronskian is now W (y) ≡ 2πi ψ 0 d dy ψ 1 − ψ 1 d dy ψ 0 = y 3 (1 − 8y)(1 + y) .(66) It's easy to check that ψ 2 satisfies L (0) 2 (y) ψ 2 (y) = 1 1 − 8y ,(67) which is an example of Eq. (60). We can also rewrite ψ 2 as an iterated integral with ψ 0 and τ as the functional degrees of freedom: ψ 2 (τ) = ψ 0 (τ) τ i∞ dτ 2 τ 2 i∞ dτ 1 (1 − 8y 1 )(1 + y 1 ) 2 y 6 1 ψ 0 (τ 1 ) 3 ,(68) where y 1 = y(τ 1 ). We now want to factorize the operator L (0) 3 completely in q-space. Although L (0) 3 is reducible and is not a genuine Calabi-Yau operator, we can borrow the treatment for the latter and define the so-called "normal forms" [37,73,74]: α 1 (y) = y d dy ψ 1 ψ 0 −1 = J(y) y , α 2 (y) = y d dy α 1 (y)y d dy ψ 2 ψ 0 −1 = y J(y) d 2 dτ 2 ψ 2 ψ 0 −1 .(69) From the above we can define one "Y -invariant": Y (τ) = α 1 α 2 = d 2 dτ 2 ψ 2 ψ 0 = ψ 0 (y) 3 (1 − 8y)W (y) 2 .(70) With the help of the Y -invariant, we can write the Picard-Fuchs operator in the factorized form in q-space: L (0) 3 (y) = ψ 0 Y J 3 Θ q 1 Y Θ 2 q 1 ψ 0 .(71) It is easy to check that the right-hand side annihilates ψ 0 , ψ 1 , and ψ 2 as expected. The above factorization pattern is essential to write down the ansatz for the canonical MIs in the top sector, and the workflow can be streamlined for other elliptic integral families. Following [35,36], we write the ansatz as: M 1 = ε 4 I 1 ψ 0 , M 2 = J(y) ε d dy M 1 − F 11 M 1 , M 3 = 1 Y (y) J(y) ε d dy M 2 − F 21 M 1 − F 22 M 2 + sub-sector integrals .(72) We demand that the differential equations for the above basis are ε-factorized under maximal cut (so the sub-sectors integrals are irrelevant at the moment). This leads to constraints on the coefficient functions F i j . Under maximal cut, the differential equation for the above ansatz takes the following form: J(y) d dy   M 1 M 2 M 3   mc = A mc   M 1 M 2 M 3   mc =   ε F 11 ε 0 ε F 21 ε F 22 εY A 31 A 32 A 33     M 1 M 2 M 3   mc .(73) The first and second rows of A mc are automatically ε-factorized. We now need to bring the third row to the ε-form by appropriate choices of the coefficient functions. We first note that J(y) and Y (y) as in Eqs. (65) and (70) satisfy the following relation: 3 J ′ (y) J(y) = 3 ψ ′ 0 (y) ψ 0 (y) + Y ′ (y) Y (y) + 1 y + 1 + 16 8y − 1 − 3 y ,(74) which makes the O(ε 0 ) terms in A 33 vanish. Combining L (0) 2 ψ 0 = 0 and Eq. (70), we can see that A 32 does not contain ε −1 terms. The absence of ε −1 terms in A 31 can be used to constrain the coefficient function F 11 . It is convenient to rewrite F 11 (y) as 5 F 11 (y) = F(y) + 28y 2 + 2y + 1 3y 4 ψ 0 (y) 2 .(75) To remove the ε −1 terms in A 31 , the function F(y) must satisfy the constraint F ′′ (y) + J ′ (y) J(y) − Y ′ (y) Y (y) F ′ (y) = 0 .(76) The solution reads F(y) = c 1 + c 2 τ i∞ dτ 1 Y (τ 1 ) = c 1 + c 2 2πi q 0 dq 1 q 1 Y (q 1 ) ,(77) where c 1 and c 2 are integration constants. Since Y (q) = 1 + O(q 2 ), the second term in the above expression is logarithmic divergent. To avoid that, we choose c 2 = 0. The constant c 1 is arbitrary and for convenience we also set it to zero. Consequently, we obtain F 11 (y) = 28y 2 + 2y + 1 3y 4 ψ 0 (y) 2 . The function F 22 (y) can be used to remove the ε 0 terms in A 32 , with the following constraint: F ′ 22 (y) − Y ′ (y) Y (y) F 22 (y) = 112y 4 + 308y 3 − 45y 2 + y − 1 y 5 (y + 1)(8y − 1) − 28y 2 + 2y + 1 3y 4 ψ ′ 0 (y) ψ 0 (y) ψ 0 (y) 2 .(79) It is easy to verify that F 22 (y) = F 11 (y) in (78) is a solution to the above equation. Further demanding A 31 to be free from ε 0 terms, we have the following constraint on F 21 (y): F ′ 21 (y) − Y ′ (y) Y (y) F 21 (y) = (2y − 1) 88y 3 + 84y 2 + 66y − 11 3y 8 ψ ′ 0 (y) ψ 0 (y) − 2 704y 6 − 64y 5 + 712y 4 − 1224y 3 − 352y 2 + 154y − 11 3y 9 (y + 1)(8y − 1) ψ 0 (y) 4 . A specific solution reads F 21 (y) = (2y − 1) 88y 3 + 84y 2 + 66y − 11 3y 8 ψ 0 (y) 4 .(81) Collecting all the entries, the 3 × 3 connection matrix in the top sector reads A mc = ε         (28y 2 +2y+1)ψ 0 (y) 2 3y 4 1 0 (2y−1)(88y 3 +84y 2 +66y−11)ψ 0 (y) 4 3y 8 (28y 2 +2y+1)ψ 0 (y) 2 3y 4 − (y+1) 2 (8y−1)ψ 0 (y) 3 y 6 64(y+1) 2 (8y−1)ψ 0 (y) 3 27y 6 0 2(y+1)(8y−1)ψ 0 (y) 2 3y 4         .(82) The structure of the above matrix hints us to perform a further rotation of the master integrals:   M ′ 1 M ′ 2 M ′ 3   =   1 0 0 0 0 3 √ 3 i 8 0 1 0     M 1 M 2 M 3   ,(83) The differential equation for the new basis takes a similar form as Eq. (73): J d dy   M ′ 1 M ′ 2 M ′ 3   mc = ε   η ′ 1,2 0 1 η ′ 3 η ′ 2,2 0 η ′ 4 η ′ 3 η ′ 1,2     M ′ 1 M ′ 2 M ′ 3   mc ,(84) The new connection matrix is persymmetric and of uniform modular weight, with the entries η ′ 1,2 = 28y 2 + 2y + 1 3y 4 ψ 0 (y) 2 , η ′ 2,2 = 2(y + 1)(8y − 1) 3y 4 ψ 0 (y) 2 , η ′ 3 = 8i(y + 1) 2 (8y − 1)ψ 0 (y) 3 3 √ 3y 6 = 8i 3 √ 3 Y , η ′ 4 = (2y − 1) 88y 3 + 84y 2 + 66y − 11 ψ 0 (y) 4 3y 8 .(85) However, this new basis is inconvenient in practice, and we won't use it in the following. Including sub-sector independence The inhomogeneous term in (56) reads: R(y, ε) = ε 2 ⃗ f 2,sub (y) + ε 3 ⃗ f 3,sub (y) · ⃗ M sub ,(86) with ⃗ M sub = (M 4 , M 5 , · · · , M 18 ) T and ⃗ f 2,sub (y) = −4 (1 + y) 2 (8y − 1) 0, −5, 5, 4, 4, 16y 2 − 58y + 1 −2(1 − 4y) 3/2 , 9 2 , −5, −6, −4, 2, 7 2 , 29, 15, 4 , ⃗ f 3,sub (y) = 4 y(1 + y) 2 (8y − 1) 0, −3 + 7y, 3 − 7y, −8y, −8y, − (y + 1) 48y 2 − 4y + 3 (1 − 4y) 3/2 , 3(y + 1)(2y − 3) 4y − 1 , 13y 2 + 10y + 5 y − 1 , −2(y + 1), −2(8y + 3), 5 8y 2 − y + 1 2(4y − 1) , 1 2 (−3y − 1), 35y 2 + 66y + 11 y − 1 , (y + 1)(13y + 15) y − 1 , − 128y 3 + 115y 2 + 18y − 9 2(y − 1)(4y − 1) . (87) Apparently, R(y, ε) is not ε-factorized due to the presence of ⃗ f 2,sub . This leads to nonfactorized ε-dependence in the differential equations for M 3 . A systematic and minimal prescription to eliminate this is subtracting certain sub-sector contributions in the definition of M 3 , as hinted in Eq. (72). Here we write, a priori, Elliptic curve and analytic continuation of the modular map As in the case of family (a), we'd like to extend the relation between y and τ to the entire phase space, given by Eq. (64). For that purpose, we study the associated elliptic curve, obtained by the maximal cut of I 1 : E : v 2 = (u − u 1 ) (u − u 2 ) (u − u 3 ) (u − u 4 ) ,(93) with the four roots u 1 = −1 , u 2 = − √ 1 − 8y + 1 2 4 , u 3 = − √ 1 − 8y − 1 2 4 , u 4 = 0 .(94) In the region 0 < y < 1/8, the four roots are ordered as u 1 < u 2 < u 3 < u 4 . We define the elliptic modulus as k 2 = (u 2 − u 1 )(u 4 − u 3 ) (u 3 − u 1 )(u 4 − u 2 ) = 1 − 4y − 8y 2 − √ 1 − 8y 1 − 4y − 8y 2 + √ 1 − 8y .(95) Similar to the family (a), we define two periods as linear combinations of two complete elliptic integrals: ψ 1 (y) ψ 0 (y) = 2 π y 2 (u 3 − u 1 )(u 4 − u 2 ) γ(y) i K(1 − k 2 ) K(k 2 ) ,(96) where the monodromy matrix is given by There are four cusps for Γ 1 (6): τ = 0, 1/3, 1/2, i∞. The first period ψ 0 is holomorphic around τ = i∞ (y = 0), but is not continuous around the cusp τ = 1/2 (y = −1), and the cusp τ = 0 (y = 1/8), which is shown as a sub-plot in the above. γ(y) =            1 2 0 1 , y < 0 or y ≥ √ 3 − 1 4 , 1 0 0 1 , 0 ≤ y < √ 3 − 1 4 .(97) One can check that the rotation above compensates for the discontinuity of the complete elliptic integral when its argument crosses the branch cut. The crossing happens twice: one at k 2 = ∞, and the other at k 2 = 2 which corresponds to y = ( √ 3 − 1)/4. Performing the series expansion around y = 0 and comparing with Eq. (62), we can identify ψ 0 =ψ 0 and ψ 1 =ψ 1 /6, and τ =ψ 1 (y) 6ψ 0 (y) . It is easy to verify that the above function is the inverse map of Eq. (64). It is also straightforward to check that the kinematic singular points are mapped to the cusps of the congruence subgroup Γ 1 (6): τ(y = −1) = 1 2 , τ(y = 0) = i∞, τ(y = 1/8) = 0, τ(y = ∞) = 1 3 .(99) Behaviors of τ and q as functions of y in the upper half complex plane are shown in Fig. 6. The analytically continued periods are shown in Fig. 7, which are smooth for all values of y except at cusps. Letters and modular forms The relevant modular group for family (b) is Γ 1 (6). Like family (a), the kinematic variable y(τ) is a modular function, and so are rational functions of y. For example, the Wronskian W (y) = y 3 (1 − 8y)(1 + y) = η(τ) 2 η(6τ) 22 η(2τ) 10 η(3τ) 14 ,(100) corresponds to the sequence A002508 [93] on OEIS. Modular forms of Γ 1 (6) have been extensively studied in the literature about Feynman integrals. The building block ψ 0 lives in the 2dimensional vector space of weight-1 modular forms. In terms of the Eisenstein basis {e 1,1 , e 2,1 }, it can be written as ψ 0 = y 2 (2e 1,1 + 4e 2,1 ) . For our purpose, it is convenient to introduce another basis: b 1,1 = 1 √ 3 ψ 0 y , b 2,1 = 1 √ 3 ψ 0 y 2 .(102) The dependence among the top-sector integrals themselves in the differential equations (91) involves the following modular form letters (the matrix elements in Eq. 82, and the modular weight is manifest by the power of ψ 0 ): η 1,2 = 28b 2 1,1 + 2b 1,1 b 2,1 + b 2 2,1 , η 2,2 = 16b 2 1,1 + 14b 1,1 b 2,1 − 2b 2 2,1 , η 1,3 = −3 √ 3 8b 3 1,1 + 15b 2 1,1 b 2,1 + 6b 1,1 b 2 2,1 − b 3 2,1 , η 4 = 3 176b 4 1,1 + 80b 3 1,1 b 2,1 + 48b 2 1,1 b 2 2,1 − 88b 1,1 b 3 2,1 + 11b 4 2,1 .(103) The dependence of the top-sector integrals on the sub-sector ones is complicated in the presence of a square root of the modular function 1 − 4y(τ) and a simple pole at y = 1. The Jacobian or the Y -invariant cancels the spurious poles at y = 0 and y = −1. Two new modular forms appearing in the mixing letters are given by: η 3,2 = 16J(y) 3y(y + 1) = 16(b 2 2,1 − 8b 1,1 b 2,1 ), η 2,3 = 2Y (y) y + 1 = 6 √ 3(b 3 2,1 − b 2 2,1 b 1,1 − 8b 2,1 b 2 1,1 ).(104) The dimension of weight-2 modular forms is 3, and the vector space is spanned by {η 1,2 , η 2,2 , η 3,2 }, for example: J(y) y = − 3 2 η 2,2 , J(y) 1 + y = − 3 2 η 2,2 − 3 16 η 3,2 .(105) Three extra letters are not modular forms: ρ = 7 − 8y √ 1 − 4y η 2,3 , ϑ = 1 + y √ 1 − 4y η 3,2 , ϖ = η 3,2 y − 1 .(106) They are well-behaved in a neighborhood of y = 0 but not globally for all y ∈ R + i0, either due to the branch cut of the square root or the pole at y = 1, which does not map to a cusp. We rewrite the ε-factorized differential equations for the top-sector integrals as follows: 1 2πi d M 1 dτ = ε η 1,2 M 1 + M 2 , 1 2πi d M 2 dτ = ε η 4 M 1 + η 1,2 M 2 + η 1,3 M 3 + 10η 2,3 M 5 − 10η 2,3 M 6 − 8η 2,3 M 7 − 8η 2,3 M 8 + ρM 9 − 9η 2,3 M 10 + 10η 2,3 M 11 + 12η 2,3 M 12 + 8η 2,3 M 13 − 4η 2,3 M 14 − 7η 2,3 M 15 − 58η 2,3 M 16 − 30η 2,3 M 17 − 8η 2,3 M 18 ,(107) 1 2πi d M 3 dτ = ε − 64 27 η 1,3 M 1 + η 2,2 M 3 − 4η 3,2 M 5 + 4η 3,2 M 6 + 2η 3,2 M 7 + 2η 3,2 M 8 − 4ϑM 9 − (4η 3,2 + 6ϖ)M 11 + 4η 3,2 M 13 − 2η 3,2 M 14 + η 3,2 M 15 − (8η 3,2 + 24ϖ)M 16 − (6η 3,2 + 12ϖ)M 17 + (5η 3,2 + 9ϖ)M 18 . Similar as family (a), letters with modular weight 2 are still algebraic functions of y, and can be casted into the d log representations: η 1,2 · 2πidτ = 1 3 d log y − d log(y + 1) − 1 2 d log(1 − 8y) , η 2,2 · 2πidτ = − 2 3 d log y, η 2,2 · 2πidτ = 16 3 d log y − d log(1 + y) , ϑ · 2πidτ = 16 3 d log 1 − √ 1 − 4y 1 + √ 1 − 4y = 16 3 d logt , ϖ · 2πidτ = 8 3 d log(1 − y) + d log(1 + y) − 2 d log y .(108) Boundary conditions The boundary point is again taken to be y = 0, and we need to calculate the asymptotic behaviors of the canonical MIs in this limit. Results for the sub-sector integrals are collected in the Appendix. For the top sector, we again employ the Mellin-Barnes representation. The 4-fold integral representation of I 1 reads I 1 = e 2ε γ E 4 ∏ i=1 d z i 2πi y 2+2ε+z 1 Γ(−z 3 )Γ(−z 4 )Γ(ε − z 2 )Γ(−ε − z 4 )Γ(1 − ε + z 2 ) × Γ(−z 1 + z 4 )Γ(−ε − z 1 + z 4 )Γ(−2ε − z 1 + z 3 )Γ(−3ε − z 1 + z 2 ) × Γ(1 + 2ε + z 1 − z 3 )Γ(−z 2 + z 3 )Γ(ε − z 2 + z 3 )Γ(ε − z 2 + z 4 )Γ(−2ε + z 3 − z 4 ) Γ(−2ε − z 1 )Γ(−3ε − z 1 )Γ(−ε − z 2 + z 3 − z 4 )Γ(−ε − z 1 − z 2 + z 3 − z 4 ) .(109) We perform the asymptotic expansion for y → 0 and the series expansion in ε. Up to order ε 2 , the result for I 1 is given by I 1 y 2 y→0 = 7 12 L 4 y − ζ 2 L 2 y − 20ζ 3 L y − 31ζ 4 2 + ε 3 20 L 5 y − 9ζ 2 L 3 y − 34ζ 3 L 2 y − 125ζ 4 2 L y + 40ζ 2 ζ 3 + 8ζ 5 + ε 2 37 360 L 6 y − 2ζ 2 L 4 y − 88ζ 3 3 L 3 y + 13ζ 4 4 L 2 y + (100ζ 2 ζ 3 − 112ζ 5 ) L y + 287 4 ζ 6 + 188ζ 2 3 + O(ε 3 ) ,(110) with L y = log y. Using the definition in Eqs. (72) and (88), it is straightforward to derive the boundary conditions for the canonical MIs in the top sector: M 1 | boundary y→0 = ε 4 7 12 L 4 q − ζ 2 L 2 q − 20ζ 3 L q − 31ζ 4 2 + ε 5 3 20 L 5 q − 9ζ 2 L 3 q − 34ζ 3 L 2 q − 125ζ 4 2 L q + 40ζ 2 ζ 3 + 8ζ 5 + O(ε 6 ) , M 2 | boundary y→0 = ε 3 7L 3 q 3 − 2ζ 2 L q − 20ζ 3 + ε 4 5 9 L 4 q − 80 3 ζ 2 L 2 q − 184ζ 3 L q 3 − 172ζ 4 3 + O(ε 5 ) , M 3 | boundary y→0 = ε 2 [0] + ε 3 112 3 ζ 3 + 16 3 ζ 2 L q − 8 9 L 3 q + O(ε 4 ) ,(111) where L q = log q. It is interesting to note that the two terms in Eq. (88) both contain order ε 2 terms. However, they cancel each other, and M 3 starts at order ε 3 . Results We can express the solutions of differential equations order-by-order in ε via iterated integrals. It turns out that the leading term in M 3 is of order ε 3 , i.e., the weight-2 part in M 3 originates entirely from sub-sectors and is canceled by the subtraction in Eq. (88). The leading terms of the three top-sector integrals are given by M 3 = ε 3 112 3 ζ 3 + ζ 2 I(6ϖ + 4ϑ +M 1 = ε 4 − 31ζ 4 2 − 20ζ 3 ln q + ζ 2 I(1, 6η 2,3 − ρ; i∞, τ) + 9 32 I (1, ρ, ϑ, η 2,2 ; i∞, τ) − 9I 1, η 2,3 , η 2,2 , η 2,2 + 9 16 η 3,2 ; i∞, τ + O(ε 5 ) .(112) Like family (a), the combination of ε-factorized differential equations and the UT boundary conditions tells us that the solutions are UT or pure functions in the elliptic sense by the definition of [69]. The convergence of the q-expansion, in this case, is again spoiled by the singularities in the sub-sector dependence. For the expansion around q = 0, the radius of convergence is fixed by the nearest singularity at y = 1. 6 We depict the convergence region as the disk bounded by the dashed circle in Fig. 8. This dashed circle intersects with the q-path at four points when varying y from −∞ + i0 to +∞ + i0. The corresponding values of y at the intersection points can be obtained by combining Eq. (98) and the equation of the circle. In terms of these values, the convergent regions correspond to y ∈ (−0.8772, 0.1248) ∪ (0.1260, 1), i.e., x ∈ (−∞, −8.013) ∪ (−7.937, −1) ∪ (1.140, +∞). Note that y = 1/8 corresponds to a cusp point, and its small neighborhood should be removed, see Eq. (99) and Fig. 7. Note that the convergence is relatively slower when the value of q is close to the boundary circle. Hence, in the following we numerically study the region x ∈ (−∞, −10) ∪ (2, ∞) In Fig. 9, we depict the numeric results from q-expansion (with the first 8 orders) for the weight-5 part of M 1 and the weight-4 parts of M 2 and M 3 . We compare them with numerical points calculated by AMFlow and find good agreements. Again, we leave the transformation of the iterated integrals to other kinematic regions for future work. Conclusions In this paper, we have investigated two Feynman integral families involving elliptic curves. Their top sectors are both two-loop non-planar triangle integrals with different configurations of internal masses. They depend on two mass scales. Hence the normalized integrals depend on only one dimensionless variable. We extend the formalism developed for equal-mass Banana integrals to these cases and derive canonical bases for all sectors that satisfy ε-factorized differential equations. Compared with equal-mass Banana integrals, the major new ingredient is the presence of non-trivial sub-sectors. We construct the sub-sector canonical bases using Baikov representations and solve the differential equations in terms of MPLs. The top-sector integrals are represented by iterated integrals with elliptic kernels. We note that the canonical bases have UT boundary conditions. As a result, the top-sector canonical integrals are pure in the elliptic sense. Our basic tool to achieve ε-factorization is the analysis of the Picard-Fuchs operators corresponding to the elliptic curve underlying the top-sector integrals. The modular variable τ defined as the ratio of the two periods of the elliptic curve, plays an essential role here. Utilizing this variable, we show that the Picard-Fuchs operator can be factorized under the maximal cut and for ε = 0. Family (b) is a more complicated example with three top-sector MIs, but its essence is still an elliptic curve. Its Picard-Fuchs operator is of order three but can be reduced (when ε = 0) to the composition of a first-order and a second-order operator. The further factorization of the second-order operator with the modular variable is then derived with the help of a new ingredient, the so-called "Y "-invariant. We emphasize that the Picard-Fuchs method is rather generic and can be applied to geometric objects beyond the elliptic curves, such as those appearing in three-and higher-loop equal-mass Banana integrals [35][36][37]. Generally speaking, the reason is that fundamental degrees of freedom related to such geometric objects behind Feynman integrals are the periods that are controlled by the Picard-Fuchs operator. After achieving ε-factorization within the top sectors, we deal with the dependence of the top-sector MIs on the sub-sectors. For family (a) with a simpler sub-sector structure, we find that this dependence is automatically ε-factorized (which is similar to the cases in Banana integrals). This is, however, not true for family (b). We introduce a systematic subtraction scheme to make the sub-sector dependence ε-factorized as well. Family (b) serves as a prototype of a large class of elliptic Feynman integrals, which involve higher-order Picard-Fuchs operators and complicated sub-sectors. We believe that our treatment for family (b) paves the way to deal with those situations. We investigate in detail the letters (i.e., kernels in the iterated integrals) in the two examples. Since the essential geometry objects are elliptic curves, it is natural to work in the modular space. We find that the letters in the top sector of the family (a) are modular forms of Γ 1 (4), while those for the family (b) are modular forms of Γ 1 (6). Letters within sub-sectors themselves are algebraic that can be rationalized with proper variable changes. The complication arises from the nontrivial mixing between top-sector MIs and sub-sector MIs. The letters are combinations of modular forms of τ and algebraic functions of y. The solutions are formally expressed in terms of iterated integrals, which are used to obtain numerical results via q-expansion. The evaluation is efficient and agrees perfectly with AMFlow. Due to the singular points in the sub-sector letters, the q-expansion around y = 0 has a finite convergence radius. Extending to other kinematic regions can be achieved by suitable transformations of the iterated integrals. We leave this to future investigations. − π 4 4 log 2 y + 8ζ 3 log y + ε 5 − 11 60 log 5 y − π 2 6 log 3 y + 31 . (114) For family (b), there are 18 master integrals in total, 3 of which are in the top sector, while the others are in sub-sectors. The pre-canonical master integrals for family (a) Figure 2 : 2The behaviors of τ and q as functions of y. The red path corresponds to y ∈ (−∞, 0), the magenta one corresponds to y ∈ [0, 1/16), and the blue one corresponds to y ∈ [1/16, +∞). Figure 3 : 3The two periods as functions of y. The solid lines represent real parts, and the dashed lines are imaginary parts. There are three cusps for Γ 1 (4): τ = 0, 1/2, i∞. With the normalization in Eq.(38), ψ 0 is holomorphic around τ = i∞ (y = 0). However, it is not continuous around the cusp τ = 1/2 (y = 1/16), shown as a zoomed sub-plot in the figure. Figure 4 : 4Radius of convergence in q-space for the series expansion around q = 0 for family (a), and the corresponding values of y. 40ζ 3 − 312ζ 2 I(ρ; i∞, τ) + 20 I(η 3 , ϕ, ϕ; i∞, τ) − 12 I(ρ, ϑ, η 2,2 ; i∞, τ) + O(ε 4 ),M 1 = ε 4 − 49ζ 4 − 40ζ 3 L q − 12ζ 2 I(1,ρ;i∞, τ) + 20 I(1, η 3 , ϕ, ϕ; i∞, τ) − 12 I(1, ρ, ϑ, η 2,2 ; i∞, τ) + O(ε 5 ). , which is constrained by the singularity at y = −1/4. The dashed circle in the figure represents the convergence region. For real values of y, this region corresponds to y ∈ (−1/4, 1/20), i.e., x = s/m 2 ∈ (−∞, −20) ∪ (4, ∞).InFig. 5, we show the numeric results from q-expansion for the integrals M 1 and M 2 in the region x ∈ (−∞, −20) ∪ (4, ∞). We give the first two orders of ε, M 1 = ∑ n≥4 ε n M Figure 5 : 5Numeric results from q-expansion (solid and dashed lines) for the integrals M 1 and M 2 of family (a) in the region. Left: a broad range for |x| > 20; right: a closer look at 4 < |x| < 20. The results are in good agreement with those from AMFlow (circles and triangles). y) is spanned by the solutions {ψ i } (i = 0, 1, . . . , n − 1) of L n (y)ψ n (y) = c y + a , 2 (y) have four regular singular points: {−1, 0, 1/8, ∞}. Again, 0, ∞ are the two points where the solutions have uniform asymptotic behaviors. Similar to family (a), we focus on the point y = 0 here. The Frobenius method tells us to write the solutions of L Figure 6 : 6The behaviors of τ and q as functions of y. The black path corresponds to y ∈ (−∞, −1), the orange one corresponds to y ∈ [−1, 0), the blue one corresponds to y ∈ [0, 1/8), and the red one corresponds to y ∈ [1/8, +∞). Figure 7 : 7The two periods after the analytic continuation. The solid lines are real parts, and the dashed lines are imaginary parts. Figure 8 : 8Radius of convergence in q-space for the series expansion around q = 0 for family (b), and the corresponding values of y. Figure 9 : 9Numeric results in family (b) from q-expansion (solid and dashed lines) for the weight-5 part of M 1 , and the weight-4 parts of M 2 and M 3 . Left: a broad range for |x| > 10; right: a closer look at 2 < x < 10. The results are in good agreement with those from AMFlow (circles and triangles). The overall factor y 2 in ψ k is determined by the indicial equation around y = 0. There are alternative ways to evaluate these integrals. See, e.g., the algorithm[78] implemented in GiNaC[79], and Ref.[80].4 The point y = 1/4 is only relevant for the sub-sector integral M 5 . This is inspired by[35] and is in agreement with the algorithm in[92]. The singularity at y = 1/4 again only affects the sub-sector integral M 9 . AcknowledgementsWe thank Stefan Weinzierl and Sebastian Pögel for detailed comments on the manuscript and fruitful discussions, Guo-Xing Wang for discussions about the Mellin-Barnes representations, and Roman Lee for providing LiteRed2. X.W is grateful for the inspiring and fruitful discussion with Christoph Nega and Lorenzo Tancredi during the work. This work was partly supported byNote that ⃗ M sub is already a UT basis for the sub-sectors. We only need to require d dy ⃗ g 2,sub (y) = (1 − 8y) ⃗ f 2,sub (y) ,to cancel the contribution from ⃗ f 2,sub in the inhomogeneous terms. It is easy to find a particular solution to the above equation and we choose it to be ⃗ g 2,sub (y) = Now, the differential equations for the three top-sector integrals are given bywhere A 31 (y), A 32 (y) and A 33 (y) have been derived in(82)andis the consequence of the subtraction term and the remaining inhomogeneous term.A Boundary conditions for sub-sector integralsThis Appendix gives the asymptotic expressions of sub-sector integrals in the limit y → 0 for families (a) and (b= ε 2 log 2 y 2 + π 2 6 + ε 3 2 log 3 y 3 + π 2 3 log y − 4ζ 3 + ε 4 5 log 4 y 12 + π 2 4 log 2 y − 8ζ 3 log y + ε 5 11 log 5 y 60log y + 32 5and for family (b) we have= ε 4 π 2 12 log 2 y − ζ 3 log y + 7π 4 360 + ε 5 π 2 12 log 3 y − π 4 360 log y − 2 3 π 2 ζ 3 + 9ζ 5= ε 4 π 2 12 log 2 y−2ζ 3 log y + 11π 4 180 +ε 5 π 2 12 log 3 y− ζ 3 2 log 2 y− π 4 18 log y + 5π 2 6 ζ 3 + 5ζ 5 +O(ε 6 , y),= ε 3 1 6 log 3 y + π 2 3 log y + 2ζ 3 + ε 4 − π 2 4 log 2 y − ζ 3 log y − 41π 4 180 + ε 5 1 120 log 5 y + π 2 12 log 3 y − ζ 3 2 log 2 y + 19π 4 72 log y + π 2 6 ζ 3 + 10ζ 5 + O(ε 6 , y) ,= ε 4 1 8 log 4 y + π 2 4 log 2 y − ζ 3 log y + π 4 30 + ε 5 1 20 log 5 y − π 2 9 log 3 y − 5ζ 3 2 log 2 y − 71π 4 360 log y − π 2 2log 3 y − π 2 3 log y + 4ζ 3 +ε 4 − 5 12 log 4 y ζ 3 log 2 y + 17π 4 180 log y + 13π 2 9log y + 2ζ 3 + ε 4 − 1 8 log 4 y − π 2 12 log 2 y + 2ζ 3 log y − π 4 72 + ε 5 − 7 120 log 5 y − π 2 18 log 3 y + 3ζ 3 log 2 y + π 4 72 log y + 2π 2 3log 3 y − π 2 6 log y + 10 3 ζ 3 +ε 4 − 13 24 log 4 y + π 2 6 log 2 y +10ζ 3 log y] + ε 5 − 29 120 log 5 y + π 2 6 log 3 y + 46 3 ζ 3 log 2 y + 53π 4 360 log y + π 2 3 ζ 3 + 198 5 = ε − 1 2 log y + ε 2 − 3 4 log 2 t + π 2 12 + ε 3 − 7 12 log 3 y + π 2 6 log y + 4ζ 3 + ε 4 − 5 16 log 4 y + π 2 6 log 2 y + 28 3 ζ 3 log y + π 4 16 + ε 5 − 31 240 log 5 y + π 2 9 log 3 y +10ζ 3 log 2 y + 17π 4 120 log y − 8π 2 9 ζ 3 + 24ζ 5 + O(ε 6 , y) , the National Natural Science Foundation of China under Grant No. 11975030 and 12147103, and the Fundamental Research Funds for the Central Universities. 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[ "Improving aircraft performance using machine learning: a review", "Improving aircraft performance using machine learning: a review", "Improving aircraft performance using machine learning: a review", "Improving aircraft performance using machine learning: a review" ]
[ "Soledad Le Clainche \nUniversidad Politécnica de Madrid\nSpain\n", "Esteban Ferrer \nUniversidad Politécnica de Madrid\nSpain\n", "Sam Gibson \nUniversity of Sheffield\nUnited Kingdom\n", "Elisabeth Cross \nUniversity of Sheffield\nUnited Kingdom\n", "Alessandro Parente \nUniversité Libre de Bruxelles\nBelgium\n", "Ricardo Vinuesa \nKTH Royal Institute of Technology\nSweden\n", "Soledad Le Clainche \nUniversidad Politécnica de Madrid\nSpain\n", "Esteban Ferrer \nUniversidad Politécnica de Madrid\nSpain\n", "Sam Gibson \nUniversity of Sheffield\nUnited Kingdom\n", "Elisabeth Cross \nUniversity of Sheffield\nUnited Kingdom\n", "Alessandro Parente \nUniversité Libre de Bruxelles\nBelgium\n", "Ricardo Vinuesa \nKTH Royal Institute of Technology\nSweden\n" ]
[ "Universidad Politécnica de Madrid\nSpain", "Universidad Politécnica de Madrid\nSpain", "University of Sheffield\nUnited Kingdom", "University of Sheffield\nUnited Kingdom", "Université Libre de Bruxelles\nBelgium", "KTH Royal Institute of Technology\nSweden", "Universidad Politécnica de Madrid\nSpain", "Universidad Politécnica de Madrid\nSpain", "University of Sheffield\nUnited Kingdom", "University of Sheffield\nUnited Kingdom", "Université Libre de Bruxelles\nBelgium", "KTH Royal Institute of Technology\nSweden" ]
[]
This review covers the new developments in machine learning (ML) that are impacting the multi-disciplinary area of aerospace engineering, including fundamental fluid dynamics (experimental and numerical), aerodynamics, acoustics, combustion and structural health monitoring. We review the state of the art, gathering the advantages and challenges of ML methods across different aerospace disciplines and provide our view on future opportunities. The basic concepts and the most relevant strategies for ML are presented together with the most relevant applications in aerospace engineering, revealing that ML is improving aircraft performance and that these techniques will have a large impact in the near future.
10.1016/j.ast.2023.108354
[ "https://export.arxiv.org/pdf/2210.11481v1.pdf" ]
253,080,460
2210.11481
67bd3ca3f936bc0e65a7cfc19777084f3fa8de66
Improving aircraft performance using machine learning: a review Soledad Le Clainche Universidad Politécnica de Madrid Spain Esteban Ferrer Universidad Politécnica de Madrid Spain Sam Gibson University of Sheffield United Kingdom Elisabeth Cross University of Sheffield United Kingdom Alessandro Parente Université Libre de Bruxelles Belgium Ricardo Vinuesa KTH Royal Institute of Technology Sweden Improving aircraft performance using machine learning: a review This review covers the new developments in machine learning (ML) that are impacting the multi-disciplinary area of aerospace engineering, including fundamental fluid dynamics (experimental and numerical), aerodynamics, acoustics, combustion and structural health monitoring. We review the state of the art, gathering the advantages and challenges of ML methods across different aerospace disciplines and provide our view on future opportunities. The basic concepts and the most relevant strategies for ML are presented together with the most relevant applications in aerospace engineering, revealing that ML is improving aircraft performance and that these techniques will have a large impact in the near future. Introduction Climate change and increasing resource scarcity are challenges that Europe needs to face in the coming decades. All this has a direct impact on air transport, which is struggling to maintain its performance and competitiveness while ensuring a development focused on sustainable mobility. Research and innovation are essential to maintain the capabilities of the aviation industry, driven by the rise of new markets and new competitors as a result of globalization. A new longterm vision for the aeronautics sector is essential to ensure its successful advancement. In this line, new requirements for the future aviation industry have been defined by the ACARE Flightpath 2050, a Group of Recognized Personalities in the aeronautic sector, including stakeholders from the aeronautics industry, air traffic management, airports, airlines, energy providers and the research community. Aeronautics and air transport comprises both: air vehicle and system technology. The future of aviation should focus on improving design, reducing manufacturing time and cost (including certification and upgrade processes), and also improving the parts forming the overall air travel system (general aviation, aircraft, airlines, airports, air traffic management and maintenance, repair and overhaul). The ACARE Flightpath 2050 has defined 5 goals that should be achieved by 2050 to guarantee the path through sustainable mobility: • Compared to the capabilities of a typical aircraft in the year 2000, by 2050 new technologies should allow 90% reduction in NOx emissions, 75% reduction in CO2 emissions per passenger/kilometre and 65% reduction in noise emission of flying aircraft. • When taxiing, aircraft movements should be emission-free. • Novel strategies to design and manufacture aerial vehicles should be developed to make them recyclable. • Sustainable alternative fuels should be developed to position Europe as the centre of excellence in the field, and sustained by a strong European energy policy. • By 2025, Europe should take the lead to establish global environmental standards, formulating and prioritizing an environmental action plan, and being at the forefront of atmospheric research. At the same time, ensuring safety and security is also a major priority, with the aim at reducing by 80% the number of accidents by 2050 compared to 2000, taking into account the rising traffic. To achieve these goals, it is extremely important to find newer eco-friendly alternatives suitable for the industry to reduce the aviation net carbon emissions and noise. To this aim, the aerospace industry is gathering efforts towards developing new aerodynamic designs, more efficient, reducing the oil consumption whilst maintaining the safety in the flight performance. Moreover, finding new alternatives to fossil fuels, improving the energy efficiency in combustion systems, or finding optimal routes for air traffic management (ATM) are also some of the key points where the aerospace industry should advance to minimize the environmental footprint. However, to achieve these objectives it is not enough to improve the 'standard' configurations. The aerospace engineering industry is aware that to go beyond the state-of-the-art it is necessary to develop novel ground-breaking disruptive technologies. Fluid and solid mechanics need to be advanced with applications in aerodynamics, acoustics, and combustion, to develop new technologies, resulting in novel aircraft designs with reduced environmental impact (see Fig. 1). Researchers in collaboration with the aeronautical industry should explore: (i) new aircraft configurations able to reduce noise and pollution emissions, (ii) cruise drag reduction by manipulating turbulent flow structures close to the aircraft surface (i.e., delaying the boundary transition from laminar to turbulent flow), using novel friendly low-risk practices, (iii) novel strategies for flow control (rising the benefits achieved by only changing the external shape) to enhance the aerodynamic performance reducing drag, noise or flow transition, rising lift or controlling unsteadiness or flow separation, and (iv) reduce the system complexity with novel aircraft materials and lighter designs (which directly results in less fuel consumption), and reduced aircraft maintenance and life cost cycle [3]. High-fidelity numerical simulations and advanced experimental techniques (i.e., wind tunnel experiments or open-air experiments as in the case of flight test, et cetera) allow collecting a large variety of data, containing relevant information about physical principles connected to the aerodynamic performance of the aircraft, the efficiency of the combustion system or the main instabilities driving the flow dynamics and the possibility of attenuating or boosting such instabilities using active or passive flow control techniques [2]. Additionally, experiments (i.e., ultrasounds, non-intrusive testing, et cetera) and simulations provide information connected to the fundamentals of solid mechanics, the presence of noise and the structural health of the aircraft, allowing for noise control and early failure detection. However, the economic and computational cost, related to the performance of experiments and simulations, encourages researchers to look for new alternatives, which allow to advance in the field i.e., developing relevant technologies for the aerospace industry, while avoiding delays in the manufacturing and time-to-market process. Aircraft development, manufacturing, maintenance and support are four critical levels that must be accurate and reliable to ensure the success of the aerospace industry. Artificial intelligence (AI) and machine learning (ML) have been introduced in the aerospace industry for various applications connected to the reduction of aircraft's environmental impact, including data interpretation [174], system management, customer service or aircraft modelling and to generate new high-fidelity databases at a reduced (economic and CPU) cost [191], solving problems of optimization, flow control, or even providing optimal sensors distributions for solid mechanics or aeroelasticity applications. In the recent review article by Brunton et al. [40], the authors summarize the new trends and perspective of ML in the aerospace industry, including its application for smart manufacturing, and in the development (and aircraft design), production and product support phases (aircraft design, manufacturing, verification and validation). The authors reveals the possibilities of ML to process data in light computations increasing the production rate, based on the idea of the use of ML techniques that are measurable, interpretable and certifiable. ML is generally understood as a branch of AI, although there are nuances in the definition: ML aims at improving systems performance using self-learning algorithms, while AI tries to mimic natural intelligence solving complex problems and enabling decision making (although not maximizing the system efficiency) [242]. Both AI and ML are connected to Big Data, a term that linked to the enormous volume of data that floods the aeronautical sector every day or that is generated from CFD simulations or experimental measurements and connected to aircraft aerodynamic performance [62,216]. Combining Big Data with ML techniques, it is possible to develop reduced dimensional systems, such as reduced order models (ROMs) [171,318], capable to accurately predict the evolution of the flow dynamics [2,110,172] (i.e., flow control, reduce cruise drag [295], boundary layer transition, etc.), or surrogate models, capable to predict the aerodynamic forces and moments acting on the aircraft as a function of some parameters (i.e., Reynolds number, Mach number, geometry shape, etc.). This review provides a state-of-the-art of AI and ML applications in the aerospace engineering field. The basic concepts and most relevant strategies for ML and AI are brought together to explain the similarities found in the nomenclature of similar techniques used in different fields, also shedding light on new applications of these algorithms, quite extended in other fields but not known to the aerospace industry. For example, the review details the use of machine learning for reduced order modelling, which can be used to accelerate numerical simulations, or for temporal and spatial forecast (including non-intrusive sensing). Additionally, we include relevant applications of the field, including flow control, acoustics, combustion, flight test and structural health monitoring. This article intends to explore the possibilities of ML, an emerging field for the aerospace industry, identifying new research lines of potential interest and bringing new ideas to developing the technologies of the future, and founded on a primary goal: to fight climate change. This article reviews the main disciplines connected to the aerodynamic performance of the aircraft (see Fig. 1): fundamental fluid dynamics, aerodynamics, acoustics, combustion, and general solid mechanics; and based on the idea of finding novel efficient designs, capable to reduce noise and pollutant emissions, while at the same time, ensuring safety and security. The article is organized as follows. Section 2 introduces the machine learning methodologies, and the literature review of the main applications in aerospace engineering is presented in Section 3 for fundamental fluid dynamics, Section 4 for aerodynamics, Section 5 for aeroacoustics, Section 6 for combustion and Section 7 for solid mechanics. The main conclusions are presented in Section 8. Machine learning methodology: a general overview Current advances in computer science are strongly related to the increasing amount of data generated and stored in the different disciplines conforming aerospace engineering. The valuable information contained in these databases encourages researchers to develop and test sophisticated algorithms to exploit such information, to gain insight and knowledge from the data and to subsequently propose and develop new commercial strategies aligned with the ideas behind the concept of sustainable aviation: developing new cleaner and safer aircraft designs. ML is a fast-growing science in the field of aerospace engineering due to its good capabilities to extract information from complex databases, which it later used to develop models, such as ROMs or surrogate models. Based on available data and the type of training carried out within the analysis, ML algorithms can be classified into unsupervised, semi-supervised or supervised learning, as presented in Fig. 2. This section briefly introduces the basic idea behind some of these ML algorithms, which have been used for different applications in the field of aerospace engineering. Figure 2: Machine learning methods: a general overview. Classification extracted from Ref. [39]. In bold, the most popular techniques in the field of aerospace engineering. Neural networks ML uses artificial neural networks (ANNs), also called as neural networks (NNs), to process and extract information from databases. The name of this computing system is inspired by the biological neural networks of the human brain. ML uses NNs to solve an optimization problem. More specifically, using back propagation and stochastic gradient descent algorithms, ML optimizes the following compound function arg min A j (f P (A P , · · · , f 2 (A 2 , f 1 (A 1 , X)) · · · ) + λg(A j )),(1) which represents a NN formed by P layers. The weights connecting years k and k + 1 are given in matrix A k . To accurately represent the data, which is the main goal of ML and NNs, it is necessary to regularize this system, which is massively undetermined, with the function bias g(A j ). Using different regularization strategies, it is possible to prevent overfitting, allowing the NNs to generalize the solution obtained from the training to different data sequences. To define the architecture of a NN, it is necessary to set the dimension and number of layers and the type of connections and mappings (linear/non-linear) between the different layers, in order to get the best system performance to represent data. Depending on all these parameter and on the way the optimization problem is defined, it is possible to define the different type of ML architectures previously presented in Fig. 2. The generic structure of a multi-layer NN is presented in Fig. 3. Figure 3: Sketch representing a NN architecture with three layers. Structure extracted from Ref. [174]. In this example, the input database X = [x 1 x 2 x 3 ] ∈ R 3 , is map Y = [y 1 y 2 ] in the output layer, with a different space dimension, R 2 . The relationship between the different layers in this NN is defined in eq. (2) as X (1) = f 1 (A 1 , X) X (2) = f 2 (A 2 , X (1) ) Y = f 3 (A 3 , X (2) ).(2) This expression is also defined by the following compound functions f j , defined for P layers as Y = f P (A P , · · · , f 2 (A 2 , f 1 (A 1 , X)) · · · ),(3) which defines the general optimization problem presented in eq. (1). This expression defines a system of equations that is highly under-determined, which requires additional constrains to be solved, where the P matrices generate the best possible mapping. The activation function f j (for the layer j), can be both linear or non-linear (i.e., sigmoid, tanh, Rectified Linear Unit -ReLU, · · · ). In aerodynamics, radial basis function neural networks (RBFNNs) have been used since the past, due to the good generalization capabilities of the RBF as activation function [130]. This NNs model, proposed by Broomhead & Lowe [37] for the first time, generally uses a non-linear activation function f j defined by a Gaussian basis function as f j (x k ) = exp − x k − v 2 j 2σ 2 j ,(4) where the two hyper-parameters v and σ are known as the center vector and the width parameter, respectively. The value set for these two parameter strongly affects the NN performance [161,162]. Depending on the type of connection between the layers and the neuron of the NNs, it is possible to identify different architectures. For instance, Recurrent Neural Networks (RNNs) process sequence of data using recurrent connections between the different layers. Among the different architectures of RNNs it is possible to distinguish Long Short-Term Memory (LSTM) networks, which in sequence prediction problems are capable of learning order dependence. Another type of architecture widely use for ML applications in fluid dynamics are Convolutional Neural Networks (CNNs). This type of networks transfer information locally between the different neurons and layers. Based on a threshold, each neuron may transfer or not transfer information to the next layer of neurons. More details describing deep learning basic concepts can be found in the book by Brunton & Kutz [38], and some applications combining NNs linear mappings with non-linear modal interactions for turbulent flows are presented by Le Clainche et al. [174]. Regression and Classification Regression and classification belong to the category of supervised learning. In supervised learning, labeled data sets are used, to guide the machine during the training process. The algorithm learns by minimizing the error with respect to the labelled target. Regression and classification are widely popular and very effective techniques, but can lead to erroneous results if data (or categories) are missing, which leads to problems during training and/or classification. The missing data or category can relate to a problem misconception leading to a category missing in the data, to data removal because of large deviations, to equipment malfunctioning, etc. [293]. To avoid problems in these supervised-learning algorithms, a good a-priori knowledge (and analysis) of the problem at hand is advisable to ensure that the labelled training data is representative of the overall dataset. In this section, we focus on the most common regression and classification algorithms, including: regression and logistic regression, decision trees, random forest, K-nearest neighbours, support-vector machines and Naive-Bayes classification. A common task in supervised learning is regression, which is used to identify patterns and relationships within a dataset. In this task, the algorithm approximates a function for a continuous output (e.g., a real value). Classic examples are linear (or quadratic) regressions, where linear (or quadratic) curves are fit to data. This type of regressions can be understood from a purely algebraic viewpoint, through least-square approximations (i.e., Ridge regression using the L2 norm) but generalises to ML using NNs. In linear regression with one variable, the output y will fulfill the equation of a straight line: y = mx + c, where m is the slope, c the intercept and x the input. The calculation of (m) and (c) will be based on the minimization of a cost function, i.e., the minimizing the error between the training data and the linear model. An example is shown in Fig. 4. The regression model can be used to interpolate missing data or to extrapolate (forecast) outside the training-data range. When considering regression with multiple dependent variables, it is necessary to analyze the influence of each variable on the overall trend of the dependent variable (system output). Multiple linear regression is a useful technique for more than one independent variable. This technique can achieves better fits than when using simple linear regression if multiple independent variables are involved. Classification also belongs to the supervised-learning family. In this problem, the output will take a discrete set of values or categories. The algorithm will use pre-set labeled data to classify the data into multiple categories. According to the type of variables, we can distinguish categorical classification from numerical classification. Logistic regression is the simplest classification algorithm that performs a binary classification (between only two categories) and can help to determine the occurrence (or not) of an event. The algorithm starts with data previously labeled in the two possible categories. These two categories are joined through a sigmoid curve, which will determine the probability that new data falls into one category or the other, see Fig. 4. The sigmoid curve P (t) = 1 1 + exp (−t) helps classify the probability of success of an event; starting with a null probability (first category) and ending at a certain event (second category). The curve shows a change of curvature when the probability of success is 50%. This point, called threshold, will differentiate the data classified into both categories, as depicted in Fig. 4. The decision tree is a conceptually simple algorithm, useful for classifying data. The algorithm uses a tree, in which each of the branches implies a decision leading to a classification. To make these decisions and determine which characteristics to discern, it is necessary to train it with data. The construction of the trees, as well as the determination of the order of the different branches alludes to the reduction of dispersion (or entropy) in the data in each step providing an optimal tree with minimal dispersion. Random forest [193] is an algorithm for both classification and regression tasks and is particularly useful for data sets in which there can be loss of information. Its development starts from the construction of multiple decision trees (hence the concept of "forest"). Each decision tree performs a classification task leading to the final result of the random forest, which is the best classification of all trees. K-Nearest Neighbours (KNN) is one of the simplest algorithms used for supervised classification. It classifies the input data based on the closeness to already classified data. The algorithm generates "neighbors" of data and measures the distance (or "similarity") of new data with respect to the classified neighboring data. The KNN algorithm is effective if the data set is not excessively large (slow learning algorithm) and the data is free of noise. The KNN concept can be understood by drawing a circle centered on the new data point, which increases its radius until there are K elements (belonging to a pre-set category) within the circle. K represents the number of elements needed within the circumference to classify the model input; this gives the name to the algorithm K-nearest neighbors, as shown in Fig. 4. The choice of the parameter K is essential to achieve the desired resolution of the problem, and needs to be large enough to avoid the influence of noisy data, but small enough to limit the computation time. A possible value is the square root of the labeled data (n), i.e., K = √ n. Support-vector machines (SVMs) are similar to K-Nearest Neighbours and can be used to perform classification and regression tasks where the transformed feature space is very large. However, for large datasets, SVMs suffer from high computational cost. SVMs create a hyperplane to delimit the neighbours or regions in the data set, see Fig. 4. To determine the position of the hyperplane, the algorithm finds support vectors, defining the distance to the neighbours. This distance must be maximum for the classification to be successful. Note that the hyperplanes may have various shapes: linear, quadratic, cubic, etc. SVMs can deal with a wide variety of classification problems including high dimensions and non-linearities. A drawback is that SVMs require calibration of the parameters to attain good results. Finally, the Naive-Bayes classifier has its mathematical basis in the conditional probability proposed by Bayes' theorem, whose formulation is: P (A|B) = P (B|A) · P (A) P (B) . The logical operation is similar to the error back-propagation algorithm. The final probability will be based on the total probability taking into account the influence of the individual events on the following events (conditional probability). The total probability theorem, used to give the final output of the model is P (B) = P (A 1 ) · P (B|A 1 ) + P (A 2 ) · P (B|A 2 ) + ... The process requires to calculate the probability of each event individually and analyze its influence on the final response. This probabilistic analysis will finally lead to a decision. The algorithm will predict events, based on present and past data, knowing the probability of occurrence of the predicted events. Some advantages of Naive-Bayes classifiers are outlined in [293] and include: (i) they remain smooth to small changes in the network, (ii) they are flexible, allowing for transfer learning between classification and regression, (iii) they can handle missing data. Gaussian-process (GP) regression is an extension of Bayesian-regression method that is used widely within the field of structural health monitoring. Rather than predicting a single deterministic function, the predictive posterior is instead a distribution of functions that are consistent with the data that the model has been conditioned by. At any finite number of points in the output space, the distribution of these functions is Gaussian. Like SVMs, GP regression uses a kernel, or covariance function to estimate the similarity between two inputs and it is the selection of this kernel that defines the form that the posterior functions take, in combination with the optimization of the model hyperparameters. The GP can be defined by its mean and this covariance; often a zero mean prior is assumed, however, knowledge of the process can be incorporated into the model relatively simply using a mean function. A commonly-used covariance function is the squared exponential: k(x, x ) = σ 2 f exp − 1 2 2 ||x − x || 2 , where x and x are the inputs to the model, σ 2 f is the signal variance hyperparameter and is the lengthscale hyperparameter. The reader is referred to Ref. [263] for an in-depth review of GP regression theory. Semi-supervised learning In this section we will discuss two types of semi-supervised algorithms: generative adversarial networks (GANs) and deep reinforcement learning (DRL). GANs [119] are a type of generative model, which comprises two different parts: the generator (G) and the discriminator (D). These two parts of the network are assigned competing tasks, and they are trained against each other using game theory. A widely-used application of GANs is that of super-resolution, i.e.,, starting from a low-resolution input image L R , the GAN architecture needs to produce a high-resolution versionH R which should be consistent with the statistical features of the original high-resolution image H R . In this setup, the generator will be tasked with creating theH R images given the L R input and the reference H R . This process is illustrated for a turbulent flow in Fig. 5. Note that this super-resolution task can also be conducted in an unsupervised manner by imposing physical properties on the data, as documented by Kim et al. [155]. The discriminator is thus trained to differentiateH R and H R , so that the generator gets progressively better at producing realistic images, and the discriminator improves its performance when identifying the ones that have been artificially generated by G. It is common to use a convolutional neural network (CNN) as generator, and the resolution increase is usually carried out at the end of the network by means of sub-pixel convolution layers [175]. A common architecture for the discriminator in this type of task [125] is to also use a CNN, followed by fully-connected layers, and using a sigmoid activation in the last one it is possible to obtain a probability value between 0 and 1 to discern whether the high-resolution image is real or not. Figure 5: Schematic representation of a GANS architecture used to increase the resolution of the quantities measured at the wall in a turbulent channel flow. The color coding for each layer is 2D convolution (beige), parametric-ReLU activation (dark green), batch normalization (blue), sub-pix convolution (pink) and ReLU activation (light green). Note that the kernel size and the number of filters are shown at the bottom of each of the layers. Figure adapted from Ref. [125] with permission of the publisher. When it comes to DRL, in this framework an agent (which can be a neural network) performs actions on an environment (which would be the flow) in a closed loop. If we consider instant t, at which the environment has a certain state s t , the agent receives a partial observation of that state denoted o t . Based on this observation, the agent decides to apply a certain action a t , which will have an impact on the system, modifying its state to s t+1 , and so forth. The quality of those actions under a certain norm is measured by the so-called reward r t , which is provided periodically to the agent. Generally in reinforcement learning, the framework is aimed at developing an optimal decision policy a t = π(o t ) with the goal of obtaining the maximum cumulative reward over a certain time horizon. This process is summarized in Fig. 6. One widely-used implementation is the proximal-policy-optimization (PPO) method [285], which has the advantage of being faster than e.g., the trust-region policy optimization (TRPO) [285]. There are other suitable options for DRL, such as deep-Q network (DQN) learning [223], which is very efficient mainly with non-continuous control. In this sense, it is important to thoroughly assess the dynamics of the problem at hand to choose the most suitable DRL implementation, see for instance the variations of DQN learning reported by Gu et al. [121]. Figure 6: Schematic of the deep-reinforcement-learning process, where the aim is to find the optimal decision policy ensuring that the cumulative reward is maximized. Figure extracted from Ref. [324] with permission of the publisher. Unsupervised learning: clustering and dimensionality reduction Compared to classification and regression approaches, unsupervised learning deals with unlabeled daya with the objective of "learning" the data structure without assistance. Unsupervised methods allow processing large amounts of data, leading to the determination of the most relevant features that can be used for a lower-dimensional representation of the original data. As such, unsupervised learning techniques are highly relevant in aerospace applications. Clustering Clustering (e.g., K-means, mean shift, Gaussian mixture models) [30], is an unsupervised-learning method that does not require pre-sampled data to cluster regions and can automatically discover grouping within the data. An overview of clustering methods and comparisons for simple problems can be found in Ref. [246]. Here, we review the popular K-means, mean shift and Gaussian mixture models. K-means is the simplest and faster clustering technique. The user inputs the number of clusters "k" and the algorithm initializes "k" centroids, assigning each point in the dataset to the closest centroid. Using these temporary clusters, new centroids are calculated employing the mean distance from all the data points. The process is repeated iteratively until the centroids do not move in the subsequent iterations. The mean-shift clustering method is a variant of K-means where instead of the centroid, a sliding circular window of radius "Kernel" are considered. The goal is to slide the windows until the location that maximises the density of points inside the window is found. Densitybased spatial clustering of applications with noise (DBSCAN) is a variant of mean-shift that considers noisy unclassified data, and can provide enhanced robustness. Gaussian-mixture models assume that the data follows a Gaussian distribution with an associated mean and standard deviation. The algorithm estimates the mean and variance of each normal distribution iteratively using the expectation-maximization method [76] to provide the optimal estimation of these parameters. There is some recent interesting in aerodynamics and CFD, which uses clustering to detect flow regions. For Reynolds Averaged Navier-Stokes (RANS) simulations, Saetta & Tognaccini [277] and Lanzetta et al. [169] showed that various viscous sensors could be used as input to a clustering ML framework. The aim was to detect the boundary layer and wake regions for turbulent flow past aeronautical applications (airfoils, aircraft). The results showed that the ML clustering outperformed the classic viscous sensors used to identify the flow regions in RANS. In the context of Large Eddy Simulations (LES), similar ideas have been proposed using Gaussian mixture to identify laminar and turbulent regions [239,306], which can also be employed to perform local mesh adaptation to increase the accuracy of simulations. Furthermore, Callaham et al. [45] used unsupervised-learning techniques to identify the dominant physical processes for different flow scenarios and in Ref. [60] the authors trained a neural network to classify different types of vortex wakes. Dimensionality reduction One of the most common tasks in unsupervised learning is dimensionality reduction. Autoencoders use NNs (with linear or non-linear activation functions) to reduce the data dimensionality and extract the main data features [87]. In fluid dynamic problems, a very rich literature exists concerning the application of proper orthogonal decomposition (POD) [23]. POD is also known with a variety of names, depending on the application field, i.e., Karhunen-Loève decomposition (KLD) [154] in image and signal processing, Principal Component Analysis (PCA) [149] in the statistics literature, Empirical Orthogonal Functions [192]in oceanography and meteorology, etc. A detailed discussion on the relation between POD, PCA and KLD, and their connection with Singular Value Decomposition (SVD) is presented in [333]. Principal Component Analysis Dimensionality reduction techniques rely on the hypothesis that a reduced and optimal basis exists to represent the data. PCA thus finds a new, orthogonal basis to represent the data set, which is approximated by retaining only a subset of fewer directions, namely the ones accounting for most of the data variance in a transformed data set. Prior to performing PCA (as in all ML approaches) the raw data set is generally transformed (centered and scaled) to allow dealing with variables of different units and sizes, and to focus on fluctuations. Many options exist for data preprocessing and optimal choices cannot be defined universally but rather depending on the problem at hand [243]. The eigenvalue decomposition of the covariance matrix yields the eigenvectors, denoted as the principal components (PCs), ordered by the magnitude of their corresponding eigenvalues. Truncating the PCs to a desired level of variance provides a lowdimensional representation of the original data ( Fig. 7), which is the best linear predictor of the data matrix X in terms of squared prediction error [149]. PCs are determined to maximise variance, not interpretability. The complex structure of the PCs and the presence of non-negligible weights for most of the original variables on an individual PC might complicate the interpretation and exploitation of the PCs identified without supervision. To alleviate that, the eigenvector matrix can be rotated to aid physical interpretation of the scores, i.e., the projection of the data onto the new basis. Many rotation methods exist, they can either be orthogonal or oblique. Orthogonal transformations, such as Varimax rotation [150], rotates the PCs rigidly while maximizing the sum of the variances of the squared coefficients within each eigenvector. The rotation generally leads to rotated PCs with few significant coefficients, thus aiding interpretation. After rotation, the amount of variance accounted for by a set of PCs is conserved but the proportion is redistributed amongst the components. Principal Component Analysis allows to identify the direction of maximum variance in the data a 1 a 2 III -Rotation Figure 7: Steps in the dimensionality reduction process using PCA. x 1 x 2 I -Original data IV -Size reduction a 1 x 1 a 1 a 2 l 1 l 2 II -PC extraction x 2 17 Local dimensionality reduction Typical flow data sets relevant to aerospace applications display non-linear relationships among variables. PCA is a multi-linear technique, i.e., additional PCs are needed to approximate the original data set with low reconstruction error to compensate for the data non-linearity. To overcome the limitations of global PCA, an alternative is to identify local principal curves [84] and locally linear embeddings (LLE) [273], and to perform local modal decompositions, i.e., applying PCA locally (LPCA). local PCA (LPCA) can be exploited, partitioning the data into clusters by either using an a priori chosen feature, or using a Vector Quantization (VQ) technique coupled to the local application of PCA within clusters. The resulting approach, denoted as VQPCA [152,245], assumes that the non-linear manifold can be locally approximated by a linear one as the data manifold will not curve too much over the extent of the local region. In the first case a supervised clustering is carried out, while the latter is fully unsupervised. In VQPCA, the data are assigned to the different clusters by minimising the reconstruction error of each point to a given set of clusters (Fig. 8). The algorithm is iterative and convergence is achieved when the centroids' change from one iteration to the next one is below a prescribed threshold. Principal Component Analysis allows to identify the direction of maximum variance in the data Kernel PCA Kernel PCA (KPCA) is a non-linear dimensionality reduction technique that makes use of kernel methods. In KPCA [284], the original data-set is transformed into an arbitrarily high-dimensional feature space where the relationship amongst the variables is considered to be linear and PCA can be effectively carried out. A non-trivial, arbitrary function is chosen but never calculated explicitly, which allows using high-dimensional feature spaces. The non-linearity of the KPCA method comes from the use of a non-linear kernel function that populates the covariance matrix (kernel trick). Thus, the eigenvectors and eigenvalues of the covariance matrix in the feature space are never actually solved. The choice of the kernel function modeling the covariance between two different points in the feature space is up to the designer. An example is the choice of a Gaussian kernel. Since no operations are directly carried out in the feature space, KPCA does not compute the PCs themselves, but the projections of our data onto those components, namely the scores. One of the main drawbacks of kernel PCA is that PCA is performed on a covariance matrix that scales with the number of observations rather than the number of variables, thus increasing the computational burden of the method. In addition, no straightforward operation is available to reconstruct the data from the KPCA scores, since the kernel is never computed explicitly. This is in contrast to linear PCA, where a direct mapping exists. To reconstruct the data in Kernel PCA, a minimisation problem must be solved, making the process computationally more expensive compared to PCA. Autoencoders An autoencoder (AE) is an unsupervised neural network (NN) aiming at learning the reduced representation (encoding) of a dataset. Along with the encoder, that projects the data into a non-linear manifold (codes), a decoder reconstructs the data from the codes side, minimising the reconstruction error. The simplest form of an AE is a feed-forward, non-recurrent neural network having an input layer, an output layer and one or more hidden layers connecting them, where the output layer has the same number of neurons as the input layer, and with the purpose of reconstructing its inputs (minimizing the difference between the input and the output) via a loss function that can accommodate regularization and sparsity terms. It is worth noting that a shallow AE with linear activation functions corresponds to PCA. Fluid mechanics Fluid mechanics is closely connected with the aerodynamic performance of airplanes, and it is therefore essential to properly model and predict the properties of the flow around the aircraft in order to carry out optimization and design tasks. Very close to the airplane surface a thin layer, called boundary layer [58], develops. This region plays a critical role in aircraft design, and it is extremely challenging to model, particularly at high Reynolds numbers, i.e., when the boundary layer is turbulent [210]. Computational fluid dynamics The numerous success stories in ML documented for a wide range of areas [325] over the past years motivate the possibility of using ML methods for simulating fluid flows. An overview of how ML can help computational fluid dynamics (CFD) is provided by Vinuesa and Brunton [319], who essentially identified potential in many areas including accelerating CFD and improving turbulence modeling. It is important to note that ML methods will not constitute a replacement for traditional CFD methods [46,118], which are based on discretizing the governing equations and integrating them numerically. Instead, it is important to identify concrete areas within CFD that can benefit from ML. For instance, interesting work has been developed by Kochkov et al. [159] on improving the accuracy of coarse simulations by means of deep learning in simplified flows. Other areas of potential are the efficient resolution of the Poisson problem [5], which is encountered in operator-splitting methods, or approaches to reduce the size of the computational domain through efficient inflow [105,341] and far-field [227] conditions. Manrique de Lara and Ferrer [206,207] have used NNs to accelerate high-order simulations by including a corrective forcing that achieves high accuracy when running a low-order case, providing a framework for accelerated simulations. When it comes to turbulence modeling, Duraisamy et al. [78] provided a review of the potential of ML to improve RANS models. Note that these models, which are based on Reynolds averaging and typically rely on an eddy-viscosity approach [36] to represent the Reynolds stresses, are not necessarily satisfactory when simply using high-fidelity data to address the closure problem [334]. The anisotropy tensor, which is an important component of RANS models, was predicted by Ling et al. [188] using a deeplearning architecture which embedded Galilean invariance, an approach leading to excellent results. On the other hand, Wu et al. [335] proposed combining the physical knowledge from the flow with deep-learning architectures, producing a framework that allowed predicting both linear and nonlinear parts of the Reynolds-stress tensor separately. In addition to improved interpretability, this approach led to very good results in a number of configurations. The interpretability [275] of the deep-learning models for RANS simulations was one of the main features of the framework proposed by Jiang et al. [145]. When it comes to interpretability, we would like to highlight that deep-learning models are generally able to provide accurate predictions based on the input data, but it is typically not possible to establish the concrete relationships relating input and output. An interpretable model allows to establish such as connection, and there are a number of research directions [321] starting to enable this property in deep-learning models. A good example of this, which relies on genetic programming, is the work by Cranmer et al. [66]. It is also worth noting that Weatheritt and Sandberg [330] developed a methodology for RANS modeling based on gene-expression programming [165] which essentially produces models that are interpretable since they rely on combinations of basic symbolic expressions. Their models exhibit excellent results in a number of complex geometries where RANS models typically fail, including cases with secondary flows and significant three-dimensionality. Along these lines, a generelization of the constitutive relations used to model the Reynolds stresses can also help to properly predict complex flows [294], and in this sense data-driven model discovery is a promising research direction. Another simulation area where ML has shown potential is that of large-eddy simulation (LES). As opposed to RANS, where all the turbulent scales are modeled through the Reynolds stresses, in LES a certain range of (the larger) scales is simulated, and low-pass filtering is employed. The scales with higher wavenumbers (i.e.,, with smaller wavelenghts) are then modeled through a so-called subgrid-scale model (SGS). There are several deep-learning-based methods to develop such SGS models, e.g., the approach based on CNNs by Beck et al. [20]. CNNs are wiely used in computer vision, and they can exploit the spatial correlations present in the data to enhance their predictions. Thus, they are suitable for making accurate predictions in the context of turbulent flows. Another CNN-based approach was proposed by Lapeyre et al. [170], in their case applied to reactive flows. Despite the potential of these approaches, in principle they rely on data of higher fidelity (e.g., direct numerical simulations, DNSs) to train the deeplearning models. The higher-fidelity data is used to develop an SGS model able to reproduce the behavior of the non-modeled simulation in the LES case. The limitation of this approach relies on the need of such high-fidelity data, which limits the applicability of these models in a wide range of cases beyond those considered in the training stage. An alternative to this approach was proposed by Novati et al. [235], who used reinforcement learning (RL) to develop an SGS model. The advantage of this method compared with the one above is that it is unsupervised, i.e., it does not require the high-fidelity data as a reference. Furthermore, it is able to develop SGS models that do not necessary emulate the behavior of a high-fidelity simulation, but rather establish a condition consistent with the coarse case being simulated. This in principle can favor the generalizability of such LESs. Another area of potential interest within ML is that of wallmodel development. Wall models are employed at very-high-Reynolds-number applications, where the whole near-wall region is replaced by a suitable boundary condition [225]. Note that at the moment there is not a clear approach to accomplish this for general cases, given the fact that the flow in the outer region (for example streamwise pressure gradients) significantly determines many of the flow features closer to the wall [255,323]. Another difficulty is the fact that in wall-bounded turbulence over roughness the outer region has no way of knowing whether the wall is smooth or rough, and therefore there is uncertainty regarding the best approach to set these boundary conditions [148]. Despite these limitations, important progress has been made based on data-driven methods relying on physical arguments [222,280]. Deep learning is also providing streategies to be able to map the flow in the outer region and at the wall-normal location where the boundary condition would be set, either through direct prediction [9] or by determining a virtual velocity [228]. Another promising approach relying on ML is that developed by Bae and Koumoutsakos [15], which is based on reinforcement learning to determine the correct boundary information close to the wall. Reduced-order models Fluid-mechanics systems involving turbulence are high dimensional and chaotic, thus it is convenient to develop frameworks to reduce their dimensionality and simplify their analysis. The so-called ROMs are not only relevant when it comes to shedding light on the physics of these systems, but also regarding other applications such as flow control or optimization [303]. Classical methods for ROM development rely on linear algebra, where a widely-used approach is the so-called proper-orthogonal decomposition (POD) [196], which relies on the singular-value decomposition (SVD) as discussed above. The POD framework, which is essentially data driven, is applied to N + 1 three-dimensional snapshots of the flow field u(x, t), where boldface denotes tensors, u is a vector containing the three velocity components at instant t and x is a vector containing the spatial coordinates. The POD of the spatio-temporal signal u(x, t) would yield: u(x, t) = u(x) + N i=1 a i (t)φ i (x),(5) where u(x) is the mean flow, a i (t) are the temporal coefficients and φ i (x) are the spatial modes. Consequently, POD separates spatial and temporal information in the flow, and because the spatial modes are ranked by their energy contribution to the original signal [196], it is possible to truncate the sum in eq. (5) at a given number of modes, defining a certain level of energy reconstruction. Another interesting related technique is the so-called dynamic-mode decomposition (DMD) [282], which also identifies a low-dimensional model of the original system, with modes ranked by frequency (instead of energy). These methods are linear, and since turbulent flows are inherently non linear, it may be interesting to to explore methods which include and exploit such non linearity. An example of this is the so-called higher-order dynamic-mode decomposition (HODMD) [173], which embeds non-linearity in the temporal dependencies among the snapshots used to develop the ROM. In this sense, ML has great potential to also help develop suitable ROMs able to exploit the massive amounts of data available and the non-linearities characterizing these flows. An early example of this is the approach by Milano and Koumoutsakos [217], who employed deep neural networks to learn the temporal dynamics of ROMs of the near-wall region of turbulent channels. In fact, they established meaningful connections between neural networks and POD, namely the fact that when nonlinearities are removed from neural networks one recovers POD when developing ROMs. Other types of ROMs purely based on HODMD (a tool generally used in fluid dynamics to identify the main flow physical patterns) have been used to predict the temporal evolution of numerical databases in transitional [171? ] and turbulent flows [172]. The advantage of these models is that they were based on physical principles (driving the main flow dynamics), although the type of models used were purely linear, limiting their application in some case. Recently, Le Clainche et al. [174] combined HODMD with NNs to develop a model capable to predict the wall shear stress in a channel with a porous wall leading to drag reducing/increasing effects (anisotropic turbulence). The model used HODMD to identify the main instabilities leading the flow dynamics. The non-linear interaction of the HODMD (also known as DMD) modes was also considered in the model to then combining this information (related to the flow physics) with simple architectures of NNs to predict the wall shear stress for long time intervals. Based on the same idea of developing predictive ROMs containing information modelling the flow physics, Abadía-Heredia et al. [2] combined POD modes with NNs using CNNs or LSTM architectures to predict the temporal evolution of the flow using data from the transitory of CFD simulations. In all these examples, using data dimensionality techniques based on the physical principles describing the flow dynamics, allowed to use one-dimensional architectures for the NNs. López-Martín et al. [191] also show that using both CNNs or LSTM it is also possible to accurately predict the temporal flow evolution from transient CFD databases, but using three-dimensional NNs architectures instead. The low-order model of the near-wall cycle of turbulence proposed by Moehlis et al. [224], which exhibits all the relevant phenomena present in the high-dimensional data, has been used as a platform to assess the feasibility of ML techniques to predict the temporal dynamics of the flow. For instance, LSTM networks were used by Srinivasan et al. [295] to model the system, obtaining excellent predictions of turbulence statistics, Poincaré maps and Lyapunov exponents (following system perturbation). Other successful approaches to model the temporal dynamics in Ref. [224] include the Koopman-based framework (with non-linear forcing) by Eivazi et al. [86] (which can provide in some cases better short-term predictions than the LSTM at a reduced cost) and the technique based on physics-constrained reservoir computing by Doan et al. [77] (which is able to accurately model extreme events). Borrelli et al. [35] employed deep learning to predict the temporal dynamics of a minimal turbulent channel, achieving very good results, in line with those achieved for low-order models of turbulent flows. Finally, AEs are another very promising method for modal decomposition, because they enable developing non-linear ROMs. CNNs have been employed for AE-based modal decomposition [106,229], and Eivazi et al. [87] have shown their applicability to complex turbulent flows. In fact, they showed significant reductions of the number of modes necessary to reconstruct the original system compared with POD, while retaining orthogonality of the modes. Experiments Experiments can also benefit from the numerous developments in the ML field. In general, exploiting the vast amount of data available in numerical simulations to improve experimental procedures is a very promising endeavor. Kim and Changhoon [156] have shown that the measured wall-shear stress can be used to predict the heat flux in a wall-bounded flow by means of deep neural networks, thus paving the way to augmented experimental data when no thermal quantities are measured. Another important area within experiments is that of non-intrusive sensing, i.e., the possibility to measure at the wall and predict the flow behavior farther from the wall. This also has important applications in the context of flow control, as discussed below. Guastoni et al. [123] showed that fully-convolutional neural networks (FCNs) can be used to predict the velocity fluctuations above the wall in a turbulent channel, using as input the wallshear-stress components and the wall pressure. Note that in FCNs all the layers throughout the network are based on convolutions, whereas this restriction does not apply in the more general CNNs. This is a very important result, because the wall quantities are easier to measure without disrupting the flow. Note that they obtained excellent predictions in the near-wall region, although their quality deteriorates farther away. Furthermore, they showed that deep neural networks are more suitable than traditional prediction methods such as linear stochastic estimation (LSE) [90] or extended POD (EPOD) [34] to predict turbulent flows. Güemes et al. [124] proposed an extension to the FCN approach by combining it with POD, such that the neural network is used to predict the temporal coefficients of the POD modes of the flow on a certain plane. Later, Güemes et al. [125] extended this work to the implementation of generative adversarial networks (GANs), which are able to perform super-resolution tasks. The GANs-based approach enabled performing accurate and robust predictions from few sparse measurements at the wall, as opposed to the previous studies, which required finely-resolved wall inputs. Another important framework for experiments is that of physics-informed neural networks (PINNs) [259]. This is a framework which exploits the machinery around the training process of neural networks (mainly the automatic differentiation) to formulate and solve partial differential equations (PDEs). Note that PINNs are currently being used to augment experimental measurements [260] and to improve the quality of the measurements in fluid-mechanics problems [85,88]. It is important to note that experiments in fluid mechanics rely on a number of empirical corrections, where data-driven methods can help to develop more general correction schemes [320,322]. And as mentioned above, ML is also helping to develop more robust flow-control strategies. Non-intrusive sensing, particularly combined with opposition control [54], is an excellent tool to enhance the experimental implementation of flow-control algorithms. Other data-driven methods for flow control include Bayesian regression based on Gaussian processes [263], as done in the context of boundary-layer control by Mahfoze et al. [200], and genetic programming, which has been recently used to control several complex flows [181,219]. Perhaps the most promising approach for control of fluid flows is DRL, which is a data-driven technique capable to discovering novel and more effective control solutions, as proved by Rabault et al. [258] in the case of the two-dimensional flow around a cylinder. Application of DRL to redue the length of separation bubbles has been documented by Guastoni et al. [122], and an overview of DRL applications to turbulent flows, including flow control, has been provided by Vinuesa et al. [324]. Aerodynamics Improving aerodynamic design in aircraft is one of the main goals of the aerospace industry. Aircraft wings, airfoil and turbine engine blades are some of the aircraft parts that need to be studied in detail to provide more efficient designs reducing drag (minimizing pollutant emissions) and noise (diminishing airframe and engine noise). The aerospace industry is already concerned about some design concepts that could result in more efficient designs reducing noise and drag. For instance, high aspect ratio wings with adaptive sections are connected to reduced vortex and wave drag in the aircraft, while to reduce friction and pressure drag it is necessary to develop flow control strategies to promote laminar flow, reducing the drag of the turbulent skin friction and controlling the unsteady flow separation and turbulent noise source. To reduce critical loads in the aircraft, in addition to control flow separation, it is important to control the presence of shock waves and unsteady loads for instance delaying the onset of buffeting, preventing flutter [216], which is induced by linear aerodynamics, or controlling other aeroelastic responses leading to limit cycle oscillations (LCO), which are induced by aerodynamic non-linearities [3]. Optimization, flow control and aeroelasticity are challenging topics that the current aerospace engineering industry need to enface during the next upcoming years to be able to develop more efficient, less pollutant and more silent aircraft designs. CFD numerical simulations and wind tunnel experiments are generally carried out to extract information connected to the aircraft performance, with the aim at using such knowledge to improve aerodynamic design. Aerodynamic data gathered from numerical and experimental sources can be divided in two groups: (i) distributed data, which includes skin friction and surface pressure distribution, and (ii) integral quantities, which include drag, lift and moment coefficients [180]. However, the high cost associated to the performance of numerical simulations (large computational times and memory resources) and experiments (elevated monetary cost), encourage researchers to use aerodynamic models to solve design and multiphysics problems. The lack of accuracy in classic aerodynamic models, which rely on empirical laws and classical theory (potential flow), combined with the rapid advance in data science and the current possibilities to generate and store large amount of high-fidelity numerical and experimental databases, motivate researchers to develop novel strategies to develop data-driven aerodynamic models. These models are more reliable and accurate than theoretical models and capable to provide relevant information connected to the physical insight of the problem under study, which opens new possibilities for applying the data-driven models to new flow control, optimization, aeroelasticity, flight dynamics and flow reconstruction applications [163]. Aerodynamic data-driven models can be divided into three groups: (i) semi-empirical models, which obtain the drag, lift and moment coefficients of airfoils as the result of linear or nonlinear differential equations in charge of describing the flow dynamics, (ii) surrogate models, which are generally used in steady applications to find unknown variables using as general input geometrical parameters or flow conditions (i.e., Mach number, Reynolds number, . . . ), and (iii) ROMs, which include models for feature extraction (i.e., POD [292], DMD [281]) that are generally used in large dimensionality flow datasets, and models for system identification based on input-output aerodynamic data (i.e., neural networks, Kriging [69,117], Volterra series [211,264], other linear and non-linear models such as Eigensystem Realization Algorithm -ERA , Auto-Regressive with Exogenous input -ARX, Non-linear ARX, Auto Regressive Moving Average -ARMA, Non-linear ARMA with exogeneous input -NARMAX, et cetera). During the last years, AI, ML and more specifically, NNs, have emerged as highly potential tools to improve and transform engineering design and analysis, offering flexible, robust and versatile tools to develop ROMs and surrogate models, both suitable for finding optimal solutions in aerodynamic optimization [178,297]. ML applications to improve aerodynamic design solving optimization problems are multiple. For instance, NNs have been used to develop models (i) capable to accelerate CFD simulations, which increases the optimization efficiency since it is possible to calculate the integral quantities and distributed data of several design geometries in a short time simplifying the possibilities of finding the optimal designs [142,316], (ii) also these models can be used to reconstruct high-fidelity data from sensor measurement [153] or (iii) to provide a general idea about the proper combination of aircraft weight, wing area and engine thrust helping in the early stages of aircraft design to take decisions [238]. ROMs can be combined with other analytical tools, can predict the aerodynamic loads and can represent aerodynamic systems, allowing several types of analysis at a reduced computational cost [160]. Even though ML is an emerging science in most of the subject connected to aerospace engineering, NNs have been used since the 90s to model aerodynamic loads for flight dynamics [189]. In the recent extensive and detailed review of Kou & Zhang [163], the authors present several data-driven modelling methods applied to improve aerodynamic performance, including several references related to NNs applications since the 90s. The authors conclude as a general observation for the field of data-driven aerodynamic modelling that the future trends in this field should focus on developing novel data fusion techniques, allowing to combine knowledge from numerical simulations with different fidelities and experiments, maximizing in this way the capabilities of the models developed. The authors also reveal the need of developing new models less dependent on data and adding information about the flow physics, which includes interpretability on the results. The need of solving more complex applications, such as turbulent flows, complex geometries, industrial problems, three-dimensional and multi-scale flows (i.e., massive flow separation), etc., encourage researchers to find novel alternatives to develop non-linear accurate models to improve aircraft designs. Evaluating the real applicability of these models by means of using uncertainty quantification tools, is critical for the industry of the future. These conclusions, generalized for data-driven modelling, can also be particularized to extend, and improve ML and NNs applications to solve aerodynamic problems. Aerodynamic coefficients estimation The performance of the ML model depends on the type of identification method, the model structure and the type of data analysed. Flight testing, experiments and numerical simulations are different data sources to obtain unsteady aerodynamic forces. Using ML tools and considering data fusion techniques, which allows to combine data from different fidelities and types (experimental and numerical), would contribute to maintain the proper balance between cost of data generation and model accuracy [163]. However, most of the models, only consider data coming from a single source. In this line, it is possible to mention the representative work of the following authors, ordered in chronological order since the 90s, who applied NNs for the identification of aerodynamic forces (lift and drag). More specifically, Linse & Stengel [189] used NNs to estimate the aerodynamic coefficients, presenting a novel methodology that was based on extracting additional training information about the first partial derivatives of the aerodynamic coefficients. The method was successfully tested in flight test data generated numerically, solving a non-linear problem of twin-jet transport, and used for non-linear control of aircraft. Suresh et al. [299] used RNNs to identify the dynamic stall effect of rotor blade on an OA212 airfoil. The authors predicted the lift coefficient in a case at high angle of attack using experimental databases and intended to show the generalization capabilities of the method presenting a way to generalize the error to model training. More recently, Secco & Mattos [286] developed a surrogate model using NNs to predict aerodynamic coefficients of transport airplanes with high accuracy. The NN-based model used data from about 100.000 cases calculated with CFD numerical simulations and the flight condition, airfoil geometry and wing planform were used as an input. The computational time to estimate the aerodynamic coefficients was ∼ 4000 times smaller than the time required by the CFD solver. This model was also successfully used for optimization design. Recently, Hou et al. [135] used NNs to study the effect of aerodynamic disturbances (due to manoeuvres or gusts) on the wing surface using pressure measurements. The database was generated by an inviscid vortex method in a two-dimensional flat plate. NNs are used to estimate the angle of attack and the leading-edge suction parameter. The authors first combine CNNs and RNNs providing an accurate estimation these variables. In a second approach, a novel and more accurate approach is presented, where ML is integrated in a dynamical system framework to further correct the prediction using a smaller number of parameters. The authors also show that adding random noise to the database it is possible to prevent overfitting. In the same year, to overcome the limitations in adapting input variables with different orders of ROMs, Wang et al. [329] proposed using a fluzzy scalar radial basis function NN, which combines fuzzy rules and radial basis function neural networks (RBFNNs), enhancing in this way, the generalization capabilities of the model to complex dynamical systems. This model was successfully tested to estimate the unsteady aerodynamics in transonic flow airfoils. RBFNNs were previously used by Ghoteyshi et al. [111] to compute the unsteady aerodynamic loads in an airfoil with pitching and plunging motion. Aeroelasticity Studying in detail aerodynamic linear and non-linear responses, for instance producing flutter (linear) or leading to LCO (non-linear), is extremely important to reduce critical loads in the aircraft. Non-linear dynamical modelling is a challenging task, since instabilities can be present in both training and simulation, additionally, purely non-linear models sometimes are unable to identify the main dynamics in systems with dominant linear behaviour. The possibilities of using NNs to combine both, linear and non-linear effects to develop models, have motivated several authors to focus their work on developing ML-based models to estimate aeroelastic responses in the aircraft. To name a few, Marques & Anderson [209] used a finite memory NN to model the nonlinear transonic aerodynamic aircraft response, motivated by the modelling difficulties generally found to control aeroelastic responses. The NN model used a supervised training process with multiple input-output sets from numerical data solving Euler equations. The finite memory NN architecture allowed to identify the memory in the hidden layer of the finite impulse response. Zhang et al. [345] developed a non-linear unsteady aerodynamic model using RBFNNs employing the aeroelastic system self-excited vibration signal as input. The non-linear aerodynamic fluid-structure interaction (FSI) model was coupled with the structural equations of motion, accurately estimating the LCO change with velocity in a transonic flow with large shock motions. Mazhar et al. [212] used NNs to successfully interpolate the pressure loads on unmanned aerial vehicles (UAVs). The aim was to present an accurate model suitable for UAV structural design, which was based on one-way FSI. The performance of NNs was superior compared to standard interpolation techniques using high order polynomials. Mannarino & Mantegazza [205] used recursive NNs to develop a model using high-fidelity numerical data solving a non-linear aerodynamic model. The NN model was successfully used to determine aeroelastic limit cycles using two different approaches: (i) through time marching simulations and (ii) through direct time collocation. The model was successfully tested in the Benchmark Active Control Technology (BACT) wing [269], which consist o a rectangular wing pitching around its mid chord axis and with a NACA0012 airfoil section. To model non-linear unsteady aerodynamics, both at constant and varying flow conditions, Kou & Zhang [161] successfully tested the performance of multi-kernel NNs, which are able to improve the generalization capabilities of the model. They also succeeded in using recursive NNs to predict non-linear aeroelastic behaviours and aerodynamic forces in a NACA 64A010 airfoil for multiple flow conditions in transonic flow. More recently, Li et al. [179] used NNs, more specifically, LSTM networks to develop a model capable to predict non-linear unsteady aerodynamics, which was successfully tested for various Mach numbers on a NACA 64A010 airfoil that was pitching and plunging in a transonic flow. The model accurately captured the aerodynamic and aeroelastic capabilities of the dynamical system, showing the generalization capabilities of this NN architecture. LSTM architectures are perfect to capture the time-delayed effects of unsteady aerodynamics without the need of including any additional information about the orders selection (in unsteady aerodynamics it is necessary to consider the time-delayed effects from the input-output dynamical system). Design optimization Design optimization is also a highly important discipline in aerospace engineering, where machine learning has recently gain strong relevance in the field. Aerodynamic design is related to airplane wing, airfoil geometry, UAVs, turbine engine blades, flight conditions, etc.. Optimization should be smart, efficient, and capable to easily balance design space (i.e., shape, morphed geometry, etc.) and modelling efforts. In the recent review article by Sun & Wang [297] the authors summarize the main contribution of the research community in the use of NNs for surrogate modelling applied to design optimization. The article discusses the effect in optimization of the type of NN selected, the type of data used for the training and if special data treatment is needed, and it reveals the potential of using NNs for optimization purposes. For instance, the authors show that multi-level surrogate modelling, where various levels of information are used, is highly effective to obtain the optimal design area with a small computational cost. NNs can be combined with several optimization algorithms and can also make use of large databases, extracting relevant information suitable for optimization purposes. In this review paper, the authors reveal the relevance of deep learning in aerodynamic design, which reduces the manual interaction in the optimization process. In the line of using NNs for design optimization, it is worth to mention the work carried out by the following authors during the last 10 years. Chen et al. [51] used NNs, used a supervised algorithm combining Self-Organizing Map networks, for selecting referenced airfoils, with Back-Propagaion, to learn the relationship between the aerodynamic performance and airfoil geometry, to estimate airfoil aerodynamic characteristics for aerodynamic design. Sun et al [298] introduced an inverse design method using NNs in trained in wing and airfoil databases. Koziel & Leifsson [166] developed a methodology for solving transonic airfoil aerodynamic shape optimization problems. The authors combined a low-fidelity CFD model (coarse mesh and relaxed convergence criteria) with NNs, in charge of correcting the low-fidelity simulations with space mapping. The method was able to maximize lift and minimize drag in a two-dimensional transonic flow. Oktay et al. [237] combined NNs with the algorithm of simultaneous perturbation stochastic approximation (SPSA) to estimate the drag coefficient optimum value of a fuselage. By this combination, the authors used SPSA without any objective function, supported by the NNs capabilities, which were trained using wind tunnel data. The NN was generalized to estimate the drag coefficient of different geometry fuselages (different fuselage shape parameters). More recently, Xu et al. [336] used NNs to develop surrogate models, later employed in a genetic algorithm to select optimized airfoil designs for transonic supercritical conditions, with the idea of minimizing time-averaged drag and reducing buffet effect (to prevent structural damages). The identification of aerodynamic forces, the estimation of linear and non-linear responses in the aircraft (aeroelasticity and FSI) and design optimization, are three main topics where researchers have put effort since the past to advance in the field of aerospace engineering and aircraft optimal design and manufacturing. Using ML in the field, allows to develop accurate ROMs and surrogate models, suitable for a wide range of problems, from incompressible to transonic compressible flows, at various flight conditions. Nevertheless, the lack of generalization in the methodologies proposed, as well as the possibility of merging different types of data (numerical and experimental), with different types of fidelities, encourage researchers to continue advancing in the field, combining ML with data fusion techniques that will also consider the physical insight of the problem under study. Developing highly accurate predictive model based on machine learning, which will be applicable to solve different problems and requiring small manual interaction, will definitely advance the field of aerospace engineering and the achievement of sustainable aviation. Aeroacoustics The recent review by Bianco et al. [26] summarises the advantages of using ML for modelling pure acoustic phenomena. In the review, the authors cover a variety of topics including source localization in speech processing, source localization in ocean acoustics, bioacoustics and environmental sounds. The main ideas in this review include localising noise sources and identifying acoustic features or patterns using ML. The references included in Ref. [26] show that MLbased methods can provide enhanced performance when compared with conventional signalprocessing methods. However, ML-based techniques are limited by the existing data, since large amounts of data are typically necessary for training. More relevant to aeronautical applications is the field of aeroacoustics and computational aero-acoustics (CAA). Aeroacoustics problems are generally more complex than pure acoustics once since the coupling between the turbulent flow interacting with the geometries, the associated acoustic generation and subsequent propagation need to be considered. The most important sources of noise in aeronautical applications include aerodynamic noise, jet noise and rotating machinery noise, see Moreau [226] for the more recent advances on these fields. To the authors' knowledge there is no review covering the use of ML to model aeroacoustics or to accelerate CAA simulations. Aeroacoustics and CAA exhibit two distinct problems: noise generation (near field) and noise propagation (far field). Although direct simulation of acoustics (near and far fields are solved together) is possible, and only for small cases to date, the hybrid approach is generally preferred to lower the cost in large complex simulations. Acoustic analogies revolutionised the field of aeroacoustics in the 1960s (see Lighthill [186,187]), and have enabled a certain degree of decoupling between the generation and propagation of noise. Following these analogies, equivalent acoustic sources can be extracted from the well-resolved near-field region. These can be subsequently modelled and inputted into a new simulation to predict their propagation. Simulations of aero-acoustics can use this decoupling to enable faster simulations, since the near field that requires highly-accurate and costly flow computations can be performed separately from the acoustic propagation (wider region but coarser mesh). The field has evolved considerably due to the development of faster computers that enable larger computations, see Lele & Nichols [176] or Moreau [226] . Following the acoustic analogies, models or correlations can be applied to either the generation or the propagation regions, or even to both regions at the same time. The model (or correlation, regression) can use NNs. When considering only propagation within the aeroacoustic context, we find only a few works with interesting ideas. For example, Alguacil et al. [7] predicted the propagation of acoustic waves using deep CNNs and developed deep-learning surrogates to characterises the transfer function for the propagation and scattering of acoustic waves in quiescent flows [6]. Tang et al. [304] studied dynamic interactive sound propagation by means of neural-network-based-learned scattered fields coupled with ray tracing to generate specular, diffuse, diffraction, and occlusion effects. Kužnar et al. [168] proposed to use linearregression models to improve vehicle acoustics. When considering generation and propagation of aeroacoustics, artificial neural networks are being deployed to model/correlate a variety of input parameters to sound pressure levels. The use of ML in acoustics is scarce and has been restricted to the automotive industry or simple flows. Beigmoradi et al. [21] used Taguchi methods and NNs to reduce the computational complexity for the aeroacoustics of a rear end simplified car model. The model was used for optimisation using genetic algorithms. Rüttgers et al. [276] trained a deep NN using a Lattice-Botlzman method to predict the aerocoustic far field of a 2D square domain that include randomly distributed rectangular and circular objects, and monopole sources. Moeen Uddin et al. [312] used acoustic analogies based on RANS and Ffowcs Williams and Hawkings, to predict the aeroacoustic tire noise at near-field and far-field receivers around the tire. Artificial neural networksbased regression were used to study the highly non-linear relationships in the system: between A-weighted sound pressure level and tire parameters (Groove depth, Groove width, Temperature and velocity). Abreu et al. [4] have used spectral POD to characterize the wave packets originated at the trailing edge of turbulent airfoils. Note that these structures have important implications in aircraft noise generation, and ROMs aimed at predicting their behavior can help to design more-silent aerodynamic solutions. Finally, Kou et al. [164] used autoencoders to improve the the optimization of airfoils where a multidisciplinary objective function combined aerodynamic and aero-acoustic targets. The main conclusions of this section is that there is huge potential in using ML methods for aeroacoustics. This field is at its infancy and has not yet been deployed to solve aeronautical applications. The main difficulty is the that the aeroacoustics transfer functions (linking flow/geometrical parameters and far field acoustics) are not always smooth and care is needed when performing tasks involving derivatives or such functions (e.g., gradient or adjoint based optimisation). NNs do not always perform well when modelling non-smooth functions [140], and therefore their use in this field needs caution. A possible way to alleviate the lack of smoothness is to decouple noise generation and propagation and use NNs separately for each part. Alternatively one may regularise the transfer function using for example smooth flow models (e.g. based on RANS equations). Combustion The ultimate goal of combustion research is to develop accurate, generalizable and predictive models to describe the phenomena occurring in combustion devices, which primarily involve turbulent flows. The recent years have witnessed significant advances in the fields of DNS and LES, as well as in the development and reduction of chemical kinetics mechanisms for hydrocarbon fuels. The currently available computing power allows to perform parametric DNS studies closer to the conditions of practical interest. LES is becoming more and more present in industry, although RANS remains the preferred option for its affordable computation cost. Despite the significant progress, existing modelling approaches generally lack generality and predictivity [42]. The main challenges are associated with the number of species involved in combustion processes, the small scales and the non-linear turbulence-chemistry interactions [29,116,254] ML methods can contribute to combustion science and treat some of the previously unmet challenges, providing interpretable feature extraction techniques, delivering generally applicable approaches to locally adapt comprehensive chemical mechanisms and sub-grid models, designing new closures to parametrise the unresolved fluctuations, and developing ROMs for fast and yet-accurate system evaluations. In the following, the current state and perspective associated to the use of ML in combustion physics are reviewed and discussed. Data analysis and feature extraction Combustion science has traditionally dealt with massive amounts of data from experiments and large-scale numerical simulations. As a matter of fact, big data has been a reality in combustion for almost three decades [52] thanks to the advent of petascale computing and, hence, the possibility of performing fine-grained simulations of canonical and laboratory-scale turbulent flames with detailed chemistry. While exascale computing will allow to investigate closer-to-reality conditions, DNS of turbulent combustion already represents a key research area for model development and validation [251,309,331]. Over the past 50 years, many techniques have been developed to handle and process combustion data from experiments and simulations. While these analyses have strongly relied on domain expertise and heuristic algorithms, they can be certainly regarded as early applications of ML. Several early studies focused on the use of dimensionality reduction techniques such as PCA [149], with the objective of identifying empirical low-dimensional manifolds in combustion systems [27, 75, 102, 198, 220, 243-245, 300, 337]. The analysis focused both on experimental [17-19, 74, 214, 283] and numerical simulation data [16,63,64,82,127,128,256,340]. The increasing availability of DNS data of complex configurations is shifting the interest towards computational data, for the possibility of mining large sets and accessing variables not available from experiments (e.g., source terms and turbulencerelated variables). The dimensionality reduction offered by PCA can be limited for combustion data. Indeed, PCA is a multi-linear technique and typical reacting flow data exhibit strong nonlinearities. This implies that additional components are identified by PCA to make up for the data non-linearity, leading to an overestimation of the true problem dimensionality. To overcome the limitations of classic PCA with non-linear systems, the application of non-linear methods has been proposed in the community, including non-linear and KPCA [63,220,221], isometric mapping (IsoMap) [108], T-distributed Stochastic Neighbor Embedding (T-SNE) [96,97] and AEs [109,344]. Recently, data-driven approaches based on CNNs were proposed for regime identification [327,347]. Moreover, non-linear regression based on Neural Networks [147] was em-ployed to identify the main features of Moderate or Intense Low-oxygen combustion (MILD) 1 and predict the co-existence of different combustion modes, paving the way to the adaptive selection of closures during numerical simulations. Non-linear methods greatly improve the dimensionality reduction potential. However, they become computationally intractable for large data sets (KPCA and IsoMap) or involve many hyper-parameters to optimise and cross validate (t-SNE, NNs and CNNs). Finally, non-linear algorithms extract features that can be more challenging to interpret and that cannot be easily converted into predictive modelling approaches. An alternative to non-linear methods is to identify local principal curves [83] and locally-linear embeddings (LLE) [273], as well as to perform local modal decomposition, i.e., applying PCA locally (LPCA) [152]. Algorithms based on the definition of an a priori prescribed (supervised) conditioning variable (such as mixture fraction), and on the iterative (unsupervised) minimization of the reconstruction error have been investigated in the context of combustion data analysis [245]. The latter has proven particularly effective and even competitive with more sophisticated approaches such as AEs for dimensionality reduction [344], as well as for clustering tasks [80,81]. Coupling vector quantization with dimensionality reduction using PCA (VQPCA) [152,245] provide robust classification algorithms, less prone to overfitting and applicable to conditions different than those met during training [73]. Indeed, VQPCA requires the determination of only two hyper-parameters, the number of eigenvalues, controlled by the amount of desired variance, and clusters, based on quantitative metrics [44,103,272,305] as well as the statistical representativeness of the clusters [81]. The classification properties of LPCA have been exploited to identify the leading principal variables (PV) and processes in different regions of the state-space [71,72,183,315], and to develop adaptive chemistry models [80,81]. Dimensionality reduction, classification and adaptive chemistry The large dimensionality of combustion systems has driven the development of dimensionality reduction approaches. Considering the impact of large kinetic mechanisms on the computational time of detailed numerical simulations [254], many different strategies have been developed in the last two decades for the systematic generation of reduced mechanisms, including Sensitivity Analysis [311], PCA [120,311], Path Flux analysis [102], graph-search methods such as Directed Relation Graph (DRG) [195], DRG-aided sensitivity analysis [279], DRG with error propagation (DRGEP) [247], lumping [136], quasi-steady state approximation (QSS) using Computational Singular Perturbation (CSP) [314], Tangential Stretch Rate [204], Level of Importance (LOI) [1], via production/consumption analysis [343], and using error estimation [308]. An effective strategy to reduce the burden of detailed chemistry is to use adaptive chemistry approaches, which adjust the mechanisms to the local flow conditions [61,184,267,268,291,346], performing the reduction on the fly. Approaches relying on pre-tabulated libraries of reduced mechanisms [73,80,81,185,230,231] have been recently proposed, thanks to the development of efficient classification algorithms. This opens the way to the use of reduction methods characterised by higher overhead and unsuited for use at runtime, i.e., error-controlled approaches [232,247,296] and methods relying on the dynamical description of the system [204,314]. In order to cope with the large number of uncertain parameters in comprehensive chemical mechanisms, uncertainty quantification and optimization have been adopted in the process of chemical mechanism development [100,218,307,328]. In particular, several methodologies have been designed to solve the so called "inverse problem" [101,290], and improve predictions [317] based on increasingly available experimental data. Notably, the use of genetic algorithms [24,89] has gained momentum to deal with the highly structured objective functions, typical of mechanism optimization, often characterised by the presence of multiple local minima/maxima [101]. Considering the increasing amount of kinetic parameters from theoretical ab initio calculations [68,218], one challenge in optimization approaches is the estimation and/or determination of plausible uncertainty ranges based on the level of theory adopted throughout the calculation protocols for electronic structures, potential energy surfaces and phenomenological reaction rate constants [24,158,289]. The availability of highly-efficient optimization algorithms and tools [107] is pushing research in the development of highly reduced chemical mechanisms for complex fuels, to allow high-fidelity simulations with realistic chemistry rather than global mechanisms [143,144]. In this line of research, new frameworks combining dimensionality reduction (e.g., with PCA) and non-linear regression (e.g., using b-spline interpolants and NNs) have been proposed to replace the tabulation of reaction rates in flamelet/progress variable [31,32,56,139] as well as in finite-rate chemistry approaches [50,99], and to simplify the chemistry of complex hydrocarbons combining data-based models for representative pyrolysis species and foundational chemical mechanisms for the remaining ones [8]. Along the same line, novel hybrid frameworks relying on CNNs [288] have been proposed to model the dynamics of particles and their size distributions, with application to soot predictions. Finally, virtual chemistry approaches consisting of virtual species and reactions, and employing optimization algorithms have been recently proposed [43] to predict quantities of interest (e.g., pollutant emissions) in large-scale simulations at a reduced computational cost. Besides rate-based methods that identify and eliminate redundant species and reactions, approaches based on the re-parameterization of the chemical state-space [95,114,236,248,249] have been further developed and combined to data-driven approaches. In particular, PCA was used to develop reduced-order models for combustion simulations based on the resolution of transport equations for the principal components [300]. To reduce the number of transported components and improve the reconstruction of the state-space as well as the associated chemical source terms, PCA was coupled to non-linear regression techniques, replacing the (multi-linear) PCA mapping between the reduced PC space and the original one. Different regressions have been used to this purpose including Multi Adaptive Regression Splines (MARS) [27,337], NNs [70,220] and Gaussian Process Regression (GPR) [141]. This modelling approach has been demonstrated for simple reactors [141,201,220] One Dimensional Turbulence (ODT) simulations [28,220], premixed and non-premixed DNS simulations [63][64][65]240], as well as in the context of RANS simulations using kernel density estimation (KDE) [261,262]. Recently, the method combining PCA and GPR (PC-GPR) was applied, for the first time, to the three-dimensional LES simulation of flames D-F [202] and of the Cabra flame [203], showing excellent predictive capabilities. The ability of the method to generalise the formulation of flamelet/progress variables approaches and facilitate the formulation of a sub-grid closures (being the selected scalars uncorrelated) appears particularly promising. Combustion closures In turbulent flows, temperature fluctuations can be as high as several hundred Kelvins. Considering that Arrhenius reaction rates are highly non-linear functions of temperature, accurate statistical closures for filtered approaches cannot be based on an expansion about mean properties [252,254]. The task of a combustion model is to provide a description of the unresolved scales based on the information available during a simulation. Combustion models have been often classified into two categories, the flamelet-like 2 [95,114,236,248,249] and PDF-like approaches 3 [98,129,253]. Besides them, we can also mention reactor-based 4 [55,94,182,199] and conditional-moment approaches (CMC) 5 [157]. Data-driven methods are significantly impacting the design of improved sub-grid models for LES and RANS simulations, beside turbulence closures [33,190]. Recently, Deep CNNs have been used by different authors, to improve the prediction of the unresolved flame surface wrinkling [170] in the context of the Thickened Flame model [59], as well as to predict the filtered progress variable source term and the unresolved scalar transport terms in LES [233,234,287]. NNs and Deep Learning (DL) were employed to develop presumed probability-density-function (PDF) models for conventional [302] and MILD combustion [53]. Data-driven approaches were recently used to assess the validity of existing reactor-based closures in MILD combustion [138,257], to obtain conditional statistics [338,339] in the context of CMC, and for the dynamic assignment of combustion closures [57]. Reduced-order models for realistic combustion systems In some applications, a very fast evaluation of a system response is required (for control, optimization, . . . ), thus limiting the use of time-consuming simulations. In this context, ROMs are used to approximate the underlying hidden relationship between inputs and outputs, using available observations to estimate the system response for unexplored conditions. Black-box approaches have been widely used in combustion to create static input-output maps [241] and for system identification [151], to predict macroscopic quantities such as exhaust gas emissions and temperature and to detect oscillatory patterns such as thermoacoustic instabilities, respectively. In combustion, NNs have been used to evaluate and tabulate reaction rates as discussed above [31,32,56,139], as well as to estimate modelling errors in LES [22,309]. Black-box approaches can be very powerful although not equipped with the guarantees of physics-based approaches. To cope with that, nonlinear system identification techniques were proposed to enlighten black box approaches by discovering the underlying physics [41,49]. An interesting approach to generate combustion ROMs is the use of reactor network models [91,197,301], which fall in the category of grey-box models, as they combine a theoretical structure, i.e., the canonical reactors, with data to generate the network. The approach can be also regarded as an example of multifidelity methods, relying on high-fidelity tools (i.e., CFD) to construct the reactor network and the lower-fidelity network model to evaluate pollutant emissions and other quantities, using detailed chemical mechanisms. These techniques are particularly useful for large and complex systems, such as furnaces or gas turbines. However, their current overall fidelity and generalizability are limited by the high-fidelity simulations required to generate the network structure, indicating the interest for data-driven approaches in this area of research. ROMs based on projection methods (such as PCA and other such techniques) have been employed for experimental and numerical combustion data [27, 75, 102, 198, 220, 243-245, 300, 337]. The direct evolution 2 Flamelet-like models are based on the re-parametrisation of the thermo-chemical state using a reduced number of variables identifying a low-dimensional manifold in the composition space on which the evolution of the system is constrained 3 PDF-like approaches do not make the assumption of a low-dimensional manifolds and treat the mean/filtered reaction rates exactly, solving the, one-point one-time, joint PDF of fluid composition. Mixing at the molecular level requires a closure in PDF approaches. 4 Reactor-based models assumes that a fraction or the entire computational cell behaves like an idealised reactor (perfectly-stirred reactor or plug-flow reactor), allowing an affordable treatment of finite-rate chemistry. 5 CMC relies on the concept of conditional manifolds and equations are derived and modeled for the conditional moments of the reactive scalars. of the modal coefficients in CFD codes (i.e., intrusive ROMs) has been quite limited for nonlinear problems like combustion, due to the high cost of the associated Galerkin projection as well as the complexity of coupling them to legacy and commercial CFD codes [213]. The application of non-intrusive projection-based ROMs has been, on the other hand, quite successful, in combination with non-linear regression approaches such as GPR [11], NNs and Polynomial Chaos Expansion (PCE) [12]. Indeed, when the number of variables of interest is high, many ROMs need to be trained. Besides, any correlation between these variables is lost in the process of training individual ROMs. Introducing PCA and similar techniques, the number of ROMs can be reduced as the original variables are encoded into a set of fewer uncorrelated scalars explaining most of the system variance. In the combustion community, the combination of dimensionality reduction (using PCA) and non-linear regression (using GPR) was used to develop ROMs for uncertainty quantification [10] as well as to design simulation-based digital twins [13] of industrial systems. Indeed, combining CFD simulations with real-time data coming from sensors is possible only if the prediction of a new system state, based on continuous incoming data, becomes instantaneous [13,167]. Structural assessment Aerospace structures are subjected to a wide spectrum of loads, operating in harsh environments [271] and phenomena such as high-velocity debris have the ability to severely impact the integrity of components [270]. Furthermore, while developments in composite materials present large advantages in terms of strength and weight, failure mechanisms are more difficult to predict as a result of an increased number of failure modes compared to metals [270]. Structural health monitoring (SHM) is the implementation of a damage-detection strategy for engineering structures [93]. Applied across many industries, the integration of SHM into the aerospace sector is particularly beneficial as a result of the difficulty of regular inspection and the very high cost of failure. Integrating SHM into aerospace structures has the capacity to reduce the downtime, operating and maintenance costs of aircraft, while also improving the safety and reliability simultaneously [342]. While vibration-based techniques are the most commonly employed [93], acoustic emission and electro-mechanical impedance (EMI) are examples of sensing techniques that are often discussed in the context of damage detection within SHM for aerospace structures. The former uses sensors to detect high-frequency stress waves that are generated during crack propagation and, while aerospace applications have been considered since the 1970s [126], significantly enhanced accuracy of crack-location prediction has been achieved for aerospace applications using ML methods such as Gaussian process regression [131] and NNs [25] in recent years. Further examples of the use of ML methods being used for damage detection classification within the aerospace industry include probabilistic neural networks [115], support-vector machines [194] and logistic regression [146]. One problem facing damage detection using SHM methods is the requirement of complex and heavy wiring requirements of the sensor networks which can be prohibitive for aerospace structures [112]. As a result, low power [112] or no power equipment [274] have been developed to enable wireless sensor networks. Such approaches require efficient methods of communicating data and, thus, new methods of interpreting the data have been developed. In Ref. [278], lowrank-approximation is used to estimate sparse data from a wireless sensor network and a Knearest-neighbour algorithm is employed to classify this imputed data. Furthermore, sensor and wiring reliability can be considered to be one of the 'weak-links' of an effective SHM system, particularly in the harsh environments aerospace structures are subjected to, and classification techniques like KNN can be used to detect sensors faults [215]. Digital twins are widely discussed across a broad range of engineering domains, where in SHM applications, they are developed with an aim of providing an accessible insight into the health of a structure or inform predictive maintenance. This is achieved by having a digital model of the structure in which measured detail can be incorporated [310]. A common example of this within the aerospace sector is performing updated finite-element modelling; however, this can be computationally costly. ML can, therefore, be used to perform surrogate modelling using methods such as Gaussian processes [48,177] to significantly reduce the computational cost. Many aerospace components such as landing gear operate a safe-life methodology, in which there is zero-crack tolerance [133]. Therefore, damage-detection methods, based e.g. on Acoustic emission, are not always relevant (although their inclusion is advocated, nonetheless). A sub-category of SHM is structural loads monitoring or virtual sensing. This involves predicting the loads, as well as the subsequent stress and strain, that a structure is subjected to using incomplete available measurements. This enables damage prognosis, which can be considered the natural next step of structural health monitoring with applications in all forms of defence and commercial aviation [92]. Fatigue analysis is of particular interest here, as more than 80% of structures fail due to fatigue [332], which occurs after the structure is subjected to a large number of stress cycles. Within the helicopter health and usage monitoring systems (HUMS) domain, there are numerous ML-based approaches to loads monitoring and fatigue analysis. In Ref. [208], the mechanical loads are predicted using a multilayer perceptron (MLP) NN. In Ref. [313], an extreme learning machine (a subclass of feed-forward NN) is used for load prediction and the results are fed through the stress-life fatigue analysis procedure to predict damage accumulation. Grey-box modelling is the integration of our knowledge of the physics of the structure into the model. The most common way of doing this, and something the authors advocate for whenever possible, is via semi-physical modelling or input augmentation, involving manipulating inputs to the model to indicate physical processes [14,104,265]. An example of this is squaring the measured airspeed before using it as an input to the model (in this case a NN) to give an indication of dynamic pressure [265,266]. Similarly, in Ref. [14], a mathematical network is used for fatigue monitoring, in which physics-inspired merging functions are used as the first layer to the network. An alternative way of integrating physics into models is via residual modelling in which ML methods such as Gaussian process regression [250,326] is used to predict the discrepancy between a physics-based or empirical prediction and the measured data. An introduction to grey-box modelling for SHM applications can be found in Ref. [67]. A data-driven model is fundamentally an estimate of the similarity of the test case compared to the training data that the model has seen. Practically, it is difficult to include all possible loading phenomena of aerospace structures in the training dataset due to magnitude of the loading spectrum and computational limits. Therefore, probabilistic methods are becoming increasingly popular. Gaussian process regression is one of the most popular ML methods for virtual sensing problems, in part, due to its inherently probabilistic nature and subsequent capacity to indicate uncertainty. In the case of Gaussian process regression, the estimate of the similarity of the training and the test data is based on the kernel, or covariance function. Based on prior knowledge of the data, an appropriate kernel can be chosen and combining kernels using addition and multiplication enables a rich language that the functional form of the model can take [79]. A novel way of inputting physical insight into Gaussian process regression models is to integrate the knowledge into the kernel itself, which has the added benefit of providing a physical interpretation to the samples drawn from the model posterior [67]. One example of Gaussian process regression used for virtual loads monitoring can be found in Ref. [132], where the loads on the landing gear of an aircraft are predicted with a high degree of accuracy. In Ref. [104], the loads on an aircraft wing are predicted and fatigue analysis is carried out using the predicted strain values. However, the probabilistic aspect of the model is often not fully utilised. In Ref. [113], Gaussian process regression is again used to predict the stress on the wing of an aircraft. By taking a large number of draws from the posterior covariance of the model, the model uncertainty can be propagated through the stress-life fatigue analysis procedure and, thereby, a probabilistic estimate for fatigue damage accumulation is developed. Such approaches can fit into probabilistic risk-based decision frameworks for structural health monitoring [137], which can be used to make risk-informed decisions for safety-critical components like those found in the aerospace industry. Adopting a probabilistic approach to fatigue in aerospace components has the potential to better control the risk level and plan maintenance more effectively [47,134]. Conclusions The continued growth of machine learning (ML) is leading to a progressively larger impact on a wide range of scientific areas, including aerospace engineering. A number of emerging MLbased technologies are already impacting every aspect of this area, e.g. in terms of simulation capabilities and enhanced physical insight. The main goal pursued by the aerospace industry is to use ML to develop several applications connected to the reduction of aircraft's environmental impact, also considering system management, customer service, data interpretation and even developing new methods based on ML-tools capable to generate new high-fidelity databases at a reduced cost, which will be later used to develop new designs, more efficient, in a fast and efficient way. These applications involve acquiring knowledge and developing new ML tools suitable to solve problems of fundamental fluid dynamics (experimental and numerical), aerodynamics, acoustics, combustion and structural health monitoring. The type of identification method, the type of data analysed, and the model structure, influences the performance of the ML model. To maintain the proper balance between the model accuracy and the cost of data generation, a suitable option would be using ML tools considering data fusion techniques, which combine data from different types (experimental and numerical) and different fidelities. The development of models that are generalizable within the different applications, would suppose and advance in the field, reducing in this way the complex process of calibration behind selecting the proper architecture composing the neural networks composing the model and extracting relevant information from the different data sources available, which is highly dependant on the type of application: aerodynamics, combustion, acoustics, solid mechanics, etc. However, most of the models currently used by the industry, only consider data coming from a single source, leaving new research opportunities to advance in the field. Also, considering models grounded in physics, which combine modern neural network architectures with other dimensionality reduction techniques based on the identification of physical patterns, would suppose and advance in the field. Some researchers are currently developing their work following these ideas, showing the good performance of these models grounded in physics: a few examples are found in the fields of fundamental fluid mechanics, aerodynamics combustion and structural health monitoring, nevertheless there are still many open questions that should be address, and also the lack of robust and generalizable schemes still leaves it as an open topic that should be address by researchers and the aerospace industry in the near future. Finally, when it comes to the application of ML to CFD, there is great potential in several areas related to improving the efficiency of numerical simulations of fluid flows. The future development of ML should not be focused on replacing CFD, but rather improving the efficiency and robustness of subproblems within CFD solvers. For instance, regarding acceleration of CFD (e.g., by accelerating the Poisson problem or improving coarse simulations), enhancing modeling (for LES and RANS) and improving ROM development. In the latter, AEs are progressively becoming a more useful technique since they enable leveraging the non-linearity while, through recent developments [87], retaining the orthogonality of the AE modes, which are ranked by their contribution to the reconstruction as in POD. One point of future improvement for AEs is the possibility to effectively integrate the temporal dynamics into the predictions, a step that currently relies on using the decoder and the latent vectors. When it comes to wind-tunnel experiments, non-intrusive sensing has greatly benefited from recent progress in deep-learning architectures for computer vision. 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[ "ENCODING NONLINEAR AND UNSTEADY AERODYNAMICS OF LIMIT CYCLE OSCILLATIONS USING NONLINEAR SPARSE BAYESIAN LEARNING", "ENCODING NONLINEAR AND UNSTEADY AERODYNAMICS OF LIMIT CYCLE OSCILLATIONS USING NONLINEAR SPARSE BAYESIAN LEARNING" ]
[ "Rimple Sandhu ", "Brandon Robinson ", "Mohammad Khalil ", "Chris L Pettit ", "Dominique Poirel ", "Abhijit Sarkar ", "\nDepartment of Civil and Environmental Engineering\nDepartment of Civil and Environmental Engineering\nCarleton University Ottawa\nONCanada\n", "\nAerospace Engineering Department US\nQuantitative Modeling & Analysis Department Sandia National Laboratories Livermore\nCarleton University Ottawa\nON, CACanada, United States\n", "\nDepartment of Mechanical and Aerospace Engineering Royal Military College of Canada Kingston\nDepartment of Civil and Environmental Engineering\nNaval Academy Annapolis\nMD, ONUnited States, Canada\n", "\nCarleton University Ottawa\nONCanada\n" ]
[ "Department of Civil and Environmental Engineering\nDepartment of Civil and Environmental Engineering\nCarleton University Ottawa\nONCanada", "Aerospace Engineering Department US\nQuantitative Modeling & Analysis Department Sandia National Laboratories Livermore\nCarleton University Ottawa\nON, CACanada, United States", "Department of Mechanical and Aerospace Engineering Royal Military College of Canada Kingston\nDepartment of Civil and Environmental Engineering\nNaval Academy Annapolis\nMD, ONUnited States, Canada", "Carleton University Ottawa\nONCanada" ]
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This paper investigates the applicability of a recently-proposed nonlinear sparse Bayesian learning (NSBL) algorithm to identify and estimate the complex aerodynamics of limit cycle oscillations. NSBL provides a semi-analytical framework for determining the data-optimal sparse model nested within a (potentially) over-parameterized model. This is particularly relevant to nonlinear dynamical systems where modelling approaches involve the use of physics-based and data-driven components. In such cases, the data-driven components, where analytical descriptions of the physical processes are not readily available, are often prone to overfitting, meaning that the empirical aspects of these models will often involve the calibration of an unnecessarily large number of parameters. While it may be possible to fit the data well, this can become an issue when using these models for predictions in regimes that are different from those where the data was recorded. In view of this, it is desirable to not only calibrate the model parameters, but also to identify the optimal compromise between data-fit and model complexity. In this paper, this is achieved for an aeroelastic system where the structural dynamics are well-known and described by a differential equation model, coupled with a semi-empirical aerodynamic model for laminar separation flutter resulting in low-amplitude limit cycle oscillations. For the purpose of illustrating the benefit of the algorithm, in this paper, we use synthetic data to demonstrate the ability of the algorithm to correctly identify the optimal model and model parameters, given a known data-generating model. The synthetic data are generated from a forward simulation of a known differential equation model with parameters selected so as to mimic the dynamics observed in wind-tunnel experiments.
10.1016/j.jsv.2023.117816
[ "https://export.arxiv.org/pdf/2210.11476v1.pdf" ]
253,080,475
2210.11476
1ddbaacfb9562e34226e4e7282806efbd3bac4af
ENCODING NONLINEAR AND UNSTEADY AERODYNAMICS OF LIMIT CYCLE OSCILLATIONS USING NONLINEAR SPARSE BAYESIAN LEARNING 18 Oct 2022 Rimple Sandhu Brandon Robinson Mohammad Khalil Chris L Pettit Dominique Poirel Abhijit Sarkar Department of Civil and Environmental Engineering Department of Civil and Environmental Engineering Carleton University Ottawa ONCanada Aerospace Engineering Department US Quantitative Modeling & Analysis Department Sandia National Laboratories Livermore Carleton University Ottawa ON, CACanada, United States Department of Mechanical and Aerospace Engineering Royal Military College of Canada Kingston Department of Civil and Environmental Engineering Naval Academy Annapolis MD, ONUnited States, Canada Carleton University Ottawa ONCanada ENCODING NONLINEAR AND UNSTEADY AERODYNAMICS OF LIMIT CYCLE OSCILLATIONS USING NONLINEAR SPARSE BAYESIAN LEARNING 18 Oct 2022* Currently at National Renewable Energy Laboratory, USA † Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA-0003525.Aeroelasticityunsteady aerodynamicsinverse problemssparse learningBayesian inferencenonlinear dynamics This paper investigates the applicability of a recently-proposed nonlinear sparse Bayesian learning (NSBL) algorithm to identify and estimate the complex aerodynamics of limit cycle oscillations. NSBL provides a semi-analytical framework for determining the data-optimal sparse model nested within a (potentially) over-parameterized model. This is particularly relevant to nonlinear dynamical systems where modelling approaches involve the use of physics-based and data-driven components. In such cases, the data-driven components, where analytical descriptions of the physical processes are not readily available, are often prone to overfitting, meaning that the empirical aspects of these models will often involve the calibration of an unnecessarily large number of parameters. While it may be possible to fit the data well, this can become an issue when using these models for predictions in regimes that are different from those where the data was recorded. In view of this, it is desirable to not only calibrate the model parameters, but also to identify the optimal compromise between data-fit and model complexity. In this paper, this is achieved for an aeroelastic system where the structural dynamics are well-known and described by a differential equation model, coupled with a semi-empirical aerodynamic model for laminar separation flutter resulting in low-amplitude limit cycle oscillations. For the purpose of illustrating the benefit of the algorithm, in this paper, we use synthetic data to demonstrate the ability of the algorithm to correctly identify the optimal model and model parameters, given a known data-generating model. The synthetic data are generated from a forward simulation of a known differential equation model with parameters selected so as to mimic the dynamics observed in wind-tunnel experiments. Introduction In this paper we demonstrate the applicability of the recently proposed nonlinear sparse Bayesian learning (NSBL) algorithm [1,2] to a single degree of freedom (SDOF) aeroelastic oscillator that is undergoing low amplitude limit cycle oscillations. This experimental setup has been studied extensively through experimentation [3,4], numerical modelling of laminar separation flutter using high-fidelity large eddy simulations (LES) [5] and unsteady Reynolds averaged Navier-Stokes (URANS) model [6]. Furthermore, the wind tunnel experiments have provided a reliable test-bed for developing Bayesian techniques for system identification and model selection for nonlinear dynamical systems [7,8,9,10]. The work in [7,8,9] use standard methods of evidence-based Bayesian model selection, which allows for the systematic comparison of a set of candidate models with varying degrees of complexity. The model evidence as a criterion for model selection ensures a balance of favouring models with superior average data-fit, while penalizing models that are overly complex and thus prone to overfitting [11]. In this context, model complexity is quantified by the KL-divergence of the parameter posterior probability density function (pdf) from the parameter prior pdf. For parameters where there exists little prior knowledge, it is typical to assign non-informative priors, however the width of the distribution used for the non-informative prior will influence the optimal complexity of the model. The issue of sensitivity to prior width is addressed in [10], whereby the problem is reposed as a sparse learning problem. Rather than non-informative priors, parameters with little prior information are assigned Gaussian automatic relevance determination (ARD) priors. The precision (inverse of the variance) of these ARD priors are determined through evidence optimization. In this re-framing of the inference problem, the optimal model is still quantified as such based on the model evidence. In contrast to the previous approach, rather than proposing an entire set of nested candidate models to determine the optimal model complexity, the current paper approaches the problem as an automatic discovery of the optimal sparse model nested within a single (potentially) over-parameterized model. Herein lies an additional benefit of approaching the problem as a sparse learning task; it is only necessary to obtain the parameter posterior for a single model, whereas standard methods require the calibration of each model in the candidate in order to then obtain an estimate of the model evidence. The shortcoming of this approach lies in the fact that the optimization process involves the use of Markov Chain Monte Carlo (MCMC) sampling at each iteration. This is addressed in the current NSBL framework, which removes the use of MCMC from within the optimization loop, resulting in significantly improved computational efficiency. The NSBL framework presented here is an extension of the sparse Bayesian learning (SBL) also known as the relevance vector machine (RVM) algorithm [12,13]. Both methods are motivated by the desire to avoid overfitting during Bayesian inversion. SBL/RVM and the similar Bayesian compressive sensing (BCS) algorithm [14] provide analytical expressions for a sparse parameter posterior distribution owing to the analytical conveniences of the semi-conjugacy that exists between the Gaussian likelihood functions, and Gaussian ARD priors that are conditioned on hyperpriors that are Gamma distributions. The SBL methodology is extended to be applicable to nonlinear-in-parameter models and for non-Gaussian prior distributions, as these both commonly arise in engineering applications. We provide the minimum required mathematical details to understand the objectives of the algorithm and to provide a complete account of all terms shown in the equations used in this paper. For the full detailed derivation and additional details, we refer the reader to [1,2]. Methodology: Nonlinear sparse Bayesian learning The NSBL methodology is applicable to general nonlinear mappings, f : φ → y where the model operator f maps the unknown model parameter vector φ ∈ R N φ to the observable model output y ∈ R Ny . In this specific application, f represents the aeroelastic model, φ are the deterministic system parameters and the stochastic parameters (relating to the model error), and y are the system output. Sensor measurements of the system output y at discrete points in time are denoted as D. The likelihood function p(D|φ) can be computed for any φ, using the observations D, and these observation may be noisy, sparse, and incomplete measurements of the system state. The purpose of the algorithm is to obtain a data-optimal sparse representation of φ using Bayesian inversion, while removing redundant parameters. NSBL operates within the following Bayesian framework; we seek the posterior distribution of the unknown model parameters φ conditioned on the data D and hyperparameters α, p(φ|D, α) = p(D|φ)p(φ|α) p(D|α) = p(D|φ)p(φ|α) p(D|φ)p(φ|α)dφ(1) for given data and hyperparameters, the denominator, which represents the model evidence (or marginal likelihood or type-II likelihood), is just a normalization constant. The parameter prior p(φ|α) is also conditional on the hyperparameter. Though the objective is not to perform full hierarchical Bayesian inference, we nevertheless define a prior on p(α) (which is notably absent in the expression above); this hyperparameter prior (or hyperprior) becomes relevant during the optimization of α. The following sections outline the three principal tasks involved in the NSBL framework, as depicted in Figure 1. Namely: (i) in section 2.1 we discuss the assignment of a hybrid prior, wherein we distinguish between a priori relevant parameters and questionable parameters, assigning known priors and ARD priors, respectively, (ii) in section 2.2, we detail the incorporation of data and the physics-based model through the construction of a Gaussian mixture model (GMM) over samples generated from the product of the likelihood function and the known prior, and (iii) in section 2.3 we discuss the optimization of the hyperparameters. The derivation of various semi-analytical entities that enable the NSBL methodology is outlined in A. Figure 1: Summary of the main steps involved in the NSBL algorithm. (i) (ii) (iii) Hybrid prior pdf The model parameter vector φ is first decomposed as φ = {φ α , φ -α }, distringuishing between the set of parameters that are known to be relevant a priori, denoted as φ α ∈ R Nα , and the parameters that the modeller has deemed to be questionable, denoted φ -α ∈ R N φ −Nα . This classification as questionable encompasses any parameter for which little or no prior knowledge exists, where a non-informative prior with large support would usually be used. The vector of questionable parameters are the set of parameters among which we will induce sparsity, as a subset of these parameters may be redundant. The mechanism for inducing sparsity follows SBL, where φ α is assigned a Gaussian ARD prior p(φ α |α) = N (φ α |0, A −1 ). This prior is a normal distribution, whose mean vector is an N α × 1 zero vector, with a covariance matrix of A −1 , where A is the precision matrix. Following SBL [15], prior independence of the questionable parameters φ α is assumed, hence, the precision matrix is diagonal, A = diag(α). Furthermore, each parameter φ i ∈ φ α has a unique variable precision α i , such that we can write p(φ i |α i ) = N (φ i |0, α −1 i ). The hyperparameter α i dictates the prior precision of parameter φ i ; where low precision (or high variance) reduces to a non-informative prior, and conversely, a high precision (or low variance) results in an informative prior with a mean of zero. In the limit where the precision tends to infinity, the ARD prior becomes a Dirac delta function centered at zero, effectively pruning the parameter. Hence, the motivation behind NSBL is that optimally selecting α, can allow us to discover the model having the optimal complexity given the available data. The optimization criteria and methodology are presented later in section 2.3. The joint prior pdf of φ is summarized as [2] p(φ|α) = p(φ -α )p(φ α |α) = p(φ -α )N (φ α |0, A −1 ).(2) This hybrid prior pdf enables sparse learning of questionable parameters in φ α through the use of an ARD prior p(φ α |α) while incorporating prior knowledge about parameters φ -α through an informative prior p(φ -α ). The ARD prior is a conditional Gaussian distribution, whose precision depends on the hyperparameter α. The marginal hyperprior pdf p(α i ) is chosen to be a Gamma distribution. Given the assumption of prior independence, the joint hyperprior p(α) is written as p(α) = Nα i=1 p(α i ) = Nα i=1 Gamma(α i |r i , s i ) = Nα i=1 s ri i Γ(r i ) α ri−1 i e −siαi ,(3) where Gamma(α i |r i , s i ) denotes a univariate Gamma distribution parameterized by shape parameter r i > 0 and rate parameter s i > 0. The use of a Gamma distribution as the hyperprior allows us to enforce the requirement that the precision parameters α be positive. Furthermore, for specific combinations of shape and rate parameters, r i and s i , the Gamma function can assume many forms of informative and non-informative priors. For instance, using values of s i ≈ 1 and r i ≈ 0 gives a flat prior over α i , or values of s i ≈ 0 and r i ≈ 0 gives Jeffreys prior, which is a flat over log α i . In fact, for reasons discussed in later sections, the NSBL algorithm operates on the natural logarithm of the hyperparameters rather than the hyperparameters directly. For this reason, we choose to use Jeffreys prior for the numerical results section. Using a univariate transformation of random variables [16], the hyperprior in Eq. (3) becomes p(log α) = Nα i=1 p(log α i ) = Nα i=1 p(α i ) d dαi log α i = Nα i=1 s ri i Γ(r i ) α ri i e −siαi .(4) Gaussian mixture-model approximation After defining the joint parameter prior distribution as in Eq. (2), we substitute the resulting expression into the conditional posterior distribution from Eq. (1), yielding [2] p(φ|D, α) = p(D|φ)p(φ -α )N (φ α |0, A −1 ) p(D|α) .(5) Given that the ARD priors are normally distributed, NSBL constructs a GMM approximation of the remaining terms in the numerator, p(D|φ)p(φ -α ), which enables the derivation of semi-analytical expressions for many entities of interest. Obtaining expressions for the model evidence and objective function (A.1), the parameter posterior (A.2), and the gradient and Hessian of the objective function (A.3) makes the optimization of the hyperparameters analytically tractable. Moreover,the use of a GMM enables the preservation of non-Gaussianity in both the likelihood function and the known prior. The construction involves the estimation of kernel parameters a (k) , µ (k) , and Σ (k) , p(D|φ)p(φ -α ) ≈ K k=1 a (k) N (φ|µ (k) , Σ (k) ),(6) where K, a (k) , and N (φ|µ (k) , Σ (k) ) are the total number of kernels, the kernel coefficient (a (k) >0), and a Gaussian pdf with mean vector µ (k) ∈ R N φ and covariance matrix Σ (k) ∈ R N φ ×N φ [2]. For a Gaussian likelihood function and a Gaussian known prior, this reduces to the case of SBL or RVM [15], and a single kernel is sufficient. Otherwise, except in the case of a Laplace approximation [17] of the likelihood function times the known prior, multiple kernels will generally be required. The construction of the GMM typically involves the use of MCMC in order to generate samples from the arbitrary distribution, which requires repeated function evaluations for different samples of the unknown parameter vector. The model itself operates as a black-box, thus, the analytical form of the model does not need to be known; the model is only needed in order to compute the likelihood function. Once samples have been generated from the posterior distribution, the estimation of the kernel parameters in Eq. (6) can be performed numerically using methods such as kernel density estimation (KDE) or expectation maximization (EM) [18]. Since the construction of the GMM involves numerous forward solves of the model, this step is the most computationally demanding component of the algorithm. Notably, the GMM itself is independent of the hyperparameters, thus this process only needs to be performed once at the onset, and does not need to be repeated during the optimization of the hyperparameters. Sparse learning optimization problem The critical step in the NSBL algorithm is the optimization of the hyperparameter, α. Within a hierarchical Bayesian framework, we seek a point estimate for the hyperparameters, rather than obtaining posterior estimates thereof. As in SBL, we perform type-II maximum likelihood, seeking the values α, which maximize the hypeperparameter posterior, p(α|D) = p(D|α)p(α) p(D) ∝ p(D|α)p(α).(7) for a fixed set of data D, the denominator p(D) is a normalization constant that is analytically intractable in general. Thus for optimization, we consider only the numerator which is the product of the model evidence p(D|α) and the hyperprior p(α) . The type-II maximum a posteriori (MAP-II) estimate α map can therefore be posed as [18] α map = arg max α {p(α|D)} = arg max α {p(D|α)p(α)}.(8) Since the natural logarithm is a strictly increasing function, we re-write the optimization problem in terms of log p(α|D), which facilitates the derivation of the semi-analytical expressions of the gradient and Hessian. Furthermore, as noted above, the optimization is performed with respect to log α instead of α. This helps to account for the potentially widespread difference in the scales of the prior precision between relevant parameters (whose α map i are finite) and irrelevant parameters (whose α map i tend to infinity). This has the added benefit of automatically enforcing the positivity constraint of α throughout the optimization procedure. Restating Eq. (8) as the optimization of the log-evidence with respect to the log of the hyperparameters α, the objective function L(log α) becomes log α map = arg max log α {L(log α)} = arg max log α {log p(log α|D)} = arg max log α {logp(D| log α) + log p(log α)}.(9) The objective function is the sum of the estimate of the model evidence and the hyperprior (where constant terms that are independent of α i are discarded). As the model evidence itself will be analytically intractable in general, it is replaced by the estimatep(D| log α) in Eq. (21) that is available in terms of the K kernels of the GMM. Substituting Eq. (4) for the hyperprior gives L(log α) = logp(D| log α) + Nα i=1 (r i log α i − s i α i ) .(10) In this form, it becomes clear that when Jeffreys prior (s i ≈ 0, r i ≈ 0) is used, the objective function reduces to the log-evidence, resulting in the common procedure known as the type-II maximum likelihood estimate or Emperical Bayes method [18], which may lead to a non-convex optimization problem [17]. The numerical examples that follow will illustrate the possibility of non-unique optima in the objective function. The possibility of multiple optima can be addressed using a multi-start optimization algorithm, where the optimization routine is initiated from an array of different coordinates of log α in an effort to discover all local optima in order to determine the global optima. Sequential estimates of {log α j } are obtained starting from the initial iterate {log α 0 }, according to [2] log α j+1 = log α j + β j p j , (11) where β j is the step-length [19]. The optimization itself is expedited by the ability to derive the gradient vector J(log α i ) and Hessian matrix H(log α i ) of the objective function from Eq. (10), as outlined in Eq. (29) and Eq. (31), respectively. This permits the convenient calculation of p j as the solution to H(log α j )p j = −J(log α j ). (12) The specific method for determining p j (e.g., modified Newton method, trust-region Newton method) must consider that the Hessian is not guaranteed to be a positive definite matrix [19]. The optimization of the objective function (the log hyperparameter posterior) with respect to log α, provides the MAP estimate log α map . As discussed previously, a large value of log α map i indicates high prior precision of parameter φ i and implies it is redundant. However, it is difficult to quantify high precision directly as this is highly parameter dependent. To remove the scale-dependence, we leverage another concept from SBL [15], wherein a relevance indicator is defined for each parameter φ i , according to γ i = 1 − α i P ii , where, P ii is the ith diagonal entry of the posterior covariance matrix and α i is the ith diagonal entry in the prior precision matrix, A. In NSBL, it is possible to define a similar metric, however, as with the semi-analytical entities described previously, a relevance indicator will be defined for each of the K Gaussian kernels. This can be re-written as a ratio of prior to posterior precision [2], γ (k) i = 1 − α i (P (k) ii ) −1 ∈ [0, 1],(13) hence, the relevance indicator provides a normalized metric on the scale of 0 to 1, where a value of 0 indicates irrelevance, while a value of 1 implies relevance. The intuition behind this conclusion is that for a parameter where the ratio of prior-to-posterior precision is close to unity, the posterior precision is dictated by the prior, rather than the likelihood, suggesting that the parameter does not learn from the data. The converse holds as well, where parameters that do learn from the data will tend to have higher posterior precision compared to the prior precision, hence the expression for the relevance indicator will approach a value of 1. Since we perform this one a kernel-by-kernel basis, to summarize the relevance indicators for a given parameter across all kernels in the GMM, we propose the use of a root-mean-square value of the relevance indicator [2] γ rms i = 1 K K k=1 (γ (k) i ) 2 1/2 = 1 K K k=1 1 − α i P (k) ii 2 1/2 .(14) 3 Application to aeroelastic oscillator In this section, we demonstrate the performance of NSBL for identifying sparsity in the unknown parameters of nonlinear stochastic differential equations. We consider a single degree-of-freedom pitching airfoil undergoing low-amplitude limit cycle oscillations. The structural model in Eq. (15a) consists of a typical mass-spring-damper system augmented by a cubic stiffness and a model for dry friction. The structural dynamics are coupled with a semi-empirical aerodynamic model in Eq. (15b), which we will refer to as an unsteady generalized Duffing-Van der Pol model [5]. The aerodynamic moment coefficient (C M ) modelled as a first-order ODE, which allows for unsteadiness in the aerodynamics, driven by the parameter B. The model features a polynomial expansion of the pitch angle (θ) and pitch rate (θ) to attempt to capture nonlinearity in the aerodynamic moment coefficient. Finally, it contains a model error term to capture the discrepancy between the modelled physics and the true, but ultimately unknown, aerodynamic loads. The resulting system of coupled ODEs are written in terms of non-dimensional time (τ ) as [8] Iθ + Cθ + Kθ + C nl sign(θ) + K nl θ 3 = 1 2 ρU 2 csC M (15a) C M B + C M = a 0 + a 1 θ + a 2θ + a 3 θ 2 + a 4 θθ + a 5θ 2 + . . . + c 6 Bθ + σξ(τ ) (15b) where the structural parameters I, C, C nl K and K nl , and aerodynamic parameters ρ, U , c, s, and c 6 are known precisely, and coefficients B, σ, and aerodynamic coefficients a i are to be estimated. In the current study, synthetic LCO data is used, in order to validate the use of NSBL in discovering the data-optimal sparse model. In this scenario, the data-generating model and the candidate model have the same underlying analytical form, though the candidate model may contain extra parameters that were not present (i.e. were equal to 0) when generating the data. Hence the algorithm should ideally recover the parameters used in the data-generating model, and identify any parameters that are not used to generate the data as irrelevant. The synthetic LCO from which the measurements of the pitch are generated have similar frequency and amplitude as are observed in the wind-tunnel experimental setup. Synthetic measurements are generated by recording the pitch deflection (representing only a partial measurement of the state). Measurements are recorded at a frequency of 1000Hz (which is consistent with the temporal density of observations available experimentally). These measurements are then corrupted by additive Gaussian noise with a noise strength that again mimics the noise levels in the wind-tunnel experiments. As in the physical experiments, a low-pass filter is used to remove contributions with a frequency above 25Hz (see the bottom pannel of Figure 2). In practice, the sensors capture contributions from the wind-tunnel motor at 30Hz. The dominant frequency of the system is 3.25Hz, so many super-harmonics remain below the cut-off frequency. The data-generating model is shown in Eq. (15). The structural model from Eq. (15a) is rewritten in a standardized form, replacing the system parameters by coefficients c 1 , . . . , c 6 . The aerodynamic model is also rewritten, retaining only a subset of the terms listed in Eq. (15b), and replacing the aerodynamic parameters by coefficients e 1 , . . . , e 4 . The terms retained are limited to: (i) linear stiffness e 1 θ, (ii) linear damping e 2θ , (iii) nonlinear Duffing-type stiffness e 3 θ 3 , and (iv) Van der Pol-type nonlinear damping term e 4 θ 2θ . Furthermore, unsteadiness is introduced through the coefficient B, and a random forcing is introduced through σξ(τ ), where ξ(τ ) is a white noise process θ = c 1 sign(θ) + c 2 θ + c 3 C M + c 4θ + c 5 θ 3 , (16a) C M B + C M = e 1 θ + e 2θ + e 3 θ 3 + e 4 θ 2θ + c 6 Bθ + σξ(τ ).(16b) The structural parameters c 1 , . . . , c 6 , from Eq. (16a) and the aeroedynamic coefficients B, e 1 , . . . , e 4 and the strength of the random forcing σ in Eq. (16b) are summarized in Table 1. The observations, D are shown in Figure 2. Next, we exploit the NSBL algorithm from Section 2 to demonstrate the algorithm's ability to correctly identify the relevance or irrelevance of parameters in one-, two-, and four-dimensional sparse learning exercises. Unidimensional sparse learning We first consider a unidimensional sparse learning problem, meaning the questionable parameter vector φ α and the associated hyperparameter vector α are both scalar entities. Given the measurements D, the proposed model for C M is chosen to be the same as the data-generating model shown in Eq. (16b). The structural equation of motion in Eq. (16a) is assumed to be known for the sake of inverse modelling. The cubic aerodynamic stiffness parameter e 3 (or term e 3 θ 3 ) in Eq. (16b) is treated as a questionable parameter, whose relevance to the LCO aerodynamics needs to be determined using the NSBL algorithm. As per Section 2.1, φ is decomposed into the questionable parameter φ α = {e 3 } and the a priori relevant parameter vector φ -α = {B, e 1 , e 2 , e 4 , σ}. The prior pdf p(φ -α ) is known, while e 3 is assigned an ARD prior N (e 3 |0, α −1 ). Table 2 summarizes this unidimensional sparse learning setup, including the known prior pdf of φ -α . Note that LN (.|r, s) represents a log-normal distribution with median at r and coefficient of variation of s; and U(.|e, f ) represents a uniform distribution with lower bound e, and upper bound f . The likelihood function p(D|φ) is computed using the extended Kalman filter (EKF) as outlined in [7,20] to propagate the joint state pdf through the nonlinear model. Given the known prior pdf p(φ -α ) from Table 2, the unnormalized pdf Aerodynamic modelĊ M B + CM = e1θ + e2θ + e3θ 3 + e4θ 2θ + c6 Bθ + σξ(τ ) φ decomposition φ α = {e3} , φ -α = {B, e1, e2, e4, σ} ARD prior, p(φ α |α) N (e3|0, α −1 ) Known prior, p(φ -α ) LN (B|0.2, 0.5)U(e1|−2, 0)U(e2|−2, 0)U(e4|−600, 0)LN (σ|0.002, 0.5) Table 2: Unidimensional sparse learning setup where parameter e 3 is treated as questionable. p(D|φ)p(φ -α ) is sampled using the MCMC sampler. A total of 5000 stationary (i.e. post burn-in) MCMC samples are generated from the six-dimensional pdf p(D|φ)p(φ -α ). These 5000 samples possess some correlation due to the Markovian nature of MCMC sampling. In an effort to alleviate the effect of correlation, every tenth sample is extracted to produce 500 independent and identically distributed (iid) samples from p(D|φ)p(φ -α ). These 500 iid samples are then decomposed to into 10 sets of 50 iid samples each. These iid sets are used to investigate the finite sample properties of NSBL algorithm. Note that this sampling of p(D|φ)p(φ -α ) is only required to be performed once since both p(D|φ) and p(φ -α ) remain unchanged during sparse learning. Moreover, generating 5000 stationary samples from a six-dimensional pdf is computationally cheap using MCMC algorithms such as delayed rejection adaptive metropolis (DRAM) [21] and transitional MCMC [22] Next, a multivariate KDE approximation is built for p(D|φ)p(φ -α ) using each of the 10 sets of 50 iid samples. When using a Gaussian kernel, the KDE approximation resembles the kernel-based approximation in Eq. (6). Henceforth, the number of kernels K is the same as the number of samples, which is 50 for the current case. The coefficient a (k) are all equal to one since the Gaussian kernels in KDE are all equally weighted. The mean vector µ (k) for each kernel is equal to the corresponding iid sample values (vector). The covariance matrix Σ (k) is computed automatically using Scott's rule for estimating KDE bandwidth [23] and is the same for all 50 kernels. Figure 3 shows the marginal parameter pdf pertaining to each of the 10 instances of KDE approximation of p(D|φ)p(φ -α ). The marginal pdfs show a reasonable variation across multiple instances considering only 50 iid samples were used. Figure 4 shows the objective function L(log α) and the log-evidence logp(D| log α) for varying log α value, pertaining to each instance of the KDE approximation. There is little-to-no variability observed in the objective function and log-evidence across the different instances of KDE. The objective function is computed using hyperprior parameters of log r i = log s i = -6.0 in Eq. (10). The similarity between the two leftmost panels implies the negligible effect of hyperprior in L(log α) for the range of log α values shown. Also, the variation in L(log α) across different KDE instances is much less than that for the marginal pdfs shown in Figure 3. This behaviour of L(log α) indicates that the relevance property of a parameter is more robust to sampling variability than the KDE approximation of p(D|φ)p(φ -α ). The optimization of the objective function (L(log α)) and the associated value of the relevance indicator (γ rms ) are shown as a function of Newton iteration count in the two right-most panels of Figure 4, respectively. The log α iterates during the multistart Newton iteration, and the corresponding value of the relevance indicator γ rms L(log α) employed in the Newton iteration was first validated using a finite-difference scheme (results not reported here for brevity). All the multistart Newton iterations converge to a unique optimum of α map = -9.1. The relevance indicator γ rms converges to a value of one, implying the strong relevance of parameter e 3 to the LCO aerodynamics. This relevance of e 3 is insensitive to the choice of tolerance γ tol as the optimal relevance indicator is very close to one. The posterior pdf of φ can be obtained as per A.2 using the optimal ARD prior N (e 3 |0, exp(9.1)). Since this optimal prior has large variance, the posterior parameter pdf obtained from NSBL is the same as that obtained by sampling p(D|φ)p(φ -α ) (the exact pdf shown in Figure 3). In other words, the optimal ARD prior did not bias the posterior pdf of e 3 . Next, consider the unidimensional sparse learning setup in Table 3 where the aerodynamic model is augmented by a higher-order nonlinear stiffness term e 5 θ 5 . Here, parameter e 5 is treated as questionable. Given that this term was not used to generate the data, we should expect the e 5 parameter to be deemed irrelevant by NSBL. Aerodynamic modelĊ Table 3: Unidimensional sparse learning setup where parameter e 5 is treated as questionable. M B + CM = e1θ + e2θ + e3θ 3 + e4θ 2θ + e5θ 5 + c6 Bθ + σξ(τ ) φ decomposition φ α = {e5} , φ -α = {B, e1, e2, e3, e4, σ} ARD prior, p(φ α |α) N (e5|0, α −1 ) Known prior, p(φ -α ) LN (B|0.2, 0.5) U(e1|−2, 0) U(e2|−2, 0) U(e3|−250, 250)U(e4|−600, 0) LN (σ|0.002, 0.5) The two left-most pannels of Figure 5 show the log-evidence logp(D| log α) and the objective function L(log α) using the hyperprior parameters of log r i = log s i = -6. Unlike the previous case, both these entities possess large sampling variability across multiple KDE instances of p(D|φ)p(φ -α ). However, the relevance of e 5 is determined by the optimum of L(log α), which remains same across multiple KDE instances. This finite sample property of L(log α) reiterates the notion that relevance is well-determined using limited iid samples from p(D|φ)p(φ -α ), even when the KDE instances possess large sampling variability. Also notice that, unlike log-evidence, the objective function L(log α) is free from flat regions for large log α values. The Hessian matrix is singular in flat regions. Therefore, the presence of flat regions in log-evidence makes it unsuitable for the application of Newton's method. The absence of flat regions in L(log α) is due to the influence of hyperprior p(α) for large log α values. This ensures L(log α) is amenable to optimization by Newton's method wherein the Hessian is non-singular at all times. The shape and rate parameters of the Gamma hyperprior must have positive values, hence to assign Jeffreys prior, they are given values that approach zero from above. If these values are sufficiently small (here we use log r i = log s i = -6.0), they should have a negligible effect of the relevance of parameters, but from Eq. (10), it is understood that they will help regularize the sparse learning optimization problem for large values of α. The two rightmost plots of Figure 5 show the log α iterates and the RMS relevance indicator γ rms ; when initiating at log α values of -15.0, -5.0 and 5.0. The multistart Newton iteration converges to a unique optimum of α map = 0 and γ rms = 0 for all multistart iterations and for all KDE instances of p(D|φ)p(φ -α ). Note that since the objective function is relatively flat close to the optimum, the estimated optimum can vary with a varying function tolerance set for terminating the Newton iteration. However, the key entity to monitor is the relevance indicator. Even with a small variation in the optimum hyperparameter value, the relevance indicator converges to the value of zero. According to Eq. (14), a γ rms value of zero implies irrelevance since the posterior pdf is entirely dictated by the Dirac-delta ARD prior centered at zero. This change in the posterior pdf of e 5 following NSBL is shown in the right pannel of Figure 6. Figure 6 shows the posterior pdf of both nonlinear stiffness parameters e 3 and e 5 before and after the inclusion of optimal ARD prior N (e 5 |0, exp(0.0)). Notice that the posterior pdf of e 3 following the removal of e 5 has a lower uncertainty and is in close agreement with the true value. This behaviour can be explained by realizing that e 3 and e 5 both quantify the nonlinear aerodynamic stiffness. When both the parameters are present in the model, overfitting occurs. When e 5 is removed, overfitting is remedied, and e 3 is estimated with an increased accuracy using D. This numerical exercise demonstrates the benefit of sparse learning in preventing overfitting and enabling robust probabilistic predictions outside the regime of measurements. The two unidimensional cases studied here instills confidence in NSBL as a practical tool to perform sparse learning among physics-based models. Next, we study two-dimensional sparse learning cases to gain a more in-depth understanding of NSBL. Two-dimensional sparse learning In this section, we consider two-dimensional sparse learning cases where the number of questionable parameters is two (i.e. N α = 2). The hyperparameter vector is α = {α 1 , α 2 }. NSBL is executed with the same settings as Section 3.1. Hence, the likelihood function p(D|φ) is computed using EKF, followed by the generation of 500 iid samples from p(D|φ)p(φ -α ) using the DRAM MCMC algorithm. NSBL is executed for each of the 10 KDE instances of p(D|φ)p(φ -α ) containing 50 iid samples each. Also, the hyperprior parameters are chosen as log r i = log s i = -6.0 for all the cases reported in this section. We consider the three possible combinations of relevant and irrelevant parameters: where the questionable parameters are both relevant, where one is relevant and the other is irrelevant, and where both are irrelevant. We first consider the Bayesian setup in Table 4 where the proposed model is same as the data-generating model from Eq. (16b), and parameters e 3 and e 4 are treated as questionable. Figure 7 shows the log α iterates during Newton iteration initiated from {-20, -20}, {-5, -20}, {-20, -5} and {-5, -5}, for each of the 10 KDE instances. Notice that all of the multistart Newton iterations converges to a unique optimum of log α map = {-9.1, -12.4}. Also, convergence to a unique optimum for varying KDE instances of p(D|φ)p(φ -α ) demonstrates that the curvature of objective function L(log α) remains fairly insensitive to sampling variability. The two rightmost panels of Figure 7 show the variation in RMS relevance indicator γ rms for parameter e 3 and e 4 , respectively. Both the relevance indicators converge to one, implying the relevance of both the parameters to the LCO physics. NSBL converges in less than 10 Newton iterations, demonstrating the power of the Hessian informed optimizer. Aerodynamic modelĊ Table 4: Two-dimensional sparse learning setup where parameters e 3 and e 4 are treated as questionable. M B + CM = e1θ + e2θ + e3θ 3 + e4θ 2θ + c6 Bθ + σξ(τ ) φ decomposition φ α = {e3, e4} , φ -α = {B, e1, e2, σ} ARD prior, p(φ α |α) N (e3|0, α −1 1 ) N (e4|0, α −1 2 ) Known prior, p(φ -α ) LN ( Next, consider the Bayesian setup in Table 5 where parameters e 3 and e 5 are treated as questionable. The data-generating model in Eq. (16b) contains the cubic aerodynamic stiffness coefficient e 3 , while the fifth-order aerodynamic stiffness coefficient e 5 is absent. Figure 8 {-20, -5} and {-5, -5}. The multistart Newton iteration converges to a unique optimum of log α map = {-9.1, 0.0}. Recall that these are the two values obtained when these parameters were studied independently in section 3.1. The two right-most panels of Figure 8 shows the RMS relevance indicators for parameter e 3 and e 5 , respectively. Parameter e 3 is determined to be relevant since γ rms 1 converges to exactly one. On the contrary, parameter e 5 is determined to be irrelevant as γ rms 2 converges to zero. These conclusion are true for all multistart locations and for all KDE instances of p(D|φ)p(φ -α ). Table 5: Two-dimensional sparse learning setup where parameters e 3 and e 5 are treated as questionable. Next, consider the Bayesian setup in Table 6, where parameters e 5 and e 6 are treated as questionable. Both these parameters were absent from the data-generating model in Eq. (16b). Figure 9 shows the log α iterates for the multistart Newton iteration beginning from {-20, -20}, {-5, -20}, {-20, -5} and {-5, -5}. The Newton iterations converge to two different optimums of {0.0, -20.5} and {0.0, 0.0}. Based on the L(log α) values at these optimums, {0.0, -20.5} was found to the global optimum (for the range of α 1 and α 2 considered here). This ability of the multistart Newton iteration to capture multiple optima was further validated using the case of bimodal likelihood functions. Table 6: Two-dimensional sparse learning setup where parameters e 5 and e 6 are treated as questionable. Figure 9 shows the RMS relevance indicator for parameter e 5 and e 6 during multistart Newton iterations. Parameter e 5 is rendered irrelevant irrespective of the optimum as γ rms 1 converges to zero for both the optima. The parameter e 6 requires a special attention as γ rms 2 in Figure 9 converges to a value in the range [0.75,0.90] for the case of the global optimum. As demonstrated previously in the unidimensional setting, γ rms i value converges to exactly one for relevant parameters. Also, based on our experience, the convergence of relevance indicator γ rms i to one is a necessary condition for a parameter to be deemed relevant. This condition implies that the posterior pdf of a relevant parameter should be largely dictated by the likelihood function and not the data-informed ARD prior. Based on this principle, the parameter e 6 should be deemed irrelevant. Alternatively, one could define a tolerance γ tol for determining the relevance of such parameters. A γ tol value of 0.5 implies parameter e 6 as relevant, whereas a γ tol value of 0.9 implies an irrelevant e 6 . This choice of an appropriate γ tol imparts greater flexibility to the modeller, thereby allowing the sparse learning process to align with the modelling goals. Figure 9: NSBL results for the relevance determination of parameter e 5 and e 6 . Figure 10 shows the marginal parameter pdf before and after the inclusion of the optimal ARD prior for questionable parameters e 5 and e 6 at both optima identified in Figure 9. The pre-NSBL pdf, shown in blue, is obtained from the KDE representation of p(D|φ)p(φ -α ) using 500 iid samples. The post-NSBL pdfs for the case where e 5 is irrelavant, but e 6 are relevant is shown in red. The case where both e 5 and e 6 are irrelevant are shown in green. These posterior pdfs are obtained using the analytical solution derived in A.2. First, in red, where only e 5 is correctly identified as irrelevant,the posterior pdf of e 5 reduces to a Dirac-delta function centered at zero. This eliminates all of the uncertainty in the estimate for e 5 , and also can be shown to reduce the uncertainty in the estimate for e 3 , which is also a nonlinear stiffness parameter. Second, in green, where both e 5 and e 6 are correctly identified as irrelevant and are both reduced to Dirac-delta function centered at zero, there is additional reduction in uncertainty observed for both parameters e 4 and e 2 , which are nonlinear and linear damping coefficients, respectively. The removal of these higher-order nonlinear terms, however, is shown here to have a minimal effect on the posterior estimate for σ. This behaviour of the marginal posterior pdf of σ indicates that the removal of redundant parameters during NSBL does not affect the data-fit property of the model. Had NSBL identified an overly-simplistic model, the model error strength would be expected to increase. Notice the decrease in posterior uncertainty of parameters e 3 and e 4 following the removal of e 5 and e 6 . This decrease is attributed to the remediation of overfitting by the sparse learning process. Figure 10 shows the corresponding marginal posterior pdfs for the case when e 5 is deemed irrelevant and e 6 is deemed relevant. The objective function L(log α) was observed to be dictated by log-evidence for the range of α values shown in Figure 9, meaning both log-evidence and L(log α) are bimodal. To our best knowledge, this is the first reporting of a multimodal log-evidence for a physics-based Bayesian inverse modelling. This multimodality also demonstrates the issue of global identifiability in the model (or hyperparameter) space, while the inverse problem may or may not be globally identifiable in the likelihood (or model parameter) space. This multimodality also warrants careful consideration of evidence-based Bayesian model comparison while dealing with nested models with closely-related nonlinear terms. Aerodynamic modelĊ M B + CM = e1θ + e2θ + e3θ 3 + e4θ 2θ + e5θ 5 + c6 Bθ + σξ(τ ) φ decomposition φ α = {e3, e5} , φ -α = {B, e1, e2, e4, σ} ARD prior, p(φ α |α) N (e3|0, α −1 1 ) N (e5|0, α −1 2 ) Known prior, p(φ -α ) LN (Aerodynamic modelĊ M B + CM = e1θ + e2θ + e3θ 3 + e4θ 2θ + e5θ 5 + e6θ 4θ + c6 Bθ + σξ(τ ) φ decomposition φ α = {e5, e6} , φ -α = {B, e1, e2, e3, e4, σ} ARD prior, p(φ α |α) N (e5|0, α −1 1 ) N (e6|0, α −1 2 ) Known prior, p(φ -α ) LN ( NSBL was also executed for cases with 100, 500, and 1000 iid samples per KDE instance of p(D|φ)p(φ -α ). These results are not reported here since they were identical to those reported in this section but with even less variation in log α iterates and relevance indicator γ rms across multiple instances. This conclusion was true for the three numerical cases studied in this section. Four-dimensional sparse learning Consider the four-dimensional sparse learning problem detailed in Table 7. Here we only consider a single case, where among four questionable parameters, two parameters are relevant, and two are irrelevant. NSBL is executed to determine the relevance of questionable parameters e 3 , e 4 , e 5 and e 6 . The NSBL setup is the same as Section 3.1. Figure 11 shows the log α iterates and relevance indicator γ rms Figure 12 shows the marginal posterior pdf before and after the inclusion of the optimal ARD prior computed using NSBL. The pre-NSBL pdf is obtained using 500 iid samples from p(D|φ)p(φ -α ), while the post-NSBL pdf is obtained using the kernel-based analytical solution derived in A.2. Once again, in Figure 12, we show the pre-NSBL results in blue, the local optimum (where e 5 is relevant and e 6 is relevant) in red, and the global optimum (where both e 5 and e 6 are irrelevant) in green. Notice the decrease in uncertainty in relevant parameters following the removal of parameter e 5 and e 6 . Note that the level of sparsity in physical models such as the LCO models is significantly lower than data-based models due to the presence of physics-based parameters. Significant benefits are realized in terms of decrease in uncertainty in the posterior pdf even with the removal of two parameters. Aerodynamic modelĊ M B + CM = e1θ + e2θ + e3θ 3 + e4θ 2θ + e5θ 5 + e6θ 4θ + c6 Bθ + σξ(τ ) φ decomposition φ α = {e3, e4, e5, e6} , φ -α = {B, e1, e2, σ} ARD prior, p(φ α |α) N (e3|0, α −1 1 )N (e4|0, α −1 2 )N (e5|0, α −1 3 )N (e6|0, α −1 4 ), Conclusion With the use of synthetic data, we have demonstrated that NSBL is a promising approach to computationally efficient model selection for nonlinear stochastic dynamical systems. In all cases studied herein, the algorithm has shown its ability to recapture the data-generating model parameters and correctly identify the relevance of the parameters. Note Figure 12: Marginal posterior pdf of model parameters before the after the inclusion of optimal ARD prior. that for this case study, the data is not particularly sparse temporally, nor is it particularly noisy, though, the data only represents a partial observation of the state. For the given data, even an overly complex model may not necessarily yield poor predictions of the response, nonetheless, it has been shown that the parameter estimates can be significantly improved by the removal of unnecessary parameters. The specific aspects of NSBL that make it particularly well suited to engineering systems is the ability to use non-Gaussian known priors, and the applicability to systems where there is a nonlinear mapping from the uncertain parameters to the observed outputs. In the aeroelastic example, the underlying physics and/or laws of statistics prevented certain parameters from assuming non-positive values, hence a Gaussian distributions would not be proper choices of prior. Likewise, the LCO being modelled is a nonlinear phenomenon, which naturally leads to a nonlinear-in-parameter model in the machine learning context. Both of these qualities are enabled by the use of a GMM approximation of the product of the likelihood function and the known prior. This GMM also enables the semi-analytical machinery that enables the efficient computation of estimates of many quantities of interest including the parameter posterior distribution, the model evidence, the objective function, Jacobian/ gradient vector and Hessian matrix. Finally, we have illustrated the need for robust optimization scheme as there is the potential for multiple optima in the objective function, corresponding to different locally optimal values of the hyperparameter, corresponding to different classifications of the parameter as relevant versus irrelevant. The two Gaussian distributions in the integral expression will generally be mismatched in dimensions (N φ and N φ α ). In order to facilitate the integration, we rewrite the kth kernel of the GMM in an expanded form as N (φ|µ (k) , Σ (k) ) = N φ α φ -α µ (k) α µ (k) -α , Σ (k) α C (k) (C (k) ) T Σ (k) -α ,(19) where µ (k) α and Σ (k) α are the mean and covariance of the kth kernel pertaining to φ α , likewise µ (k) -α and Σ (k) -α are the mean and covariance of the kth kernel pertaining to φ -α , and C (k) is the cross-covariance of φ α and φ -α . These entities are obtained from the construction of the GMM (whether by KDE, EM, or other means). Now, rewriting this joint distribution of φ α and φ -α as the product of the conditional distribution of φ -α given φ α and the marginal distribution of φ α , we obtain [2] N (φ|µ (k) , Σ (k) ) = N (φ -α |μ (k) -α ,Σ (k) -α )N (φ α |µ (k) α , Σ (k) α ),(20a)µ (k) -α = µ (k) -α + (C (k) ) T (Σ (k) α ) −1 (φ α − µ (k) α ),(20b)Σ (k) -α = Σ (k) -α − (C (k) ) T (Σ (k) α ) −1 C (k) ,(20c) whereμ (k) -α andΣ (k) -α are the mean and covariance of the kth kernel, pertaining to φ -α conditioned on a given value of φ α . Consequently, the estimate of the model evidence becomes [2] p (D|α) = K k=1 a (k) N (φ -α |μ (k) -α ,Σ (k) -α )N (φ α |µ (k) α , Σ (k) α )N (φ α |0, A −1 )dφ,(21) where the product of the two left-most Gaussians can be rewritten as N (φ α |µ (k) α , Σ (k) α )N (φ α |0, A −1 ) = N (µ (k) α |0, B (k) α )N (φ α |m (k) α P (k) α ) (22a) m (k) α = µ (k) α − Σ (k) α (B (k) α ) −1 µ (k) α (22b) P (k) α = Σ (k) α − Σ (k) α (B (k) α ) −1 Σ (k) α (22c) B (k) α = Σ (k) α + A −1 (22d) where m (k) α and P (k) α are the mean and covariance of the kth kernel of the posterior of φ α . Note that N (µ (k) α |0, B (k) α ) is now independent of the parameters. Substituting and integrating over all parameters gives [2] p(D|α) = K k=1 a (k) N (φ -α |μ (k) -α ,Σ (k) -α )N (µ (k) α |0, B (k) α )N (φ α |m (k) α P (k) α )dφ = K k=1 a (k) N (µ (k) α |0, B (k) α ).(23) A.2 Posterior parameter pdf As the estimate of the model evidence is now available in Eq. (21), it is now possible to obtain an estimate of the parameter posterior pdfp(φ|D, α) making the same substitutions from above to obtain [2] p(φ|D, α) = K k=1 a (k) N (φ|µ (k) , Σ (k) )N (φ α |0, A −1 ) p(D|α) . Substituting N (φ|µ (k) , Σ (k) ) from Eq. (20a) reduces Eq. (24) to [2] p(φ|D, α) = K k=1 a (k) N (φ -α |μ (k) -α ,Σ (k) -α )N (φ α |µ (k) α , Σ (k) α )N (φ α |0, A −1 ) p(D|α) = K k=1 a (k) N (µ (k) α |0, B (k) α ) K r=1 a (r) N (µ (r) α |0, B (r) α ) w (k) N (φ -α |μ (k) -α ,Σ (k) -α )N (φ α |m (k) α , P (k) α ) (25) = K k=1 w (k) N φ α φ -α m (k) α m (k) -α , P (k) α D (k) (D (k) ) T P (k) -α (26) where 0 ≤ w (k) ≤ 1, k w (k) = 1 is the weight coefficient of kernel k, and m (k) and P (k) are the mean and covariance of the kth kernel of the the posterior of φ, respectively. Note that m (k) α and P (k) α are known from Eq. (22b), and Eq. (22c). Furthermore, m (k) -α and P (k) -α are the mean and covariance of the kth kernel of the posterior of φ -α and D (k) is the cross-covariance of φ α and φ -α . These entities are given by m (k) -α = µ (k) -α − (C (k) ) T (B (k) α ) −1 µ (k) α (27a) P (k) -α = Σ (k) -α − (C (k) ) T (B (k) α ) −1 C (k) (27b) D (k) = C (k) − Σ (k) α (B (k) α ) −1 µ (k) α ((C (k) ) T (B (k) α ) −1 µ (k) α ) T (27c) A.3 Gradient vector and Hessian matrix Let J i (log α) denote the ith element of the Jacobian vector J(log α). Differentiating the objective function in Eq. (10) with respect to the log(α i ) gives [2] J i (log α) = ∂L(log α) ∂ log α i = ∂ ∂ log α i logp(D| log α) + Nα i=1 (r i log α i − s i α i ) = ∂ logp(D| log α) ∂ log α i + r i − s i α i = 1 p(D| log α) ∂ ∂ log α i K k=1 a (k) N (µ (k) α |0, B (k) α ) + r i − s i α i = 1 p(D| log α) K k=1 a (k) ∂ ∂ log α i exp log N (µ (k) α |0, B (k) α ) + r i − s i α i = K k=1 a (k) N (µ (k) α |0, B (k) α ) p(D| log α) w (k) ∂ log N (µ (k) α |0, B (k) α ) ∂ log α i v (k) i +r i − s i α i (28) = K k=1 w (k) v (k) i + r i − s i α i ,(29) where w (k) is known from Eq. (25), and factor v (k) i is given as v (k) i = − 1 2 −1 + α i P (k) ii + α i (m (k) i ) 2 .(30) Now, let H ij (log α) denote element (i, j) of the Hessian matrix H(log α). Differentiating the jth element of the Jacobian in Eq. (29) with respect to the ith element α gives [2] H ij (log α) = ∂ 2 L(log α) ∂ log α i ∂ log α j = ∂J j (log α) ∂ log α i = ∂ ∂ log α i K k=1 w (k) v (k) j + r j − s j α j = K k=1 w (k) ∂v (k) j ∂ log α i + v (k) j ∂w (k) ∂ log α i − δ ij s i α i ,(31)with ∂v (k) j ∂ log α i = α i α j (P (k) ij ) 2 2 + m (k) i m (k) j P (k) ij + δ ij v (k) i − 1 2 ,(32)∂w (k) ∂ log α i = w (k) v (k) i −v i ,(33) where δ ij is the Kronecker delta function, andv i is the average of v (k) i over all K kernels. Figure 2 : 2Synthetic pitch measurements and its PSD. The data used for ARD computations is shown in red. (Reprinted from Sandhu et al.[10], with permission from Elsevier). Figure 3 : 3Marginal parameter pdf obtained using 10 instances of 50 iid samples from p(D|φ)p(φ -α ). The baseline (converged) pdf is obtained by using all of the 500 iid samples. i from Eq. (14) are shown. The log α iterates and γ rms i are shown for each of the 10 KDE instances of p(D|φ)p(φ -α ). The Newton iteration is initiated from three log α values of -20.0, -10.0 and 0.0. The analytical solution for the gradient and Hessian of Figure 4 : 4NSBL results for the relevance determination of an a priori relevant parameter e 3 . Figure 5 : 5NSBL results for the relevance determination of an a priori irrelevant parameter e 5 . Figure 6 : 6NSBL results for the relevance determination of an a priori irrelevant parameter e 5 . shows the log α iterates pertaining to starting log α values of {-20, -20}, {-5, -20}, Figure 7 : 7NSBL results for the relevance determination of parameter e 3 and e 4 . Figure 8 : 8NSBL results for the relevance determination of parameter e 3 and e 5 . Figure 10 : 10Marginal posterior pdf of model parameters before and after sparse learning, for the case when e 5 and e 6 are deemed irrelevant. i for multistart Newton iterations beginning from {-20, -20, -20, -20}, {-20, -20, -5, -5}, {-5,-5, -20, -20} and {-5, -5, -5, -5}. Notice that all but log α 4 converges to a unique optimum. The log α 4 converges to two different optimums depending on the starting point. This type of behaviour has been previously reported in Figure 9. Based on the L(log α) values at these optimums, log α map 4 = 0.0 was found to the global optimum. This produces the solution log α map = {-9.1, -12.4, 0.0, 0.0}, and the corresponding value of relevance indicator as {1.0, 1.0, 0.0, 0.0}. This optimal value of γ rms i indicates that parameter e 3 and e 4 are important, while parameter e 5 and e 6 are redundant. This conclusion is in agreement with the data-generating model in Eq. (16b) that was used to generate D. Known prior, p(φ -α ) LN (B|0.2, 0.5) U(e1|−2, 0) U(e2|−2, 0)LN (σ|0.002, 0.5) Figure 11 : 11NSBL results for the four-dimensional sparse learning. Table 1 : 1Structural and aerodynamic model parameters used to generate the synthetic data. Table 7 : 7Four-dimensional sparse learning setup. A Semi-analytical calculation of Bayesian entities A.1 Model evidenceThe model evidence was first introduced as the normalization factor in Eq. (1). Substituting the expression for the joint prior pdf in Eq. (2), the model evidence can be rewritten as[2]Now, substituting the GMM approximation of the product of the likelihood function and the known prior from Eq. (6), and the expression for the ARD prior, p(φ|α) = N (φ α |0, A −1 ), Eq. (17) becomeŝa (k) N (φ|µ (k) , Σ (k) )N (φ α |0, A −1 )dφ. Model comparison and sparse learning of nonlinear physics-based models using Bayesian inference. Rimple Sandhu, Carleton UniversityPhD thesisRimple Sandhu. Model comparison and sparse learning of nonlinear physics-based models using Bayesian inference. PhD thesis, Carleton University, 2020. Nonlinear sparse bayesian learning for physics-based models. Rimple Sandhu, Mohammad Khalil, Chris Pettit, Dominique Poirel, Abhijit Sarkar, Journal of Computational Physics. 426109728Rimple Sandhu, Mohammad Khalil, Chris Pettit, Dominique Poirel, and Abhijit Sarkar. Nonlinear sparse bayesian learning for physics-based models. Journal of Computational Physics, 426:109728, 2021. Self-sustained aeroelastic oscillations of a NACA0012 airfoil at low-tomoderate Reynolds numbers. D Poirel, Y Harris, A Benaissa, Journal of Fluids and Structures. 245D. Poirel, Y. Harris, and A. Benaissa. Self-sustained aeroelastic oscillations of a NACA0012 airfoil at low-to- moderate Reynolds numbers. Journal of Fluids and Structures, 24(5):700 -719, 2008. Detection of laminar flow separation and transition on a naca-0012 airfoil using surface hot-films. Daniel Rudmin, Azemi Benaissa, Dominique Poirel, Journal of Fluids Engineering. 13510Daniel Rudmin, Azemi Benaissa, and Dominique Poirel. Detection of laminar flow separation and transition on a naca-0012 airfoil using surface hot-films. Journal of Fluids Engineering, 135(10), 2013. Aerodynamics of laminar separation flutter at a transitional Reynolds number. D Poirel, W Yuan, Journal of Fluids and Structures. 267-8D. Poirel and W. Yuan. Aerodynamics of laminar separation flutter at a transitional Reynolds number. Journal of Fluids and Structures, 26(7-8):1174-1194, October 2010. Computational aeroelastic simulations of self-sustained pitch oscillations of a naca0012 at transitional reynolds numbers. D Poirel, G Metivier, Dumas, Journal of Fluids and Structures. 278D Poirel, V Metivier, and G Dumas. Computational aeroelastic simulations of self-sustained pitch oscillations of a naca0012 at transitional reynolds numbers. Journal of Fluids and Structures, 27(8):1262-1277, 2011. Bayesian model selection for nonlinear aeroelastic systems using wind-tunnel data. Rimple Sandhu, Mohammad Khalil, Abhijit Sarkar, Dominique Poirel, Computer Methods in Applied Mechanics and Engineering. 282Rimple Sandhu, Mohammad Khalil, Abhijit Sarkar, and Dominique Poirel. Bayesian model selection for nonlinear aeroelastic systems using wind-tunnel data. Computer Methods in Applied Mechanics and Engineering, 282:161 - 183, 2014. Bayesian inference of nonlinear unsteady aerodynamics from aeroelastic limit cycle oscillations. Rimple Sandhu, Dominique Poirel, Chris Pettit, Mohammad Khalil, Abhijit Sarkar, Journal of Computational Physics. 316Rimple Sandhu, Dominique Poirel, Chris Pettit, Mohammad Khalil, and Abhijit Sarkar. Bayesian inference of nonlinear unsteady aerodynamics from aeroelastic limit cycle oscillations. Journal of Computational Physics, 316:534 -557, 2016. Combined selection of the dynamic model and modeling error in nonlinear aeroelastic systems using bayesian inference. Philippe Bisaillon, Rimple Sandhu, Chris Pettit, Mohammad Khalil, Dominique Poirel, Abhijit Cs Manohar, Sarkar, Journal of Sound and Vibration. 522116418Philippe Bisaillon, Rimple Sandhu, Chris Pettit, Mohammad Khalil, Dominique Poirel, CS Manohar, and Abhijit Sarkar. Combined selection of the dynamic model and modeling error in nonlinear aeroelastic systems using bayesian inference. Journal of Sound and Vibration, 522:116418, 2022. Bayesian model selection using automatic relevance determination for nonlinear dynamical systems. Rimple Sandhu, Chris Pettit, Mohammad Khalil, Dominique Poirel, Abhijit Sarkar, Computer Methods in Applied Mechanics and Engineering. 320Rimple Sandhu, Chris Pettit, Mohammad Khalil, Dominique Poirel, and Abhijit Sarkar. Bayesian model selection using automatic relevance determination for nonlinear dynamical systems. Computer Methods in Applied Mechanics and Engineering, 320:237 -260, 2017. 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[]
[ "Contactless determination and parametrization of charge carrier mobility in silicon as a function of injection level and temperature using time resolved THz spectroscopy", "Contactless determination and parametrization of charge carrier mobility in silicon as a function of injection level and temperature using time resolved THz spectroscopy" ]
[ "Sergio Revuelta \nIMDEA Nanociencia\nCampus Universitario de Cantoblanco, Faraday 928049MadridSpain\n", "Enrique Cánovas *[email protected] \nIMDEA Nanociencia\nCampus Universitario de Cantoblanco, Faraday 928049MadridSpain\n" ]
[ "IMDEA Nanociencia\nCampus Universitario de Cantoblanco, Faraday 928049MadridSpain", "IMDEA Nanociencia\nCampus Universitario de Cantoblanco, Faraday 928049MadridSpain" ]
[ "Phys. Rev. B" ]
Here, we analyze in a non-contact fashion charge carrier mobility as a function of injection level and temperature in silicon by time resolved THz spectroscopy (TRTS) and parametrize our data by the classical semi-empirical models of Klaassen and Dorkel & Leturcq. Our experimental results are in very good agreement with the pioneering works of Krausse and Dännhauser analyzing this phenomena by employing contact-based methods. This agreement, that validates our methodology, can only be achieved by considering charge carrier diffusion effects following above bandgap near-surface pump photo-excitation of the sample. From our results, obtained over a large range of injection levels, we conclude that the model of Klaassen is the best on describing the collected data at room temperature. Furthermore, we analyze by TRTS the dependence of charge carrier mobility with temperature for a fixed injection level. Once more, the parametrization made by the classical model of Klaassen describe our data appropriately even without the necessity of applying any fitting parameters (just with the charge carrier density as an input). In this respect, our work supports the validity of the model and parametrization proposed by Klaassen, and also illustrate how TRTS can be reliably employed for the quantitative determination of mobility in semiconductors as a function of key parameters as injection level and temperature.
10.1103/physrevb.107.085204
[ "https://export.arxiv.org/pdf/2303.04524v1.pdf" ]
257,168,330
2303.04524
cda35b1c006dc3e21a16439a4e9aac756d6dc9b3
Contactless determination and parametrization of charge carrier mobility in silicon as a function of injection level and temperature using time resolved THz spectroscopy 2023 Sergio Revuelta IMDEA Nanociencia Campus Universitario de Cantoblanco, Faraday 928049MadridSpain Enrique Cánovas *[email protected] IMDEA Nanociencia Campus Universitario de Cantoblanco, Faraday 928049MadridSpain Contactless determination and parametrization of charge carrier mobility in silicon as a function of injection level and temperature using time resolved THz spectroscopy Phys. Rev. B 10785204202310.1103/PhysRevB.107.0852041 Here, we analyze in a non-contact fashion charge carrier mobility as a function of injection level and temperature in silicon by time resolved THz spectroscopy (TRTS) and parametrize our data by the classical semi-empirical models of Klaassen and Dorkel & Leturcq. Our experimental results are in very good agreement with the pioneering works of Krausse and Dännhauser analyzing this phenomena by employing contact-based methods. This agreement, that validates our methodology, can only be achieved by considering charge carrier diffusion effects following above bandgap near-surface pump photo-excitation of the sample. From our results, obtained over a large range of injection levels, we conclude that the model of Klaassen is the best on describing the collected data at room temperature. Furthermore, we analyze by TRTS the dependence of charge carrier mobility with temperature for a fixed injection level. Once more, the parametrization made by the classical model of Klaassen describe our data appropriately even without the necessity of applying any fitting parameters (just with the charge carrier density as an input). In this respect, our work supports the validity of the model and parametrization proposed by Klaassen, and also illustrate how TRTS can be reliably employed for the quantitative determination of mobility in semiconductors as a function of key parameters as injection level and temperature. Charge carrier mobility is a fundamental figure of merit determining the performance of opto-electronic devices. This variable depends critically on few parameters as the charge carrier concentration via doping, the temperature and the injection level [1,2]. To our knowledge, pioneering studies of the charge-carrier mobility dependence vs injection level in silicon were made in the 70s from contact-based techniques [3,4] and these results still represents nowadays the basis for semi-empirical models employed in modelling charge carrier mobility as a function of charge carrier density in silicon [5][6][7][8][9]. These initial studies analyzed the interplay of mobility and injection level by contact-based approaches, and hence required the manufacturing of a fully functional working device (e.g., a PIN or PN diode structure). Over the years all-optical non-contact methods have been employed as alternatives for scrutinizing the charge carrier mobility-charge carrier injection relationship, most notably photoconductance decay (PCD) measurements. This technique is very powerful for estimating the mobility, recombination processes and carrier lifetimes in semiconductors as a function of injection level, however it cannot disentangle (from conductance) the contributions of carrier density and mobility [10][11][12][13] unless it is assisted by a second set of measurements, e.g. time-resolved photoluminescence (TRPL) [13]. Furthermore, PCD techniques are often able to monitor a relatively narrow range of injection levels (i.e. charge carrier densities). A powerful non-contact alternative to PCD is Time-Resolved THz spectroscopy (TRTS) [14][15][16]. Following an optical pump-THz probe scheme it is possible to retrieve the frequency-resolved complex photoconductivity of a given sample at any pump probe delay, from which mobility and carrier density can be independently inferred. A TRTS study as a function of impingingabove band gap -pump photon flux, and hence injection level, offers an avenue for determining the interplay between photo-generated charge carrier density and charge carrier mobility. Although several studies have been already done analyzing by TRTS the mobility dependence on charge carrier density in materials as silicon [17,18], gallium arsenide [19,20], titanium oxide [18] or zinc oxide [21], to our knowledge none of these studies have attempted, nor reached a quantitative agreement with classical estimates made by contact methods; a singular aspect that represents by itself a goal between the THz and metrology research communities [22]. In this work, we have employed TRTS to analyze the dependence of sample's charge-carrier mobility with injection level and temperature as controlled by photodoping in silicon. From TRTS data at room temperature, we retrieve the expected reduction in the charge carrier mobility as a function of injection level in a trend that closely matches the one obtained in classical works employing contact-based methods. Notably, this agreement can only be reached by considering diffusion effects impacting the TRTS data and analysis, which, we show here can critically play a role on determining unambiguously bulk charge carrier densities under our TRTS conventional experimental conditions (i.e. near surface photoexcitation with UV photons). Furthermore, we have analyzed the dependence of charge carrier mobility with temperature at low injection levels and compared our data to the predictions made by the classical semi-empirical models of Klaassen [6,7] and Dorkel & Leturcq [5]. Our results, retrieved over a large range of injection levels and temperatures reveal that the model of Klaassen is the best on describing the data at room temperature and as a function of temperature. For the latter variable even without the necessity of applying any fitting parameters (just the charge carrier density as an input). In this respect, our work supports the validity of the model and parametrization proposed by Klaassen, and also demonstrates that TRTS can be employed as powerful tool for the quantitative determination of mobility in semiconductors, an aspect validated by the good agreement found between our data and the classical results made by Krausse and Dännhauser by contact methods [3,4]. I. EXPERIMENTAL The sample analysed in this work consisted of a 0.5mm thick semi-insulating silicon float zone (FZ-Si) wafer with <100> orientation (Sigma-Aldrich ID: 646687, resistivity 100-3000 Ω·cm) with native oxide passivation. A Ti:Sapphire amplified laser system providing 775 nm wavelength output (~150 fs pulse width at 1 KHz repetition rate) was employed to run the optical pump-THz probe experiment [14,18]. For optical pump excitation of the sample, we employed the 387.5 nm output generated by a Beta Barium Borate (BBO) crystal. The employed ~1THz bandwidth probe was generated via optical rectification on In a TRTS measurement we are able to monitor the ~1THz width freely propagating probe in the time domain, as such, changes in amplitude and phase induced in the transmitted THZ probe by the pump excitation can be recorded. From this data, the time dependent sample´s sheet frequency resolved complex photoconductivity, Δ ℎ ( , ) can be retrieved at any pump-probe delay time ( ). Under the employed photoexcitation conditions (387.5 nm) the optical penetration depth in silicon is estimated to be ~80 nm [23], much smaller than the sample´s thickness. This allows us to employ the Tinkham approximation [14,23] from which we can retrieve the frequency resolved complex photoconductivity as: Δ ℎ ( , ) = −( 1 + 2 ) 0 Δ ( , ) ( ) (eq. 1) Where 0 is the intrinsic impendance of free space, 1 is the refractive index of the medium before the sample and 2 is the refractive index of the photo-excited material, is the transmitted THz electric field through the unexcited sample and Δ is the pump-induced change in the THz waveform. II. RESULTS In figure Right after charge carrier photogeneration ( (~0) ), the penetration depth for the photo-excited slab in silicon is estimated to be ~80 nm [24] under the 387.5 nm above bandgap pump beam. As neither hot carrier cooling [25], nor charge carrier surface/bulk recombination substantially affect our dynamics within the analysed ~1ns refers to the initial carrier density and is the ambipolar diffusivity in silicon [27,28]. The ambipolar diffusivity is parametrized with the scattering time of the electron-hole plasma retrieved from the frequency-resolved complex sheet photoconductivity (see eq. S2 in Supplemental Material [26]). Once the effective penetration depth is accurately estimated, we can obtain the In order to infer the variation of charge carrier mobility vs injection level, we performed TRTS experiments for a broad range of 387.5 nm pump excitation conditions ranging between ~0.15 and ~70 μJ/cm 2 and at pump-probe delays higher than 10 ps, thereby ensuring the complete cooling of hot carriers [25] (see Fig. S3 in Supplemental Material [26]). The employed upper threshold photon flux was simply limited by our setup specs. For small photo-excitation conditions (below ~1 μJ/cm 2 ) the recorded frequency resolved complex photoconductivity was found to be invariant vs photon flux and pump-probe delay. Beyond this threshold, an increase in photon flux alone for a given pump-probe delay was sufficient to modulate the charge carrier scattering rate, in good agreement with a previous TRTS study made in silicon [18]. Furthermore, for each given fluence above the mentioned threshold of ~1 μJ/cm 2 , we found the same temporal trend shown in figure 1, i.e., an improvement in charge carrier scattering rate as a function of pump-probe delay. Taking into account diffusion processes for each analysed fluence (and pump-probe delay), an accurate bulk carrier density can be inferred and linked to the obtained scattering rate from TRTS analysis. In agreement with a previous study conducted in GaAs which considered diffusion effects following photoexcitation [20], we found a substantial overlap between the data obtained from different optical pump-THz probe measurements made at different pump fluences (Figure 2a and Table S3 in Supplemental Material [26]), an aspect that supports the validity for the diffusion model depicted in Figure 1b. Figure 2b we also include as solid circles the classical data obtained in the 70s by Krausse and Dännhauser [3,4] representing the sum mobility (μe + μh) as a function of injection level measured by contact-based methods on working devices. Remarkably, the optically retrieved THz mobility as function of photo-induced carrier density agrees quite well with the electrical carrier mobility obtained via contact methods. To our knowledge this is the first time such correlation is made by TRTS, and over such a large range of injection levels (higher than 1·10 16 cm -3 ) when compared to e.g. photoconductance based non-contact methods [13,29]. Here, we warn the reader that this agreement between TRTS data and the one obtained by contact methods is only reached if diffusion of photogenerated charge carriers from the surface towards the bulk is considered, otherwise bulk charge carrier densities are strongly overestimated (see Figure S4 [26]). In order to parametrize the observed experimental results summarized in Figure 2b we fit the data by the classical models developed by Dorkel & Leturcq [5] (dashed line) and Klaassen (solid line) [6,7]. In brief, Dorkel & Leturcq approach considers the mobility contributed by phonon-carrier, impurity-carrier and carrier-carrier interactions. On the other hand, Klaassen model employs a more refined model by considering phonon-carrier, electron-hole, effects of the majority and minority impurity scattering while including a screening effect and an increase of the system temperature with charge carrier density. A summary and of the parameters employed in the models can be found in Table S1 and Table S2 in the Supplemental Material [26]. From our results, we can conclude that the Klaassen model offers a better description of the experimental data retrieved by TRTS. A conclusion that agrees with non-contact PCD data collected over a narrower injection level window [9,13,29,30]. At present we cannot rationalize the slight deviation between TRTS data and those retrieved by contact methods for concentrations above 1·10 16 cm -3 , in this respect, Klassen suggests that this data might be underestimated by an increase of temperature in the system during the measurements [6]. Once we have analysed the dependence of the mobility vs injection level by TRTS at room temperature, we study the dependence of the mobility as a function of temperature for a fixed sheet charge carrier injection. To study this dependence, we selected a photon flux of ~1.55 μJ/cm 2 ; low enough to prevent strong diffusion effects in the samples. Figure 3 shows exemplary measurements of the frequency resolved complex sheet photoconductivity under 387.5 nm excitation at 1 ns of pump-probe delay and different temperatures: 90, 150, 220 and 300 K. All the plots can be properly described by the Drude model. Considering a temperature dependent ambipolar diffusivity (see Figure S1 in Supplemental Material [26] 16 cm -1735 ± 164 cm 2 V -1 s -1 respectively for the above-mentioned temperatures. As anticipated by the selection of the low photon flux, the carrier concentration values are almost identical as a function of temperature, ranging from ~2·10 16 to ~1.4·10 16 cm -3 , this aspect is critical for parametrizing the collected mobility data for a fixed carrier concentration. In this sense, the obtained variation of mobility vs T, truly represents the impact of temperature on the monitored scattering rates for a given charge carrier injection. Figure 3b summarizes, as open diamonds, the charge carrier mobility inferred from TRTS analysis for a set of temperatures ranging between 77 and 300 K (1.55 μJ/cm 2 , 387.5 nm and 1 ns of pump-probe delay, see Figure S5 in Supplemental Material for the rest of the frequency-resolved data [26]). Figure 3b also includes a parametrization of the obtained data following the models of Klaassen [6,7] (solid line) and Dorkel & Leturcq [5] (dashed lines); where remarkably the only variable input is the charge carrier concentration figure of 1.73·10 16 cm -3 (i.e. the median value of the retrieved carrier densities from the Drude fits). It is evident form the plot that the mobility inferred by TRTS matches nicely the curve predicted by Klaassen [6,7] while the Dorkel & Leturcq [5] model is unable to correctly describe the experimental data. The validation made here for the Klaassen model as a function of temperature for a given injection level is to our knowledge unique. We are only aware of a couple of PCD works that have previously attempted such study. In these works they found that Klaassen model failed to describe the experimentally resolved trend for temperatures below ~150 K [9,29]. The reason for this disagreement is unclear to us, however, we note that for low temperatures, one should consider an extra effect decreasing charge carrier density in silicon, an effect linked to the condensation of photogenerated e-h pairs into excitons. This effect indeed has been previously reported and modelled to occur at temperatures around the ~100 K onset [31][32][33]. [5] and the solid line corresponds to the model described by Klaassen [6,7]. The grey areas represent the lower and maximum error deviations on the avg inferred charge-carrier density within the analyzed range of temperatures. III. CONCLUSSIONS In this work we have investigated the dependence of the charge carrier mobility vs injection density and temperature in silicon using time resolved THz spectroscopy. Our results agree well with previously reported studies in silicon measured by contact methods [3,4] and validate the semi-empirical model developed by Klaassen [6,7]. Notably, the agreement with previous data and modelling can only be achieved when charge carrier diffusion effects following near surface photo-excitation are considered. Otherwise, the charge carrier density may be over-estimated for a given charge carrier mobility. The retrieved dependency of the carrier mobility for a fixed injection level with temperature further supports the validity of the Klaassen model against the one proposed by Dorkel & Leturcq. While in principle our approach can be generalized to any semiconductor, surface recombination effects might complicate or make not possible determining unambiguously charge carrier density as a function of time after excitation. Furthermore, a pre-knowledge of the diffusion constant vs injection and temperature seems required for properly modelling the retrieved TRTS data. Nevertheless, our results highlight the strength of TRTS as a powerful non-contact method for analyzing the mobility in semiconductors as a function of key variables as the injection level and temperature. 1a we show the frequency resolved complex sheet photoconductivity obtained for silicon for three pump-probe delays (from top to bottom: 60, 460 and 1260 ps) at a fluence of 10.58 μJ/cm 2 at 387.5 nm of pump wavelength. Black solid and blue open circles are the experimental real and imaginary components of the sheet complex conductivity within the probed THz window. Each plot also shows black solid and blue dashed lines representing respectively the best fits to real and imaginary components of the frequency resolved complex conductivity by using the Drude model as: Δ̂ℎ ( , ) = Δ̂( , ) · = ℎ 2 * (1− ) (eq. 2) Here, Δ̂( , ) represents the complex frequency resolved bulk photoconductivity, is the penetration depth of the impinging photon flux, is the electron charge, is the average scattering time, * is the effective mass and ℎ is the sheet charge carrier density in units of cm -2 (i.e. ℎ = [1 − ] ℎ ; where and ℎ are the reflectivity and pump photon flux respectively). From the best Drude fits to the data we can directly obtain values for the sheet carrier density (by considering an optical effective mass of 0.16 [8]), ℎ , of 7.97·10 16 , 7.55·10 16 and 7.08·10 16 m -2 and averaged scattering times, , of 95, 123 and 141 fs respectively for the analysed pump-probe delays of 60, 460 and 1260 ps. These figures show an improvement inscattering rate upon pump-probe delay that has been previously interpreted by some authors as a diffusion process of photo-generated charge carriers from near the surface towards the bulk[20]. In this respect, an increase in scattering rate is linked with a reduction in bulk charge carrier density of the electron-hole plasma as modulated by a time dependent penetration depth. Then, in eq. 2 should be defined as ( ) (seeFigure 1bwhere ( ) represents the distance from the sample´s surface where the initial carried density amplitude drops a value of 1/e ). ( probed time window (see Fig. S2 in Supplemental Material [26]), we can find a numerical solution for the effective time dependent thickness ( ) of the expanding electronhole plasma following the expression ( , ) = which corresponds to the photo-injected density of charge-carriers.From the results shown inFigure 1a, we retrieve charge carrier concentrations of(2.0±0.3)·10 17 , (6.6±1.4)·10 16 and (3.5+0.5)·10 16 cm -3 and charge carrier mobilities of 640 ± 22, 831 ± 43 and 956 ± 38 cm 2 V -1 s -1 for the pump-probe delays of 60, 460 and 1260 ps respectively (notice here that mobility is defined as [ ] = · [ ] * ⁄ ). Figure 1 . 1(a) Complex sheet photoconductivity at 300 K and under 387.5 nm of pump excitation with a fluence of 10.58 μJ/cm 2 at different pump-probe delays (60, 460 and 1260 ps). Filled and open symbols refer to the real and imaginary components of the sheet photoconductivity. Solid and dashed lines correspond to the best fit by the Drude model for the real and imaginary components respectively. (b) Numerical solution for the effective thickness of the photo-generated electron-hole gas for the considered pump-probe delays. The effective thickness for the photo-excited slab is defined as the 1/e drop from the carrier density near the air-solid interface. Figure 2b summarizes our findings as open diamonds regarding charge carrier mobility vs injection level. From the figure, it is clear that injection level modulates charge carrier mobility in the sample for carrier concentrations above ~10 16 cm -3 . In Figure 2 . 2(a) Scattering time as function of injection level. Different symbols correspond to different fluences as indicated. Solid line is the best fit to a hyperbolic function as described in the text. (b) Room temperature charge-carrier mobility as function of injection level. Open diamonds refer to experimental points retrieved from TRTS. Black solid dots correspond to the charge-carrier mobility measured by Krausse and Dännhauser [3,4]. Solid dashed line indicates the theoretical value predicted by Dorkel & Leturcq[5] while solid line corresponds to the model described byKlaassen [6,7]. Figure 3 . 3(a) Complex sheet photoconductivity for silicon measured with different temperatures at a fluence of 1.55 μJ/cm 2 and fixed pump-probe delay of 1 ns. Filled and open symbols refer to the real and imaginary components of the sheet photoconductivity. Solid and dashed lines represent the best fit of the Drude model for the real and imaginary parts respectively. (b) Temperature dependence on chargecarrier mobility retrieved from the signatures of the frequency-resolved photoconductivity. Open black symbols refer to experimental points while the solid dashed line indicates the carrier mobility predicted by Dorkel & Leturcq mm thick ZnTe crystal cut along the <110> axis. 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E. Rougieux, D. MacDonald, and A. Cuevas, Measurement and Parameterization of Carrier Mobility Sum in Silicon as a Function of Doping, Temperature and Injection Level, IEEE J Photovolt 4, 560 (2014). Contactless Determination of Current-Voltage Characteristics and Minority-Carrier Lifetimes in Semiconductors from Quasi-Steady-State Photoconductance Data. R A Sinton, A Cuevas, Appl Phys Lett. 692510R. A. Sinton and A. Cuevas, Contactless Determination of Current-Voltage Characteristics and Minority-Carrier Lifetimes in Semiconductors from Quasi-Steady-State Photoconductance Data, Appl Phys Lett 69, 2510 (1996). Quasi-Steady-State Photoconductance, a New Method for Solar Cell Material and Device Characterization. R A Sinton, S Consulting, S Jose, A Cuevas, M Stuckings, Conference Record of the Twenty Fifth IEEE Photovoltaic Specialists Conference. R. A. Sinton, S. Consulting, S. Jose, A. Cuevas, and M. Stuckings, Quasi-Steady-State Photoconductance, a New Method for Solar Cell Material and Device Characterization, in Conference Record of the Twenty Fifth IEEE Photovoltaic Specialists Conference (1996), pp. 457-460. A Comparison between Quasi-Steady State and Transient Photoconductance Lifetimes in Silicon Ingots: Simulations and Measurements. M Goodarzi, R Sinton, D Chung, B Mitchell, T Trupke, D Macdonald, IEEE 44th Photovoltaic Specialist Conference. PVSCM. Goodarzi, R. Sinton, D. Chung, B. Mitchell, T. Trupke, and D. Macdonald, A Comparison between Quasi-Steady State and Transient Photoconductance Lifetimes in Silicon Ingots: Simulations and Measurements, in IEEE 44th Photovoltaic Specialist Conference (PVSC) (2017), pp. 2707-2710. Contactless Determination of the Carrier Mobility Sum in Silicon Wafers Using Combined Photoluminescence and Photoconductance Measurements. Z Hameiri, F Rougieux, R Sinton, T Trupke, Appl Phys Lett. 104Z. Hameiri, F. Rougieux, R. Sinton, and T. Trupke, Contactless Determination of the Carrier Mobility Sum in Silicon Wafers Using Combined Photoluminescence and Photoconductance Measurements, Appl Phys Lett 104, (2014). Carrier Dynamics in Semiconductors Studied with Time-Resolved Terahertz Spectroscopy. R Ulbricht, E Hendry, J Shan, T F Heinz, M Bonn, Rev Mod Phys. 83543R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, Carrier Dynamics in Semiconductors Studied with Time-Resolved Terahertz Spectroscopy, Rev Mod Phys 83, 543 (2011). Terahertz Spectroscopy of Nanomaterials: A Close Look at Charge-Carrier Transport. P Kužel, H Němec, Adv Opt Mater. 81P. Kužel and H. Němec, Terahertz Spectroscopy of Nanomaterials: A Close Look at Charge- Carrier Transport, Adv Opt Mater 8, 1 (2020). Exploring Dynamics in the Far-Infrared with Terahertz Spectroscopy. C A Schmuttenmaer, Chem Rev. 1041759C. A. Schmuttenmaer, Exploring Dynamics in the Far-Infrared with Terahertz Spectroscopy, Chem Rev 104, 1759 (2004). Temperature and Carrier-Density Dependence of Electron-Hole Scattering in Silicon Investigated by Optical-Pump Terahertz-Probe Spectroscopy. T Terashige, H Yada, Y Matsui, T Miyamoto, N Kida, H Okamoto, Phys Rev B Condens Matter Mater Phys. 91T. Terashige, H. Yada, Y. Matsui, T. Miyamoto, N. Kida, and H. Okamoto, Temperature and Carrier-Density Dependence of Electron-Hole Scattering in Silicon Investigated by Optical- Pump Terahertz-Probe Spectroscopy, Phys Rev B Condens Matter Mater Phys 91, (2015). Reduction of Carrier Mobility in Semiconductors Caused by Charge-Charge Interactions. E Hendry, M Koeberg, J Pijpers, M Bonn, Phys Rev B Condens Matter Mater Phys. 754E. Hendry, M. Koeberg, J. Pijpers, and M. Bonn, Reduction of Carrier Mobility in Semiconductors Caused by Charge-Charge Interactions, Phys Rev B Condens Matter Mater Phys 75, 4 (2007). Transient Photoconductivity in GaAs as Measured by Time-Resolved Terahertz Spectroscopy. 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Trupke, Contactless Determination of the Carrier Mobility Sum in Silicon Wafers Using Combined Photoluminescence and Photoconductance Measurements, Appl Phys Lett 104, (2014). Exciton Mott Transition in Si Revealed by Terahertz Spectroscopy. T Suzuki, R Shimano, Phys Rev Lett. 1091T. Suzuki and R. Shimano, Exciton Mott Transition in Si Revealed by Terahertz Spectroscopy, Phys Rev Lett 109, 1 (2012). Time-Resolved Formation of Excitons and Electron-Hole Droplets in Si Studied Using Terahertz Spectroscopy. T Suzuki, R Shimano, Phys Rev Lett. 1031T. Suzuki and R. Shimano, Time-Resolved Formation of Excitons and Electron-Hole Droplets in Si Studied Using Terahertz Spectroscopy, Phys Rev Lett 103, 1 (2009). Cooling Dynamics of Photoexcited Carriers in Si Studied Using Optical Pump and Terahertz Probe Spectroscopy. T Suzuki, R Shimano, Phys Rev B Condens Matter Mater Phys. 83T. Suzuki and R. 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[]
[ "F 3 Net: Fusion, Feedback and Focus for Salient Object Detection", "F 3 Net: Fusion, Feedback and Focus for Salient Object Detection" ]
[ "Jun Wei [email protected] \nKey Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS)\nInstitute of Computing Technology\nCAS\n100190BeijingChina\n\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "Shuhui Wang [email protected] \nKey Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS)\nInstitute of Computing Technology\nCAS\n100190BeijingChina\n", "Qingming Huang [email protected] \nKey Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS)\nInstitute of Computing Technology\nCAS\n100190BeijingChina\n\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n" ]
[ "Key Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS)\nInstitute of Computing Technology\nCAS\n100190BeijingChina", "University of Chinese Academy of Sciences\n100049BeijingChina", "Key Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS)\nInstitute of Computing Technology\nCAS\n100190BeijingChina", "Key Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS)\nInstitute of Computing Technology\nCAS\n100190BeijingChina", "University of Chinese Academy of Sciences\n100049BeijingChina" ]
[]
Most of existing salient object detection models have achieved great progress by aggregating multi-level features extracted from convolutional neural networks. However, because of the different receptive fields of different convolutional layers, there exists big differences between features generated by these layers. Common feature fusion strategies (addition or concatenation) ignore these differences and may cause suboptimal solutions. In this paper, we propose the F 3 Net to solve above problem, which mainly consists of cross feature module (CFM) and cascaded feedback decoder (CFD) trained by minimizing a new pixel position aware loss (PPA). Specifically, CFM aims to selectively aggregate multilevel features. Different from addition and concatenation, CFM adaptively selects complementary components from input features before fusion, which can effectively avoid introducing too much redundant information that may destroy the original features. Besides, CFD adopts a multi-stage feedback mechanism, where features closed to supervision will be introduced to the output of previous layers to supplement them and eliminate the differences between features. These refined features will go through multiple similar iterations before generating the final saliency maps. Furthermore, different from binary cross entropy, the proposed PPA loss doesnt treat pixels equally, which can synthesize the local structure information of a pixel to guide the network to focus more on local details. Hard pixels from boundaries or error-prone parts will be given more attention to emphasize their importance. F 3 Net is able to segment salient object regions accurately and provide clear local details. Comprehensive experiments on five benchmark datasets demonstrate that F 3 Net outperforms state-of-the-art approaches on six evaluation metrics. Code will be released at https://github.com/weijun88/F3Net.
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[ "https://arxiv.org/pdf/1911.11445v1.pdf" ]
208,291,093
1911.11445
b86cf0116da79318b07062b1d3456711fdf808aa
F 3 Net: Fusion, Feedback and Focus for Salient Object Detection Jun Wei [email protected] Key Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS) Institute of Computing Technology CAS 100190BeijingChina University of Chinese Academy of Sciences 100049BeijingChina Shuhui Wang [email protected] Key Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS) Institute of Computing Technology CAS 100190BeijingChina Qingming Huang [email protected] Key Laboratory of Intelligent Information Processing of Chinese Academy of Sciences (CAS) Institute of Computing Technology CAS 100190BeijingChina University of Chinese Academy of Sciences 100049BeijingChina F 3 Net: Fusion, Feedback and Focus for Salient Object Detection Most of existing salient object detection models have achieved great progress by aggregating multi-level features extracted from convolutional neural networks. However, because of the different receptive fields of different convolutional layers, there exists big differences between features generated by these layers. Common feature fusion strategies (addition or concatenation) ignore these differences and may cause suboptimal solutions. In this paper, we propose the F 3 Net to solve above problem, which mainly consists of cross feature module (CFM) and cascaded feedback decoder (CFD) trained by minimizing a new pixel position aware loss (PPA). Specifically, CFM aims to selectively aggregate multilevel features. Different from addition and concatenation, CFM adaptively selects complementary components from input features before fusion, which can effectively avoid introducing too much redundant information that may destroy the original features. Besides, CFD adopts a multi-stage feedback mechanism, where features closed to supervision will be introduced to the output of previous layers to supplement them and eliminate the differences between features. These refined features will go through multiple similar iterations before generating the final saliency maps. Furthermore, different from binary cross entropy, the proposed PPA loss doesnt treat pixels equally, which can synthesize the local structure information of a pixel to guide the network to focus more on local details. Hard pixels from boundaries or error-prone parts will be given more attention to emphasize their importance. F 3 Net is able to segment salient object regions accurately and provide clear local details. Comprehensive experiments on five benchmark datasets demonstrate that F 3 Net outperforms state-of-the-art approaches on six evaluation metrics. Code will be released at https://github.com/weijun88/F3Net. Introduction Salient object detection (SOD) aims to estimate the visual significant regions of images or videos and often serves as the pre-processing step for many downstream vision tasks (Wang et al. 2019a). Earlier SOD algorithms mainly rely on heuristic priors (e.g., color, texture and contrast) to generate saliency maps. However, these hand-craft features can hardly capture high-level semantic relations and context information, thus they are not robust enough to complex scenarios. Recently, convolutional neural networks (CNNs) have demonstrated its powerful feature extraction capability in visual feature representation (Zhang et al. 2018c;Szegedy et al. 2015;Huang et al. 2017;Hu, Shen, and Sun 2018). Many CNNs-based models Qin et al. 2019;Chen et al. 2018;Wu, Su, and Huang 2019;Li et al. 2018;Zhang et al. 2018b;Feng, Lu, and Ding 2019;Wang et al. 2017b;Zhang et al. 2017) have achieved remarkable progress and pushed the performance of SOD to a new level. These models adopt the encoder-decoder architecture, which is simple in structure and computationally efficient. The encoder usually is made up of a pretrained classification model (e.g. ResNet (He et al. 2016) and VGG (Simonyan and Zisserman 2015)), which can extract multiple features of different semantic levels and resolutions. In the decoder, extracted features are combined to generate saliency maps. However, there still remains two big challenges in accurate SOD. First, features of different levels have different distribution characteristics. High level features have rich se-mantics but lack accurate location information. Low level features have rich details but full of background noises. To generate better saliency maps, multi-level features are combined. However, without delicate control of the information flow in the model, some redundant features, including noises from low level layers and coarse boundaries from high level layers will pass in and possibly result in performance degradation. Second, most of existing models use binary cross entropy that treats all pixels equally. Intuitively, different pixels deserve different weights, e.g., pixels at the boundary are more discriminative and should be attached with more importance. Various boundary losses (Qin et al. 2019;Feng, Lu, and Ding 2019) have been proposed to enhance the boundary detection accuracy, but considering only the boundary pixels is not comprehensive enough, since there are lots of pixels near the boundaries prone to wrong predictions. These pixels are also important and should be assigned with larger weights. In consequence, it is essential to design a mechanism to reduce the impact of inconsistency between features of different levels and assign larger weights to those truly important pixels. To address above challenges, we proposed a novel SOD framework, named F 3 Net, which achieves remarkable performance in producing high quality saliency maps. First, to mitigate the discrepancy between features, we design cross feature module (CFM), which fuses features of different levels by element-wise multiplication. Different from addition and concatenation, CFM takes a selective fusion strategy, where redundant information will be suppressed to avoid the contamination between features and important features will complement each other. Compared with traditional fusion methods, CFM is able to remove background noises and sharpen boundaries, as shown in Fig. 1. Second, due to downsampling, high level features may suffer from information loss and distortion, which can not be solved by CFM. Therefore, we develop the cascaded feedback decoder (CFD) to refine these features iteratively. CFD contains multiple sub-decoders, each of which contains both bottom-up and top-down processes. For bottom-up process, multi-level features are aggregated by CFM gradually. For top-down process, aggregated features are feedback into previous features to refine them. Third, we propose the pixel position aware loss (PPA) to improve the commonly used binary cross entropy loss which treats all pixels equally. In fact, pixels located at boundaries or elongated areas are more difficult and discriminating. Paying more attention to these hard pixels can further enhance model generalization. PPA loss assigns different weights to different pixels, which extends binary cross entropy. The weight of each pixel is determined by its surrounding pixels. Hard pixels will get larger weights and easy pixels will get smaller ones. To demonstrate the performance of F 3 Net, we report experiment results on five popular SOD datasets and visualize some saliency maps. We conduct a series of ablation studies to evaluate the effect of each module. Quantitative indicators and visual results show that F 3 Net can obtained significantly better local details and improved saliency maps. Codes has been released. In short, our main contributions can be summarized as follows: • We introduce the cross feature module to fuse features of different levels, which is able to extract the shared parts between features and suppress each other's background noises and complement each other's missing parts. • We propose the cascaded feedback decoder for SOD, which can feedback features of both high resolutions and high semantics to previous ones to correct and refine them for better saliency maps generation. • We design pixel position aware loss to assign different weights to different positions. It can better mine the structure information contained in the features and help the network focus more on detail regions. • Experimental results demonstrate that the proposed model F 3 Net achieves the state-of-the-art performance on five datasets in terms of six metrics, which proves the effectiveness and superiority of the proposed method. Related Work Early SOD methods mainly rely on intrinsic cues, such as color contrast (Cheng et al. 2015), texture (Yan et al. 2013) and center prior (Jiang and Davis 2013) to extract saliency maps, which mainly focus on low-level information and ignore rich contextual semantic information. Recently, CNNs has been used to extract multi-level features from original images and aggregate the extracted features to produce saliency maps. Among these methods, ) introduced short connections in fully convolutional networks (Long, Shelhamer, and Darrell 2015) to integrate features from different layers. (Deng et al. 2018) and (Wang et al. 2017b) adopted an iterative strategy to refine the saliency maps step-bystep, using features both from deep layers and shallow layers. (Liu, Han, and Yang 2018) proposed to generate attention over the context regions for each pixel, which can help suppress the interference of background noises. (Chen et al. 2018) and (Zhang et al. 2018b) used attention-guided network to select and extract supplementary features and integrate them to enhance saliency maps. (Qin et al. 2019) designed hybrid loss to make full use of boundary information and (Feng, Lu, and Ding 2019) used a two-branch network to simultaneously predict the contours and saliency maps. (Zhang et al. 2018a) designed a bi-directional message passing model for better feature selection and integration. (Liu et al. 2019) utilized simple pooling and feature aggregation module to build fast and high performance model. (Zhao and Wu 2019) introduced the channel-wise attention and spatial attention to extract valuable features and suppress background noise. However, the discrepancy between features of different levels has not been comprehensively studied. How to design more effective fusion strategies to reduce this discrepancy has become an important problem in SOD. In addition, apart from boundaries, there are lots of hard pixels deserving more attention. Increasing their weights in loss function can further improve the discriminating ability. Based on above mentioned problems, we design F 3 Net to generate saliency maps accurately and efficiently. Proposed Method We design cross feature module to selectively integrate features, which can prevent the introduction of redundant features. To refine the saliency maps, we propose a cascaded feedback decoder to refine multi-level features by multiple iterations. To guide the network to focus more on local details, we introduce pixel position aware loss which assigns different weights to different pixels. See Fig.2 for details. Cross Feature Module We propose cross feature module (CFM) to refine both high level features f h ∈ R H×W ×C and low level features f l ∈ R H×W ×C . f l preserves rich details as well as background noises, due to the restriction of the receptive field. These features have clear boundaries, which are important to generate accurate saliency maps. In contrast, f h is coarse in boundaries, because of multiple downsamplings. Despite of losing too much detailed information, f h still has consistent semantics and clear background. There exists big statistical discrepancy between these two kinds of features. Some examples have been shown in Fig. 1. CFM performs feature crossing to mitigate the discrepancy between features. It firstly extracts the common parts between f l and f h by element-wise multiplication and then combines them with original f l and f h respectively by element-wise addition. Compared with direct addition or concatenation employed in existing study, CFM avoids redundant information introduced to f l and f h , which may "pollute" the original features and bring adverse effect to the generation of saliency maps. By multiple feature crossings, f l and f h will gradually absorb useful information from each other to complement themselves, i.e., noises of f l will be suppressed and boundaries of f h will be sharpened. Specifically, CFM contains two branches, one for f l and the other for f h , as shown in Fig. 2. At first, one 3x3 convolutional layer is applied to f l and f h respectively to adapt them for follow-up processing. Then these features are transformed and fused by multiplication. The fused features share the properties of both f l and f h , i.e., clear boundaries and consistent semantics. Finally, the fused features will be added to the original f l and f h for refine representations. The whole process could be shown as follows. f l = f l + M l (G l (f l ) * G h (f h )) (1) f h = f h + M h (G l (f l ) * G h (f h )) (2) where each of M h (·), M l (·), G h (·), G l (·) is the combination of convolution, batchnorm and relu. After getting the refined features, 3x3 convolution is applied to restore the original dimensions. The whole module presents a completely symmetric structure, where f l embeds its details to f h and f h filters the background noises of f l . Cascaded Feedback Decoder Cascaded feedback decoder (CFD) is built upon CFM which refines the multi-level features and generate saliency maps iteratively. For SOD, traditional methods aim to directly aggregate multi-level features to produce the final saliency maps. In fact, features of different levels may have missing or redundant parts because of downsamplings and noises. Even with CFM, these parts are still difficult to identify and restore, which may hurt the final performance. Considering the output saliency map is relatively complete and approximate to ground truth, we propose to propagate the features of the last convolution layer back to features of previous layers to correct and refine them. Fig. 2 shows the architecture of CFD which contains multiple decoders. Each decoder consists of two processes, i.e., bottom-up and top-down. For bottom-up process, features are gradually aggregated by CFM from high level to low level. The aggregated features will be supervised and produce a coarse saliency map. For top-down process, features aggregated by last process are directly downsampled and added to previous multi-level features exported by CFM to refine them. These refined features will be sent to the next decoder to go through the same processes. In fact, inside CFD, two processes of multiple decoders are linked one-byone and form a grid net. Multi-level features are flowing and refined in this net iteratively. At last, these features will be complete enough to generate finer saliency maps. Specifically, we build CFD on ResNet-50 (He et al. 2016), a widely used backbone in SOD tasks. For an input image with size HxW , ResNet-50 will extract its features at five levels, denoted as {f i |i = 1, ..., 5} with resolu- tions [ H 2 i−1 , W 2 i−1 ] . Because low level features bring too much computational cost but little performance improvement (Wu, Su, and Huang 2019), we only use features of the last four levels f 2 , f 3 , f 4 , f 5 , which have lower resolutions and cost less computation. The whole process of CFD can be formulated as Alg. 1, where De i (·) is the i-th sub-decoder and Ds i (·) means the downsampling operation. Algorithm 1: Cascaded Feedback Decoder Input: multi-level features {f i |i = 2, ..., 5} iteration times N Output: saliency map {m i |i = 1, ..., N } 1 f2, f3, f4, f5, p ← De1(f2, f3, f4, f5); 2 m1 ← Conv1(p); 3 for i = 2; i ≤ N ; i ← i + 1 do 4 p2, p3, p4, p5 ← Ds2(p), Ds3(p), Ds4(p), Ds5(p); 5 f2, f3, f4, f5, p ← Dei(f2+p2, f3+p3, f4+p4, f5+p5); 6 mi ← Convi(p); 7 end 8 return {mi|i = 1, ..., N }; Pixel Position Aware Loss In SOD, binary cross entropy (BCE) is the most widely used loss function. However, BCE loss has three drawbacks. First, it calculates the loss for each pixel independently and ignores the global structure of the image. Second, in pictures where the background is dominant, loss of foreground pixels will be diluted. Third, it treats all pixels equally. In fact, pixels located on cluttered or elongated areas (e.g., pole and horn) are prone to wrong predictions and deserve more attention and pixels located areas, like sky and grass, deserve less attention. So we propose a weighted binary cross en- tropy (wBCE) loss as shown in Eq. 3 L s wbce = − H i=1 W j=1 (1+γαij) 1 l=0 1(g s ij = l)logPr(p s ij = l|Ψ) H i=1 W j=1 γαij(3) where 1(·) is the indicator function and γ is a hyperparameter. The notation l ∈ {0, 1} indicates two kinds of the labels. p s ij and g s ij are prediction and ground truth of the pixel at location (i, j) in an image. Ψ represents all the parameters of the model and Pr(p s i,j = l|Ψ) denotes the predicted probability. In L s wbce , each pixel will be assigned with a weight α. Hard pixel corresponds to larger α and simple pixel will be assigned a smaller one. α could be regarded as the indicator of pixel importance, which is calculated according to the difference between the center pixel and its surroundings, Eq. 4. where A ij represents the area that surrounds the pixel (i, j). For all pixels, α s ij ∈ [0, 1]. If α s ij is large, pixel at (i, j) is very different from its surroundings. So it is an important pixel (e.g., edge or hole) and deserves more attention. On the contrary, if α s ij is small, we think it is a plain pixel and deserve less attention. Fig. 3 has shown some examples. Compared with BCE, L s wbce pays more attention hard pixels. In addition, local structure information has been encoded into L s wbce , which may help the model focus on a larger receptive field rather than on a single pixel. To further make the network focus on global structure, we introduce weighted IoU (wIoU) loss, as shown in Eq. 5. L s wiou = 1 − H i=1 W j=1 (gt s ij * p s ij ) * (1 + γα s ij ) H i=1 W j=1 (gt s ij + p s ij − gt s ij * p s ij ) * (1 + γα s ij )(5) IoU loss has been widely used in image segmentation (Rahman and Wang 2016). It aims to optimize the global structure instead of focusing on single pixel and it is not affected by the unbalanced distribution. Recently, it has been introduced into SOD (Qin et al. 2019) to make up for the deficiency of BCE. But it still treats all pixels equally and ignores the difference between pixels. Different from IoU loss, our wIoU loss assigns more weights to hard pixels to emphasize their importance. Based on above discussion, the pixel position aware loss is shown in Eq. 6. It synthesizes local structure information to generate different weights for all pixels and introduce both pixel restriction (L s wbce ) and global restriction (L s wiou ), which can better guide the network learning and produce Each sub-decoder in CFD corresponds to one L s ppa . Besides, multi-level supervision (MLS) is added as an auxiliary loss to facilitate sufficient training, as shown in Fig. 2. Given N sub-decoders in CFD and M levels in total, the whole loss is defined in Eq. 7 L s = 1 N N i=1 L si ppa + 5 j=2 1 2 j−1 L sj ppa (7) The first item corresponds to the mean of all sub-decoders' loss and the second corresponds to the weighted sum of auxiliary loss where high level loss has smaller weight because of its larger error. Experiments Datasets and Evaluation Metrics The performance of F 3 Net is evaluated on five popular datasets, including ECSSD (Yan et al. 2013) with 1000 images, PASCAL-S (Li et al. 2014) with 850 images, DUT-OMRON (Yang et al. 2013) with 5168 images, HKU-IS (Li and Yu 2015) with 4,447 images and DUTS (Wang et al. 2017a) with 15,572 images. All datasets are human-labeled with pixel-wise ground-truth for quantitative evaluations. DUTS is currently the largest SOD dataset, which are divided into 10,553 training images (DUTS-TR) and 5,019 testing images (DUTS-TE). We follow (Wu, Su, and Huang 2019;Qin et al. 2019) to use DUTS-TR as the training dataset and others as testing datasets. In addition, six metrics are used to evaluate the performance of F 3 Net and existing state-of-the-art methods. The first metric is the mean absolute error (MAE), as shown in Eq. 8, which is widely adopted in Liu, Han, and Yang 2018). Mean F-measure (mF ), structural similarity measure (S α , α = 0.5) (Fan et al. 2017) and E-measure (E ξ ) (Fan et al. 2018) are also widely used to evaluate salient maps. In addition, precision-recall (PR) and F-measure curves are drawn to show the whole performance. M AE = 1 H × W H i=1 W j=1 |P (i, j) − G(i, j)| (8) where P is the predicted map and G is the ground truth. Implementation Details DUTS-TR is used to train F 3 Net and other above mentioned datasets are used to evaluate F 3 Net. For data augmentation, we use horizontal flip, random crop and multi-scale input images. ResNet-50 (He et al. 2016), pre-trained on Ima-geNet, is used as the backbone network. Maximum learning rate is set to 0.005 for ResNet-50 backbone and 0.05 for other parts. Warm-up and linear decay strategies are used to adjust the learning rate. The whole network is trained end-to-end, using stochastic gradient descent (SGD). Momentum and weight decay are set to 0.9 and 0.0005, respectively. Batchsize is set to 32 and maximum epoch is set to 32. We use Pytorch 1.3 to implement our model. An RTX 2080Ti GPU is used for acceleration. During testing, we resized each image to 352 x 352 and then feed it to F 3 Net to predict saliency maps without any post-processing. Codes has been released at https://github.com/weijun88/F3Net. Ablation Studies Before analyzing the influence of each module, there are two hyper parameters (i.e., γ and N ) to be determined. γ is used in PPA loss to adjust the proportion of hard pixels. Tab.3 lists the scores of M AE, mF , S α and E ξ when γ is given different values. As can be seen, when γ equals 5, these indicators reach highest scores. In addition, N represents the number of sub-decoders in CFD. We increase N gradually from 1 to 4 and measure the corresponding scores of above metrics, as shown in Tab. 4. When N =2, the model achieves the best performance. Both of these experiments are conducted on DUT-OMRON and DUTS. To investigate the importance of different modules in F 3 Net, we conduct a series of controlled experiments on DUTS, as shown in Tab.2. First, we test the effect of different loss functions, inlcuding BCE, IoU and PPA. Among them, PPA loss achieves the best performance on three evaluation metrics. Furthermore, we keep adding the multi-level supervision, cross feature module and cascaded feedback decoder to evaluate their performance. As we can see, all these modules boost the model performance. When these modules are combined, we can get the best SOD results. It demonstrates that all components are necessary for the proposed framework. Comparison with State-of-the-arts Quantitative Comparison. To demonstrate the effectiveness of the proposed F 3 Net, we compare it against 12 stateof-the-art SOD algorithms, including AFNet (Feng, Lu, and 1: Performance comparison with 12 state-of-the-art methods over 5 datasets. MAE (smaller is better), mean Fmeasure (mF , larger is better), Smeasure (S α , larger is better) and Emeasure (E ξ , larger is better) are used to measure the model performance. The best results are highlighted in bold. Our model ranks first on all datasets and metrics. 046 .886 .917 .913 .075 .798 .849 .833 .051 .759 .869 .862 .043 .870 .904 .936 .065 .717 .832 .841 BMPM(CVPR2018) .044 .894 .911 .914 .073 .803 .840 .838 .049 .762 .861 .859 .039 .875 .906 .937 .063 .698 .809 .839 DGRL(CVPR2018) .043 .903 .906 .917 .074 .807 .834 .836 .051 .764 .846 .863 .037 .881 .896 .941 .063 .709 .810 .843 PAGE(CVPR2019) .042 .906 .912 .920 .077 .810 .835 .841 .052 .777 .854 .869 .037 .882 .903 .940 .062 .736 .824 .853 AFNet(CVPR2019) .042 .908 .913 .918 .070 .821 .844 .846 .046 .792 .867 .879 .036 .888 .905 .942 .057 .738 .826 .853 TDBU(CVPR2019) .041 .880 .918 .922 .071 .779 .844 .852 .048 .767 .865 .879 .038 .878 .907 .942 .061 .739 .837 .854 PoolNet(CVPR2019) .039 .915 .921 .924 .074 .822 .845 .850 .040 .809 .883 .889 .032 .899 .916 .949 .055 .747 .835 .863 BASNet(CVPR2019) .037 .880 .916 .921 .076 .775 .832 .847 .048 .791 .866 .884 .032 .895 .909 .946 .056 .756 .836 .869 CPD-R(CVPR2019) .037 .917 .918 .925 .072 .824 .842 .849 .043 .805 .869 .886 .034 .891 .905 .944 .056 .747 .825 .866 F 3 Net(ours . For fair comparison, we use all saliency maps provided by the authors and evaluate them with the same code. As shown in Tab.1, our approach achieves the best scores across five datasets with respect to four metrics, compared with other counterparts. It demonstrates the superior performance of the proposed F 3 Net. In addition, Fig. 4 shows the precision-recall curves of above mentioned algorithms on five datasets, which can evaluate the holistic performance of models. From these curves, we can observe that F 3 Net consistently outperforms all other models under different thresholds, which means that our method have a good capability to detect salient regions as well as generate accurate saliency maps. Visual Comparison. In order to evaluate the proposed F 3 Net, we visualize some saliency maps produced by our model and other approaches in Fig. 5. We observe that the proposed method not only highlights the salient object regions clearly, but also well suppresses the background noises. It excels in dealing with various challenging scenarios, including cluttered backgrounds (row 2 and 6), small objects (row3), inverted reflection in water (row1) and occlusion (row 2). Compared with other counterparts, the saliency maps produced by our method are clearer and more accurate. Most importantly, our method achieves these results without any post-processing. Image Label Ours PiCA-R BMPM DGRL PAGE AFNet TDBU PoolNet BASNet CPD-R Figure 5: Visual comparison of the proposed model with nine state-of-the-art methods. Apparently, saliency maps produced by our model are clearer and more accurate than others and our results are more consistent with the ground truths. Table 2: Ablation study for different modules. BCE and IoU are two kinds of loss functions above mentioned. MSL means multi-level supervision. CFM and CFD are the main modules in F 3 Net. PPA is the proposed loss function. Conclusion In this paper, we propose a novel SOD framework named F 3 Net. First, considering the difference between features of different levels, we propose CFM to selectively integrate features, which prevents the improper influence of redundant features. To further get finer details, we introduce CFD to refine multi-level features iteratively with feedback mechanisms. Besides, we design PPA loss to pay more attention to hard pixels and guide the network focus more on error-prone parts. The whole framework demonstrates remarkable feature extraction capability, which makes it robust and effective in various challenging scenarios. Experimental results on five datasets demonstrate that F 3 Net outperforms state- of-the-art methods under six evaluation metrics. Figure 1 : 1Fusing features of different levels. (c) represents the low level features. (d) means the high level features. (e) is the fused features by F 3 Net. Clearly, the fused features have clear boundaries as well as few background noises. Figure 2 : 2An overview of proposed F 3 Net. ResNet-50 is used as the backbone encoder. Cross feature module (CFM) is used as the basic module to fuse features of different layers. Cascaded feedback decoder (CFD) contains multiple sub-decoders to feedback and refine multi-level features. Multi-level supervision (MLS) helps to ease the optimization of F 3 Net. Figure 3 : 3Some examples of the calculated weight α. (d) shows the superposition of original image and its corresponding α. We can see that pixels located at boundaries, elongated areas or holes, have larger α. Ding 2019), BASNet (Qin et al. 2019), CPD-R (Wu, Su, and Huang 2019), BMPM (Zhang et al. 2018a), R 3 Net (Deng et Figure 4 : 4Performance comparison with 12 state-of-the-art methods over 5 datasets. The first row shows comparison of precision-recall curves. The second row shows comparison of F-measure curves over different thresholds. As the figure shows, F 3 Net achieves the best performance on all datasets.al. 2018), PiCA-R (Liu, Han, and Yang 2018), DGRL (Wang et al. 2018), TDBU (Wang et al. 2019b), PoolNet (Liu et al. 2019), PAGE (Wang et al. 2019c), RAS (Chen et al. 2018) and C2SNet Table MAE mF Sα E ξ MAE mF Sα E ξ MAE mF Sα E ξ MAE mF Sα E ξ MAE mF Sα E ξAlgorithm ECSSD PASCAL-S DUTS-TE HKU-IS DUT-OMRON 1,000 images 850 images 5,019 images 4,447 images 5,168 images C2SNet(ECCV2018) .059 .853 .882 .906 .086 .761 .822 .835 .066 .710 .817 .841 .051 .839 .873 .919 .079 .664 .780 .817 RAS(ECCV2018) .055 .890 .894 .916 .102 .782 .792 .832 .060 .750 .838 .861 .045 .874 .888 .931 .063 .711 .812 .843 R 3 Net(IJCAI2018) .051 .883 .910 .914 .101 .775 .809 .824 .067 .716 .837 .827 .047 .853 .894 .921 .073 .690 .819 .814 PiCA-R(CVPR2018) . BCE IoU PPA MLS CFM CFD DUTS-TE MAE mF S α E ξ .051 .779 .861 .871 .047 .783 .864 .874 .045 .789 .867 .875 .043 .808 .872 .880 .040 .812 .875 .882 .036 .831 .884 .893 .035 .840 .888 .902 DUT-OMRON DUTS-TE MAE mF S α E ξ MAE mF S α E ξ γ=3 .058 .755 .835 .857 .038 .835 .888 .898 γ=4 .057 .758 .837 .859 .037 .837 .888 .900 γ=5 .053 .766 .838 .870 .035 .840 .888 .902 γ=6 .060 .752 .833 .855 .038 .834 .887 .897 Table 3 : 3Comparison with different γ. When γ = 5, the model achieves the best results.DUT-OMRON DUTS-TE MAE mF S α E ξ MAE mF S α E ξ N =1 .055 .760 .834 .866 .037 .838 .886 .897 N =2 .053 .766 .838 .870 .035 .840 .888 .902 N =3 .057 .762 .837 .867 .036 .837 .887 .900 N =4 .059 .758 .833 .863 .038 .835 .885 .896 Table 4 : 4The effect of sub-decoder number. When N = 2, the model achieves the best results. 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[ "https://github.com/weijun88/F3Net.", "https://github.com/weijun88/F3Net." ]
[ "The Largest Entry in the Inverse of a Vandermonde Matrix", "The Largest Entry in the Inverse of a Vandermonde Matrix" ]
[ "Carlo Sanna [email protected] \nDepartment of Mathematical Sciences Politecnico di Torino Corso Duca degli Abruzzi\nSchool of Computer Science\nUniversity of Waterloo\n24 10129, N2L 3G1Torino, WaterlooONItaly, Canada\n", "Jeffrey Shallit [email protected]@uwaterloo.ca \nDepartment of Mathematical Sciences Politecnico di Torino Corso Duca degli Abruzzi\nSchool of Computer Science\nUniversity of Waterloo\n24 10129, N2L 3G1Torino, WaterlooONItaly, Canada\n", "Shun Zhang \nDepartment of Mathematical Sciences Politecnico di Torino Corso Duca degli Abruzzi\nSchool of Computer Science\nUniversity of Waterloo\n24 10129, N2L 3G1Torino, WaterlooONItaly, Canada\n" ]
[ "Department of Mathematical Sciences Politecnico di Torino Corso Duca degli Abruzzi\nSchool of Computer Science\nUniversity of Waterloo\n24 10129, N2L 3G1Torino, WaterlooONItaly, Canada", "Department of Mathematical Sciences Politecnico di Torino Corso Duca degli Abruzzi\nSchool of Computer Science\nUniversity of Waterloo\n24 10129, N2L 3G1Torino, WaterlooONItaly, Canada", "Department of Mathematical Sciences Politecnico di Torino Corso Duca degli Abruzzi\nSchool of Computer Science\nUniversity of Waterloo\n24 10129, N2L 3G1Torino, WaterlooONItaly, Canada" ]
[]
We investigate the size of the largest entry (in absolute value) in the inverse of certain Vandermonde matrices. More precisely, for every real b > 1, let M b (n) be the maximum of the absolute values of the entries of the inverse of the n × n matrix [b ij ] 0≤i,j<n . We prove that lim n→+∞ M b (n) exists, and we provide some formulas for it. * C. Sanna is a member GNSAGA of the INdAM and of CrypTO, the Group of Cryptography and Number Theory of Politecnico di Torino.
10.1080/03081087.2021.1922337
[ "https://arxiv.org/pdf/2008.01012v1.pdf" ]
220,935,725
2008.01012
93a834e6dcfeccf807b742f20292fe948f2e16f9
The Largest Entry in the Inverse of a Vandermonde Matrix Aug 2020 Carlo Sanna [email protected] Department of Mathematical Sciences Politecnico di Torino Corso Duca degli Abruzzi School of Computer Science University of Waterloo 24 10129, N2L 3G1Torino, WaterlooONItaly, Canada Jeffrey Shallit [email protected]@uwaterloo.ca Department of Mathematical Sciences Politecnico di Torino Corso Duca degli Abruzzi School of Computer Science University of Waterloo 24 10129, N2L 3G1Torino, WaterlooONItaly, Canada Shun Zhang Department of Mathematical Sciences Politecnico di Torino Corso Duca degli Abruzzi School of Computer Science University of Waterloo 24 10129, N2L 3G1Torino, WaterlooONItaly, Canada The Largest Entry in the Inverse of a Vandermonde Matrix Aug 2020 We investigate the size of the largest entry (in absolute value) in the inverse of certain Vandermonde matrices. More precisely, for every real b > 1, let M b (n) be the maximum of the absolute values of the entries of the inverse of the n × n matrix [b ij ] 0≤i,j<n . We prove that lim n→+∞ M b (n) exists, and we provide some formulas for it. * C. Sanna is a member GNSAGA of the INdAM and of CrypTO, the Group of Cryptography and Number Theory of Politecnico di Torino. Introduction Let a = (a 0 , a 1 , . . . , a n−1 ) be a list of n real numbers. The classical Vandermonde matrix V (a) is defined as follows: V (a) :=      1 a 0 a 2 0 · · · a n−1 0 1 a 1 a 2 1 · · · a n−1 1 a n−1 a 2 n−1 · · · a n−1 n−1      . As is well-known, the Vandermonde matrix V (a) is invertible if and only if the a i are pairwise distinct. See, for example, [3]. In what follows, n is a positive integer and b > 1 is a fixed real number. Let us define the entries c i,j,n by [c i,j,n ] 0≤i,j<n = V (b 0 , b 1 , b 2 , . . . , b n−1 ) −1 , and let M b (n) = max 0≤i,j<n |c i,j,n |, the maximum of the absolute values of the entries of V (1, b, b 2 , . . . , b n−1 ) −1 . The size of the entries of inverses of Vandermonde matrices have been studied for a long time (e.g., [1]). Recently, in a paper by the first two authors and Daniel Kane [2], we needed to estimate M 2 (n), and we proved that M 2 (n) ≤ 34. In fact, even more is true: the limit lim n→∞ M 2 (n) exists and equals 3 i≥2 1 + 1 2 i −1 . = 5.19411992918 · · · . In this paper, we generalize this result, replacing 2 with any real number greater than 1. Our main results are as follows: Theorem 1. Let b > 1 and n 0 = ⌈log b (1 + 1 b )⌉. Then |c i,j,n | ≤ |c n 0 ,n 0 ,n | for i, j ≥ n 0 . Hence M b (n) ∈ {|c i,j,n | : 0 ≤ i, j ≤ n 0 }. Preliminaries For every real number x, and for all integers 0 ≤ i, j < n, let us define the power sum σ i,j,n (x) := 0≤h 1 <···<h i <n h 1 ,...,h i =j x h 1 +···+h i . The following lemma will be useful in later arguments. Lemma 4. Let i, j, n be integers with 0 ≤ i < n, 0 ≤ j < n − 1, and let x be a positive real number. (a) If x > 1, then σ i,j,n (x) ≥ σ i,j,n+1 (x). (b) If x < 1, then σ i,j,n (x) ≤ σ i,j+1,n (x). Proof. We have σ i,j+1,n (x) − σ i,j,n (x) = (h 1 ,...,h i )∈S i,j,n x h 1 +···+h i − (h 1 ,...,h i )∈T i,j,n x h 1 +···+h i , where S i,j,n := {0 ≤ h 1 < · · · < h i < n : j ∈ {h 1 , . . . , h i }, j + 1 / ∈ {h 1 , . . . , h i }} and T i,j,n := {0 ≤ h 1 < · · · < h i < n : j / ∈ {h 1 , . . . , h i }, j + 1 ∈ {h 1 , . . . , h i }}. Now there is a bijection S i,j,n → T i,j,n given by (h 1 , . . . , h i ) → (h 1 , . . . , h i 0 −1 , h i 0 + 1, h i 0 +1 , . . . , h i ), where i 0 is the unique integer such that h i 0 = j. Hence, it follows easily that σ i,j,n (x) ≥ σ i,j+1,n (x) for x > 1, and σ i,j,n (x) ≤ σ i,j+1,n (x) for x < 1. Recall the following formula for the entries of the inverse of a Vandermonde matrix (see, e.g., [4, §1.2.3, Exercise 40]). Lemma 5. Let a 0 , . . . , a n−1 be pairwise distinct real numbers. If V (a 0 , a 1 , . . . , a n−1 ) = [c i,j ] 0≤i,j<n then c n−1,j X n−1 + c n−2,j X n−2 + · · · + c 0,j X 0 = 0≤i<n i =j X − a i a j − a i . For 0 ≤ i, j < n define π j,n := 0≤h<n h =j |b j − b h |. We now obtain a relationship between the entries of V (b 0 , b 1 , . . . , b n−1 ) −1 and σ i,j,n and π j,n . Lemma 6. Let V (b 0 , b 1 , . . . , b n−1 ) −1 = [c i,j,n ] 0≤i,j<n . Then |c i,j,n | = σ n−i−1,j,n π j,n(1) for 0 ≤ i, j < n. Proof. By Lemma 5, we have 0≤h<n h =j X − b h b j − b h = 0≤i<n c i,j,n X i . which in turn, by Vieta's formulas, gives c n−i−1,j,n = (−1) i     0≤h<n h =j 1 b j − b h     0≤h 1 <···<h i <n h 1 ,...,h i =j b h 1 +···+h i(2) for 0 ≤ i < n. The result now follows by the definitions of σ and π. Next, we obtain some inequalities for π. Lemma 7. Define n 0 = ⌈log b (1 + 1 b )⌉. Then π j,n ≤ π j+1,n for n 0 ≤ j < n. Proof. For 0 ≤ j < n − 1, we have π j+1,n := 0≤h<n h =j+1 |b j+1 − b h | = b n−1 0≤h<n h−1 =j |b j − b h−1 | = b n+j−1 − b n−2 b n−1 − b j π j,n . A quick computation shows that the inequality b n+j−1 − b n−2 b n−1 − b j ≥ 1 is equivalent to b j ≥ b n−1 + b n−2 b n−1 + 1 . Let n 0 be the minimum positive integer such that b n 0 ≥ 1 + 1 b . Then n 0 = ⌈log b (1 + 1 b )⌉. Hence, for n 0 ≤ j < n, we have b j ≥ 1 + 1 b > b n−1 + b n−2 b n−1 + 1 , so that π j,n ≤ π j+1,n for n 0 ≤ j < n. Finally, we have the easy Lemma 8. For 0 ≤ i, j < n we have c i,j,n = c j,i,n . Proof. V (b 0 , b 1 , . . . , b n−1 ) is a symmetric matrix, so its inverse is also. Proof of Theorem 1 Proof. Suppose i, j ≥ n 0 . Then |c i,j,n | = σ n−i−1,j,n π j,n (by (1)) ≤ σ n−i−1,n 0 ,n π j,n (by Lemma 4 (a)) ≤ σ n−i−1,n 0 ,n π n 0 ,n (by Lemma 7) = |c i,n 0 ,n | (by (1)), and so we get |c i,j,n | ≤ |c i,n 0 ,n |. But c i,n 0 ,n = c n 0 ,i,n (5) by Lemma 8. Make the substitutions n 0 for i and i for j in (4) to get |c n 0 ,i,n | ≤ |c n 0 ,n 0 ,n |. The result now follows by combining Eqs. (4), (5), and (6). Proof of Theorem 2 Proof. Since b ≥ τ , it follows that b ≥ 1 + 1/b. Hence in Theorem 1 we can take n 0 = 1, and this gives M b (n) ∈ {|c 0,0,n |, |c 1,0,n |, |c 0,1,n |, |c 1,1,n |}. However, by explicit calculation, we have σ n−1,1,n = b n(n−1)/2−1 σ n−2,1,n = b n(n−1)/2−1 + (n−1)(n−2)/2−1≤i≤n(n−1)/2−3 b i , so that σ n−1,1,n ≤ σ n−2,1,n . Hence |c 1,0,n | = |c 0,1,n | (by Lemma 8) = σ n−1,1,n π 1,n (by (1)) ≤ σ n−2,1,n π 1,n (by (7)) = |c 1,1,n | (by (1)), and the result follows. Proof of Theorem 3 Proof. We have |c i,j,n | = σ n−i−1,j,n π j,n = σ n−i−1,j,n (b) 0≤h<n h =j |b j − b h | = σ n−i−1,j,n (b) 0≤h<n h =j (b h · |b j−h − 1|) = σ n−i−1,j,n (b) b n(n−1)/2−j · 1 0≤h<n h =j |b j−h − 1| = σ i,j,n (b −1 ) 1 0≤h<n h =j |b j−h − 1| , where the equality σ n−i−1,j,n (b) b n(n−1)/2−j = σ i,j,n (b −1 ) arises from the one-to-one correspondence between the subsets of {0, 1, . . . , n − 1} − {j} of cardinality i and those of cardinality n − 1 − i. For x < 1 define σ i,j,∞ (x) = 0≤h 1 <···<h i <∞ h 1 ,...,h i =j 1 x h 1 +···+h i . Hence the limits From this theorem we can explicitly compute lim n→+∞ M b (n) for b ≥ τ . ℓ i,j := lim n→+∞ |c i,j,n | = lim n→+∞ σ i,j,n (b −1 ) 1 0≤h<n h =j |b j−h − 1| = σ i,j,∞ (b −1 ) 1≤s≤j 1 b s − 1 t≥1 1 1 − b −t(8) Corollary 9. Let α=2.324717957 be the real zero of the polynomial X 3 − 3X 2 + 2X − 1. (a) If b ≥ α, then lim n→∞ M b (n) = t≥1 (1 − b −t ) −1 . (b) If τ ≤ b ≤ α, then lim n→∞ M b (n) = b 2 −b+1 b(b−1) 2 t≥1 (1 − b −t ) −1 . Proof. From Theorem 2 we know that for b ≥ τ we have lim n→∞ M b (n) ∈ {ℓ 0,0 , ℓ 1,1 }. Now an easy calculation based on (8) shows that ℓ 0,0 = t≥1 (1 − b −t ) −1 ℓ 1,1 = b 2 − b + 1 b(b − 1) 2 t≥1 (1 − b −t ) −1 . By solving the equation b 2 −b+1 b(b−1) 2 = 1, we see that for b ≥ α we have ℓ 0,0 ≥ ℓ 1,1 , while if τ ≤ b ≤ α we have ℓ 1,1 ≥ ℓ 0,0 . This proves both parts of the claim. Remark 10. The quantity M b (n) converges rather slowly to its limit when b is close to 1. The following table gives some numerical estimates for M b (n). Final remarks We close with a conjecture we have been unable to prove. Conjecture 11. Let b > 1 and n 0 = ⌈log b (1 + 1 b )⌉. Then, for all sufficiently large n, we have M b (n) = |c i,i,n | for some i, 0 ≤ i ≤ n 0 . Theorem 2 . 2Let b ≥ τ = (1 + √ 5)/2 and n ≥ 2. Then M b (n) ∈ {|c 0,0,n |, |c 1,1,n |}. Theorem 3 . 3For all real b > 1 the limit lim n→∞ M b (n) exists. On inverses of Vandermonde and confluent Vandermonde matrix. W Gautschi, Numer. Mathematik. 4W. Gautschi. On inverses of Vandermonde and confluent Vandermonde matrix. Numer. Mathematik 4 (1962), 117-123. Waring's theorem for binary powers. D M Kane, C Sanna, J Shallit, Combinatorica. 39D. M. Kane, C. Sanna, and J. Shallit. Waring's theorem for binary powers. Combinatorica 39 (2019), 1335-1350. The Vandermonde matrix. A Klinger, Amer. Math. Monthly. 74A. Klinger. The Vandermonde matrix. Amer. Math. Monthly 74 (1967), 571-574. D E Knuth, The Art of Computer Programming. Addison-Wesley1Fundamental Algorithms. third editionD. E. Knuth. The Art of Computer Programming, Vol. 1, Fundamental Algorithms. Addison-Wesley, third edition, 1997.
[]
[ "GAN-Based Data Augmentation and Anonymization for Skin-Lesion Analysis: A Critical Review", "GAN-Based Data Augmentation and Anonymization for Skin-Lesion Analysis: A Critical Review" ]
[ "Alceu Bissoto \nInstitute of Computing (IC)\n\n", "Eduardo Valle \nSchool of Electrical and Computing Engineering (FEEC) RECOD Lab\nUniversity of Campinas (UNICAMP)\nBrazil\n", "Sandra Avila \nInstitute of Computing (IC)\n\n" ]
[ "Institute of Computing (IC)\n", "School of Electrical and Computing Engineering (FEEC) RECOD Lab\nUniversity of Campinas (UNICAMP)\nBrazil", "Institute of Computing (IC)\n" ]
[]
Despite the growing availability of high-quality public datasets, the lack of training samples is still one of the main challenges of deep-learning for skin lesion analysis. Generative Adversarial Networks (GANs) appear as an enticing alternative to alleviate the issue, by synthesizing samples indistinguishable from real images, with a plethora of works employing them for medical applications. Nevertheless, carefully designed experiments for skin-lesion diagnosis with GAN-based data augmentation show favorable results only on out-of-distribution test sets. For GAN-based data anonymization -where the synthetic images replace the real ones -favorable results also only appear for out-ofdistribution test sets. Because of the costs and risks associated with GAN usage, those results suggest caution in their adoption for medical applications.
10.1109/cvprw53098.2021.00204
[ "https://arxiv.org/pdf/2104.10603v1.pdf" ]
233,324,158
2104.10603
b6d60127144011a51780328e50266fce8fdcafb0
GAN-Based Data Augmentation and Anonymization for Skin-Lesion Analysis: A Critical Review Alceu Bissoto Institute of Computing (IC) Eduardo Valle School of Electrical and Computing Engineering (FEEC) RECOD Lab University of Campinas (UNICAMP) Brazil Sandra Avila Institute of Computing (IC) GAN-Based Data Augmentation and Anonymization for Skin-Lesion Analysis: A Critical Review Despite the growing availability of high-quality public datasets, the lack of training samples is still one of the main challenges of deep-learning for skin lesion analysis. Generative Adversarial Networks (GANs) appear as an enticing alternative to alleviate the issue, by synthesizing samples indistinguishable from real images, with a plethora of works employing them for medical applications. Nevertheless, carefully designed experiments for skin-lesion diagnosis with GAN-based data augmentation show favorable results only on out-of-distribution test sets. For GAN-based data anonymization -where the synthetic images replace the real ones -favorable results also only appear for out-ofdistribution test sets. Because of the costs and risks associated with GAN usage, those results suggest caution in their adoption for medical applications. Introduction The lack of training images is perhaps the main challenge faced by medical deep learning, deep skin-analysis being no exception. Although the availability of high-quality public datasets has mitigated the issue [4,10,22,26,33,41], the total number of annotated skin-lesion images available to researchers is still 1-2 orders of magnitude smaller than the size of general-purpose computer vision datasets [34,39]. More training images translate to better results, especially for deeper network architectures [27,42]. However, annotating skin-lesion images to increase datasets process is very costly, depending on the scarce time of medical specialists. Generative Adversarial Networks (GANs) [15] appear as an alternative to increase the amount of training data without incurring those costs. GANs aim to artificially synthesize samples that are indistinguishable from real images. They may be employed as a complement to traditional data augmentation [30], artificially increasing the amount of training samples. A plethora of existing works, which we discuss in our literature sur-vey, suggest applying them for that purpose, but -as we will show -obtaining reliable improvements from GANbased data augmentation is far from obvious: in our carefully designed experiments, GAN-based augmentation failed to reliably improve the classification performance of skin-lesion diagnosis, although we obtained good results for selected special cases. Another potential application of GANs is data anonymization, where the synthetic images are used to replace, instead of augment, the original training set. Here again, our results recommend caution, although they were promising for out-of-distribution tests. The main contribution of this work is a detailed study of the factors that can impact GAN-based augmentation, including GAN architectures, amount of real images used, proportion of real and synthetic images, and method for sampling synthetic images. The procedure adopted, from GAN checkpoint selection until classification network evaluation, can serve as a guideline to increase the reliability of future works using GAN-based augmentation. In addition, we carry out a systematic literature review, where we summarize the techniques used for GAN-based augmentation. In that review, we list issues in experimental design that may lead to overoptimistic results. The text follows the usual organization, with the literature review next, proposed approach in Section 3, followed by results in Section 4. We close the paper with a discussion of the main findings, the risks of using GAN-based augmentation in medical applications, as well as cautious avenues for continuing their use in this context. Literature review The review in this section started to seed an attempt to obtain reliable performance improvements from GAN-based data augmentation. Since several existing works reported measurable gains, we explored literature to understand which factors previous authors had tested in their experiments. For more details over different GAN methods, we direct the reader to surveys in the area [37,44]. However, as our experiments progressed without revealing reliable improvements, we returned to literature to subsidize a larger-scale experiment. Our review grew, in scope and formality, proportionally to our experimental ambitions. Although this review does not intend to be a formal metaanalysis, we took inspiration from Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) [28] to gather a representative sample of existing art. Our starting point was all GAN-related works published in the past ISIC Workshops. To that we added a database search, in Google Scholar, with the query "GAN generative adversarial networks medical image synthetic image data augmentation classification OR segmentation -NLP -temporal -tabular", which gave 251 results. Notice that we did not restrict our query to skin-lesion analysis but to all medical-image applications. We excluded from the sample all works outside our scope (i.e., no GAN data augmentation, no test on a medical dataset, no classification or segmentation), as well as works without experimental results. We excluded surveys and reviews from our sample as well. To get a manageable sample of papers to study, we kept our sample only works from top conferences (CVPR, NeurIPS, MICCAI), and their respective workshops, or published in journals of impact factor 3.0 or higher. For the same reason, we did not include unpublished preprints. The resulting collection of 18 works appears in Table 1. The table details which GANs each work evaluated, which deep-learning model was the target of the data augmentation, what was the medical application domain, which datasets were used on the evaluation, and which improvements the authors reported over which metrics. Depending on the domain, dataset, and task at hand, different families of GANs may be more appropriate, or conversely, completely unfeasible. Translation-based GANs, which include pix2pix [18], pix2pixHD [43], and SPADE [29], learn to translate between different types of images, e.g., from a segmentation mask into a new synthesized input, or from a non-contrast to a contrast CT-scan. Their main advantage is that adherence to the mask tends to improve their biological/medical coherence. In contrast, noise-based generation models, like DC-GAN [32] and PGAN [20] offer flexibility, the latter being able to generate high-resolution images. Noise-based generation may suffer, however, if the training sets are too small, requiring mitigating techniques such as patch extractions and traditional data augmentation. During the review, we took freehand notes about the experimental protocol of each paper, to understand which factors literature considered the most important to evaluate, and which varied the most among works. We found the proportion of real training images to images synthesized by the GAN to be one of the foremost factors, as well as the technique used to sample the images from the GAN. In addition to requiring an input to serve as a guide for generation, translation-based GANs tend to have a limited range of outputs, each input being able to create a single synthetic image (which is why most of them are marked as fixed for sampling in Table 1). For that same reason, works using them tend to employ a 1:1 proportion of real:synthesized images, although 1: < 1 ratios are also possible. Works using noise-based GANs tend to display more diverse choices of sampling and image ratios. In our freehand comments, we also noticed possible issues with experimental protocols that we wished to avoid in our large-scale experiments. The main issues we found amidst existing art were: (1) giving to the GAN-augmented models better data-access than to the baseline model, especially by choosing hyperparameters directly on the test set; (2) (hyper)optimizing the GAN-augmented models more thoroughly than the baseline models; (3) failing to use best current training practices on the baseline model, e.g. best available (conventional) data-augmentation, learning-rate choice, normalization, etc.; (4) ignoring performance fluctuations, e.g., by performing a single run, or by failing to report the deviation statistics. Not all works suffer from all those issues, of course, but we believe they may explain the discrepancy between the results we report next, and those found in current art. We decided to limit the scope of this review to works that apply GANs to medical applications, and not to the GANs themselves. We remark, however, that choosing the GAN model for medical-image augmentation puts stringent requirements on the model. First, the model must be able to generate high-resolution images, to accommodate the visual patterns that characterize medical images. For skinlesion analyses, the patterns that differentiate benign or malignant skin lesions are rather fine-grained, and state-of-theart classification networks have inputs from 224 × 224 to 1024×1024 pixels. Second, the model must able to generate class-conditional samples, i.e., to create synthetic samples which convincingly belong for each of the dataset classes, so those may join the supervised training dataset coherently. Experiment design When we started this study, the initial goal was to maximize the performance of GAN data-augmentation for skin-lesion analysis. As our preliminary experiments progressed, we found our results to be extremely noisy. Performance improvement, when it happened at all, was completely random: the choice of GAN model or other factors had no explanatory power. We changed our research question to a more fundamental one: can GAN data-augmentation actually improve the performance of skin-lesion analysis? Failing that, can GANs be used to anonymize the training data? The latter application -using synthetic samples instead of the actual real data -would be profitable even with a small, tolerable drop in accuracy, since it could make feasible for different countries and institutions to share knowledge in situations where direct patient images could not be exchanged (e.g., due to incompatible privacy laws). We design two experiments ( Table 2) to answer those questions, which we detail below. GAN-based data augmentation. As seen in the previous section, literature shows no consensus on how to perform GAN-based data augmentation. Our experimental design attempted to reflect the diversity of approaches found in existing art, contemplating a diversity of GANs, real:synthetic image ratios, real training dataset sizes, and synthetic sampling techniques. We present those choices next. The GAN models investigated were pix2pixHD [43], PGAN [20], SPADE [29], and StyleGAN2 [21]. We chose pix2pixHD and PGAN because they are known to work on skin-lesion data augmentation [7], while SPADE and Style-GAN2 are considered the state of the art on image generation. Pix2pixHD and SPADE are translation-based, while PGAN and StyleGAN2 are noise-based techniques. While the former tend to generate very high-quality images, they have stringent limitations on the amount of images they can generate, due to the requirement of using segmentation masks or different image modalities as inputs. Noise-based techniques have no such limitations, but tend to generate images with lower visual quality, and risk reproducing artifacts (e.g., vignettes, rulers) that may reinforce biases in the dataset [6]. For pix2pixHD and SPADE, we use the whole training set's mask to generate our synthetic set. For PGAN and Style-GAN2, we sample enough images to keep a 1: 1 /2 ratio. In Figure 1, we show lesions generated by the GANs used in our investigation. For pix2pixHD and SPADE, we use a mask from the training set as input, causing the synthetic [22] 872 Dermoscopic melanocytic nevus, melanoma, seborrhoeic keratosis out-of-distribution, fewer classes than train derm7pt-clinic [22] 839 Clinical melanocytic nevus, melanoma, seborrhoeic keratosis out-of-distribution, fewer classes than train dermofit [4] 973 Clinical melanocytic nevus, melanoma, seborrhoeic keratosis, actinic keratosis, pyogenic granuloma, haemangioma, dermatofibroma out-of-distribution benign classes Table 3: Description of the test sets used in the evaluation of the classification networks trained with the augmented training set. images to be almost identical to the real ones for the human eye (although features from deep networks may be still considerably different between the two kinds of images). For real:synthetic ratios, we considered the ratios 1: 1 /4, 1: 1 /2, and 1:1, where 1 is the size of our original realimage training set (14, 805 samples). We considered additional experiments varying the proportion of benign and malignant synthetic images proportion, to evaluate the opportunity of using GAN-based data augmentation to correct class-imbalance [14]. Those experiments are notated (real:synthetic benign:synthetic malignant), and we considered the ratios (1: 1 /2: 1 /2) (which is the same as 1: 1 /2 and has no balancing effect), (1: 1 /2: 3 /4), and (1: 1 /2:1). The baseline for all the experiments in this group is the ratio 1:0, i.e., the experiment with the entire real training set and no synthetic data augmentation. We varied the amount of real images employed in training mainly as an evaluation of GAN-based anonymization (see below). Those experiments may also be interpreted as the impact of GAN-based data-augmentation for different training-set sizes. We used real-image training set sizes with fractions of 1 /16, 1 /8, 1 /4, 1 /2, and 1 /1 (the baseline) of the whole dataset. Finally, we investigate different ways of sampling the synthetic images from the generated pool. That only applies to the noise-based GANs (PGAN and StyleGAN2), which can generate a limitless amount of images. We generated 100, 000 images for each of the 2 classes (benign and melanoma), and evaluated different methods to select the ones to compose the training set: choosing them at random, choosing them at random but with a criterion of diversity inspired on the perceptual-sensitive hash (pHash) to exclude near-duplicates, or choosing the ones best-classified (lowest error) or worst-classified (highest error) by an ancillary skin-lesion classification model. GAN-based anonymization. In anonymization, instead of using GANs to augment the training set, synthetic images replace real images. That application has received less attention in literature [25], but could be invaluable for researchers and institutions wishing to share knowledge while having to navigate issues of patient confidentiality. For that experimental design, we evaluate a single GAN (StyleGAN2) and a single sampling technique (random), explained above. In contrast, we evaluate many more real:synthetic ratios in this experiment, varying the amount of both kinds of images, in order to evaluate the situation where an institution was training a model with its own real images, adding synthetic images from a GAN provided by another institution. For each group of experiments with a fraction of the initial real training-set 1 /x, we evaluated experiments with ratios 1 /x:0 (the baseline for the group, with no synthetic images), 1 /x: 1 /x (doubling the training set), and 1 /x: x−1 /x (topping up the training set). We included the 1:0 ratio as a reference for the expected upper bound on the accuracy for those experiments. Datasets For all experiments, the reference training set of real images was based on the training set of the ISIC 2019 challenge [10,41]. We split that dataset into a training set (14, 805 samples) and a validation set (1, 931 samples) used in all our experiments, and a test set, with 3, 863 samples, added to our collection of test sets (isic19 on Table 3). We trained the noise-based PGAN and StyleGAN2 with the entire training set, but pix2pixHD and SPADE require semantic segmentation masks to guide the generation. We employ the clinical attribute semantic masks of the ISIC Challenge 2018 Task 2, which are available for only 2, 594 images, or about 1 /6 of the training set. When fine-tuning the target models, fractions of real images refer to selections of that same training dataset (e.g., 1 /4:0 would refer to a training set of 3, 701 randomly selected samples from the 14, 805 real ones). Fractions of the synthetic dataset refer to the same size of the real dataset, for a selection on the synthetic generated images (e.g., 0: 1 /4 would refer to a training set of 3, 701 randomly selected while keeping the real dataset class ratio, sampled from the 200, 000 synthetic images, half benign, and half malignant). We perform our tests in five gold-standard datasets (Table 3), selecting the classes to always perform a melanomavs.-benign task (carcinomas, if present in the dataset, are discarded from both training and testing). Having an array of test sets, both similar to our training set ("in-distribution") and very different ("out-of-distribution") is an attempt to mitigate the effect of dataset bias [6,8,13] and measure the models' generalization ability. For derm7pt, we remove near duplicates, and keep only classes present in the ISIC 2018 Challenge Task 2 dataset (melanoma, sebhorreic keratosis, and nevus). Those alterations result in 872 samples for derm7pt-derm and 839 samples for derm7pt-clinic. For dermofit, we remove the carcinomas, leaving 973 samples. Analyses. We replicated each experiment ten times, varying the selection of the real images from the training set and repeating the fine-tuning of the target model. In all experiments, the metric was the area under the ROC-curve of the target model (AUC) for melanoma-vs.-benign classification. A visual analysis of the results is given by blotting the individual data-points superimposed with a box-plot that, as usual, reveals the medians, quartiles, and range. We also plot the arithmetic means (red dots). There are two sets of plots, one for each application (augmentation and anonymization), plots were separated per test dataset, and within each plot, experiments were grouped (blue and black colors) to facilitate comparison. The labels of the experiments reveal the proportion of real:synthetic -or (real:synthetic benign:synthetic malignant) -images used in the training set used to fine-tune the target model, with additional information for the choice of GAN (p2p: pix2pixHD, spd: SPADE, pgn: PGAN2, sgn: StyleGAN2, all: samples from all GANs together), and the choice of sampling method (wst: worst, bst: best, div: diverse). When omitted, the GAN is StyleGAN2 and the sampling method is random. Implementation details For PGAN and pix2pixHD, we follow Bissoto et al.'s [7] implementation, modifications, and hyperparameters. We also adopt their pix2pixHD generation procedure to SPADE. For StyleGAN2, we do not need to adapt the original GAN model implementation 1 , as it could generate classconditioned samples from the start. Evaluating the quality of synthetic images is an open problem. Available metrics often fail to evaluate diversity, and both broad-and fine-grained details. Most popular metrics (e.g., FID [17], Inception Score [35]) rely on ImageNettrained networks to extract characteristics, and are, thus, questionable for applications where classes are very different, such as medical applications. We address that issue using a broad analysis, comprising traditional GAN metrics, metrics on classifier models, and visualization techniques of the trained classifier. To select the best training checkpoint for the GAN model, we considered both the time spent on training, and the FID metric [17] preferring later checkpoints (longer training) for similar FID (< 0.3 difference). The exact training lengths, and corresponding FIDs, appear in Table 4 Table 4: Amount of epochs and FIDs for each of the generative models used in this research. To select the checkpoint used to generate the samples that compose our classification model's training dataset, we consider both FID, and time spent training the GAN. The target model for both the data-augmentation and anonymization experiments is a skin-lesion classification Inception-v4 [40], pre-trained on ImageNet. We chose this model based on its well-known performance on skin-lesion analysis, including its ability to profit from larger training sets [42]. The full set of weights of the target model is finetuned with stochastic gradient descent with momentum of 0.9, weight decay of 0.001 and learning rate of 0.001, which we reduce by 10 on each validation plateau after a 10-epoch patience until a minimum of 10 −5 . We use a batch size of 32, shuffling the data before each epoch. We resize input images to 299 × 299 pixels, and z-normalize the input with Ima-geNet's training set mean and standard deviation. We train for a maximum of 100 epochs, and an early stopping with a patience of 22 epochs. We apply (conventional) data augmentation to all experiments both during train and test, with random horizontal and vertical flips, resized crops containing 75-100% of the original image area, rotations from -45-45 • degrees, and hue changes of −10-10%. We apply the same augmentations on both train and test. For test augmentation we average the predictions of 50 augmented replicas of each sample [30]. We run each experiment 10 times, and in each one, we vary the real images, but keep the synthetics the same (following the sampling criteria). Full-reproducible source code is available in our repository https://github.com/alceubissoto/ganaug-analysis. Results In this section, we evaluate GAN-based augmentation procedures for skin lesion analysis. In all the following experiments, we want to make the comparisons as fair as possible, giving equal opportunity to all models to be at their peak performance. Augmentation The experiments for GAN-based data augmentation appear on the top row of Figure 2. Comments on how to read and interpret those plots appear in Section 3, under the heading "Analyses". The leftmost experiment in each plot, with label 1:0, is the baseline with no synthetic-data augmentation. The plots reveal that, for dermoscopic test images, augmented-train sets are unable to confer a significant advantage, with augmented models showing lower -or at best similar -to the baseline. The experiments suggest that the more synthetic images we add to the training set, the worse the results are. The performance of different GANs fluctuated across datasets, but translation-based GANs tended to work better than noise-based GANS -but please notice that this factor is slightly confounded with image proportion in our tests due to the very limited generation ability of translationbased GANs. Of the two noise-based GANs, StyleGAN2 performed better (or at least, less worse). The scenario was less clear for the derm7pt-clinic dataset, where most experiments significantly improved the results. Those results departed from the other datasets also by showing StyleGAN2 ahead of all other GANs. However, the results on the dermofit dataset, also with clinical images, were more similar to the results on the dermoscopic datasets than to the ones on derm7pt-clinic. We remind that the experiments on anonymization (explained next), may also be interpreted as experiments on dataaugmentation for small training datasets, i.e., an anonymization experiment with ratio 1 /x:y can be reinterpreted as a data augmentation experiment with ratio 1: x /y for a test dataset with 1 /x of the samples of our main one. As we will show, those experiments failed to reveal the advantage of synthetic augmentation, even for small datasets. Anonymization The experiments for GAN-based anonymization appear on the bottom row of Figure 2. Comments on how to read and interpret those plots appear in Section 3, under the heading "Analyses". Those experiments are organized in groups of alternating blue and black colors, the rightmost experiment in each group, with label 1 /x:0, being the baseline for the group. GAN-based anonymization only has interest if the performance of the experiment is significantly above that baseline -otherwise, it can be trivially obtained by simply discarding the problematic samples. The rightmost experiment shows the model trained with all real samples, and gives an expected upper bound of accuracy for the plot. Unfortunately, for dermoscopic test images, no experiment appeared above that baseline. The derm7pt-clinic was again an exception, showing three experiments above the baseline, and one even slightly above the expected upper bound. In this set of experiments, the results on the (also clinical-image) dermofit dataset were also positive, with several configurations above the baseline. , and the choice of sampling method (wst: worst, bst: best, div: diverse). When omitted, the GAN is StyleGAN2 and the sampling method is random. Experiments showcasing a factor are grouped together in alternating blue/black color. Discussion Our results suggest future authors interested in GANbased data augmentation to be conservative about expected results, and cautious about evaluation protocols. GAN-based augmentation is a technique extensively explored in the literature for mitigating the scarcity of training images, being particularly interesting for the medical images community. However, we found our preliminary experiments to be excessively noisy, and noticed flaws in some experimental protocols during our literature review. Reliably transforming synthetic images into reliable performance gains is far from obvious. We will not go as far as condemning GAN-based data augmentation. Our experiments suggest that for some specific out-of-distribution scenarios (e.g., training on isic19 and testing on derm7pt-clinic) the technique may provide reliable improvements. Characterizing exactly which scenarios are those is, however, still an open question, as experiments on the also clinical-image dermofit dataset did not confirm those findings. Because training GANs requires a huge computation investment -a single training of StyleGAN2 takes up to a month of GPU time -researchers and practitioners should carefully evaluate whether their application to data-augmentation is justifiable, considering, among other things, their energetic footprint [47]. The finicky nature of GAN training also brings other risks: missing modes [2] may aggravate dataset biases, reinforcing the disparity of over/underrepresented groups instead of correcting them. GANs may also fixate on artifacts (such as vignettes, rulers, gel bubbles, etc.) and introduce or reinforce spurious correlations on the data [6]. Our results for GAN-based anonymization show modest results, but here, at least, there seems to be a trend, with results for out-of-distribution data being generally favorable, and results for in-distribution data being generally unfavorable. Those data suggest that using GANs may be possible at least as an ancillary method for sharing knowledge while preserving patient privacy. For making that application safe, however, we need further studies on how much the GAN "remembers" each original training sample and on its ability to (purposefully or accidentally) reconstructing original samples. Possible avenues for unblocking GAN-based data augmentation point towards attempting to conciliate the advantages of translation-based and noise-based techniquesobtaining the high-quality of the former and the limitless sampling availability of the latter -but such conciliation is a hard open problem. More achievable may be enhanced sampling methods, able to select the highest-quality, or better yet most relevant for decision, samples from the limitless sample of a noise-based technique. Even if future works fail to improve the ability of GANs to reliably provide data-augmentation or anonymization, a better characterization of the cases they are able to improve may provide interesting insights on the fundamental workings of deep learning. Figure 1 : 1Synthetic samples for different GAN-based approaches: (a) Real, (b) pix2pixHD, (c) SPADE, (d) PGAN, (e) StyleGAN2. In the first row, we present the full image, while in the second, we zoom-in to focus on the details. Figure 2 : 2Results for GAN-based data augmentation (left) and GAN-based anonymization (right), separated by test dataset. Box-plots display medians, quartiles, and range, as usual; red dots show arithmetic means. Y-axis labels show real:synthetic or (real:synthetic benign:synthetic malignant) training image proportion, with additional information for the choice of GAN (p2p: pix2pixHD, spd: SPADE, pgn: PGAN2, sgn: StyleGAN2, all: samples from all GANs together) Table 1 : 1The 18 works selected for our analysis of existing art. -: unclear or missing. *: missing details. †All gains in percentage points over the metric, min-max: ranges of gains found in experiments, ood:out of distribution, acc: accuracy., Table 2 : 2Summary of the two proposed experiments, with the main factors and their levels. Test Dataset DatasetSize Imaging Tech. Diagnostic ClassesNotesisic19 [10] 3, 863 Dermoscopic melanocytic nevus, melanoma, benign keratosis, actinic keratosis, dermatofibroma, vascular lesion in-distribution, same classes as train isic20 [33] 1, 743 Dermoscopic melanocytic nevus, melanoma, benign keratosis, actinic keratosis, lentigo, benign unknown mainly in-distribution, potential out-of-distribution samples among the 'benign unknown' samples derm7pt-derm .GAN Architecture Epochs FID SPADE 300 16.62 pix2pixHD 400 19.27 PGAN 890 39.57 StyleGAN2 565 15.98 https://github.com/NVlabs/stylegan2 AcknowledgementsA. Bissoto is partially funded by FAPESP 2019/19619-7. S. Avila is partially funded by CNPq PQ-2 315231/2020-3, and FAPESP 2013/08293-7. A. Bissoto and S. Avila are also partially funded by Google LARA 2020. E. Valle is funded by CNPq 315168/2020-0. This project is partially funded by a CNPq Universal grant 424958/2016-3. 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[ "https://github.com/alceubissoto/ganaug-analysis.", "https://github.com/NVlabs/stylegan2" ]
[ "Accessing the Full Superconducting Dome in Pristine YBa 2 Cu 3 O 6+x Under Pressure", "Accessing the Full Superconducting Dome in Pristine YBa 2 Cu 3 O 6+x Under Pressure" ]
[ "P L Alireza \nCavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K\n", "G H Zhang \nCavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K\n\nDepartment of Physics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n", "W Guo \nCavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K\n", "J Porras \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany\n", "T Loew \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany\n", "Y. -T Hsu \nCavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K\n", "G G Lonzarich \nCavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K\n", "M Le Tacon \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany\n\nKarlsruher Institut für Technologie\nInstitut für Festkörperphysik\nHermann-v.-Helmholtz-Platz 1D-76344Eggenstein-LeopoldshafenGermany\n", "B Keimer \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany\n", "Suchitra E Sebastian \nCavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K\n" ]
[ "Cavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K", "Cavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K", "Department of Physics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA", "Cavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K", "Max-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany", "Max-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany", "Cavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K", "Cavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K", "Max-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany", "Karlsruher Institut für Technologie\nInstitut für Festkörperphysik\nHermann-v.-Helmholtz-Platz 1D-76344Eggenstein-LeopoldshafenGermany", "Max-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany", "Cavendish Laboratory\nCambridge University\nJJ Thomson AvenueCB3 OHECambridgeU.K" ]
[]
We attain the previously unaccessed full superconducting dome in a pristine high temperature cuprate superconductor by applying pressures up to 280 kbar to samples of near stoichiometric YBa2Cu3O7. The obtained superconducting phase boundary as a function of hole doping closely follows the superconducting dome in La2−xSrxCuO4. Measurements are now enabled to trace the evolution of various entangled phases and the Fermi surface from the underdoped to overdoped regime in a single high purity cuprate superconducting family of materials. arXiv:1610.09790v1 [cond-mat.supr-con]
10.1103/physrevb.95.100505
[ "https://arxiv.org/pdf/1610.09790v1.pdf" ]
56,029,261
1610.09790
f4f8d873bf5a6430b792fe58e4811aaff2ea67a6
Accessing the Full Superconducting Dome in Pristine YBa 2 Cu 3 O 6+x Under Pressure (Dated: October 29, 2016) 31 Oct 2016 P L Alireza Cavendish Laboratory Cambridge University JJ Thomson AvenueCB3 OHECambridgeU.K G H Zhang Cavendish Laboratory Cambridge University JJ Thomson AvenueCB3 OHECambridgeU.K Department of Physics Massachusetts Institute of Technology 02139CambridgeMassachusettsUSA W Guo Cavendish Laboratory Cambridge University JJ Thomson AvenueCB3 OHECambridgeU.K J Porras Max-Planck-Institut für Festkörperforschung Heisenbergstrasse 1D-70569StuttgartGermany T Loew Max-Planck-Institut für Festkörperforschung Heisenbergstrasse 1D-70569StuttgartGermany Y. -T Hsu Cavendish Laboratory Cambridge University JJ Thomson AvenueCB3 OHECambridgeU.K G G Lonzarich Cavendish Laboratory Cambridge University JJ Thomson AvenueCB3 OHECambridgeU.K M Le Tacon Max-Planck-Institut für Festkörperforschung Heisenbergstrasse 1D-70569StuttgartGermany Karlsruher Institut für Technologie Institut für Festkörperphysik Hermann-v.-Helmholtz-Platz 1D-76344Eggenstein-LeopoldshafenGermany B Keimer Max-Planck-Institut für Festkörperforschung Heisenbergstrasse 1D-70569StuttgartGermany Suchitra E Sebastian Cavendish Laboratory Cambridge University JJ Thomson AvenueCB3 OHECambridgeU.K Accessing the Full Superconducting Dome in Pristine YBa 2 Cu 3 O 6+x Under Pressure (Dated: October 29, 2016) 31 Oct 2016 We attain the previously unaccessed full superconducting dome in a pristine high temperature cuprate superconductor by applying pressures up to 280 kbar to samples of near stoichiometric YBa2Cu3O7. The obtained superconducting phase boundary as a function of hole doping closely follows the superconducting dome in La2−xSrxCuO4. Measurements are now enabled to trace the evolution of various entangled phases and the Fermi surface from the underdoped to overdoped regime in a single high purity cuprate superconducting family of materials. arXiv:1610.09790v1 [cond-mat.supr-con] Copper oxide high temperature superconductors, while showing apparently conventional Fermi liquid behaviour in the overdoped region of the phase diagram, exhibit a mysterious plethora of phases in the underdoped region of the phase diagram. In addition to d-wave superconductivity and Mott insulating antiferromagnetism, phases identified in the underdoped region include various forms of charge density wave, spin density wave, electronic nematic order, and a puzzling pseudogap region [1]. Recent studies have also suggested the potential deviation of superconducting behaviour in the overdoped regime from the conventional Bardeen-Cooper-Schrieffer (BCS) theory [2,3]. It is as yet an open question as to the relevance of each of the identified superconducting and density wave phases to the pseudogap region, and the evolution of each of these phases from the underdoped to the overdoped Fermi-liquid-like region. An understanding of this evolution will shed light on the nature of the ill-understood pseudogap region, yet this has proved challenging thus far given the unavailability of pristine cuprate materials families that can be tuned across the entire superconducting phase diagram from underdoped to overdoped. Among the high-T c superconductors, YBa 2 Cu 3 O 6+x (YBCO) (see inset of figure 1), is a material recognized for being clean and well-ordered, which leads to its prevalence in quantum oscillation experiments and other measurements [4,5]. It has become one of the most extensively studied materials in the cuprate family, having well-identified critical points [6] and electronic orders [7][8][9][10][11]. While quantum oscillations and transport have been thoroughly studied in underdoped YBCO [12][13][14], similar measurements cannot be extended to the overdoped regime in the same material, despite having been successfully used to study the Fermi surface of a different cuprate, Tl 2 Ba 2 CuO 6+x (Tl-2201) [15,16]. This is due to the fact that chemical doping in YBCO can only achieve a maximum hole concentration of 19.4% with oxygen tuning, and up to 22% with calcium substitution [17]. It is important to find a control parameter that can tune YBCO from the underdoped to the overdoped region in a pristine way in order to track the evolution of the various forms of order and their relation to the pseudogap across the full range of doping, as well as to investigate the overdoped region itself. Hydrostatic pressure has been used as a tuning parameter [18,19] in past experiments to tune between adjoining electronic phases in materials where the doping or bandwidth was found to change significantly with pressure [20]. For instance, pressure has both induced superconductivity and suppressed competing orders in magnetic materials such as UGe 2 [21][22][23][24][25][26], and revealed in materials such as CePd 2 Si 2 and CeIn 3 critical points as a function of pressure (doping) that are relevant to superconductivity [27,28]. We report that by starting near YBa 2 Cu 3 O 7 and applying high pressures of up to 280 kbar, we have tuned the material into the heavily overdoped region up to a hole doping of ≈ 26%, where the superconducting transition temperature becomes almost fully suppressed. By using applied pressure as the control parameter in YBCO, therefore, we have achieved unprecedented access to the entire superconducting dome in a pristine high temperature superconducting material. Previous studies of YBCO under lower applied pressures have shown that increasing hydrostatic pressure decreases the distance between the BaO and CuO 2 planes, compressing the softer Cu(plane)-O bond and elongating the stiffer Cu(chain)-O bond (see inset of figure 1), and thus increasing the number of holes per Cu atom in the CuO 2 plane (n h ) through hole transfer from the CuO chains [29,30]. While pressure has also been suggested to have intrinsic effects on T c through induced structural phase transitions, oxygen ordering and changes in the effective interaction strength [31][32][33], such effects are in practice expected to be negligible above the slightly overdoped region [34][35][36]. Thus, we assume that in the region which is of interest to us, from the optimally doped region to the overdoped region, the effect of pressure is principally to tune the properties of YBCO through dop-ing. Recent experiments have shown pressure to be an effective control parameter for cuprates; resistivity measurements in Bi 1.98 Sr 2.06 Y 0.68 CaCu 2 O 8+x have demonstrated pressure-induced superconductivity [37], while pressure-tuned Raman and x-ray diffraction data have indicated quantum critical points corresponding to electronic transitions [38]. For YBCO specifically, hydrostatic pressure has been applied up to 170 kbar on an initially underdoped sample with x = 0.66 and up to 110 kbar on an initially optimally doped sample with x = 0.95, showing pressure to affect the superconducting critical temperature in both cases. In the former, the effects of pressure, applied in the underdoped region, are complicated by oxygen ordering and thus difficult to analyze while the latter achieves a maximum effective n h of around 0.2 which ventures into the slightly overdoped region [34,35]. Here, we start near stoichiometric YBa 2 Cu 3 O 7 in which the hole fraction is intrinsically just above optimal, in order to most effectively access the highly overdoped regime with applied pressure as a tuning parameter. Single crystals of YBa 2 Cu 3 O 6+x with x = 0.98 were grown out of solution [39] and detwinned under uniaxial pressure. The nominal oxygen content was determined by relating the superconducting temperature to the hole doping and oxygen concentration using the previously determined relationship for this material [17]. Polycrystalline YBa 2 Cu 3 O 7 was grown by conventional solid state reaction followed by sintering and cooling, according to methods described in previous literature [40]. To determine the effect of pressure on the superconducting transition temperature, we measured ac-susceptibility in a diamond anvil cell (DAC) using a micro-coil system [41]. Pressures of up to 300 kbar were generated using 0.6 mm culet diamonds with glycerol as the pressure medium. The sample space was a 200 µm hole drilled into a MP35 gasket, in which the signal coil, a 130 µm diameter, 4 turn micro-coil made with 10 µm insulated copper wire, was placed. The drive coil, a 130 turn coil made with 30 µm insulated copper wire, was placed outside the pressure region. Pressure inside the sample space was determined by the ruby fluorescence method at room temperature and in some instances verified by the superconducting transition temperature of a lead sample at low temperatures. Additional measurements on DC magnetization were performed using a SQUID anvil cell specifically developed to be used in a Quantum Design MPMS system [42,43]. The cell, gasket, and pressure medium were set up similarly to the one described above, using 0.6 mm diamond anvils, MP35 gasket, and glycerol as the pressure medium. Ac-susceptibility curves measured for both single crystal and polycrystalline YBCO at several values of applied pressure are shown in figure 1, where the susceptibilities have been normalised. We define the superconducting temperature as corresponding to the midpoint of the ac-susceptibility curve. We find, as shown in table I and figure 2, that up to applied pressures of around 90 kbar, the evolution in superconducting temperature is modest, falling by ≈ 20%, similar to that previously reported [35]. However, at higher values of applied pressures beyond those previously reported, the superconducting transition temperature begins to drop rapidly. For significantly high pressure up to 280 kbar, we find that superconductivity is almost destroyed, with the superconducting temperature falling to below 10 K. It is Pressure (kbar) clear that in the case of YBCO, the large energy scales that characterise this material necessitate extremely high applied pressures far beyond those achieved in previous studies to access the overdoped region of the phase diagram. Superconducting transition temperatures as a function of applied pressure are found to be similar for single crystal and polycrystalline YBCO, demonstrating the suppression of superconductivity to be primarily a volume effect rather than a uniaxial effect. The width of the superconducting transition under pressure reflects both the gasket condition during the run and the level of pressure homogeneity within the sample. The remarkable sharpness of the superconducting transitions we observe at the highest applied pressures in the vicinity of 250 kbar (figure 1) demonstrates the high homogeneity of pressure conditions achieved even at these very high pressures. In order to map the YBCO phase diagram with T c as a function of n h for our applied pressure measurements, we need to translate applied pressure into hole doping. Early pressure studies have shown the effect of pressure on the change in n h to closely correspond to a linear dependence on the relative decrease in cell volume [45][46][47]. We follow similar analysis; denoting the relative decrease in the unit cell volume as ξ(P ) ≡ (1 − V (P )/V 0 ), where V is the cell volume, P is the applied pressure, and V 0 is the cell volume at zero pressure. Using the first-order previous Calcium-doped data (hollow circle) [44], and previous zero-pressure data (filled circle) [17]. Translation from pressure to doping n h was done using relations 1 and 2 while considering constraints in accordance with previous data on Y1−xCaxBa2Cu3O6.96 [44], for which n h was estimated using a combination of copper and oxygen bond valence sums (BVS). The effects of the different constraints are reflected in the horizontal error bars, and the dashed curve indicates the LSCO superconducting dome, 1 − Tc/Tc,max = 82.6(n h − 0.16) 2 , with Tc,max = 94.3 K for YBCO single crystals [17]. Murnaghan equation [48], we assume, dn h (P ) dξ(P ) = n h (P )B 0 1 + B 0 B 0 P 1+1/B 0 (1) ξ(P ) = 1 − 1 + B 0 B 0 P −1/B 0 ,(2) where n h (P ) = dn h (P )/dP and n h,0 , B 0 , and B 0 are the hole doping, bulk modulus −V (dP/dV ), and pressure derivative of the bulk modulus respectively at zero pressure. The initial doping n h (0) = 0.19 holes/Cu was found by locating T c on the LSCO superconducting parabola, 1−T c /T c,max = 82.6(n h −0.16) 2 with T c,max = 94.3 K for YBCO single crystals [17]. Values considered for B 0 and B 0 were taken from previous pressure studies using X-ray analysis, on samples with oxygen content close to that of our single crystals x = 0.98; as such, we assume a range of B 0 = 1306±11 kbar and B 0 = 4.8±0.2 to B 0 = 1500 kbar and B 0 = 4, obtained for x = 0.95 [49] and x = 1 [50], respectively. Starting with an assumption of parabolic dependence of the superconducting temperature as a function of hole doping up to n h = 0.22 similar to previous work [44], we apply relations 1 and 2 (See Supplementary Material for further details of the calculation) to yield the dependence of the superconducting temperature as a function of hole doping at higher dopings. The full superconducting phase diagram thus obtained as a function of hole doping is shown in figure 3. We are therefore able to map the full superconducting dome in a pristine way in a high temperature cuprate superconductor by applied pressure tuning starting near stoichiometric YBa 2 Cu 3 O 7 . Importantly, we find that the superconducting temperature dependence on hole doping is in good agreement with the form of the superconducting dome mapped in the LSCO cuprates [51][52][53]. Crucially, our applied pressure measurements provide unprecendented access beyond the optimally doped region in YBCO, allowing for measurements of quantum oscillations, electrical and thermal transport, magnetisation and other thermodynamic properties in the overdoped regime of this family of high-T c materials. In particular, we expect to trace the evolution of the Fermi surface pockets from small to large across the entire phase diagram, and to resolve each of the entangled density wave and d-wave superconducting phases from the underdoped to the overdoped region, and their relation to each other and the pseudogap region in this family of high-T c cuprates. Another important question to be addressed pertains to the robustness of superconductivity with increasing temperature and applied magnetic field in the underdoped region compared to the overdoped region. Controversy surrounds whether superconductivity persists to high magnetic fields in the underdoped regime, as would be characteristic of strongly coupled superconductivity in proximity to the Mott transition [54], or whether superconductivity in the underdoped regime is suppressed by modest critical magnetic fields as would be characteristic of conventional BCS superconductivity [55]. We can shed light on this issue by a comparative study of the robustness of superconductivity in the overdoped regime of YBCO we now access, which is expected to be closer in behaviour to conventional BCS superconductivity. P Denoting ξ(P ) ≡ (1 − V (P )/V 0 ) as the relative decrease in cell volume, we start with n h (P ) = n h,0 + dn h (P ) dξ(P ) ξ(P ),(3) where n h,0 = n h (P = 0). Using the first-order Murnaghan equation of state [48], V (P )/V 0 = (1 + B 0 P/B 0 ) −1/B 0 , where B 0 is the bulk modulus −V (dP/dV ), B 0 is the pressure derivative of B 0 , and V 0 is the cell volume at zero pressure, gives the following expressions, as presented in the main text, dn h (P ) dξ(P ) = n h (P )B 0 1 + B 0 B 0 P 1+1/B 0 (4) ξ(P ) = 1 − 1 + B 0 B 0 P −1/B 0 ,(5) where n h (P ) = dn h (P ) dP . Substitution of expressions 4 and 5 into equation 3 gives a first-order differential equation of n h (P ) with constant parameters B 0 , B 0 , and n h,0 , which can be solved without further approximation with the specified boundary condition. The initial doping at zero applied pressure n h (0) = 0.19 holes/Cu was found by locating T c on the La 2−x Sr x CuO 4 (LSCO) superconducting parabola, 1 − T c /T c,max = 82.6(n h − 0.16) 2 with T c,max = 94.3 K for YBa 2 Cu 3 O 6+x (YBCO) single crystals [17]. Values considered for B 0 and B 0 were taken from previous pressure studies using X-ray analysis, on samples with oxygen content close to that of our single crystals x = 0.98; as such, we assumed a range of B 0 = 1306 ± 11 kbar and B 0 = 4.8 ± 0.2 to B 0 = 1500 and B 0 = 4 kbar, obtained for x = 0.95 [49] and x = 1 [50], respectively. We separately set these values as parameters in expressions 4 and 5 to estimate the contribution to uncertainty in doping as a function of pressure, the horizontal error bars in figure 3 in the main text, due to errors in the exact values of B 0 and B 0 for the oxygen content of x = 0.98. A median set of values was determined in this range to use for translation of the square points in figure 3. Based on the phase diagram of YBCO up to the slightly overdoped region of n h ≈ 0.22 by calcium substitution [44], we assume that the progression of T c from the optimally doped region initially follows the LSCO parabola. Considering these constraints as the boundary condition, we solve equation 3 numerically and find that regardless of which single point we choose to constrain in accordance with this condition, the remaining data consistently appear to closely follow the LSCO parabola. Horizontal error bars in figure 3 for higher dopings reflect considerations of the possible doping points at which any possible deviation from the parabola could start to occur, as well as the uncertainty in values of B 0 and B 0 for our exact doping, though the effect of the latter is small in comparison. Calculation using the numerical result obtained for n h (P ) shows dn h (P )/dξ(P ) to be approximately constant along P , where denoting ∆[dn h (P )/dξ(P )] ≡ dn h (P )/dξ(P ) − dn h (P = 0)/dξ(P ), we find that ∆[dn h (P )/dξ(P )]/[dn h (P = 0)/dξ(P )] ∼ 0.01% for P = 0 kbar to 300 kbar, the full pressure range considered; observing the behavior of the two factors in equation 4, it is apparent that the decrease in n h (P ) from the numerical solution for n h (P ) effectively cancels the increase in (1 + B 0 P/B 0 ) 1+1/B 0 . This confirms the starting assumption of n h (P ) being linearly dependent on the relative decrease in cell volume. FIG. 1 . 1Progression of ac-susceptibility curves for single crystal YBa2Cu3O6.98 with increasing applied pressure. Measurements were taken over multiple runs, by increasing pressure in the same setup or by remounting the sample at a higher pressure in a new setup. The varied widths of each transition are likely due to specific conditions of the setup at each run, including pressure homogeneity and potential deformation of the gasket. The inset shows the YBCO unit cell. FIG. 2 . 2Determination of Tc from ac-susceptibility curves shows the decrease of transition temperature with increasing pressure for both single crystal and polycrystalline YBCO. Each point has a horizontal error of ±2 kbar or ±3%, the greater of which indicates uncertainty in the determination of pressure inside the sample space as described above. The vertical error bar corresponds to uncertainty in Tc, which is related to the width of the transition. 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[ "Three pion nucleon coupling constants *", "Three pion nucleon coupling constants *" ]
[ "E Ruiz Arriola ", "J E Amaro [email protected]érez ", "\nDepartamento de Física Atómica\nMolecular y Nuclear\nNuclear and Chemical Science Division\nInstituto Carlos I de Física Teórica y Computacional. Universidad de Granada\nE-18071GranadaSpain\n", "\nLawrence Livermore National Laboratory\n94551LivermoreCAUSA\n" ]
[ "Departamento de Física Atómica\nMolecular y Nuclear\nNuclear and Chemical Science Division\nInstituto Carlos I de Física Teórica y Computacional. Universidad de Granada\nE-18071GranadaSpain", "Lawrence Livermore National Laboratory\n94551LivermoreCAUSA" ]
[]
There exist four pion nucleon coupling constants, f π 0 pp , −f π 0 nn , f π + pn / √ 2 and f π − np / √ 2 which coincide when up and down quark masses are identical and the electron charge is zero. While there is no reason why the pion-nucleon-nucleon coupling constants should be identical in the real world, one expects that the small differences might be pinned down from a sufficiently large number of independent and mutually consistent data. Our discussion provides a rationale for our recent determination f 2 p = 0.0759(4) , f 2 0 = 0.079(1) , f 2 c = 0.0763(6) , based on a partial wave analysis of the 3σ self-consistent nucleon-nucleon Granada-2013 database comprising 6713 published data in the period 1950-2013.
10.1142/s0217732316300275
[ "https://arxiv.org/pdf/1606.02171v2.pdf" ]
119,229,346
1606.02171
44e7b87e0ff975525001ea0c2fd2f2cdb0d83af1
Three pion nucleon coupling constants * September 16, 2016 E Ruiz Arriola J E Amaro [email protected]érez Departamento de Física Atómica Molecular y Nuclear Nuclear and Chemical Science Division Instituto Carlos I de Física Teórica y Computacional. Universidad de Granada E-18071GranadaSpain Lawrence Livermore National Laboratory 94551LivermoreCAUSA Three pion nucleon coupling constants * September 16, 2016Received (Day Month Year) Revised (Day Month Year)1:32 WSPC/INSTRUCTION FILE Ruiz-Arriola Modern Physics Letters A c World Scientific Publishing CompanyNN interactionPartial Wave AnalysisOne Pion Exchange PACS Nos:0365Nk1110Gh1375Cs2130Fe2145+v There exist four pion nucleon coupling constants, f π 0 pp , −f π 0 nn , f π + pn / √ 2 and f π − np / √ 2 which coincide when up and down quark masses are identical and the electron charge is zero. While there is no reason why the pion-nucleon-nucleon coupling constants should be identical in the real world, one expects that the small differences might be pinned down from a sufficiently large number of independent and mutually consistent data. Our discussion provides a rationale for our recent determination f 2 p = 0.0759(4) , f 2 0 = 0.079(1) , f 2 c = 0.0763(6) , based on a partial wave analysis of the 3σ self-consistent nucleon-nucleon Granada-2013 database comprising 6713 published data in the period 1950-2013. Introduction Four score and a year ago Yukawa brought forth a new theory of nuclear forces and dedicated to the proposition that protons and neutrons exchange pions 1 . But created and anihilated pions are all not equal. The quest for isospin violations in particle and nuclear physics has been a permanent goal ever since Kemmer 2 invented the concept and generalized the Pauli principle. A readable account of the early developpments can be found in Ref. 3 . Actually, the neutral pion was sought and found because isospin symmetry required it. While the mass of the pion may be deduced directly from their decays π 0 → γγ and π ± → µ ± + ν µ the determination of the coupling constant to nucleons is more intrincate and needs further theoretical ellaboration. Although this is not a fundamental constant of QCD, the pion-nucleonnucleon coupling constant is the strong hadronic charge of neutrons and protons, which appear as the effective constituents of atomic nuclei. In 1940 Bethe obtained the value f 2 = 0.077 − 0.080 from the study of deuteron properties 4 . Subsequent determinations based on a variety of processes can be traced from recent compilations 5,6 . Attempts to make a microscopic distinction via radiative vertex corrections aiming at predictive power have been made in the past (see 7 for pre-QCD account). At the hadronic level and within the meson exchange picture there have been many attempts to determine the many possible causes of isospin breaking, ρ 0 − ω and π − η mixing, pion mass differences in the two-pion exchange interaction, πγ exchange, etc. (see e.g. 8,9 for post QCD comprehensive accounts). All these many complications contribute to the belief that isospin violations in strong interactions remains one of the least understood issues in the nuclear force. In this contribution we provide our point of view on this subject and the elements underlying ongoing work on the determination of the three coupling constants. Although many of the issues we address here are well known for the experts after 60 years of NN Partial Wave Analysis fits, we try in this short account to be pedagogical for the non-experts. For comprehensive and concise reviews we suggest Ref. 5,6 where the latest NN based determination f 2 = 0.0750(9) is recommended. Charge Dependent One Pion Exchange (CD-OPE) The coupling constant is rigurously defined as the πN N three point vertex function when all three particles are on the mass shell, a condition that cannot generally be satisfied for real momenta. At the hadronic level the pion-nucleon-nucleon vertex is described by the Lagrangian (we use the standard convention 10 ) L = √ 4π f e,ab m π +B a γ µ γ 5 B b ∂ µ φ e ,(1) whereB a , B b and φ e are antibaryon, baryon and pseudoscalar meson field respectively and γ µ and γ 5 are Dirac matrices. This yields four possible vertices p → π + n , n → π − p , p → π 0 p , n → π 0 n , so their amplitudes are, The couplings are assumed to be in the isospin limit g = g π 0 pp = g π ± np / √ 2 = −g π 0 nn . and so on. The relevant relationships between the pseudoscalar pion coupling constants, g πN N , and the pseudovector ones, f πN N , are given by A(p → π + n) = f π + npūn (p n , s n )γ µ γ 5 u p (p p , s p )q µ ,(3) Ruiz-Arriola g π 0 pp √ 4π = 2M p m π ± f π 0 pp , g π 0 nn √ 4π = 2M n m π ± f π 0 nn , g π ± pn √ 4π = M p + M n m π ± f π ± pn . (4) with M p = 938.272 MeV the proton mass, M n = 939.566 MeV the neutron mass, and m π ± = 139.570 MeV the mass of the charged pion. From these vertices one may obtain the NN scattering amplitude to lowest order in perturbation theory, see Fig. 1. For instance, the unpolarized differential nn-cross section reads dσ nn dt = g 4 π 0 nn 32πs(s − 4M 2 n ) t 2 (t − m 2 π 0 ) 2 − 2( t 2 − M 2 n ) 2 − ( s 2 − M 2 n ) 2 − 4M 2 n (t − m 2 π 0 )(u − m 2 π 0 ) + (u ↔ t) ,(5) where s = 4(p 2 + M 2 n ), t = −q 2 = −4p 2 sin(θ/2) 2 < 0 and s + t + u = 4M 2 n with (p, θ) the CM momentum and scattering angle respectively. This is perhaps the most straightforward way of checking OPE from data, i.e. extrapolating (t − m 2 π 0 ) 2 dσ/dt from the physical t < 0 kinematics to the unphysical limit t → m 2 π 0 . Unfortunately, the Born approximation violates elastic unitarity and there are many ways to restore it. For reasons to be explained below we use the phenomenological potential approach and a partial wave expansion analysis 11 . The concept of a potential is essentially non-relativistic and the procedure to obtain it is to match perturbatively the non-relativistic quantum mechanical potential to the same scattering amplitude obtained in quantum field theory for the direct term, namely f Born QFT (θ, E) = − 2µ 4π d 3 xe −i k · x V ( x, p)e i k· x ,(6) where the on-shell condition is understood k = k = p. This already incorporates an ambiguity, since one may add terms which vanish on the mass-shell. In the static limit of heavy nucleons the ambiguity dissapears and the potential deduced from field theory becomes local, which is of great advantage since we obtain a Schrödinger equation. The CD-OPE potential in the pp and np channels so obtained reads V pp→pp (r) = f 2 π 0 pp V m π 0 (r),(7)V np→np (r) = V pn→pn (r) = −f π 0 nn f π 0 pp V m π 0 (r) (8) V pn→np (r) = V np→pn (r) = f π − pn f π + np V m π ± (r) (9) V nn→nn (r) = f 2 π 0 nn V m π 0 (r),(10) where V m,OPE is given by V m,OPE (r) = m m π ± 2 1 3 e −mr r σ 1 · σ 2 + 1 + 3 mr + 3 (mr) 2 S 12 ,(11) with σ 1 and σ 2 the single nucleon Pauli matrices and S 12 = 3σ 1 ·rσ 2 ·r − σ 1 · σ 2 the tensor operator. Using the standard notation we make the identifications f 2 p = f π 0 pp f π 0 pp , f 2 0 = −f π 0 nn f π 0 pp , 2f 2 c = f π − pn f π + np .(12) In many derivations of the OPE potential some emphasis is placed on the contact and singular piece which is proportional to a Dirac delta function located at the origin. We omit them here, since they will play no role in our subsequent discussion. The static nuclear potential The saturation property of nuclei suggests that there is an equilibrium distance between two nucleons. While one talks about nuclear forces, the truth is that they have never been measured directly in experiment. From a purely classical viewpoint, this would require to pull two nucleons apart at distances larger than their size and measure the necessary force, similarly as Coulomb and Cavendish did for the electric and gravitational forces 250 years ago. For such an ideal experiment the behaviour of the system at shorter distances would be largely irrelevant. This situation would naturally occurr if nucleons were truely infinitely heavy sources of baryon charge. From a fundamental point of view the nucleon force is defined from the static energy between two nucleons which in QCD are made of three quarks and any number of quark-antiquark pairs and gluons. Nucleons and pions are composite and extended particles which can be characterized by any gauge invariant combination of interpolating fields with the proper quantum numbers. This generates some ambiguity except for heavy quarks. In any case the static energy reads E N N (R) = 2M N + q∈N,q ∈N V q,q ( x q − x q ) .(13) Here, the quarks in each nucleon are located at the same point x q = R/2 and x q = − R/2. On the lattice static baryon sources have been placed at a fixed distance 12,13 , and in fact there exist lattice calculations addressing the pion-nucleon coupling constant. Momentum dependence of the strong form factor and coupling constant determination in lattice QCD gives g πN N ∼ 10 − 12 14 and more recently g πN N ∼ 13(1) for m π ∼ 560MeV 15 using QCD sum rules. On the lattice in the quenched approximation it has been found that for a pion mass of m π = 380 MeV the value g 2 πN N /(4π) = 12.1 ± 2.7 which is encouraging 16 . These are courageous efforts which still are far from the 1% accuracy needed for witnessing isospin breaking in the couplings (see below). Some intuition may be gathered from a chiral quark model picture. At scales above the confinement radius we expect exchange of purely hadronic states and at long distances the OPE mechanism will dominate 17 , but because the quarks in each nucleon are on top of each other, they will contribute coherently to the couplings. Of course, u and d quarks are not heavy, but spontaneous chiral symmetry breaking will provide them with a constituent mass M 0 which will give a total mass M q = M 0 + m q . Thus, neglecting em contributions, the proton and neutron masses are M p = 3M 0 +2m u +m d and M n = 3M 0 +m u +2m d , so that M p −M n = m d −m u , a reasonable value. Within this picture, relative corrections δg/g at the nucleon or quark level are the same. The πqq coupling is the residue of the Bethe-Salpeter equation at the pion pole. In a model where the pion is composite such as the NJL model (see e.g. 18 for a review) one has for m u , m d M 0 (Λ is the NJL cut-off) δg g πqq = A(M 0 , Λ) m q + mq M 0 + . . .(14) with A(M 0 , Λ) > 0 so the coupling to the pion grows with the quark mass. Thus δg g π 0 nn > δg g π + np > δg g π 0 pp .(15) Scattering and analytical properties Unfortunately we cannot carry out the classical Cavendish experiment for nucleons in the laboratory. We may instead analyze the about 8000 pp and np scattering data collected in accelerators below LAB energy 350MeV in the period 1950-2013. Quantum mechanically wave-particle duality implies a finite wavelength resolution for a given relative momentum, ∆r ∼ 1/p. This will allow to sample the potential at coarse grained distance scales, r n ∼ n∆r. Fluctuations of the interaction below that scale will be unobservable and manifest as correlations among the interaction at the sampled points V (r n ). The "force" is defined as the average change of V (r n ) over the resolution scale ∆r, for instance F (r n ) = −(V (r n + ∆r/2) − V (r n − ∆r/2))/∆r. The NN scattering amplitude has five independent complex components which are a function of energy and scattering angle, M = a + m(σ 1 · n)(σ 2 · n) + (g − h)(σ 1 · m)(σ 2 · m) + (g + h)(σ 1 · l)(σ 2 · l) + c(σ 1 + σ 2 ) · n .(16) We use the three unit vectors (k f and k i are relative final and initial momenta), The LAB energy complex energy plane, showing the partial waves left cut structure due to multipion and σ, ρ, ω exchange and the right cut structure due to pion production. Right panel: NN Potential as a function of distance, compared with a free wave, sin(pr), with p = 2k F , the relative momentum corresponding to back-to-back scattering in nuclear matter; the most energetic process inside a heavy nucleus. l = k f + k i |k f + k i | , m = k f − k i |k f − k i | , n = k f ∧ k i |k f ∧ k i | .(17) For this amplitude the partial wave expansion in this case reads M s m s ,ms (θ) = 1 2ik J,l ,l 4π(2l + 1)Y l m s −ms (θ, 0) × C l ,S,J ms−m s ,m s ,ms i l−l (S J,S l,l − δ l ,l )C l,S,J 0,ms,ms ,(18) where S is the unitary coupled channel S-matrix, and the C s are Clebsch-Gordan coefficients. One has that S JS = (M JS − i1)(M JS + i1) −1 with (M JS ) † = M JS a hermitian coupled channel matrix (also known as the K-matrix). At the level of partial waves the multipion exchange diagrams generate left hand cuts in the complex s-plane, which come in addition to the NN elastic right cut and the πN N , 2πN N etc., pion production cuts. At low energies for |p| ≤ m π /2 we have 19 p l+l +1 M JS l,l (p) = −(α −1 ) JS l,l + 1 2 (r) JS l,l p 2 + (v) JS l,l p 4 + . . .(19) which is the coupled channels effective range expansion. We sketch the situation in Fig. 2 (left panel) where a possible contour for a dispersion relations study is also depicted. When the cuts are explicitly taken into account there still remains the question on the number of subtraction constants (see e.g. Ref. 20 and references therein). While one can pursue such an analysis, comparison with experimental data goes beyond just partial waves and generates complications coming from long range effects, which are most efficiently treated in coordinate space within the potential approach preferred by nuclear physicists. The magnetic moment interaction which decreases as 1/r 3 is crucial to describe the data but remains a challenge in momentum space. Even in the friendly coordinate space these terms usually need summing about 1000-2000 partial waves. This is a serious bottleneck for any analysis aiming at a direct comparison to experimental data. Fortunately, any pion cut generates a contribution to the potential which at long distances falls off as e −nmπr . The potential in coordinate space, sketched in Fig. 2 (right panel), has the same analytical structure (left panel) but permits incorporating these otherwise difficult long distance effects by simply solving the Schrödinger equation and using the sampled, coarse grained potential V (r n ), as fitting parameters themselves. An elementary determination of the coupling constant One motivation to study the NN force is to apply it to nuclear structure and nuclear reactions. Many studies are conducted with this application in mind, when not specifically designed to produce a potential friendly to some preferred computational many-body method. This introduces a bias and hence a systematic error in the analysis of nuclear forces which is often forgotten . Thus, fits rarely go much beyond the pion production region, since the inelastiticy becomes important above 350 MeV LAB energy and the addition of πN N channel does not improve the description. On the other hand, being composite particles made out of quarks and gluons, nucleons have a finite size which can be determined by a variety of methods, mostly by electron and neutrino scattering. To what extent this finite size is relevant for nuclear structure calculations is not completely obvious, but in practice nucleons in nuclear physics are treated as elementary and pointlike. Excitations such as the ∆ resonance are explicitly included themselves as elementary point like particles as well. The most dense known nuclear system is nuclear matter where an average equilibrium separation distance between nucleons is d N N ∼ 1.8fm. Thus, nuclear binding is obviously related to this mid-range distance. On the other hand the pion production threshold happens at LAB energy ∼ 2m π = 280MeV. An extreme situation corresponds to assume that the elementarity radius r e is arbitrarily small. The lightest nucleus, the deuteron, is a bound np system with J C = 1 + , and corresponds to a 3 S 1 − 3 D 1 mixed state. The tensor part of the OPE potential diverges as 1/r 3 as already pointed out by Bethe 4 . Thus, the deuteron equations are singular at short distances, but they can be renormalized 21 by imposing physical renormalization conditions. In Fig. 3 we show results for a number of renormalized observables and requiring the state to be normalizable at short distances, showing that f 2 = 0.072 − 0.074, not far from Bethe's venerable value. NN interaction and Effective elementarity of the Nucleon The elementarity size r e of the nucleon can operationally be characterized by looking at departures from point-like behaviour. If we look for instance at the electromagnetic interaction, Fig. 4 (left panel), it clearly suggests that protons interact as point-like charges, 1/r, above r > r e = 2fm. The electric charge screening feature we see in the electromagnetic case also holds for the strong interaction but here, it corresponds to axial charge screening. This can be illustrated within a Quark Cluster picture using a Chiral Quark model 25 (TPE) for r > r e . The result is shown in Fig. 5 and, as we see, the elementarity radius turns out, again, to be about r e = 2fm ∼ d N N the average distance of nucleons in nuclear matter, suggesting that nucleon compositeness should be play a marginal role in nuclei. The Quark model estimate suggests that we may pin down the interaction as if nucleons were elementary particles down to the elementarity radius r e = 2fm. The practical utility of this informaton is that we can compare different contributions stemming from meson exchange above the elementarity radius r e , showing that (unregularized) OPE is the main contribution for distances larger than a cut-off scale r > r c = 3fm below which TPE starts contributing significantly. This pattern occurs also with the exchange of heavier mesons including σ, ρ, ω, η, δ, a 1 etc. which allows us to discard them in our analysis as well as TPE. In fact, one common feature of all the high quality interactions is that they contain unregularized OPE above 3 fm. Thus, we will take V N N (r)| strong = V OPE N N (r) , r > r c = 3fm .(20) Until 1993 most "high quality fits" carried out by many groups in Bonn, Paris Washington and Nijmegen never improved over a χ 2 /ν ∼ 2. From the point of view of statistical analysis this is undesirable; one finds the closest theory to experiment but the difference between theory and experiment cannot be identified as a random fluctuation, precluding any sensible error analysis. The great achievement of the Nijmegen group in the 1990's was to pursue all effects which could explain the very accurate pp and np measurements. Besides Coulomb this includes three main effects: vacuum polarization (VP), relativistic corrections and magnetic moments (MM) interactions. For instance VP dominates over MM for r < 1000fm whereas beyond the effect is the opposite, see Fig. 4. The em piece becomes, V N N (r)| em = V C N N (r) + V M M N N (r) + V V P N N (r) + V C Rel (r) , r > r c = 3fm . (21) As mentioned before we cannot pin down details below ∆r = 1/p max , so we explicitly coarse grain the interaction in the innermost unknown region, which means in practice sampling the "unknown" original potential at equidistant values separated by a distance ∆r = 0.6fm, r n = n∆r, using Dirac delta-shells. This was an idea suggested by Aviles long ago 28 and rediscovered recently 29 which means V (r)| strong = V (r) DS ≡ ∆r 5 n=1 V ∆r (r n )δ(r − r n ) r ≤ r c = fm(22) where V ∆r (r n ) become the fitting parameters which depend on the resolution scale ∆r and whose total number can be estimated a priori 30 . This can be seen by inspecting Fig. 2 (right panel) by counting how many partial waves and how many points r n per partial wave sample the interaction with resolution ∆r. This viewpoint provides a rationale for the number of parameters, N Par , (typically about 40-50) that were traditionally needed in the past for high quality fits and allows at the same time the most succesful pp and np fit to date. Fitting and selecting data Fitting and selecting data are intertwined, particularly when there are incompatible data, as it is the case in NN scattering. We have collected N Dat = 8000 np+pp scattering data, O exp i , measured between 1950 and 2013 below E LAB = 350MeV, with given experimental uncertainty ∆O i . On the other hand the NN potential V NN (r) = V DS (r)θ(r c − r) + {V NN (r)| OPE + V NN (r)| em } θ(r − r c )(23) will generate O th i , and we can minimize the distance between theory and experiment by tuning the fitting parameters, the potential at the coarse grained distances, [V (r n )] JS l,l , and the pion-nucleon coupling constants f 2 0 , f 2 p , f 2 c . Even in the case of mutually consistent data, we can never be sure that the phenomenological theory is correct, so one poses the classical (and admittedly twisted) statistical question as follows: Assuming that the theory is correct, what is the probability q that the data are not described by the theory ?. This corresponds to find the probability that for the measured observables, we cannot say that the true values fulfill the relation O i = O th i + ξ i ∆O i ,(24) with i = 1, . . . , N Dat and ξ i are independent random normal variables with vanishing mean value ξ i = 0 and unit variance ξ i ξ j = δ ij , implying that O i = O th i . If this probability is large then we can discard the phenomenological theory and look for a better one. But if it turns out to be small, there is no good reason to discard it, and moreover we can vary the parameters in such a way that they cover the fluctuations in the data. The p-value is p = 1 − q and determines the confidence level we have on the theory. This is the standard set up for the χ 2 least square fitting, since the sum of ν-gaussians, ν i=1 ξ 2 i has a χ 2 -distribution with ν-degrees of freedom. Of course, one can only check this assumption after the optimal fit has been carried out, and determining whether the outcoming residuals, R i = O exp i − O th i ∆O i(25) belong to a gaussian distribution as we initially assumed. If this is the case the fit is self-consistent. The most popular χ 2 -test provides a p-value of 68% when 1 − 2/ν ≤ χ 2 min /ν ≤ 1 + 2/ν(26) where ν = N Dat − N Par . In previous works we extensively discuss more stringent tests as well as the conditions allowing a legitimate global scaling of errors 31,32,33 . One important issue here is the role played by the number of fitting parameters, which we claim to be optimally fixed by the maximum scattering LAB energy. Obviously, if we have too few parameters a successfull fit will not be acomplished a . On the other hand, although there is no limit in principle to include more parameters, we expect strong correlations among them which display explicitly a undesirable parameter redundancy and no real fit improvement b . In our case we found about 50 parameters to be realistic and with moderate correlations 31,32 . All this is fine provided we have a collection of mutually compatible data. When this is not the case, we may ask which experiment or datum including its error estimate is correct O exp i ± ∆O i . This may not necessarily mean genuinely wrong experiments, but rather unrealistic error estimates. Note that the main purpose of a fit is to determine the true values of certain parameters with a given and admissible confidence level, so we search for a maximization of experimental consensus by excluding data sets inconsistent with the rest of the database within the fitting model. We extend the Nijmegen 3σ criterion 41 by the following selection process: (1) Fit to all data. If χ 2 /ν < 1 you can stop. If not proceed further. (2) Remove data sets with improbably high or low χ 2 (3σ criterion) (3) Refit parameters for the remaining data. (4) Re-apply 3σ criterion to all data (5) Repeat until no more data are excluded or recovered a Of course with decreasing energy the number of essential parameters decreases; for instance at threshold only two scattering lengths are needed for the only non-vanishing S-wave contributions. b In addition, the covariance matrix size increases and, furthermore, may become numerically singular, preventing both efficiently finding an optimum and making an assesment of uncertainties. The effect of the selection criterion is to go from χ 2 /ν| all = 1.41 to χ 2 /ν| selected = 1.05 with a reduction in the number of data from N Data = 8173 to N Data = 6713. While this seems a drastic rejection of data it allows to perform the largest selfconsistent fit to date below 350 MeV. For such a large number of data this is not a minor improvement; it makes the difference between having p 1 or p ∼ 0.68. In the process of selecting and fitting we have learned some features of the phenomenological interaction within the coarse grained aproach where no a priori condition on the fitting parameters was imposed. We can fit the pp database independently. However, the isovector phases in the pn system are largely uncontrolled by the np data. Therefore it is preferable to fit the pp first, and to refit the pp+np system simultaneously by making some statistically testable isospin assumptions. We have also tried to use normality of the residuals as a rejection tool, without much success. The reason is that a normality test checks the excess or deficit of residuals as compared to the guess distribution, but does not indicate specifically which data are responsible for normality deviations. Likewise, we have analyzed the robustness of the database with respect to restricting or enlarging the rejection level from 3σ to 2σ or 4σ, respectively. In the first case, it was not possible to find a self-consistent database with the number of accepted data fluctuating from subsequent fits. This is probably related to the grouping of data in a common experiment, preventing a stable decision of accepting/rejecting data groups. In the 4σ-rejection level case, the self-consistent database exists but does not comply to the normality test at the imposed significance level. Why are these apparently small details important?. One reason is that nuclear structure calculations are insensitive to long range potential, but quite dependent on the least known mid range part, so that errors are propagated to binding energies or matrix elements 22,23,24,34,35,36 . The role played by χTPE and determination of chiral constants c 1 , c 3 c 4 37,38 (where N Par = 30 and r c = 1.8) and inclusion ∆-resonance 39 have been analyzed with the Granada-2013 database. . In all, the present situation regarding both the selection of data and the normality of residuals is highly satisfactory. In our view, this combined consistency of the statistical assumptions and the model analyzing them provides a good starting point to proceed further to determine the pion-nucleon coupling constants. Determination of πNN coupling constants The charge symmetry breaking is restricted to mass differences by setting f p = −f n = f c = f and the value f 2 | Nij = 0.0750(9) recommended by the Nijmegen group 40 has been used in most of the potentials since the seminal 1993 partial wave analysis 41 . In their 1997 status report 5 the Nijmegen group wrote: "The present accuracies in the determination of the various coupling constants are such, that with a little improvement in the data and in the analyses these charge-independence breaking effects could be checked.". The Granada-2013 database has 6713 data compared to the 4313 of Nijmegen-1993. Can this be the invoked little improvement ?. We try to answer this by recalling that electroweak corrections scale with the fine structure constant α = 1/137 and the light quark mass differences. Thus δg g = O α, m u − m d Λ QCD = O α, M p − M n Λ QCD(27) for the relative change around a mean value. These are naturally at the 1 − 2% level, a small effect. The question is on how many independent measurements are needed to achieve this desired accuracy. According to the central limit theorem, for N direct independent measurements the relative standard deviation scales as ∆g g = O 1 √ N and δg ∼ ∆g for N = 7000 − 10000. We cannot carry out these direct measurements of g but we can proceed indirectly by considering a set of mutually consistent NN scattering measurements O exp i with i = 1, . . . , N Dat and use a model with g and λ = (λ 1 , . . . , λ Npar ) parameters, which produces O th i (g, λ). We can then eliminate the parameters λ, in favor of N Par experiments and we are left with N = N Dat − N Par independent observables which depend just on g providing N independent determinations. Of course, these measurements will have some statistical error, so that O i =Ō i ± ∆O i , which means that O i is a random variable, and a χ 2 -fit is nothing but a democratic way of eliminating the parameters. Since N Dat N Par , we need about N Dat ∼ 7000 − 10000 to witness isospin breaking with the coarse grained interaction, and in our recent work we do 42 . From our full covariance matrix analysis we get for the g s g 2 n /(4π) = 14.91(39) , g 2 c /(4π) = 13.81(11), g 2 p /(4π) = 13.72 (7) . (28) We thus confirm the premonition of the Nijmegen group, although further "little improvements" are still needed to confirm an ordering pattern, such as e.g. Eq. (15). In Fig. 6 we show a chronological recreation of f 2 p and f 2 determinations using at any rate the NN data of the complete database measured up to a given year, which expectedly resembles the historic plot 6 . We consider in any case and, when needed, the corresponding 3σ-consistent database. We also plot several Nijmegen determinations including f 2 | Nij 5 . Assuming a unique pion-nucleon coupling constant we obtain f 2 = 0.07611(33) which is 1σ compatible but almost three times more accurate. The latest most accurate πN scattering determinations has are based on the GMO rule 43 with g 2 c /(4π) = 14.11 (20), use fixed-t dispersion relations 44 with g 2 c /(4π) = 13.76 (8) and are based on πN scattering lengths and π − d scattering and the GMO sum rule yielding g 2 c /(4π) = 13.69 (19) 45 . Our value, Eq. (28) is compatible with this last determination, but twice more accurate. Conclusions According to our analysis neutrons interact more strongly than protons above r c = 3fm, but we cannot check what is the influence on neutron-neutron scattering as we have not determined the nn-interaction below r c . The traditional and compelling explanation that the nn scatering length is larger than the strong contribution to the pp-scattering length (a model dependent quantity) would rest on the assumption that there is no relevant isospin breaking below 3 fm, a fact that is not supported by our analysis and requires further understanding. Isospin breaking at short distances has always been a difficult subject. We suggest to cut the gordian knot by separating the NN interaction in two distinct regions marked by a short distance radius r c and assuming a charge dependent one pion exchange above r c , as the unique strong contribution. Below this radius we purposely ignore the specific form of the interaction by coarse graining it down to ∆r = 0.6fm, the shortest deBroglie wavelength before pions are produced. This distance turns out to be r c = 3fm and we can describe 6727 NN sattering data with a total χ 2 = 6907 and 55 short distance parameters plus f 2 p , f 2 0 and f 2 c . Fig. 1 . 1Feynman diagrams contributing to the charge dependent one pion exchange interaction. Fig. 2 . 2Two complementary views of the anatomy of the NN interaction. Left panel: Fig. 3 . 3Renormalized OPE deuteron properties using the physical binding energy, average pion mass and asymptotic D/S ratio η = 0.0256(4) as a function f 2 πN N compared with experiment. Fig. 4 . 4Left panel: Proton-proton electrostatic interaction as a function of the separation distance. We compare point like protons with the charge density deduced from the electric form factor. Right panel: : Electromagnetic interactons (note the log-log scale): We show the Coulomb interaction, Magnetic Moments, Relativistic Coulomb, Vacuum polarization. Fig. 5 . 5NN Potentials in the Born-Oppenheimer, including OPE and TPE via ∆ excitation, as a function of NN distance. We show the effect of axial-vector (transition) form factors, generating an axial screening of the interaction below the elementarity radius re ∼ 2fm. Fig. 6 . 6(Color online) Chronological recreation of pion-nucleon coupling constants determinations from NN data compared to several Nijmegen determinations.(see main text) ,26 . At long distances one can determine the NN potential in the Born-Oppenheimer approximation, allowing for ∆ isobar intermediate states, but include the πN ∆ transition form factor. In the Chiral Quark model, where PCAC holds the vertices πN N , πN ∆ and π∆∆ are proportional to the Axial Form Factors, for which we may use axial-vector meson dominance27 . This provides a dominating contribution to the Two Pion ExchangeThree pion nucleon coupling constants 9 -10 -8 -6 -4 -2 0 1.6 1.8 2 2.2 2.4 2.6 2.8 3 V C (r) [MeV] r [fm] BO-TPE no-FF BO-TPE axial-FF 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1.6 1.8 2 2.2 2.4 2.6 2.8 3 V S (r) [MeV] r [fm] BO-TPE no-FF BO-TPE axial-FF -1 -0.8 -0.6 -0.4 -0.2 1.6 1.8 2 2.2 2.4 2.6 2.8 3 V T (r) [MeV] r [fm] BO-TPE no-FF BO-TPE axial-FF -1 -0.8 -0.6 -0.4 -0.2 0 1.6 1.8 2 2.2 2.4 2.6 2.8 3 W C (r) [MeV] r [fm] BO-TPE no-FF BO-TPE axial-FF 0 0.2 0.4 0.6 0.8 1 1.2 1.6 1.8 2 2.2 2.4 2.6 2.8 3 W S (r) [MeV] r [fm] OPE no-FF OPE axial-FF BO-TPE no-FF BO-TPE axial-FF -1 0 1 2 3 4 5 6 7 8 1.6 1.8 2 2.2 2.4 2.6 2.8 3 W T (r) [MeV] r [fm] OPE no-FF OPE axial-FF BO-TPE no-FF BO-TPE axial-FF AcknowledgmentsOne of us (E.R.A.) would like to thank J.L. Goity and J. Ruiz de Elvira for discussions, the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support and the organizers I. Caprini, K. Chetyrkin, C. 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[ "A comparison of estimators of mean and its functions in finite populations", "A comparison of estimators of mean and its functions in finite populations" ]
[ "Anurag Dey \nIndian Statistical Institute\nKolkata\n", "Probal Chaudhuri \nIndian Statistical Institute\nKolkata\n" ]
[ "Indian Statistical Institute\nKolkata", "Indian Statistical Institute\nKolkata" ]
[]
Several well known estimators of finite population mean and its functions are investigated under some standard sampling designs. Such functions of mean include the variance, the correlation coefficient and the regression coefficient in the population as special cases. We compare the performance of these estimators under different sampling designs based on their asymptotic distributions. Equivalence Let us begin by providing the expressions (seeTable 8below) of those estimators of Table 8: Estimators of Y Estimator ExpressionY , which are considered in this paper. InTable 8, {π i } N i=1 denote inclusion probabilities, and G i is the total of the x values of that randomly formed group from which the i th population unit is selected in the sample by RHC sampling design (cf. Chaudhuri et al. (2006)). In the case of the GREG estimator,Ŷ * = i∈s d(i, s)Y i / i∈s d(i, s),where {d(i, s) : i ∈ s} are sampling design weights. Finally, the c i 's (> 0) in the PEML estimator are obtained by maximizing i∈s d(i, s) log(c i ) subject to i∈s c i =1 and i∈s c i (X i − X)=0. Following Chen and Sitter (1999), we consider both the GREG and the PEML estimators with d(i, s)=(N π i ) −1 under SRSWOR, LMS sampling design and any HEπPS sampling design, and with d(i, s)=G i /N X i under RHC sampling design.Let us denote the HT, the RHC, the Hájek, the ratio, the product, the GREG and the PEML estimators of population means of h(y) byĥ HT ,ĥ RHC ,ĥ H ,ĥ RA ,ĥ P R ,ĥ GREG andĥ P EM L , respectively. Now, we give the proofs of Theorems 1, 4 and 5. The proofs of Remark 1 and Theorems 2, 3, 6 and 7 are given in Section S3 of the supplement.AbstractIn this supplement, we discuss conditions C1 through C4 from the main paper and demonstrate situations, where these conditions hold. Then, we state and prove some additional mathematical results. We also give the proofs of Remark 1 and Theorems 2, 3, 6 and 7 of the main text. The biased estimators considered in the main paper are then compared empirically with their bias-corrected versions based on jackknifing in terms of MSE. Finally, we provide the numerical results related to the analysis based on both synthetic and real data.
10.5705/ss.202022.0181
[ "https://export.arxiv.org/pdf/2305.15019v1.pdf" ]
258,012,516
2305.15019
6fd7623c52fcb8398ebfd4182c6caeedd88e5303
A comparison of estimators of mean and its functions in finite populations 24 May 2023 Anurag Dey Indian Statistical Institute Kolkata Probal Chaudhuri Indian Statistical Institute Kolkata A comparison of estimators of mean and its functions in finite populations 24 May 20231and phrases: Asymptotic normalityEquivalence classes of estimatorsHigh entropy sampling designsInclusion probabilityLinear regression modelRejective sampling designRelative efficiencySuperpopulation models Several well known estimators of finite population mean and its functions are investigated under some standard sampling designs. Such functions of mean include the variance, the correlation coefficient and the regression coefficient in the population as special cases. We compare the performance of these estimators under different sampling designs based on their asymptotic distributions. Equivalence Let us begin by providing the expressions (seeTable 8below) of those estimators of Table 8: Estimators of Y Estimator ExpressionY , which are considered in this paper. InTable 8, {π i } N i=1 denote inclusion probabilities, and G i is the total of the x values of that randomly formed group from which the i th population unit is selected in the sample by RHC sampling design (cf. Chaudhuri et al. (2006)). In the case of the GREG estimator,Ŷ * = i∈s d(i, s)Y i / i∈s d(i, s),where {d(i, s) : i ∈ s} are sampling design weights. Finally, the c i 's (> 0) in the PEML estimator are obtained by maximizing i∈s d(i, s) log(c i ) subject to i∈s c i =1 and i∈s c i (X i − X)=0. Following Chen and Sitter (1999), we consider both the GREG and the PEML estimators with d(i, s)=(N π i ) −1 under SRSWOR, LMS sampling design and any HEπPS sampling design, and with d(i, s)=G i /N X i under RHC sampling design.Let us denote the HT, the RHC, the Hájek, the ratio, the product, the GREG and the PEML estimators of population means of h(y) byĥ HT ,ĥ RHC ,ĥ H ,ĥ RA ,ĥ P R ,ĥ GREG andĥ P EM L , respectively. Now, we give the proofs of Theorems 1, 4 and 5. The proofs of Remark 1 and Theorems 2, 3, 6 and 7 are given in Section S3 of the supplement.AbstractIn this supplement, we discuss conditions C1 through C4 from the main paper and demonstrate situations, where these conditions hold. Then, we state and prove some additional mathematical results. We also give the proofs of Remark 1 and Theorems 2, 3, 6 and 7 of the main text. The biased estimators considered in the main paper are then compared empirically with their bias-corrected versions based on jackknifing in terms of MSE. Finally, we provide the numerical results related to the analysis based on both synthetic and real data. It is also shown that for the variance, the correlation coefficient and the regression coefficient of the population, the plug-in estimators based on the PEML estimator have the lowest asymptotic MSEs among all the estimators considered in this paper under SRSWOR. On the other hand, for any high entropy πPS (HEπPS) sampling design, which uses the auxiliary information, the plug-in estimators of those parameters based on the Hájek estimator have the lowest asymptotic MSEs among all the estimators considered in this paper. Introduction Suppose that P={1, 2, . . . , N } is a finite population of size N , s is a sample of size n (< N ) from P, and S is the collection of all possible samples having size n. Then, a sampling design P (s) is a probability distribution on S such that 0 ≤ P (s) ≤ 1 for all s ∈ S and s∈S P (s)=1. In this paper, simple random sampling without replacement (SRSWOR), Lahiri-Midzuno-Sen (LMS) sampling design (see Lahiri (1951), Midzuno (1952) and Sen (1953)), Rao-Hartley-Cochran (RHC) sampling design (see Rao et al. (1962)) and high entropy πPS (HEπPS) sampling designs (see Section 2) are considered. Note that all of the above sampling designs except SRSWOR are implemented utilizing some auxiliary variable. Let (Y i , X i ) be the value of (y, x) for the i th population unit, i=1, . . . , N , where y is a univraite or multivariate study variable, and x is a positive real valued size/auxiliary variable. Suppose that Y = N i=1 Y i /N is the finite population mean of y. The Horvitz-Thompson (HT) estimator (see Horvitz and Thompson (1952))) and the RHC (see Rao et al. (1962)) estimator are commonly used design unbiased estimators of Y . Other well known estimators of Y are the Hájek estimator (see Hájek (1971), Särndal et al. (2003) and references therein), the ratio estimator (see Cochran (1977)), the product estimator (see Cochran (1977)), the generalized regression (GREG) estimator (see Chen and Sitter (1999)) and the pseudo empirical likelihood (PEML) estimator (see Chen and Sitter (1999)). However, these latter estimators are not always design unbiased. For the expressions of the above estimators, the reader is referred to the Appendix. Now, suppose that y is a R d -valued (d ≥ 1) study variable, and g( N i=1 h(Y i )/N ) is a population parameter. Here, h: R d → R p is a function with p ≥ 1 and g: R p → R is a continuously differentiable function. All vectors in Euclidean spaces will be taken as row vectors and superscript T will be used to denote their transpose. Examples of such a parameter are the variance, the correlation coefficient, the regression coefficient, etc. associated with a finite population. For simplicity, we will often write h(Y i ) as h i . Then, g(h)=g( N i=1 h i /N ) is estimated by plugging in the estimatorĥ of h. In this article, our objective is to find asymptotically efficient (in terms of mean squared error (MSE)) estimator of g(h). In Section 2, based on the asymptotic distribution of the estimator of g(h) under above sampling designs, we construct equivalence classes of estimators such that any two estimators in the same class have the same asymptotic MSE. We consider the special case, when g(h)=Y , and compare equivalence classes of estimators under superpopulations satisfying linear models in Section 3. Among different estimators under different sampling designs considered in this article, the PEML estimator of the population mean under SRSWOR turns out to be the estimator with the lowest asymptotic MSE. Also, the PEML estimator has the same asymptotic MSE under SRSWOR and LMS sampling design. Interestingly, we observe that the performance of the PEML estimator under RHC and any HEπPS sampling designs, which use auxiliary information is worse than its performance under SRSWOR. Earlier, it was shown that the GREG estimator is asymptotically at least as efficient as the HT, the ratio and the product estimators under SRSWOR (see Cochran (1977)). It will follow from our analysis that the PEML estimator is asymptotically equivalent to the GREG estimator under all the sampling designs considered in this paper. In Section 3, we consider the cases, when g(h) is the variance, the correlation coefficient and the regression coefficient in the population. Note that if the estimators of the population variance are constructed by plugging in the HT, the ratio, the product or the GREG estimators of the population means, then the estimators of the variance may become negative. For this reason, one also faces problem with the plug-in estimators of the correlation coefficient and the regression coefficient as these estimators require estimators of population variances. On the other hand, if the estimators of the above-mentioned parameters are constructed by plugging in the Hájek or the PEML estimators of the population means, such a problem does not occur. Therefore, for these parameters, we compare only those equivalence classes, which contain the plug-in estimators based on the Hájek and the PEML estimators. From this comparison under superpopulations satisfying linear models, we once again conclude that for any of these parameters, the plug-in estimator based on the PEML estimator has asymptotically the lowest MSE among all the estimators considered in this article under SRSWOR as well as LMS sampling design. Moreover, under any HEπPS sampling design, which uses the auxiliary information, the plug-in estimator based on the Hájek estimator has asymptotically the lowest MSE among all the estimators considered in this article. Scott and Wu (1981) proved that the ratio estimator has the same asymptotic distribution under SRSWOR and LMS sampling design. Chen and Sitter (1999) showed that the PEML estimator is asymptotically equivalent to the GREG estimator under some conditions on the sampling design, which are satisfied by SRSWOR and RHC sampling design. However, asymptotic equivalence classes as in Table 2 in Section 2, which consist of several estimators of a function of the population means under several sampling designs, were not constructed by any earlier author. Raj (1954) compared the sample mean under the simple random sampling with replacement with the usual unbiased estimator of the population mean under the probability proportional to size sampling with replacement, when the study variable and the size variable are exactly linearly related. Avadhani and Sukhatme (1970) compared the ratio estimator of the population mean under SRSWOR with the RHC estimator under RHC sampling design, when an approximate linear relationship holds between the study variable and the size variable. Avadhani and Srivastava (1972) carried out the comparison of the ratio estimator of the population mean under LMS sampling design and the RHC estimator under RHC sampling design, when the study variable and the size variable are approximately linearly related. It was shown that the GREG estimator of the population mean is asymptotically at least as efficient as the HT, the ratio and the product estimators under SRSWOR (see Cochran (1977)). However, the above comparisons included neither the PEML estimator nor HEπPS sampling designs. Some empirical studies carried out in Section 4 using synthetic and real data demonstrate that the numerical and the theoretical results corroborate each other. We make some remarks on our major findings in Section 5. Proofs of the results are given in the Appendix. Comparison of different estimators of g(h) In this section, we compare the estimators of g(h), which are obtained by plugging in the estimators of h mentioned in Table 1 below. First, we find equivalence classes of such that any two estimators in the same class are asymptotically normal with the same mean g(h) and the same variance. We define our asymptotic framework as follows. Let {P ν } be a sequence of nested populations with N ν , n ν → ∞ as ν → ∞ ( see Isaki and Fuller (1982), Wang and Opsomer (2011), Conti and Marella (2015), Boistard et al. (2017), Han and Wellner (2021) and references therein), where N ν and n ν are, respectively, the population size and the sample size corresponding to the ν th population. Henceforth, we shall suppress the subscript ν that tends to ∞ for the sake of simplicity. Throughout this paper, we C 0. n/N → λ as ν → ∞, where 0 ≤ λ < 1. Before we state the main results, let us discuss the HEπPS sampling design and some conditions on {(X i , h i ) : 1 ≤ i ≤ N } (recall that h i =h(Y i )). A sampling design P (s) satisfying the condition, D(P ||R)= s∈S P (s) log (P (s)/R(s)) → 0 as ν → ∞, for some rejective sampling design (see Hájek (1964)) R(s) is known as the high entropy sampling design ( see Berger (1998), Conti (2014), Cardot et al. (2014), Boistard et al. (2017) and references therein). A sampling design P (s) is called the HEπPS sampling design if it is a high entropy sampling design, and its inclusion probabilities satisfy the condition π i =nX i / N i=1 X i for i=1, . . . , N . An example of the HEπPS sampling design is the Rao-Sampford (RS) sampling design (see Sampford (1967) and Berger (1998)). We now state the following conditions. (1) as ν → ∞. Further, lim ν→∞ h exists, and X= N i=1 X i /N and S 2 x = N i=1 (X i − X) 2 /N are bounded away from 0 as ν → ∞. Moreover, ∇g(µ 0 ) ̸ = 0, where µ 0 =lim ν→∞ h and ∇g is the gradient of g. C 1. {P ν } is such that N i=1 ||h i || 4 /N =O(1) and N i=1 X 4 i /N =OC 2. max 1≤i≤N X i / min 1≤i≤N X i =O(1) as ν → ∞. Let V i be one of h i , h i − h, h i − hX i /X, h i + hX i /X and h i − h − S xh (X i − X)/S 2 x for i=1, . . . , N , h= N i=1 h i /N and S xh = N i=1 X i h i /N − h X. Define T= N i=1 V i (1 − π i )/ N i=1 π i (1 − π i ), where π i is the inclusion probability of the i th population unit. Also, in the case of RHC sampling design, define 1), where N i is the size of the i th group formed randomly in RHC sampling design (see Rao et al. (1962)), i=1, . . . , n. Now, we state the following conditions on the population values and the sampling designs. V= N i=1 V i /N , X= N i=1 X i /N and γ= n i=1 N i (N i − 1)/N (N −C 3. P (s) is such that nN −2 N i=1 (V i − Tπ i ) T (V i − Tπ i )(π −1 i − 1) converges to some positive definite (p.d.) matrix as ν → ∞. Boistard et al. (2017), andF2 andF3 in Han andWellner (2021)). Conditions C1 and C4 hold (almost surely), whenever {(X i , h i ) : 1 ≤ i ≤ N } are generated from a superpopulation model satisfying appropriate moment conditions (see Lemma S2 in the supplement). The condition N i=1 ||h i || 4 /N =O(1) holds, when h is a bounded function (e.g., h(y)=y and y is a binary study variable). Condition C2 implies that the variation in the population values X 1 , . . . , X N cannot be too large. Under any πPS sampling design, C2 is equivalent to the condition that L ≤ N π i /n ≤ L ′ for some constants L, L ′ > 0, any i=1, . . . , N and all sufficiently large ν ≥ 1. This latter condition was considered earlier in the literature (see (C1) in Boistard et al. (2017) and Assumption 2-(i) in Wang and Opsomer (2011)). Condition C2 holds (almost surely), when C 4. nγXN −1 N i=1 (V i − X i V/X) T (V i − X i V/X)/X i{X i } N i=1 are generated from a superpopulation distribution, and the support of the distribution of X i is bounded away from 0 and ∞. Condition C3 holds (almost surely) for SRSWOR, LMS sampling design and any πPS sampling design under appropriate superpopulation models (see Lemma S2 in the supplement). For the RHC sampling design, we also assume that {N i } n i=1 are as follows. N i =              N/n, for i = 1, · · · , n, when N/n is an integer, ⌊N/n⌋, for i = 1, · · · , k, and ⌊N/n⌋ + 1, for i = k + 1, · · · , n, when N/n is not an integer, where k is such that n i=1 N i =N . Here, ⌊N/n⌋ is the integer part of N/n. Rao et al. (1962) showed that this choice of {N i } n i=1 minimizes the variance of the RHC estimator. Now, we state the following theorem. Theorem 1. Suppose that C0 through C3 hold. Then, classes 1, 2, 3 and 4 in Table 2 describe equivalence classes of estimators for g(h) under SRSWOR and LMS sampling design. For next two theorems, we assume that n max 1≤i≤N X i / N i=1 X i < 1. Note that this condition is required to hold for any without replacement πPS sampling design. Theorem 2. (i) If C0 through C3 hold, then classes 5, 6 and 7 in Table 2 describe equivalence classes of estimators for g(h) under any HEπPS sampling design. (ii) Under RHC sampling design, if C0 through C2 and C4 hold, then classes 8 and 9 in Table 2 describe equivalence classes of estimators for g(h). Remark 1. It is to be noted that if C1 through C3 hold, and C0 holds with λ=0, then in Table 2, class 8 is merged with class 5, and class 9 is merged with class 6. For details, see Section S3 in the supplement. HEπPS class 5 2 class 6 class 7 2 class 6 2 class 6 RHC class 8 class 9 1 The HT and the Hájek estimators coincide under SRSWOR. 2 The HT, the ratio and the product estimators coincide under HEπPS sampling designs. Next, suppose that W i =∇g(h)h T i for i=1, . . . , N , W = N i=1 W i /N , S xw = N i=1 W i X i /N − W X, S 2 w = N i=1 W 2 i /N −W 2 , S 2 x = N i=1 X 2 i /N − X 2 and ϕ=X − (n/N ) N i=1 X 2 i /N X. Now, we state the following theorem. Theorem 3. Suppose that the assumptions of Theorems 1 and 2 hold. Then, Table 3 gives the expressions of asymptotic MSEs, ∆ 2 1 , . . . , ∆ 2 9 , of the estimators in equivalence classes 1, . . . , 9 in Table 2, respectively. Remark 2. It can be shown in a straightforward way from Table 3 that ∆ 2 1 ≤ ∆ 2 i for i=2, 3 and 4. Thus, both the plug-in estimators of g(h) that are based on the GREG and the PEML estimators are asymptotically as good as, if not better than, the plug-in estimators based on the HT (which coincides with the Hájek estimator), the ratio and the product estimators under SRSWOR, and the plug-in estimators based on the HT, the Hájek, the ratio and the product estimators under LMS sampling design. Let us now consider some examples of g(h) in Table 4 below. Conclusions of Theorems 1 through 3, and Remarks 1 and 2 hold for all the parameters in Table 4. Here, we recall from the introduction that for the variance, the correlation coefficient and the regression coefficient, we consider only the plug-in estimators that are based on the Hájek and the PEML estimators. Table 3: Asymptotic MSEs of estimators for g(h) (note that for simplifying notations, the subscript ν is dropped from the expressions on which limits are taken.) ∆ 2 1 =(1 − λ) lim ν→∞ S 2 w − (S xw /S x ) 2 ∆ 2 2 =(1 − λ) lim ν→∞ S 2 w ∆ 2 3 =(1 − λ) lim ν→∞ S 2 w − 2W S xw /X + W /X 2 S 2 x ∆ 2 4 =(1 − λ) lim ν→∞ S 2 w + 2W S xw /X + W /X 2 S 2 x ∆ 2 5 = lim ν→∞ (1/N ) N i=1 W i − W − (S xw /S 2 x )(X i − X) 2 × (X/X i ) − (n/N ) ∆ 2 6 = lim ν→∞ (1/N ) N i=1 W i + ϕ −1 X −1 X i (n/N ) N i=1 W i X i /N − W X 2 × (X/X i ) − (n/N ) ∆ 2 7 = lim ν→∞ (1/N ) N i=1 W i − W + (n/N ϕX)X i S xw 2 × (X/X i ) − (n/N ) ∆ 2 8 = lim ν→∞ nγ(X/N ) N i=1 W i − W − (S xw /S 2 x )(X i − X) 2 /X i ∆ 2 9 = lim ν→∞ nγ (X/N ) N i=1 W 2 i /X i − W 2 3 Comparison of estimators under superpopulation models In this section, we derive asymptotically efficient estimators for the mean, the variance, the correlation coefficient and the regression coefficient under superpopulations satisfying linear regression models. Earlier, Raj (1954) Murthy (1967, Avadhani and Sukhatme (1970), Avadhani and Srivastava (1972) and Cochran (1977) used the linear relationship between the Y i 's and the X i 's for comparing different estimators of the mean. However, they did not use any probability distribution for the (Y i , X i )'s. Subsequently, Rao (2003), Fuller (2011), Chaudhuri (2014) (see chap. 5) and some other authors considered the linear relationship between the Y i 's and the X i 's and a probability distribution for the (Y i , X i )'s for constructing different estimators and studying their behaviour. However, the problem of finding asymptotically the most efficient estimator for the mean among a large class of estimators as considered in this paper was not done earlier in the literature. Correlation 2 5 h(z 1 , z 2 )=(z 1 , z 2 , g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − s 1 s 2 )/ coefficient z 2 1 , z 2 2 , z 1 z 2 ) ((s 3 − s 2 1 )(s 4 − s 2 2 )) 1/2 Regression 2 4 h(z 1 , z 2 )=(z 1 , z 2 , g(s 1 , s 2 , s 3 , s 4 , s 5 )= coefficient z 2 2 , z 1 z 2 ) (s 4 − s 1 s 2 )/(s 3 − s 2 2 ) Also, large sample comparisons of the plug-in estimators of the variance, the correlation coefficient and the regression coefficient considered in this paper were not carried out in the earlier literature. Suppose that {(Y i , X i ) : 1 ≤ i ≤ N } are i.i.d. random vectors defined on a probability space (Ω, F, P). Without any loss of generality, for convenience, we take σ 2 x =E P (X i − E P (X i )) 2 =1 . This might require rescaling the variable x. Here, E P denotes the expectation with respect to the probability measure P. Recall that the population values X 1 , . . . , X N are used to implement some of the sampling designs. In such a case, we consider a function P (s, ω) on S × Ω so that P (s, ·) is a random variable on Ω for each s ∈ S, and P (·, ω) is a probability distribution on S for each ω ∈ Ω (see Boistard et al. (2017)). Note that P (s, ω) is the sampling design for any fixed ω in this case. Then, the ∆ 2 j 's in Table 3 can be expressed in terms of superpopulation moments of (h(Y i ), X i ) by strong law of large numbers (SLLN). In that case, we can easily compare different classes of estimators in Table 2 under linear models. Let us first state the following conditions on superpopulation distribution P. C 5. X i ≤ b a.s. [P] for some 0 < b < ∞, E P (X i ) −2 < ∞, and max 1≤i≤N X i / min 1≤i≤N X i =O(1) as ν → ∞ a.s. [P]. Also, the support of the distribution of (h(Y i ), X i ) is not a subset of a hyper-plane in R p+1 . The condition, X i ≤ b a.s. [P] for some 0 < b < ∞, in C5 and C0 along with 0 ≤ λ < E P (X i )/b ensure that n max 1≤i≤N X i / N i=1 X i < 1 for all sufficiently large ν a.s. [P], which is required to hold for any without replacement πPS sampling design. On the other hand, the condition, max 1≤i≤N X i / min 1≤i≤N X i =O(1) as ν → ∞ a.s. [P], in C5 implies that C2 holds a.s. [P]. Further, C5 ensures that C4 holds a.s. [P] (see Lemma S2 in the supplement). C5 also ensures that C3 holds under LMS and any πPS sampling designs a.s. [P] (see Lemma S2 in the supplement). Let us first consider the case, when g(h) is the mean of y (see the 2 nd row in Table 4). Further, suppose that Y i =α + βX i + ϵ i for α, β ∈ R and i=1, . . . , N , where {ϵ i } N i=1 are i.i.d. random variables and are independent of {X i } N i=1 with E P (ϵ i )=0 and E P (ϵ i ) 4 < ∞. Then, we have the following theorem. Theorem 4. Suppose that C0 holds with 0 ≤ λ < E P (X i )/b, and C5 holds. Then, a.s. [P], the PEML estimator under SRSWOR as well as LMS sampling design has the lowest asymptotic MSE among all the estimators of the population mean under different sampling designs considered in this paper. Remark 3. Note that for SRSWOR, the PEML estimator of the population mean has the lowest asymptotic MSE among all the estimators considered in this paper a.s. [P], when C0 holds with 0 ≤ λ < 1 and C5 holds (see the proof of Theorem 4). Theorem 5. Suppose that C0 holds with 0 ≤ λ < E P (X i )/b, and C5 holds. Then, a.s. [P], the performance of the PEML estimator of the population mean under RHC and any HEπPS sampling designs, which use auxiliary information, is worse than its performance under SRSWOR. Recall from the introduction that for the variance, the correlation coefficient and the regression coefficient, we compare only those equivalence classes, which contain the plugin estimators based on the Hájek and the PEML estimators. We first state the following condition. C 6. ξ > 2 max{µ 1 , µ −1 /(µ 1 µ −1 − 1)}, where ξ=µ 3 − µ 2 µ 1 is the covariance between X 2 i and X i and µ j =E P (X i ) j , j=−1, 1, 2, 3. The above condition is used to prove part (ii) in each of Theorems 6 and 7. This condition holds when the X i 's follow well known distributions like Gamma (with shape parameter value larger than 1 and any scale parameter value), Beta (with the second shape parameter value greater than the first shape parameter value and the first shape parameter value larger than 1), Pareto (with shape parameter value lying in the interval (3, (5 + √ 17)/2) and any scale parameter value), Log-normal (with both the parameters taking any value) and Weibull (with shape parameter value lying in the interval (1, 3.6) and any scale parameter value). Now, consider the case, when g(h) is the variance of y (see the 3 rd row in Table 4). Recall the linear model Y i =α + βX i + ϵ i from above and assume that E P (ϵ i ) 8 < ∞. Then, we have the following theorem. Theorem 6. (i) Let us first consider SRSWOR and LMS sampling design and suppose that C0 and C5 hold. Then, a.s. [P], the plug-in estimator of the population variance based on the PEML estimator has the lowest asymptotic MSE among all the estimators considered in this paper. (ii) Next, consider any HEπPS sampling design and suppose that C0 holds with 0 ≤ λ < E P (X i )/b, and C5 and C6 hold. Then, a.s. [P], the plug-in estimator of the population variance based on the Hájek estimator has the lowest asymptotic MSE among all the estimators considered in this paper. Now, suppose that y=(z 1 , z 2 ) ∈ R 2 and consider the case, when g(h) is the correlation coefficient between z 1 and z 2 (see the 4 th row in Table 4). Let us also consider the case, when g(h) is the regression coefficient of z 1 on z 2 (see the 5 th row in Table 4). Further, suppose that Y i =α + βX i + ϵ i for Y i =(Z 1i , Z 2i ), α, β ∈ R 2 and i=1, . . . , N , where {ϵ i } N i=1 are i.i.d. random vectors in R 2 independent of {X i } N i=1 with E P (ϵ i )=0 and E P ||ϵ i || 8 < ∞. Then, we have the following theorem. Theorem 7. (i) Let us first consider SRSWOR and LMS sampling design and suppose that C0 and C5 hold. Then, a.s. [P], the plug-in estimator of each of the correlation and the regression coefficients in the population based on the PEML estimator has the lowest asymptotic MSE among all the estimators considered in this paper. (ii) Next, consider any HEπPS sampling design and suppose that C0 holds with 0 ≤ λ < E P (X i )/b, and C5 and C6 hold. Then, a.s. [P], the plug-in estimator of each of the above parameters based on the Hájek estimator has the lowest asymptotic MSE among all the estimators considered in this paper. Data analysis In this section, we carry out an empirical comparison of the estimators of the mean, the variance, the correlation coefficient and the regression coefficient, which are discussed in this paper, based on both real and synthetic data. Recall that for the above parameters, we have considered several estimators and sampling designs, and conducted a theoretical comparison of those estimators in Sections 2 and 3. For empirical comparison, we exclude some of the estimators considered in theoretical comparison so that the results of the comparison become concise and comprehensive. The reasons for excluding those estimators are given below. (i) Since the GREG estimator is well known to be asymptotically better than the HT, the ratio and the product estimators under SRSWOR (see Cochran (1977)), we exclude these latter estimators under SRSWOR. (ii) Since the MSEs of the estimators under LMS sampling design become very close to the MSEs of the same estimators under SRSWOR as expected from Theorem 1, we do not report these results under LMS sampling design. Moreover, SRSWOR is a simpler and more commonly used sampling design than LMS sampling design. Thus we consider the estimators mentioned in Table 5 below for the empirical comparison. Recall from Table 1 that the HT, the ratio and the product estimators of the mean coincide under any HEπPS sampling design. We draw I=1000 samples each of sizes n=75, 100 and 125 using sampling designs mentioned in Table 5. We use the R software for drawing samples as well as computing different estimators. For RS sampling design, we use the 'pps' package in R, and for the PEML estimator, we use R codes in Wu (2005). Two estimators g(ĥ 1 ) and g(ĥ 2 ) of g(h) under sampling designs P 1 (s) and P 2 (s), respectively, are compared empirically by means of the relative efficiency defined as RE(g(ĥ 1 ), P 1 |g(ĥ 2 ), P 2 ) = M SE P 2 (g(ĥ 2 ))/M SE P 1 (g(ĥ 1 )), where M SE P j (g(ĥ j ))=I −1 I l=1 (g(ĥ jl ) − g(h 0 )) 2 is the empirical mean squared error of g(ĥ j ) under P j (s), j=1, 2. Here,ĥ jl is the estimate of h based on the j th estimator and the l th sample, and g(h 0 ) is the true value of the parameter g(h), j=1, 2, l=1, . . . , I. g(ĥ 1 ) under P 1 (s) will be more efficient than g(ĥ 2 ) under P 2 (s) if RE(g(ĥ 1 ), P 1 |g(ĥ 2 ), P 2 ) > 1. and it is easier to implement than other HEπPS sampling designs. Next, for each of the parameters considered in this section, we compare average lengths of asymptotically 95% confidence intervals (CIs) constructed based on several estimators used in this section. In order to construct asymptotically 95% CIs, we need an estimator of the asymptotic MSE of √ n(g(ĥ) − g(h)) and we shall discuss it in detail now. If we consider SRSWOR or RS sampling design, it follows from the proofs of Theorems 1 and 2 that the asymptotic MSE of √ n(g(ĥ) − g(h)) is∆ 2 1 =lim ν→∞ nN −2 ∇g(h) N i=1 (V i − Tπ i ) T (V i − Tπ i )(π −1 i − 1)∇g(h) T , where T= N i=1 V i (1 − π i )/ N i=1 π i (1 − π i ). Moreover, V i is h i or h i − h or h i − h − S xh (X i − X)/S 2 x ifĥ isĥ HT orĥ H orĥ P EM L (as well aŝ h GREG ) with d(i, s)=(N π i ) −1 , respectively. Recall from the paragraph following C2 that S xh = N i=1 X i h i /N − X h.∆ 2 1 = nN −2 ∇g(ĥ) i∈s (V i −Tπ i ) T (V i −Tπ i )(π −1 i − 1)π −1 i ∇g(ĥ) T ,(2) whereT= i∈sV i (π −1 i − 1)/ i∈s (1 − π i ),ĥ=ĥ HT in the case of the mean, the variance and the regression coefficient, andĥ=ĥ H in the case of the correlation coefficient. Here, V i is h i or h i −ĥ HT or h i −ĥ HT −Ŝ xh,1 (X i −X HT )/Ŝ 2 x,1 ifĥ isĥ HT orĥ H orĥ P EM L (as well asĥ GREG ) with d(i, s)=(N π i ) −1 . Further,Ŝ xh,1 = i∈s (N π i ) −1 X i h i −X HTĥHT and Ŝ 2 x,1 = i∈s (N π i ) −1 X 2 i −X 2 HT . We estimate h in ∇g(h) byĥ HT in the case of the mean, the variance and the regression coefficient becauseĥ HT is an unbiased estimator, and it is easier to compute than the other estimators of h considered in this paper. On the other hand, different estimators of the correlation coefficient that are considered in this paper may become undefined if we estimate h by any estimator other thanĥ H andĥ P EM L (see the 4 th paragraph in the introduction). In this case, we chooseĥ H because it is easier to compute thanĥ P EM L . Next, if we consider RHC sampling design, it follows from the proof of Theorem 2 that the asymptotic MSE of √ n(g(h) − g(ĥ)) is∆ 2 2 =lim ν→∞ nγXN −1 × ∇g(h) N i=1 (V i − X i V/X) T (V i − X i V/X)X −1 i ∇g(h) T , where γ and V are as in the paragraph following C2. Moreover, V i is h i or h i − h − S xh (X i − X)/S 2 x ifĥ isĥ RHC orĥ P EM L (as well aŝ h GREG ) with d(i, s)=G i /N X i , respectively. We estimate∆ 2 2 bŷ ∆ 2 2 = nγXN −1 ∇g(ĥ) i∈s (V i − X iVRHC /X)× (V i − X iVRHC /X)(G i X −2 i )∇g(ĥ) T ,(3) whereV RHC = i∈sV i G i /N X i ,ĥ=ĥ RHC in the case of the mean, the variance and the regression coefficient, andĥ=ĥ P EM L in the case of the correlation coefficient. Here,V i is h i or h i −ĥ RHC −Ŝ xh,2 (X i − X)/Ŝ 2 x,2 ifĥ isĥ RHC orĥ P EM L (as well asĥ GREG ) with d(i, s)=G i /N X i . Further,Ŝ xh,2 = i∈s h i G i /N − Xĥ RHC andŜ 2 x,1 = i∈s X i G i /N − X 2 . In the case of the mean, the variance and the regression coefficient, we estimate h in ∇g(h) byĥ RHC for the same reason as discussed in the preceding paragraph, where we discuss the estimation of h byĥ HT under SRSWOR and RS sampling design. On the other hand, in the case of the correlation coefficient, we estimate h in ∇g(h) byĥ P EM L under RHC sampling design so that the estimator of the correlation coefficient appeared in the expression of ∇g(h) in this case becomes well defined. We draw I=1000 samples each of sizes n=75, 100 and 125 using sampling designs mentioned in Table 5. Then, for each of the parameters, the sampling designs and the estimators mentioned in Table 5, we construct I many asymptotically 95% CIs based on these samples and compute the average and the standard deviation of their lengths. Analysis based on synthetic data In this section, we consider the population values {(Y i , X i ) : 1 ≤ i ≤ N } on (y, x) generated from a linear model as follows. We choose N =5000 and generate the X i 's from a gamma distribution with mean 1000 and standard deviation (s.d.) 200. Then, Y i is generated from the linear model Y i =500 + X i + ϵ i for i=1, . . . , N , where ϵ i is generated independently of {X i } N i=1 from a normal distribution with mean 0 and s.d. 100. We also generate the population values {(Y i , X i ) : 1 ≤ i ≤ N } from a linear model, when y=(z 1 , z 2 ) is a bivariate study variable. The population values {X i } N i=1 are generated in the same way as in the earlier case. Then, Y i =(Z 1i , Z 2i ) is generated from the linear model Z ji =α j + X i + ϵ ji for i=1, . . . , N , where α 1 =500 and α 2 =1000. The ϵ 1i 's are generated independently of the X i 's from a normal distribution with mean 0 and s.d. 100, and the ϵ 2i 's are generated independently of the X i 's and the ϵ 1i 's from a normal distribution with mean 0 and s.d. 200. We consider the estimation of the mean and the variance of y for the first dataset and the correlation and the regression coefficients between z 1 and z 2 for the second dataset. The results of the empirical comparison based on synthetic data are summarized as follows. For each of the mean, the variance, the correlation coefficient and the regression coefficient, the plug-in estimator based on the PEML estimator under SRSWOR turns out to be more efficient than any other estimator under any other sampling design (see Tables 2 through 6 in the supplement) considered in Table 5 when compared in terms of relative efficiencies. Also, for each of the above parameters, asymptotically 95% CI based on the PEML estimator under SRSWOR has the least average length (see Tables 7 through 11 in the supplement). Thus the empirical results stated here corroborate the theoretical results stated in Theorems 4 through 7. Analysis based on real data In this section, we consider a dataset on the village amenities in the state of West Bengal in India obtained from the Office of the Registrar General & Census Commissioner, India (https://censusindia.gov.in/nada/index.php/catalog/1362). Relevant study variables for this dataset are described in Table 6 below. We consider the following estimation problems for a population consisting of 37478 villages. For these estimation problems, we use the number of people living in village x as the size variable. It can be shown from the scatter plot and the least square regression line in Figure 1 in the supplement that y 1 and x have an approximate linear relationship. Also, the correlation coefficient between y 1 and x is 0.72. On the other hand, y 2 and x do not seem to have a linear relationship (see the scatter plot and the least square regression line in Figure 2 in the supplement). (ii) Next, we consider the estimation of the correlation and the regression coefficients of y 1 and y 3 as well as of y 2 and y 4 . The scatter plot and the least square regression line in Figure 3 in the supplement show that y 3 does not seem to be dependent on x. Further, we see from the scatter plot and the least square regression line of y 4 and x (see Figure 4 in the supplement) that y 4 and x do not seem to have a linear relationship. The results of the empirical comparison based on real data are summarized in Table 7 below. For further details see Tables 12 through 31 in the supplement. The approximate linear relationship between y 1 and x (see the scatter plot and the least square regression line in Figure 1 in the supplement) could be a possible reason why the plug-in estimator based on the PEML estimator under SRSWOR becomes the most efficient for each of the mean and the variance of y 1 among all the estimators under different sampling designs considered in this section. Also, possibly for the same reason, the plug-in estimators of the correlation and the regression coefficients between y 1 and y 3 based on the PEML On the other hand, any of y 2 , and y 4 does not seem to have a linear relationship with x (see the scatter plots and the least square regression lines in Figures 2 and 4 in the supplement). Possibly, because of this reason, the plug-in estimators of the parameters related to y 2 and y 4 based on the PEML estimator are not able to outperform the the plug-in estimators of those parameters based on the HT and the Hájek estimators. Next, we observe that there are substantial correlation present between y 2 and x (correlation coefficient=0.67), and y 4 and x (correlation coefficient=0.25). Possibly, because of this, under RS sampling design, which uses the auxiliary information, the plug-in estimators of the parameters related to y 2 and y 4 based on the HT and the Hájek estimators become the most efficient among all the estimators under different sampling designs considered in this section. Concluding discussion and remarks It follows from Theorem 4 that the PEML estimator of the mean under SRSWOR becomes asymptotically either more efficient than or equivalent to any other estimator under any other sampling design considered in this paper. It also follows from Theorems 1 and 2 that the GREG estimator of the mean is asymptotically equivalent to the PEML estimator under different sampling designs considered in this paper. However, our numerical studies (see Section 4) based on finite samples indicate that the PEML estimator of the mean performs slightly better than the GREG estimator under all the sampling designs considered in Section 4 (see Tables 2, 12 and 14 in the supplement). Moreover, as pointed out in the introduction, if the estimators of the variance, the correlation coefficient and the regression coefficient are constructed by plugging in the GREG estimator of the mean, then the estimators of the population variances involved in these parameters may become negative. On the other hand, if the estimators of these parameters are constructed by plugging in the PEML estimator of the mean, then such a problem does not occur. Further, for these parameters, depending on sampling designs, the plug-in estimator based on either the PEML or the Hájek estimator turns out to be asymptotically best among different estimators that we have considered (see Theorems 6 and 7). We see from Theorem 4 that for the population mean, the PEML estimator, which is not design unbiased, performs better than design unbiased estimators like the HT and the RHC estimators. Further, as pointed out in the introduction, the plug-in estimators of the population variance based on the HT and the RHC estimators may become negative. This affects the plug-in estimators of the correlation and the regression coefficients based on the HT and the RHC estimators. It follows from Table 2 that under LMS sampling design, the large sample performances of all the estimators of functions of means considered in this paper are the same as their large sample performances under SRSWOR. The LMS sampling design was introduced to make the ratio estimator of the mean unbiased. It follows from Remark 2 in Section 2 that the performance of the ratio estimator of the mean is worse than several other estimators that we have considered even under LMS sampling design. The coefficient of variation is another well known finite population parameter, which can be expressed as a function of population mean g(h). We have d=1, p=2, h(y)=(y 2 , y) and g(s 1 , s 2 )= s 1 − s 2 2 /s 2 in this case. Among the estimators considered in this paper, the plug-in estimators of g(h) that are based on the PEML and the Hájek estimators of the mean can be used for estimating this parameter since it involves the finite population variance (see the 4 th paragraph in the introduction). We have avoided reporting the comparison of the estimators of the coefficient of variation in this paper because of complex mathematical expressions. However, the asymptotic results stated in Theorems 6 and 7 also hold for this parameter. An empirical comparison of the biased estimators considered in this paper and their bias-corrected versions are carried out based on jackknifing in Section S4 in the supplement. It follows from this comparison that for all the parameters considered in this paper, the bias-corrected estimators become worse than the original biased estimators in the cases of both the synthetic and the real data. This is because, although bias-correction results in reduction of biases in the original biased estimators, the variances of these estimators increase substantially after bias-correction. Supplementary material In the supplement, we discuss some conditions from the main paper and demonstrate situations, where these conditions hold. Then, we state and prove some additional mathematical results. We also give the proofs of Remark 1 and Theorems 2, 3, 6 and 7. The biased estimators considered in this paper are then compared empirically with their biascorrected versions based on jackknifing in terms of MSE. Finally, we provide the numerical results related to the analysis based on both synthetic and real data (see Section 4). for several discussions about Section 4.2 of the paper. Proof of Theorem 1. Let us consider SRSWOR and LMS sampling design. It follows from (i) in Lemma S6 in the supplement that √ n(ĥ−h) L − → N (0, Γ) as ν → ∞ for some p.d. matrix Γ, whenĥ is one ofĥ HT ,ĥ H ,ĥ RA ,ĥ P R , andĥ GREG with d(i, s)=(N π i ) −1 under any of these sampling designs. Now, note that max i∈s |X i − X|=o p ( √ n), and i∈s π −1 i (X i − X)/ i∈s π −1 i (X i − X) 2 =O p (1/ √ n) as ν → ∞ under the above sampling designs (see Lemma S8 in the supplement). Then, by applying Theorem 1 of Chen and Sitter (1999) to each real valued coordinate ofĥ P EM L andĥ GREG , we get √ n(ĥ P EM L −ĥ GREG )=o p (1) as ν → ∞ for d(i, s)=(N π i ) −1 under these sampling designs. This implies thatĥ P EM L andĥ GREG with d(i, s)=(N π i ) −1 have the same asymptotic distribution. Therefore, ifĥ is one ofĥ HT ,ĥ H ,ĥ RA ,ĥ P R , andĥ GREG andĥ P EM L with d(i, s)=(N π i ) −1 , we have √ n(g(ĥ) − g(h)) L − → N (0, ∆ 2 ) as ν → ∞ under any of the above-mentioned sampling designs for some ∆ 2 > 0 by the delta method and the condition ∇g(µ 0 ) ̸ = 0 at µ 0 =lim ν→∞ h. It can be shown from the proof of (i) in Lemma S6 in the supplement that ∆ 2 =∇g(µ 0 )Γ 1 (∇g(µ 0 )) T , where Γ 1 =lim ν→∞ nN −2 × N i=1 (V i −Tπ i ) T (V i −Tπ i )(π −1 i −1) . It can also be shown from Table 1 in the supplement that under each of the above sampling designs, V i in Γ 1 is h i or h i − h or h i − hX i /X or h i + hX i /X or h i − h − S xh (X i − X)/S 2 x ifĥ isĥ HT orĥ H orĥ RA orĥ P R , orĥ GREG with d(i, s)=(N π i ) −1 , respectively. Now, by (i) in Lemma S7 in the supplement, we have σ 2 1 = σ 2 2 = (1 − λ) lim ν→∞ N i=1 (A i −Ā) 2 /N.(5) where σ 2 1 and σ 2 2 are as defined in the statement of Lemma S7, and A i =∇g(µ 0 )V T i for different choices of V i mentioned in the preceding paragraph. Note that g(ĥ GREG ) and g(ĥ P EM L ) have the same asymptotic distribution under each of SRSWOR and LMS sampling design since √ n(ĥ P EM L −ĥ GREG )=o p (1) for ν → ∞ under these sampling designs as pointed out earlier in this proof. Further, (5) implies that g(ĥ GREG ) with d(i, s)=(N π i ) −1 has the same asymptotic MSE under SRSWOR and LMS sampling design. Thus g(ĥ GREG ) and g(ĥ P EM L ) with d(i, s)=(N π i ) −1 under SRSWOR and LMS sampling design form class 1 in Table 2. Next, (5) yields that g(ĥ HT ) has the same asymptotic MSE under SRSWOR and LMS sampling design. It also follows from (5) that g(ĥ H ) has the same asymptotic MSE under SRSWOR and LMS sampling design. Now, note that g(ĥ HT ) and g(ĥ H ) coincide under SRSWOR. Thus g(ĥ HT ) under SRSWOR, and g(ĥ HT ) and g(ĥ H ) under LMS sampling design form class 2 in Table 2. Next, (5) implies that g(ĥ RA ) has the same asymptotic MSE under SRSWOR and LMS sampling design. Further, (5) implies that g(ĥ P R ) has the same asymptotic MSE under SRSWOR and LMS sampling design. Thus g(ĥ RA ) under SRSWOR and LMS sampling design forms class 3 in Table 2, and g(ĥ P R ) under those sampling designs forms class 4 in Table 2. This completes the proof of Theorem 1. Table 3 can be expressed in terms of superpopulation moments of (Y i , X i ) a.s. [P] by SLLN since E P (ϵ i ) 4 < ∞. Recall from the beginning of Section 3 that we have taken σ 2 W i =∇g(h)h T i =Y i . Also, note that the ∆ 2 i 's in x =1. Then, we have ∆ 2 2 − ∆ 2 1 =(1 − λ)σ 2 xy , ∆ 2 3 − ∆ 2 1 =(1 − λ)(σ xy − E P (Y i )/µ 1 ) 2 and ∆ 2 4 − ∆ 2 1 =(1 − λ)(σ xy + E P (Y i )/µ 1 ) 2 a.s. [P] , where µ 1 =E P (X i ) and σ xy =cov P (X i , Y i ). Hence, ∆ 2 1 < ∆ 2 i a.s. [P] for i=2, 3, 4. Next consider the case of 0 ≤ λ < E P (X i )/b. Note that nγ → c as ν → ∞ for some c ≥ 1−λ by Lemma S1 in the supplement. Also, note that a.s. [P], C4 holds in the case of RHC sampling design and C3 holds in the case of any HEπPS sampling design (see Lemma S2 in the supplement). Then, under RHC and any HEπPS sampling designs, conclusions of Theorems 2 and 3 hold a.s. [P] for d=p=1, h(y)=y and g(s)=s. Further, we have ∆ 2 5 − ∆ 2 1 = E P Y i − E P (Y i )) 2 µ 1 /X i − λ − µ 2 1 σ xy σ xy cov P (X i , 1/X i ) − 2cov P (Y i , 1/X i ) + λσ 2 xy −(1−λ) σ 2 y −σ 2 xy , ∆ 2 6 −∆ 2 5 = E P Y 2 i µ 1 /X i −λ − λE P (Y i X i )−E P (Y i )µ 1 2 /χµ 1 − E P Y i − E P (Y i ) − σ xy (X i − µ 1 ) 2 µ 1 /X i − λ , ∆ 2 7 − ∆ 2 5 = µ 2 1 σ xy σ xy cov P (X i , 1/X i ) − 2cov P (Y i , 1/X i ) −λσ 2 xy −λ 2 σ 2 xy /µ 1 χ , ∆ 2 8 −∆ 2 1 =c µ 1 E P (Y i −E P (Y i )) 2 /X i −µ 2 1 σ xy (σ xy cov P (X i , 1/X i ) − 2cov P (Y i , 1/X i )) − (1 − λ) σ 2 y − σ 2 xy and ∆ 2 9 − ∆ 2 1 =c µ 1 E P (Y 2 i /X i ) − E 2 P (Y i ) − (1 − λ) σ 2 y − σ 2 xy a.s. [P], where σ 2 y =var P (Y i ), χ=µ 1 − λ(µ 2 /µ 1 ) and µ 2 =E P (X i ) 2 . Here, we note that χ=E P X 2 i (µ 1 /X i − λ) /µ 1 > 0 because C5 holds and C0 holds with 0 ≤ λ < E P (X i )/b. Moreover, from the linear model set up, we can show that ∆ 2 5 −∆ 2 1 =σ 2 (µ 1 µ −1 − 1) > 0, ∆ 2 6 − ∆ 2 5 =E P (α + βX i ) − χ −1 X i (α + βµ 1 − λα − λβµ 2 /µ 1 ) 2 µ 1 /X i − λ ≥ 0, ∆ 2 7 −∆ 2 5 =β 2 E P (X i −µ 1 )−λχ −1 X i (µ 1 −µ 2 /µ 1 ) 2 µ 1 /X i −λ ≥ 0, ∆ 2 8 −∆ 2 1 =σ 2 cµ 1 µ −1 − (1 − λ) ≥ cσ 2 (µ 1 µ −1 − 1) > 0 and ∆ 2 9 − ∆ 2 1 =σ 2 cµ 1 µ −1 − (1 − λ) + cα 2 (µ 1 µ −1 − 1) > 0 a.s. [P], where σ 2 =E P (ϵ i ) 2 . Note that ∆ 2 6 − ∆ 2 5 ≥ 0 and ∆ 2 7 − ∆ 2 5 ≥ 0 because C5 holds and C0 holds with 0 ≤ λ < E P (X i )/b. Therefore, ∆ 2 1 < ∆ 2 i a.s. [P] for i=2, . . . , 9. This completes the proof of Theorem 4. Supplementary material for "A comparison of estimators of mean and its functions in finite populations" Lemma S 1. Suppose that C0 holds. Then, nγ → c for some c ≥ 1 − λ > 0 as ν → ∞, where λ is as in C0. Proof. Let us first consider the case of λ=0. Note that n(N/n − 1)(N − n)/(N (N − 1)) ≤ nγ ≤ n(N/n + 1)(N − n)/(N (N − 1))(1) by (1) in Section 2 of the main text. Moreover, n(N/n + 1)(N − n)/(N (N − 1))=(1 + n/N )(N −n)/(N −1) → 1 and n(N/n−1)(N −n)/(N (N −1))=(1−n/N )(N −n)/(N −1) → 1 as ν → ∞ because C0 holds and λ=0. Thus we have nγ → 1 as ν → ∞ in this case. Next, consider the case, when λ > 0 and λ −1 is an integer. Here, we consider the following sub-cases. Let us first consider the sub-case, when N/n is an integer for all sufficiently large ν. Then, by (1), we have nγ=(N − n)/(N − 1) for all sufficiently large ν. Now, since C0 holds, we have (N − n)/(N − 1) → 1 − λ as ν → ∞.(2) Further, consider the sub-case, when N/n is a non-integer and N/n − λ −1 ≥ 0 for all sufficiently large ν. Then by (1) in Section 2 of the main text, we have nγ = (N/(N − 1))(n/N )⌊N/n⌋ 2 − (n/N )⌊N/n⌋ − (n/N ) for all sufficiently large ν. Now, since C0 holds, we have 0 ≤ N/n − λ −1 < 1 for all sufficiently large ν. Then, ⌊N/n⌋=λ −1 for all sufficiently large ν, and hence (N/(N − 1))(n/N )⌊N/n⌋ 2 − (n/N )⌊N/n⌋ − (n/N ) → 1 − λ(4) as ν → ∞. Next, consider the sub-case, when N/n is a non-integer and N/n − λ −1 < 0 for all sufficiently large ν. Then, the result in (3) holds by (1), and −1 ≤ N/n − λ −1 < 0 for all sufficiently large ν by C0. Therefore, ⌊N/n⌋=λ −1 − 1 for all sufficiently large ν, and hence the result in (4) holds. Thus, in the case of λ > 0 and λ −1 being an integer, nγ converges to 1 − λ as ν → ∞ through all the sub-sequences, and hence nγ → 1 − λ as ν → ∞. Thus we have c=1 − λ in this case. Finally, consider the case, when λ > 0, and λ −1 is a non-integer. Then, N/n must be a non-integer for all sufficiently large ν, and hence nγ=(N/(N − 1))(n/N )⌊N/n⌋ 2 − (n/N )⌊N/n⌋ − (n/N ) for all sufficiently large ν by (1) in Section 2 of the main text. Note that in this case, N/n − ⌊λ −1 ⌋ → λ −1 − ⌊λ −1 ⌋ ∈ (0, 1) as ν → ∞ by C0. Therefore, ⌊λ −1 ⌋ < N/n < ⌊λ −1 ⌋ + 1 for all sufficiently large ν, and hence ⌊N/n⌋=⌊λ −1 ⌋ for all sufficiently large ν. Thus nγ → λ⌊λ −1 ⌋(2 − λ⌊λ −1 ⌋ − λ) as ν → ∞ by C0. Now, if m=⌊λ −1 ⌋ and λ −1 is a non-integer, then (m + 1) −1 < λ < m −1 . Therefore, λ⌊λ −1 ⌋(2 − λ⌊λ −1 ⌋ − λ) − 1 + λ=− 1 − (2m + 1)λ + m(m + 1)λ 2 =−(1 − mλ)(1 − (m + 1)λ) > 0. Thus we have c=λ⌊λ −1 ⌋(2 − λ⌊λ −1 ⌋ − λ) > 1 − λ in this case. This completes the proof of the Lemma. Next, recall {V i } N i=1 from the paragraph preceding the condition C3 and b from the condition C5 in the main text. Let us define Σ 1 =nN −2 N i=1 (V i − Tπ i ) T (V i − Tπ i )(π −1 i − 1) and Σ 2 =nγXN −1 N i=1 (V i − X i V/X) T (V i − X i V/X)/X i , where T= N i=1 V i (1 − π i )/ N i=1 π i (1 − π i ) , the π i 's are inclusion probabilities and V= N i=1 V i /N . Now, we state the following lemma. Lemma S 2. (i) Suppose that C0 and C5 hold, and {(h(Y i ), X i ) : 1 ≤ i ≤ N } are generated from a superpopulation distribution P with E P ||h(Y i )|| 4 < ∞.Σ 2 =cE P (X i )E P [(h i − (E P (X i )) −1 X i E P (h i )) T (h i − (E P (X i )) −1 X i E P (h i )) X −1 i ] a.s. [P] for V i =h i , h i − hX i /X and h i + hX i /X because nγ → c as ν → ∞ by Lemma S1. Similarly, lim ν→∞ Σ 2 =cE P (X i )E P [(h i −E P (h i )) T (h i − E P (h i ))/X i ] a.s. [P] for V i =h i − h, and lim ν→∞ Σ 2 =cE P (X i )E P [ (h i − E P (h i ) − C xh (X i − E P (X i ))) T (h i − E P (h i ) − C xh (X i − E P (X i )))/X i ] a.s. [P] for V i =h i − h − S xh (X i − X)/S 2 x . Here, C xh =(E P (h i X i ) − E P (h i )E P (X i ))/ (E P (X i ) 2 − (E P (X i )) 2 ) . Note that the above limits are p.d. matrices because C5 holds. Therefore, C4 holds a.s. [P]. This completes the proof of (i) in Lemma S2. Next, note that Σ 1 =(1 − n/N )( N i=1 V T i V i /N − V T V)=E P h i + χ −1 (E P (X i )) −1 X i λE P (h i X i ) − E P (h i )E P (X i ) T h i + χ −1 (E P (X i )) −1 X i λE P (h i X i ) − E P (h i )E P (X i ) E P (X i )/X i − λ a.s. [P] for V i =h i , h i −hX i /X and h i +hX i /X under any πPS sampling design (i.e., a sampling design with π i =nX i / N i=1 X i ) by SLLN because C0 and C5 hold, and E P ||h i || 2 < ∞. Here, χ=E P (X i ) − λ(E P (X i ) 2 /E P (X i )). Moreover, under any πPS sampling de- sign, we have lim ν→∞ Σ 1 =E P h i − E P (h i ) + λχ −1 (E P (X i )) −1 X i C xh T h i − E P (h i ) + λχ −1 (E P (X i )) −1 X i C xh × E P (X i )/X i − λ a.s. [P] for V i =h i − h and lim ν→∞ Σ 1 = E P h i −E P (h i )−C xh (X i −E P (X i )) T h i −E P (h i )−C xh (X i −E P (X i )) E P (X i )/X i −λ a.s. [P] for V i =h i − h − S xh (X i − X)/S 2 x . Note that the above limits are p.d. matrices because C5 holds and C0 holds with 0 ≤ λ < E P (X i )/b. Therefore, C3 holds a.s. [P] under any πPS sampling design. This completes the proof of (ii) in Lemma S2. Additional mathematical details In this section, we state and prove some technical results, which will be required to prove the theorems stated in the main text. Lemma S 3. Suppose that C2 holds. Then, LMS sampling design is a high entropy sampling design. Moreover, under LMS sampling design, there exist constants L, L ′ > 0 such that L ≤ min 1≤i≤N (N π i /n) ≤ max 1≤i≤N (N π i /n) ≤ L ′(5) for all sufficiently large ν . The condition (5) was considered earlier in Wang andOpsomer (2011), Boistard et al. (2017), etc. However, the above authors did not discuss whether LMS sampling design satisfies (5) or not. Proof. Suppose that P (s) and R(s) denote LMS sampling design and SRSWOR, respectively. Note that SRSWOR is a rejective sampling design. Then, P (s)=(x/X)/ N C n and R(s)=( N C n ) −1 , where x= i∈s X i /n and s ∈ S. By Cauchy-Schwarz inequality, we have D(P ||R)=E R ((x/X) log(x/ X)) ≤ K 1 E R |x/X − 1| ≤ K 1 E R (x/X − 1) 2 for some K 1 > 0 since C2 holds, and log(x) ≤ |x − 1| for x > 0. Here E R denotes the expectation with respect to R(s). Therefore, nD(P ||R) ≤ K 1 (1 − f )(N/(N − 1))(S 2 x /X 2 ) ≤ 2K 1 ( N i=1 X 2 i /N X 2 ) ≤ 2K 1 (max 1≤i≤N X i / min 1≤i≤N X i ) 2 =O(1) as ν → ∞, where f =n/N . Hence, D(P ||R) → 0 as ν → ∞. Thus LMS sampling design is a high entropy sampling design. Next, suppose that {π i } N i=1 denote inclusion probabilities of P (s). Then, we have π i =(n − 1)/(N − 1) + (X i / N i=1 X i )((N − n)/(N − 1)) and π i − n/N =−(N − n)(N (N − 1)) −1 (X i /X − 1). Further, |π i − n/N | n/N = N − n n(N − 1) X i X − 1 ≤ N − n n(N − 1) max 1≤i≤N X i min 1≤i≤N X i + 1 . Therefore, max 1≤i≤N |N π i /n − 1| → 0 as ν → ∞ by C2. Hence, K 2 ≤ min 1≤i≤N (N π i /n) ≤ max 1≤i≤N (N π i /n) ≤ K 3 for all sufficiently large ν and some constants K 2 > 0 and K 3 > 0. Thus (5) holds under LMS sampling design. Next, suppose that {V i } N i=1 , V, Σ 1 and Σ 2 are as in the previous Section 1. Let us defineV 1 = i∈s (N π i ) −1 V i andV 2 = i∈s G i V i /N X i , where G i 's are as in the paragraph containing√ n(V 1 − V) L − → N (0, Γ 1 ) as ν → ∞, where Γ 1 =lim ν→∞ Σ 1 . Further, suppose that C0 through C2 and C4 hold. Then, we have √ n(V 2 − V) L − → N (0, Γ 2 ) as ν → ∞ under RHC sampling, where Γ 2 =lim ν→∞ Σ 2 . Proof. Note that SRSWOR is a high entropy sampling design since it is a rejective sampling design. Also, (5) in Lemma S3 holds trivially under SRSWOR. It follows from Lemma S3 that LMS sampling design is a high entropy sampling design, and (5) holds under this sampling design. Further, any HEπPS sampling design satisfies (5) since C2 holds. Now, fix ϵ > 0 and m ∈ R p . Suppose that L(ϵ, m)=(n −1 N 2 mΣ 1 m T ) −1 i∈G(ϵ,m) (m (V i − Tπ i ) T ) 2 (π −1 i − 1) for G(ϵ, m)={1 ≤ i ≤ N : |m(V i − Tπ i ) T | > ϵπ i N (n −1 mΣ 1 m T ) 1/2 }, T= N i=1 V i (1 − π i )/ N i=1 π i (1 − π i ) and Z i =(n/N π i )V i −(n/N )T, i=1, . . . , N . Then, given any η > 0, L(ϵ, m) ≤ (mΣ 1 m T ) −(1+η/2) n −η/2 ϵ −η N −1 N i=1 (||m||||Z i ||) 2+η (N π i /n) since |mZ T i |/( √ nϵ(mΣ 1 m T ) 1/2 ) > 1 for any i ∈ G(ϵ, m). It follows from Jensen's in- equality that N −1 N i=1 ||Z i || 2+η (N π i /n) ≤ 2 1+η (N −1 N i=1 ||V i (n/N π i )|| 2+η (N π i /n) + ||(n/N )T|| 2+η ) since N i=1 π i =n. It also follows from C1, C2 and Jensen's inequality that (1) as ν → ∞ for any 0 < η ≤ 2. Further, N i=1 π i (1 − π i )/n is bounded away from 0 as ν → ∞ under SRSWOR, LMS sampling design and any HEπPS sampling design because (5) holds under these sampling designs, and C0 holds. (1) and ||(n/N )T|| 2+η =O (1), and hence (1) as ν → ∞ under the above sampling designs. Then, L(ϵ, m) → 0 as ν → ∞ for any ϵ > 0 under all of these sampling designs since C3 holds. Therefore, inf{ϵ > 0 : L(ϵ, m) ≤ ϵ} → 0 as ν → ∞, and consequently the Hájek-Lindeberg condition holds for {mV T i } N i=1 under each of the above sampling designs. Also, N i=1 π i (1 − π i ) → ∞ as ν → ∞ under these sampling designs. Then, from Theorem 5 in Berger (1998), N i=1 ||V i || 2+η /N =OTherefore, N −1 N i=1 ||V i (n/N π i )|| 2+η (N π i /n)=ON −1 N i=1 ||Z i || 2+η (N π i /n)=O√ nm(V 1 − V) T L − → N (0, mΓ 1 m T ) as ν → ∞ under each of the above sampling designs for any m ∈ R p and Γ 1 =lim ν→∞ Σ 1 . Hence, √ n(V 1 − V) L − → N (0, Γ 1 ) as ν → ∞ under the above-mentioned sampling designs. (1) and (max 1≤i≤N X i )/X =O(1) since C1 and C2 hold. Now, recall from Section 2 in the main text that the N i 's are considered as in (1). Then, under C0, we have ( (1) as ν → ∞. Therefore, L(m) → 0 as ν → ∞ since C4 holds. Next, define L(m)=nγ(max 1≤i≤N X i )(N −1 n i=1 N 3 i (N i −1) N i=1 (m(V i X/ X i −V) T ) 4 X i ) 1/2 ×(X 3/2 n i=1 N i (N i − 1)mΣ 2 m T ) −1 , where γ= n i=1 N i (N i − 1)/N (N − 1) as before. Note that as ν → ∞, (N −1 N i=1 (m(V i X/X i − V) T ) 4 (X i /X)) 1/2 =On i=1 N 3 i (N i − 1)) 1/2 ( n i=1 N i (N i − 1)) −1 =O(1/ √ n) and nγ=O This implies that condition C1 in Ohlsson (1986) holds for {mV T i } N i=1 . Therefore, by Theorem 2.1 in Ohlsson (1986), √ nm(V 2 − V) T L − → N (0, mΓ 2 m T ) as ν → ∞ under RHC sampling design for any m ∈ R p and Γ 2 =lim ν→∞ Σ 2 . Hence, √ n(V 2 − V) L − → N (0, Γ 2 ) as ν → ∞ under RHC sampling design. Next, suppose that W= N i=1 W i /N ,Ŵ 1 = i∈s (N π i ) −1 W i andŴ 2 = i∈s G i W i /N X i for W i =(h i , X i h i , X 2 i ), i=1, . . . , N . Let us also defineX 1 = i∈s (N π i ) −1 X i . Now, we state the following lemma. Proof. We first show that as ν → ∞,Ŵ 1 −W=o p (1), √ n(X 1 −X)=O p (1) and √ n( i∈s ( N π i ) −1 − 1)=O p (1) under a high entropy sampling design P (s) satisfying (5) in Lemma S3. Fix m ∈ R 2p+1 . Suppose thatR(s) is a rejective sampling design with inclusion probabilities equal to those of P (s) (cf. Berger (1998)). UnderR(s), var(m( (5) holds under P (s), and C0 holds. Then, (1) as ν → ∞ for any m ∈ R 2p+1 underR(s). Now, s∈E P (s) ≤ s∈ER (s)+ s∈S |P (s)−R(s)| ≤ s∈ER (s)+(2D(P ||R)) 1/2 ≤ s∈ER (s)+ (2D(P ||R)) 1/2 (see Lemmas 2 and 3 in Berger (1998)), where E={s ∈ S : || √ n(Ŵ 1 − W)|| > δ} for δ > 0 and R(s) is any other rejective sampling design. Let us consider a rejective sampling design R(s) such that D(P ||R) → 0 as ν → ∞. Therefore, given any ϵ > 0, there exists a δ > 0 such that s∈E P (s) ≤ ϵ for all sufficiently large ν. Hence, √ n(Ŵ 1 − W) T ))=m(nN −2 N i=1 (W i − Tπ i ) T (W i − Tπ i )(π −1 i − 1))m T (1 + e) (see Theorem 6.1 in Hájek (1964)), where T= N i=1 W i (1−π i )/ N i=1 π i (1−π i ), and e → 0 as ν → ∞ whenever N i=1 π i (1 − π i ) → ∞ as ν → ∞. Note that (5) holds underR(s), and hence N i=1 π i (1 − π i ) → ∞ as ν → ∞ underR(s) becausem(nN −2 N i=1 (W i − Tπ i ) T (W i − Tπ i )(π −1 i − 1))m T ≤ nN −2 N i=1 (mW T i ) 2 /π i =O(1) underR(s) since C1 holds. Therefore, √ n(Ŵ 1 − W)=O p (1) as ν → ∞ underR(s) since var(m( √ n(Ŵ 1 − W) T ))=Oas ν → ∞, √ n(Ŵ 1 − W)=O p (1) andŴ 1 − W=o p (1) under P (s). Similarly, we can show that as ν → ∞, √ n(X 1 − X)=O p (1) and √ n( i∈s (N π i ) −1 − 1)=O p (1) under P (s). Now, recall from the proof of Lemma S4 that SRSWOR and LMS sampling design are high entropy sampling designs, and they satisfy (5). Also, any HEπPS sampling design satisfies (5). Therefore, as ν → ∞,Ŵ 1 − W=o p (1), √ n(X 1 − X)=O p (1) and √ n( i∈s ( thatĥ denotes one ofĥ HT ,ĥ H ,ĥ RA ,ĥ P R , andĥ GREG with d(i, s)=(N π i ) −1 . Then, a Taylor type expansion ofĥ − h can be obtained asĥ − h=Θ(V 1 − V) + Z, wherê V 1 = i∈s (N π i ) −1 V i , and the V i 's, Θ and Z are as described in Table 1 below. On the Table 1: Expressions of V i , Θ and Z for differentĥ'ŝ h V i Θ Ẑ h HT h i 1 0 h H h i − h ( i∈s (N π i ) −1 ) −1 0 h RA h i − hX i /X X/X 1 0 h P R h i + hX i /XX 1 /X −(1 −X 1 /X)) 2 ĥ h GREG with h i − h− ( i∈s (N π i ) −1 ) −1 (X 2 − X)× d(i, s)=(N π i ) −1 S xh (X i − X)/S 2 x (S xh /S 2 x −β 1 ) h RHC h i 1 0 h GREG with h i − h− ( i∈s G i /N X i ) −1 X(( i∈s G i /N X i ) −1 d(i, s)=G i /N X i S xh (X i − X)/S 2 x −1)(S xh /S 2 x −β 2 ) other hand, ifĥ is eitherĥ RHC orĥ GREG with d(i, s)=G i /N X i , a Taylor type expansion ofĥ − h can be obtained asĥ − h=Θ(V 2 − V) + Z. Here,V 2 = i∈s G i V i /N X i , the G i 's are as in the paragraph containing Table 8 in the main text, and the V i 's, Θ and Z are once again described in Table 1. In Table 1, X 1 = i∈s (N π i ) −1 X i ,X 2 =X 1 / i∈s (N π i ) −1 , β 1 =( i∈s (N π i ) −1 i∈s (N π i ) −1 h i X i −ĥ HTX 1 )/( i∈s (N π i ) −1 i∈s (N π i ) −1 X 2 i − (X 1 ) 2 ) andβ 2 =( i∈s (G i /N X i ) i∈s (G i h i /N )−ĥ RHC X)/( i∈s (G i /N X i ) i∈s (G i X i /N ) −X 2 ).=lim ν→∞ nN −2 × N i=1 (V i − Tπ i ) T (V i − Tπ i )(π −1 i − 1) with T= N i=1 V i (1 − π i )/ N i=1 π i (1 − π i ) . Note that Γ 1 is a p.d. matrix under each of the above sampling designs as C3 holds under these sampling designs. Let us now consider from Table 1 various choices of Θ and Z corresponding toĥ HT ,ĥ H ,ĥ RA ,ĥ P R , andĥ GREG with d(i, s)=(N π i ) −1 . Then, it can be shown from Lemma S5 that for each of these choices, √ nZ=o p (1) and Θ − 1=o p (1) as ν → ∞ under the above-mentioned sampling designs. Therefore, (6) holds under those sampling designs with Γ=Γ 1 . This completes the proof of (i) in Lemma 6 We can show from Lemma S4 that Let {V i } N i=1 be as described in Table 1. Recall Σ 1 and Σ 2 from the paragraph preceding Lemma S2 in this supplement. Note that the expression of Σ 1 remains the same for different HEπPS sampling designs. Also, recall from the paragraph preceding Theorem 3 in the main text that ϕ=X − (n/N ) N i=1 X 2 i /N X. Now, we state the following lemma. Lemma S 7. (i) Suppose that C0 through C3 hold. Further, suppose that σ 2 1 and σ 2 2 denote lim ν→∞ ∇g(µ 0 )Σ 1 ∇g(µ 0 ) T under SRSWOR and LMS sampling design, respectively, where √ n(V 2 − V) L − → N (0, Γ 2 ) as ν → ∞ under RHC sampling design, where Γ 2 =lim ν→∞ nγ XN −1 N i=1 (V i − X i V/X) T (V i − X i V/X)/X i with γ= n i=1 N i (N i − 1)/N (N − 1µ 0 =lim ν→∞ h. Then, we have σ 2 1 =σ 2 2 =(1−λ) lim ν→∞ N i=1 (A i −Ā) 2 /N for A i =∇g(µ 0 )V T i , i=1, . . . , N . (ii) Next, suppose that C4 holds, and σ 2 3 =lim ν→∞ ∇g(µ 0 )Σ 2 ∇g(µ 0 ) T in the case of RHC sampling design. Then, we have σ 2 3 =lim ν→∞ nγ((X/N ) N i=1 A 2 i /X i −Ā 2 ). On the other hand, if C0 through C3 hold, and σ 2 4 =lim ν→∞ ∇g(µ 0 )Σ 1 ∇g(µ 0 ) T under any HEπPS sampling design, then we have σ 2 4 = lim ν→∞ (1/N ) N i=1 A 2 i (X/X i ) − (n/N ) − ϕ −1 X −1 × (n/N ) N i=1 A i X i /N − AX 2 . Further, if C0 holds with λ=0 and C1 through C3 hold, then we have σ 2 4 =σ 2 3 =lim ν→∞ ((X/N ) N i=1 A 2 i /X i −Ā 2 ). Proof. Let us first note that the limits in the expressions of σ 2 1 and σ 2 2 exist in view of C3. Also, note that ∇g (µ 0 )Σ 1 ∇g(µ 0 ) T =nN −2 N i=1 (A i −T A π i ) 2 (π −1 i −1)=nN −2 [ N i=1 A 2 i (π −1 i − 1) − ( N i=1 A i (1 − π i )) 2 / N i=1 π i (1 − π i )], where T A = N i=1 A i (1 − π i )/ N i=1 π i (1 − π i ) and A i =∇g(µ 0 )V T i . Now, substituting π i =n/N in the above expression for SRSWOR, we get σ 2 1 = lim ν→∞ nN −2 [ N i=1 A 2 i (N/n − 1) − ( N i=1 A i (1 − n/N )) 2 /n(1 − n/N )]=lim ν→∞ (1 − n/N ) N i=1 (A i −Ā) 2 /N . Since C0 holds, we have σ 2 1 =(1 − λ) lim ν→∞ N i=1 (A i − A) 2 /N . Let {π i } N i=1 be the inclusion probabilities of LMS sampling design. Then, σ 2 2 − σ 2 1 =lim ν→∞ nN −2 [ N i=1 A 2 i (π −1 i − N/n) − (( N i=1 A i (1 − π i )) 2 / N i=1 π i (1 − π i ) − ( N i=1 A i (1 − n/N )) 2 /n(1 − n/N ))] . Now, it can be shown from the proof of Lemma S3 that max 1≤i≤N |N π i /n − 1| → 0 as ν → ∞. Therefore, using C1, we can show that lim ν→∞ nN −2 N i=1 A 2 i (π −1 i − N/n)=0 and lim ν→∞ nN −2 [( N i=1 A i (1 − π i )) 2 / N i=1 π i (1 − π i ) − ( N i=1 A i (1 − n/N )) 2 /n(1 − n/N )]=0 , and consequently σ 2 1 =σ 2 2 . This completes the proof of (i) in Lemma 7. Next, consider the case of RHC sampling design and note that the limit in the expres- sion of σ 2 3 exists in view of C4. Also, note that ∇g(µ 0 )Σ 2 ∇g(µ 0 ) T =nγ(X/N ) N i=1 ( A i − AX i /X) 2 /X i =nγ((X/N ) N i=1 A 2 i / X i −Ā 2 ), whereĀ= N i=1 A i /N and γ= n i=1 N i (N i − 1)/N (N − 1). Thus we have σ 2 3 =lim ν→∞ nγ((X/N ) N i=1 A 2 i /X i −Ā 2 ). Next, note that the limit in the expression of σ 2 4 exists in view of C3. Substi- tuting π i =nX i / N i=1 X i in ∇g(µ 0 )Σ 1 ∇g(µ 0 ) T for any HEπPS sampling design, we get σ 2 4 =lim ν→∞ nN −2 [ N i=1 A 2 i ( N i=1 X i /nX i −1)−( N i=1 A i (1−nX i / N i=1 X i )) 2 / N i=1 (nX i / N i=1 X i )(1−nX i / N i=1 X i )]= lim ν→∞ (1/N ) N i=1 A 2 i (X/X i )−(n/N ) −ϕ −1 X −1 (n/N ) × N i=1 A i X i /N − A X 2 . Further, we can show that σ 2 4 =lim ν→∞ ((X/N ) N i=1 A 2 i /X i − A 2 ) , when C1 and C2 hold, and C0 holds with λ=0. It also follows from Lemma S1 that nγ → 1 as ν → ∞, when C0 holds with λ=0. Thus we have σ 2 3 =σ 2 4 =lim ν→∞ ((X/N ) N i=1 A 2 i / X i −Ā 2 ) . This completes the proof of (ii) in Lemma 7. Lemma S 8. Suppose that C0 through C2 hold. Then under SRSWOR, LMS sampling design and any HEπPS sampling design, we have (i) u * = max i∈s |Z i | = o p ( √ n), and (ii) i∈s π −1 i Z i / i∈s π −1 i Z 2 i = O p (1/ √ n) as ν → ∞, where Z i =X i − X for i=1, . . . , N Proof. Let P (s) be any sampling design and E P be the expectation with respect to P (s). Then, (1) Using similar arguments as in the first paragraph of the proof of Lemma S5, it can be shown that E P (u * / √ n) ≤ (max 1≤i≤N X i + X)/ √ n ≤ X(max 1≤i≤N X i / min 1≤i≤N X i + 1)/ √ n=o√ n( i∈s Z i /N π i − Z)= √ n i∈s Z i /N π i =O p (1) and i∈s Z 2 i /N π i − N i=1 Z 2 i /N =o p (1) as ν → ∞ under a high entropy sampling design P (s) satisfying (5) in Lemma S3. Therefore, 1/( i∈s Z 2 i /N π i ) =O p (1) as ν → ∞ under P (s) since N i=1 Z 2 i /N is bounded away from 0 as ν → ∞ by C1. Thus under P (s), i∈s π −1 i Z i / i∈s π −1 i Z 2 i =O p (1/ √ n) as ν → ∞. It follows from Lemma S3 that SRSWOR and LMS sampling design are high entropy sampling designs and satisfy (5) Also, any HEπPS sampling design satisfies (5) since C2 holds. Therefore, the result in (ii) holds under the above-mentioned sampling designs. 3 Proofs of Remark 1 and Theorems 2, 3, 6 and 7 In this section, we give the proofs of Remark 1 and Theorems 2, 3, 6 and 7 of the main text. Proof of Theorem 2. Let us first consider a HEπPS sampling design. Then, it can be shown in the same way as in the 1 st paragraph of the proof of Theorem 1 that √ n(ĥ P EM L − h GREG )=o p (1) for d(i, s)=(N π i ) −1 under this sampling design. It can also be shown in the same way as in the 1 st paragraph of the proof of Theorem 1 that ifĥ is one ofĥ HT ,ĥ H , andĥ GREG andĥ P EM L with d(i, s)=(N π i ) −1 , then (4) in the proof of Theorem 1 holds under the above-mentioned sampling design. Here, we recall from Table 2 in the main text that the HT, the ratio and the product estimators coincide under any HEπPS sampling design. Further, the asymptotic MSE of √ n(g(ĥ) − g(h)) is ∇g(µ 0 )Γ 1 (∇g(µ 0 )) T , where µ 0 =lim ν→∞ h, Γ 1 =lim ν→∞ nN −2 N i=1 (V i − Tπ i ) T (V i − Tπ i )(π −1 i − 1), and V i in Γ 1 is h i or h i − h or h i − h − S xh (X i − X)/S 2 x ifĥ isĥ HT orĥ H , orĥ GREG with d(i, s)=(N π i ) −1 , respectively. Now, since √ n(ĥ P EM L −ĥ GREG )=o p (1) for ν → ∞ under any HEπPS sampling design, g(ĥ GREG ) and g(ĥ P EM L ) have the same asymptotic distribution under this sampling design. Thus under any HEπPS sampling design, g(ĥ GREG ) and g(ĥ P EM L ) with d(i, s)=(N π i ) −1 form class 5, g(ĥ HT ) forms class 6, and g(ĥ H ) forms class 7 in Table 2 of the main text. This completes the proof of (i) in Theorem 2. Let us now consider the RHC sampling design. We can show from (ii) in Lemma S6 that √ n(ĥ − h) L − → N (0, Γ) as ν → ∞ for some p.d. matrix Γ, whenĥ is either h RHC orĥ GREG with d(i, s)=G i /N X i under RHC sampling design. Further, √ n(ĥ P EM L − h GREG )=o p (1) as ν → ∞ for d(i, s)=G i /N X i under RHC sampling design since C2 holds, and S 2 x is bounded away from 0 as ν → ∞ (see A2.2 of Appendix 2 in Chen and Sitter (1999)). Therefore, ifĥ is one ofĥ RHC , andĥ GREG andĥ P EM L with d(i, s)=G i /N X i , then we have √ n(g(ĥ) − g(h)) L − → N (0, ∆ 2 ) as ν → ∞(7) for some ∆ 2 > 0 by the delta method and the condition ∇g(µ 0 ) ̸ = 0 at µ 0 =lim ν→∞ h. Moreover, it follows from the proof of (ii) in Lemma S6 that Table 1 ∆ 2 = ∇g(µ 0 )Γ 2 (∇g(µ 0 )) T , where Γ 2 =lim ν→∞ nγXN −1 N i=1 (V i − X i V/X) T (V i − X i V/X)/X i . It further follows fromin this supplement that V i in Γ 2 is h i ifĥ isĥ RHC . Also, V i in Γ 2 is h i −h−S xh (X i −X)/S 2 x ifĥ isĥ GREG with d(i, s)=G i /N X i . Now, g(ĥ GREG ) and g(ĥ P EM L ) have the same asymptotic distribution under RHC sampling design since √ n(ĥ P EM L − h GREG )=o p (1) for ν → ∞ under this sampling design as pointed out earlier in this paragraph. Thus g(ĥ GREG ) and g(ĥ P EM L ) with d(i, s)=G i /N X i under RHC sampling design form class 8, and g(ĥ RHC ) forms class 9 in Table 2 of the main article. This completes the proof of (ii) in Theorem 2. Proof of Remark 1. It follows from (ii) in Lemma S7 that in the case of λ=0, σ 2 3 = σ 2 4 = lim ν→∞ ((X/N ) N i=1 A 2 i /X i −Ā 2 ),(8) where σ 2 3 and σ 2 4 are as defined in the statement of Lemma S7, and A i =∇g(µ 0 )V T i for different choices of V i mentioned in the proof of Theorem 2 above. Thus g(ĥ GREG ) with d(i, s)=(N π i ) −1 under any HEπPS sampling design, and with d(i, s)=G i /N X i under RHC sampling design have the same asymptotic MSE. Therefore, class 8 is merged with class 5 in Table 2 of the main text. Further, (8) implies that g(ĥ HT ) under any HEπPS sampling design and g(ĥ RHC ) have the same asymptotic MSE. Therefore, class 9 is merged with class 6 in Table 2 of the main text. This completes the proof of Remark 1. Proof of Theorem 3. Recall the expression of A i 's from the proofs of Theorem 1 and Table 1 in this supplement since ∇g(h) → ∇g(µ 0 ) as ν → ∞. Here, ϕ=X − Remark 1. Note that lim ν→∞ (A i −Ā) 2 /N =lim ν→∞ (B i −B) 2 /N , lim ν→∞ nγ (X/N ) × N i=1 A 2 i /X i −Ā 2 =lim ν→∞ nγ (X/N ) N i=1 B 2 i /X i −B 2 and lim ν→∞ (1/N ) N i=1 A 2 i × (X/X i ) − (n/N ) − ϕ −1 X −1 (n/N ) N i=1 A i X i /N −ĀX 2 =lim ν→∞ (1/N ) N i=1 B 2 i × (X/X i ) − (n/N ) − ϕ −1 X −1 (n/N ) N i=1 B i X i /N −BX 2 for B i =∇g(h)V T i and V i as in(n/N ) N i=1 X 2 i /N X. Then, from Lemma S7 and the expressions of asymptotic MSEs of √ n(g(ĥ) − g(h)) discussed in the proofs of Theorems 1 and 2 , the results in Table 3 of the main text follow. This completes the proof of Theorem 3. Proof of Theorem 6. Using similar arguments as in the 1 st paragraph of the proof of Theorem 4, we can say that under SRSWOR and LMS sampling design, conclusions of Theorems 1 and 3 hold a.s. [P] for d=1, p=2, h(y)=(y, y 2 ) and g(s 1 , s 2 )=s 2 − s 2 1 in the same way as conclusions of Theorems 1 and 3 hold a.s. [P] for d=p=1, h(y)=y and g(s)=s in the 1 st paragraph of the proof of Theorem 4. Note that W i =Y 2 i − 2Y i Y for the above choices of h and g. Further, it follows from SLLN and the condition E P (ϵ i ) 8 < ∞ that the ∆ 2 i 's in Table 3 in the main text can be expressed in terms of superpopulation moments of (Y i , X i ) a.s. [P]. Note that ∆ 2 2 − ∆ 2 1 =cov 2 P (W i , X i ) a.s. [P], whereW i =Y 2 i − 2Y i E P (Y i ). Then, ∆ 2 1 < ∆ 2 2 a.s. [P] . This completes the proof of (i) in Theorem 6. Next consider the case of 0 ≤ λ < E P (X i )/b. Using the same line of arguments as in the 2 nd paragraph of the proof of Theorem 4, it can be shown that under RHC and any HEπPS sampling designs, conclusions of Theorems 2 and 3 hold a.s. [P] for d=1, p=2, h(y)=(y, y 2 ) and g(s 1 , s 2 )=s 2 −s 2 1 in the same way as conclusions of Theorems 2 and 3 hold a.s. [P] for d=p=1, h(y)=y and g(s)=s in the 2 nd paragraph of the proof of Theorem 4. Note that ∆ 2 7 − ∆ 2 5 = µ 2 1 cov P (W i , X i ) cov P (W i , X i )cov P (X i , 1/X i ) − 2cov P (W i , 1/X i ) − λ 2 cov 2 P (W i , X i )/χµ 1 − λcov 2 P (W i , X i ) ≤ µ 2 1 × cov P (W i , X i ) cov P (W i , X i )cov P (X i , 1/X i ) − 2cov P (W i , 1/X i ) a.s. [P] because χ > 0. Recall from C6 that ξ=µ 3 − µ 2 µ 1 and µ j =E P (X i ) j for j=−1, 1, 2, 3. Then, from the linear model set up, we have µ 2 1 cov P (W i , X i )× cov P (W i , X i )cov P (X i , 1/X i ) − 2cov P (W i , 1/X i ) =(β 2 µ 1 ) 2 (ξ − 2µ 1 )((ξ + 2µ 1 )ζ 1 − 2ζ 2 ). Here, ζ 1 =1 − µ 1 µ −1 and ζ 2 =µ 1 − µ 2 µ −1 . Note that (ξ + 2µ 1 )ζ 1 − 2ζ 2 =ξζ 1 + 2µ −1 and ζ 1 < 0. Therefore, µ 2 1 cov P (W i , X i ) cov P (W i , X i )cov P (X i , 1/X i ) − 2cov P (W i , 1/X i ) < 0 if ξ > 2 max{µ 1 , µ −1 /(µ 1 µ −1 − 1)}. Hence, ∆ 2 7 − ∆ 2 5 < 0 a.s. [P] . This completes the proof of (ii) in Theorem 6. Proof of Theorem 7. Using the same line of arguments as in the 1 st paragraph of the proof of Theorem 4, it can be shown that under SRSWOR and LMS sampling design, conclusions of Theorems 1 and 3 hold a.s. [P] for d=2, p=5, h(z 1 , z 2 )=(z 1 , z 2 , z 2 1 , z 2 2 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 −s 1 s 2 )/((s 3 −s 2 1 )(s 4 −s 2 2 )) 1/2 in the case of the correlation coefficient between z 1 and z 2 , and for d=2, p=4, h(z 1 , z 2 )=(z 1 , z 2 , z 2 2 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , s 4 )= (s 4 − s 1 s 2 )/(s 3 − s 2 2 ) in the case of the regression coefficient of z 1 on z 2 in the same way as conclusions of Theorems 1 and 3 hold a.s. [P] for d=p=1, h(y)=y and g(s)=s in the case of the mean of y in the 1 st paragraph of the proof of Theorem 4. Further,if C0 holds with 0 ≤ λ < E P (X i )/b, then using similar arguments as in the 2 nd paragraph of the proof of Theorem 4, it can also be shown that under RHC and any HEπPS sampling designs, conclusions of Theorems 2 and 3 hold a.s. [P] for d=2, p=5, h(z 1 , z 2 )=(z 1 , z 2 , z 2 1 , z 2 2 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − s 1 s 2 )/((s 3 − s 2 1 )(s 4 − s 2 2 )) 1/2 in the case of the correlation coefficient between z 1 and z 2 , and for d=2, p=4, h(z 1 , z 2 )=(z 1 , z 2 , z 2 2 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ) in the case of the regression coefficient of z 1 on z 2 in the same way as conclusions of Theorems 2 and 3 hold a.s. [P] for d=p=1, h(y)=y and g(s)=s in the case of the mean of y in the 2 nd paragraph of the proof of Theorem 4. Note that W i =R 12 [(Z 1 /S 2 1 − Z 2 /S 12 )Z 1i + (Z 2 /S 2 2 − Z 1 /S 12 )Z 2i − Z 2 1i /2S 2 1 − Z 2 2i /2S 2 2 + Z 1i Z 2i /S 12 ] for the correlation coefficient, and Table 3 in the main text can be expressed in terms of superpopulation moments of (h(Z 1i , Z 2i ), X i ) a.s. [P] for both the parameters by SLLN. Further, for the above parameters, we have W i =(1/S 2 2 )[−Z 2 Z 1i − (Z 1 − 2S 12 Z 2 /S 2 2 )Z 2i − S 12 Z 2 2i /S 2 2 + Z 1i Z 2i ] for the regression coefficient. Here, Z 1 = N i=1 Z 1i /N , Z 2 = N i=1 Z 2i /N , S 2 1 = N i=1 Z 2 1i /N − Z 2 1 , S 2 2 = N i=1 Z 2 2i /N − Z 2 2 , S 12 = N i=1 Z 1i Z 2i /N − Z 1 Z 2 and R 12 =S 12 /S 1 S 2 . Also, note that since E P ||ϵ i || 8 < ∞, the ∆ 2 i 's in∆ 2 2 − ∆ 2 1 =cov 2 P (W i , X i ) > 0 and ∆ 2 7 − ∆ 2 5 = µ 2 1 cov P (W i , X i ) cov P (W i , X i )cov P (X i , 1/X i ) − 2cov P (W i , 1/X i ) −λ 2 cov 2 P (W i , X i )/χµ 1 −λcov 2 P (W i , X i ) ≤ µ 2 1 cov P (W i , X i ) cov P (W i , X i )× cov P (X i , 1/X i )−2cov P (W i , 1/X i ) a.s. [P] , whereW i is the same as W i with all finite population moments in the expression of W i replaced by their corresponding superpopulation moments. Also, from the linear model set up, we have µ 2 1 cov P (W i , X i ) cov P (W i , X i )cov P (X i , 1/X i ) − 2cov P (W i , 1/X i ) =K(ξ − 2µ 1 )((ξ + 2µ 1 )ζ 1 − 2ζ 2 ) for some constant K > 0 in the case of the correlation coefficient, and µ 2 1 cov P (W i , X i )× cov P (W i , X i )cov P (X i , 1/X i ) − 2cov P (W i , 1/X i ) =K ′ (ξ − 2µ 1 )((ξ + 2µ 1 )ζ 1 − 2ζ 2 ) for some constant K ′ > 0 in the case of the regression coefficient. Thus proofs of both the parts of the theorem follow in the same way as the proof of Theorem 6. Comparison of estimators with their bias-corrected versions In this section, we empirically compare the biased estimators considered in Table 5 in Section 4 of the main text with their bias-corrected versions based on both synthetic and real data used in Section 4. Following the idea in Stefan and Hidiroglou (2022), we consider the bias-corrected jackknife estimator corresponding to each of the biased estimators considered in Table 5 Suppose that s is a sample of size n drawn using one of the sampling designs given in Table 5 of the main text. Further, suppose that s −i is the subset of s, which excludes the i th unit for any given i ∈ s. Now, for any i ∈ s, let us denote the estimator g(ĥ) constructed based on s −i by g(ĥ −i ). Then, we compute the bias-corrected jackknife estimator of g(h) corresponding to g(ĥ) as ng(ĥ) − (n − 1) i∈s g(ĥ −i )/n (cf. Stefan and Hidiroglou (2022)). Recall from Section 4 in the main article that we draw I=1000 samples each of sizes n=75, 100 and 125 from some synthetic as well as real datasets using sampling designs mentioned in Table 5 and compute MSEs of the estimators considered in Table 5 based on these samples. Here, we compute MSEs of the above-mentioned bias-corrected jackknife estimators using the same procedure and compare them with the original biased estimators in terms of their MSEs. We observe from the above analyses that for all the parameters considered in Section 4 of the main text, the bias-corrected jackknife estimators become worse than the original biased estimators in the cases of both the synthetic and the real data (see Tables 2 through 6 and 12 through 21 in Sections 5 and 6 below). Despite reducing the biases of the original biased estimators, bias-correction increases the variances of these estimators significantly. This is the reason why the biascorrected jackknife estimators have larger MSEs than the original biased estimators in the cases of both the synthetic and the real data. Analysis based on synthetic data The results obtained from the analysis carried out in Section 4.1 of the main paper and Section 4 in this supplement are summarized in these sections. Here, we provide some tables that were mentioned in these sections. Tables 2 through 6 contain relative efficiencies of estimators for the mean, the variance, the correlation coefficient and the regression coefficient in the population. Tables 7 through 11 contain the average and the standard deviation of lengths of asymptotically 95% CIs of the above parameters. Table 4 in Section 2 that for variance of y, h(y)=(y 2 , y) and g(s 1 , s 2 )=s 1 − s 2 2 .````````````````````R elative efficiency Sample size n=75 n=100 n=125 Table 4 in Section 2 that for correlation coefficient between z 1 and z 2 , h(z 1 , z 2 )=(z 1 , z 2 , z 2 1 , z 2 2 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − s 1 s 2 )/((s 3 − s 2 1 )(s 4 − s 2 2 )) 1/2 .````````````````````R elative efficiency Sample size n=75 n=100 n=125 Table 4 in Section 2 that for regression coefficient of z 1 on z 2 , h(z 1 , z 2 )=(z 1 , z 2 , z 2 2 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , Table 6: Relative efficiencies of estimators for regression coefficient of z 2 on z 1 . Recall from Table 4 in Section 2 that for regression coefficient of z 2 on z 1 , h(z 1 , z 2 )=(z 2 , z 1 , z 2 1 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ).````````````````````R elative efficiency Sample size n=75 n=100 n=125 3 It is to be noted that in the cases of PEML and GREG estimators under any given sampling design, we have the same asymptotic MSE and hence the same asymptotic CI. RE(g(ĥ P EM L ), SRSWOR | g(ĥ H ),RE(g(ĥ P EM L ), SRSWOR | g(ĥ H ),s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ).````````````````````R elative efficiency Sample size n=75 n=100 n=125 RE(g(ĥ P EM L ), SRSWOR | g(ĥ H ),RE(g(ĥ P EM L ), SRSWOR | g(ĥ H ), Therefore, the average and the standard deviation of CIs are not reported for the GREG estimator. Table 4 in Section 2 that for variance of y, h(y 1 )=(y 2 , y) and g(s 1 , s 2 )=s 1 − s 2 2 . Average length (Standard deviation) P P P P P P P P P P P P P P P P P P P P P P Table 9: Average and standard deviation of lengths of asymptotically 95% CIs for correlation coefficient between z 1 and z 2 . Recall from Table 4 in Section 2 that for correlation coefficient between z 1 and z 2 , h(z 1 , z 2 )=(z 1 , z 2 , z 2 1 , z 2 2 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − Table 4 in Section 2 that for regression coefficient of z 1 on z 2 , h(z 1 , z 2 )=(z 1 , z 2 , z 2 2 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , Table 4 in Section 2 that for regression coefficient of z 2 on z 1 , h(z 1 , z 2 )=(z 2 , z 1 , z 2 1 , z 1 z 2 ) and g(s 1 , s 2 , s 3 , s 1 s 2 )/((s 3 − s 2 1 )(s 4 − s 2 2 )) 1/2 . Average length (Standard deviation)````````````````````````````s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ). Average length (Standard deviation)````````````````````````````s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ). Average length (Standard deviation)```````````````````````````` Analysis based on real data The results obtained from the analyses carried out in Section 4.2 of the main paper and Section 4 in this supplement are summarized in these sections. Here, we provide some scatter plots and tables that were mentioned in these sections. Figures 1 through 4 present scatter plots and least square regression lines between different study and size variables drawn based on all the population values. Tables 12 through 21 contain relative efficiencies of estimators for the mean, the variance, the correlation coefficient and the regression coefficient in the population. Tables 22 through 31 contain the average and the standard deviation of lengths of asymptotically 95% CIs of the above parameters. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q qqq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q qq q qq q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q qq q q q q qq q qqqq q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq qqq q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q q q q q qq q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q qq qq q q q q q q q q q qq q q qq q q q q qq q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q qq q q qq q q q q q qqq q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q qq 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q q q q qq q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qq q qq q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q qq q q q q q q q q qq q q qqq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q qq qq q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q qqqq q q q q qq q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q qq q q q q q qqq q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q qq q q q q q q q q q q qq q q q q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q qq q q qq q q qq q q q q q q q q q q q qqq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq qq q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q qq q q q q q q q q qq q qq q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qqq q q q q q q q q q q q q q q q qq qq q q q q q q qq q q q q q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qqq q qq q qqq qq q q q q qq q q q q q qq q q q q q q q q q q q qq q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q qq q qq q q q qq q q q q q q q qq q q q q qq qq q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q qqqq q q q q qq q q q q qqq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qqq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq qq qq q q q q q q q q q qq q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q qq q qqq q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q qq q q q q q q q q q q q q q qq q q q q qq q q q q q q q qq qq q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q qq q qqq q q q qq q q q q q q q q q q q q q q q q q q qq qq q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qqq q qq q q q qq qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q qq q q qq qq q q q q q q q qq qq qq q q q q q q q q q qq q q qq q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q qqqq q qq q q q q qq q q q q qq qq q q q q qq qq qq qq q q q q q q q q q q q q q q q q qqq q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q qq q q q q q qq q q q qq q q q q q q q q q qqq q q q q q qq qq q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q qq q q qqq q q q q qq q q q qq q qq q q q q q q q qqqqq q q q q qq q qq q qq q q q qq q q q q q q q q q q q q q q qq q q qq qq q q qq q qq q q qq qq q q q q qq q q q q q q q q q q q q q q q q q q q qqq q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q qqq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q qq q q q q q q q qq q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q qq qq q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q qq qqq q qq q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q qq qq q q q qqq q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q qq q q q q q q q q q qqq q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q qqq q qq qq q q q q q qqqq q q q q q q q q qq q q qq q q q qq q q q q q q q q q q q qq qq q q q q q q q q q q q qq q q q qq q q qqq q q q q qq q q q q q q q q qq q q qq q q q qq q qq qq q q q qqq q q q q q qqq q q q q q q qq qqq qqq q q q q q q q qq q q q q q q q q q q q q q q qqq qq q q q q qq q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qqq q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q qq qq q q q qqq q q qq q q q q q q q q q q q q q qq qq q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q qq q q qq qq qq q q q q q q q qqq q q q q qq q q qq q q q q q q q q q q qq q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q qqq q q q qq q q q q qq q q q q qq q q q q q q q q q q q q q qq q q q q qq q q q qq q q q q q q q q q q q qq q q q q q q q q qq q q q q q q qq qq q q q q qq q q qq qq q q qq q q q q q q qq q q q q qqq q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q qq q q qq q q q q q q q q q qq q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q qq q q qq q q qq q q q qq q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q qq q qq qq q q qq q q q q q q q q q q qq q q q q q qq q q q q q q qq q q q q q q qqq qq q q q q q qq q qqq q q q q q q qq q q q q q q qqq q q q q q qq q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q qq q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q Table 4 in Section 2 that for regression coefficient of y 1 on y 3 , h(y 1 , y 3 )=(y 1 , y 3 , y 2 3 , y 1 y 3 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ).````````````````````R elative efficiency Sample size n=75 n=100 n=125 Table 4 in Section 2 that for variance of y 1 , h(y 1 )=(y 2 1 , y 1 ) and g(s 1 , s 2 )=s 1 − s 2 2 . Average length (Standard deviation)`````````````````````````````È stimator and sampling design based on which CI is constructed Table 4 in Section 2 that for variance of y 2 , h(y 2 )=(y 2 2 , y 2 ) and g(s 1 , s 2 )=s 1 − s 2 2 . Average length (Standard deviation) P P P P P P P P P P P P P P P P P P P P P P Estimator and sampling design based on which CI is constructed Sample size n=75 n=100 n=125 g(ĥ H ), SRSWOR 1498664 1588740 2418155 Table 26: Average and standard deviation of lengths of asymptotically 95% CIs for correlation coefficient between y 1 and y 3 . Recall from Table 4 in Section 2 that for correlation coefficient between y 1 and y 3 , h(y 1 , y 3 )=(y 1 , y 3 , y 2 1 , y 2 3 , y 1 y 3 ) and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − s 1 s 2 )/((s 3 − s 2 1 )(s 4 − s 2 2 )) 1/2 . Average length (Standard deviation)`````````````````````````````È stimator and sampling design based on which CI is constructed Table 4 in Section 2 that for regression coefficient of y 1 on y 3 , h(y 1 , y 3 )=(y 1 , y 3 , y 2 3 , y 1 y 3 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ). Average length (Standard deviation)`````````````````````````````È stimator and sampling design based on which CI is constructed Table 4 in Section 2 that for regression coefficient of y 3 on y 1 , h(y 1 , y 3 )=(y 3 , y 1 , y 2 1 , y 1 y 3 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ). Average length (Standard deviation)`````````````````````````````È stimator and sampling design based on which CI is constructed Table 29: Average and standard deviation of lengths of asymptotically 95% CIs for correlation coefficient between y 2 and y 4 . Recall from Table 4 in Section 2 that for correlation coefficient between y 2 and y 4 , h(y 2 , y 4 )=(y 2 , y 4 , y 2 2 , y 2 4 , y 2 y 4 ) and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − s 1 s 2 )/((s 3 − s 2 1 )(s 4 − s 2 2 )) 1/2 . Average length (Standard deviation)`````````````````````````````È stimator and sampling design based on which CI is constructed Table 4 in Section 2 that for regression coefficient of y 2 on y 4 , h(y 2 , y 4 )=(y 2 , y 4 , y 2 4 , y 2 y 4 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ). Average length (Standard deviation)`````````````````````````````È stimator and sampling design based on which CI is constructed Table 4 in Section 2 that for regression coefficient of y 4 on y 2 , h(y 2 , y 4 )=(y 4 , y 2 , y 2 2 , y 2 y 4 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ). Average length (Standard deviation)`````````````````````````````È stimator and sampling design based on which CI is constructed classes of estimators under different sampling designs are constructed so that estimators in the same class have equivalent performance in terms of asymptotic mean squared errors (MSEs). Estimators in different equivalence classes are then compared under some superpopulations satisfying linear models. It is shown that the pseudo empirical likelihood (PEML) estimator of the population mean under simple random sampling without replacement (SRSWOR) has the lowest asymptotic MSE among all the estimators under different sampling designs considered in this paper. consider the following condition (cf. Assumption 1 in Cardot & Josserand (2011), A4 in Conti (2014), A1 in Cardot et al. (2014) A4 in Conti and Marella (2015) and (HT3) in Boistard et al. (2017)) converges to some p.d. matrix as ν → ∞. Similar conditions like C1, C3 and C4 are often used in sample survey literature (see Assumption 3 in Cardot & Josserand (2011), A3 and A6 in both Conti (2014) and Conti and Marella (2015), (HT2) in Following the idea of Cardot et al. (2014), we estimate∆ 2 1 bŷ y 1 1Number of primary schools in village y 2 Scheduled castes population size in village y 3 Number of secondary schools in village y 4 Scheduled tribes population size in village (i) First, we consider the estimation of the mean and the variance of each of y 1 and y 2 . Number of peopl e l i vi ng i n vi l l age y1 = Number of primary schools in village Figure 1 : 1Scatter plot and least square regression line for variables y 1 and x Scheduled castes population size in village Figure 2 : 2Scatter plot and least square regression line for variables y 2 and x Figure 3 : 3Scatter plot and least square regression line for variables y 3 and x Scheduled tribes population size in village Figure 4 : 4Scatter plot and least square regression line for variables y 4 and x Table 1 : 1Estimators of hSampling Estimators designs SRSWOR HT (which coincides with Hájek estimator), ratio, product, GREG and PEML estimators LMS HT, Hájek, ratio, product, GREG and PEML estimators HEπPS HT (which coincides with ratio and product estimators), Hájek, GREG and PEML estimators RHC RHC, GREG and PEML estimators estimators of g(h) Table 2 : 2Disjoint equivalence classes of estimators for g(h)Estimators of h Sampling GREG and HT RHC Hájek Ratio Product design PEML SRSWOR class 1 1 class 2 1 class 2 class 3 class 4 LMS class 1 class 2 class 2 class 3 class 4 Table 4 : 4Examples of g(h)Parameter d p h g Mean 1 1 h(y)=y g(s)=s Variance 1 2 h(y)=(y 2 , y) g(s 1 , s 2 )=s 1 − s 2 2 Table 5 : 5Estimators considered for the empirical comparisonParameters Estimators Mean GREG and PEML estimators under SRS- WOR; HT, Hájek, GREG and PEML estimators under 3 RS sampling design; and RHC, GREG and PEML estimators under RHC sampling design Variance, correlation Obtained by plugging in Hájek and PEML coefficient and regression estimators under SRSWOR and 1 RS coefficient sampling design, and PEML estimator under RHC sampling design 3 We consider RS sampling design since it is a HEπPS sampling design, Table 6 : 6Description of study variables Table 7 : 7Most efficient estimators in terms of relative efficiencies (it follows from Tables 22 through 31 in the supplement that asymptotically 95% CIs based on most efficient estimators have least average lengths.) estimator under SRSWOR become the most efficient among all the estimators under different sampling designs considered in this section.Parameters Most efficient estimators Mean and variance of y 1 The plug-in estimator based on the the PEML estimator under SRSWOR Mean of y 2 The HT estimator under RS sampling design Variance of y 2 The plug-in estimator based on the Hájek estimator under RS sampling design Correlation and regression The plug-in estimator based on the PEML coefficients of y 1 and y 3 estimator under SRSWOR Correlation and regression The plug-in estimator based on the Hájek coefficients of y 2 and y 4 estimator under RS sampling design Then, C1, C2 and C4 hold a.s. [P]. (ii) Further, if C0 and C5 hold, and E P ||h(Y i )|| 2 < ∞, then C3 holds a.s. [P] under SRSWOR and LMS sampling design. Moreover, if C0 holds with 0 ≤ λ < E P (X i )/b, C5 holds, and E P ||h(Y i )|| 2 < ∞, then C3 holds a.s. [P] under any πPS sampling design. Proof. As before, for simplicity, let us write h(Y i ) as h i . Under the conditions C5 and E P ||h(Y i )|| 4 < ∞, C1 holds a.s. [P] by SLLN. Also, under C5, C2 holds a.s. [P]. Next, by SLLN, lim ν→∞ under SRSWOR. Then, C3 holds a.s. [P] by directly applying SLLN. Under LMS sampling design, C3 can be shown to hold a.s.[P] in the same way as the proof of the result σ 2 1 =σ 2 2 in the proof of Lemma 2 in the Appendix. Next, we have lim ν→∞ Σ 1 Table 8 8in the main article. Now, we state the following lemma. Lemma S 4. Suppose that C0 through C3 hold. Then, under SRSWOR, LMS sampling design and any HEπPS sampling design, we have Now, we state the following Lemma. Next, suppose that C0 through C2 and C4 hold, andĥ isĥ RHC orĥ GREG with d(i, s)=G i /N X i . Then, (6) holds under RHC sampling design. SWOR, LMS sampling design and any HEπPS sampling design, where Γ 1Lemma S 6. (i) Suppose that C0 through C3 hold. Further, suppose thatĥ is one ofĥ HT , h H ,ĥ RA ,ĥ P R , andĥ GREG with d(i, s)=(N π i ) −1 . Then, under SRSWOR, LMS sampling design and any HEπPS sampling design, √ n(ĥ − h) L − → N (0, Γ) as ν → ∞ (6) for some p.d. matrix Γ. (ii) Proof. It can be shown from Lemma S4 that √ n(V 1 −V) L − → N (0, Γ 1 ) as ν → ∞ under SR- ). Note that Γ 2 is a p.d. matrix since C4 holds. Let us now consider from Table 1 different choices of Θ and Z corresponding toĥ RHC , and h GREG with d(i, s)=G i /N X i . Then, it follows from Lemma S5 that for each of these choices, √ nZ=o p (1) and Θ − 1=o p (1) as ν → ∞ under RHC sampling design. Therefore,(6)holds under RHC sampling design with Γ=Γ 2 . This completes the proof of (ii) inLemma 6 as ν → ∞ since C1 and C2 hold. Therefore, (i) holds under P (s) by Markov inequality. Thus (i) holds under SRSWOR, LMS sampling design and any HEπPS sam-pling design. of the main article. For the mean, we consider the bias-corrected jackknife estimators corresponding to the GREG and the PEML estimators under each of SRSWOR, RS and RHC sampling designs, and the Hájek estimator under RS sampling design. On the other hand, for each of the variance, the correlation coefficient and the regression coefficient, we consider the bias-corrected jackknife estimators corresponding to the estimators that are obtained by plugging in the Hájek and the PEML estimators under each of SRSWOR and RS sampling design, and the PEML estimator under RHC sampling design. Table 2 : 2Relative efficiencies of estimators for mean of y.````````````````````Relative efficiency Sample size n=75 n=100 n=125 RE(Ŷ P EM L , SRSWOR |Ŷ GREG , SRSWOR) 1.049985 1.020252 1.035038 RE(Ŷ P EM L , SRSWOR |Ŷ H , RS) 4.870516 5.370899 4.987635 RE(Ŷ P EM L , SRSWOR |Ŷ HT , RS ) 2.026734 2.061607 2.027386 RE(Ŷ P EM L , SRSWOR |Ŷ P EM L , RS) 1.144439 1.124697 1.170224 RE(Ŷ P EM L , SRSWOR |Ŷ GREG , RS) 1.144455 1.124975 1.170267 RE(Ŷ P EM L , SRSWOR|Ŷ RHC , RHC ) 2.022378 1.978623 2.143015 RE(Ŷ P EM L , SRSWOR |Ŷ P EM L , RHC) 1.089837 1.030332 1.094067 RE(Ŷ P EM L , SRSWOR |Ŷ GREG , RHC) 1.089853 1.030587 1.094108 RE(Ŷ P EM L , SRSWOR | 1Ŷ BCP EM L , SRSWOR) 1.050461 1.021275 1.038282 RE(Ŷ GREG , SRSWOR | 1Ŷ BCGREG , SRSWOR) 1.002649 1.003156 1.005397 RE(Ŷ H , RS | 1Ŷ BCH , RS) 1.036379 1.006945 1.12841 RE(Ŷ P EM L , RS | 1Ŷ BCP EM L , RS) 1.016953 1.013402 1.011762 RE(Ŷ GREG , RS | 1Ŷ BCGREG , RS) 1.016692 1.011597 1.011493 RE(Ŷ P EM L , RHC | 1Ŷ BCP EM L , RHC) 1.01914 1.02292 1.024689 RE(Ŷ GREG , RHC | 1Ŷ BCGREG , RHC) 1.011583 1.052311 1.023058 1 BCPEML=Bias-corrected PEML estimator, BCH=Bias-corrected Hájek estima- tor, and BCGREG=Bias-corrected GREG estimator. Table 3 : 3Relative efficiencies of estimators for variance of y. Recall from Table 4 : 4Relative efficiencies of estimators for correlation coefficient between z 1 and z 2 . Recall from Table 5 : 5Relative efficiencies of estimators for regression coefficient of z 1 on z 2 . Recall from Table 7 : 7Average and standard deviation of lengths of asymptotically 95% CIs for mean of y.Average length (Standard deviation)`````````````````````````````È Table 8 : 8Average and standard deviation of lengths of asymptotically 95% CIs for variance of y. Recall from Table 10 : 10Average and standard deviation of lengths of asymptotically 95% CIs for regression coefficient of z 1 on z 2 . Recall from Table 11 : 11Average and standard deviation of lengths of asymptotically 95% CIs for regression coefficient of z 2 on z 1 . Recall from Table 12 : 12Relative efficiencies of estimators for mean of y 1 .````````````````````R elative efficiency Sample size n=75 n=100 n=125 RE(Ŷ P EM L , SRSWOR |Ŷ GREG , SRSWOR) 1.008215 1.005233 1.020408 RE(Ŷ P EM L , SRSWOR |Ŷ H , RS) 3.503939 3.880443 4.175886 RE(Ŷ P EM L , SRSWOR |Ŷ HT , RS) 1.796937 2.182675 1.8311 RE(Ŷ P EM L , SRSWOR |Ŷ P EM L , RS) 1.20961 1.228022 1.50233 RE(Ŷ P EM L , SRSWOR |Ŷ GREG , RS) 1.21831 1.237737 1.553863 RE(Ŷ P EM L , SRSWOR |Ŷ RHC , RHC) 3.274031 2.059141 2.030995 RE(Ŷ P EM L , SRSWOR |Ŷ P EM L , RHC) 1.088166 1.388563 1.51547 RE(Ŷ P EM L , SRSWOR |Ŷ GREG , RHC) 1.097934 1.398241 1.567545 RE(Ŷ P EM L , SRSWOR | 1Ŷ BCP EM L , SRSWOR) 1.070226 1.019958 1.007533 RE(Ŷ GREG , SRSWOR | 1Ŷ BCGREG , SRSWOR) 1.146007 1.116225 1.117507 RE(Ŷ H , RS | 1Ŷ BCH , RS) 1.240493 1.012969 1.155246 RE(Ŷ P EM L , RS | 1Ŷ BCP EM L , RS) 1.374578 1.046986 1.055930 RE(Ŷ GREG , RS | 1Ŷ BCGREG , RS) 1.466647 1.138300 1.205053 RE(Ŷ P EM L , RHC | 1Ŷ BCP EM L , RHC) 1.566827 1.083589 1.132790 RE(Ŷ GREG , RHC | 1Ŷ BCGREG , RHC) 1.460886 1.037045 1.028358 Table 14 : 14Relative efficiencies of estimators for mean of y 2 .````````````````````Relative efficiency Sample size n=75 n=100 n=125 Table 17 : 17Relative efficiencies of estimators for regression coefficient of y 1 on y 3 . Recall from Table 22 : 22Average and standard deviation of lengths of asymptotically 95% CIs for mean of y 1 .Average length (Standard deviation)`````````````````````````````È Table 23 : 23Average and standard deviation of lengths of asymptotically 95% CIs for variance of y 1 . Recall from Table 24 : 24Average and standard deviation of lengths of asymptotically 95% CIs for mean of y 2 .Average length (Standard deviation)`````````````````````````````È Table 25 : 25Average and standard deviation of lengths of asymptotically 95% CIs for variance of y 2 . Recall from Table 27 : 27Average and standard deviation of lengths of asymptotically 95% CIs for regression coefficient of y 1 on y 3 . Recall from Table 28 : 28Average and standard deviation of lengths of asymptotically 95% CIs for regression coefficient of y 3 on y 1 . Recall from Table 30 : 30Average and standard deviation of lengths of asymptotically 95% CIs for regression coefficient of y 2 on y 4 . Recall from Table 31 : 31Average and standard deviation of lengths of asymptotically 95% CIs for regression coefficient of y 4 on y 2 . Recall from AcknowledgmentsThe authors gratefully acknowledge careful reading of an earlier version of the paper by an anonymous reviewer and an associate editor. Critical comments and constructive suggestions from the reviewer and the associate editor led to significant improvement of the paper. The authors would also like to thank Prof. Aloke Kar and Prof. Sandip MitraAppendix Relative efficiencies of estimators for variance of y 1 . Recall from Table 4 in Section 2 that for variance of y 1 , h(y 1 )=(y 2 1 , y 1 ) and g(s 1 , s 2 )=s 1 − s 2 2. Table. 13Table 13: Relative efficiencies of estimators for variance of y 1 . Recall from Table 4 in Section 2 that for variance of y 1 , h(y 1 )=(y 2 1 , y 1 ) and g(s 1 , s 2 )=s 1 − s 2 2 .````````````````````R SRSWOR | g(ĥ H ), RS) 2.5303 1.6656. 15374RE(g(ĥ P EM L )RE(g(ĥ P EM L ), SRSWOR | g(ĥ H ), RS) 2.5303 1.6656 1.5374 RE(g(ĥ H ), SRSWOR | 2 BC g(ĥ H ), SRSWOR). RE(g(ĥ H ), SRSWOR | 2 BC g(ĥ H ), SRSWOR) RE(g(ĥ H ), RS | 2 BC g(ĥ H ), RS) 1.115 1.1239. 12269RE(g(ĥ H ), RS | 2 BC g(ĥ H ), RS) 1.115 1.1239 1.2269 . R S Re(ŷ Ht, R S |ŷ H, RE(Ŷ HT , RS |Ŷ H , RS) . R S Re(ŷ Ht, |ŷ P Em L, RS) 1.148074 1.082488 1.088804RE(Ŷ HT , RS |Ŷ P EM L , RS) 1.148074 1.082488 1.088804 . R S Re(ŷ Ht, |ŷ Greg, RS) 1.216958 1.115967 1.154132RE(Ŷ HT , RS |Ŷ GREG , RS) 1.216958 1.115967 1.154132 . R S Re(ŷ Ht, Rhc |ŷ Rhc, RE(Ŷ HT , RS |Ŷ RHC , RHC) . R S Re(ŷ Ht, |ŷ P Em L, RHC) 1.230884 1.0937 1.207308RE(Ŷ HT , RS |Ŷ P EM L , RHC) 1.230884 1.0937 1.207308 . R S Re(ŷ Ht, |ŷ Greg, RHC) 1.304737 1.127526 1.279746RE(Ŷ HT , RS |Ŷ GREG , RHC) 1.304737 1.127526 1.279746 . R S Re(ŷ Ht, |ŷ P Em L, SRSWOR) 2.440441 2.305339 2.350916RE(Ŷ HT , RS |Ŷ P EM L , SRSWOR) 2.440441 2.305339 2.350916 . R S Re(ŷ Ht, |ŷ Greg, SRSWOR) 2.58687 2.376638 2.49197RE(Ŷ HT , RS |Ŷ GREG , SRSWOR) 2.58687 2.376638 2.49197 . Rs | 1ŷ Re(ŷ H, Bch, RS) 1.252123 1.325047 1.241809RE(Ŷ H , RS | 1Ŷ BCH , RS) 1.252123 1.325047 1.241809 . Rs | 1ŷ Re(ŷ P Em L, Bcp Em L, RS) 1.988105 2.146357 2.260343RE(Ŷ P EM L , RS | 1Ŷ BCP EM L , RS) 1.988105 2.146357 2.260343 . Rs | 1ŷ Re(ŷ Greg, Bcgreg, RS) 2.055588 2.018015 2.287817RE(Ŷ GREG , RS | 1Ŷ BCGREG , RS) 2.055588 2.018015 2.287817 . Rhc | 1ŷ Re(ŷ P Em L, Bcp Em L, RHC) 1.831377 2.083210 2.006134RE(Ŷ P EM L , RHC | 1Ŷ BCP EM L , RHC) 1.831377 2.083210 2.006134 . Rhc | 1ŷ Re(ŷ Greg, Bcgreg, RHC) 1.925938 1.983984 2.091003RE(Ŷ GREG , RHC | 1Ŷ BCGREG , RHC) 1.925938 1.983984 2.091003 . Re(ŷ P Em L, Srswor | 1ŷ, RE(Ŷ P EM L , SRSWOR | 1Ŷ . Bcp Em L, SRSWOR) 1.001786 1.004973 1.060588BCP EM L , SRSWOR) 1.001786 1.004973 1.060588 . Srswor | 1ŷ Re(ŷ Greg, Bcgreg, SRSWOR) 1.021103 1.008525 1.003390RE(Ŷ GREG , SRSWOR | 1Ŷ BCGREG , SRSWOR) 1.021103 1.008525 1.003390 Relative efficiencies of estimators for variance of y 2 . Recall from Table 4 in Section 2 that for variance of y 2 , h(y 2 )=(y 2 2 , y 2 ) and g(s 1 , s 2 )=s 1 − s 2 2. Table. 15Table 15: Relative efficiencies of estimators for variance of y 2 . Recall from Table 4 in Section 2 that for variance of y 2 , h(y 2 )=(y 2 2 , y 2 ) and g(s 1 , s 2 )=s 1 − s 2 2 .````````````````````R RS) 11.893 6.967 34.691RS | g(ĥ P EM L ). RE(g(ĥ H ), RS | g(ĥ P EM L ), RS) 11.893 6.967 34.691 RHC) 5.0093 19.456 21.919RS | g(ĥ P EM L ). RE(g(ĥ H ), RS | g(ĥ P EM L ), RHC) 5.0093 19.456 21.919 RS | g(ĥ H ), SRSWOR) 9. 8232763RE(g(ĥ H ), RS | g(ĥ H ), SRSWOR) 9.8232 10.27 16.763 SRSWOR) 2.4768 4.8093 6.2264RS | g(ĥ P EM L ). RE(g(ĥ H ), RS | g(ĥ P EM L ), SRSWOR) 2.4768 4.8093 6.2264 RS) 13.301 6.3589 33.579RE(g(ĥ H ), RS | 2 BC g. RE(g(ĥ H ), RS | 2 BC g(ĥ H ), RS) 13.301 6.3589 33.579 RE(g(ĥ H ), SRSWOR | 2 BC g(ĥ H ), SRSWOR). RE(g(ĥ H ), SRSWOR | 2 BC g(ĥ H ), SRSWOR) Recall from Table 4 in Section 2 that for correlation coefficient between y 1 and. 16h(y 1 , y 3 )=(y 1 , y 3 , y 2 1 , y 2 3and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − s 1 s 2 )/((s 3 − s 2Table 16: Relative efficiencies of estimators for correlation coefficient between y 1 and y 3 . Recall from Table 4 in Section 2 that for correlation coefficient between y 1 and y 3 , h(y 1 , y 3 )=(y 1 , y 3 , y 2 1 , y 2 3 , y 1 y 3 ) and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − s 1 s 2 )/((s 3 − s 2 SRSWOR | g(ĥ P EM L ). RE(g(ĥ P EM L ). RSRE(g(ĥ P EM L ), SRSWOR | g(ĥ P EM L ), RS) SRSWOR) 90.769 163.74 154.97RE(g(ĥ H ), SRSWOR | 2 BC g. RE(g(ĥ H ), SRSWOR | 2 BC g(ĥ H ), SRSWOR) 90.769 163.74 154.97 RS) 72.604 79.355 163.03RE(g(ĥ H ), RS | 2 BC g. RE(g(ĥ H ), RS | 2 BC g(ĥ H ), RS) 72.604 79.355 163.03 SRSWOR | g(ĥ H ), RS) 1.8046 1.2304. 13482RE(g(ĥ P EM L )RE(g(ĥ P EM L ), SRSWOR | g(ĥ H ), RS) 1.8046 1.2304 1.3482 SRSWOR | g(ĥ P EM L ). RE(g(ĥ P EM L ). RHCRE(g(ĥ P EM L ), SRSWOR | g(ĥ P EM L ), RHC) 63.933 77.049 184.45RE(g(ĥ H ), RS | 2 BC g(ĥ H ), RS). RE(g(ĥ H ), RS | 2 BC g(ĥ H ), RS) 63.933 77.049 184.45 Recall from Table 4 in Section 2 that for regression coefficient of y 3 on y 1 , h(y 1 , y 3 )=(y 3 , y 1 , y 2 1 , y 1 y 3 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ). Relative efficiencies of estimators for regression. 18Table 18: Relative efficiencies of estimators for regression coefficient of y 3 on y 1 . Recall from Table 4 in Section 2 that for regression coefficient of y 3 on y 1 , h(y 1 , y 3 )=(y 3 , y 1 , y 2 1 , y 1 y 3 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ).````````````````````R RS) 100.3 110.15 196.34RE(g(ĥ H ), RS | 2 BC g. RE(g(ĥ H ), RS | 2 BC g(ĥ H ), RS) 100.3 110.15 196.34 Recall from Table 4 in Section 2 that for correlation coefficient between y 2 and. Table 19: Relative efficiencies of estimators for correlation coefficient between y 2 and y 4. h(y 2 , y 4 )=(y 2 , y 4 , y 2 2 , y 2 44and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − s 1 s 2 )/((s 3 − s 2Table 19: Relative efficiencies of estimators for correlation coefficient between y 2 and y 4 . Recall from Table 4 in Section 2 that for correlation coefficient between y 2 and y 4 , h(y 2 , y 4 )=(y 2 , y 4 , y 2 2 , y 2 4 , y 2 y 4 ) and g(s 1 , s 2 , s 3 , s 4 , s 5 )=(s 5 − s 1 s 2 )/((s 3 − s 2 RS | g(ĥ P EM L ), RHC) 1.491 2.135. 227RE(g(ĥ H ), RS | g(ĥ P EM L ), RHC) 1.491 2.135 2.27 2.39 2.521 2.849RS | g(ĥ H ), SRSWOR). RE(g(ĥ H ), RS | g(ĥ H ), SRSWOR) 2.39 2.521 2.849 SRSWOR) 2.185 2.396 2.594RS | g(ĥ P EM L ). RE(g(ĥ H ), RS | g(ĥ P EM L ), SRSWOR) 2.185 2.396 2.594 SRSWOR) 87.942 36.363 97.432RE(g(ĥ H ), SRSWOR | 2 BC g. RE(g(ĥ H ), SRSWOR | 2 BC g(ĥ H ), SRSWOR) 87.942 36.363 97.432 Recall from Table 4 in Section 2 that for regression coefficient of y 2 on. Table 20: Relative efficiencies of estimators for regression. h(y 2 , y 4 )=(y 2 , y 4 , y 2 4 , y 2 y44 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 )Table 20: Relative efficiencies of estimators for regression coefficient of y 2 on y 4 . Recall from Table 4 in Section 2 that for regression coefficient of y 2 on y 4 , h(y 2 , y 4 )=(y 2 , y 4 , y 2 4 , y 2 y 4 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ).````````````````````R RS) 1.8158 2.3771 3.2021RS | g(ĥ P EM L ). RE(g(ĥ H ), RS | g(ĥ P EM L ), RS) 1.8158 2.3771 3.2021 RHC) 2.5985 2.6002 3.4744RS | g(ĥ P EM L ). RE(g(ĥ H ), RS | g(ĥ P EM L ), RHC) 2.5985 2.6002 3.4744 SRSWOR) 3.3278 4.5041 6.312RS | g. RE(g(ĥ H ), RS | g(ĥ H ), SRSWOR) 3.3278 4.5041 6.312 SRSWOR) 2.9788 3.9417 6.0391RS | g(ĥ P EM L ). RE(g(ĥ H ), RS | g(ĥ P EM L ), SRSWOR) 2.9788 3.9417 6.0391 RS | 2 BC g(ĥ H ), RS) 125. 256.45 260.15RE(g(ĥ H ). 17RE(g(ĥ H ), RS | 2 BC g(ĥ H ), RS) 125.17 256.45 260.15 Recall from Table 4 in Section 2 that for regression coefficient of y 4 on y 2. Table 21: Relative efficiencies of estimators for regression. h(y 2 , y 4 )=(y 4 , y 2 , y 2 2 , y 2 y4 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 )Table 21: Relative efficiencies of estimators for regression coefficient of y 4 on y 2 . Recall from Table 4 in Section 2 that for regression coefficient of y 4 on y 2 , h(y 2 , y 4 )=(y 4 , y 2 , y 2 2 , y 2 y 4 ) and g(s 1 , s 2 , s 3 , s 4 )=(s 4 − s 1 s 2 )/(s 3 − s 2 2 ).````````````````````R RS | g(ĥ P EM L ). RE(g(ĥ H ). RSRE(g(ĥ H ), RS | g(ĥ P EM L ), RS) RHC) 1.652 2.7715 2.0362RS | g(ĥ P EM L ). RE(g(ĥ H ), RS | g(ĥ P EM L ), RHC) 1.652 2.7715 2.0362 SRSWOR) 3.8248 2.4388 3.4371RS | g. RE(g(ĥ H ), RS | g(ĥ H ), SRSWOR) 3.8248 2.4388 3.4371 SRSWOR) 3.1843 2.3399 3.038RS | g(ĥ P EM L ). RE(g(ĥ H ), RS | g(ĥ P EM L ), SRSWOR) 3.1843 2.3399 3.038 RS) 47.3317 73.749 52.592RE(g(ĥ H ), RS | 2 BC g. RE(g(ĥ H ), RS | 2 BC g(ĥ H ), RS) 47.3317 73.749 52.592 126.42 323.82RS | 2 BC g(ĥ P EM L ), RS) 105. 87RE(g(ĥ H ), RS | 2 BC g(ĥ P EM L ), RS) 105.87 126.42 323.82 SRSWOR) 394.29 156.27 164.7 (3236118) (2694726) (3205532RS | 2 BC g. RE(g(ĥ HRE(g(ĥ H ), RS | 2 BC g(ĥ H ), SRSWOR) 394.29 156.27 164.7 (3236118) (2694726) (3205532) P EM L ). 1035032g(ĥ P EM L ), SRSWOR 1035032 1077345 1002397 (1472036) (1376947) (1573834) P EM L ). 1385778g(ĥ P EM L ), RS 1385778 1168689 1055339 (1584677) (1339377) (1177054) . P Em L, RHC 1319413 1134532 1072290 (1473379) (1384754g(ĥ P EM L ), RHC 1319413 1134532 1072290 (1473379) (1384754) (1472584) Rate of convergence to normal distribution for the Horvitz-Thompson estimator. Y G Berger, J. Statist. Plann. Inference. 67Berger, Y. G. (1998). Rate of convergence to normal distribution for the Horvitz- Thompson estimator, J. Statist. Plann. Inference 67, 209-226. Functional central limit theorems for single-stage sampling designs. H Boistard, H P Lopuhaä, A Ruiz-Gazen, Ann. Stat. 45Boistard, H., Lopuhaä, H. P. and Ruiz-Gazen, A. (2017). Functional central limit theorems for single-stage sampling designs. Ann. Stat. 45, 1728-1758. A pseudo empirical likelihood approach to the effective use of auxiliary information in complex surveys. J Chen, R R Sitter, Statist. Sinica. 9Chen, J. and Sitter, R. R. (1999). A pseudo empirical likelihood approach to the effective use of auxiliary information in complex surveys. Statist. Sinica 9, 385-406. Asymptotic theory of rejective sampling with varying probabilities from a finite population. J Hájek, Ann. Math. Stat. 35Hájek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a finite population. Ann. Math. Stat. 35, 1491-1523. Asymptotic normality of the Rao-Hartley-Cochran estimator: an application of the martingale CLT. Scand. E Ohlsson, J. Stat. 13Ohlsson, E. (1986). Asymptotic normality of the Rao-Hartley-Cochran estimator: an application of the martingale CLT. Scand. J. Stat. 13, 17-28. On a simple procedure of unequal probability sampling without replacement. J N Rao, H Hartley, W Cochran, J. R. Stat. Soc. Ser. B Methodol. 24Rao, J. N., Hartley, H. and Cochran, W. (1962). On a simple procedure of unequal probability sampling without replacement. J. R. Stat. Soc. Ser. B Methodol. 24, 482- 491. Jackknife Bias-Corrected Generalized Regression Estimator in Survey Sampling. M Stefan, M A Hidiroglou, Journal of Survey Statistics and Methodology. Stefan, M. and Hidiroglou, M. A. (2022). Jackknife Bias-Corrected Generalized Regression Estimator in Survey Sampling. Journal of Survey Statistics and Methodology. On asymptotic normality and variance estimation for nondifferentiable survey estimators. J C Wang, J D Opsomer, Biometrika. 98Wang, J. C. and Opsomer, J. D. (2011). On asymptotic normality and variance estimation for nondifferentiable survey estimators. Biometrika 98, 91-106.
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[ "One-Class Learned Encoder-Decoder Network with Adversarial Context Masking for Novelty Detection", "One-Class Learned Encoder-Decoder Network with Adversarial Context Masking for Novelty Detection" ]
[ "John Taylor Jewell [email protected] \nWestern University London\nONCanada\n", "Reza Vahid \nWestern University London\nONCanada\n", "Khazaie [email protected] \nWestern University London\nONCanada\n", "Yalda Mohsenzadeh \nWestern University London\nONCanada\n" ]
[ "Western University London\nONCanada", "Western University London\nONCanada", "Western University London\nONCanada", "Western University London\nONCanada" ]
[]
Novelty detection is the task of recognizing samples that do not belong to the distribution of the target class. During training, the novelty class is absent, preventing the use of traditional classification approaches. Deep autoencoders have been widely used as a base of many novelty detection methods. In particular, context autoencoders have been successful in the novelty detection task because of the more effective representations they learn by reconstructing original images from randomly masked images. However, a significant drawback of context autoencoders is that random masking fails to consistently cover important structures of the input image, leading to suboptimal representations -especially for the novelty detection task. In this paper, to optimize input masking, we introduce a Mask Module that learns to generate optimal masks and a Reconstructor that aims to reconstruct masked images. The networks are trained in an adversarial setting in which the Mask Module seeks to maximize the reconstruction error that the Reconstructor is minimizing. When applied to novelty detection, the proposed approach learns semantically richer representations compared to context autoencoders and enhances novelty detection at test time through more optimal masking. Novelty detection experiments on the MNIST and CIFAR-10 image datasets demonstrate the proposed approach's superiority over cutting-edge methods. In a further experiment on the UCSD video dataset for novelty detection, the proposed approach achieves a frame-level Area Under the Curve (AUC) of 99.02% and an Equal Error Rate (EER) of 5.4%, exceeding recent state-of-the-art models.
10.1109/wacv51458.2022.00291
[ "https://arxiv.org/pdf/2103.14953v3.pdf" ]
232,404,108
2103.14953
8b22a58da3c1a469b09806d1734a4b029d0b1bbc
One-Class Learned Encoder-Decoder Network with Adversarial Context Masking for Novelty Detection John Taylor Jewell [email protected] Western University London ONCanada Reza Vahid Western University London ONCanada Khazaie [email protected] Western University London ONCanada Yalda Mohsenzadeh Western University London ONCanada One-Class Learned Encoder-Decoder Network with Adversarial Context Masking for Novelty Detection Code available at https://github.com/jewelltaylor/OLED. Novelty detection is the task of recognizing samples that do not belong to the distribution of the target class. During training, the novelty class is absent, preventing the use of traditional classification approaches. Deep autoencoders have been widely used as a base of many novelty detection methods. In particular, context autoencoders have been successful in the novelty detection task because of the more effective representations they learn by reconstructing original images from randomly masked images. However, a significant drawback of context autoencoders is that random masking fails to consistently cover important structures of the input image, leading to suboptimal representations -especially for the novelty detection task. In this paper, to optimize input masking, we introduce a Mask Module that learns to generate optimal masks and a Reconstructor that aims to reconstruct masked images. The networks are trained in an adversarial setting in which the Mask Module seeks to maximize the reconstruction error that the Reconstructor is minimizing. When applied to novelty detection, the proposed approach learns semantically richer representations compared to context autoencoders and enhances novelty detection at test time through more optimal masking. Novelty detection experiments on the MNIST and CIFAR-10 image datasets demonstrate the proposed approach's superiority over cutting-edge methods. In a further experiment on the UCSD video dataset for novelty detection, the proposed approach achieves a frame-level Area Under the Curve (AUC) of 99.02% and an Equal Error Rate (EER) of 5.4%, exceeding recent state-of-the-art models. Introduction Novelty detection involves determining whether or not an unknown sample belongs to the distribution of the training data. In the case the sample is similar to the training data, it is referred to as an inlier or normal sample. Alternatively, if the sample does not follow the distribution defined in the training examples, it is referred to as an outlier or anomaly. Novelty detection differs from other machine learning tasks in that the outlier class is poorly sampled or nonexistent. Due to the unavailability of outlier samples, traditional classification approaches are not suitable. Within computer vision, novelty detection is ubiquitous with subtasks that have widespread applications such as marker discovery in biomedical data [1] and video surveillance [2]. Anomaly detection in images is one such task that involves identifying whether an image is an inlier or an outlier based on training data that mostly consists of inlier images. To compensate for the unavailability of outlier samples, one-class classification methods aim to model the distribution of the inlier data [3]. New samples that do not conform to the target distribution are considered outliers. However, it is often hard to model the distribution of image data with conventional methods because of the high dimensionality in which the data points reside [3]. With the advent of deep learning, methods have been proposed that are able to effectively produce representations for high dimensional data [4]. Autoencoders (AE) are an unsupervised class of approaches that are well suited for modeling image data [5]. At a high level, an AE consists of two modules: an encoder and a decoder. The encoder learns a mapping from an image to a lower-dimensional latent space, and the decoder learns a mapping from the latent space back to the original image. In this way, AEs are trained in an unsupervised manner by minimizing the error between the original image and the reconstruction. As a powerful unsupervised method for learning representations, AEs are the basis of many one-class classification approaches [6]. To detect anomalous images, the AE is first trained on a set of primarily normal images. At test time, the reconstruction error of a sample is used as an anomaly score. The underlying intuition is that the reconstruction error will be lower for inlier samples than outlier samples [7]. This follows from the fact that the AE is trained solely on inlier samples. However, this assumption is often violated, and the AE generalizes well to construct images outside of the distribution of the training data [8,9]. This is especially evident in cases where anomalous images share similar compositional patterns as inlier images. Recent methods introduce additional complexity into the autoencoders reconstruction task so that outliers are not reconstructed well [10,11,12]. To this end, denoising autoencoders (DAE) have been used. DAEs learn to reconstruct unperturbed images from images that have been perturbed by noise [13]. Beyond yielding more robust representations, the denoising task of the AE has been shown to induce a reconstruction error that approximates the local derivative of the log-density with respect to the input [14]. Thus, a sample's global reconstruction error reflects the norm of the derivative of the log-density with respect to the input. In this way, DAEs provide a more interpretable and theoretically grounded anomaly score. Context autoencoders (CAE) [15], a specific type of DAE, have shown strong performance in the anomaly detection task [12,16]. Instead of being perturbed by noise, input images are subjected to random masking. Consequently, the CAE learns to inpaint a randomly masked region of the input image in conjunction with the reconstruction task. This random masking is similar to using salt-and-pepper noise, which has been shown to yield better representations by implicitly enforcing the AE to learn semantic information about the distribution of the training data [15]. Despite these strengths, in some cases CAEs suffer from suboptimal representations leading to poor performance in the anomaly detection task. Inspired by the drawbacks of CAEs [15], we proposed One-Class Learned Encoder-Decoder (OLED) Network with Adversarial Context Masking. OLED introduces a Mask Module M M that produces masks applied to images input into the Reconstructor R. The masks generated by M M are optimized to cover the most important parts of the input image, resulting in a comparable reconstruction score across samples. The underlying intuition is that the loss of the masked region will be low in the case of inlier images and high in the case of outlier images. This stems from the fact that the Reconstructor learns to inpaint masked regions using mostly inlier samples. Thus, the inpainted regions of outlier images will consist of patterns present in the inlier images, yielding a high reconstruction error. At a high level, the Mask Module is a convolutional autoencoder, and the Reconstructor is a convolutional encoder-decoder. They are trained in an adversarial manner, where the Mask Module is trying to generate masks that yield higher reconstruction errors, and the Reconstructor is trying to minimize the reconstruction error of the masked image. The architecture of the proposed approach is shown in Figure 1. We applied OLED to several benchmark datasets for anomaly detection in addition to providing a formal analysis of the efficacy of the Mask Module. Experimental results demonstrate that OLED is able to outperform a variety of recent state-of-the-art methods and hints at the broader usefulness of the mask module in other core computer vision tasks. In this paper our contributions are the following: • We proposed a novel approach for finding the most important parts of images for novelty detection. • Our framework is optimized through adversarial setting which yields more efficient representations for novelty detection. • Our method provides several anomaly scores which capture different aspects of normality • Due to effectiveness of our method in masking important parts of the image, we can leverage it at the test time which yields better anomaly scores. Related Works One-class classification is primarily associated with the domain of novelty, outlier, and anomaly detection. In these types of problems, a model attempts to capture the distribution of the inlier class to finally detect the unknown outliers or novel concepts. The conventional methods in the anomaly detection field utilized one-class SVM [17,18] and Principal Component Analysis (PCA) and its variations [19,20] to find a subspace that best represents the distribution of normal samples. Unsupervised clustering techniques like k-means [3] and Gaussian Mixture Models (GMM) [21] also have been used to formulate the distribution of normal data for identifying the anomalies, but they normally fail in dealing with high-dimensional data. Several other proposed methods benefit from selfrepresentation learning, such as reconstruction-based approaches. They usually rely on the hypothesis that the outlier samples cannot be reconstructed precisely by a model that only learned the distribution of inlier samples. For example, Cong et al. [22] suggested a model for video anomaly and outlier detection by learning sparse representations for distinguishing between inlier and outlier samples. In [23,24], test samples are reconstructed using the representations learned from inlier samples, and the reconstruction error is employed as a metric for novelty detection. Most of the deep learning-based models with encoderdecoder architecture [25,26,27,8,28] also used this score to detect anomalies. Although effective, these methods are limited by the under-designed representation of their latent space. Gong et al. [9] proposed a deep autoencoder augmented with a memory module to encode the input to a latent space with the encoder. The resulting latent vector is used as a query to retrieve the most relevant memory item for reconstruction with the decoder. In [1], a deep convolutional generative adversarial network (GAN) is leveraged to learn a manifold of normal images with a novel anomaly score based on the mapping from image space to a random distribution. Sabokrou et al. [29] took advantage of Generative Adversarial Networks (GAN) [30] along with denoising autoencoders to use the discriminator's score for the reconstructed images for the novelty detection task. Zaheer et al. [31] redefined the adversarial one-class classifier training setup by modifying the role of the discriminator to distinguish between good and bad quality reconstructions and improved the results even further. Perera et al. used denoising auto-encoder networks to enforce the normal samples to be distributed uniformly across the latent space [11]. Abati et al. suggested a deep autoencoder model with a parametric density estimator that learns the probability distribution underlying its latent representations through an autoregressive procedure [10]. Some recent works [32], [33] have tried to leverage pretrained deep neural networks by distilling the knowledge. In [32], they utilized a VGG16 [34] to compute a multilevel loss for training the student network to calculate the anomaly score and perform anomaly segmentation. Even though these methods could achieve high performance, they benefit from the knowledge attained by training on millions of labeled images and also may not work well on other modalities of data. As our proposed method does not leverage pretrained networks, we consider our work complimentary, and thus do not compare against this class of approaches. Method Motivation Previous works have demonstrated that the reconstruction error of an Autoencoder (AE) acts as a good indicator of whether or not a sample conforms to the distribution defined in the training examples [7]. As such, AEs are commonly used for anomaly detection. To this end, Denoising Auotoencoders (DAE) have often been used because of the more robust representations they offer [14]. Context Autoencoders (CAEs), a subclass of DAEs, have been particularly successful in the anomaly detection task by offering representations that capture the semantics of the underlying training distribution [12]. However, CAEs have a number of disadvantages. The first drawback of CAEs is that they learn suboptimal representations by failing to consistently mask important parts of the image during training. Furthermore, they perform poorly at test time if they include random masking. This is because the mask placement is closely related to the reconstruction score. An outlier with a simple part of the image masked may have a lower reconstruction error than an inlier image with a difficult part of the image masked. Thus, random masking cannot be effectively used at test time for more robust anomaly detection. Conversely, our approach avoids these drawbacks by learning to mask intelligently. Experimental results from the ablation study in section 4.6 support this conclusion. In order to mitigate these shortcomings while leveraging the benefits offered by CAEs, we propose a One-Class Learned Encoder-Decoder Network with adversarial context masking, which we call OLED. Overview Our proposed framework, OLED, consists of two modules: the Reconstructor R and the Mask Module M M . An overview of the architecture is available in 1. R and M M are trained in an adversarial manner, where R seeks to reconstruct images that have been covered by masks generated by M M . Masks have the same spatial resolution as input images with a single channel of 0 or 1 activations. As such, a masked image is easily obtained by taking an element-wise product of an image and its corresponding mask. Through the adversarial training process, R learns representations that encode semantic information of the train-ing distribution through the inpainting task. Alternatively, M M learns to mask the most important parts of the input image by maximizing the reconstruction error of R. At test time, new samples are subjected to masks generated by M M and fed to R where the reconstruction error is used as an anomaly score. Accordingly, the reconstruction error will be low for the inlier class because R is trained to reconstruct and inpaint inlier samples. However, in the case of anomalies, the reconstruction error will be higher primarily. This stems from the fact that R learns to reconstruct and inpaint masked regions using mostly inlier samples. Reconstructor R is a convolutional encoder-decoder network that is trained to reconstruct masked images. Following some of the previous works [16], a dense bottleneck is used. The full connectivity of the dense layer is helpful for the inpainting task, especially for shallow networks with low receptive fields. Moreover, R does not include max-pooling layers for greater stability in training. To further promote stability, Leaky ReLU and batch normalization are used in each convolutional block. The values after the last convolution layer are clipped to in between -1 and 1. Mask Module M M consists of a mask generator M G followed by a threshold unit T that generates masks of the same resolution as the input image. These masks are applied to the corresponding input image prior to being fed into R. M M seeks to produce a mask that maximizes the reconstruction error of the input. In this way, it learns to mask the most optimal parts of the image. Thus, masks generated by M M yield more comparable anomaly scores across samples in contrast to random masking. Mask Generator M G is a convolutional autoencoder that takes an input image and generates a corresponding activation map. This activation map is input into the threshold unit to produce a binary mask. Similar to R, M G avoids the use of max pooling. Additionally, batch normalization and Leaky ReLU are used in each convolutional block, with the exception of the last convolution block that uses ReLU. In contrast to R, M G has a spatial bottleneck and contains much fewer parameters. This reflects the fact that R has a substantially more complex task than M G. Threshold Unit Activation maps generated by M G are input into T to generate a mask. T requires a threshold hyperparameter that determines what percentage of the pixels in the image will not be masked. In this way, the same amount of pixels are masked in each image, ensuring that the reconstruction errors are comparable between samples. For each activation map, pixels with activations in the top 1 -t percent are set to 0. The final mask is obtained by setting the remaining activations to one. More formally, given an activation map A and a scalar s that represents the numeric value of the pixel with the t highest activations: A ij = 0, if A ij ≥ s 1, otherwise(1) As it stands, this is a discontinuous function, which is known to have less stable optimization. In order to eliminate this problem, the threshold operation is reformulated in terms of continuous ReLU activation function: A ij = max(A ij * −1 + s, 0) max(A ij * −1 + s, 0) + ϵ(2) where max(A ij , 0) represents the ReLU activation, and ϵ is an infinitesimal positive scalar. The above formulation ensures continuity over the entire domain of the function enabling backpropagation through T into M G. Masking Procedure M G and T sequentially process an input image to create a mask. Masks generated by M M are single-channel binary images with the same spatial resolution as input images. The masked image is obtained by applying the mask to its corresponding image for each channel. More precisely, given an image x, the corresponding masked image x m is defined as: x m = x ⊙ M M (x)(3) where ⊙ denotes element-wise multiplication. In this way, activations in the mask that are 0 set the corresponding pixel in the input image to 0, otherwise the pixel remains unchanged. It is important to note that input images, and thus the reconstructions generated by R, are scaled between -1 and 1. Because of this, masked pixels are set to the midpoint of the color range. Adversarial Training Adversarial training is a learning mechanism in which two networks compete in a minmax game that iteratively enhances the ability to model the underlying distribution of the data. Following this intuition, Generative Adversarial Networks (GANs) [30] have been proposed and shown immense success in generating samples with similar distribution of the training data. In order to do so, a generator network G and discriminator network D are trained in this manner. G takes as input a noise vector and seeks to produce samples that follow the distribution of the training data. Alternatively, D takes as input real samples from the training set along with fake samples generated by G and seeks to discriminate between the two. More formally, given an image x sampled from p data and a random latent vector z sampled from p z the objective of a GAN is: min G max D E x∼pdata(x) [log D(x)]+ E z∼pz(z) [log(1 − D(G(z))](4) G(z) is a sample generated by G with input z. D(x) and D(G(z)) are the discriminator's classification scores for a real and generated sample, respectively. Similarly, we train M M and R adversarially. M M seeks to generate masks that yield the highest reconstruction error from R. The total reconstruction error L tot consists of an L2 loss of the masked image L mask , contextual loss of the masked region L cont and an L2 loss of an unperturbed image L recon . Given an inlier image x and the corresponding masked image x m , L mask , L cont and L recon are defined as: L mask = ∥x − R(x m )∥ 2 (5) L cont = ∥x c − R(x c )∥ (6) L rec = ∥x − R(x)∥ 2(7) where x c is the masked region of the input image and R(x c ) is the reconstruction of the masked region. R(x m ) denotes the reconstruction of the masked image x m . R(x) is the reconstruction of the intact image x with the Reconstructor. The following are the components of the objective: • L mask : Forces the network to form a semantic understanding of characteristic elements of inlier samples. • L cont : Emphasizes that the masked region of the image is reconstructed properly to avoid blurry reconstructions of the masked region. • L rec : Helps the network learn the distribution of unmasked inliers. As such, the objective function of OLED is given by: min R max M M L mask + γL cont + λL rec(8) where γ and λ are hyperparameters that weigh L cont and L rec , respectively. Since M M has no bearing on L rec , it is not included in the error of M M . Anomaly Scoring The three distinct loss terms in the OLED objective present the opportunity for three anomaly scores to be defined: s mask , s cont and s rec . s mask , s cont and s rec are obtained through scaling L mask , L cont and L rec between 0 and 1. By virtue of being derived from the respective losses, each anomaly score captures a different element of normality. s cont and s mask capture normality local to the masked region which tends to cover the most characteristic parts of the image. s rec captures the global normality of the image, taking into account how good the entire reconstruction of the image is. s avg is obtained by taking the average of s mask , s cont and s rec . Experiments This section contains a detailed analysis of the proposed method, OLED. In particular, we evaluated OLED on three datasets that are benchmarks in the novelty/anomaly detection literature, and the results are compared to recent stateof-the-art methods. Additionally, we presented a formal analysis exploring the effectiveness of masks generated by M M . Implementation Details OLED is implemented in Python using the TensorFlow [35]. A detailed overview of the architecture of R and M M is available in Section 3.3 and Section 3.4, respectively. t, λ and γ are set to 87.5, 1 and 50 respectively. These hyperparameters were set based on experimentation and an ablation study showing the stability of the performance across different settings. The threshold parameter can be adjusted based on the difficulty of the dataset; where larger values of the threshold are more suitable for more complex datasets. The weights of the the loss function listed as the defaults are to balance out the effect of reconstruction losses since they are on different scales. R and M M use an Adam optimizer with a learning rate of 5e −4 , b 1 of .5 and b 2 of .9. The networks are trained for 300 epochs. Following [29], a small validation set containing 150 samples from inliers and 150 samples from outliers from the training set is used to determine the best epoch to select models R and M M . Datasets The three datasets chosen for the experiments are MNIST [36], CIFAR-10 [37] and UCSD [38]. These particular datasets were chosen based on their popularity as benchmarks in the anomaly detection literature. The setups were chosen in a way that enables OLED to be compared to a variety of recent state-of-the-art methods. MNIST: MNIST is a dataset that contains 60,000 images of handwritten digits from 0 to 9. Images in MNIST are grayscale with a resolution of 28 x 28. Method AUCROC OCSVM [17] 0.9499 AE [5] 0.9619 VAE [40] 0.9643 PixCNN [41] 0.6141 DSEBM [26] 0.9554 MemAE [9] 0.9751 OLED (Ours) s rec 0.9772 OLED (Ours) s mask 0.9851 OLED (Ours) s cont 0.9650 OLED (Ours) s avg 0.9845 CIFAR-10: CIFAR-10 is a dataset that contains 60,000 natural images of objects from across ten classes. Images in CIFAR-10 are RGB with a resolution of 32 x 32. Similar to MNIST, CIFAR-10 is also used widely as a benchmark in the anomaly detection literature. However, CIFAR-10 presents more of a challenge because images differ substantially across classes, and the background of images are not aligned. UCSD: This dataset [39] consists of two subsets (Ped1 and Ped2) with different outdoor scenes. Available objects in the frames are pedestrians, cars, skateboarders, wheelchairs, and bicycles. Pedestrians are dominant in nearly all frames and considered as the normal class, while other objects are anomalies. We assessed our method on Ped2, which includes 2,550 frames in 16 training and 2,010 frames in 12 test videos, all with a resolution of 240×360 pixels. Following [31], we calculated frame-level area under the receiver operating characteristic (AUCROC) and Equal Error Rate (EER) to evaluate performance and compare against both patch-based and full-frame setups. Novelty Detection in Image Datasets MNIST: OLED is evaluated on MNIST using the protocol defined in [9]. This protocol involves dividing the dataset into ten different anomaly detection datasets corresponding to the ten predefined classes in MNIST. In each anomaly detection dataset, the inliers are sampled from 1 class, and the outliers are sampled from the remaining 9 classes. The normal data is split into train and test sets with a ratio of 2:1, and the anomaly proposition is set to be 30%. Following [9], AUCROC is the evaluation metric for this experiment. Given the protocol in [9], OLED is compared against MemAE [9] and other methods [17,40,41,26]. The results are reported in Table 1. OLED yields excellent results, surpassing MemAE and other approaches. In particular, s rec , s mask and s avg exceed all other identified approaches, Method AUCROC OCSVM [17] 0.5856 DAE [13] 0.5358 VAE [40] 0.5833 PixCNN [41] 0.5506 GAN [1] 0.5916 AND [10] 0.6172 AnoGAN [1] 0.6179 DSVDD [42] 0.6481 OCGAN [11] 0.6566 OLED (Ours) s rec 0.6622 OLED (Ours) s mask 0.6711 OLED (Ours) s avg 0.6683 OLED (Ours) s cont 0.6673 Table 2. One-class novelty detection Average AUCROC results on CIFAR-10 image dataset following the protocol in [11]. recording an AUCROC of 0.977, 0.985 and 0.984, respectively. A visualization of OLED applied to both inlier and outlier samples for MNIST is available in Figure 2. Additionally, in Figure 3, the reconstructions of OLED are compared to that of a normal AE, further demonstrating the superiority of the representations offered by OLED for the anomaly detection task. CIFAR-10: OLED is evaluated on CIFAR-10 using the protocol defined in [11]. This protocol involves dividing the dataset into ten different anomaly detection datasets corresponding to the ten predefined classes in CIFAR-10. In each anomaly detection dataset, the inliers are sampled from 1 class, and the outliers are sampled from the remaining 9 classes. The predefined train and test splits are used to conduct the experiments. Testing data of all classes are used for testing. Following [11], AUCROC is the evaluation metric for this experiment. OLED is compared to OCGAN [11] and other recently Method AUCROC EER TSC [43] 0.922 -FRCN action [44] 0.922 -AbnormalGAN [45] 0.935 0.13 MemAE [9] 0.941 -GrowingGas [46] 0.941 -FFP [47] 0.954 -ConvAE+UNet [48] 0.962 -STAN [49] 0.965 -Object-centric [50] 0.978 -Ravanbakhsh [51] -0.14 ALOCC [29] -0.13 Deep-cascade [52] -0.09 Old is gold [31] 0 proposed methods for anomaly detection [10,13,1,42]. The results are reported in Table 2. OLED outperforms the compared methods, including OCGAN, by a considerable margin. Particularly, s rec , s mask , s avg and s cont exceed all other identified approaches, recording an AUCROC of 0.662, 0.671, 0.6683 and 0.667, respectively. A visualization of OLED applied to both inlier and outlier samples for CIFAR-10 is available in Figure 2. Video Novelty Detection One of the common use cases of one-class classification is in the domain of novelty detection for surveillance purposes [9,27,29]. Nonetheless, this task is more difficult in the video domain because of the variations of mobile objects across the frames. In this experiment, each frame of the dataset is divided into patches of size 30×30 pixels following [29]. Training patches only include scenes of walking pedestrians, while in the testing phase, patches are extracted from outlier frames that contain abnormal as well as normal objects. Frame-level AUROC and EER are the two metrics used to compare our method with state-ofthe-art methods in recent years. As depicted in Table 3, our method outperforms recent state-of-the-art models in the video novelty detection task. More specifically, our approach achieves an AUCROC performance of 99.02% and an EER of 5.4%. The visualization in Figure 4 demonstrates the separability of anomaly scores for inliers and outliers. Mask Module Evaluation The results from the experiments in Section 4.3 and Section 4.4 are a clear indication that OLED is a strong method for anomaly detection. In every case, anomaly scores that leveraged masking, and by extension M M , yielded the highest performance. Visual results in Figure 2 and 3 support the initial hypothesis that M M generates masks that cover important structures in the input image. Furthermore, this is the case for both inlier and outlier images. The following section seeks to solidify these observations more formally. To quantitatively assess the effectiveness of M M in masking important parts of images, M M is re-purposed to perform a binary segmentation task that involves identifying whether or not each pixel in the input image is important. Specifically, the activation maps A generated by M M serve as the predicted semantic maps for images. A is used instead of M M (x) to avoid the threshold constraint imposed by T . Using A and the ground truth semantic maps, the pixelwise AUCROC score is computed for both inlier and outlier images. The aforementioned analysis is realized by evaluating the M M trained on digit class 8 from the MNIST experiments in Section 4.3 on the corresponding test set. MNIST is well suited for this experiment because we are able to make the assumption that nonzero pixels are part of the digit and thus important. The ground truth semantic maps for the test set are obtained by setting non zero activations to 1 otherwise 0. The former signals the pixel corresponds to part of the written digit, and the latter signals the pixel is part of the background. The results for the experiment are displayed in Table 5. M M is able to segment important pixels in both inlier and outlier images with a high degree of accuracy with no modifications to the original architecture. This is a testament to the usefulness of M M in the anomaly detection task and hints at broader use cases in computer vision. Ablation Study In order to further assess the value of the proposed learned masking approach, OLED is compared to the baseline method context autoencoders (CAE). As CAEs employ random masking during training, the following section seeks to compare the learned masking proposed by OLED with random masking utilized in CAEs. To realize this comparison, a CAE was implemented and evaluated on the MNIST dataset using the protocol outlined in Section 4.3. The CAE shared the same architecture as R. The CAE is given input images with a random 10 x 10 region cropped out during training, keeping the number of masked pixels relatively consistent with R. The results from the above experiment are displayed in 4. Similar to OLED, s rec , s mask , s avg and s cont are reported for the CAE. OLED is able to substantially outperform CAE, despite having identical architectures for the base reconstruction module. This is a clear indication that the learned masking approach proposed in OLED outperforms random masking for the anomaly detection task. Additionally, masking at test time enhances the performance of OLED but substantially decreases the performance of the CAE. This supports our intuition that the wrong placement of the masks by CAEs leads to suboptimal representations and introduce unwanted variations in the reconstruction error of samples that is detrimental to novelty detection performance. Discussion The results presented in Section 4 are a clear indication of the effectiveness of OLED for the anomaly detection task. In all three anomaly detection experiments on MNIST, CIFAR-10 and UCSD, OLED outperformed state-of-the-art methods by a large margin. Additional experiments evalu- As initially hypothesized, OLED is able to reconstruct samples from the inlier class with ease but struggles to reconstruct samples from the outlier class. This addresses one of the fundamental problems AE face when applied to the anomaly detection task; reconstructing outliers too well. OLED accomplishes this by offering representations that are optimized for reconstructing important parts of the inlier samples through the adversarial training of R and M M . Beyond this, anomaly detection is enhanced through the use of masking at test time. OLED also presents the benefit of being trained endto-end, resulting in a less cumbersome training procedure than some of the identified methods. In this way, M M can be included seamlessly into existing AE-based anomaly detection methods. There are also no constraints that prevent OLED from being applied to other modalities of data. Furthermore, the core innovation proposed in this research, learned optimal masking, has the potential to be applied to other tasks in computer vision and beyond. Conclusion In this paper, we proposed an adversarial framework for novelty detection in both images and videos. More specifically, our method includes a Mask Module and a Reconstructor; the Mask Module is a convolutional autoencoder that learns to cover the most important parts of images, and the Reconstructor is a convolutional encoder-decoder that strives to reconstruct the masked images. The mask module will learn to mask the parts of input in a way to increase the reconstruction loss while the Reconstructor tries to minimize it. The proposed approach allows semantically rich representations and improves novelty detection at test time by covering the most important parts of the context. We have applied our method to a variety of tasks, including outlier and anomaly detection in images and videos. The results illustrate the superiority of OLED in identifying samples related to the outlier class compared to recent state-ofthe-art models. Figure 1 . 1An overview of the architecture in OLED. The Mask Module adversarially learns to cover the important parts of the input image; it consists of an autoencoder that generates an activation map and a threshold unit to produce the binary mask. The Reconstructor aims to minimize the reconstruction error and the Mask Module aims to maximize the reconstruction error. Figure 2 . 2OLED Reconstructions. For both MNIST and CIFAR-10, the original image, perturbed image after applying the mask generated by M M (masks are illustrated in gray) and the final reconstruction are shown. Inlier samples are in the top two rows and outlier samples are in the bottom two rows. Figure 3 . 3AE vs OLED Reconstructions for the MNIST dataset. Figure 4 . 4Sample of anomaly scores for both the inlier and outlier class for the UCSD dataset. ating the performance of M M demonstrated strong performance in segmenting the most important parts of samples for both the inlier and outlier class. Table 1 . 1Average AUCROC values on all 10 classes sampled from MNIST image dataset. Table 4. Comparison between our method (OLED) vs. Context Autoencoder (CAE) on MNIST image dataset.Table 5. Segmentaion performance of mask generator M on MNIST dataset.Method Score Type AUCROC CAE s rec 0.9209 CAE s mask 0.8936 CAE s cont 0.6869 CAE s avg 0.8768 OLED (Ours) s rec 0.9772 OLED (Ours) s mask 0.9851 OLED (Ours) s cont 0.9650 OLED (Ours) s avg 0.9845 Data AUCROC Inlier 0.8499 Outlier 0.8472 Unsupervised anomaly detection with generative adversarial networks to guide marker discovery. Thomas Schlegl, Philipp Seeböck, Ursula Sebastian M Waldstein, Georg Schmidt-Erfurth, Langs, International conference on information processing in medical imaging. 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[ "https://github.com/jewelltaylor/OLED." ]
[ "BOREx: Bayesian-Optimization-Based Refinement of Saliency Map for Image-and Video-Classification Models ⋆", "BOREx: Bayesian-Optimization-Based Refinement of Saliency Map for Image-and Video-Classification Models ⋆" ]
[ "Atsushi Kikuchi \nKyoto University\n\n", "Kotaro Uchida \nKyoto University\n\n", "Masaki Waga \nKyoto University\n\n" ]
[ "Kyoto University\n", "Kyoto University\n", "Kyoto University\n" ]
[]
Explaining a classification result produced by an image-and video-classification model is one of the important but challenging issues in computer vision. Many methods have been proposed for producing heat-map-based explanations for this purpose, including ones based on the white-box approach that uses the internal information of a model (e.g., LRP, Grad-CAM, and Grad-CAM++) and ones based on the black-box approach that does not use any internal information (e.g., LIME, SHAP, and RISE). We propose a new black-box method BOREx (Bayesian Optimization for Refinement of visual model Explanation) to refine a heat map produced by any method. Our observation is that a heat-map-based explanation can be seen as a prior for an explanation method based on Bayesian optimization. Based on this observation, BOREx conducts Gaussian process regression (GPR) to estimate the saliency of each pixel in a given image starting from the one produced by another explanation method. Our experiments statistically demonstrate that the refinement by BOREx improves low-quality heat maps for image-and video-classification results.
10.48550/arxiv.2210.17130
[ "https://export.arxiv.org/pdf/2210.17130v1.pdf" ]
253,237,736
2210.17130
d8728e387dae528fd11b493b4e0cf14c07dbb9a3
BOREx: Bayesian-Optimization-Based Refinement of Saliency Map for Image-and Video-Classification Models ⋆ Atsushi Kikuchi Kyoto University Kotaro Uchida Kyoto University Masaki Waga Kyoto University BOREx: Bayesian-Optimization-Based Refinement of Saliency Map for Image-and Video-Classification Models ⋆ [0000−0001−9360−7490] , and Kohei Suenaga [0000−0002−7466−8789] Explaining a classification result produced by an image-and video-classification model is one of the important but challenging issues in computer vision. Many methods have been proposed for producing heat-map-based explanations for this purpose, including ones based on the white-box approach that uses the internal information of a model (e.g., LRP, Grad-CAM, and Grad-CAM++) and ones based on the black-box approach that does not use any internal information (e.g., LIME, SHAP, and RISE). We propose a new black-box method BOREx (Bayesian Optimization for Refinement of visual model Explanation) to refine a heat map produced by any method. Our observation is that a heat-map-based explanation can be seen as a prior for an explanation method based on Bayesian optimization. Based on this observation, BOREx conducts Gaussian process regression (GPR) to estimate the saliency of each pixel in a given image starting from the one produced by another explanation method. Our experiments statistically demonstrate that the refinement by BOREx improves low-quality heat maps for image-and video-classification results. Introduction Many image-and video-classification methods based on machine learning have been developed and are widely used. However, many of these methods (e.g., DNN-based ones) are not interpretable to humans. The lack of interpretability is sometimes problematic in using an ML-based classifier under a safety-critical system such as autonomous driving. To address this problem, various methods to explain the result of image and video classification in the form of a heatmap called saliency map [28,17,21,5,22,10,16] have been studied. Fig. 1 shows examples of saliency maps synthesized by several methods, including ours. A saliency map for an image-classification result is an image of the same size as the input image. Each pixel in the saliency map shows the contribution of the corresponding pixel in ⋆ We thank Atsushi Nakazawa for his fruitful comments on this work. KS is partially supported by JST, CREST Grant Number JPMJCR2012, Japan. MW is partially supported by JST, ACT-X Grant Number JPMJAX200U, Japan. , RISE [17] (in column (c)), and the Bayesian-optimization-based method [16] (in column (d)); the input images to each method are presented in column (a). The classification label used in the saliency maps in each row is "dining table", "Labrador retriever", and "folding chair" from the first row. the input image to the classification result. In each saliency map, the part that positively contributes to the classification result is shown in red, whereas the negatively-contributing parts are shown in blue. The notion of saliency maps is extended to explain the results produced by a video-classification model, e.g., in [5] and [23]. These saliency-map generation techniques can be classified into two groups: the white-box approach and the black-box approach. A technique in the former group uses internal information (e.g., gradient computed inside DNN) to generate a saliency map; Grad-CAM [21] and Grad-CAM++ [5] are representative examples of this group. A technique in the latter group does not use internal information. Instead, it repeatedly perturbs the input image by occluding several parts randomly and synthesizes a saliency map based on the change in the outputs of the model to the masked images from that of the original one. The representative examples of this group are LIME [20], SHAP [15], and RISE [17]. Although these methods provide valuable information to interpret many classification results, the generated saliency maps sometimes do not correctly localize the regions that contribute to a classification result [3,25,9]. Such a low-quality saliency map cannot be used to interpret a classification result correctly. Mokuwe et al. [16] recently proposed another black-box saliency map generation method using Bayesian optimization based on the theory of Gaussian processes regression (GPR) [19]. Their method maintains (1) the estimated saliency value of each pixel and (2) the estimated variance of the saliency values during an execution of their procedure, assuming that a Gaussian process can approximate the saliency map; this assumption is indeed reasonable in many cases Refinement Fig. 2: Our saliency map generation scheme via refinement. Starting from a potentially low-quality saliency map i 0 generated by an existing method, we refine the saliency map using Bayesian optimization and obtain a better-quality saliency map i N . because a neighbor of an important pixel is often also important. Using this information, their method iteratively generates the most effective mask to refine the estimations and observes the saliency value using the generated mask instead of randomly generating masks. Then, the estimations are updated with the observation using the theory of Gaussian processes. Inspired by the method by Mokuwe et al., we propose a method to refine the quality of a (potentially low-quality) saliency map. Our idea is that the GPRbased optimization using a low-quality saliency map i 0 as prior can be seen as a procedure to iteratively refine i 0 . Furthermore, even if a saliency map i 0 generated by certain method is of low quality, it often captures the characteristic of the real saliency of the input image; therefore, using i 0 as prior is helpful to guide the optimization. Based on this idea, we extend their approach so that it uses i 0 as prior information for their Bayesian optimization; see Fig. 2 for an overview of our saliency map generation scheme via refinement. Our method can be applied to a saliency map i 0 generated by any method; by the iterative refinement conducted by GPR, i 0 is refined to a better-quality saliency map as Fig. 1 presents. Each saliency map in Fig. 1b is generated by refining the one generated by RISE [17] presented in Fig. 1c; each saliency map in Fig. 1 generated by our method localizes important parts better than that by RISE. In addition to this extension, we improve their method to generate better saliency maps in a nontrivial way; these improvements include the way a saliency value is observed using a mask and the way a saliency map is generated from the final estimation of GPR. With these extensions, our method BOREx (Bayesian Optimization for Refinement of visual model Explanation) can generate betterquality saliency maps as presented in Fig. 1. We also present an extension of BOREx to video-classification models. Given a video-classification result, the resulting extension produces a video that indicates the saliency of each pixel in each frame using colors. Combined with a naively extended RISE for video-classification models, BOREx can generate a saliency map for a video-classification result without using any internal information of the classification model. We implemented BOREx and experimentally evaluated the effectiveness of BOREx. The result confirms that BOREx effectively improves the quality of low-quality saliency maps, both for images and for videos, in terms of several standard metrics for evaluating saliency maps with statistical significance (p < 0.001). We also conducted an ablation study, which demonstrates that the additional improvements to the method by Mokuwe et al. [16] mentioned above are paramount for this effectiveness. Our contribution can be summarized as follows. -We propose a new black-box method to refine a saliency map generated by any method. BOREx is an extension of the method by Mokuwe et al. [16] so that it uses a saliency map to be refined as prior in its Bayesian-optimization phase. Besides the extension to take a saliency map as a prior, BOREx also enhances Mokuwe et al. [16] in several features, including how saliency values are evaluated using masks and how a saliency map is calculated from the final estimation obtained by the Bayesian optimization. -We present an extension of BOREx to explain video-classification results. The resulting extended BOREx produces a saliency map in the form of a video in a black-box manner. -We implemented BOREx and empirically evaluated its effectiveness. The experimental results statistically confirm the effectiveness of BOREx as a method for refining saliency-map-based explanation for image and video classifiers. We also conducted an ablation study, which demonstrates that the enhancement added to the method by Mokuwe et al. [16] is essential for the effectiveness. Related Work. For both white-box and black-box approaches, various techniques have been proposed to explain a classification result of an image classifier by generating a saliency map. The white-box approach exploits the internal information of the classifier, e.g., the network architecture and the parameters, and generates a saliency map, typically without using the inference result. Zhou et al. [28] introduce class activation maps (CAM) that generate a saliency map exploiting the global average pooling layer in the classification model. Grad-CAM [21] and Grad-CAM++ [5] generalize CAM by focusing on the gradient during back propagation to relax the requirements on the architecture of the classification model. Zoom-CAM [22] is a variant of Grad-CAM that utilizes the feature map of the intermediate convolutional layers as well as the last convolutional layer. Although these techniques are efficient since an inference is not necessary, gradient-based methods do not always generate a faithful explanation because the inference result is ignored in a saliency-map generation [1,12,7,24]. In contrast, the black-box approach treats a classifier as a black-box function without using its internal information. These techniques typically perturb the given image and explain the classifier utilizing the difference in inference results between the original and the perturbed images. For example, RISE [17] and PN-RISE [10] randomly generate a mask by the Monte-Carlo method and perturb the image by occluding the pixels using the mask. Although these techniques can be applied to a model whose internal information is not available, it requires many inferences to obtain a high-quality saliency map. As shown in Fig. 2, our technique, saliency map refinement by Bayesian optimization, requires an initial saliency map i 0 generated by an explanation technique mentioned above and refines it to improve its quality. Thus, our technique allows combining one of the techniques above and the Bayesian optimization to balance various tradeoffs. Typically, one can balance the tradeoff between the number of inferences and quality by feeding a saliency map that is not necessarily of high quality but requires less number of inferences. Saliency-based explanation methods have also been investigated for video classifiers. Stergiou et al. [23] propose an explanation of a 3D CNN model for video classification by generating a saliency tube that is a 3D generalization of a saliency map. They use the white-box approach based on the idea of CAM [28]. Chattopadhyay et al. [5] show that Grad-CAM++ outperforms in the explanation of a 3D CNN model for action recognition compared to Grad-CAM. Bargal et al. [2] propose an explanation technique for recurrent neural networks (RNNs) with convolutional layers utilizing excitation backpropagation [27]. Perturbationbased black-box approaches have also been investigated to explain a video classifier by presenting salient frames [18] or a 3D generalization of a saliency map [14]. Same as the explanation of image classifiers, our technique allows combining the techniques above and the Bayesian optimization to balance various tradeoffs. The rest of the paper is organized as follows. Section 2 defines saliency maps and reviews the saliency-map generation method by Mokuwe et al. [16]; Section 3 introduces BOREx and an extension for video classifiers; Section 4 explains the experiments; Section 5 concludes. We write Λ for a set of pixels; we write λ for an element of Λ. An image is a map from Λ to N 3 ; we write i for an image and I for the set of images. The value i(λ) represents the RGB value of pixel λ in image i. We write L for the finite set of labels. A classification model is a function from I to a probability distribution over L; we write M for a model. For a model M and an image i, the distribution M (i) represents the confidence of M in classifying i to each label. We write M (i, l) for the confidence of M classifying i to l. Background Saliency Petsiuk et al. [17] define the saliency of each part in an image i based on the following idea: A part in i is important for a model M classifying i as l if the confidence remains high even the other part in i is masked. This intuition is formulated as follows by using the notion of masks. A mask m is a function m : Λ → {0, 1} that expresses how the value of each pixel of an image i is diminished; the value of pixel λ in the masked image-written i⊙m-is obtained by occluding the pixel λ if m(λ) = 0. Then, given a model M , an image i, and a label l, the saliency S i,l (λ) of pixel λ in image i in M classifying i to l is defined as follows: S i,l (λ) := E[M (i ⊙ m, l) | m(λ) = 1].(1) In the above definition and in the following, the expectation E[M (i ⊙ m, l) | m(λ) = 1] is taken over a given distribution M of masks. Notice that the above formula defines saliency only by the input-output relation of M . We call S i,l a saliency map. In (1), m is randomly taken from a distribution M over masks that models the assumption on how a salient part tends to distribute in an image. M is typically designed so that it gives higher probabilities to a mask in which masked regions form lumps, rather than the one in which masked pixels are scattered around the image; this design reflects that if a pixel is salient in an image, then the neighborhoods of the pixel are often also salient. The definition of saliency we use in this paper is the refinement of S i,l by Hatakeyama et al. [10] so that it takes negative saliency into account. Concretely, their definition of saliency S PN i,l is as follows. S P i,l (λ) := S i,l (λ). (2) S N i,l (λ) := E[M (i ⊙ m, l) | m(λ) = 0]. (3) S PN i,l (λ) := S P i,l (λ) − S N i,l (λ).(4) Their saliency S PN i,l (λ) is defined as the difference between the positive saliency S P i,l (λ) and the negative saliency S N i,l (λ). The latter is the expected confidence M (i ⊙ m, l) conditioned by m(λ) = 0; therefore, a pixel λ is negatively salient if masking out λ contributes to increasing confidence in classifying the image as l. Hatakeyama et al. [10] show that the saliency of an irreverent pixel calculated by S PN i,l (λ) is close to 0, making the generated saliency map easier to interpret. Evaluating S i,l and S PN i,l requires exhausting all masks, which is prohibitively expensive. Petsuik et al. [17] and Hatakeyama et al. [10] propose a method to approximate these saliency values using the Monte-Carlo method. Their implementations draw masks {m 1 , . . . , m N } from M and approximate S i,l and S PN i,l using the following formulas, which are derived from the definitions of S i,l and S PN i,l [10,17] where p = P [m(λ) = 1]: S i,l (λ) ≈ 1 N n m n (λ) p M (i ⊙ m n , l) (5) S PN i,l (λ) ≈ 1 N n m n (λ) − p p(1 − p) M (i ⊙ m n , l).(6) Saliency Map Generation using Gaussian Process Regression Mokuwe et al. [16] propose another approach to generate saliency maps for blackbox classification models. Their approach uses Bayesian optimization, in particular Gaussian process regression (GPR) [19] for this purpose. We summarize the theory of GPR and how it serves for saliency-map generation in this section; for a detailed exposition, see [19]. In general, a Gaussian process is a set of random variables, any finite number of which constitute a joint Gaussian distribution. In our context, Gaussian Algorithm 1 GPR-based saliency-map generation [16]. The function k is used in Line 9, which is kept implicit there. 1: D ← [] 2: Set µ(λ, r) ← 0 for every pixel λ and r ∈ L 3: j ← 0 4: while j < N do 5: (λ, r) ← arg max u µ,D 6: Set m to a square mask whose center is λ, whose side length is r, and m(λ ′ ) = 0 if λ ′ is in the square 7: s ← M (i, l) − M (i ⊙ m, l) 8: Add (λ, s) at the end of D 9: Update µ using Bayes' law 10: j ← j + 1 11: end while 12: i sal (λ) ← 1 q i µ(λ, ri) for every λ. 13: return i sal process is a distribution over functions; each f drawn from a Gaussian process maps (λ, r) to a saliency value f (λ, r) ∈ R, where r ∈ R p is a vector of auxiliary parameters for determining a mask. The r expresses, for example, the position and the size of a generated mask. A Gaussian process is completely determined by specifying (1) a mean function µ(λ, r) that maps a pixel λ and mask parameters r to their expected value E[f (λ, r)] and (2) a covariance function k((λ, r), (λ ′ , r ′ )) that maps (λ, r) and (λ ′ , r ′ ) to their covariance E[(f (λ, r) − µ(λ, r))(f (λ ′ , r ′ ) − µ(λ ′ , r ′ ))]. We write GP(µ, k) for the Gaussian process with µ and k. GPR is a method to use Gaussian processes for regression. Suppose we observe the saliency at several points in an image as D := {((λ 1 , r 1 ), s 1 ), . . . , ((λ n , r n ), s n )}. For an unseen (λ, r), its saliency conditioned by D is obtained as a Gaussian distribution whose mean and variance can be computed by D, µ, and k. Furthermore, once a new observation is obtained, the optimization procedure can update µ using the Bayes' law. These properties allow Gaussian processes to explore new observations and predict the saliency at unseen points. Using these properties of GPs, Mokuwe et al. [16] propose Algorithm 1 for saliency-map generation. Their method models a saliency map as a Gaussian process with mean function µ and covariance function k. Under this model, Algorithm 1 iteratively chooses (λ, r) (Line 5), observe the saliency evaluated with (λ, r) by using a mask whose center is at λ and with side length r (Lines 6 and 7), and update µ using Bayes' law (Line 9). To detect the most positively salient part with a small number of inferences, Algorithm 1 uses an acquisition function u µ,D (λ, r). This function is designed to evaluate to a larger value if (1) |µ(λ, r)| or (2) the expected variance of the saliency at λ estimated from D is high; therefore, choosing λ and r such that u µ,D (λ, r) is large balances exploiting the current estimation of the saliency value µ(λ, r) and exploring pixels whose saliency values are uncertain. To keep the search space reasonably small, we keep 1: D ← [] 2: Set µ(λ, r) ← i0(λ) for every pixel λ and side size r ∈ L. 3: j ← 0 4: while j < N do 5: (λ, r) ← arg max u µ,D 6: Set m to a square mask with side length r, whose center is λ, and m(λ ′ ) = 0 if λ ′ is inside the rectangle 7: s ← M (i ⊙ m, l) − M (i ⊙ m, l) 8: Add ((λ, r), s) at the end of D 9: Update µ using Bayes' law 10: j ← j + 1 11: end while 12: i sal (λ) ← 1 p i 1 r 2 i µ(λ, ri) for every λ. 13: return i sal the shape of the generated masks simple; in Algorithm 1, to a finite set of square masks. Various functions that can be used as a covariance function k have been proposed; see [19] for detail. Mokuwe et al. [16] use Matérn kernel [19]. Algorithm 1 returns the saliency map i sal by i sal (λ) := 1 q i µ(λ, r i ). The value of i sal at λ is the average of µ(λ, r) over r ∈ L. BOREx GPR-based Refinement of Saliency Map Algorithm 2 is the definition of BOREx. The overall structure of the procedure is the same as that of Algorithm 1. The major differences are the following: (1) the input given to the procedures; (2) how the saliency is evaluated; and (3) how a saliency map is produced from the resulting µ. We explain each difference in the following. Input to the algorithm. Algorithm 2 takes the initial saliency map i 0 , which is used as prior information for GPR. Concretely, this i 0 is used to initialize µ(λ, r) in Line 2. To generate i 0 , one can use any saliency-map generation methods, including ones based on black-box approach [10,17,20,15] and ones based on white-box approach [21,5]. m(λ) = 0, the value of M (i ⊙ m, l) − M (i ⊙ m, l) is expected to be close to S PN i,l (λ ′ ) if λ ′ is near λ. Another reason of using a flipped mask in the saliency observations of Algorithm 2 is to handle images in which there are multiple salient parts. For example, the image of goldfish in Fig. 3a has multiple salient regions, namely, multiple goldfish. If we apply Algorithm 1, which does not use flipped masks, to this image, we obtain the saliency map in Fig. 3b; obviously, the saliency map does not capture the salient parts in the image. This is because the value of M (i, l) − M (i ⊙ m, l) in Line 7 of Algorithm 1 is almost same everywhere; this value becomes high for this image only if m hides every goldfish in the image, which is difficult using only a single mask. Our method generates the saliency map in Fig. 3c; an observed saliency value M (i ⊙ m, l) − M (i ⊙ m, l) in Algorithm 2 is higher if m hides at least one goldfish than if m does not hide any goldfish. Generating saliency map from resulting µ. Algorithm 2 returns the saliency map i sal defined by i sal (λ) = 1 p i 1 r 2 i µ(λ, r i ). Instead of the saliency map computed by taking the simple average over every mask in Algorithm 1, the saliency map map returned by Algorithm 2 is the average weighted by the inverse of the area 1 r 2 i of each mask with the side size r. This weighted average gives more weight to the saliency values obtained by smaller masks. Using the weighted average helps a saliency map produced by Algorithm 2 localizes salient parts better than Algorithm 1. Extension for Video-Classification Models Algorithm 2 can be naturally extended for a video classifier with the following changes. -The set of masks is extended, from 2D squares specified by their side length, to 3D rectangles specified by the side length of the square in a frame, and the number of frames that they hide. Suppose a mask m with side length r and the number of frames t is applied to the pixel λ at coordinate (x, y) and at n-th frame of a video i. Then, i ⊙ m is obtained by hiding the pixel at (x, y) in each of the n-th to (n + t − 1)-th frame with the 2D square mask specified by r. -The type of functions drawn from the Gaussian process is changed to f (λ, r, t) from f (λ, r) in Algorithm 2 reflecting the change of the definition of masks. -The algorithm takes T := {t 1 , . . . , t k } in addition to L; the set T expresses the allowed variation of parameter t of a mask. -The expression to update i sal in Line 12 of Algorithm 2 is changed to 1 pk i s 1 r 2 i ts µ(λ, r i , t s ); the weight is changed to the reciprocal of the volume of each mask. Experiments We implemented Algorithm 2 and conducted experiments to evaluate the effectiveness of BOREx.Due to the limited space, we report a part of the experimental results. See the appendix for the experimental environment and more results and discussions, particularly on video classification. The research questions that we are addressing are the following. RQ1: Does BOREx improve the quality of an input saliency map? This is to evaluate that BOREx is useful to refine a potentially low-quality saliency map, which is the main claim of this paper. RQ2: Does Algorithm 2 produce a better saliency map than one produced by Algorithm 1 by Mokuwe et al. [16]? This is to demonstrate the merit of BOREx over the algorithm by Mokuwe et al. The insertion and the deletion metrics are introduced by [17] to quantitatively evaluate how well a saliency map localizes a region that is important for a decision by a model. The higher value of the insertion metric is better; the lower value of the deletion metric is better. The higher insertion implies that i sal localizes regions in i that are enough for classifying i to l. The lower deletion implies that i sal localizes regions that are indispensable for classifying i to l. The F-measure is an extension of their pointing-game metric also to consider recall, not only the precision. The higher value of F-measure is better, implying i sal points out more of an important region correctly. In what follows, we use a statistical hypothesis test called the one-sided Wilcoxon signed-rank test [26] (or, simply Wilcoxon test). This test is applied to matched pairs of values {(a 1 , b 1 ), . . . , (a n , b n )} sampled from a distribution and can be used to check whether the median of {a 1 , . . . , a n } can be said to be larger or smaller than that of {b 1 , . . . , b n } with significance. To compare saliency generation methods X and Y , we calculate the pairs of the values of metrics evaluated with a certain dataset, the first of each are of the method X and the second are of Y ; then, we apply the Wilcoxon test to check the difference in the metrics. For further details, see [26]. To address these RQs, we conducted the following experiments: RQ1: We used RISE [17] and Grad-CAM++ [5] to generate saliency maps for the images in PascalVOC dataset [8]; we write D RISE and D GradCAM++ for the set of saliency maps generated by RISE and Grad-CAM++, respectively. Then, we applied BOREx with these saliency maps as input; we write D BOREx RISE (resp., D BOREx GradCAM++ ) for the saliency maps generated using D RISE (resp., D GradCAM++ ) as input. We check whether the quality of the saliency maps in D BOREx − is better than D − by the one-sided Wilcoxon signed-rank test. If so, we can conclude that BOREx indeed improves the saliency map generated by other methods. RQ2: We generated saliency maps for the PascalVOC dataset using Mokuwe et al. [16] presented in Algorithm 1; we write D BO for the generated saliency maps. We check if the quality of the saliency maps in D BOREx RISE is better than D BO by one-sided Wilcoxon signed-rank test. If so, we can conclude the merit of BOREx over the method by Mokuwe et al. RQ3: We generated saliency maps for the dataset in Kinetics-400 using an extension of GradCAM++ and RISE for video classification implemented by us; let the set of saliency maps D M,GradCAM++ and D M,RISE , respectively. Then, we applied BOREx with these saliency maps as input; we write D BOREx M,RISE (resp., D BOREx M,GradCAM++ ) for the saliency maps generated using D M,RISE (resp., D M,GradCAM++ ) as input. We check whether the quality of the saliency maps in D BOREx M,− is better than D M,− by one-sided Wilcoxon signed-rank test. If so, we can conclude the merit of BOREx as an explanation method for a video-classification result. As the model whose classification behavior to be explained, we used ResNet-152 [11] obtained from torchvision.models 1 , which is pre-trained with Ima-geNet [6], for RQ1 and RQ2; and i3D [4] obtained from TensorFlow Hub 2 , which is pre-trained with Kinetics-400 [13]. Notice that the datasets PascalVOC and Kinetics-400 provide human-annotated bounding regions for each label and each image, enabling computation of the F-measure. Results and Discussion RQ1. Table 1 shows that BOREx improved the quality of the saliency maps generated by RISE and Grad-CAM++ in several metrics with statistical significance (p < 0.001). Therefore, we conclude that BOREx successfully refines an input saliency map. This improvement is thanks to the Gaussian process regression that successfully captured the locality of the salient pixels. For example, the saliency maps in Fig. 1 suggest that BOREx is better at generalizing the salient pixels to the surrounding areas than RISE. The time spent for GPR-based optimization was 9.26±0.26 seconds in average for each image. We believe this computation time pays off if we selectively apply BOREx to saliency maps whose quality needs to be improved. To investigate the effect of the features of BOREx presented in Section 3.1 (i.e., flipped masks and the saliency-map computation from the result of GPR by weighted average in its performance), we conducted an ablation study; the result is shown in Table 2. We compared BOREx with (1) a variant that does not use flipped masks (no-flip), (2) a variant that uses simple average instead of the average weighted by the inverse of the area of masks (simple-avg), and (3) a variant that does not use prior (no-prior). The statistical test demonstrates that flipped masks and weighted averages are effective in the performance of BOREx. However, the effectiveness over the no-prior variant is not confirmed. This is mainly because, if the quality of a given prior is already high, the effectiveness of BOREx is limited. Indeed, BOREx is confirmed to be effective over the no-prior case if the insertion metric of the priors is less than 0.6; see the row "no-prior (base insertion < 0.6)" in Table 2. The statistical test did not demonstrate the improvement in the deletion metric for a saliency map generated by RISE and the F-measure for a saliency map generated by Grad-CAM++. Investigation of several images for which BOREx degrades the metrics reveals that this is partly because the current BOREx allows only square-shaped masks; this limitation degrades the deletion metric for an image with multiple objects with the target label l. For example, a single square-shaped mask cannot focus on both chairs simultaneously in the image in Fig. 4a. For such an image, BOREx often focuses on only one of the objects, Table 1: Result of the experiments. "Image/Video": The kind of the classifier; "Compared with": the baseline method; "Metric": evaluation metric; "p-value": the p-value. The null hypothesis of each test expresses that the average of the metric of BOREx is not better than that of baseline. One asterisk indicates p < 0.05; two asterisks indicates p < 0.001. Image/Video Compared with Metric p-value Image RISE F-measure 8.307e-21 * * insertion 1.016e-23 * * deletion 8.874e-01 Grad-CAM++ F-measure 1.000 insertion 5.090e-08 * * deletion 6.790e-04 * * BO F-measure 1.800e-05 * * insertion 6.630e-11 * * deletion 3.111e-01 Video RISE F-measure 4.988e-07 * * insertion 8.974e-01 deletion 8.161e-18 * * Grad-CAM++ F-measure 9.9980e-01 insertion 3.636e-01 deletion 2.983e-07 * * generating the saliency map in Fig. 4b. Even if we mask the right chair in Fig. 4a, we still have the left chair, and the confidence of the label "chair" does not significantly decrease, which degrades the deletion metric of the BOREx-generated saliency map. RQ2. The last three rows of Table 1 show that the use of an initial saliency map improved the quality of the saliency maps generated by Bayesian optimization in terms of several metrics with statistical significance compared to the case where the initial saliency map is not given (p < 0.001). Therefore, we conclude that BOREx produces a better saliency map than the one produced by Mokuwe et al. in terms of the insertion metric and F-measure. The improvement was not concluded in terms of the deletion metrics. Investigation of the generated saliency maps suggests that such degradation is observed when the quality of a given initial saliency map is too low; if such a saliency map is given, it misleads an execution of BOREx, which returns a premature saliency map at the end of the prespecified number of iterations. RQ3. Table 1 shows the result of the experiment for RQ3. It shows that the saliency maps generated by the extensions of RISE and Grad-CAM++ for video classifiers are successfully refined by BOREx in terms of at least one metric with statistical significance (p < 0.001). Therefore, we conclude that a saliency map produced by BOREx points out regions in a video that are in- dispensable to explain the classification result better than the other methods. The improvement in the insertion metric over RISE and Grad-CAM++, and in F-measure over Grad-CAM++ were not concluded. The investigation of saliency maps whose quality is degraded by BOREx reveals that the issue is essentially the same as that of the images with multiple objects discussed above. A mask used by BOREx occludes the same position across several frames; therefore, for a video in which an object with the target label moves around, it is difficult to occlude all occurrences of the object in different frames. This limitation leads to a saliency map generated by BOREx that tends to point out salient regions only in a part of the frames, which causes the degradation in the insertion metric. The improvement in deletion metric seems to be due to the mask shape of BOREx. To improve the deletion metric for a video-classifier explanation, a saliency map must point out a salient region across several frames. The current mask shape of BOREx is advantageous, at least for a video in which there is a single salient object that does not move around, to cover the salient object over several frames. Conclusion This paper has presented BOREx, a method to refine a potentially low-quality saliency map that explains a classification result of image and video classifiers. Our refinement of a saliency map with Bayesian optimization applies to any existing saliency-map generation method. The experiment results demonstrate that BOREx improves the quality of the saliency maps, especially when the quality of the given saliency map is neither too high nor too low. We are currently looking at enhancing BOREx by investigating the optimal shape of masks to improve performance. Another important research task is making BOREx more robust to an input saliency map with very low quality. A The detail of experimental environment We implemented Algorithm 2using Python 3.6.12 and PyTorch 1.6.0. The experiments on image classification are conducted on a GPU workstation with 3.60 GHz Intel Core i7-6850K, 12 CPUs, NVIDIA Quadro P6000, and 32GB RAM that runs Ubuntu 20.04.2 LTS (64 bit) and CUDA 11.0. The experiments on video classification are conducted on a GPU workstation with 3.00 GHz Intel Xeon E5-2623 v3, 16 CPUs, NVIDIA Tesla P100, and 500GB RAM that runs Ubuntu 16.04.3 LTS (64 bit). In the implementation of Algorithm 2, we used Matérn kernel defined as follows: k((λ 1 , r 1 ), (λ 2 , r 2 )) := 2 1−ν Γ (ν) √ 2νd ′ l ν K ν √ 2νd ′ l , where d ′ is the Euclidean distance between (λ 1 , r 1 ) and (λ 2 , r 2 ) (i.e., d(λ 1 , λ 2 ) 2 + (r 1 − r 2 ) 2 where d(λ 1 , λ 2 ) is the Euclidean distance between λ 1 and λ 2 in the input image); ν and l are positive parameters that control the shape of the function; Γ is the gamma function; and K ν is a modified Bessel function [?]. Mokuwe et al. [16] used ν = 2.5 and l = 12 in their implementation. We use the Matérn kernel with ν = 1.5 and l = 12. B Results of additional experiments B.1 Effect of increasing N in Algorithm 2 We executed Algorithm 2with different N , the number of iterations for Gaussian process regression. With more iterations, the estimated µ is expected to be more precise. However, the time spent in one iteration in the later iterations tends to be longer because there are more observations to fit. Tables 3 and 4 show the result. We observe the significant improvement in the insertion metric only when we increased N from 10 to the other values; in the other cases, improvement nor degradation is not concluded (Table 4). For the other metrics, increasing N from 50 to the other values is also concluded to be effective. In any metrics, increasing N from 80 to 100 was not concluded to be effective. This result suggests that increasing N , which incurs time for executing Algorithm 3, is effective; however, the merit of increasing the number beyond certain number (here 50) is limited. B.2 Effect of the quality of an input saliency map on the output of Algorithm 2 We executed Algorithm 2with input maps generated by RISE with different numbers of masks. This experiment is to study the effect of the quality of an input saliency map on the output because the quality of an input saliency map is expected to be higher with more masks. Each result uses an input saliency map generated by RISE with 100 masks. "BOn" represents Algorithm 2with N = n. Each column represents the following: "Metric" for the metric; "Base." for the baseline; "Comp." for the compared method; "p-val." for the p-value. One asterisk indicates p < 0.05; two asterisks indicates p < 0.001. Table 5 shows the result. Significant improvement is observed (1) in the insertion metric when we increased the number of masks from 0 to 300 or more and (2) in the F-measure when we increased the number of masks from 0 to 300. We cannot conclude the significant improvement in the other cases. The two-sided test we conducted, whose result is presented in Table 6, does not conclude that the quality of the saliency maps measured in these metrics differ among the input saliency maps generated by RISE with different numbers of masks, which implies that the quality is not degraded by increasing the number of masks. These results back our conclusion of BOREx is effective to improve a low-quality saliency map in terms of the insertion metric. by RISE, the saliency maps generated by BOREx localize the important regions better and less noisy. C Examples of saliency maps C.1 Examples in which BOREx successfully improves input images C.2 Examples in which BOREx degraded the quality of input images Figure 9 presents the examples in which BOREx degraded the input images measured in the quantitative metrics. We add explanations for each example. The first row in Figure 9, which BOREx degraded the insertion metric, presents an example in which RISE successfully identifies the aeroplane, whereas BOREx wrongly identifies the ground in addition to the aeroplane. This is caused by the saliency map produced by RISE used as the prior; in the prior, the saliency of the ground in the image is high, which misled 2. The saliency maps in the second row of Figure 9 are generated by the label "chair". BOREx identifies one of the chairs in the image, but not the other chair, degrading the insertion metric because identifying two chairs are needed to recognize a park bench. This is due to the issue of the limited shape of the masks used by BOREx discussed in Section 4.1. RISE looks successfully identifies both chairs. BOREx degraded the F-measure metric for the image in the third row of Figure 9. The important region identified by BOREx concentrates around the lid of the bottle, whereas the region identified by RISE exists also on the body of the bottle. The PascalVOC dataset specifies the entire bottle as the correct answer, which leads to the poor value in the F-measure metric for the saliency map generated by BOREx. Deciding from the insertion and the deletion metrics, we guess that the model indeed considers the lid part as the salient region. D Comparison with Grad-CAM++ Although the statistical tests in Section 4do not conclude the effectiveness of BOREx measured in the insertion and the F-measure metrics to refine the saliency map produced by Grad-CAM++, there are some instances that indeed benefit from BOREx. Figure 10 presents several examples of the saliency maps in which BOREx outperforms Grad-CAM++. Interestingly, if we let BOREx and Grad-CAM++ produce saliency maps for the same image with different labels, the saliency maps produced by Grad-CAM++ are often less sensitive to the change in the label than BOREx. For example, the first (resp., the second) row in Figure 10 are the saliency maps produced by BOREx and Grad-CAM++ with label "sofa" (resp., "chair"). The saliency maps produced by BOREx correctly identify the region that is important for classifying the image to the given label; however, the saliency maps produced by Grad-CAM++ is less focused to the given label than BOREx. examples of saliency maps for a video classifier produced by BOREx and a naive extension of RISE. Compared with the saliency maps produced by RISE, the saliency maps generated by BOREx better localize the important parts. It is also observed that the salinecy maps produced by BOREx are comparable to those produced by Grad-CAM++ in spite of the black-box nature of BOREx. In Figure 12, we can observe that BOREx follows the skier better than Grad-CAM++. Table 5: Comparison of the results of Algorithm 2with input saliency maps produced by RISE with different number of masks. N is set to 50 in each execution of Algorithm 2. "RISEn" represents input salinecy maps produced by RISE with n masks are used. Each column represents the following: "Metric" for the metric; "Base." for the baseline; "Comp." for the compared method; "p-val." for the p-value. One asterisk indicates p < 0.05; two asterisks indicates p < 0.001. Table 6: Two-sided test to compare Algorithm 2with input saliency maps produced by RISE with different number of masks. Examples of saliency maps that are successfully refined by BOREx. The labels used in each explanation are: "bird", "motor bike", "dining table", "bird", "bird", "sheep", and "bicycle", from the first row. Fig. 6: Examples of saliency maps that are successfully refined by BOREx. The labels used in each explanation are: "aeroplane", "cow", "dining table", "cat", "horse", "bird", and "TV monitor", from the first row. Fig. 7: Examples of saliency maps that are successfully refined by BOREx. The labels used in each explanation are: "sofa", "car", "dining table", "dog", "dining table", "cow", and "aero plane", from the first row. E Examples of saliency maps for video classifiers (a) Input (b) BOREx (c) RISE Fig. 8: Examples of saliency maps that are successfully refined by BOREx. The labels used in each explanation are: "dog", "dining table", "horse", "bird", and "cow", from the first row. (a) Input (b) BOREx (c) RISE Fig. 9: Examples of saliency maps that BOREx degraded the quantitative metric of the input image synthesized by RISE. The labels used in each explanation are: "warplane", "park bench", and "water bottle", from the first row. The labels used in each explanation are: "sofa", "chair", "bird", "cow", "chair", "TV monitor", and "potted plant", from the first row. Fig. 1 : 1Example of the saliency maps generated by our method BOREx (in column (b)) Input: Model M ; Image i; Label l; Function k; Upperbound of iterations N ; Set of mask size L := {r1, . . . , rq}. Output: Saliency map that explains the classification of i to l by M . Algorithm 2 2GPR-based refinement of a saliency map. Input: Model M ; Image i; Initial saliency map i0; Label l; Function k((λ, r), (λ ′ , r ′ ); Upperbound of iterations N ; List of the side length of a mask L := {r1, . . . , rp}. Output: Refined saliency map obtained with GP. Saliency evaluation. Algorithm 1 evaluates the saliency by calculating M (i, l) − M (i ⊙ m, l). This value corresponds to the value of −S N i,l around the pixel λ defined in Section 2.1 since it computes how much the confidence drops if a neighborhood of λ is masked out. To estimate S PN i,l instead of −S N i,l , Algorithm 2 calculates M (i⊙m, l)−M (i⊙ m, l) in Line 7, where m is the flipped mask obtained by inverting the value at each pixel (i.e., m(λ ′ ) = 1 − m(λ ′ ) for any λ ′ ). Since m(λ) = 1 if and only if Fig. 3 : 3Example of an image with multiple salient regions. RQ3: Does the extension in Section 3.2 useful as a saliency-map generation for video classifiers? This is to evaluate the competency of BOREx to explain a video-classification result.Evaluation metrics. To quantitatively evaluate the quality of a saliency map, we used the following three measures.Insertion: For a saliency map i sal explaining a classification of an image i to label l, the insertion metric is defined as k M (i (k) , l), where i (k) is the image obtained by masking all the pixels other than those with top-k saliency values in i sal to black. Deletion: The deletion metric is defined as k M (i (−k) , l), where i (−k) is the image obtained by masking all the pixels with top-k saliency values in i sal to black. F-measure: The F-measure in our experiments is defined as kF (i (k) , l, B i,l ),where F (i (k) , l, B i,l ) is the F-measure calculated from the recall and the precision of the pixels in i (k) against the human-annotated bounded region B i,l in i that indicates an object of label l. Fig. 4 : 4Image of chairs and saliency maps to explain it. Figures 5 - 58 present examples of saliency maps generated by BOREx. We picked several examples in which BOREx successfully refines the quality of input images measured in the quantitative metrics. Compared to the input images generated Fig. 5: Examples of saliency maps that are successfully refined by BOREx. The labels used in each explanation are: "bird", "motor bike", "dining table", "bird", "bird", "sheep", and "bicycle", from the first row. Fig. 10 : 10Examples of saliency maps in which BOREx outperforms GradCAM++. Fig. 11 : 11Examples of saliency maps for video classifiers with label "surfing". Fig. 12 : 12Examples of saliency maps for video classifiers with label "skiing". Fig. 13 : 13Examples of saliency maps for video classifiers with label "horseriding". Table 2 : 2The result of ablation study.Compared with Metric p-value no-flip F-measure 2.987e-29 * * insertion 1.416e-04 * * deletion 2.024e-03 * simple-avg F-measure 2.026e-03 * insertion 1.184e-46 * * deletion 4.871e-03 * no-prior F-measure 4.514e-02 * insertion 3.84624e-01 deletion 2.2194e-01 no-prior (base insertion < 0.6) F-measure 2.4825e-02 * insertion 3.219e-03 * deletion 6.47929e-01 Table 3 : 3Comparison of the results produced by different number of N in Algorithm 2. BO50 BO100 1.132e-04 * * BO80 BO100 3.721e-02 * F-measure BO10 BO50 4.825e-32 * * BO10 BO80 1.442e-34 * * BO10 BO100 1.010e-34 * * BO50 BO80 1.606e-03 * BO50 BO100 7.387e-04 * *Metric Base. Comp. p-val. Insertion BO10 BO50 7.042e-36 * * BO10 BO80 4.023e-38 * * BO10 BO100 2.261e-38 * * BO50 BO80 8.156e-02 BO50 BO100 7.542e-02 BO80 BO100 3.066e-01 Deletion BO10 BO50 2.937e-32 * * BO10 BO80 1.838e-38 * * BO10 BO100 2.007e-41 * * BO50 BO80 4.454e-03 * BO80 BO100 5.806e-01 Table 4 : 4Two-sided test to compare the results produced by different number of N in Algorithm 2. Each result uses an input saliency map generated by RISE with 100 masks.Metric Base. Comp. p-val. Insertion BO10 BO50 1.408e-35 * * BO10 BO80 8.046e-38 * * BO10 BO100 4.522e-38 * * BO50 BO80 1.631e-01 BO50 BO100 1.508e-01 BO80 BO100 6.131e-01 Deletion BO10 BO50 5.875e-32 * * BO10 BO80 3.677e-38 * * BO10 BO100 4.013e-41 * * BO50 BO80 8.907e-03 * BO50 BO100 2.264e-04 * * BO80 BO100 7.442e-02 F-measure BO10 BO50 9.650e-32 * * BO10 BO80 2.884e-34 * * BO10 BO100 2.020e-34 * * BO50 BO80 3.212e-03 * BO50 BO100 1.477e-03 * BO80 BO100 8.388e-01 https://pytorch.org/vision/stable/models.html 2 https://tfhub.dev/deepmind/i3d-kinetics-400/1 Sanity checks for saliency maps. J Adebayo, J Gilmer, M Muelly, I J Goodfellow, M Hardt, B Kim, Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems. NeurIPS; Montréal, CanadaAdebayo, J., Gilmer, J., Muelly, M., Goodfellow, I.J., Hardt, M., Kim, B.: Sanity checks for saliency maps. 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[ "arXiv:hep-lat/0603010v2 12 May 2006 Properties of Color-Coulomb String Tension", "arXiv:hep-lat/0603010v2 12 May 2006 Properties of Color-Coulomb String Tension" ]
[ "Y Nakagawa \nResearch Center for Nuclear Physics\nOsaka University\n567-0047IbarakiOsakaJapan\n", "A Nakamura \nResearch Institute for Information Science and Education\nHiroshima University\nHigashi-Hiroshima739-8521Japan\n", "T Saito \nResearch Center for Nuclear Physics\nOsaka University\n567-0047IbarakiOsakaJapan\n", "H Toki \nResearch Center for Nuclear Physics\nOsaka University\n567-0047IbarakiOsakaJapan\n", "D Zwanziger \nPhysics Department\nNew York University\n10003New YorkNYUSA\n" ]
[ "Research Center for Nuclear Physics\nOsaka University\n567-0047IbarakiOsakaJapan", "Research Institute for Information Science and Education\nHiroshima University\nHigashi-Hiroshima739-8521Japan", "Research Center for Nuclear Physics\nOsaka University\n567-0047IbarakiOsakaJapan", "Research Center for Nuclear Physics\nOsaka University\n567-0047IbarakiOsakaJapan", "Physics Department\nNew York University\n10003New YorkNYUSA" ]
[]
We study the properties of the color-Coulomb string tension obtained from the instantaneous part of gluon propagators in Coulomb gauge using quenched SU (3) lattice simulation. In the confinement phase, the dependence of the color-Coulomb string tension on the QCD coupling constant is smaller than that of the Wilson loop string tension. On the other hand, in the deconfinement phase, the color-Coulomb string tension does not vanish even for T /T c = 1 ∼ 5, the temperature dependence of which is comparable with the magnetic scaling, dominating the high temperature QCD. Thus, the color-Coulomb string tension is not an order parameter of QGP phase transition.
10.1103/physrevd.73.094504
[ "https://export.arxiv.org/pdf/hep-lat/0603010v2.pdf" ]
119,435,543
hep-lat/0603010
d31460c9e2a064ed50266647c0cf6a92d50a2ee9
arXiv:hep-lat/0603010v2 12 May 2006 Properties of Color-Coulomb String Tension Y Nakagawa Research Center for Nuclear Physics Osaka University 567-0047IbarakiOsakaJapan A Nakamura Research Institute for Information Science and Education Hiroshima University Higashi-Hiroshima739-8521Japan T Saito Research Center for Nuclear Physics Osaka University 567-0047IbarakiOsakaJapan H Toki Research Center for Nuclear Physics Osaka University 567-0047IbarakiOsakaJapan D Zwanziger Physics Department New York University 10003New YorkNYUSA arXiv:hep-lat/0603010v2 12 May 2006 Properties of Color-Coulomb String Tension numbers: 1238Aw1238Gc1115Ha Keywords: lattice QCDcolor confinementCoulomb gaugequark-gluon plasma 1 We study the properties of the color-Coulomb string tension obtained from the instantaneous part of gluon propagators in Coulomb gauge using quenched SU (3) lattice simulation. In the confinement phase, the dependence of the color-Coulomb string tension on the QCD coupling constant is smaller than that of the Wilson loop string tension. On the other hand, in the deconfinement phase, the color-Coulomb string tension does not vanish even for T /T c = 1 ∼ 5, the temperature dependence of which is comparable with the magnetic scaling, dominating the high temperature QCD. Thus, the color-Coulomb string tension is not an order parameter of QGP phase transition. I. INTRODUCTION Understanding color confinement in quantum chromodynamics (QCD) is one of the most challenging problems in quantum field theory, and also provides essential knowledge for low temperature hadron physics. There are many approaches to understand color confinement dynamics: dual superconductor scenario, center vortex model, the infrared behavior of gluon propagators, etc. have been widely studied and a large amount of information on color confinement has been accumulated. See reviews in Refs. [1,2,3]. In those scenarios, topological objects and gauge-dependent quantities, bringing out the properties of QCD vacuum, may play an important role. A key issue is the choice of gauge in which the confinement scenario is realized. Recently, there has been considerable interest in the Coulomb gauge color confinement scenario. This scenario was originally discussed by Gribov [4], and in recent years, Zwanziger has advocated the importance of a color-Coulomb potential in Coulomb gauge for color confinement [5]. He and his collaborators showed that, in Coulomb gauge, the time-time component of gluon propagators, g 2 D 00 , including an instantaneous color-Coulomb potential plus a noninstantaneous vacuum polarization, is invariant under renormalization [5,6,7]. It has been found by perturbative analysis [7] that the instantaneous part in Coulomb gauge QCD causes antiscreening, while the vacuum polarization part causes screening. Hence, one expects that the instantaneous color-Coulomb potential represents a linearly rising behavior for large quark separations. Moreover, Zwanziger pointed out that there is an inequality [10], V phys (R) ≤ V coul (R), where V phys (R) means a physical heavy-quark-antiquark potential and V coul (R) the Coulomb heavy-quark potential corresponding to the instantaneous part of D 00 . This inequality indicates that if the physical heavy-quark potential is confining, then the Coulomb heavy-quark potential is also confining. See Ref. [8] for a review. In order to verify the Coulomb gauge color confinement scenario, one needs a nonperturbative technique to describe the low-energy color confinement. Therefore, nonperturbative verifications have been tried in lattice gauge simulation. In the SU(2) lattice numerical simulation carried out by Cucchieri and Zwanziger [9], it was found that g 2 D 00 ( k) is strongly enhanced at k = 0. In SU (2) and SU(3) lattice simulations [11,12,13], furthermore, it was reported that the Coulomb heavy-quark potential grows linearly at large quark separations in the confinement phase. On the lattice, it is essential to study the magnitude and the scaling for the string tension, which is a characteristic quantity for confinement physics. Numerical lattice calculations [11,12,13] indicate that the color-Coulomb string tension has 2 − 3 times larger value in comparison with the case of a gauge invariant Wilson loop, as expected by V phys (R) ≤ V coul (R). In addition, the SU(2) lattice numerical data in Ref. [12] show the possibility that an asymptotic scaling violation for the color-Coulomb string tension may be less than the usual Wilson loop string tension. Accordingly, the dependence of the color-Coulomb string tension on a gauge coupling or a lattice cutoff ought to be extensively investigated in SU (3) lattice gauge theory. The lattice simulations mentioned above have shown the linearity of the instantaneous color-Coulomb potential at large distances in the confinement regions. At the same time, the lattice calculations at finite temperature in the deconfinement phase indicate that the Coulomb string tension remains after the quark-gluon plasma (QGP) phase transition [12,13]. One possible explanation is that the color-Coulomb potential, determined by the spatiallike and time (temperature) independent Faddeev-Popov operator, is not sensitive to the system temperature. In addition, we note that the potential obtained from a spatial Wilson loop above T c behaves as a linearly rising function [17,18,19,20,21,22,23,24]. Both the color-Coulomb and the spatial Wilson potentials have a common feature that they are defined by spatial variables. However, the higher temperature lattice simulations than the previous calculations [12,13] are indispensable. Nevertheless, the noninstantaneous retarded part with the vacuum polarization still gives a color-screened potential [13,25,26]. In Coulomb gauge QCD there are no unphysical degrees of freedom for gauge fields; namely, Coulomb gauge is a physical gauge. In contrast, Lorentz covariant gauges generate a negative spectral function due to the indefinite metric of Fock space. This is very convenient on discussing a physical hadron, and a lot of attempts have been made to construct models based on the Coulomb gauge Hamiltonian to describe color confinement [27,28,29,30,31] and hadrons [32,33]. In this paper, we will perform more extensive lattice QCD studies on the Coulomb gauge confinement scenario comparing with the previous calculation [13]. In the confinement phase, we investigate the scaling behavior of the color-Coulomb string tension by varying a lattice cutoff or a coupling constant β = 6/g 2 . In the deconfinement phase, we discuss the relation between the thermal color-Coulomb string tensions, which are calculated at high temperatures, T /T c = 1.5 ∼ 5.0, and the magnetic scaling that is believed to dominate the high temperature QCD. In Sec. II we briefly review the partition function in Coulomb gauge and describe the instantaneous color-Coulomb potential and the noninstantaneous vacuum polarization part. In Sec. III, we give the definition of the partial-length Polyakov line correlator [11,12] to evaluate the instantaneous part. Section IV is devoted to show the numerical results. Section V gives conclusions. II. INSTANTANEOUS COLOR-COULOMB POTENTIAL The construction of the partition function in Coulomb gauge through the Faddeev-Popov technique and the derivation of the instantaneous color-Coulomb potential were done in Ref. [6]. The Hamiltonian of QCD in Coulomb gauge can be written as H = 1 2 d 3 x(E tr2 i ( x) + B 2 i ( x)) + 1 2 d 3 xd 3 y(ρ( x)V( x, y)ρ( y)),(1) where E tr i , B i and ρ are the transverse electric field, the magnetic field and the color charge density, respectively. The function V in the second term is made by the Faddeev-Popov (FP) operator in the spatial direction, M = − D ∂ = −( ∂ 2 + g A × ∂), V( x, y) = d 3 z 1 M( x, z) (− ∂ 2 ( z) ) 1 M( z, y) .(2) From the partition function with Hamiltonian Eq. (1), one can evaluate the time-time gluon propagator composed of the following two parts: g 2 A 0 (x)A 0 (y) = g 2 D 00 (x − y) = V (x − y) + P (x − y),(3) where V (x − y) = g 2 V( x, y) δ(x 4 − y 4 ).(4) The equation (4) corresponds to the instantaneous color-Coulomb potential at equal time and causes antiscreening; namely, it is the most important quantity in Coulomb gauge confinement scenario, and is constructed by the spatial FP matrix. Therefore, if the potential V is a linearly rising potential for large quark separations, then color confinement is attributed to an enhancement of the low-lying mode of FP eigenvalues [4,5]. Note that Eq. (4) in the case of quantum electrodynamics (QED) as a non-confining theory is identified as a Coulomb propagator −1/∂ 2 i or a Coulomb potential 1/r. Simultaneously, the quantity P in Eq. (3) is a vacuum polarization term, P (x − y) = −g 2 V( x, z)ρ( z, x 4 )d 3 z V( y, z ′ )ρ( z ′ , y 4 )d 3 z ′ ,(5) which causes color-screening effect owing to the minus sign of this equation, and produces the reduction of a color-confining force and a quark-pair creation from vacuum when dynamical quarks exist. Moreover, this perturbative argument is also satisfied at one-loop order [5]. III. PARTIAL-LENGTH POLYAKOV LINE In this section, we give the definition of a static heavy quark-antiquark potential in the color-singlet channel as a function of distance, R, and summarize how to fix the gauge on the lattice. We introduce a partial-length Polyakov line (PPL) defined as [11,12] L ( x, n t ) = nt ns=1 U 0 ( x, n s ), n t = 1, 2, · · · , L t .(6) Here U 0 ( x, t) = exp(iagA 0 ( x, t)) is an SU(3) link variable in the temporal direction and a, g, A 0 ( x, t) and L t represent the lattice cutoff, the gauge coupling, the time component of gauge potential and the temporal-lattice size. A PPL correlator in color-singlet channel is given by G(R, n t ) = 1 3 T r[L(R, n t )L † (0, n t )] ,(7) where R stands for | x|. From Eq. (7) one evaluates the color-singlet potential on the lattice, V (R, n t ) = log G(R, n t ) G(R, n t + 1) .(8) In the case of n t = 0, we define V (R, 0) = − log[G(R, 1)].(9) Greensite et al. argued that this function V (R, 0) in Coulomb gauge corresponds to an instantaneous color-Coulomb potential V coul (R) [11,12]. The potential V (R, n t ) in the limit n t → ∞ is expected to correspond to a physical potential, V phys (R), usually calculated from the Wilson loops in the same limit. Both potentials are known to satisfy Zwanziger's inequality, V phys (R) ≤ V coul (R) [10]. Since the color-decomposed potential defined by the PPL correlator such as Eq. (7), do not have a gauge invariant form, we must fix the gauge. One can realize the Coulomb gauge on the lattice to maximize the measurement x 3 i=1 ReTrU † i ( x, t),(10) by repeating the gauge rotations: U i ( x, t) → U ω i ( x, t) = ω † ( x, t)U i ( x, t)ω( x +î, t),(11) where ω ∈ SU(3) [46] is a gauge rotation matrix and U i ( x, t) are link variables for the spatial direction. Thus, each lattice configuration thermalized after the Monte Carlo quantization can be gauge fixed iteratively [34]. IV. RESULTS AND DISCUSSIONS We carried out SU(3) lattice gauge simulations in the quenched approximation to calculate the instantaneous color-Coulomb qq potential in the confinement and deconfinement phases. The lattice gauge configurations were generated by the standard heat-bath Monte Carlo technique with a simple plaquette Wilson gauge action. A. Linearity of instantaneous color-Coulomb potential An example of the variation of the instantaneous color-Coulomb potential V (R, 0) with distances is shown in Fig. 1, which demonstrates that the potential V (R, 0) behaves as a linearly rising function with increasing distance R and can be described in terms of the Coulomb term plus linear term with a nonzero string tension, V (R, 0) = c 0 + KR + e/R,(12) where e is fixed to −π/12 for a two-parameter fit, and K = σ c a 2 is the color-Coulomb string tension. Thus we find that the instantaneous potential V (R, 0) is a confining potential. In contrast, the vacuum polarization (retarded) part causes color screening, which weakens the confining force as reported in Refs. [11,13]. Consequently, the color-Coulomb potential in the limit n t → ∞ is expected to approach the Wilson loop potential. The slope of the potential V (R, n t ) with finite n t decreases as displayed in Fig. 1. Note that the numerical result in Fig. 1 was obtained in the previous work [13], and in the present study, we will not enter into details on the vacuum polarization part any further. [13]. The solid and dashed curves stand for the fitted result for the potential V (R, 0) and the Wilson loop potential reported in Ref. [16], respectively. B. Color-Coulomb string tension The string tension is a characteristic quantity in discussing confinement physics and thus one should investigate the scaling behavior of the color-Coulomb string tension √ σ c = √ Ka −1 , obtained by the lattice simulation with a finite cutoff. Here we can introduce a two-loop asymptotic scaling of QCD with the mass parameter Λ and lattice cutoff a as aΛ = exp − 1 2b 0 g 2 (b 0 g 2 ) − b 1 2b 2 0 = f (g),(13) where b 0 and b 1 are universal first two coefficients of β function. Since the quantity √ σ c is expected to be proportional to the scale of QCD Λ in asymptotic regions, it makes sense to consider the following relation: √ σ c Λ = √ K f (g) ,(14) which would be reduced to a constant in the continuum (weak coupling) limit. In the present study, we carried out calculations at β = 6.1 − 6.4 on a 18 4 lattice and used 300 gauge configurations measured every 100 sweeps after sufficient thermalization. The β = 6/g 2 dependence of the color-Coulomb string tensions is plotted in Fig. 2, in which we additionally employed the data at β = 5.85 − 6.00 reported in the previous calculation [13]. For the two-parameter fitting by Eq. (12), we employed the data over R ∼ 0.2 fm up to R ∼ 0.5 fm for β = 6.1 − 6.4, which are R ∼ 3 − 6 in lattice units, restricted due to the periodic boundary condition. Although the results of the high β regions have large errors in our calculations [47], the variations of the color-Coulomb string tensions as β varies seem to be smaller than the case of the Wilson loop string tension ( √ σ w ) [35], included for comparison. The relative fluctuation (the ratio of the minimum and maximum) of those data is within ∼ 6(2) %. Such tendency was also observed in the SU(2) lattice simulations [12]. Moreover, in the range of β used here, the value of √ σ c still remains approximately 2 times as large as that of √ σ w , and monotonically varies with β. This tendency seems to be unchanged in the continuum limit. However, the larger lattice simulation at higher β is required to realize the asymptotic scaling. C. Behavior of instantaneous color-Coulomb potential at T = 0 In the confinement phase, as seen in the previous section, the instantaneous color-Coulomb part gives a confining potential. However, it is reported in Refs. [12,13] that the linearity of V (R, 0) is not lost even after the QGP phase transition. Therefore, in the present work, we carried out lattice simulations at T /T c = 1.5 − 5.0 on the fixed lattice size 24 3 × 6. Here, the critical temperature of the QGP phase transition T c is approximately 256 MeV for N t = 6 [14]. We fixed the lattice temperature T = 1/N t a to vary the lattice cutoff a (β) [15] and 300 gauge configurations measured every 100 sweeps were used. In Fig. 3, we show the temperature dependence of the thermal color-Coulomb potential with a nonvanishing string tension. The potentials V (R, 0) in the deconfinement phase still behave like a linear-confining potential, and furthermore the slope and magnitude of those potentials become larger. It is found in Fig. 4 that the main temperature dependence of the thermal Table III in Ref. [35]. color-Coulomb string tension is apparently given by the linear relation, √ σ c ∼ T,(15) being directly proportional to the temperature. D. Spatial Wilson loop potential and magnetic scaling In order to interpret the confining phenomenon caused by the instantaneous color-Coulomb part in the deconfinement phase, it may be instructive to review studies on the thermal behavior of the spatial Wilson loop above T c [17,18,19,20,21,22,23,24]. The spatial Wilson loop, W (R, S), is constructed by the only spatial links (or spatial gluon fields), where R and S are spatial extents on a lattice. If the loop W (R, S) follows area law as a function of S, then the spatial Wilson loop potential V s (R) would be given by V s (R) = lim S→∞ ln W (R, S) W (R, S + 1) .(16) The potential V s on the confined T = 0 (symmetric hypercube) lattices can be identical with the usual Wilson loop potential. However, in the deconfinement phase, it is known that the potential V s is a linearly rising potential at large distances rather than a color-screened potential. Consequently, the nonvanishing spatial string tension, √ σ s , exists even in the deconfinement phase. In a series of studies on the thermal spatial Wilson loop, the spatial string tension √ σ s at high temperature has been discussed in terms of the magnetic scaling. According to perturbation theory of thermal QCD (TQCD) [36], the temporal gluon propagator yields the electric mass, m e ∼ g(T )T,(17) usually referred to as a color-Debye screening mass, while the spatial gluon propagator also yields the magnetic mass m m ∼ g 2 (T )T,(18) which must be introduced as the cutoff factor to solve an infrared divergence that appears in TQCD perturbation. Thus, the magnetic scaling somewhat has a nonperturbative origin and is closely relevant to longer range physics than the electric scale. The infrared sensitivity in TQCD is known to survive in the high temperature limit (T → ∞) through the argument of 3-dimensional reduction [38,39]. This approach enables us to obtain the effective theory that is defined by integrating out a nondynamical heavy mode in the high temperature limit; this theory proves that the long-range properties of TQCD are dominated by the magnetic scaling. E. Thermal color-Coulomb string tension To obtain the thermal color-Coulomb string tension σ c (T ) for the deconfining phase, we employed an ansatz of the Coulomb plus linear terms. The actual fitting analyses by the use of the same function as Eq. (12) give χ 2 /ndf 1 for the data of the fixed range of R = 3 − 7 for T /T c = 1.50 − 5.00. The thermal color-Coulomb string tensions do not vanish for those temperatures, the values of which increase with temperature, and the rate of increase of the temperature is more rapid than those of the string tension σ c (T ). This tendency is acceptable if the color-Coulomb string tension at finite temperature is regarded as a thermal quantity, such as the electric and magnetic scaling described in Eqs. (17) and (18). From Eq. (4), it is clear that the instantaneous part made by the Faddeev-Popov matrix is independent of time and a spatial-like quantity although the temporal and spatial gluon fields are correlated by the self-interaction in QCD. This situation is very similar to the case of the spatial Wilson loop at finite temperature as reviewed in the previous section. Therefore, we shall describe these data by the magnetic scaling: T σ c (T ) = 1 c 1 g 2 (T ) ,(19) and the running coupling depending on the system temperature, 1 g 2 (T ) = 2b 0 ln T Λ + b 1 b 0 ln(2 ln T Λ ),(20) where c and Λ are free parameters for fitting. Using the data for T /T c = 1.5 − 5.0 we obtained the fitted result, listed in TABLE I, and in particular, the fitted line using the data for T /T c = 2.0 − 4.0 is shown as the solid line in Fig. 5. It is found that the color-Coulomb string tension in the thermal phase is described by the magnetic scaling. The fitted results in the present lattice simulation seem to depend significantly on the fitting condition. However, the fitted value of the coefficient by the magnetic scaling is comparable with that reported by the following studies: c = 0.566 (13) and Λ/T c = 0.104 (9) by the lattice calculation of the spatial Wilson loop in Ref. [22], c = 0.554(04) by the numerical study of the 3-D SU(3) gauge theory in Ref. [24], and c = 0.482(31) − 0.549 (16) from the lattice calculation of the spatial gluon propagator in Ref. [37]. Furthermore, the analysis of the magnetic mass in a self-consistent way of high QCD theory leads to c ∼ 0.569 [40]. If one requires more precise data for magnetic (spatial) or long-range quantities, which are expected to be sensitive to the volume size [37], then the computation on the larger lattice is necessary. In addition, if we assume the electric scaling given by Eq. (17) shown as the dashed line in Fig. 5. It seems that the electric scaling also yields a good description. In the perturbation theory of the leading order, the coefficient of the electric scaling is known to be c = 1 [36], and moreover the nonperturbative lattice simulation gives the electric scaling with c > 1 [37]. Thus, this procedure may not be as proper a way as the analysis based on magnetic scaling. Nonetheless, this implies that in the temperature range of T /T c = 1.5 − 5.0, the magnitude of the coupling constant is of order 1, i.e. there still remains a strong nonperturbative effect. As a result, the distinction between g 2 (T ) and g(T ) is not so clear from the present numerical data. In the confinement phase, the instantaneous color-Coulomb potential behaves as a linearly rising potential at large distances. As a consequence, there exists the nonvanishing color-Coulomb string tension for several coupling constants (β's) investigated in this work, the values of which are approximately 2 times as large as that of the Wilson loop string tensions. The variation of color-Coulomb string tension on the lattice cutoff is also found to be small although we are still far from the continuum limit. These results are qualitatively consistent with those obtained by the same analysis in the SU(2) lattice calculations in Refs. [11,12]. Note that if one employs the gluon propagator itself to extract the instantaneous part, then the value of the color-Coulomb string tension tends to become smaller than that measured by using the Polyakov loop correlator as used in the present work [42]. However, it is concluded in Ref. [43] that the emergence of a large string tension is not ruled out. Even in the deconfinement phase, it is observed that the color-Coulomb string tension remains finite. This may be an acceptable result because the instantaneous part is constructed in terms of the Faddeev-Popov matrix with the derivative operator for the spatial direction; i.e., the instantaneous part is not sensitive to the system temperature. Nevertheless, we should also note that the thermal fluctuation still produces a color-screened dynamics as has been reported in Ref. [13]. The remarkable feature that the color-Coulomb string tension does not disappear in the deconfinement phase was first shown in the SU(2) lattice calculation [12] done by Greensite, Olejník and Zwanziger. It is confirmed in the present study that the SU(3) gauge theory has the same feature, and we investigated extensively the temperature dependence of the color-Coulomb string tension, which is found to be in proportion to the temperature. Note that this issue is supported through the discussion of the remnant symmetry in Coulomb gauge [12,44]. The occurrence of a confining force in the deconfinement phase was observed in other studies. The existence of the spatial string tension above T c is well known. In addition, as reported in Ref. [23], by the SU(2) lattice simulation in maximally abelian gauge, the spatial Wilson loop can almost be reproduced by the wrapped monopole loops. In particular, the 3-D reduction arguments support these phenomena. For the case of the Coulomb gauge QCD, the confining linearity in the deconfinement phase is caused by the instantaneous part. In both confinement and deconfinement phases, there is no qualitative change in the behavior of the instantaneous part. Therefore, it is evident that the color-Coulomb string tension obtained from that potential is not an order parameter for the QGP phase transition. It is found that the thermal behavior of the color-Coulomb string tension is understood by assuming the magnetic scaling, ∼ g 2 (T )T , which is actually identified as an infrared regulator or a pole mass of the spatial gluon propagator. If this is a possible interpretation, then we can mention the following two points. Firstly, we conclude that the color-Coulomb string tension in the deconfinement phase is a kind of the thermal quantity and survives in the high temperature limit as being the same as the case of the spatial Wilson loop. Secondly, because the magnetic scaling originates in the infrared sensitivity of the thermal QCD, in the case of Coulomb gauge, the instantaneous part of the gluon propagators reveals such infrared behavior. In the Coulomb gauge confinement scenario discussed by Gribov and Zwanziger [4,5], the linearity of the instantaneous part for large quark separations is conjectured to be ascribed to a singularity emerging from the gauge configurations with a low-lying eigenvalue of the Faddeev-Popov operator. Hence, it is an important task that the distribution of the eigenvalues of the Faddeev-Popov operator in Coulomb gauge is investigated by the lattice simulation. The SU(2) lattice simulation in Coulomb gauge performed in Ref. [41] proves the indication of the enhancement of the low-lying eigenvalues. The SU(3) lattice study along this line is also being undertaken. In the present work, we focused the calculation of the instantaneous part only and did not deal with the vacuum polarization (retarded) part, being of little significance in the view of understanding color confinement in Coulomb gauge. However, the vital change concerning the QGP phase transition seems to be relevant to the vacuum polarization part, the role of which ought to be discussed in a subsequent study. In a phenomenological point of view, it is interesting that there remains the thermal string tension. If this is regarded as the indication that confining features survive above the critical temperature, then this observation may provide some insight into understanding the strongly correlated QGP. It has been tried in Ref. [45] the description of equation of state in the quasiparticle model with the dispersion relation of Gribov type. However, in other cases, it is not obvious how the confining property in the thermal phase affects physical spectroscopy. Nevertheless, in Coulomb gauge, it is significant that these findings are achieved by classifying the time-time gluon propagator into the instantaneous and noninstantaneous parts. VI. ACKNOWLEDGMENTS The simulation was performed on an SX-5(NEC) vector-parallel computer at the RCNP of Osaka University. We appreciate the warm hospitality and support of the RCNP administrators. This work is supported by Grants-in-Aid for Scientific Research from Monbu-Kagaku-sho (Nos. 13135216 and 17340080). FIG. 1 : 1An example of the R-dependence of the instantaneous color-Coulomb potential (in dimensionless lattice units) at β = 6.0 (a ∼ 0.1 fm) on a 18 3 × 32 lattice in the confinement phase. This result was obtained in the previous work FIG. 2 : 2The dependence of the color-Coulomb string tensions on β = 6/g 2 . The triangle symbols with error bar stand for the color-Coulomb string tension in the deconfinement phase. The Wilson loop string tensions, represented by the symbols of the diamond shape, are also plotted for comparison taken from FIG. 3 : 3The dependence of the instantaneous color-Coulomb potential on the temperature in the deconfinement phase. The dashed curves stand for the fitted results. FIG. 4 : 4The temperature dependence of the color-Coulomb string tension in units of T c in the deconfinement phase. The color-Coulomb string tension is proportional to T . to describe the thermal color-Coulomb string tension, then the function 1/cg(T ) is employed. The results are listed in TABLE I and the resultant line obtained in the fitted range of T /T c = 2.0 − 4.0 is FIG. 5 : 5The dependence of the color-Coulomb string tension as a function of T /T c . The circle symbols are numerical data and the solid line represents the fitted result by the magnetic scaling, while the dashed curve (line) represents that by the electric scaling.V. CONCLUSIONSWe have investigated the scaling behavior of the color-Coulomb string tension in the confinement and deconfinement phases using quenched SU(3) lattice gauge simulations. 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However we have not plotted the result of the fit because of the large uncertainty, particularly at high β.
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[ "Fast Beam Alignment via Pure Exploration in Multi-armed Bandits", "Fast Beam Alignment via Pure Exploration in Multi-armed Bandits" ]
[ "Yi Wei ", "Zixin Zhong ", "Senior Member, IEEEVincent Y F Tan " ]
[]
[]
The beam alignment (BA) problem consists in accurately aligning the transmitter and receiver beams to establish a reliable communication link in wireless communication systems. Existing BA methods search the entire beam space to identify the optimal transmit-receive beam pair. This incurs a significant latency when the number of antennas is large. In this work, we develop a bandit-based fast BA algorithm to reduce BA latency for millimeter-wave (mmWave) communications. Our algorithm is named Two-Phase Heteroscedastic Track-and-Stop (2PHT&S). We first formulate the BA problem as a pure exploration problem in multi-armed bandits in which the objective is to minimize the required number of time steps given a certain fixed confidence level. By taking advantage of the correlation structure among beams that the information from nearby beams is similar and the heteroscedastic property that the variance of the reward of an arm (beam) is related to its mean, the proposed algorithm groups all beams into several beam sets such that the optimal beam set is first selected and the optimal beam is identified in this set after that. Theoretical analysis and simulation results on synthetic and semi-practical channel data demonstrate the clear superiority of the proposed algorithm vis-à-vis other baseline competitors.Index Terms-Beam alignment, beam selection, mmWave communication, multi-armed bandits.Y. Wei is with the College
10.1109/isit50566.2022.9834431
[ "https://export.arxiv.org/pdf/2210.12625v1.pdf" ]
251,325,358
2210.12625
f9604b163603092af5ac90d65f49ae25e0d5adbf
Fast Beam Alignment via Pure Exploration in Multi-armed Bandits Yi Wei Zixin Zhong Senior Member, IEEEVincent Y F Tan Fast Beam Alignment via Pure Exploration in Multi-armed Bandits 1Index Terms-Beam alignmentbeam selectionmmWave com- municationmulti-armed bandits The beam alignment (BA) problem consists in accurately aligning the transmitter and receiver beams to establish a reliable communication link in wireless communication systems. Existing BA methods search the entire beam space to identify the optimal transmit-receive beam pair. This incurs a significant latency when the number of antennas is large. In this work, we develop a bandit-based fast BA algorithm to reduce BA latency for millimeter-wave (mmWave) communications. Our algorithm is named Two-Phase Heteroscedastic Track-and-Stop (2PHT&S). We first formulate the BA problem as a pure exploration problem in multi-armed bandits in which the objective is to minimize the required number of time steps given a certain fixed confidence level. By taking advantage of the correlation structure among beams that the information from nearby beams is similar and the heteroscedastic property that the variance of the reward of an arm (beam) is related to its mean, the proposed algorithm groups all beams into several beam sets such that the optimal beam set is first selected and the optimal beam is identified in this set after that. Theoretical analysis and simulation results on synthetic and semi-practical channel data demonstrate the clear superiority of the proposed algorithm vis-à-vis other baseline competitors.Index Terms-Beam alignment, beam selection, mmWave communication, multi-armed bandits.Y. Wei is with the College I. INTRODUCTION In millimeter-wave (mmWave) communications, the beams at both the transmitter and receiver are narrow directional. The beam alignment (BA) problem consists in ensuring that the transmitter and receiver beams are accurately aligned to establish a reliable communication link [2], [3]. An optimal transmitter (receiver) beam or a pair of transmitter and receiver beam is required to be selected to maximize the overall capacity, throughput, SNR or diversity. To achieve this goal, a naïve exhaustive search method scans all the beam space and hence causes significant BA latency. Specifically, there are two fundamental challenges to implement BA: (1) the amount of additional channel state information (CSI) corresponding to each beam (pair) is large; (2) the frequency of the channel measurement [4], [5] is high. These challenges become even more difficult to overcome when a large number of antennas are employed at both of the transmitter and receiver. Moreover, the change of antenna orientation, the effect of transmitter/reciever mobility, and the dynamic nature of the wireless channel will result in previously found optimal beam pairs to be suboptimal over time, which further exacerbates the latency problem [6], [7]. Therefore, the design of fast BA algorithms is of paramount importance, and has stimulated intense research interest. By utilizing the sparsity of the mmWave channel, Marzi et al. [8] incorporated compressed sensing techniques in the BA problem to reduce the beam measurement complexity. Wang et al. [9] developed a fast-discovery multi-resolution beam search method, which first probes the wide beam and continues to narrow beams until the optimal arm is identified. However, the beam resolution needs to be adjusted at each step. Besides, various forms of side information, e.g., location information [10], out-of-band measurements [11], and dedicated shortrange communication [12], have also been used to reduce the required number of probing beams. Due to its inherent ability to balance between exploration and exploitation, multi-armed bandit (MAB) theory has been recently utilized in wireless communication field, such as channel access in cognitive network [13], channel allocation [14] and adaptive modulation/coding [15], etc., and harnessed to address the BA problems; see [16]- [22]. In a single-user MIMO system, the work [16] employed the canonical upper confidence bound (UCB) bandit algorithm in beam selection, where the instantaneous full CSI is not required. The work [17] applied linear Thompson sampling (LTS) to address the beam selection problem. For mmWave vehicular systems, Sim et al. [18] developed an online learning algorithm to address the problem of beam selection with environment-awareness based on contextual bandit theory. The proposed method explores different beams over time while accounting for contextual information (i.e., vehicles' direction of arrival). Similarly, Hashemi et al. [19] proposed to maximize the directivity gain (i.e., received energy) of the BA policy within a time period using the contextual information (i.e., the inherent correlation and unimodality properties), and formulate the BA problem as contextual bandits. Wu et al. [20] provided a method to quickly and accurately align beams in multi-path channels for a point-to-point mmwave system, which takes advantage of the correlation structure among beams such that the information from nearby beams is extracted to identify the optimal beam, instead of searching the entire beam space. In [21], Va et al. used a UCB-based framework to develop the online learning algorithm for beam pair selection and refinement, where the algorithm first learns coarse beam directions in some predefined beam codebook, and then fine-tunes the identified directions to match the peak of the power angular spectrum at that position. Hussain et al. [22] also designed a novel beam pair alignment scheme based on Bayesian multi-armed bandits, with the goal of maximizing the alignment probability and the data-communication throughput. In this paper, different from existing methods, we first formulate the BA problem as a pure exploration problem in the theory of MABs, called bandit BA problem, with the objective of minimizing the required time steps to find the optimal beam pair with a prescribed probability of success (confidence level). Second, we derive a lower bound on the expected time complexity of any algorithm designed to solve the BA problem. Third, we present a bandit-based fast BA algorithm, which we name Two-Phase Heteroscedastic Track-and-Stop (2PHT&S). We do so by exploiting the correlation among beams and the heteroscedastic property that the variance of the reward of an arm (beam) is linearly related to its mean. Instead of searching over the entire beam space, the proposed algorithm groups all beams into several beam sets such that the optimal beam set is first selected and the optimal beam is identified within this set during the second phase. We derive an upper bound on the time complexity of the proposed 2PHT&S algorithm. Finally, extensive numerical results demonstrate that 2PHT&S significantly reduces the required number of time steps to identify the optimal beam compared to existing algorithms in both simulated and semi-practical channel data scenarios. The remainder of this paper is organized as follows. The system model and problem setup are presented in Section II. Section III proposes our bandit-based fast BA algorithm and its time complexity upper bound. Simulation results are given in Section IV, and Section V concludes this paper. II. PROBLEM FORMULATION In this section, we first describe the system model and propose our problem setup, after that we provide a generic lower bound of the time complexity of the problem. A. System Model As shown in Fig. 1, we consider a massive mmWave MISO system, where a base station (BS) equipped with N transmit antennas serves a single-antenna user. According to [23], each transmitting beam is selected from a predefined analog beamforming codebook C of size K, which can be defined as C {f k = a(−1 + 2k/K) | k = 0, 1, 2, . . . , K − 1} with a(·) denoting the array response vector. For a typical uniform linear array, it holds that a(x) = 1 √ N 1, e j 2π λ dx , e j 2π λ 2dx , . . . , e j 2π λ (N −1)dx H ∈ C N ×1 , where λ is the wavelength and d is the antenna spacing. Note that if K ≤ N , the beams in the codebook C are linearly independent of one another. Based on channel measurement studies done in [24], [25], the mmWave channel follows a multipath channel model, and the number of propagation paths is small. We consider a quasi-static channel, where the channel stays unchanged for a period covariance interval which consists of T time slots. As shown in Fig. 2, each covariance interval can be divided into two phases, the BA phase (which consists of T B time slots) and the data transmission phase (which consists of T D = T − T B time slots). With the selected beam f , the effective achievable rate (EAR) of each covariance interval R eff := 1 − T B T D log 1 + p|h H f | 2 σ 2 is widely accepted as a metric to measure the throughput performance, where p and σ 2 represent the transmit power and noise variance respectively. We observe that the throughput performance increases with the decreasing T B /T D and increasing |h H f * | 2 . As such, the time allocated for selecting the optimal beam f * = arg max f ∈C |h H f | 2 should be minimized for higher system throughput, which is the motivation of this work. In the BA phase, the BS selects one beam from the codebook C and transmits pilot signals to the user with this beam. Without loss of generality, we set the pilot signal to be 1. Then the received power can be expressed as R(f k ) = | √ ph H f k + n| 2 = p|h H f k | 2 + |n| 2 + 2 √ p (h H f k n * ), where n * represents the complex conjugate of the complex Gaussian noise n ∼ CN (0, σ 2 ). Note that the BS can observe the received power from the user through a feedback link, and this assumption is also adopted in other studies such as [20], [21]. We can see that R(f k ) is a random variable which is the sum of a Gamma random variable |n| 2 ∼ Γ(1, 1/σ 2 ) and a Gaussian random variable p|h Fig. 3 shows the average received power associated with different beams over the codebook space when K = 120, N = 64, p = 26 dBm and the power of LoS path is around 3 dB larger than those of the NLoS paths. It can be observed that all the received powers become larger as the increase of the noise variance, and the improved value is the difference between the noise variances. Since the noise variance is much smaller than the transmit power, the variable R(f k ) is approximately a Gaussian random variable with mean µ k = p|h H f k | 2 and variance σ 2 H f k | 2 + 2 √ p (h H f k n * ) ∼ N (p|h H f k | 2 , 2p|h H f k | 2 σ 2 ).k = 2p|h H f k | 2 σ 2 , i.e., r k = p|h H f k | 2 + 2 √ p (h H f k n * ).(1) The optimal beam is defined as the beam that has the largest value of r k , i.e., k * = arg max k p|h H f k | 2 . Let J be the correlation length of arms, which is related to the number of beams N and the size of the codebook K as follows: J = 2 K N − 1. Then the received power using the beam f k and the nearby beam f i , |k − i| ≤ J/2 are similar. A new 1 J -lower resolution beam codebook C (J) can be constructed by grouping the nearby beams in the codebook C, i.e., C (J) b g = Jg k=J(g−1)+1 f k | Jg k=J(g−1)+1 f k | : g = 0, 1, . . . , G − 1 , where the beams b g 's are normalized (to unit 2 norm) to mimic reality in a phased array. If we choose a beam b g in the codebook C (J) , the received power can be expressed as R g = p|h H b g | 2 + 2 √ p (h H b g n * ) . Since the received powers associated with the nearby beams in the codebook C have means that are close to one another, R g has large mean if r k , k ∈ {(g − 1)J + 1, . . . , gJ} have large means. B. Problem Setup To introduce the idea of bandit to address the BA problem, we consider a bandit model where the beam f k is regarded as the base arm k, and the received power using this beam is regarded as the reward of the base arm k. The means of the rewards when arms are pulled have the following property: Property 1. Let µ = (µ 1 , . . . , µ K ), and let µ (1) ≥ µ (2) ≥ µ (3) ≥ . . . ≥ µ (K) be the sorted sequence of means. Then we assume that these means have the following properties: 1. The means of the reward associated with arms k and i, where |i − k| ≤ J/2, are close. 2. The rewards are sparse. In particular, there are only LJ arms that have high rewards; the other K − LJ arms have rewards that are close to zero. 3. The variance of the reward of an arm is related to its mean as follows: σ 2 k = 2µ k σ 2 . As shown in Fig. 3, Property 1.1 holds because the beam codebook is designed such that beams that are nearby have rewards that are close. Property 1.2 is due to the fact that there are only a few main paths in the considered channel model, which has been corroborated by studies in channel measurements [26]. Finally, Property 1.3 holds because the mean of reward associated with base arm k follows a heteroscedastic Gaussian distribution N (µ k , 2µ k σ 2 ) (see (1)). Furthermore, in this work, we focus on a challenging case with large noise, which satisfies that σ 2 > max k∈[K] (µ (1) − µ (k) ) 2 4µ (k) − 2µ (1) − 2µ (k) ln µ (k) µ (1) ,(2) and this condition is assumed to hold in our theoretical analyses. In the simulations, we also numerically check that this condition holds. Since a large number of base arms have mean rewards that are close to zero (Property 1.3), it is clearly a waste of BA overhead to search for the optimal arm among all the base arms. To overcome this problem, we propose to group the base arms by utilizing Property 1.1 which says that the means of the reward associated with nearby arms are similar. As such, we can formulate the BA problem as a novel MAB problem, called the bandit BA problem. In a bandit BA problem with K base arms, the base arm k is associated with the beam f k . We let [K] = {1, . . . , K} and [K] ≤J consec be the set of all non-empty consecutive tuples of length ≤ J whose element belongs to the set [K], e.g., if J = 2 and K = 6, [6] ≤ 2 consec = {1}, {1, 2}, {2}, {2, 3}, {3}, {3, 4}, {4}, {4, 5}, {5}, {5, 6}, {6} , and we consider each tuple as a (K, J)-super arm. Moreover, the super arm associated with b g ∈ C (J) is a subset of [K] ≤J consec , e.g., {{1, 2}, {3, 4}, {5, 6}} ⊂ [6] ≤2 consec . At time step t, we choose an action (or a (K, J)-super arm) A(t) ∈ [K] ≤J consec , and observe the reward R(t) which is related to the base arms included in A(t), i.e., R(t) = F k∈A(t) f k | k∈A(t) f k | , p, h, n t ,(3) where F (f , p, h, n) = p|h H f | 2 + 2 √ p (h H f n * ) , n is a complex Gaussian random variable with mean zero and variance σ 2 . We can conclude that for an arbitrary (K, J)-super arm A ∈ [K] ≤J consec , the empirical mean R A also follows a heteroscedastic Gaussian distribution, i.e., R A ∼ N (µ A , 2µ A σ 2 ), where µ A = p|h H k∈A f k | 2 . To identify the optimal base arm, an agent uses an algorithm π that decides which super arms to pull, the time τ π to stop pulling, and which base arm k π to choose eventually. An algorithm consists of a triple π := {(π t ) t , τ π , ψ π , J} in which: • a sampling rule π t determining the (K, J)-super arm A(t) ∈ [K] ≤J consec to pull at time step t based on the observation history and the arm history {A(1), R(1), A(2), R(2), . . . , A(t − 1), R(t − 1)}; • a stopping rule leading to a stopping time τ π satisfying P(τ π < ∞) = 1; • a recommendation rule ψ π that outputs a base arm k π ∈ [K]. We define the time complexity of π as τ π . In the fixedconfidence setting, a maximum risk δ ∈ (0, 1) is fixed. We say an algorithm π is (δ, J)-probably approximately correct (PAC) if P(k π = k * ) ≥ 1 − δ. The purpose is to design a (δ, J)-PAC algorithm π that minimizes the expected stopping time E[τ π ]. C. Lower Bound on the Time Complexity Denote by ν a K-armed heteroscedastic Gaussian bandit instance, ν = N (µ ν 1 , 2µ ν 1 σ 2 ), . . . , N (µ ν K , 2µ ν K σ 2 ) . Let V represent the set of K-armed heteroscedastic Gaussian bandit instances such that each bandit instance ν ∈ V has a unique optimal arm: for each ν ∈ V, there exists an arm A * (ν) such that µ ν A * (ν) > max{µ ν k : k = A * (ν)}. Define W K = w ∈ R K + : K k=1 w k = 1 , Alt(ν) := {u ∈ V : A * (u) = A * (ν)}, and T k (t) as the times the base arm k is pulled until time step t. Moreover, the KL divergence between two heteroscedastic Gaussian distributions, i.e., N (µ i , 2µ i σ 2 ) and N (µ j , 2µ j σ 2 ), can be calculated as D HG (µ i , µ j ) = 1 2 ln µ j µ i + µ i 2µ j + (µ j − µ i ) 2 4µ j σ 2 − 1 2 . According to Theorem 1 of [27], we can obtain the general lower bound of this problem as follows. Theorem 1. For any (δ, J)−PAC algorithm where δ ∈ (0, 1), it holds that E π [τ ] ≥ c * (ν) ln 1 4δ , where c * (ν) −1 = sup w∈W K inf u∈Alt(ν) K k=1 w k D HG (µ ν k , µ u k ) .(4) III. OUR ALGORITHM: 2PHT&S In this section, we first state the performance baseline of this work, i.e., the original Track-and-Stop (T&S) algorithm [27], then we elaborate the design of the proposed 2PHT&S. We then upper bound its time complexity in Theorem 2. A. A Baseline Algorithm Note that the original Track-and-Stop (T&S) [27] is the state-of-the-art best arm identification algorithm with fixed confidence for exponential bandits, which can be modified to solve the bandit BA problem with low sample complexity, resulting in a fast BA solution. Let ν represent a K-armed Gaussian bandit instance, ν = N µ ν 1 , (σ ν 1 ) 2 , N µ ν 2 , (σ ν 2 ) 2 , . . . , N µ ν K , (σ ν K ) 2 . Given the fixed δ, the time complexity of T&S, E τ T&S , satisfies lim δ→0 E τ T&S log(1/δ) = T * (ν), where T * (ν) −1 := sup w∈W K inf u∈Alt(ν) K k=1 w k D N µ ν k , (σ ν k ) 2 , N µ u k , (σ u k ) 2 . and the KL-divergence D N µ ν k , (σ ν k ) 2 , N µ u k , (σ u k ) 2 = ln σ u k σ ν k + (σ ν k ) 2 +(µ ν k −µ u k ) 2 2(σ u k ) 2 − 1 2 . Furthermore, when we apply T&S to the K-armed heteroscedastic Gaussian bandit instance ν ∈ V under consideration, we have lim sup δ→0 E τ T&S log(1/δ) ≤ T * u (ν),(5) where T * u (ν) = 8µ ν A * (ν) σ 2 ∆ 2 ν,min + k =A * (ν) 8µ ν A * (ν) σ 2 ∆ 2 ν,k , and the gaps are ∆ ν,k := µ ν A * (ν) − µ ν k , k = A * (ν) and ∆ ν,min := min k =A * (ν) µ ν A * (ν) − µ ν k . B. 2PHT&S For our bandit BA problem, we propose the 2PHT&S algorithm inspired by the original T&S algorithm [27] to achieve a smaller time complexity by leveraging the structure of the BA problem. The main idea of the proposed algorithm is to exploit the prior knowledge which have not been considered by existing algorithms, i.e., Property 1 and the fact that neighboring beams can be grouped and the corresponding "grouped" reward can be observed in one time step. As shown in Fig. 4, 2PHT&S consists of two phases. We describe them in the following. • In Phase I, we first group K base arms into G arm sets and accordingly construct a set of super arms as G = {S g = {(g − 1)J + 1, . . . , gJ}|g ∈ [G]}.(6) According to (3), the reward associated with super arm S g can be written as R g (t) = F ( k∈Sg f k , p, h, n t ). Second, we search for the optimal super arm, g * = arg max g∈[G] E[R g (t)] , with a high probability of at least 1 − δ 1 , where δ 1 denotes the maximum allowable risk in Phase I. • In Phase II, we construct a base arm set (including two super arms) and select the optimal base arm from this base arm set. Considering the case shown in Fig. 5, where the optimal base arm lies in the right edge of the super arm S g and the second optimal base arm lies in the left edge of the super arm S g+2 , the mean of the reward of the super arm S g+1 may be higher than that of the super arm S g . This observation results in a higher difficulty of identifying the optimal super arm. Therefore, to improve the selection accuracy, we construct a base arm set, i.e., the combination of the optimal super arm and either one of its (left or right) neighbors that has the larger mean. Next, we select the optimal base arm in this base arm set with a certain fixed confidence probability of at least 1 − δ 2 . Given the fixed maximum risk δ, risks δ 1 and δ 2 should satisfy the constraint δ = δ 1 + δ 2 . 1 The proposed 2PHT&S is detailed in Algorithm 1, whereμ s = [μ s 1 , . . . ,μ s G ] and µ b = [μ b S f (1) , . . . ,μ b S f (2J) ] represent the empirical means of super arms and base arms included in S f , respectively. In both of Phase I and Phase II of 2PHT&S, an improved T&S algorithm exploiting the heteroscedastic property, which is referred to as Heteroscedastic Track-and-Stop (HT&S), is leveraged to search for the optimal super arm and the optimal base arm, respectively. 1 Note that since the two phases are independent, the probability of selecting the optimal base arm can be expressed as (1 − δ 1 )(1 − δ 2 ) = 1 − δ 1 − δ 2 + δ 1 δ 2 . Then, the error probability is δ 1 + δ 2 − δ 1 δ 2 , which is less than δ 1 + δ 2 since the two phases are independent. For simplicity, for the given fixed confidence δ, we set δ 1 + δ 2 = δ. Algorithm 1 2PHT&S Input: Maximum risk δ and a set of base arms k ∈ K. Output: Optimal base arm k * , required number of time steps τ . 1: Choose δ 1 and δ 2 which satisfy δ 1 + δ 2 = δ, # Phase I: search for the optimal super arm. 2: Group the arms k ∈ K into G arm sets and formulate the super arms G according to (6), 3: [τ 1 ,μ s ] = HT&S(δ 1 , G) (defined in Algorithm 2), # Phase II: search for the optimal base arm. 11: return k * = argmax k∈S fμ b k and τ = τ 1 + τ 2 . 4: Select g * = argmax g∈Gμ s g , 5: ifμ s g * +1 (t) ≥μ s g * −1 (t) then 6: S f = S g * ∪ S g * +1 , 7: else 8: S f = S g * ∪ S g * −1 , 9: end if 10: [τ 2 ,μ b ] = HT&S δ 2 , S f (defined in Algorithm 2), We now introduce the HT&S algorithm. Define a set of I ∈ N super arms I = {S 1 , . . . , S I } where the reward R i (t) of super arm S i is related to the beams k ∈ S i as follows: R i (t) = F ( k∈Si f k , p, h, n t ) . Given I and the fixed δ, Pull the super arm i, observe the reward, then update the T i (0) andμ ν i (0) according to (7). 3: end for 4: Initializeŵ * (0) according to (12) Observe the reward R(t), updateμ ν i (t) and T i (t) according to (7), update Z(t) according to (9), updatê w * i (t) according to (12), 12: usingμ ν (0), t = 1. 5: while Z(t) ≤ β(t, δ, α) do (The definition of Z(t) is given in (9)) 6: if argmin i∈I T i (t − 1) ≤ ( √ t − I 2 ) + then 7: A(t) = argmin i∈I T i (t − 1), 8:A(t) = argmax i∈I (tŵ * i (t − 1) − T i (t − 1)),t = t + 1, 13: end while 14: τ δ = t,μ ν = [μ ν i (t), . . . ,μ ν I (t)], 15: return τ = I + τ δ ,μ ν . HT&S outputs all the empirical means,μ ν = [μ ν 1 , . . . ,μ ν I ], and the required number of time steps τ . Let A(t) represent the index of the super arm pulled in time step t, i.e., in time step t the super arm S A(t) is selected by the sampling rule. We observe the following reward associated with the super arm S A(t) is R(t) = F k∈S A(t) f k , p, h, n t . Accordingly, we update the number of times each arm is selected up to time step t and their empirical mean as T i (t) = t a=1 1 {A(a) = i} and µ i (t) = 1 T i (t) t a=1 R(a)1 {A(a) = i} .(7) The search is terminated when the stopping rule (delineated in (8) below) is satisfied. Define the heteroscedastic Gaussian bandit instance at time step t asν t = N μ ν 1 (t), 2μ ν 1 (t)σ 2 , . . . , N μ ν I (t), 2μ ν I (t)σ 2 . The stopping rule and sampling rule, which exploit the heteroscedastic property, form the core of HT&S: 1) Stopping rule: We choose the threshold as β(t, δ, α) = ln(αt/δ), and the stopping rule as τ δ = min t ∈ N : Z(t) ≥ β(t, δ, α) ,(8) where Z(t) is defined in (9) at the top of the next page, and q i (t) = T A * (νt) (t) + T i (t) T A * (νt) (t)μ ν i (t) + T i (t)μ ν A * (νt) (t)μ ν A * (νt) (t)μ ν i (t), A (ν t ) = arg max i∈[I] {μ ν i (t) : i = A * (ν t )}.(10) Please refer to (16)- (17) in Appendix C for more details of the derivation of (a) in (9). Lemma 1. Let ν be a heteroscedastic Gaussian bandit instance. Let δ ∈ (0, 1) and α > 1. Using the above stopping rule in (8) with the threshold β(t, δ, α) = ln( αt δ ) ensures that P s τ δ < ∞, A τ δ = A * (ν) ≤ δ. Proof. Please refer to Appendix A. 2) Sampling rule: The sampling rule can be summarized by Q(t) defined in (11) on the top of the next page, wherê w * (t) ∈ W I is the set of parameters to be updated at time step t. Define f Y,i,t (x i ) := 1 2 ln μ ν i (t) + x iμ ν A * (ν) (t) (1 + x i )μ ν i (t) + x i 2 ln μ ν i (t) + x iμ ν A * (ν) (t) (1 + x i )μ ν A * (ν) (t) + x i (μ ν A * (ν) (t) +μ ν i (t))(μ ν A * (ν) (t) −μ ν i (t)) 2 4σ 2 (μ ν i (t) + x iμ ν A * (ν) (t)) , and f X,i,t (y) f −1 Y,i,t (x i ). The value ofŵ * i (t) is set to bê w * i (t) = f X,i,t (y * (t)) I a=1 f X,a,t (y * (t)) ,(12) where y * (t) is the unique solution of the equation i =A * (µ) yf X,i,t (y) − f X,i,t (y) = 1. Note that the choice ofŵ * (t) is determined according to the optimization problem in (4); see the technical details in Appendix C. It is worth noting that the stopping rule and sampling rule of HT&S are different from those of original T&S, and both of them are derived from the lower bound given in Theorem 1 based on the heteroscedastic property. C. Time Complexity Analysis of 2PHT&S Theorem 2. Let s = N (µ s 1 , 2µ s 1 σ 2 ), . . . , N (µ s G , 2µ s G σ 2 ) and b = N (µ b S f (1) , 2µ b S f (1) σ 2 ), . . . , N (µ b S f (2J) , 2µ b S f (2J) σ 2 ) be heteroscedastic Gaussian bandit instances studied in Phases I and II of 2PHT&S, where µ s g = p|h H ( k∈Sg f k )| 2 . Using the proposed stopping rule and the sampling rule, we have lim sup δ→0 E[τ ] ln(1/δ) ≤ c * u (s) + c * u (b), where c * u (s) = µ s A * (s) 2 G i=1 µ s i ln 2µ s A * (s) µ s A * (s) + µ s A (s) + µ s A (s) 2 G i=1 µ s i ln 2µ s A (s) µ s A * (s) + µ s A (s) + (µ s A * (s) − µ s A (s) ) 2 8σ 2 G i=1 µ s i − (µ s A * (s) + µ s A (s) ) 2 G i=1 µ s i −1 , Z(t) = min i =A * (νt) inf u∈Alt(νt) T A * (νt) (t)D HG (μ ν A * (νt) (t), µ u A * (νt) ) + T i (t)D HG (μ ν i (t), µ u i ) (a) = min i =A * (νt) T A * (νt) (t)D HG μ ν A * (νt) (t), q i (t) + T i (t)D HG μ ν i (t), q i (t) = T A * (νt) (t)D HG μ ν A * (νt) (t), q A (νt) (t) + T A (νt) (t)D HG μ ν A (νt) (t), q A (νt) (t) ,(9)Q(t) = argmin i T i (t − 1), if min i T i (t − 1) ≤ √ t, argmax i tŵ * i (t − 1) − T i (t − 1), otherwise.(11) and c * u (b) = µ b A * (b) 2 i∈S f µ b i ln 2µ b A * (b) µ b A * (b) + µ b A (b) + µ b A (b) 2 i∈S f µ b i ln 2µ b A (b) µ b A * (b) + µ b A (b) + (µ b A * (b) − µ b A (b) ) 2 8σ 2 i∈S f µ b i − (µ b A * (b) + µ b A (b) ) 2 i∈S f µ b i −1 . Proof. We first obtain the upper bound of the time complexity achieved by HT&S in Lemma 2. Lemma 2. For a heteroscedastic Gaussian bandit instance ν, Algorithm 2 ensures that for any α ≥ 1 t 0 > 0, > 0 and δ > 0, there exists three constants Γ 1 = Γ 1 ( , t 0 , I), Γ 2 = Γ 2 (ν, ) and Γ 3 = Γ 3 (ν, ) such that E[τ ] ≤ Γ 1 + αc * u (ν) ln αec * u (ν) δ + ln ln αc * u (ν) δ + ∞ T =1 Γ 2 T exp −Γ 3 T 1/8 . Proof. See the proof in Appendix D. Using HT&S in Phase I and Phase II, the overall required number of time steps of Algorithm 1, i.e., τ , can be expressed as E[τ ] = E[τ 1 ] + E[τ 2 ] ≤ α 1 c * u (s) ln 1 δ 1 +α 2 c * u (b) ln 1 δ 2 +M s +M b ,(13) where M s = Γ s 1 + α 1 c * u (s) ln(α 1 e) + ln ln α 1 c * u (s) δ 1 + ∞ T =1 Γ s 2 T exp −Γ s 3 T 1/8 , M b = Γ b 1 + α 2 c * u (b) ln(α 2 e) + ln ln α 2 c * u (b) δ 2 + ∞ T =1 Γ b 2 T exp −Γ b 3 T 1/8 .(14) For simplicity, we fix α 1 = α 2 = 1 and δ 1 = δ 2 = 1 2 δ, then (13) can be rewritten as E[τ ] ≤ c * u (s) ln 1 2δ + c * u (b) ln 1 2δ + M s + M b . When we normalize by ln(1/δ) and let δ tend to zero, we obtain (24) as desired. When δ tends to 0, we obtain lim sup δ→0 E[τ ] ln(1/δ) ≤ c * u (s) + c * u (b), which concludes the proof. To summarize, we present in Table I the time complexities of the proposed 2PHT&S algorithm and its baseline, the original T&S algorithm [27], with fixed α = α 1 = α 2 = 1. In Table I, the terms c * u (s) and c * u (b) result from the time complexities of Phase I and Phase II, respectively. These constants respectively represent the difficulties of finding the optimal super arm in Phase I and the optimal base arm in Phase II. There are two main differences between 2PHT&S and T&S. First, the proposed 2PHT&S algorithm contains two phases. In the first phase, the BS searches for the optimal super arm (beam set), and in the second phase, the BS searches for the optimal base arm (beam) among the beams in the optimal beam set and its neighbor with higher mean reward. However, the vanilla T&S algorithm directly searches for the optimal base arm (beam) among the whole beam space, which may be prohibitively large. Second, the heteroscedastic property is explicitly considered in the design of the sampling rule and the stopping rule of the 2PHT&S algorithm, but it is not exploited by the original T&S algorithm. Furthermore, we would like to note that the computational complexity of T&S algorithm scales linearly with the number of beams K, while that of the proposed 2PHT&S algorithm scales linearly with the number of beam sets G and the number of beams in each beam set J. Because G and J are usually much smaller than K, in general, the proposed 2PHT&S algorithm can significantly reduce the computational complexity as compared to the original T&S algorithm. IV. SIMULATION RESULTS We consider a massive mmWave MISO system, where a BS equipped with N = 64 transmit antennas serves a singleantenna user. The size of codebook is set as K = 120 and the correlation length is set to J = 2 K N − 1. Note that the mmWave channel is sparse, and hence we set the maximum number of channel paths as 3, which consists of one dominant LoS path and two NLoS paths. In addition, according to practical in-field measurements, NLoS paths suffer around 1 dB more path loss than the LoS path. The noise variance is fixed to σ 2 = −80 dBm. The code to (14)) Lower bound of E[τ ] c * (ν) ln 1 4δ (see (4)) reproduce all the experiments can be found at this Github link (https://github.com/YiWei0129/Fast-beam-alignment). T * u (ν) (see (5)) c * u (s) + c * u (b) (see In the simulation, we compare the proposed 2PHT&S with three other bandit algorithms, i.e., the original T&S [27], twophase T&S (2PT&S), where the search process is also divided into two phases as the proposed algorithm and the original T&S is used in each phase, and the proposed HT&S. Note that 2PT&S also exploits the correlation between the nearby beams (Property 1.1), but it does not take the heteroscedascity into account. The algorithms we consider are all designed to search the optimal beam/base arm with high probability using as few time steps as possible. Moreover, we also investigate the 2PHT&S algorithm with an overlapping scheme, i.e., in Phase II, we construct the base arm set with 2J base arms where first J base arms overlap with the last J base arms of the super arm to the left of the selected super arm in Phase I, and the last J base arms overlap with the first J base arms of the super arm to the right of the selected super arm. We call this scheme 2PHT&S (overlapping). Finally, we also consider the classical BA algorithms in wireless communications, i.e., exhaustive BA (EBA) algorithm and the hierarchical exhaustive BA (HEBA) algorithm as performance baselines. 2 Additionally, note that in a typical mmWave communication setting, one may experience channel coherence times of 35 µs when the system is deployed at the carrier frequency 28 GHz. In this setting, T = 35 µs, each time slot is 2.5 ns, and there are approximately 14000 time slots in each coherence time period. In practice, another link with different frequency can be used for feedback, thus no delay will result. First, we fix δ = 0.1, and δ 1 = δ 2 = δ 2 for Phase I and Phase II. We consider a scenario where the following widely-used channel model is considered, i.e., h = N L β (1) a θ (1) + L l=2 β (l) a θ (l) , and the angle of departure (AoD) and path losses of three paths are set to [0.7546π, 0.3489π, 0.6971π] and [0, −3, −3] dB, respectively. Fig. 6 illustrates the means of the rewards associated with each base arm and super arm when p = 0 dBm. In this setting, we have K = 120, N = 64, J = 2 K N − 1 = 3, G = K J = 40. As can be seen, in this scenario, the AoDs of the three paths are different, such that the index of beams with non-zero means are separated by a certain distance. Furthermore, the base arm 18 is the optimal arm, and the super arm 6, which is related to the base arms 16, 17 and 18, also has the largest mean of the reward. Table II presents the estimated 2 IIn the HEBA algorithm, the search phase is divided into two phases. In each phase, the EBA algorithm is employed. Table II, the time complexities of the algorithms under consideration in Scenario 2 are higher. Note that if the amplitudes of the largest path and the second largest path are close, the time complexities of all algorithms will increase. Third, we consider a more realistic channel model from the 3GPP TR 38.901 standard, i.e., Scenario 3. The channel coefficients are generated by the QuaDRiGa simulator [28], which extends the popular Wireless World Initiative for New Radio (WINNER) channel model with new features to enhance its realism. In Fig. 7, the base arm 86 is the optimal arm to be chosen, and the super arm 29, which is related to the base arms 85, 86 and 87, has the largest means of the reward among super arms of size 3. is clearly superior in terms of time complexity compared to the other competing algorithms. Finally, to validate the effectiveness of the proposed 2PHT&S, we investigate a practical scenario in a city, i.e., Scenario 4, which is shown in Fig. 8, where the BS and user are located at (573m, 622m, 41m) and (603m, 630m, 43m), respectively. For this scenario, we generate the semi-practical channel data in the ray-tracing setups using the software Wireless InSite. Note that Wireless InSite is a professional electromagnetic simulation tool that models the physical characteristics of irregular terrain and urban building features, performs the electromagnetic calculations, and then evaluates the signal propagation characteristics. As can be seen in Fig. 9, the base arm 119 is the optimal arm, and the super arm 40, which is related to the base arm 118, 119 and 120, has the largest means of the rewards among super arms of size 3. Table V presents the estimated average time complexities of the considered algorithms in Scenario 4 when SNR ∈ {70, 74, 78, 82} dB. It can be observed that similar to the previous simulated scenarios, the proposed 2PHT&S outperforms the other algorithms in this more practical scenario, which implies that the proposed algorithm is also effective in practice. V. CONCLUSION In this work, we developed a bandit-based fast BA algorithm 2PHT&S to reduce BA latency for mmWave communications. By taking advantage of the correlation structure among beams that the information from nearby beams are similar and the heteroscedastic property that the variance of the reward of an arm (beam) is related to its mean, the proposed algorithm groups all beams into several beam sets such that the optimal beam set is first selected and the optimal beam is identified in this set. The proposed 2PHT&S is shown to theoretically and empirically perform much better than its baseline competitors. APPENDIX A PROOF OF LEMMA 1 Proof. Let µ ν i be the mean of the reward of the arm i in the heteroscedastic Gaussian bandit instance ν. First, for every arm i 1 and i 2 where µ ν i1 < µ ν i2 andμ ν i1 (t) >μ ν i2 (t), we define Z i1,i2 (t) = inf µ u i 1 <µ u i 2 T i1 (t)D HG (μ ν i1 (t), µ u i1 ) + T i2 (t)D HG (μ ν i2 (t), µ u i2 ) ≤ T i1 (t)D HG (μ ν i1 (t), µ ν i1 ) + T i2 (t)D HG (μ ν i2 (t), µ ν i2 ). Then, according to the definition of Z(t) given in (9), we can rewrite Z(t) as Z(t) = min i =A * (νt) inf u∈Alt(νt) T A * (νt) (t)D HG (μ ν A * (νt) (t), µ u A * (νt) ) + T i (t)D HG (μ ν i (t), µ u i ) = min i∈[I] Z A * (νt),i (t). Lemma 3. For all ψ ≥ I + 1 and t ≥ 0, it holds that P I i=1 T i (t)D HG μ ν i (t), µ ν i ≥ ψ ≤ e I+1−ψ ψ ln t ψ I I . Proof. See the proof in Appendix B. Now consider P τ δ < ∞, A τ δ = A * (ν) ≤ P ∃ i ∈ G\A * (ν), ∃ t > 0 :μ ν i (t) >μ ν A * (νt) , Z A * (νt),i (t) > β(t, δ 1 , α 1 ) ≤ P ∃ t > 0 : ∃ i ∈ [I]\A * (ν) : T A * (νt) (t)D HG μ ν A * (νt) (t), µ ν A * (ν) + T i (t)D HG (μ ν i (t), µ ν i ) > β(t, δ, α) ≤ P ∃ t > 0 : i∈[I] T i (t)D HG μ ν i (t), µ ν i ≥ β(t, δ, α) (a) ≤ ∞ t=1 e I+1 β(t, δ, α) 2 ln(t) I I e −β(t,δ,α) where (a) holds using the union bound and Lemma 3. Hence, with β(t, δ, α) = ln(αt/δ), by choosing an α satisfying ∞ t=1 e I+1 tI I ln 2 (αt/δ) ln(t) I ≤ α, we obtain P τ δ < ∞, A τ δ = A * (ν) ≤ δ. APPENDIX B PROOF OF LEMMA 3 In this section, we prove Lemma 3 assuming that Lemma 4 holds, and then we complete the argument by proving Lemma 4. E = I i=1 {t i ≤ T i (t) ≤ (1 + η)t i }.(15) For ψ ≥ (1 + η)I, it holds that P 1 E I i=1 T i (t)D HG (μ ν i (t), µ ν i ) ≥ ψ ≤ ψe I I e −ψ/(1+η) . Proof. See the proof in Appendix B-B. A. Proof of Lemma 3 Proof. Let ψ ≥ I + 1 and η > 0. Define M = ln(t)/ ln(1 + η) , and set M = {1, . . . , M } I and events A and B m as Proof. Define the event A = I i=1 T i (t)D HG μ ν i (t), µ ν i ≥ ψ , B m = I i=1 (1 + η) m−1 ≤ T i (t) ≤ (1 + η) m , ∀ m ∈ M. Since A = ∪ m∈M (A ∩ B m ),F = 1 E I i=1 T i (t)D HG (μ ν i (t), µ ν i ) ≥ ψ . Let ζ i ∈ (R + ) I , T ∈ N and x i (t) such that (i) if there exists 0 ≤ x ≤ µ ν i such that T D HG (x, µ ν i ) = ζ i , then x i (T ) = x, (ii) else x i (T ) = 0. Since dD HG (x, µ ν i )/dx ≤ 0 for 0 ≤ x ≤ µ ν i , D HG (x, µ ν i ) is a monotonically decreasing function of x and furthermore x i (T ) is a monotonically increasing function of T . As a result, if T i (t)D HG (μ ν i (t), µ ν i ) ≥ ζ i , we havê µ ν i (t) (a) ≤ x i (T i (t)) (b) ≤ x i ((1 + η)t i ) where (a) holds due to the monotonicity of x i (T ), and (b) is due to the definition of event E in (15). Lemma 5. Fix an I-armed heteroscedastic Gaussian bandit ν and let 1 ≤t i ≤ t. For any collection of 0 ≤ b i ≤ µ ν i , i ∈ [I], we have P ∩ i∈[I] μ ν i (t) ≤ b i ,t i ≤ T i (t) ≤ I i=1 e −tiDHG(bi,µ ν i ) . Proof. See Appendix B-C. With this lemma, we can deduce that P ∩ i∈[I] 1 E T i (t)D HG (μ ν i (t), µ ν i ) ≥ ζ i ≤ P ∩ i∈[I] μ ν i (t) ≤ x i (T i (t)), E ≤ P ∩ i∈[I] μ ν i (t) ≤ x i ((1 + η)t i ), E (a) ≤ I i=1 e −tiDHG(xi((1+η)ti),µ ν i ) (b) = I i=1 e −ζi/(1+η) where (a) is obtained by Lemma 5 with b i = x i ((1 + η)t i ) and (b) is becauset i D HG (x i ((1 + η)t i ), µ i ) = ζ i /(1 + η). Lastly, by applying [29,Lemma 8], we have P(F) ≤ ψe I(1 + η) I e −ψ/(1+η) ≤ ψe I I e −ψ/(1+η) , which concludes the proof. C. Proof of Lemma 5 Proof. According to the Gibbs variational principle [30], one has the following dual characterization of the KL divergence: D(P 1 , P 2 ) = sup f (E P1 [f (X 1 )] − ln(E P2 [e f (X2) ])), where the variables X 1 ∼ P 1 and X 2 ∼ P 2 . If we let f (x) = λx, λ ≤ 0, for 0 ≤ x ≤ µ i , we have D HG (x, µ ν i ) = sup λ≤0 {λx − φ i (λ i )}, where φ i (λ i ) = ln E[e λiXi ] = ln(µ ν i e λ + 1 − µ ν i ) and X i ∼ N (µ ν i , 2µ ν i σ 2 ). Define the event Q = Q 1 ∩ Q 2 , where Q 1 = ∩ i∈[I] {t i ≤ T i (t)} and Q 2 = ∩ i∈[I] {μ ν i (t) ≤ b i }. For all i, let λ i ≤ 0 and define C(t) = exp i∈[I] λ i S i (t) − T i (t)φ i (λ i ) , where S i (t) = T i (t)μ ν i (t). For all i, we set λ i = arg max λ≤0 {λb i − φ i (λ)} and λ i b i − φ i (λ i ) = D HG (b i , µ ν i ). Then, we have P(Q) = P ∩ i∈[I] {S i (t) ≤ T i (t)b i , Q 1 } ≤ P i∈[I] λ i S i (t) ≥ i∈[I] λ i T i (t)b i , Q 1 ≤ P 1 Q1 e i∈[I] λiSi(t) ≥ e i∈[I] λiTi(t)bi = P 1 Q1 C(t) ≥ e i∈[I] Ti(t)(λibi−φi(λi)) = P 1 Q1 C(t) ≥ e i∈[I] Ti(t)DHG(bi,µ ν i ) ≤ P 1 Q1 C(t) > e i∈[I]t iDHG(bi,µ ν i ) . Then, it holds that P(Q) ≤ P 1 Q1 C(t) > e i∈[I]t iDHG(bi,µ ν i ) (a) ≤ E[1 Q1 C(t)] exp − i∈[I]t i D HG (b i , µ ν i ) (b) ≤ exp − i∈[I]t i D HG (b i , µ ν i ) , where (a) is due to Markov's inequality, (b) is due to the fact that E[1 Q1 C(t)] ≤ E[C(t)] and E[C(t)] = 1 is shown in the proof of [29,Lemma 7]. This concludes the proof. APPENDIX C In this section, we first study the optimization problem (4), so as to better understand the choice of w * , then we provide an efficient method for computing w * . By minimizing the KL divergence between the reward distributions of two bandits associated with arm A * (ν) and arm i = A * (ν), (4) can be transformed into the following optimization problem c * (ν) −1 = sup w∈W I inf u∈Alt(ν) I i=1 w k D HG (µ ν i , µ u i ) = sup w∈W I inf u∈V: i =A * (ν),µ u i ≥µ u A * (ν) a∈{A * (ν),i} w a D HG (µ ν a , µ u a ) .(16) It follows that w * = arg max w∈W I inf u∈V: i =A * (ν),µ u i ≥µ u A * (ν) a∈{A * (ν),i} w a D HG (µ ν a , µ u a ) . By using Lagrange multipliers, we can solve the optimization problem min µ u A * (ν) ,µ u i w A * (ν) 1 2 ln( µ ν A * (ν) µ u A * (ν) ) + µ u A * (ν) 2µ ν A * (ν) + (µ u A * (ν) − µ ν A * (ν) ) 2 4µ ν A * (ν) σ 2 − 1 2 + w i 1 2 ln( µ ν i µ u i ) + µ u i 2µ ν i + (µ u i − µ ν i ) 2 4µ ν i σ 2 − 1 2 , s.t. 0 < µ u A * (ν) ≤ µ u i , and obtain the optimal values of µ u A * (ν) , µ u i , i.e., µ u A * (ν) = µ u i = w A * (ν) + w i w A * (ν) µ ν i + w i µ ν A * (ν) µ ν A * (ν) µ ν i .(17) By substituting (17) into (16), we have c * (ν) −1 = sup w∈W I inf i =A * (ν) w A * (ν) 2 ln w A * (ν) µ ν i + w i µ ν A * (ν) (w A * (ν) + w i )µ ν i + w i 2 ln w A * (ν) µ ν i +w i µ ν A * (ν) (w A * (ν) +w i )µ ν A * (ν) + w A * (ν) w i (µ ν A * (ν) −µ ν i ) 2 ) 4σ 2 (w A * (ν) µ ν i +w i µ ν A * (ν) ) − 1 2 (w A * (ν) + w i ) .(18)With x * i = µ ν i µ ν A * (ν) , we have w * A * (ν) = 1 1 + a =A * (ν) x * a , w * i = x * i 1 + a =A * (ν) x * a . As a result, according to (18) 1 1 + a =A * (ν) x a f Y,i (x i ).(19) where f Y,i (x i ) = 1 2 ln µ ν i + x i µ ν A * (ν) (1 + x i )µ ν i + x i 2 ln µ ν i + x i µ ν A * (ν) (1 + x i )µ ν A * (ν) + x i (µ ν A * (ν) − µ ν i ) 2 4σ 2 (µ ν i + x i µ ν A * (ν) ) − 1 + x i 2 . By differentiating f Y,i (x i ) with respect to x i , we have df Y,i dx i = µ ν A * (ν) − µ ν i 2(µ ν i + x i µ ν A * (ν) ) + 1 2 ln µ ν i + x i µ ν A * (ν) (1 + x i )µ ν A * (ν) + µ ν i (µ ν A * (ν) − µ ν i ) 2 4σ 2 (µ ν i + x i µ ν A * (ν) ) 2 − 1 2 . Since d 2 f Y,i dx 2 i ≤ 0, df Y,i dxi is monotonically decreasing for all x i ∈ [0, ∞). Furthermore, it holds that lim xi→∞ df Y,i dxi = −1/2. Because of (2), we have df Y,i dxi | xi=0 ≤ 0. As such, df Y,i dxi ≤ 0 holds for all x i ∈ [0, ∞), and f Y,i (x i ) is a monotonically decreasing function. Then, we let y * = f Y,1 (x * 1 ) = f Y,2 (x * 2 ) = . . . = f Y,I (x * I ), y * ∈ [0, y u ], based on which we then introduce the inverse function of f Y,i (x), i.e. f X,i = f −1 Y,i . According to (19), y * belongs to the set arg min y∈[0,yu] F Y (y) with F Y (y) = y 1 + g =A * (ν) f X,i (y * ) . By taking the derivative of F Y (y), we have dF Y dy = 1 1+ i =A * (ν) f X,i (y) − i =A * (ν) yf X,i (y) (1+ i =A * (ν) f X,i (y)) 2 . By setting dF Y dy = 0, we obtain y such that F Y (y) achieves its minimum, i.e., i =A * (ν) yf X,i (y) − f X,i (y) = 1. Then we let F ν (y) = i =A * (ν) yf X,i (y) − f X,i (y). Since f X,i (y) = 1 f Y,i (y) and d 2 f Y,i dy 2 ≤ 0, we have dF ν (y) dy = i =A * (ν) yd 2 f X,i (y) dy 2 = i =A * (ν) − y (f Y,i (y)) 2 d 2 f Y,i dy 2 ≥ 0. Therefore, since F ν (y) is strictly increasing and F ν (y u ) = ∞, F ν (0) = 0, F ν (y) = 1 must have a unique solution y * , which can be solved using the bisection method. Then, we can obtain w * i as w * i = f X,i (y * ) I a=1 f X,a (y * ) . Then, by replacing the mean µ ν i with the empirical mean µ ν i (t), we can obtainŵ * i (t) as in (12). APPENDIX D A. Proof of Lemma 2 Proof. Assume a heteroscedastic Gaussian bandit instance ν with µ ν 1 ≥ µ ν 2 ≥ . . . ≥ µ ν I . There exists ξ = ξ( ) ≤ (µ ν 1 − µ ν 2 ) /4 such that I := [µ ν 1 − ξ, µ ν 1 + ξ] × . . . × [µ ν I − ξ, µ ν I + ξ] . Then, for a bandit modelν t ∈ I and t 0 > 0 sup t≥t0 max i |ŵ * i (t) − w * i | ≤ . Furthermore, for allν t ∈ I , the empirical optimal arm is A * (ν t ) = 1. Let define h(T ) = T 1/4 and the event E T ( ) = T t=h(T ) {ν t ∈ I } in which it holds for t ≥ h(T ), A * (ν t ) = 1. Then, let us rewrite Z(t) in (9) as Z(t) = min i =1 T 1 (t)D HG (μ ν 1 (t), q i (t))+T i (t)D HG (μ ν i (t), q i (t)) = t f Z ν t , T i (t) t I i=1 , where q i (t) is defined in (10), f Z ν , w = min i =1 w 1 D HG (µ 1 , q i ) + w i D HG (µ i , q i ) and q i = w 1 + w i w 1 µ i + w i µ 1 µ 1 µ i . Lemma 6. The sampling rule ensures that T i (t) ≥ √ t − 1 and that for all > 0 and t 0 > 0, there exists a constant t = max t0 3 , 1 3 2 , 1 12 3 such that sup t≥t0 max i |ŵ * i (t) − w * i | ≤ =⇒ sup t≥t max i T i (t) t − w * i ≤ 3(I − 1) . Proof. See the proof in Appendix D-B. According to Lemma 6 and the definition of E T , when T ≥ t = max{ t0 3 , 1 3 2 , 1 12 3 }, we define C * (ν) = inf ν∈I ,ŵ:|ŵi−w * i |≤3(I−1) f Z ν,ŵ ,(20) then on the event E T it is holds that Z(t) ≥ tC * (ν), ∀ t ≥ √ T . When T ≥ t , and on the event E T , it holds that min τ δ , T ≤ √ T + T t= √ T I (τ δ >t) ≤ √ T + T t= √ T I (Z(t)≤β(t,δ,α)) ≤ √ T + T t= √ T I (tC * (ν)≤β(T,δ,α)) = max √ T , β(T, δ, α) C * (ν) Define T 0 (δ) = inf T ∈ N : max √ T , β(T, δ, α) C * (ν) ≤ T = inf T ∈ N : β(T, δ, α) C * (ν) ≤ T = inf T ∈ N : C * (ν)T ≥ ln(αT /δ) . Using [27,Lemma 18], we have T 0 (δ) ≤ α C * (ν) ln αe δC * (ν) + ln ln α δC * (ν) . Lemma 7. There exist two constants Γ 2 and Γ 3 (which depend on ν and ) such that P(E c T ) ≤ Γ 2 T exp(−Γ 3 T 1/8 ). Proof. See the proof in Appendix D-C. Using Lemma 7, for every T ≥ max(T 0 (δ), t ), one has E T ⊂ {τ δ ≤ T }, therefore P (τ δ > T ) ≤ P (E c T ) ≤ Γ 2 T exp −Γ 3 T 1/8 . According to the definition of C * (ν) in (20) and c * (ν) −1 in (4), it holds that C * (ν) ≤ c * (ν) −1 . As a result, we have E[τ δ ] = ∞ T =1 P(τ δ ≥ T ) = max(T0(δ),t ) T =1 P(τ δ ≥ T )+ ∞ T =max(T0(δ),t )+1 P(τ δ ≥ T ) ≤ t + T 0 (δ) + ∞ T =1 Γ 2 T exp −Γ 3 T 1/8 ≤ t + αc * (ν) ln αec * (ν) δ + ln ln αc * (ν) δ + ∞ T =1 Γ 2 T exp −Γ 3 T 1/8 . Lemma 8. Let ν represent a heteroscedastic Gaussian bandit instance. When ν has a unique optimal arm, then we have c * (ν) ≤ c * u (ν), where c * u (ν) = µ ν A * (ν) 2 I i=1 µ ν i ln 2µ ν A * (ν) µ ν A * (ν) + µ ν A (ν) + µ ν A (ν) 2 I i=1 µ ν i ln 2µ ν A (ν) µ ν A * (ν) + µ ν A (ν) + (µ ν A * (ν) − µ ν A (ν) ) 2 8σ 2 I a=1 µ ν a − µ ν A * (ν) + µ ν A (ν) 2 I a=1 µ ν a −1 . Proof. Please refer to Appendix D-D. Let V represent a set of I-armed heteroscedastic Gaussian bandit instances. Note that c * (ν) −1 = sup w∈W I inf u∈V: i =A * (ν),µ u i >µ u A * (ν) a∈{A * (ν),i} w i D HG (µ ν a , µ u a ) , which is related to the choice of u and w. According to Lemma 8,it holds that E[τ δ ] ≤ t + αc * (ν) ln αec * (ν) δ + ln ln αc * (ν) δ + ∞ T =1 Γ 2 T exp(−Γ 3 T 1/8 ) ≤ t + αc * u (ν) ln αec * u (ν) δ + ln ln αc * u (ν) δ + ∞ T =1 Γ 2 T exp(−Γ 3 T 1/8 ). Since τ = I +τ δ according to Line 15 of Algorithm 2, it holds that E[τ ] = I + E[τ δ ] ≤ I + t + αc * u (ν) ln αec * u (ν) δ + ln ln αc * u (ν) δ + ∞ T =1 Γ 2 T exp(−Γ 3 T 1/8 ). The proof is concluded by letting A = I + T . B. Proof of Lemma 6 Proof. Fix t 0 such that when t 0 ≥ t 0 , it holds that ∀ t ≥ t 0 , √ t ≤ 2t and 1/t ≤ . To satisfy the requirement √ t ≤ 2t , we fix t ≥ max{t 0 , 1 4 2 }. Thus, we let t 0 := max{t 0 , 1 , 1 4 2 }. According to [27,Lemma 17] and its proof in [27,Appendix B.2], by choosinĝ λ(i) =ŵ * i (t) and λ * = w * , we have sup i T i (t) t − w * i (ν) ≤ (I − 1) max 2 + 1 t , t 0 t ≤ (I − 1) max 3 , t 0 t . As a result, when t ≥ t0 3 , it holds that sup i Ti(t) t − w * i (ν) ≤ 3(I − 1) . Let t = max t0 3 , 1 3 2 , 1 12 3 , which concludes the proof. I i=1 [P(μ ν i (t) ≤ µ ν i − ξ) + P(μ ν i (t) ≥ µ ν i + ξ)]. According to Lemma 6, for each arm, we have T i (t) > √ t − I, ∀ t ≥ h(T ).(21) Then, we have . P (μ ν i (t) ≤ µ ν i − ξ) (a) = P(μ ν i (t) ≤ µ ν i − ξ, T i (t) ≥ √ t − I) (b) ≤ t m= √ t−I P (μ ν i (m) ≤ µ ν i − ξ)( where (a) is obtained due to (21), (b) and (c) holds due to the union bound and the Chernoff inequality. Due to the same reason, it holds that P (μ ν i (t) ≥ µ ν i + ξ) ≤ min D HG (µ ν i − ξ, µ ν i ), D HG (µ ν i + ξ, µ ν i ) , we obtain . P(E c T ) ≤ T t=h(T ) I i=1 P(μ ν i (t) ≤ µ ν i − ξ) + P(μ ν i (t) ≤ µ ν i + ξ) ≤ T t=h(T ) Γ 2 exp(− √ tΓ 3 ) ≤ Γ 2 T exp(− h(T )Γ 3 ) = Γ 2 T exp − Γ 3 T 1/8 , Since dFµ(µ ν i ) dµ ν i < 0 holds, F µ (µ ν i ) is monotonically decreasing. Therefore, we conclude that when a suboptimal arm A (ν) = arg min i =A * (ν) µ ν i is selected, (22) can be rewritten as c * (ν) −1 ≥ µ ν A * (ν) 2 I a=1 µ ν a ln 2µ ν A * (ν) µ ν A * (ν) + µ ν A (ν) + µ ν A (ν) 2 I a=1 µ ν a ln 2µ ν A (ν) µ ν A * (ν) + µ ν A (ν) + (µ ν A * (ν) − µ ν A (ν) ) 2 8σ 2 I a=1 µ ν a − µ ν A * (ν) + µ ν A (ν) 2 I a=1 µ ν a , as such we have c * (ν) ≤ µ ν A * (ν) 2 I a=1 µ ν a ln 2µ ν A * (ν) µ ν A * (ν) + µ ν A (ν) + µ ν A (ν) 2 I a=1 µ ν a ln 2µ ν A (ν) µ ν A * (ν) + µ ν A (ν) + (µ ν A * (ν) − µ ν A (ν) ) 2 8σ 2 I a=1 µ ν a − µ ν A * (ν) + µ ν A (ν) 2 I a=1 µ ν a −1 , which concludes the proof. Fig. 1 : 1A mmWave massive MISO system system. Fig. 3 : 3The average received power associated different beams over the codebook space in a three-path channel (K = 120, N = 64, p = 26 dBm and the power of line-of-sight (LoS) path is around 3 dB larger than that of the non-line-of-sight (NLoS) path.) Fig. 4 : 4Illustration of the proposed algorithms. Fig. 5 : 5Illustration of a case that the optimal base arm does not lie in the optimal super arm. Fig. 6 : 6Mean of the reward of each base arm and super arm in Scenario 1 (p = 0 dBm).average time complexities of the considered algorithms in this scenarios when the transmit signal-to-noise is chosen from the set {66, 70, 74, 78} dB. It can be observed that HT&S and 2PHT&S outperform T&S and 2PT&S, since the former algorithms explicitly exploit the heteroscedastic property in the considered bandit BA problem. Furthermore, the number of time steps of 2PHT&S (2PT&S) is much smaller than that of HT&S (T&S). This improvement is a consequence of dividing the search process into two phases.Second, we investigate a scenario in which the AoD of the LoS path and that of one NLoS path are close, i.e., the AoDs and path losses of three paths are [0.3352π, 0.3521π, 0.8125π] and [0, −3, −3] dB. In this scenario, since the AoDs of the main LoS path and one NLoS path are very close, the optimal base arm (base arm 91) is the neighbor of the suboptimal base arm (base arm 90), then the optimal super arm (super arm 31) is also the neighbor of the suboptimal super arm (super arm 30) and they have similar means of the rewards. Fig. 7 : 7Mean of the reward of each base arm and super arm in Scenario 3 (p = 0 dBm). Fig. 8 : 8Scenario 4: A practical scenario in a city. Fig. 9 : 9Means of the rewards associated with each base arm and super arm in Scenario 4 (p = 10 dBm). Lemma 4 . 4Fix an I-armed heteroscedastic Gaussian bandit ν and let 1 ≤t i ≤ t. Let η > 0. Define the event: Since we have P(A) ≤ m∈M P(A ∩ B m ). Using Lemma 4, we obtain for all m ∈ M: P(A ∩ B m ) |M| = M I , we have P(A) ≤ M I P(A ∩ B m ) With the choice η = 1/(ψ − 1) and the inequality ln(1 + η) ≥ 1 − 1/(1 + η) = 1/ψ, it holds that P(A) ≤ e −ψ ψ ψ ln(t) I I e I+1 , which concludes the proof. B. Proof of Lemma 4 (−mD HG (µ ν i − ξ, µ ν i )) ≤ e −( √ t−I)DHG(µ ν i −ξ,µ ν i ) ν a − (µ ν A * (ν) − µ ν i ) 2 8σ 2 ( Algorithm 2 HT&SInput: Maximum risk δ, super arm set I. Output: The required number of time steps τ , the empirical mean of the rewardμ ν .1: for i ∈ I do 2: TABLE I : ITime complexitiesAlgorithm T&S 2PHT&S Upper bound of lim δ→0 E[τ ] ln(1/δ) Table III displays the estimated average time complexities of the algorithms we consider in the there scenarios when SNR ∈ {70, 74, 78, 82} dB. As compared to the results in Table IV presents the estimated average time complexities of the considered algorithms in Scenario 3 when SNR ∈ {78, 80, 82, 84} dB. It can be observed that similar to the previous channel models, the proposed 2PHT&S algorithm TABLE II : IIAverage time complexities in Scenario 1 for δ = 0.1, averaged over 100 experiments.SNR (dB) 66 70 74 78 EBA 4044.0 ±1089.7 1620.0 ±630.6 744.0 ±308.8 348.0 ±68.1 T&S 3941.6 ±1177.8 1409.4 ±504.1 495.3 ±197.1 155.1±49.6 HT&S 1631.7 ±1028.6 577.8 ±349.0 213.2 ±110.3 128.6±23.9 HEBA 1562.6 ±356.4 630.4 ±136.3 234.4 ±38.4 124.0 ±16.9 2PT&S 1387.8 ±397.0 496.2 ±136.8 155.9 ±50.4 58.9 ±12.3 2PHT&S (overlapping) 452.1 ±322.3 270.8 ±124.3 82.7 ±42.9 51.2 ±4.3 2PHT&S 384.0 ±302.9 161.4 ±82.4 93.0 ±48.7 49.2 ±2.5 TABLE III : IIIAverage time complexities in Scenario 2 for δ = 0.1, averaged over 100 experiments.SNR (dB) 70 74 78 82 EBA 6518.4 ±1889.7 2373.6±904.0 912.0 ±309.5 441.6 ±124.7 T&S 5981.1 ±1919.3 2119.5 ±1063.6 806.7 ±358.6 295.6 ±113.4 HT&S 3752.0 ±2029.4 1569.9 ±866.5 529.5 ±355.8 233.5 ±109.3 HEBA 1599.0 ±616.7 570.0 ±172.2 249.6 ±65.8 134.6 ±34.2 2PT&S 1437.6 ±427.7 509.0 ±203.2 172.7 ±69.7 82.5±31.6 2PHT&S 713.6 ±543.3 329.4 ±203.6 135.1 ±64.1 70.8 ±29.1 0 0.005 0.01 0.015 0.02 0.025 0.03 Reward/Recieved signal power (W) 0 20 40 60 80 100 120 Base arm index 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Reward/Recieved signal power (W) 0 10 20 30 40 Super arm index TABLE IV : IVAverage time complexities in Scenario 3 for δ = 0.1, averaged over 100 experiments.SNR (dB) 78 80 82 84 EBA 478.8 ±123.6 351.6 ±80.5 289.2 ±61.7 244.8 ±23.6 T&S 357.2 ±147.7 219.2 ±78.0 163.4 ±52.9 128.0±23.9 HEBA 320.1 ±100.8 222.4 ±56.2 161.6 ±43.8 126.0 ±27.5 2PT&S 246.5 ±91.9 160.2 ±61.7 109.5 ±43.8 70.8±21.9 HT&S 232.0 ±115.6 175.4 ±77.5 130.5 ±32.85 121.8±8.6 2PHT&S 159.7 ±84.5 122.6 ±59.1 78.2 ±29.1 65.0 ±18.9 TABLE V : VAverage time complexities in Scenario 4 for δ = 0.1, averaged over 100 experiments. 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Sun, "Wide- band millimeter-wave propagation measurements and channel models for future wireless communication system design," IEEE Trans. Commun., vol. 63, no. 9, pp. 3029-3056, Sept. 2015. Optimal best arm identification with fixed confidence. A Garivier, E Kaufmann, Journal of Machine Learning Research. 49A. Garivier and E. Kaufmann, "Optimal best arm identification with fixed confidence," Journal of Machine Learning Research, vol. 49, pp. 998-1027, 2016. Quadriga: A 3-D multi-cell channel model with time evolution for enabling virtual field trials. S Jaeckel, L Raschkowski, K Borner, L Thiele, IEEE Trans. Antennas Propag. 626S. Jaeckel, L. Raschkowski, K. Borner, and L. Thiele, "Quadriga: A 3-D multi-cell channel model with time evolution for enabling virtual field trials," IEEE Trans. Antennas Propag., vol. 62, no. 6, pp. 3242-3256, Jun. 2014. Lipschitz bandits: Regret lower bounds and optimal algorithms. S Magureanu, R Combes, A Proutiere, PMLR35S. Magureanu, R. Combes, and A. Proutiere, "Lipschitz bandits: Regret lower bounds and optimal algorithms," in PMLR, vol. 35, no. 975-999, 2014. Elementary principles in statistical mechanics developed with special reference to the rational foundations of thermodynamics. J W Gibbs, Franklin Classics. J. W. Gibbs, Elementary principles in statistical mechanics developed with special reference to the rational foundations of thermodynamics. Franklin Classics, 2018. She is currently working as a Research Engineer at China Aerospace Science and Technology Corporation. Her research interests include signal processing for wireless communications, algorithm design for advanced MIMO. she was a Visiting Scholar at the Department of Electrical and Computer Engineering. SingaporeYi Wei received the B.Eng. and the Ph.D. degrees in Information and Communication Engineering from Zhejiang University in ; National University of Singaporedeep learning for wireless communicationsYi Wei received the B.Eng. and the Ph.D. degrees in Information and Communication Engineering from Zhejiang University in 2017 and 2022, respectively. From May 2021 to Apr 2022, she was a Visiting Scholar at the Department of Electrical and Com- puter Engineering, National University of Singapore, Singapore. She is currently working as a Research Engineer at China Aerospace Science and Tech- nology Corporation. Her research interests include signal processing for wireless communications, algo- rithm design for advanced MIMO and deep learning for wireless communications. She is supervised by Prof. Csaba Szepesvári. Dr. Zhong received her PhD degree from the Department of Mathematics of National University of. Prof. Vincent Y. F. Tan and ProfSingapore (NUSShe is currently a postdoctoral fellow at the Department of Computing Science of University of Alberta (UofA)Zixin Zhong was born in China in 1995. She is currently a postdoctoral fellow at the Department of Computing Science of University of Alberta (UofA). She is supervised by Prof. Csaba Szepesvári. Dr. Zhong received her PhD degree from the Department of Mathematics of National University of Singapore (NUS) in October 2021. Dr. Zhong was privileged to be supervised by Prof. Vincent Y. F. Tan and Prof. Wang Chi Cheung during her PhD study, and she worked with them as a research fellow between. Wang Chi Cheung during her PhD study, and she worked with them as a research fellow between June 2021 and July 2022. Zhong's research interests are in reinforcement learning, online machine learning and, in particular, multi-armed bandits. Her work has been presented at top machine learning (ML) conferences including ICML and AISTATS, and also in top ML journals such as the Journal of Machine Learning Research (JMLR). ; Dr, Iclr, Icml, Neurips, Tsp Tit, Tmlr , She also serves as a reviewer for several conferences and journals including AISTATS. She was recognized as a top reviewer of NeurIPS 2022Dr. Zhong's research interests are in reinforcement learning, online machine learning and, in particular, multi-armed bandits. Her work has been presented at top machine learning (ML) conferences including ICML and AISTATS, and also in top ML journals such as the Journal of Machine Learning Research (JMLR). She also serves as a reviewer for several conferences and journals including AISTATS, ICLR, ICML, NeurIPS, TIT, TSP, and TMLR. She was recognized as a top reviewer of NeurIPS 2022. He received the B.A. and M.Eng. degrees in Electrical and Information Sciences from Cambridge University in 2005 and the Ph.D. degree in Electrical Engineering and Computer Science (EECS) from the Massachusetts Institute of Technology (MIT) in 2011. His research interests include information theory, machine learning, and statistical signal processing. Dr. Tan received the MIT EECS Jin-Au Kong outstanding doctoral thesis prize in 2011, the NUS Young Investigator Award in 2014, the Singapore National Research Foundation (NRF) Fellowship (Class of 2018) and the NUS Young Researcher Award in 2019. He was also an IEEE Information Theory Society Distinguished Lecturer. Y F Vincent, Tan, He is currently serving as a Senior Area Editor of the IEEE Transactions on Signal Processing and an Associate Editor of Machine Learning for the IEEE Transactions on Information Theory. Department of Electrical and Computer Engineering at the National University of Singapore (NUSS'07-M'11-SM'15) was born in Singapore in 1981. He is currently an Associate Professor in the Department of Mathematics and the. He is a member of the IEEE Information Theory Society Board of GovernorsVincent Y. F. Tan (S'07-M'11-SM'15) was born in Singapore in 1981. He is currently an Associate Professor in the Department of Mathematics and the Department of Electrical and Computer Engineering at the National University of Singapore (NUS). He received the B.A. and M.Eng. degrees in Electrical and Information Sciences from Cambridge Univer- sity in 2005 and the Ph.D. degree in Electrical Engineering and Computer Science (EECS) from the Massachusetts Institute of Technology (MIT) in 2011. His research interests include information theory, machine learning, and statistical signal processing. Dr. Tan received the MIT EECS Jin-Au Kong outstanding doctoral thesis prize in 2011, the NUS Young Investigator Award in 2014, the Singapore National Research Foundation (NRF) Fellowship (Class of 2018) and the NUS Young Researcher Award in 2019. He was also an IEEE Information Theory Society Distinguished Lecturer for 2018/9. He is currently serving as a Senior Area Editor of the IEEE Transactions on Signal Processing and an Associate Editor of Machine Learning for the IEEE Transactions on Information Theory. He is a member of the IEEE Information Theory Society Board of Governors.
[ "https://github.com/YiWei0129/Fast-beam-alignment)." ]
[ "A new stochastic framework for ship capsizing", "A new stochastic framework for ship capsizing" ]
[ "Manuela L Bujorianu [email protected] \nMaritime Safety Research Centre\nMathematics Institute\nUniversity of Strathclyde\nUK\n", "Robert S Mackay [email protected] \nMathematics Institute\nUniversity of Warwick\nUK\n", "Tobias Grafke [email protected] \nSchool of Mathematics\nUniversity of Warwick\nUK\n", "Shibabrat Naik [email protected] \nMaritime Safety Research Centre\nUniversity of Bristol\nUK\n", "Evangelos Boulougouris [email protected] \nUniversity of Strathclyde\nUK\n" ]
[ "Maritime Safety Research Centre\nMathematics Institute\nUniversity of Strathclyde\nUK", "Mathematics Institute\nUniversity of Warwick\nUK", "School of Mathematics\nUniversity of Warwick\nUK", "Maritime Safety Research Centre\nUniversity of Bristol\nUK", "University of Strathclyde\nUK" ]
[]
We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands of transition state theory. The first is an extension of deterministic transition state theory to dissipative non-autonomous systems, together with a probability distribution over the forcing functions. The second is stochastic reachability and large deviation theory for transition paths in Markovian systems. In future work we aim to bring these together to make a tool for predicting capsize rate in different stochastic sea states, suggesting control strategies and improving designs.
null
[ "https://arxiv.org/pdf/2105.05965v1.pdf" ]
234,482,742
2105.05965
5bd0817c24fa26ba4e98755753f749f2a63350e1
A new stochastic framework for ship capsizing Manuela L Bujorianu [email protected] Maritime Safety Research Centre Mathematics Institute University of Strathclyde UK Robert S Mackay [email protected] Mathematics Institute University of Warwick UK Tobias Grafke [email protected] School of Mathematics University of Warwick UK Shibabrat Naik [email protected] Maritime Safety Research Centre University of Bristol UK Evangelos Boulougouris [email protected] University of Strathclyde UK A new stochastic framework for ship capsizing Transition State TheoryTransition Path TheoryFlux-over-saddleMarkov ModelsStochastic ReachabilityCapsize ProbabilityLarge Deviations We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands of transition state theory. The first is an extension of deterministic transition state theory to dissipative non-autonomous systems, together with a probability distribution over the forcing functions. The second is stochastic reachability and large deviation theory for transition paths in Markovian systems. In future work we aim to bring these together to make a tool for predicting capsize rate in different stochastic sea states, suggesting control strategies and improving designs. INTRODUCTION A new stochastic framework for studying ship capsize is presented for a general class of sea states exceeding mere regular waves. It has two strands, both starting from transition state theory (Truhlar et al., 1996;Waalkens et al., 2008) (in which we include transition path theory (Vanden-Eijnden, 2006)). The common outcomes are survivability probabilities (Long et al., 2010), the probability rate for capsize and the most likely paths to capsize. First is a formulation of capsize for given forces and moments as functions of time and state, leading from given initial condition to a deterministic time to capsize (infinite if no capsize) and hence from a probability distribution on initial conditions to a distribution of times to capsize. This is based on a proposed extension of the "flux over a saddle" paradigm (Mackay, 1990) to include dissipation and non-autonomous forcing. To take into account uncertainty about the forcing, we consider probability distributions over forcing functions (together with initial conditions) and aim to deduce the survivability probability, the probability rate for capsize per unit time as a function of time, and the most likely paths to capsize. The second is stochastic reachability theory (Bujorianu, 2012) and large deviation theory (Dembo and Zeitouni, 2010) for transitions of Markovian processes in continuous state-space. Defining unsafe regions to be avoided in state space, we can formally write down the probability of observing trajectories that start at normal conditions and reach an unsafe set. We compute this probability asymptotically in various limits via large deviation theory to avoid inefficient sampling problems. This allows us to efficiently explore stochastic capsize events and obtain the probability rate for capsize per unit time (the reachability from stochastic reachability), the survivability probability (the viability from stochastic reachability) and the most likely path to capsize (the large deviation minimiser). An important strand that we do not address here is how to pass from a given incident field of wave, wind and current to the resulting forces and moment on the ship. Another is how to formulate safety conditions for the operation or design of a ship, because that would depend on the above mapping. Another is the formulation of control strategies for a ship's captain to avoid capsize, such as change of speed or course. We contrast our framework with previous approaches, represented for example by (Umeda et al., 1995;Thompson, 1997;McCue, 2011;Almeida Santos Neves et al., 2011;Fossen and Nijmeijer, 2011). One is the study of response to periodic forcing, including the resulting bifurcations between attractors, e.g. (Spyrou, 1996); this gives very useful insights but real-world forcing is not periodic. Periodic forcing has been combined with white noise (Lin and Yim, 1995;Jamnongpipatkul et al., 2011) but this is still a limited perspective. Rough seas are typically modelled as a train of random waves from some probability distribution and hence capsize in rough sea requires inclusion of more general stochastic processes (Perez, 2006). Statistical approaches include extreme value theory (Leadbetter et al., 2012;Belenky, Glotzer, Pipiras and Sapsis, 2019), where a universal form is derived for extreme values from various types of stochastic process, but the known results require quite strong hypotheses and the approach to the asymptotic regime can be very slow. Here is the structure of the paper. We begin with a rapid statement about our ship models in section 2. In section 3 we explain the flux over a saddle paradigm and its adaptation here. Then we summarise the use of stochastic reachability theory and large deviation theory in section 4. We bring these two strands together into a synthesis in section 5 and conclude in section 6. Following standard practice (Belenky, Spyrou, Walree, Neves and Umeda, 2019;Lu et al., 2020), we consider a ship as a rigid body with six degrees of freedom: roll, pitch, yaw, heave, surge and sway, subject to external forces and moments, as sketched in Fig. 1. Each degree of freedom consists of a configuration variable and a velocity or momentum. The ship has an associated 6 × 6 inertia matrix, giving the kinetic energy as a function of the state of the ship (including added mass effects for the surrounding fluid). In addition, we use phenomenological damping forces and moments. The result is a coupled system of 6 second-order differential equations, or equivalently of 12 first-order equations. SHIP MODELS FLUX OVER A SADDLE Just as a continental divide separates points from which water flows to different oceans, and it consists in a set of points whose gradient trajectories flow to a saddle, the set of points whose trajectories flow to a saddle plays a key role in understanding capsize. The starting point for the "flux over a saddle" paradigm is an autonomous Hamiltonian system with a saddle point having just one downhill dimension for the energy function (Mackay, 1990). The saddle then possesses a "centre manifold" of codimension-2 in the state space (codimension-2 means it has 2 dimensions less than the total state space), representing the set of initial conditions whose trajectories remain close to the saddle. The centre manifold has a forwards contracting manifold + of codimension-1 (commonly called its "stable manifold") representing states whose forward trajectories converge to trajectories on the centre manifold. It also has a backwards contracting manifold − ("unstable manifold") representing states whose backwards trajectories converge to trajectories on the centre manifold. Also the centre manifold can be spanned by a codimension-1 manifold, separated into two parts by the centre manifold (in the same way that the equator can be spanned by the surface of the earth, separated into two hemispheres). This manifold divides the state space into two parts, corresponding to the two sides of the saddle. To get from one side to the other, a trajectory has to cross it. The two parts correspond to the two directions of crossing. There is some arbitrariness in the choice of the dividing manifold, but it makes only a minor difference to when a trajectory is declared to have crossed. The manifold + separates the region that will make the transition from the region that will not. Thus to find the region that will capsize one has to follow + backwards in time. If it avoids a core around the upright state of the ship, then the ship can be considered safe from capsize. Use of the flux over a saddle picture in the context of ship capsize was suggested by (Naik and Ross, 2017). Although originally developed in the context of Hamiltonian systems (for transition state theory of chemical reactions), the above picture persists for weak dissipation. Furthermore, it generalises from systems with a saddle to ones with what we call a "saddle manifold", being a "normally hyperbolic" submanifold of codimension-2 with one forwards and one backwards contracting dimension (normally hyperbolic means all tangential contraction in either direction of time is slower than normal contraction in that direction of time). We believe this is the case for a large range of realistic parameters for the standard ship models introduced in section 2, with the saddle manifold being specified roughly as zero roll-velocity and a critical roll-angle as a function of all the other variables and their velocities (actually, two saddle manifolds, for port and starboard roll, and the interaction of their contracting manifolds is important). Lastly, the framework has a version for non-autonomous systems, as is needed for periodic or more general time-dependent forcing. To describe this, we extend the 12-dimensional state space by adding time as a 13th variable. Then, if the time-dependence is not too strong, the centre manifold of the saddle has a locally unique continuation as a normally hyperbolic manifold of dimension 11 in the extended state space, that we denote by . Its backwards and forwards contracting manifolds persist too, denoted by ± . The dividing manifold can be continued to a dividing manifold in the extended state-space. Hence capsize for a dissipative, non-autonomous system is described by passage over this generalised saddle. A 3D sketch of the situation is given in Figure 2. We define the time to capsize to be the time until the first intersection with the dividing manifold, with the convention that = ∞ if it is never reached. Thus from a probability distribution over initial states, we obtain a probability distribution for the time to capsize. Its derivative is the probability rate for capsize at time . In addition to probability distribution over initial conditions, we are interested in taking probability distributions over the forcing functions. Then Figure 2: Schematic of the geometry in extended state space: the curve represents a bundle of trajectories of dimension 11 that remain at the capsize threshold; deviations from it are denoted Δ in position/attitude and Δ in velocity; has codimensionone forwards and backwards contracting submanifolds ± , dividing the space into four sectors, labelled for capsize, for near-capsize, for righting, and for failed righting. we want to compute features of the probability distribution of the time to capsize, in particular what is its rate as a function of , and what is the probability of eventual capsize? More broadly, what are the most likely routes to capsize? How do all these depend on the probability distributions for the forcing functions and initial conditions, and on the parameters of the ship model? STOCHASTIC REACHABILITY AND LARGE DEVIATIONS A complementary approach to the above formalism is the perspective of stochastic reachability and large deviations. Stochastic reachability is a technique used in engineering and computer science to deal with safety issues in a quantitative manner. The objective of stochastic reachability analysis is to evaluate the probabilities associated with dynamic optimization problems. This technique can be used for optimal control under uncertainty, for risk assessment of technical systems, and for safety verification. Formally, the system is modelled using a stochastic process (e.g., a Markov chain/process, Wiener process, Gaussian process, or diffusion process) and the unsafe region is modelled as a set in its phase space. Stochastic reachability aims to estimate the probability measure of the set of the trajectories that start in a given set of initial states and reach a target set (a possible unsafe set for the system) in a given time interval. There is a close connection to the terminology of chemical reaction kinetics: a chemical reaction can be viewed a transition from one locally stable position in state space to another, driven by the system's stochasticity (for example thermal noise) and against its typical short-time behaviour. The picture is that of a random walk in an energy landscape, where a barrier must be overcome for a reaction to happen. Such transition events are generally very rare on the timescale intrinsic to the stochasticity, but waiting long enough one will eventually observe them. A body of literature is concerned with transition events (Truhlar et al., 1996), their dynamics (Vanden-Eijnden, 2006) and likelihood (Freidlin and Wentzell, 2012). The ultimate question is, of course, an estimate of the probability of observing a transition, or equivalently, the transition rate. In the situation of ship capsize, a ship in its upright position can similarly be considered only locally stable: while a large enough perturbation will topple it into a capsize, there generally is a generous region in its 12 dimensional state space where restoring mechanisms, such as its righting moments, keep it afloat most of the time. A transition trajectory or reactive trajectory for ship capsize, thus, describes the movement of a ship in time that, starting in an upright position, will eventually hit an unsafe region and subsequently capsize, due to a rare influence of its stochastic components, and generally against its restoring forces. In this sense, ship capsize can be seen as a first hitting problem, or stochastic reachability problem. Analytical characterizations of the stochastic reachability use equations that link the hitting distribution of the unsafe set with the occupation measure of the safe basin. This is based on the operator methods and Dynkin formula associated to Markov processes. Martingale characterization can be also derived from this equation. In general the fact that reactive trajectories are rare outliers in a usually mechanically stable system renders their observation by experiment or numerical sampling quite hard. Crucially, though, their rareness often paradoxically makes them predictable, which is the core idea behind sample path large deviations. In this paper, we propose a large deviation methodology to deal with stochastic reachability to provide asymptotic estimates for the probabilities of rare events (Freidlin and Wentzell, 2012). To make the above more concrete, we consider the motion of the ship ( ) ∈ R 12 as introduced in section 2 to be a continuous-time Markov process with stationary distribution ( ). Denote by a neighbourhood of the upright ship state, and by the unsafe region ultimately leading to capsize, for example as specified in section 3. We can define by + ( ) the forward committor, i.e. the probability density over state space that we will visit (capsize) before (righting), or in other words the probability that we have committed to a capsizing event when being located at . Similarly, the backwards committor is the probability density that the process at originates from rather than . Given these, the density of reactive trajectories is immediately available as = + − , as can be intuited by reading the formula as the combined probability of coming from , being at , and going to . From committor functions and the density of reactive trajectories, one can finally deduce quantities such as the probability flux towards capsize, and the capsize rate , with specific formulas depending on the nature of the process. The above quantities are generally not accessible for any system of interest as they necessitate the solution of Dirichlet boundary value problems (similar to the Fokker-Planck equation) in high dimensions. Fortunately, this becomes drastically better in the presence of a large deviation principle (LDP). Intuitively speaking, one obtains the probability of observing an outcome by integrating (or summing) over all possible ways this outcome can occur, weighted by its respective probability. The same is true for reactive trajectories by defining an appropriate path measure. In the presence of an LDP, this integral can be replaced in an appropriate limit (such as thermodynamic limit, low temperature limit, or small noise limit) by the value of the integrant at the most likely path realizing the outcome. In essence, the integral is computed by a Laplace method, exchanging a costly transition sampling problem with an optimisation problem. Knowledge of the large deviation optimal path allows the computation of transition rates in the large deviation limit, and the optimal path can be computed quite efficiently by numerical means even for rather complex systems (Grafke and Vanden-Eijnden, 2019). This is exemplified in figure 3: Here, we consider a toy model for ship capsize for the roll angle and its velocity, ( , ) ∈ R 2 , under Gaussian forcing. We want to consider only trajectories leading to capsize, i.e. starting upright (green set) and ending capsized (red set). While the direction of the righting moment in phase space is depicted as streamlines, the density of reactive trajectories is shown as shading, and the large deviation minimizing trajectory is highlighted in white. It is clear from the picture that the capsize trajectories concentrate around the optimal path predicted by large deviation theory. INTERCONNECTIONS The two strands are closely related. They both represent the uncertainties in forcing by probability distributions. They formulate capsize as transition across some distinguished set, random (but highly correlated with the forcing function) in the first approach, and fixed or not needing to be specified precisely in the second. They both aim to produce estimates or bounds on the capsize rate, particularly in the regime where it is expected to be rare. The two descriptions overlap if the forcing functions are assumed to be the result of filtering a white noise, as is often assumed in the literature (Chai et al., 2015). This means there is a filter state ∈ R , some ∈ N, satisfying in the simplest case = + , where is an asymptotically stable matrix, is a multidimensional white noise (say stationary Gaussian) with autocorrelation ( ) ( ) = ( − ) for some positive semi-definite matrix and a small parameter . Then the ship dynamics can be taken to be of the form = ( , ), where represents the 12 dimensions of the ship state-space. The probability distribution on the functions is easy to handle (linear stochastic process), so one could hope to obtain probabilistic results for the flux over a saddle approach. Considered as a system on ( , ) the model also fits in the Markovian context of the second approach. Thus the two can be directly compared. Our hope is that further understanding will allow development of large deviation theory to more general probability distributions over forcing functions, thereby escaping the Markovian restriction of the second approach. CONCLUSION We have presented a new stochastic framework for studying ship capsize. It has two parallel strands, both based on transition state theory, one starting from a deterministic view, the other from a Markovian view. For filtered white noise models of forcing, the two approaches can in principle be carried to conclusion. A synthesis is required to treat more general probability distributions for forcing functions. Extensions are required to pass from probability distributions for sea states to those for forcing functions. Once established, this framework could be used as a building-block for the formulation of safety criteria, optimizing vessel design, and control strategies for the captain to avoid capsize. Figure 1 : 1Degrees of freedom for ship motion. Figure 3 : 3Toy model of a ship capsize as 2-dimensional stochastic system for the roll angle and corresponding angular velocity . Trajectories from upright (green) to capsized (red) correspond to transition paths out of the stable basin. The deterministic dynamics are shown as vector field, the density of reactive trajectories as shading, and the large deviation minimiser as white line. Almeida Santos Neves, M Belenky, V L De Kat, J O Spyrou, K Umeda, N , 2011. Contemporary Ideas on Ship Stability and Capsizing in Waves. NetherlandsSpringer97Dordrechtof Fluid Mechanics and Its ApplicationsAlmeida Santos Neves, M., Belenky, V. L., de Kat, J. O., Spyrou, K. and Umeda, N. (eds), 2011. Contemporary Ideas on Ship Stability and Cap- sizing in Waves, Vol. 97 of Fluid Mechanics and Its Applications, Springer Netherlands, Dor- drecht. Distribution tail structure and extreme value analysis of constrained piecewise linear oscillators. V Belenky, D Glotzer, V Pipiras, T P Sapsis, Probabilistic Engineering Mechanics. 5701Belenky, V., Glotzer, D., Pipiras, V. and Sapsis, T. P., 2019. Distribution tail structure and extreme value analysis of constrained piecewise linear oscillators, Probabilistic Engineering Mechanics 57(01): 1-13. V L Belenky, K J Spyrou, F V Walree, M A S Neves, N Umeda, Contemporary Ideas on Ship Stability: Risk of Capsizing. SpringerBelenky, V. L., Spyrou, K. J., Walree, F. v., Neves, M. A. S. and Umeda, N., 2019. Contemporary Ideas on Ship Stability: Risk of Capsizing, Springer. Stochastic reachability analysis of hybrid systems. L M Bujorianu, Springer Science & Business MediaBujorianu, L. M., 2012. Stochastic reachability analysis of hybrid systems, Springer Science & Business Media. Filter models for prediction of stochastic ship roll response. W Chai, A Naess, B J Leira, Probabilistic Engineering Mechanics. 41Chai, W., Naess, A. and Leira, B. J., 2015. Filter models for prediction of stochastic ship roll response, Probabilistic Engineering Mechanics 41: 104-114. Large deviations techniques and applications. A Dembo, O Zeitouni, Springer-VerlagBerlinDembo, A. and Zeitouni, O., 2010. Large devi- ations techniques and applications, Springer- Verlag, Berlin. Parametric resonance in dynamical systems. T I Fossen, H Nijmeijer, Springer Science & Business MediaFossen, T. I. and Nijmeijer, H. (eds), 2011. Paramet- ric resonance in dynamical systems, Springer Science & Business Media. Random perturbations of dynamical systems. M I Freidlin, A D Wentzell, Springer260Freidlin, M. I. and Wentzell, A. D., 2012. Random perturbations of dynamical systems, Vol. 260, Springer. Numerical computation of rare events via large deviation theory. T Grafke, E Vanden-Eijnden, An Interdisciplinary Journal of Nonlinear Science. 29663118ChaosGrafke, T. and Vanden-Eijnden, E., 2019. Nu- merical computation of rare events via large deviation theory, Chaos: An Interdisciplinary Journal of Nonlinear Science 29(6): 063118. Nonlinear ship rolling motion subjected to noise excitation. A Jamnongpipatkul, Z Su, J M Falzarano, Ocean Systems Engineering. 0103Jamnongpipatkul, A., Su, Z. and Falzarano, J. M., 2011. Nonlinear ship rolling motion subjected to noise excitation, Ocean Systems Engineering 01(03): 249-261. Extremes and Related Properties of Random Sequences and Processes. M R Leadbetter, G Lindgren, H Rootzen, Springer Science & Business MediaLeadbetter, M. R., Lindgren, G. and Rootzen, H., 2012. Extremes and Related Properties of Ran- dom Sequences and Processes, Springer Sci- ence & Business Media. Chaotic roll motion and capsize of ships under periodic excitation with random noise. H Lin, S C Yim, Applied Ocean Research. 173Lin, H. and Yim, S. C., 1995. Chaotic roll motion and capsize of ships under periodic excitation with random noise, Applied Ocean Research 17(3): 185-204. Estimation of survival probability for a ship in beam seas using the safe basin. Z J Long, S K Lee, J Y Kim, Ocean Engineering. 374Long, Z. J., Lee, S. K. and Kim, J. Y., 2010. Estima- tion of survival probability for a ship in beam seas using the safe basin, Ocean Engineering 37(4): 418-424. Model experiments and direct stability assessments on pure loss of stability in stern quartering waves. J Lu, M Gu, E Boulougouris, Ocean Engineering. 216108035Lu, J., Gu, M. and Boulougouris, E., 2020. Model experiments and direct stability assessments on pure loss of stability in stern quartering waves, Ocean Engineering 216: 108035. Flux over a saddle. R S Mackay, Physics Letters A. 1458Mackay, R. S., 1990. Flux over a saddle, Physics Letters A 145(8): 425-427. Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas. L S Mccue, Contemporary Ideas on Ship Stability and Capsizing in Waves. M. Almeida Santos Neves, V. L. Belenky, J. O. de Kat, K. Spyrou and N. UmedaNetherlands, DordrechtSpringer97McCue, L. S., 2011. Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas, in M. Almeida Santos Neves, V. L. Belenky, J. O. de Kat, K. Spyrou and N. Umeda (eds), Contemporary Ideas on Ship Stability and Capsizing in Waves, Vol. 97, Springer Netherlands, Dordrecht, pp. 433-447. Geometry of escaping dynamics in nonlinear ship motion. S Naik, S D Ross, Communications in Nonlinear Science and Numerical Simulation. 47Naik, S. and Ross, S. D., 2017. Geometry of escaping dynamics in nonlinear ship motion, Communications in Nonlinear Science and Nu- merical Simulation 47: 48-70. Ship Motion Control: Course Keeping and Roll Stabilisation Using Rudder and Fins. T Perez, SpringerPerez, T., 2006. Ship Motion Control: Course Keeping and Roll Stabilisation Using Rudder and Fins, Springer. Dynamic Instability in Quartering Seas-Part I1: Analysis of Ship Roll and Capsize for Broaching. K J Spyrou, 11Spyrou, K. J., 1996. Dynamic Instability in Quar- tering Seas-Part I1: Analysis of Ship Roll and Capsize for Broaching, Journal of Ship Research p. 11. Designing Against Capsize in Beam Seas: Recent Advances and New Insights. J M T Thompson, Applied Mechanics Reviews. 505Thompson, J. M. T., 1997. Designing Against Capsize in Beam Seas: Recent Advances and New Insights, Applied Mechanics Reviews 50(5): 307-325. Current Status of Transition-State Theory. D G Truhlar, B C Garrett, S J Klippenstein, The Journal of Physical Chemistry. 10031Truhlar, D. G., Garrett, B. C. and Klippenstein, S. J., 1996. Current Status of Transition-State Theory, The Journal of Physical Chemistry 100(31): 12771-12800. Model experiments of ship capsize in astern seas. N Umeda, M Hamamoto, Y Takaishi, Y Chiba, A Matsuda, W Sera, S Suzuki, K Spyrou, K Watanabe, Journal of the Society of Naval Architects of Japan. 177Umeda, N., Hamamoto, M., Takaishi, Y., Chiba, Y., Matsuda, A., Sera, W., Suzuki, S., Spyrou, K. and Watanabe, K., 1995. Model experi- ments of ship capsize in astern seas, Journal of the Society of Naval Architects of Japan 1995(177): 207-217. Transition Path Theory. E Vanden-Eijnden, Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology. M. Ferrario, G. Ciccotti and K. Binder1SpringerLecture Notes in PhysicsVanden-Eijnden, E., 2006. Transition Path Theory, in M. Ferrario, G. Ciccotti and K. Binder (eds), Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 1, Lecture Notes in Physics, Springer, Berlin, Heidelberg, pp. 453-493. Wigner's dynamical transition state theory in phase space: classical and quantum. H Waalkens, R Schubert, S Wiggins, Nonlinearity. 211Waalkens, H., Schubert, R. and Wiggins, S., 2008. Wigner's dynamical transition state theory in phase space: classical and quantum, Nonlinear- ity 21(1): R1-R118.
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[ "OMNIFORCE: ON HUMAN-CENTERED, LARGE MODEL EMPOWERED AND CLOUD-EDGE COLLABORATIVE AUTOML SYSTEM", "OMNIFORCE: ON HUMAN-CENTERED, LARGE MODEL EMPOWERED AND CLOUD-EDGE COLLABORATIVE AUTOML SYSTEM" ]
[ "Chao Xue [email protected] \nDacheng Tao\n\n", "Wei Liu \nDacheng Tao\n\n", "Shuai Xie \nDacheng Tao\n\n", "Zhenfang Wang \nDacheng Tao\n\n", "Jiaxing Li \nDacheng Tao\n\n", "Xuyang Peng \nDacheng Tao\n\n", "Liang Ding \nDacheng Tao\n\n", "Shanshan Zhao \nDacheng Tao\n\n", "Qiong Cao \nDacheng Tao\n\n", "Yibo Yang \nDacheng Tao\n\n", "Fengxiang He \nDacheng Tao\n\n", "Bohua Cai \nDacheng Tao\n\n", "Rongcheng Bian \nDacheng Tao\n\n", "Yiyan Zhao \nDacheng Tao\n\n", "Heliang Zheng \nDacheng Tao\n\n", "Xiangyang Liu \nDacheng Tao\n\n", "Dongkai Liu \nDacheng Tao\n\n", "Daqing Liu \nDacheng Tao\n\n", "Li Shen \nDacheng Tao\n\n", "Chang Li \nDacheng Tao\n\n", "Shijin Zhang \nDacheng Tao\n\n", "Yukang Zhang \nDacheng Tao\n\n", "Guanpu Chen \nDacheng Tao\n\n", "Shixiang Chen \nDacheng Tao\n\n", "Yibing Zhan \nDacheng Tao\n\n", "Jing Zhang \nDacheng Tao\n\n", "Chaoyue Wang [email protected] \nDacheng Tao\n\n" ]
[ "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n", "Dacheng Tao\n" ]
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Automated machine learning (AutoML) seeks to build ML models with minimal human effort. While considerable research has been conducted in the area of AutoML in general, aiming to take humans out of the loop when building artificial intelligence (AI) applications, scant literature has focused on how AutoML works well in open-environment scenarios such as the process of training and updating large models, industrial supply chains or the industrial metaverse, where people often face open-loop problems during the search process: they must continuously collect data, update data and models, satisfy the requirements of the development and deployment environment, support massive devices, modify evaluation metrics, etc. Addressing the open-environment issue with pure data-driven approaches requires considerable data, computing resources, and effort from dedicated data engineers, making current AutoML systems and platforms inefficient and computationally intractable. Humancomputer interaction is a practical and feasible way to tackle the problem of open-environment AI. In this paper, we introduce OmniForce, a human-centered AutoML (HAML) system that yields both human-assisted ML and ML-assisted human techniques, to put an AutoML system into practice and build adaptive AI in open-environment scenarios. Specifically, we present OmniForce in terms of ML version management for data, labels, models, algorithms and search spaces; pipeline-driven development and deployment collaborations; a flexible search strategy framework; and widely provisioned and crowdsourced application algorithms, including large models. Our proposed cloudnative OmniForce method can be run either on a public/private cloud or in an on-premise environment. Furthermore, the (large) models constructed by OmniForce can be automatically turned into remote services in a few minutes; this process is dubbed model as a service (MaaS). Experimental results obtained in multiple search spaces and real-world use cases demonstrate the efficacy and efficiency of OmniForce.Keywords Human-Centered Automated Machine Learning (HAML) · Cloud-Edge Collaborations · Large Model · Model-as-a-Service (MaaS)
10.48550/arxiv.2303.00501
[ "https://export.arxiv.org/pdf/2303.00501v1.pdf" ]
257,255,201
2303.00501
1049bc831ff08c43700d664edb321af908d223eb
OMNIFORCE: ON HUMAN-CENTERED, LARGE MODEL EMPOWERED AND CLOUD-EDGE COLLABORATIVE AUTOML SYSTEM 1 Mar 2023 Chao Xue [email protected] Dacheng Tao Wei Liu Dacheng Tao Shuai Xie Dacheng Tao Zhenfang Wang Dacheng Tao Jiaxing Li Dacheng Tao Xuyang Peng Dacheng Tao Liang Ding Dacheng Tao Shanshan Zhao Dacheng Tao Qiong Cao Dacheng Tao Yibo Yang Dacheng Tao Fengxiang He Dacheng Tao Bohua Cai Dacheng Tao Rongcheng Bian Dacheng Tao Yiyan Zhao Dacheng Tao Heliang Zheng Dacheng Tao Xiangyang Liu Dacheng Tao Dongkai Liu Dacheng Tao Daqing Liu Dacheng Tao Li Shen Dacheng Tao Chang Li Dacheng Tao Shijin Zhang Dacheng Tao Yukang Zhang Dacheng Tao Guanpu Chen Dacheng Tao Shixiang Chen Dacheng Tao Yibing Zhan Dacheng Tao Jing Zhang Dacheng Tao Chaoyue Wang [email protected] Dacheng Tao OMNIFORCE: ON HUMAN-CENTERED, LARGE MODEL EMPOWERED AND CLOUD-EDGE COLLABORATIVE AUTOML SYSTEM 1 Mar 2023* Correspondence author Automated machine learning (AutoML) seeks to build ML models with minimal human effort. While considerable research has been conducted in the area of AutoML in general, aiming to take humans out of the loop when building artificial intelligence (AI) applications, scant literature has focused on how AutoML works well in open-environment scenarios such as the process of training and updating large models, industrial supply chains or the industrial metaverse, where people often face open-loop problems during the search process: they must continuously collect data, update data and models, satisfy the requirements of the development and deployment environment, support massive devices, modify evaluation metrics, etc. Addressing the open-environment issue with pure data-driven approaches requires considerable data, computing resources, and effort from dedicated data engineers, making current AutoML systems and platforms inefficient and computationally intractable. Humancomputer interaction is a practical and feasible way to tackle the problem of open-environment AI. In this paper, we introduce OmniForce, a human-centered AutoML (HAML) system that yields both human-assisted ML and ML-assisted human techniques, to put an AutoML system into practice and build adaptive AI in open-environment scenarios. Specifically, we present OmniForce in terms of ML version management for data, labels, models, algorithms and search spaces; pipeline-driven development and deployment collaborations; a flexible search strategy framework; and widely provisioned and crowdsourced application algorithms, including large models. Our proposed cloudnative OmniForce method can be run either on a public/private cloud or in an on-premise environment. Furthermore, the (large) models constructed by OmniForce can be automatically turned into remote services in a few minutes; this process is dubbed model as a service (MaaS). Experimental results obtained in multiple search spaces and real-world use cases demonstrate the efficacy and efficiency of OmniForce.Keywords Human-Centered Automated Machine Learning (HAML) · Cloud-Edge Collaborations · Large Model · Model-as-a-Service (MaaS) Introduction In recent decades, machine learning (ML) has achieved great success in the fields of computer vision [1,2], natural language processing (NLP) [3,4], speech recognition [5,6], content generation [7,8] and tabular data processing [9,10]. The rapid development of ML technology has given birth to highly popular artificial intelligence (AI) products, such as Tesla Autopilot [11], Google Translate [12], Siri [13], and ChatGPT [14]. With ML requirements growing exponentially in terms of both the amount of training data and the number of models/neural networks tailored to different tasks, the design of tailored hyperparameters and identifying neural networks for training in a fully automatic fashion without human intervention, which is referred to as automated ML (AutoML), has yielded great achievements. The recent progress of AutoML has been characterized by algorithms and systems. Regarding the former, considerable studies have used methods based on genetic algorithms [15], random search [16], Bayesian optimization [17], reinforcement learning [18] and differentiable techniques [19]. To achieve the latter, numerous frameworks such as Optuna [20], Ray-Tune [21], HyperOpt [22], NNI [23] and Orion [24] for hyperparameter optimization have been developed to support scalable trials and customizable search algorithms. Auto-sklearn [25] also supports the use of meta-learning to leverage historical records to warm-start the search procedure. Unlike other frameworks that require additional effort to support Kubernetes [26], Katib [27] is a cloud-native framework and can realistically be run in a production environment. Compared to the exploration of an open-source AutoML system, many companies offer their AutoML products to the market, such as Google Cloud AutoML [28], IBM Watson AutoAI [29], Amazon SageMaker [30], and H2O Driverless AI [31]. Such platforms are targeted at building AI models in a short period of time for developers with limited ML expertise. While numerous frameworks have been proposed for AutoML, as described above, we have not seen the expected widespread adoption of AutoML systems in industry. We presume the following reasons for the low adoption rate of these frameworks. • Only targeting closed-loop problems -Most AutoML frameworks only focus on closed-loop problems, where the data, algorithms, and metrics are deterministic; thus, their design concept is to take humans out of the loop when building AI applications. However, AI-related problems are often open-loop tasks in practice, especially in the process of training and updating large models or in industrial supply chains, where people need to collect data continuously, update the versions of data and models, and modify the evaluations and rewards produced during the production process. It would be inefficient and even computationally intractable to use current data-driven AutoML systems to address open-loop problems since they require considerable data to learn domain knowledge and business logic. • Lack of deployment considerations -Most AutoML frameworks only focus on the search and training phases, ignoring the inference and deployment phases. However, massive devices with different deployment (inference) requirements are encountered in real industrial scenarios or industrial metaverses, where simulation or XR 2 technology is used to reduce the risk of failure in the physical production process and to build a highly efficient supply chain, including the design, development, manufacturing, pricing, sales, storage, transportation, and after-sale service phases. • Limited application algorithms -Most AutoML frameworks only have some predefined or built-in application algorithms and their corresponding search spaces, which are often transparent to users for ease of use. However, a large number of various AI applications are available in practice, and an AutoML system is limited to a fixed number of built-in application algorithms. Considering this large number of various applications, especially those with different development and deployment requirements, e.g., Cloud-Edge collaborations, the wide use of AutoML systems will be restricted if only predefined application algorithms are provided for the search process. In an attempt to address the above issues, we develop OmniForce for supporting AutoML in open environments; OmniForce is centered on the following ideas. • Human-centered and adaptive AutoML (HAML) -We design OmniForce for both human-assisted ML and ML-assisted humans. Thus, users can efficiently deal with their business logic and data collection processes by interacting with OmniForce. • Cloud-edge collaborations in practice -We propose a pipeline-driven AutoML framework with collaborative development and deployment to search AI applications with different training and deployment requirements. • Crowdsourced application algorithms -We introduce the crowdsourcing concept to integrate the various observed application algorithms into the OmniForce platform. By standardizing the data abstraction paradigm, application algorithm, and search space, we can easily integrate and reuse the application algorithms and search space. We illustrate the concept of HAML in Figure 1. The user interacts with the AutoML system in terms of human-assisted ML and ML-assisted humans. In particular, HAML tasks have the elements of data collection and annotation, features, application algorithms, search spaces, searching, training and deployment, and visualization. For data collection and annotation, on the one hand, users collect data to make ML algorithms accurate by using active learning; on the other hand, ML algorithms help users label data efficiently. Moreover, data privacy protection and security play important roles in interactions between human and AutoML systems. OmniForce supports the differential privacy technique for protecting users' data. Regarding features, OmniForce supports customized feature pipelines through user interaction and SQL. Additionally, users can view and analyze the statistical meta-data information of data and features. In terms of application algorithms, given the generality of crowdsourcing and super-deep models, OmniForce is more widely applicable than many other AutoML systems with only built-in small-scale application algorithms. OmniForce hides the details used to set the search space by default, but users can configure the search space by tuning priors or preferences when they obtain knowledge from visualizations. For searching, training, and deployment, users define the development and deployment environment, set their requirements and constraints, and perform single/multiple-objective optimizations. OmniForce supports Cloud-Edge collaboration to address the different requirements of the training and deployment environments by means of its powerful search ability. For the visualization part, which is the core of HAML, users learn the knowledge of their AutoML pipeline from OmniForce, obtain explanations of the searched architecture and hyperparameters, and acquire advice that can be used to guide the next steps of their work. For example, the advisor may suggest that the user update the search space based on the statistical distribution of the sampling candidates. Additionally, it may encourage the user to collect more data from a specific class or relax some latency constraints and power restrictions. The HAML cycle enables users to fully participate in human-computer cooperation and achieve the purposes of both using machines to enhance human abilities and leveraging human experiences and operations to improve machine intelligence. Our contributions are as follows. 1) OmniForce is a cutting-edge human-centered and adaptive AutoML system that supports for open-environment scenarios such as the process of training and updating large models, industrial supply chains and industrial metaverses. It includes a set of novel search strategies, a search space update policy, and large model algorithms for computer vision (CV), NLP, and AI-generated content (AIGC). As such, OmniForce caters to both developers with limited ML expertise and data scientists. 2) OmniForce is a cloud-native AutoML system that is scalable, fault tolerant and cloud-edge collaborative; thus, it can be run in a production environment. 3) OmniForce follows the model-as-a-service (MaaS) pattern and fully connects the search, training, inference, and deployment processes. At the moment when OmniForce successfully completes the model construction process, users will not only obtain the model but also have the inference and deployment service of the model. This enables users to transform the model into a remote service that can be deployed in the cloud or on the edge in a few minutes, helping users quickly build cutting-edge applications with AI capabilities. The rest of this paper is arranged as follows. Section 2 describes the system architecture and workflow of OmniForce. Section 3 describes the detailed design concepts, and Section 4 shows the supported features, followed by evaluations in Section 5. We compare the related work in Section 6 and finally conclude in Section 7. System Architecture As we will see, OmniForce is implemented with several components such as a job estimator, workers, a training task estimator (TTE), a task deployment estimator (DTE), a task sidecar, a scheduler, and a manager. Figure 2 shows a simplified block diagram of the system overview of OmniForce. The application layer is concerned with processing business logic, such as uploading the data, choosing the application algorithms and pretrained models, and setting the search space to start the search process. OmniForce provides widely provisioned data and application algorithms that contain large model-based methods for users who focus on their business logic regardless of the details of the ML algorithms. Users can also utilize and contribute to the crowdsourced resources that are integrated into OmniForce to satisfy the growing demand for ML applications. All data, algorithms, and search spaces of the application layer interact with the search engine via uniform abstractions, meaning that these resources are represented and organized as uniform views. Further changes are reflected in the Figure 1: HAML. The ring represents the pipeline of HAML. Users interact with the key steps/nodes in the loop. Unlike most AutoML frameworks that only focus on the searching and training parts, HAML pays attention to the whole ML pipeline, where one needs to consider data collection; updating the data version, feature pipelines, algorithms, and search spaces; modifying the evaluations and rewards; and obtaining knowledge from the visualization that can be used to guide his or her work in the next steps. different versions of the resources. Privacy and security are also involved in the interface layer. More details regarding the interface layer can be found in Section 3.1. The search engine is implemented via a search strategy framework with multiobjective collaboration, which attaches to a semisynchronized controller for multiple programs/multiple data (MPMD) dispatch, a scheduler that assigns jobs and tasks to the cloud resources, and a formatter for learning the historical knowledge to generate the AutoML pipeline (serving as a meta-learner). In particular, we implement a flexible parallel Bayesian optimization (BO) framework that fully runs on PyTorch, which supports BO with a variety of surrogate models and acquisition functions, including our novel model, to deal with large discrete spaces (which form the scenario of neural architecture search (NAS)). We also support a revised hyperband [32], MF-NAS [33], and a novel evolution approach in the search strategy framework. The search process becomes more difficult when the search space is large. We involve multiple workers to find good candidates in parallel. Considering the tradeoff between parallel efficiency and inevitable synchronization that some strategies need to guarantee performance, we implement a semisynchronized MPMD dispatcher. Similar to Pathways [34], which has demonstrated the limits of the single program/multiple data (SPMD) paradigm for ML computations, we express the NAS process as MPMD. More details regarding the search engine can be found in Section 3.4. OmniForce runs on Kubernetes and Kubeflow [35] to support scalability, fault tolerance, and multitenancy. OmniForce is a product-ready, cloud-native system that can be deployed as a service either in a public/private cloud or in an on-premise environment. Components In this section, we explain some fundamental components of OmniForce in detail, including the job estimator, workers, TTE, DTE, task sidecar, scheduler (formatter and resource scheduler), manager and advisor. Figure 2: OmniForce system overview. Based on Kubernetes and Kubeflow, OmniForce builds an AutoML system with widely provisioned ML algorithms and uniform interfaces for crowdsourcing algorithms, models, and search spaces. Job Estimator The job estimator is a semisynchronized controller of the AutoML process that determines which tasks should be evaluated next, how to handle their rewards, and when to start the next iteration. Two key components are contained in the job estimator: a user-defined search space from which the candidates are generated and a search strategy that conducts the sequential and parallel search processes via advanced search algorithms. The details of the search space and search strategy can be found in Sections 3.2.2 and 3.4, respectively. Some AutoML frameworks tend to be fully synchronized, where the next iteration will not start until all tasks in the current round are completed. However, synchronized frameworks cannot guarantee high efficiency in MPMD settings. In contrast, our job estimator adopts a semisynchronized mechanism that controls the starting of the next iteration through an adaptive maximum waiting time. The statuses of tasks that exceed this time are set to timeout, and the job estimator ignores these tasks in the current iteration and generates new tasks via the completed observations. The waiting time calculation is flexible. For example, we can use the mean and variance of the execution times of previously completed tasks to estimate the waiting time. Considering the massive execution time differences between tasks in the MPMD setting, we can also build a time cost-aware surrogate model provided by our BO framework to give an estimation for each task. Worker A worker is a group of processes that reserve and evaluate candidate tasks. In our design, the job estimator and workers do not communicate directly but rather through a task broker, which is usually middleware. The existing large-scale distributed AutoML systems always have large computational demands and place a high value on scalability and fault tolerance. Our design carefully considers these requirements and decouples the job estimator and workers, achieving scalability through which users can freely increase or decrease the number of utilized workers based on their practical resources. Furthermore, if some worker nodes break down unexpectedly, our search process adapts to this new circumstance without any effort and continues to search as long as one worker is alive. In addition, workers can handle many common exceptions, such as GPU memory exhaustion and middleware connection loss. TTE The TTE is the actual entity used to evaluate candidate tasks. After a worker reserves a task, a specific TTE will be launched to evaluate the task and report the result to the worker when finished. OmniForce develops a task sidecar service accompanied by the TTE to proxy the communication process. OmniForce supports crowdsourcing, and users can easily upgrade their training code to a searchable code in the TTE by implementing the algorithm interface defined in Section 3.2.1. More importantly, as it runs on Kubernetes and Kubeflow, OmniForce facilitates elastic distributed training with cloud-native workloads such as PytorchJob [36] and MPIJob [37]. This feature helps to efficiently use the available cluster resources and to handle variable training settings such as a large model. DTE The DTE is a deployment entity that cooperates with the TTE. When the TTE finishes the model training process, the trained model is sent to the DTE to assess the model's performance in the deployment environment. The communication between the DTE and TTE is carried out by task sidecars. For example, in the cloud-edge collaboration scenario, the cloud side is responsible for model searching and training with its powerful cloud computing capability, while the edge side is the practical environment for production. OmniForce cares about this requirement and provides a multiobjective optimization service to jointly evaluate model performance in the training and deployment environments. The design takes advantage of cloud computing and edge deployment, bridging the gap between the development and production environments. Task Sidecar A sidecar [38] is a design pattern in a cloud-native system that is quite useful when running two tightly coupled processes together. OmniForce builds task sidecar components to decouple the search logic and the connection logic to adjust to complex training and deployment environments. In our AutoML workflow, the task sidecar behaves as a middleman to connect the workers, TTE, and DTE. This component handles the trivial operations between components and expands the boundary of the service, making it applicable to more downstream scenarios. Scheduler The scheduler estimates and schedules resources for jobs and consists of two parts. The first part formats a new job as an appropriate configuration including a search space and estimated resources, and this component is dubbed the "formatter" in our system. The second part, called the resource scheduler, assigns the actual computing resources to the given job based on the estimated resources and the current payload of the AutoML system. To estimate the time slot, memory, and computing resources, the formatter builds a knowledge base containing the historical experiments that have been searched before. We obtain knowledge from previous jobs, such as hyperparameters, metrics configurations, the multifidelity of the searching process, memory, and utilized computations. Due to its use of an off-the-shelf knowledge base, the formatter finds a proper search algorithm and search space for an entering job and provides the resource scheduler with the estimated resources and the parallelism of workers. Additionally, the formatter defines the structure of the search pipeline. The resource scheduler manages all the cluster resources and prevents deadlocks or pending exceptions due to the preemption of resources. Specifically, the resource scheduler allocates the resources and adjusts the parallelism of the search and training stage according to the job details given by the formatter and the current payload of the system. It divides jobs into search, training, inference, and deployment phases and schedules their resources separately. The resource scheduler processes one job of a certain type at a time. Manager The manager is a housekeeper service that enables OmniForce to interact with cloud-native Kubernetes and Kubeflow resources. In addition to the basic create, read, update and delete (CRUD) operations, deferring to the ML operation philosophy, the OmniForce manager automates the AutoML workflow with a Kubeflow pipeline [39], which is a cloud-native service for building portable and scalable ML workflows. In addition, we build some specific AutoML pipelines and integrate various cloud-native workloads, such as Kubernetes jobs, Kubeflow PytorchJobs, and KServe InferenceServices, into one pipeline. Thus, OmniForce can manage these cloud-native resources in a unified form, which simplifies the process of automating and reproducing an AutoML workflow. Moreover, considering the flexible requirements of AutoML workflows and the unpredictability of cluster computing resources, the OmniForce manager abstracts the cloud-native workloads into modular pipeline elements so that the OmniForce scheduler and formatter can orchestrate these elements into pipelines with specific architectures and resource parallelism. After a pipeline is generated, the OmniForce manager starts the related components in topological order. We present two AutoML pipeline examples in Figure 3; one is called the NoCode pipeline, and the other is called the HPO pipeline. In the NoCode pipeline, the job estimator and workers are launched simultaneously to search for the best architecture and hyperparameters. Then, the task estimator builds the model with collaboration between training and deployment. Finally, the advisor gathers the meta-data generated in the workflow and provides informative insights for users. The start and end components here are responsible for some preparation and cleaning steps. In the HPO pipeline, the smoke job estimator and workers are added to verify the correctness of the crowdsourcing algorithms before entering the actual search phase. Then, the default job estimator and workers are added to reproduce the default algorithm performance for a comparison with the AutoML search results. Advisor An advisor is a component that gives users comprehensive insights into their datasets and crowdsourcing algorithms. The OmniForce advisor visualizes the search process during task execution. In addition to basic metrics such as the loss and accuracy, the advisor provides a visualized map of the candidates in the search space and a bar map of the hyperparameter importance levels to demonstrate their correlations with the selected metrics, aiming to untangle the complicated interactions between the hyperparameters. This information helps users understand which hyperparameters matter the most to the performance of their models. Moreover, the advisor supplies valuable suggestions about the search space and datasets. For example, the advisor may suggest that the user update the search space or encourage the user to collect more data on a specific class. System Workflow In this section, we illustrate the overall AutoML workflow that a user would interact with in the OmniForce system, best viewed in Figure 4. Adhering to the human-centric concept, OmniForce designs informative and friendly user interfaces for ease of use. In addition, OmniForce operates the services in a cloud-native, stable and cross-platform manner by leveraging Kubernetes and Kubeflow. After users set their objectives, OmniForce relays the subsequent model construction and analysis process. The major steps are as follows. 1. The user's input is translated to a standard model requirement and sent to the application server. 2. The application server verifies the user's identity and privileges, converts the model requirement to a specific AutoML job, and stores the job information in a database. Finally, this job is submitted to the scheduler. 3. The scheduler orchestrates the pipeline based on the job information and current cluster resources. This process can be divided into two steps. First, the formatter generates a search space, selects the proper search strategy based on the job information and historical records, and then organizes the modular pipeline elements into a logical pipeline. Second, the resource scheduler computes a rational resource allocation solution based on the current cluster resources. Notably, this step converts the logical pipeline to a specific resource pipeline that is ready to be launched by the manager. 4. The manager converts the resource pipeline into a standard format supported by the Kubeflow pipeline controller and starts the pipeline components in logical order, as illustrated in Figure 3. 5. The job estimator parses the search space and generates search candidates based on specific search strategies such as hyperband search and BO. 6. The candidates generated in the last step are stored in the middleware and ready to be reserved by the workers. 7. The workers reserve the candidates from the middleware in a mutually exclusive mode and launch specific task estimators to evaluate the candidates with the help of the task sidecar. 8. The task sidecar bridges the candidates reserved by the workers and instantiates them to cloud-native workloads such as jobs and PytorchJobs to conduct an evaluation. After these workloads are complete, the evaluation results are fetched and reported to the workers and job estimator for the next round of candidate generation. 9. The task sidecar also plays a vital role in training and deployment collaboration. When the training process is finished, the task sidecar on the training side relays the trained model to the deployment side to evaluate the model performance (such as latency and power) in the deployment environment. This step is an essential part of OmniForce's multiobjective optimization design. 10. During the model search process, the advisor comprehensively analyzes the meta-data and generates useful suggestions for users such as datasets, algorithms or configuration analyses. Design Concept Abstraction and Management for Human-Centered AI As a human-centered AI platform, OmniForce reduces the complexity of ML operation processes and lets users focus on their most relevant work, such as business logic. Specifically, OmniForce adopts a project concept to uniformly manage data, features, models, and other meta-data . A project contains all the materials needed to build a specific business solution. Below the project, the fundamental abilities required for AI in production, such as data privacy, version control, feature engineering, model training, serving and monitoring, and ML pipeline automation, are well abstracted and convenient to use. In this section, we detail the data, feature pipeline, and model management of OmniForce to illustrate its AI life cycle management capability. Data Management Data form the cornerstone for building successful AI solutions. When people choose ML platforms, they usually have the following central concerns. Is my data safe here? Will my privacy be carefully protected? Is the platform compatible with my diverse data and able to give full play to the data value? Can the platform handle rapid data iteration and update solutions in time? OmniForce provides a comprehensive data management service, covering privacy, data accessibility and versioning, data annotation, and lifelong learning. Data Accessibility and Versioning Most AI platform developers agree that data access is the main problem when connecting users to a platform. Diverse data derived from all walks of life increase the burden of data standardization and governance. To tackle this problem, OmniForce develops a group of uniform data accessing and fetching application programming interfaces (APIs). With little effort, users can upload diverse data from multiple sources and transform raw data into a standard format with the data accessing API. The data fetching API abstracts the complexity of data sources and helps developers and data scientists retrieve data freely. After standardizing the data, OmniForce performs data version control with DVC [40], which is used to handle large files, datasets, models, configurations, and codes. When users upload data, OmniForce converts all the media files (except tabular data files) to a metadata file that maintains the references to the original data. Users can freely perform operations on meta-data and save these changes to a new version. These operations do not change the previous meta-data, and all the meta-data are traceable and reusable. These operations mainly include the following: 1. Adding or modifying the annotations. 2. Data filtering, e.g., filtering 5 classes from the whole dataset. 3. Merging two versions of a dataset. Notably, all the data uploaded to OmniForce are safely hosted with our privacy protection algorithm. Active Learning and Intelligent Annotation Active learning (AL) [41] is a practical technology for addressing a lack of annotation data. The fundamental problem in AL is to develop a cost-effective data ranking strategy to find the most informative samples among a vast unlabeled data pool. In a typical AL loop, a model is used to recommend samples with appropriate ranking strategies. Then, professional annotators evaluate these recommendations and actively annotate the samples that the model urgently needs. When the annotation budget is satisfied, the model updates using the newly labeled dataset, typically with more samples. In return, the better-trained model will be used to recommend better data for the next round of recommendations. Data are the fuel moving the AI wave. AL helps collect high-quality fuel and quickly build high-performance AI models, which is significant in industrial fields. AL involves humans in the ML life cycle and quickly builds high-quality datasets with mutual human-machine assistance. OmniForce combines this ability with our other services and devises an intelligent data annotation service. This service has the following main features. 1. Abundant data annotations support the most common AI tasks. 2. Human-machine collaborative annotation. OmniForce supports two annotation modes, online and offline. In the online mode, users can correct the machine annotation results in real time, benefitting from the convenient model deployment service of OmniForce. In the offline mode, users can submit an annotation task with a large amount of unlabeled data at one time and review all the results when the automated annotation process is finished, which benefits from the batch inference service of OmniForce. 3. Customizable machine annotation capability. Unlike the other counterparts in the market that only support provisioned models with limited annotation ability, OmniForce benefits from its crowdsourcing design and supports users in deploying their annotation models. These models can be directly uploaded by users or generated by OmniForce. 4. AL-driven data recommendation service. OmniForce integrates many practical and advanced AL algorithms to help users find the most informative data and quickly build high-quality datasets. Differential Privacy The differential privacy (DP) technique was first proposed to guarantee the privacy of database querying operations [42]. Recently, it has also been extended to measure the privacy preservation level of an algorithm [43,44]. Suppose two adjacent datasets (S, S ), where S and S only differ by at most one sample, an arbitrary subset H of the model hypothesis space, and an algorithm A are given. The DP of A is defined as the change in A's output hypothesis when A is applied to S and S . In particular, (ε, δ)-DP is mathematically defined as log P A(S) (A(S)∈H)−δ P A(S ) (A(S )∈H) ≤ ε. This means that an algorithm with small differential privacy (ε, δ) is robust to changes in the individual training samples. Thus, the value of (ε, δ) indexes the DP level, i.e., the ability to resist differential attacks that use individual samples as probes to attack ML algorithms; then, the individual privacy is inferred via the changes in the output hypotheses. Generally, the privacy preservation level of an iterative algorithm degrades along with the number of iterations since the amount of leaked information accumulates as the algorithm progresses [43,45,46]. Deep neural networks have been practically demonstrated to have good generalization abilities. However, a deep neural network is an overparameterized model and is difficult for the existing statistical learning theory to explain [47]. This has attracted the community's interest in studying it, and many works have found that a DP model usually also has a guaranteed generalization ability [48,49,50,51,46]. Feature Pipeline Management ML models are highly dependent on the quality of the input data, and raw data preprocessing is often a crucial part of the ML pipeline. To assist data scientists and engineers in efficiently and accurately infusing their experience into AI products, OmniForce provides data processing capabilities such as exploratory data analysis, interactive feature engineering, and advanced SQL processing. Custom feature pipelines can ease the burden of data preparation in an automated and low-code manner, enabling more focus on data collection, feature design, and model selection innovations. Feature Pipeline Version Control Distinguished from data version control, feature pipeline version control emphasizes tracing users' feature engineering operations. A feature pipeline contains a series of operation steps. When users perform operations on their data in the interface, they are cast to a small part of the data and stored for future processing in a lazy loading manner. Thus, we provide an elegant compromise between real-time user operation feedback and operation version control. Based on the historical operation versions, users can build a new feature pipeline with little effort by modifying or merging the operation steps. Customized Feature Pipeline Users can employ interactive feature engineering and write SQL statements to generate features based on human experience, enabling human-assisted ML to achieve satisfactory results more quickly. Specifically, interactive feature engineering is divided into the following categories: temporal feature extraction, single-column calculation, intercolumn calculation, and specific condition processing, which covers atomic operations that are commonly used in data processing procedures. We also support users in performing high-level operations on data through custom SQL statements, dramatically improving their feature engineering efficiency. In addition, we use a lazy processing strategy to perform operations to ensure real-time interaction, making user clicks smoother when dealing with big data. Model Management In this section, we introduce the model management technique in OmniForce, which includes three main parts: model construction and versioning, model deployment, and monitoring in production. The core idea of model management is to realize automatic model iteration with a closed loop of training, deployment, monitoring, and optimization for releasing the model to production in an agile manner. Model Construction and Versioning OmniForce constructs models with AutoML pipelines in an automated manner, and all the produced models are version-controlled. We design a three-layer model management architecture to demonstrate the relationships among models. In our design, the top layer is called the model family, a collection of models with strong correlations. For example, when using large models for downstream tasks, OmniForce manages derived models as a family. The middle layer is called the model register, pointing out the model that is currently in use. The bottom layer is called the model asset, representing the raw model generated by the AutoML pipeline. Model Deployment Deploying a trained model into a production environment is a crucial part of model management and plays a large role in the MaaS paradigm because many developers leverage AI capabilities to build applications through provided APIs after their models are deployed. OmniForce applies KServe [52] for deploying a model into production and quickly building an MaaS architecture. The model runs on a Kubernetes cluster in a serverless form, and users can access the model through the provided API. OmniForce is designed with a crowdsourced code specification, and the models produced by the code written by algorithm developers according to these guidelines can be quickly deployed into production. The deployed models will have the following properties: 1. Support rolling updates and rollbacks. 2. Support high concurrency and low latency. 3. Support autoscaling, including scaling to 0, to resolve the conflict between latency sensitivity and demand predictability. 4. Support advanced deployment methods such as canary release, blue-green release, and A/B testing. 5. Support the simultaneous deployment of multiple models and cascaded models. 6. Support multistage conditional model inference. 7. Support model explanation. 8. Support model monitoring. Model Monitoring Model monitoring is an operational stage in the ML lifecycle that comes after model deployment. Since the production environment changes all the time, a production model will result in performance loss, which is called model drift. Model drift comes in two forms: 1. Data drift. Data drift is caused by data distribution changes. Since AI models are sensitive to the given data distribution, as the data distribution changes more drastically, the performance of the model drops rapidly. 2. Concept drift. Concept drift refers to the situation when a model is no longer applicable to its environment due to changes in the properties of the dependent variable. Model monitoring is used to monitor whether the model has drifted in the current environment. Our model monitoring system can monitor models in production in real time, send out warnings when model drift occurs, and initiate model retraining using recently collected data to iteratively update models and keep their performance at an acceptable level. New models generated by model retraining are automatically archived in the original model family and updated with a new version. Informative Visualization and Insights As the core of human-centered AI, visualization helps users understand their data and algorithms. OmniForce provides rich visualization information, such as data distributions, search spaces, hyperparameter importance levels, and inference statistics. Users can join the solution construction loop by analyzing this information and helping OmniForce build more versatile models. For example, users can redesign the search space or adjust the data distribution to construct better models. Hyperparameter Importance Analysis The performance of a deep learning model highly depends on its hyperparameter settings. Some modern optimization methods have been proposed and have successfully optimized hyperparameters automatically. However, these methods do not interpret how the specific hyperparameters affect the resulting model performance. To provide users with a more apprehensible hyperparameter report, OmniForce extracts the relationships between hyperparameters and metrics with a hyperparameter importance assessment method, named functional analysis of variance (fANOVA) [53]. Given the model metrics and corresponding hyperparameter settings, fANOVA fits a random forest to approximate the mapping between the hyperparameter space and the performance space. Then, fANOVA is applied to assess the importance of each hyperparameter. Furthermore, we make some improvements to adapt this method to hyperparameters with hierarchical dependencies. Real-Time Model Inference Explanation Currently, most deep models work in a black-box way, which lacks explainability and hinders the application of deep models in many fields. Following the idea of HAML, OmniForce provides a real-time model explanation service to explain the model's output. Specifically, OmniForce builds an explainable model and deploys this model along with the target model using KServe [52]. When users make an inference online, both models operate, and the explainable model analyzes the results of the target model. Based on the explanatory information, users can understand why their model outputs such a result. Additionally, considering that interpretable methods should be compatible with the various heterogeneous models supported by our platform, we mainly choose black-box model interpretation algorithms, such as anchors [54]. This method can approximate the decision-making process of a neural network model to a rule-based discrimination process. Its local interpretation characteristics facilitate the interpretation of a single sample predicted by the model and help users intuitively understand the interpretation process. Multiobjective Model Performance Evaluation Training and Deploying Comprehensive Metrics Most existing AutoML platforms only focus on the model performance achieved during the training phase, ignoring the deployment phase. OmniForce bridges this gap with the task sidecar. The model developed in the training phase is sent to deployment environments such as Qualcomm A650 [55] and the NVIDIA Jetson Develop Kit [56] to benchmark the model performance in production. The metrics on the deployment side, such as latency and power, are collected to compute a comprehensive reward for the next round of model searching. Model Robustness Model robustness is the ability to resist external disturbances, which is a prerequisite for widely using AI. A robust model can produce stable outputs and adapt to various environments in production. This feature is quite critical in some applications, such as healthcare, finance, and security. Model robustness is closely related to the underlying data distribution. Recent research has found that a slight data deviation may causing the associated model to give completely different results, highlighting the fact that current AI models are too sensitive to data. In production, these problems may be encountered accidentally, such as by a natural data distribution shift. However, they may also be intentional, such as hacking attacks. OmniForce provides a model robustness evaluation service to evaluate a model's ability to resist noise attacks. Users can choose the model to be deployed in production according to its comprehensive accuracy and robustness performance. Additionally, OmniForce provides a robustness evaluation tool, which includes two popular evaluation methods: model adversarial attack evaluation [57,58] and model privacy evaluation based on membership inference [59,46,58]. The former evaluates the model's robustness under adversarial examples, and the latter evaluates the model's data privacy under membership inference. Pipeline-Driven Training and Deployment Collaboration for Crowd-sourcing After uploading and processing the given dataset on OmniForce, to generate an industrial application, users need to upload their own models or choose a crowdsourcing model recommended by the OmniForce formatter. Then, OmniForce organizes the entire process from training to deployment through an automated pipeline. In particular, based on the pipeline-driven approach, the adaptation and miniaturization of large models can be achieved spontaneously. When searching for a new model, an engineer or data scientist completes the algorithm interface according to the document and configures the corresponding search space design, running device (CPU, GPU), and optimization metrics. Application Algorithm Interface A new program tracer is implemented in our design, as shown in Listing 1. To crowdsource a new model, users need to complete the interface and wrap several functional blocks of Python code. Considering the desire for a user-friendly interface, complex message sending and receiving operations are implemented internally. Listing 1: Python user application algorithm interface for crowdsourcing with the decorator. Application Algorithm Configuration Search Space Configuration Hyperparameter tuning is a crucial step when generating and deploying AI algorithms for industrial applications. For the new model search task, OmniForce allows users to tune their hyperparameters and customize the search space. In our configuration workflow, search spaces can be configured via user-friendly interactions on the front end or by uploading YAML files for complex spaces. Various tasks may have very different definitions of search spaces. Generally, in deep learning models, there are always dependencies among the parameters. For example, when searching the network structure of a neural network, we usually want to explore the network's depth and width. Depth indicates how many convolutional blocks or layers are in the backbone network, while width always refers to the number of channels per block or layer. Therefore, the length of the channel array to be sampled depends on the sampled depth value. If the depth is three, three channel values are sampled from the search space, as shown in Listing 2. Another complication concerns conditional space. For example, suppose that different kinds of blocks with various parameters are searched in a deep model. In that case, the parameters to be sampled are determined after sampling the types of the corresponding blocks. These situations frequently occur in the search spaces of various algorithms. Therefore, OmniForce supports tree-and DAG-based search space sampling rules, modeling the dependencies among the parameters as a graph. Inference Configuration OmniForce requires users to provide corresponding configurations during the batch inference phase of crowdsourcing, including the number of inference devices and inference resource usage. These two metrics ensure that OmniForce can build an inference environment for large amounts of data. Deployment Configuration During the deployment phase, OmniForce also requires users to provide appropriate configurations to quickly deploy their models into production. These configurations include the deployment devices, deployment resource usage, and single-sample inference latency. The latency can help us deduce the QPS after the model is deployed. Cloud-Edge Collaborative Training Environments and Requirements OmniForce proposes a novel AutoML practice on the basis of a cloud-edge collaborative framework. In this way, users are able to develop the most suitable models for their specific devices under both performance and latency metrics by installing the OmniForce cloud-edge collaborative python package and registering their devices. In addition, when training large models, OmniForce can support the interaction between the production environment and the supercomputing environment. Inference Optimization Due to the lack of deployment considerations, many AutoML frameworks consider only the performance of the resulting model on the search side and ignore the performance of the model in the actual deployment environment. On edge devices such as ARM [60] and ROCm [61], OmniForce addresses this issue by using TVM [62], a deep learning compiler that enables high-performance ML anywhere. We incorporate TVM to establish a connection between the training and deployment environments through the task sidecar and complete the collaborative search process of the AutoML task during the training and deployment tests through the relay method. This comprehensive search strategy helps OmniForce find models that excel in production. OmniForce also uses different tools for specific devices, such as TensorRT [63] for NVIDIA GPUs and OpenVINO [64] for Intel CPUs. Moreover, to convert our model between different machine learning frameworks, OmniForce uses ONNX [65] as an intermediary. In these scenarios, we convert the model of the high-level framework such as PyTorch into ONNX format, a common file format for machine learning models, and then further convert it into TensorRT or OpenVINO for targeted optimization of deep learning inference on different devices. Application Algorithm Register After users implement their application algorithm with our standard interface and prepare the configurations well, this algorithm can be sealed in a self-contained docker image and registered in the OmniForce crowdsourcing system. Our system conducts a smoke test to validate the completeness of the algorithm, which has three main stages: searching, inference, and deployment. 1. The search test verifies whether the given algorithm can be searched using OmniForce's AutoML search strategy. The search space configuration is checked during this phase. 2. The inference test verifies whether the algorithm can perform offline batch inference. The inference configuration is checked in this phase. 3. The deployment test verifies whether the algorithm can be deployed and generates the corresponding API. The deployment configuration is checked in this phase. In general, OmniForce evaluates four capabilities of the tested algorithm: searchability, batch inference, cloud deployment and edge-side optimization. Notably, all the above tests are not required to pass, but passing as much as possible is recommended so that the algorithm can be applied in more scenarios. For example, if an algorithm only passes the searching test, it can only be used for hyperparametric optimization but is not able to be deployed in a production environment. After passing the smoke test, users' algorithms are crowdsourced on OmniForce with version control. Both docker images and configurations are versions controlled for agile updating in the future. Pipeline Generation Analogous to the AutoML pipeline described in Section 2.1.7, the OmniForce crowdsourcing system adopts a pipelinedriven method to automatically apply the provisioned and crowdsourced algorithms. The generated pipeline can flexibly support different tasks. For example, a user might need a model that is empowered by a large model to run on devices with limited resources, such as edge devices. OmniForce may use a pipeline that consists of two steps, including large model adaptation and multiobjective optimization, to generate efficient models. (Large) MaaS Leveraging the power of large models, recent trends in the AI community have shed light on the performance improvement yielded by scaling. Large models have achieved success in AI products such as ChatGPT [14] and Pathways [34]. However, large models bear massive memory and computation consumption burdens. Practicability may be hindered when large models are deployed on edge devices or applied to situations with limited computation and memory resources. Especially when the application scenario requires low latency, such as autonomous driving, a large model with low inference speed cannot guarantee accurate online prediction, which inevitably induces safety problems. Due to their large capacities and state-of-the-art performance, large models exhibit high generalization across a large number of tasks. In practice, this usually comes with a high pretraining cost, making these models unsuitable for applications involving frequent model adaptation and update steps in industrial cases. Therefore, when using large models in these domains, there is a need for adaptation and miniaturization procedures that increase the iteration and inference speeds of the models while retaining their performance to the maximum extent possible. OmniForce supports large model adaptation and miniaturization technologies, mainly including automatic adaptation, filtering, and knowledge distillation with model inference optimization, as shown in Figure 5. Figure 6. According to the system meta knowledge and the user's interactions, OmniForce decides whether to use large or small models and when to reuse or update models in practice. Large Model Support Adaptation With the development of high-performance accelerators, the sizes of models have grown exponentially [4]. Due to the lack of labeled resources, to train such large models, some self-supervised methods have achieved great success, such as MAE [66] and masked language modeling [4]. The resulting self-supervised pretrained model is then fine-tuned to adapt to the specific task and dataset. Some well-known pretrained checkpoints, such as bidirectional encoder representations from transformers (BERT) [67] and Clip [68], have become popular, facilitating a range of downstream applications. However, under this trend, given a realistic downstream dataset, it is difficult to train a model from scratch or frequently fine-tune an entire large model for each task due to the massive computational and storage costs required. To solve these problems, some effective parameter tuning methods have aroused the interest of researchers. Three of the most impressive branches of this field are Prompt [4], Adapter [69], and low-order decomposition [70], which are supported in OmniForce. In addition, with the help of the knowledge base maintained by the system, the formatter 2.1.6 can automatically create a suitable pipeline to select and search for adaptation modules that are appropriate for the downstream tasks and the large model. After the adaptation process, the large model serves users through a simple API. Miniaturization While large models have shown great power in many tasks, certain situations, such as real industrial environments, require running models with limited resources, such as Internet of Things (IoT) edge devices. In this case, the inference time of a large model may take up a large part of the overall system, leading to a large response time. Therefore, developers prefer a model that forms a tradeoff between performance and speed. OmniForce supports two ways to generate efficient models powered by large models. One way is to minimize a high-capacity model under downstream tasks, and the other way is to transfer knowledge during the pretraining (upstream) stage from the large models. Miniaturization in the Downstream Stage Filtering has been a popular technique for reducing the parameters and computation costs of large models. Current methods mainly learn to drop some unimportant connections or channels, which is also known as neural network pruning or compression [71]. The filtering variable can be represented by a binary mask, where 1 indicates that the corresponding connection or channel is kept; otherwise, it should be pruned. A mask is learned to induce sparsity, which leads to smaller numbers of parameters and computations. After filtering the unimportant connections and channels, the smaller model is usually fine-tuned on the original training task to restore Given input meta-data and the corresponding data, OmniForce automatically creates an AutoML pipeline from selecting an algorithm to deploying the final output model, which may be a large model or an efficient model. Large models and their derived adaptive models can be served through the API. To meet the needs of different objectives, the derived adaptation model can be automatically generated by a combination of one or more adaptation methods. The miniaturization approaches enable efficient model generation for edge devices. OmniForce provides the optimized and accelerated model as an API to developers, lowering the barrier to using AI technology. the performance drop. Recently, filtering studies on transformers have learned a sparse mask to keep only a small portion of the total tokens to induce acceleration [72]. Knowledge distillation is developed to distill the rich knowledge of a pretrained model, named the teacher, into a new model, named the student [73]. Generally, the teacher model is a large model with a large number of parameters. It has a slow inference speed but enjoys high performance. The student model is a small-capacity model formed by handcrafted design or NAS [74,75,76,77]. It runs fast, but directly training it will not yield a satisfactory performance. Adopting the knowledge distillation technique [78], we use the large model as the teacher and train the small model with extra supervision from the representation output obtained from the last layer of the large model. The small model trained with knowledge distillation performs better than its original version. We can enjoy low latency and high performance when deploying the small model on edge devices. Miniaturization in the Upstream Stage Small models are always limited by their low capacity to absorb knowledge from large datasets, while large models trained with massive data have better transfer capabilities for downstream tasks. However, pretraining a large model with large datasets is costly. Performing distillation during the pretraining stage is a novel way to tackle this problem [79,80]. OmniForce supports the miniaturization method by using the knowledge distillation technique in the pretraining phase without training large models from scratch. Such a pipeline is set up in OmniForce to facilitate the sharing of large models and generate fast and frequent iterative models. Flexible Search Strategy Framework The search strategy is one of the most important parts of an AutoML technique, as it conducts the entire search process given a search space. A well-designed search strategy tends to be the key to efficiency and efficacy. In this subsection, we introduce the interface of our search strategy to show what role it plays in our framework as well as a flexible search framework in OmniForce. Search Strategy Interface In our framework, without any other additional restrictions, the interface of the search strategy consists of two functions, generate_tasks and handle_rewards, and a search space object, as Listing 3 shows. The former function is used to generate the hyperparameters that are most appropriate for the given space according to observations. When all tasks are completed or timed out (see Section 2.1.1) during the current iteration, the latter function is called to handle the rewards of these tasks and update the observations for the next iteration. Listing 3: Search strategy interface. c l a s s S e a r c h S t r a t e g y ( metaclass= ABCMeta ) : """ This is the abstract base class for OmniForce search strategies. """ Search Strategy Framework OmniForce supports a variety of search strategies such as revised hyperband [32], MF-NAS [33], novel BO and evolution approaches. BO is a sample-efficient method that aims to find x * = arg min x∈χ f (x), where f is a black-box function that is expensive to evaluate and χ is the search space or domain [81]. BO consists of two main components: a surrogate model for modeling the response surface of f and an acquisition function forming an exploitation-exploration tradeoff. While many libraries have been developed for BO, such as Spearmint [82], GPyOpt [83], scikit-optimize [84], RoBO [85], ProBO [86], GPyTorch [87] and BoTorch [88], they all focus on exploiting a certain aspect, and there is no one inclusive framework that can hold them all. For example, some advanced batch BO methods, such as local penalization (LP) [89], cannot be implemented easily in the methods mentioned above except GPyOpt. To accommodate increasingly advanced BO algorithms, OmniForce utilizes a composable BO framework that maintains five main component sets, including a surrogate model, an acquisition function, an acquisition optimizer, a candidate generator, and a suggester. Similar to BoTorch, our framework is built on Pytorch [90] and benefits from autodifferentiation and GPU acceleration, and the overview of our framework is shown in Figure 7. For the surrogate model, we implement the most popular methods, such as GPs [82], SMAC [91], BNNs [92,93], and a novel method (OF) to address discrete optimization problems. Our proposed OF surrogate model with the OF acquisition function achieves state-of-the-art accuracy on NAS-Bench-201 [94]. Moreover, considering multitasks and high-dimensional optimization problems, we design OF-Trans and OF-HD, respectively, and implement some existing methods, which is unavailable in other BO libraries. Regarding the acquisition functions, in addition to popular basic functions such as the expected improvement (EI) [95], lower confidence bound (LCB) [96] and entropy search (ES) [97] functions, we also support some powerful batch acquisition techniques such as LP and Monte Carlo acquisition functions. To provide a batch of new quires, some additional logic is usually needed, which inspires us to define a new component in the BO framework called the suggester. In the inference stage, the suggester takes over all other components and conducts the generation process for new quires. For example, in most batch BO settings, we need to look ahead to those pending tasks and fine-tune the surrogate model using MC samples [82] or constant liars [98] to obtain new quires. To optimize the acquisition function, we introduce a candidate generator to sample sufficient candidates as the starting points for the lbfgs optimizer, which is the most common choice in the BO context. Different sampling methods can be easily plugged in, and the sampling space changes with the search space to adapt to some search space shrinking methods. During the training stage, we use the observations to fit the surrogate model via MCMC or gradient-based methods. When a trained surrogate model is given, the recommender controls the generator to generate candidates and optimize the acquisition function according to the posterior and the acquisition optimizer. Then, with the new queries suggested by the requester sent to the database, the parallel workers reserve these new tasks for evaluation purposes and send the observations back for the next iteration until the budget is exhausted. In conclusion, compared to other existing BO libraries, our contributions to the BO framework are highlighted as follows. • Flexible and composable. • Novel methods for several problems. • Autodifferentiation and GPU acceleration. Widely Provisioned Application Algorithm AI applications can be found in industrial production and our daily lives. This ubiquity of AI is reflected not only in the variety of available application scenarios, including cloud-edge collaborations, VR integration, and open-environment AI, but also in the diversity of the utilized application algorithms, such as table data analysis and processing, NLP, computer vision, AIGC, and graph representation learning. As scenario diversity has been introduced above, this chapter mainly focuses on the widely provisioned application algorithm in OmniForce. Tabular Data Tabular data analysis is a long-standing topic that performs association analysis on structured data and mines complex business relationships through feature combination and feature extraction. As an AI application under the HAML framework, OmniForce encourages users to focus more on data collection, custom feature pipelines, and model design, which can yield improved performance through the human experience. For fundamental exploratory data analysis, data cleaning, null filling, category coding, and other tedious but necessary procedures, meta-learning-based feature engineering automatically completes the above tasks to reduce the burden imposed on humans. In general, automatic feature engineering is mainly divided into three parts: cleaning and filling, feature transformation, and feature combination. OmniForce provides various standard processing methods for different Figure 7: The BO framework of OmniForce. Five key component sets are contained in our framework: a surrogate model, an acquisition function, an acquisition optimizer, a candidate generator, and a suggester. In addition, we divide the BO procedure into three steps. During the training stage, we train the surrogate model to fit the observations via gradient descent or MCMC methods. Then, during the inference stage, we suggest finding the optimal candidates as new queries with the trained surrogate model and the acquisition function. During the evaluation stage, as the new queries are sent to the database, we use the parallel workers in OmniForce to evaluate these candidates and update the observations for the next iteration. types of features as search spaces and then obtains the most suitable processing methods for the given data. For example, for numerical features, the methods used to filling empty values include taking the mean, median, upper quartile, and lower quartile; feature transformation methods include max-min, z score, and log scale normalization; feature combination methods include multiplication and division and other conventional mathematical transformations. Ensemble models based on decision trees, such as random forests [105] and the LightGBM [106], have always been favored by the industry due to their low computational costs and high interpretability. In recent years, breakthroughs in neural networks have drawn attention to deep learning models that have achieved impressive results in tasks such as recommender systems and click-through rate prediction [103] [107]. We use both tree-based ensemble models and deep learning models to construct a search space to ensure excellent performance on various challenging datasets. Large Tabular Data In the current era of big data, ML algorithms are used to analyze massive amounts of data, serving human life and industrial production and bringing tremendous value [108]. Therefore, the ability to efficiently train models on large tabular datasets is a highly competitive issue for IT companies. Compared to training a deep model on a GPU, reading hundreds of GB of data from a disk into memory is redundant and highly time-consuming for each worker/task estimator. Therefore, we propose a systematic design named double shared memory to speed up the process of training on large tabular data in a node. On the one hand, we save one piece of training data to /dev/shm of the Linux system, mount it into multiple worker containers, and use the high-speed access memory to quickly load the data. In this way, the task estimators of the same node can share the same data in memory, which we call the outer shared memory. On the other hand, due to automatic feature engineering, the data used by each task estimator are different. To avoid the time cost of feature engineering, we enable a sharing mechanism called inner shared memory. The first epoch performs feature engineering and model training in a batch-by-batch manner and writes the processed data into the container, which allows subsequent epochs to save time during their file loading and feature engineering processes. This efficient memory sharing approached used for data loading can be widely applied to deep learning models such as multilayer perceptrons (MLPs). In addition, for nodes with small amounts of memory, the scheduler can use only inner or outer shared memory to flexibly accelerate the training process. Time Series Data As one of the most challenging tabular data problems with numerous applications, time series forecasting has been one of the primary research areas that the AI community has attempted to solve with ML and deep learning [109] [110]. Adhering to the human-centric philosophy, we open up interactions such as time series intervals and forecasting time lengths. Users can create customized models based on data characteristics and business scenarios according to their experience. Likewise, automated feature engineering cleans and fills the data, identifies multiseries cases, groups them, and generates temporal features such as sliding time windows and lags. The search space includes traditional autoregressive models [111] tree-based models[9] [106], and the recently proposed transformer-based models [112]. The formatter in subsection 2.1.6 assigns weights to the models for datasets with different statistics according to the available historical information to ensure the optimal performance. Computer Vision Computer vision, as the main research field of AI, aims to extract, process, and understand the information contained in digital images and videos. Traditional computer vision algorithms, such as support vector machines (SVMs) [113], usually solve specific tasks via handcrafted feature engineering. Benefiting from their data-driven learning scheme, deep learning algorithms have achieved great success in various computer vision tasks. On the one hand, end-to-end training and inference strategies enable deep learning models to be easily adapted to different computer vision tasks. On the other hand, with the development of deep neural network architectures (e.g., AlexNet [114] and ResNet [1]), the capability of deep learning models is growing quickly, and an increasing number of deep learning models are being employed in real-world applications. Although deep learning models, especially deep convolutional neural networks (CNNs), have demonstrated promising performance in computer vision tasks, there still exists a relatively large gap for these models to become robust and generic computer vision models. Recently, supported by the rapid growth of computational resources and massive amounts of visual data, super-deep models have attracted increasing attention from the computer vision community [115,116,117,118,119,120]. Relying on their powerful learning capacity, super-deep models can learn general and discriminative representations. Additionally, their strong modeling capacity also enables super-deep models to adapt to a new scenario with a small amount of labeled data [117,118,120]. Such an ability is essential in real-world applications. For example, we can validate the feasibility of technical solutions with fewer costs and thus accelerate the development cycle. To help users quickly develop and deploy models for specific applications, we provide various super-deep vision models for different vision tasks, e.g., 2D/3D object detection [121,122], semantic segmentation [123,124], road and lane detection [125,126], image matting [127,128], keypoint detection [129,118], scene text detection and spotting [130,131,132]. By providing task descriptions and some labeled samples, users can easily obtain high-performance task-specific models from the platform. Moreover, super-deep models usually require large resources for deployment, while users prefer lightweight models with fewer parameters and higher inference efficiency. To satisfy such requirements, we provide several solutions to compress and speed up these models. More specifically, we develop efficient model compression techniques, such as quantization-based and pruning-based techniques, to compact and accelerate the models. Moreover, relying on a designed search space containing various operations, blocks, loss functions, etc., our platform can automatically search a lightweight model and improve the performance of the searched model with the guidance of the super-deep models via knowledge distillation. Natural Language Processing Natural language processing (NLP) is one of the major branches of AI that aims to automatically process human languages (both spoken and written) with computers, and the tasks involved are often classified as cognitive intelligence. NLP has evolved from several disciplines, e.g., computer science, AI, and linguistics. NLP can be basically divided into two categories: 1) Natural language understanding (NLU). NLU explores the strategies that enable computers to grasp textual instructions provided by human users. The most common NLU tasks include text classification [133], sentiment analysis [134,135,136], question answering [137,138], named entity recognition [139,140], etc. 2) Natural language generation (NLG). NLG allows computers to generate textual outputs after understanding user inputs in natural languages such as English and Chinese. The common NLG tasks include machine translation [141,142,143,144,145], summarization [146,147], dialogue [148,149], etc. Although deep learning-based NLP models have demonstrated promising performance on a series of tasks, the existing learning paradigm lacks the capacity to leverage many tasks and data, leading to a series of issues, e.g., model training redundancy for different tasks, data island problems, complex deployment problems, and poor learning ability in low-resource scenarios. To help users efficiently and effectively develop and deploy models for different applications, our system consists of a series of foundation models, including a super model for general language understanding and generation [150,151,152], a super model for cross-lingual generation [153,154,155,156,157], and their efficient tuning and distillation versions [158], distillation-based prompt learning [159] and PESF-KD [160], respectively. By providing the task description and a few labeled samples, the users can obtain a high-performing model/small adapter tuned by our built-in foundation models from the platform. Encouragingly, with our efficient tuning strategy of distillation-based prompt learning, our server-end foundation model requires finetuning only 0.5% of the parameters, which is on par with the parameters of the original foundation model, while achieving comparable or even better performance. In this way, the users only need to deploy their small prompt/adapter, which is an efficient and private way to incrementally update the user-end small prompt/adapter without uploading their precious data to our platform. Learning Multimodal Deep Generative Models One of the main objectives of artificial intelligence and machine learning is to learn and manipulate high-dimensional probability distributions of real-world data [161]. By doing so, these technologies can extract valuable insights from data that can be used to improve many related tasks [162]. In recent years, deep generative models have emerged as a powerful means of learning data distributions. These models, which include generative adversarial networks (GANs) [163,164,165,166], Vector-Quantized Variational Autoencoders (VQ-VAEs) [167,168], autoregressive models [169,170], and diffusion models [171,172], have demonstrated impressive capabilities in a wide range of applications. By learning the underlying probability distribution that generated the data, researches can gain insights into the underlying mechanisms of the data-generating process. Furthermore, well-trained generative models can be widely used in content generation-related tasks. Our system consists of a series of built-in deep generative models, which have been designed to improve the realism of generated content and deliver a generative model that can handle general content generation tasks. By learning the feature alignment between different modalities, these models can generate more diverse, high-quality content.To achieve these goals, we have developed a set of advanced algorithms that can train and apply these generative models to a variety of real-world applications. These algorithms have been designed to enhance the performance of the generative models in tasks such as visual concept exploration, generation controllability, and content diversity. Overall, our built-in models and algorithms have been used in several artificial intelligence-generated content (AIGC) tasks, including vision-language generation [173], complex scene generation [174], portrait animation [175,176], 3D object rendering [177], etc. We believe that our system has the potential to revolutionize the field of content generation and pave the way for new, innovative applications of AIGC technologies in the future. Graph Representation Learning Graph data are all around us; examples include social graphs, knowledge graphs, and protein structures. Typically, a graph consists of nodes and edges that connect the nodes. Even sentences and images can be represented by graphs: the words in a sentence and the patches of images can be treated as nodes, and the connections between nodes represent their edges. Considering that graphs are ubiquitous, it is important to analyze graph data and learn graph representations for solving node-level, edge-level, and graph-level applications. Graph neural networks (GNNs) are neural networks that operate in the graph domain. Our systems provide a variety of built-in GNNs, such as graph convolutional networks (GCNs) [178], graph attention networks (GATs) [179], and graph transformers [180]. For example, our systems provide a type of efficient graph transformer, i.e., Gapformer, that deeply incorporates graph pooling into a graph transformer. By using Gapformer, the negative impact of having several unrelated nodes is minimized while long-range information is preserved, and the quadratic complexity of message passing is reduced to linear complexity. In addition, our systems develop many diverse plugin modules to improve the capabilities of GNNs. For instance, our systems adopt SkipNode [181], which samples graph nodes in each convolutional layer to skip the convolution operation, thereby alleviating the oversmoothing and gradient vanishing problems of GCN-based networks. Our systems also use a plug-and-play scheme for graph pooling, referred to as MID, with a multidimensional score space and two score operations, to explore the diversity of the node features and graph structures in graphs to achieve improved graph-level representations. Furthermore, our systems also provide typical graph applications, including network structures that are designed for learning on signed network embeddings [182], GCNs with multilevel learning for hyperspectral image classification [183], and heterophily networks for scene graph generation [184]. Features Ease of Use for Development-Deployment Collaboration Committed to building models that are suitable for production, OmniForce devises a veritable development-deployment collaborative model construction framework. Unlike many AI platforms that only have the ability to release the model in production in an agile manner with a CI/CD pipeline, OmniForce bridges the development and deployment environments and adopts a multiobjective optimization method to construct more practical and versatile models in the searching and training phases. OmniForce aims to enable both developers with limited ML expertise and data scientists to deploy their own model services with only a few clicks. Industrial Availability of Open-Environment Adaptation Unlike conventional AutoML platforms, we propose OmniForce to study open-environmental and open-loop problems because data, labels, features, models, evaluations, and metrics usually change during the learning process in practice [185]. Our intuition is that we need to involve people in the loop for leveraging human knowledge and enhancing human capabilities based on the smooth interactions shown in Figure 1 to achieve the goal of HAML. Cloud-Native Production and (Large) MaaS An increasing number of large-scale systems are being built through containers and equipped with a container orchestration system to manage all components, as these system generally have the advantages of high resource utilization, strong isolation, and continuous delivery. OmniForce is designed based on Kubernetes, which means that OmniForce is a fully cloud-native AutoML system and can leverage many excellent cloud-native tools. Based on Kubernetes and Kubeflow, OmniForce supports multitenancy, high scalability, strong disaster recovery capabilities, and automated transformation from a trained model into a deployment service. Crowdsourcing OmniForce supports large-scale algorithms and extends the set of applied algorithms. With a system that was widely used among a group of engineers, it was previously possible to directly search and deploy a new task on the new dataset through crowdsourced application algorithms. To inspire the concept of crowdsourcing, we start with ML version management for data, labels, models, algorithms, and search spaces; pipeline-driven development and deployment collaboration; a flexible search strategy framework; and a broad offering of applied algorithms that include super-deep (large) model-based methods. Evaluation This section contains three parts. The first subsection gives a brief introduction to the innovations provided by the industrial metaverse. The second part presents a set of use cases in a human-centered real-world industrial metaverse scenario, showing the practical operation of XR simulation, continuous data acquisition, crowdsourcing, and cloud-edge collaboration to solve open-loop AI problems. Finally, we demonstrate the capabilities of OmniForce through some experiments on scalability, fault tolerance, search performance, algorithm performance, and a human-centered AutoML practice. Innovations of the Industrial Metaverse Standard assembly lines have employed AI technologies to improve the production efficiency of a single factory. Recent advancements in industrial metaverse technologies will further the application of AI to the next level and change the structure of the entire supply chain. The industrial metaverse has formed a new manufacturing paradigm, encouraged collaboration between factories, and accelerated the connection between the upstream and downstream parts of the industrial chain. For example, in the conventional customer-to-manufacturer (C2M) business model, manufacturers produce small market testing batches before products' final releases and then improve their product designs based on market feedback or increase production when products sell well. However, the emergence of the metaverse will change the structure of the conventional C2M business model. With the development of immersive experiences in the virtual world, digital content from the real economy will be used as primary data to help construct the digital world. Technology has reconstructed the form of the existing industrial chain. The metaverse allows consumers to experience products and make purchasing decisions during the product design stage, allowing manufacturers to obtain more detailed feedback in this stage. They improve their product designs based on customers' feedback and even sell digital content services (new revenue streams). In addition, the emergence of the metaverse will cause changes in the supply and demand structure. It will shorten the distance between manufacturers and customers by eliminating the intermediate steps. In the next part, we provide a case of using OmniForce to help manufacture self-driving vehicles in the context of the industrial metaverse. In the design stage, user experience and feedback are involved. In the development and manufacturing stage of the metaverse, OmniForce uses cloud-edge collaboration technology to conduct rapid product iteration as well as feedback to improve the performance and user experience of the product. Additionally, OmniForce assists in automatic order disassembly, automatic order placement, supply chain sourcing, and intelligent stocking in the whole supply chain service. User Case of the Industrial Supply Chain and Industrial Metaverse Design In this case, manufacturers aim to design self-driving cars that are suitable for different scenarios. For example, citizens from different cities may prefer diverse colors and patterns in their cars. Some small shapes may be better suited for delivering packages and driving around communities than massive trucks. This mission can rely on OmniForce's AIGC capabilities. In this generation task, the client wants the tool to automatically generate car models. After a car model dataset is fed into the system, OmniForce outputs a model to generate content. The resulting car model can be simulated and checked by an XR system with customer interaction. This step shortens the process of consumer feedback and is a way of implementing industrial metaverse design programs, saving time and budget resources. If the obtained results do not meet the requirements of consumers, the user can adjust the reward of the model on OmniForce, triggering the model production loop again until the results can be used for the next step. Development and Manufacturing In this case, the customer is from an advanced delivery company with new technology equipment empowered by AI. They want to run a set of models across different scenarios on their smart vehicle, which involves massive devices with different deployment (inference) requirements in this real industrial scenario. For example, they needed three models with different requirements: • a truck traveling between cities; • a pickup truck driving on the city's main streets; • and a microdelivery van traveling across communities and buildings; Specifically, when driving on highways between cities, trucks can reach speeds of 80 km/h or higher in relatively clean road conditions. Therefore, the constructed model needs to respond quickly and responsively to obstacles in front of the truck. While running on main streets, the model deployed on the pickup truck should handle complex road conditions, such as pedestrians, bicycles, motorcycles, and pets. In contrast, tiny courier vehicles usually travel across communities and buildings at low speeds but may drive on icy roads under extreme weather. To address these challenges, OmniForce constantly applies an automated pipeline including data collection and model searching to extract values from the development, deployment, and maintenance phases. To collect the data, we can collect a set of example images from public datasets or the real world. Usually, we need some data requirements to make the AI model learn well. One requirement is that the scenes and the types of cars, pedestrians, and trees on the road should be divergent. After uploading the data to OmniForce and obtaining the resulting dataset, we may find that "sedans" or "city roads on sunny days" appear much more frequently in the images than other types of cars or scenes, leading to a long-tailed distribution problem. Therefore, the human-centered open loop is triggered, and the cue from OmniForce is that users need to constantly collect other types of data, such as sports cars, vans, and limousines. OmniForce can also generate images in different types of scenes that include highways, crowded bicycles, motorcycles, wet roads on rainy days, shadows of trees, and children in neighborhoods. As new data are acquired, OmniForce updates the version of the data to satisfy the imposed balance requirements. To search for models under different constraints, OmniForce supports cloud-edge collaboration for developers with limited ML expertise and data scientists to adjust multiple objectives such as accuracy, recall, power, and latency, modifying the evaluations and rewards to automatically produce various models. After generating the model, we can validate the model in a simulated system where data are collected by drivers in XR applications and then test the model in some small-scale scenarios before deployment. When model drift occurs, OmniForce involves people in the loop for checking the data, collecting data if needed, re-searching or retraining the model, and updating the model's version. Supply Chain Service In the process of the industrial metaverse, intelligent warehousing and stocking realize a fast supply chain and reduce the raw material budget and production time. That is, people can use automated ML technology to obtain an efficient supply chain service, including ordering, storage, transportation, and marketing. OmniForce supports the analysis and prediction of tabular data and time series data. After automatically generating models with Figure 9: Four trials concerning fault tolerance. These trials achieve comparable performance with various fault tolerances. Yellow bars show defective candidates, while blue bars represent surviving candidates. We mark the number of surviving candidates and the resulting accuracy next to the bars. OmniForce, some simulation systems, such as cellular automata and small-scale tests, can be used to validate the model at a relatively low cost. Based on the feedback, visualizations and explanations of the searched architectures and hyperparameters, OmniForce provides a convenient interface to bring people in the loop, guiding them to tune the search space, refine the metrics, collect the combined data, adjust the simulation system, trigger the next model production cycle, and update the version of the data and model. During this process, algorithms can be implemented by a crowdsourced knowledge base. By standardizing the data abstraction processes, application algorithms, and search spaces, OmniForce makes it easy to integrate and reuse application algorithms and search spaces. Furthermore, users can learn the knowledge of ML pipelines from OmniForce. Empirical Results Scalability and Fault Tolerance OmniForce supports scalable search jobs assigned by the scheduler. As shown in Figure 8, the experiments run with different scalable resources and parameters on the same dataset and use two search algorithms: an evolutionary algorithm [186] and Hyperband [187]. In large-scale testing cases, larger groups with more computational resources can achieve the same accuracy in less time than those with fewer resources. For example, experiments conducted on 64 GPUs take approximately one hour to achieve above 90% test accuracy. In contrast, experiments conducted on 16 GPUs take more than four hours to obtain the same accuracy. Hence, the models trained on 16 GPUs cannot reach good performance under the time constraint shown in Figure 8 (b). Additionally, experiments conducted on 16 GPUs and 64 GPUs take similar amounts of time to achieve low accuracy since reaching such an inferior level of performance is not challenging for a search job. It can be seen that in Figure 9, different failure rates have little effect on performance. The four experiments achieve comparable performance but with different candidate fault rates. In these experiments, we add perturbations to kill some surviving candidates. With our carefully designed semisynchronization scheme, jobs can continue to run under a limited number of dead candidates or be paused and restarted during a search. Search Performance Based on our proposed BO framework, we design a novel BO method that is well-suited for discrete optimization problems. We compare our method with various NAS algorithms and BO methods on a popular benchmark (NAS-Bench-201 [94]), which contains 15625 network architectures and their evaluations on three visual classification datasets. Following the setting of [94], we search on the CIFAR-10 validation set after 12 epochs of training and then directly look up the evaluations in other datasets. We run these BO methods for 80 iterations with 12 initial points and report the mean and standard deviation of the best observation encountered during the search process across 10 duplicated runs. Table 1 illustrates that our method achieves the best performance on all three datasets. Algorithmic Performance Here, we provide the performance of the models provisioned by OmniForce. First, we show our state-of-the-art scaled-up vision models (ViTAE [197,117]) and a comparison with other transformer-based deep models in Table 2. We find that with over one hundred million parameters, the models are able to achieve impressive performance. Relying on such powerful capacity, the models can perform well on new scenarios after being fine-tuned with few labeled data. As a result, users can validate new methods and quickly deploy models. In addition, we also report the NLP performance of our built-in platform models on NLU (i.e., performance on the GLUE benchmark) and NLG (i.e., machine translation tasks). Table 3 shows the contrastive results obtained on 9 NLU tasks with one model, showing that our method can leverage any existing fine-tuned prompt to achieve better transfer learning performance. Importantly, with our built-in efficient approach, even better performance than that achieved with full model tuning can be attained by tuning only 0.5% of the original parameters, which is extremely critical for users to perform low-resource/low-cost training and deployment with only a few (or even zero) labeled data. Figure 10 shows the performance of our Vega-MT translation models, where we participate in 10 shared tasks, including Chinese↔English (Zh↔En), German↔English (De↔En), Czech↔English (Cs↔En), Russian↔English (Ru↔En), and Japanese↔English (Ja↔En). With our multilingual foundation model, we achieve 7 championships, 2 runners-up finishes, and 1 third-place result with respect to the BLEU points. A platform with Vega-MT can empower users with the ability to easily understand and generate any cross-lingual content. Notably, our platform also could speed up the language generation process by switching to our developed non-autoregressive generation algorithms [204,205,206]. Table 3: Results (%) of part of the cross-task efficient prompt transfer experiment based on foundation language models. Note that our method is model-agnostic and therefore can be used to enhance any foundation model. Here, we use BERT-Large as an example. In groups (a) and (b), each cell denotes the target task performance achieved when transferring the prompt from the source task (row) to the associated target task (column). "AVG." denotes the average performance across all target tasks. Notably, positive prompt transfers are in bold, and numbers in the subscripts indicate relative improvements. Finally, we report the comparison results of a complex scene generation experiment conducted with OmniForce. The quantitative results produced by the involved competitors on both the COCO-stuff and Visual Genome datasets are reported in Table 4. For a fair comparison, we adopt their officially released pretrained models or the officially reported scores in their papers. Compared with both CNN-based and transformer-based complex scene generation methods, TwFA [207] achieves significant improvements in terms of all metrics. Furthermore, since we employ the same texture tokenization strategy utilized in the transformer-based approach, HCSS [208], the generation performance demonstrates how well a transformer can model the compositions of complex scenes with focal attention. HAML Practice As mentioned above, both human-assisted ML and ML-assisted humans play important roles in an AutoML system. This section takes a Kaggle competition-Test Time Cost Forecasting for Mercedes-Benz-as an example to introduce how the OmniForce platform implements the human-centric concept and achieves efficient human-machine interaction. The purpose of the competition is to forecast the time required for the user-defined auto to pass a safety test according to the provided anonymous dataset, thereby helping the Mercedes-Benz team optimize the test system. The dataset is automatically recognized as a tabular dataset by OmniForce after the user uploads it to the platform. Furthermore, 1st! 1st! 1st! 1st! 1st! 1st! 1st! 2nd" 2nd" exploratory data analysis reports are generated, which helps humans further explore the statistics of the data from all aspects. The relevant interfaces are shown in Figure 11 and Figure 12. Under variegated datasets and limited candidates over a complex search space, pure machine-based AutoML has difficulty finding the optimal solution in one iteration. Therefore, OmniForce introduces a multiround optimization process with human-machine interaction. In the first round, we directly input the raw data into the OmniForce platform and run the AutoML lifecycle, including automatic merge table generation, automatic data processing, automatic feature engineering, model search, and hyperparameter tuning, to obtain the baseline result. Regarding the lack of human intervention, the formatter described in section 2.1.6 configures the search space for regression tasks according to the knowledge base and then generates the AutoML pipeline to be run. An interactive search detail page is shown in Figure 13, including model performance, candidates' performance, parallel hyperparameter coordinates, etc. After obtaining the result by batch inference, we evaluate it on the Kaggle website to simulate the model's online service scenario. Our primitive results obtain an r2 score of 0.54956 on the private leaderboard, and the corresponding ranking is 1540 out of a total of 3823 teams (Late Submission). In the second round, we modify the search space by following the OmniForce advisor presented in Section 2.1.8, which provides suggestions for changing the search space and performing feature engineering. Humans can significantly improve the machine's efficiency based on the above suggestions and their own experience. Search process visualization is a long-standing topic in AutoML systems and has recently attracted widespread attention recently [215] [216][217] [218]. Furthermore, we argue that efficient ways to display and modify the search space are necessary prerequisites for humans to assist machines in HAML systems. On the search detail page, OmniForce shows the current candidates and search space in three different diagrams, which are shown in Figure 14 (full view), Figure 15 (dimensionality reduction view), and Figure 16 (pairwise view). Figure 11: Data detail page. The meta-data, data preview and version management interface are shown here. The meta-data contain data IDs, related feature pipeline IDs and file sizes. The uploaded data can be previewed on this page for further checking. Version management is used to handle rapid data iteration and update solutions in time. The initial state of the editing interface is the configuration suggested by the advisor. The OmniForce advisor summarizes some optimization schemes based on data statistics, the current search results, and the knowledge base. Most suggestions are general and portable; users can choose whether to adopt them or further improve them according to their experience. After interacting with the figure, a comparison between the modified and original search spaces is shown in Figure 17, which can be previewed and further edited. We adjust the search configuration above and obtain an r2 score of 0.5511 on the private leaderboard, obtaining a ranking of 820/3823, which illustrates the effectiveness of the advisor. Additionally, people can pay attention to every aspect of the search procedure to enhance the capability of the machine. The above three diagrams related to the search space can be directly edited by clicking or using the lasso tool. Namely, we can further reduce the max_depth of the tree and the ratio of row sampling at the cost of a few clicks, and the process for doing so is shown in Figure 18. After absorbing the valuable experience of humans, the search performance of the machine is further improved, with an r2 score of 0.55184 on the private leaderboard, and the corresponding ranking is 338. Next, the advisor provides suggestions for feature engineering based on the observed information by analyzing the training data and the importance levels of the features in the search procedure. Specifically, the suggested features are shown in Figure 19, and users can choose whether to apply the suggestions based on their experience. After completing feature configuration, the platform creates a new version of the feature pipeline and automatically applies the selected changes to the original data, as shown in Figure 20. After making the above modification based on the machine's suggestion, we obtain a score of 0.55284, and the corresponding ranking is 76. Figure 14: Parallel hyperparameter coordinate diagrams and the machine's suggestions. Users can obtain insights from the relationships between the hyperparameters and model performance. For the convenience of adjusting the hyperparameters, OmniForce provides a multilevel search space configuration interface next to these parameters. In the last round, users can unleash their feature engineering abilities to further achieve improved performance. Specifically, humans can enter the feature engineering interface through the 'Go to feature pipeline' button and modify the data based on experience by executing SQL statements or preset methods, as shown in Figure 21. Finally, we obtain a score of 0.55394, and the corresponding ranking is 11 out of 3823. As shown in Figure 22, all of the derived improvement points come from human experience and the machine suggestions, which illustrates the advancement of the human-machine interaction feature proposed by the OmniForce system. The above attempts are saved as different feature pipeline versions, search configuration versions, and model versions for users to switch between as needed. Related Work In recent years, some commercial companies have released their own AutoML platforms for industrial applications. Some identify market segments and develop different products for different tasks. For example, Amazon SageMaker Canvas [219] allows business analysts to build AI models and gives accurate predictions for tabular data in a no-code manner, while Amazon Forecast [220] employs a time series prediction service. IBM Watson AutoAI [29] supports the construction of classification and prediction tasks for tabular data, while Watson Natural Language Understanding [221] focuses on advanced text analytics. Others such as Abacus [222], Microsoft Azure AutoML [223] and Google Cloud Vertex AI [224] support different types of data and tasks in one platform. OmniForce chooses the latter option and targets a general and reusable AI application production pipeline. To break the isolated island of task automation and single-point optimization, OmniForce emphasizes human-centered operations; involves business people in the process Figure 15: Search space heatmap and the machine's suggestions. OmniForce maps the high-dimensional and multilevel search space to a two-dimensional space, which can clearly show the relationships between the candidates and performance. Incumbents represent candidates for achieving large performance gains, and regions selected by lasso regression represent suggested search spaces. Figure 3 : 3AutoML pipeline examples of OmniForce. (a) An example of a NoCode pipeline. (b) An example of a HPO pipeline. Each pipeline depicts a specific AutoML workflow in a directed acyclic graph (DAG) format, where every component executes in topological order. Each component launches appropriate cloud-native workloads to complete its work. Figure 4 : 4The AutoML workflow of OmniForce. The components of the OmniForce system interact with each other by following this workflow. ABCMeta ) : """ This is the abstract base class for OmniForce task estimators. """ @estimator.wrap d e f _ _ i n i t _ _ ( s e l f , run_epochs : i n t , data_path : s t r , i s _ t r a i n e r : bool , * * model_args , # Parameters customized for your algorithm. r a n s f o r m e r : m l p _ e x p e n d _ r a t i o : t y p e : c h o i c e range: { 1 , 2 , 4 , 8} Figure 5 : 5Illustration of the large MaaS paradigm in OmniForce. Figure 6 : 6Illustration of the large model workflow in OmniForce. Five blocks and different types of data interact in the workflow, including the model drift detector, large model training and updating, active learning, miniaturization and adaptation. A surrogate model is designed for efficiently training and updating large models. OmniForce provides an elegant and consistent AutoML technology to solve the complicated problems in each block. Figure 8 : 8Scalability of OmniForce with the evolutionary algorithm and Hyperband. The red lines represent the experiments conducted using 64 GPUs, while the blue lines represent the experiments conducted using 16 GPUs. Lines with various patterns show experiments with different precision results. The experiments conducted using 64 GPUs take less time to achieve the same accuracy than the experiments conducted using 16 GPUs. 3rd# Figure 10 : 3rd#10Vega-MT achieves state-of-the-art BLEU points on 7 out of 10 high-resource translation tasks among all constrained systems in WMT-2022 and significantly outperforms the competitive transformer-BIG baselines in terms of BLEU scores. Figure 12 : 12Data analysis page. OmniForce can help users gain more insights into data in an efficient manner by analyzing data based on automatically displayed statistical features such as histograms and means. Figure 13 : 13Search detail page. The details of the job, such as the data version and feature pipeline version, are illustrated on this page. The performance preview and detailed candidate information are also displayed in the charts. Figure 16 : 16Hyperparameter binary heatmap and the machine's suggestions. The binary hyperparameter heatmap reflects the interaction relationship between the two selected hyperparameters, which can help humans make finer adjustments. Figure 17 : 17Comparison between the modified and original search spaces. Row-by-row comparisons can clearly draw attention to search space modifications. Furthermore, users can directly modify the search space by using the syntax of YAML files. Figure 18 : 18The process of modifying the search space. We use balls to represent categorical hyperparameters and click them to change their value ranges; sliders are used to represent numerical hyperparameters, and the search space can be modified by dragging the sliders. Figure 19 : 19Feature importance and the machine's suggestions. The importance of all features is shown on the search detail page, and the advisor provides suggestions for further adjustments. Humans can choose whether to apply the suggestions based on their own experience. Figure 20 : 20Suggested feature construction method in the feature pipeline. All applied feature construction methods are automatically executed and updated to the new feature pipeline version, which is described in 3.1.2. Figure 21 : 21Customized feature pipeline. After inheriting the feature pipeline of the previous version, users can further create features based on their insight into feature importance. The version of the feature pipeline can be flexibly switched to compare the final performance or adapt to different scenarios. Figure 22 : 22Illustration of human-machine interaction in OmniForce. Based on the human-machine interactions in both the search space and feature pipelines, the HAML system can achieve better performance. Users can search and train either their own large models through the crowdsourcing interface or the large model provided by OmniForce. The method of invoking a model through the simple API and service of the large model lowers the barrier for users to access, thus shortening the cycle of developing and iterating AI products.The large model workflow is demonstrated inOmniForce supports large model technology. At present, AI faces a variety of industries and business scenarios and their needs; for example, people need to design neural architectures, adjust hyperparameters, and deploy models based on the hardware requirements for each specific scenario. The large model concept is a breakthrough technology for general-purpose AI that aims to solve the fragmentation problem of AI applications. OmniForce supports large model technology with highly efficient and uniform adaptation in computer vision 3.5.2 and NLP 3.5.3 tasks, as well as automated adaptation 3.3.2 and miniaturization 3.3.3. Table 1 : 1Top-1 test accuracy (%) for classification on NAS-Bench-201. The first block shows the results of parameter sharing-based NAS methods. The second block shows the results of nonparameter sharing algorithms and various BO methods. The third block shows the results of our proposed BO method. The † symbol means that the performance of the corresponding method is directly obtained from NAS-Bench-201 [94]. RSPS † [188] 80.42±3.58 84.07±3.61 52.12±5.55 52.31±5.77 27.22±3.24 26.28±3.09 DARTS-V1 † [189] 39.77±0.00 54.30±0.00 15.03±0.00 15.61±0.00 16.43±0.00 16.32±0.00 DARTS-V2 † [189] 39.77±0.00 54.30±0.00 15.03±0.00 15.61±0.00 16.43±0.00 16.32±0.00 GDAS † [190] 89.89±0.08 93.61±0.09 71.34±0.04 70.70±0.30 41.59±1.33 41.71±0.98 SETN † [191] 84.04±0.28 87.64±0.00 58.86±0.06 59.05±0.24 33.06±0.02 32.52±0.21 ENAS † [74] 37.51±3.19 53.89±0.58 13.37±2.35 13.96±2.33 15.06±1.95 14.84±2.10 GibbsNAS † [192] 90.02±0.60 92.72±0.60 68.88±1.43 69.20±1.40 42.31±1.69 42.08±1.95 REA † [193] 91.19±0.31 93.92±0.30 71.81±1.12 71.84±0.99 45.15±0.89 45.54±1.03 RS † [16] 90.93±0.36 93.70±0.36 70.93±1.09 71.04±1.07 44.45±1.10 44.57±1.25 REINFORCE † [194] 91.09±0.37 93.85±0.37 71.61±1.12 71.71±1.09 45.05±1.02 45.24±1.18 BOHB † [195] 90.82±0.53 93.61±0.52 70.74±1.29 70.85±1.28 44.26±1.36 44.42±1.49 TPE [196] 91.30±0.18 94.07±0.17 71.93±0.89 72.08±0.83 45.71±0.68 45.94±0.83 SMAC [91] 91.23±0.21 94.05±0.23 72.17±0.61 72.21±0.76 45.51±0.33 46.08±0.74 BOHAMIANN [93] 91.36±0.16 94.13±0.23 72.36±0.82 72.38±0.81 45.93±0.66 46.18±0.60 OF-BO 91.52±0.05 94.35±0.03 73.21±0.29 73.25±0.18 46.27±0.36 46.54±0.19Methods CIFAR-10 CIFAR-100 ImageNet-16-120 valid test valid test valid test Table 2 : 2The performance of scaled-up ViTAE models on the ImageNet1K dataset. † indicates that ImageNet22K is used to further fine-tune the models with 224×224 resolution for 90 epochs.#Params Test size ImageNet Top-1 Real Top-1 Swin-L † [198] 197 M 384 87.3 90.0 SwinV2-L † [199] 197 M 384 87.7 - CoAtNet-4 † [200] 275 M 384 87.9 - CvT-W24 † [201] 277 M 384 87.7 - ViT-L * [66] 304 M 224 85.5 90.1 ViT-L [202] 304 M 224 85.7 - ViTAE-L [117] 311 M 224 86.0 90.3 ViTAE-L † [117] 311 M 224 87.5 90.8 ViTAE-L † [117] 311 M 384 88.3 91.1 SwinV2-H [203] 658 M 224 85.7 - SwinV2-H [203] 658 M 512 87.1 - ViTAE-H [117] 644 M 224 86.9 90.6 ViTAE-H [117] 644 M 512 87.8 91.2 ViTAE-H † [117] 644 M 224 88.0 90.7 ViTAE-H † [117] 644 M 448 88.5 90.8 MethodCB COPA WSC RTE WIC CoLA MRPC STSB Conll 04 AVG.model tuning 94.6 69.0 68.3 75.8 74.9 60.6 88.0 90.0 85.6 78.5 prompt tuning 87.5 76.0 64.4 76.2 66.9 63.8 86.8 90.5 85.5 77.5 (a) Transfer with the vanilla prompt transfer approach MNLI 96.4 71.0 67.3 80.9 66.5 58.9 88.2 91.0 83.0 78.1 QNLI 89.3 76.0 65.4 76.2 70.4 63.7 88.5 90.7 83.5 78.2 Record 78.6 63.0 65.4 53.8 51.7 0.0 77.7 85.0 82.7 62.0 SQuAD 87.5 74.0 66.3 71.8 51.7 6.0 87.3 89.3 82.5 68.5 CoNLL03 73.2 64.0 63.5 60.3 51.9 0.0 71.3 16.4 84.8 53.9 Ontonotes 78.6 65.0 66.3 56.7 54.1 59.3 82.4 84.5 86.1 70.3 CoNLL05 87.5 65.0 64.4 69.3 68.3 61.3 88.7 88.4 83.8 75.2 CoNLL12 89.3 62.0 67.3 63.2 67.4 58.7 90.4 88.5 83.6 74.5 SST2 92.9 74.0 64.4 71.8 66.8 60.1 87.0 89.6 84.3 76.8 (b) Transfer with Our built-in efficient approach MNLI 92.9 77.0 67.3 78.0 68.8 66.3 88.5 90.6 85.4 79.4 1.3 QNLI 92.9 77.0 66.3 77.3 70.8 63.9 87.5 90.8 86.6 79.2 1.0 Record 87.5 76.0 66.3 77.3 68.5 62.4 87.5 90.7 84.9 77.9 15.9 SQuAD 89.3 75.0 66.3 75.5 69.3 63.1 87.3 88.9 85.7 77.8 9.3 CoNLL03 91.1 72.0 68.3 76.9 67.4 63.6 86.5 90.6 85.6 78.0 24.1 Ontonotes 89.3 74.0 66.3 76.2 69.1 64.2 88.0 90.8 85.7 78.2 7.8 CoNLL05 87.5 79.0 65.4 77.6 69.6 63.7 87.5 90.8 84.8 78.4 3.2 CoNLL12 87.5 76.0 66.3 74.4 68.5 63.7 87.5 90.8 85.0 77.7 3.3 SST2 92.9 77.0 68.3 76.5 70.1 64.8 88.5 90.7 86.3 79.5 2.7 Table 4 : 4Comparisons among the results obtained on COCO-stuff[209] and Visual Genome (VG)[210]. All the results are taken from the original papers and are based on a 256 × 256 resolution. '-' means that the related value is unavailable in the corresponding papers.FID ↓ SceneFID ↓ Inception Score ↑ Diversity Score ↑ COCO VG COCO VG COCO VG COCO VG LostGAN-V2 [211] 42.55 47.62 22.00 18.27 18.01±0.50 14.10±0.38 0.55±0.09 0.53±0.09 OCGAN [212] 41.65 40.85 - - - - - - HCSS [208] 33.68 19.14 13.36 8.61 - - - - LAMA [213] 31.12 31.63 18.64 13.66 - - 0.48±0.11 0.54±0.09 Frido [214] 37.14 - 14.91 - 18.62±0.54 - - - TwFA [207] 22.15 17.74 11.99 7.54 24.25±1.04 25.13±0.66 0.67±0.00 0.64±0.00 Table 5 : 5Comparison of the features in open-source and commercial AutoML platforms. We compare them based on XR is an umbrella term covering virtual reality (VR), augmented reality (AR), and mixed reality (MR). AcknowledgmentsWe thank Dr. Xiaodong He for providing valuable support and constructive comments. This work was (partially) done during Dr. Jing Zhang's visit and Jiaxing Li's internship at JD Explore Academy.the provisioned application and the system point of view. * means that we did not find enough published information to verify the corresponding item. V. G. represents "very good". An overview of the ecosystem of AutoML systems is presented inTable 5, where some open-source platforms such as HyperOpt[22], Katib[27], NNI[23], Ray-Tune[21]and commercial platforms such as Abacus[222], Google Vertex AI[224], IBM Watson AutoAI[29], Amazon SageMaker Canvas[219], Microsoft Azure AutoML[223]are compared with OmniForce. OmniForce is one of the rare systems to support crowdsourcing, cloud-edge collaboration, super-deep models (large models), and human-centric systems. OmniForce is a cloud-native AutoML system that can be run in a production environment for commercial use.ConclusionDue to the proposed human-centered and cloud-edge collaborative AutoML method and the widely provisioned applications that enable large models and their highly efficient transfer strategies, OmniForce builds a practicable and powerful system for people and companies that want to benefit from AI technology in open-environment scenarios, such as the industrial supply chain and industrial metaverse, where people often face open-loop problems and need to train or update large models and deploy models on massive devices with different constraints and requirements. OmniForce has a uniform user interface, a seamless and flexible search strategy framework, and cloud-native features, making it an accessible, versatile and product-ready system for helping people establish and improve AI engineering in practice and continuously extract business value from AI technology. 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[ "https://github.com/microsoft/nni,", "https://github.com/Epistimio/", "https://github.com/automl/auto-sklearn,", "https://github.com/kubeflow/kubeflow/,", "https://github.com/kubeflow/training-operator,", "https://github.com/kubeflow/mpi-operator,", "https://github.com/kubeflow/pipelines,", "https://github.com/kserve/kserve," ]
[ "Hydrodynamic and hydrostatic limit for a contact process with random slowdowns in finite volume and with slow boundaries", "Hydrodynamic and hydrostatic limit for a contact process with random slowdowns in finite volume and with slow boundaries" ]
[ "M Mourragui \nLMRS\nUMR 6085\nUniversité de Rouen\nAvenue de l'Université\nBP.12, Technopôle du MadriletF76801Saint-Etienne-du-RouvrayFrance\n", "E Saada \nUMR 8145\nLaboratoire MAP5\nCNRS\nUniversité Paris Cité\n45 rue des Saints-Pères75270Paris Cedex 06France\n", "S Velasco \nLaboratoire MAP5\nUniversité Paris Cité\n45 rue des Saints-Pères75270, Cedex 06ParisFrance\n" ]
[ "LMRS\nUMR 6085\nUniversité de Rouen\nAvenue de l'Université\nBP.12, Technopôle du MadriletF76801Saint-Etienne-du-RouvrayFrance", "UMR 8145\nLaboratoire MAP5\nCNRS\nUniversité Paris Cité\n45 rue des Saints-Pères75270Paris Cedex 06France", "Laboratoire MAP5\nUniversité Paris Cité\n45 rue des Saints-Pères75270, Cedex 06ParisFrance" ]
[]
We consider an interacting particle system which models the sterile insect technique. It is the superposition of a generalized contact process with exchanges of particles on a finite cylinder with open boundaries (seeKuoch et al., 2017 ). We show that when the system is in contact with reservoirs at different slow-down rates, the hydrodynamic limit is a set of coupled non linear reaction-diffusion equations with mixed boundary conditions. We also prove the hydrostatic limit when the macroscopic equations exhibit a unique attractor.
null
[ "https://export.arxiv.org/pdf/2212.07762v1.pdf" ]
254,685,891
2212.07762
cf5b7945a4c9d9c8ab1287d552c33a935710c387
Hydrodynamic and hydrostatic limit for a contact process with random slowdowns in finite volume and with slow boundaries M Mourragui LMRS UMR 6085 Université de Rouen Avenue de l'Université BP.12, Technopôle du MadriletF76801Saint-Etienne-du-RouvrayFrance E Saada UMR 8145 Laboratoire MAP5 CNRS Université Paris Cité 45 rue des Saints-Pères75270Paris Cedex 06France S Velasco Laboratoire MAP5 Université Paris Cité 45 rue des Saints-Pères75270, Cedex 06ParisFrance Hydrodynamic and hydrostatic limit for a contact process with random slowdowns in finite volume and with slow boundaries Hydrodynamic limitHydrostatic limitRandom reservoirsSystem of reaction-diffusion equationsGeneralized contact processStirring processMixed boundary conditionsMSC : 60K3582C22 We consider an interacting particle system which models the sterile insect technique. It is the superposition of a generalized contact process with exchanges of particles on a finite cylinder with open boundaries (seeKuoch et al., 2017 ). We show that when the system is in contact with reservoirs at different slow-down rates, the hydrodynamic limit is a set of coupled non linear reaction-diffusion equations with mixed boundary conditions. We also prove the hydrostatic limit when the macroscopic equations exhibit a unique attractor. Introduction In this paper, we consider the interacting particle system which was introduced in [17] to model the sterile insect technique. This technique was developed, among others, by E. Knipling (see [16]) to eradicate New World screw worms in the 1950's, a serious pest for warm blooded animals. The method is still used today, for instance in France, to protect crops from the very invasive Mediterranean flies, and it is also being tested to fight mosquitoes which transmit dengue in countries like Panama or Brazil. The sterile insect technique works as follows: male insects are sterilized in captivity using gamma rays. They are then released in the wild population, where females mate only once, giving rise to no off springs if they mate with a sterile male. When enough sterile individuals are released, the wild population eventually becomes extinct. From a mathematical perspective, the sterile insect technique has mainly been modeled in a deterministic way through the study of partial differential equations (see [1]). The sterile insect technique was studied from a probabilistic perspective in [17] and [18] using interacting particle systems. In [17], a phase transition result is proved at the microscopic level. Recently, another probabilistic model was studied in [14], also at the microscopic level. In [18], the study is carried at the macroscopic level (hydrodynamic limit) in finite volume with reservoirs, in order to account for the migration/immigration mechanism. Here, we aim at studying the hydrodynamic limit and hydrostatic limit of that interacting particle system under the effect of slow reservoirs. The slow-down mechanism models the fact that beyond the boundary through which insects arrive into the system or leave it, there are very few insects (the exterior of the system might be a territory which is much less favorable to the development of these insects). In the perspective of interacting particle systems, the sterile insect technique is modeled as follows: insects evolve on a d-dimensional finite set B N = {−N, ..., N } × T d−1 N , where N ≥ 1, and the evolution of the population is described by a continuous time Markov process (η N t ) t≥0 with state space E B N where E is a countable set. The quantity of interest here is not the number of insects per site but the types of insects present at a given site. Precisely, E = {0, 1, 2, 3} and for x in B N , The dynamics of the Markov process is the superposition of three Markovian jump processes: η(x) =        (i) A process which models the fact that insects move in an isotropic way within B N and which is parameterized by a diffusivity constant D > 0. Precisely, for a configuration η and x, y two sites in B N , the states of sites x and y in η are exchanged at rate D. (ii) A birth and death dynamics which models births of individuals due to the mating of a wild females with wild or sterile insects, as well as deaths of individuals. This is parameterized by a release rate r > 0 and growth rates λ 1 , λ 2 > 0. Sterile males are injected on a site at rate r independently of everything else. The rate at which wild males give birth (to wild males) on neighbouring sites is λ 1 at sites in state 1, and λ 2 at sites in state 3. Sterile males do not give birth. We take λ 2 < λ 1 to reflect the fact that fertility is reduced at sites in state 3. Deaths for each type of male insects occur independently and at rate 1. (iii) A boundary dynamics which models the slow migration/immigration mechanism. This is parameterized by a function b = (b 1 , b 2 , b 3 ) : {−1, 1} × T d−1 → [0, 1] 3 and two constants θ and θ r in R + . For i ∈ {0, 1, 2, 3}, a particle of type i ∈ {0, 1, 2, 3} is injected in the system through x ∈ {−N } × T d−1 N , resp. x ∈ {N } × T d−1 N at rate N −θ b i (x/N ), resp. N −θr b i (x/N ) and is expelled from the system through x at rate N −θ (1 − b i (x/N )), resp. N −θr ( 1 − b i (x/N )), with b 0 = 1 − b 1 − b 2 − b 3 . The birth and death mechanism is referred to as a contact process with random slowdowns (or CPRS). Indeed, without the presence of sterile insects, it would be a basic contact process (as defined for instance in [20]) with parameter λ 1 , and the presence of sterile insects can be interpreted as a random decrease of the fertility rate due to the presence of sites containing sterile and wild individuals. In [17], the microscopic study of the birth and death dynamics alone leads to the following phase transition result: for certain values of λ 1 and λ 2 , when r is large enough, the healthy population almost surely becomes extinct and survives otherwise. In [18], the hydrodynamic limit of the superposition of the three dynamics above, where the first and the third one are accelerated in the diffusive scaling N 2 , and where θ = θ r = 0, is proven to be a system of non linear reaction-diffusion equations with Dirichlet boundary conditions. In this paper, we prove the finite volume hydrodynamic limit of this interacting particle system for any values of θ , θ r ≥ 0. The hydrodynamic equation obtained has mixed boundary conditions which depend on the values of θ , resp. θ r . Precisely, for θ ∈ [0, 1), resp. θ r ∈ [0, 1), we get a Dirichlet type boundary condition at the left hand side, resp. right hand side of the system. For θ = 1, resp. θ r = 1, we get a Robin type boundary condition at the left hand side, resp. right hand side of the system. For θ > 1, resp. θ r > 1, we get a Neumann type boundary condition at the left hand side, resp. right hand side of the system. We then prove the finite volume hydrostatic limit of the interacting particle system for a specific class of parameters regarding the dynamics. Within that class of parameters, the sequence of invariant measures of the interacting particle system is associated to a profile which is the stationary solution of the hydrodynamic equation with corresponding mixed boundary conditions. Our paper is, up to our knowledge, the first one regarding the effect of mixed reservoirs in and out of equilibrium (hydrodynamic and hydrostatic limit) for a multi species process in finite volume. The effect of reservoirs on a one dimensional conservative system has been widely studied in finite volume (see for instance [5], [8]). Much is now known both at the microscopic and macroscopic level. Recently, the effect of slow reservoirs has aroused much interest for the symmetric simple exclusion process in one dimension (see for instance [2], [10], [11], [12] and references therein). In [9], authors proved a hydrostatic principle for a boundary driven gradient symmetric exclusion process using the fact that the stationary profile is a global attractor for the hydrodynamic equation. This method inspired our proof for the hydrostatic limit. However, the coupled equations obtained for the hydrodynamic limit, and the fact that we work in any dimension make the analysis more subtle. The proof of the hydrodynamic limit for each of these regime is established in Section 3 via the Entropy Method. Among other things, as we work in arbitrary dimension, some care must be taken to define and characterize the solution of the hydrodynamic limit at the boundary, through the use of the Trace Operator (see subsection 3.4). The proof of the hydrostatic limit, established in Section 4 and inspired by [9] relies on the use of a change of coordinates for the coupled equations. Under this change of coordinates (inspired by some simulations see Appendix B ), a comparison principle holds. It allows us to find a unique attractor when some conditions on the parameters are satisfied. Outside that class of parameters, although uniqueness of the invariant measure holds, we do not even know whether there is uniqueness of the stationary solution of the hydrodynamic equation and simulations show (see Appendix B) that for Neumann type boundary conditions there are several stationary profiles. However, we believe that a more general hydrostatic principle in the spirit of the one proved in [19] is valid. Notations and results The microscopic model The dynamics of our interacting particle system is given by three generators, one for the diffusive dynamics, one for the contact dynamics and one for the boundary dynamics. In order to explicit each one of those generators, let us give a few notations. Let N ∈ N. Denote B N = {−N, ..., N } × T d−1 N the bulk and Γ N = {−N, N } × T d−1 N , resp. Γ + N = {N } × T d−1 N , resp. Γ − N = {−N } × T d−1 N the boundary, resp. left hand side boundary, resp. right hand side boundary of the bulk. Denote B = (−1, 1) × T d−1 the continuous counter part of the bulk, B = [−1, 1] × T d−1 its closure, Γ = {−1, 1} × T d−1 , Γ − = {−1} × T d−1 and Γ + = {1} × T d−1 . The microscopic state space is denoted Ω N := {0, 1, 2, 3} B N and its elements, also called configurations, are denoted η. Therefore, for x ∈ B N , η(x) ∈ {0, 1, 2, 3}. To describe the dynamics of our model, we will use the correspondence introduced in [18] between the state space Ω N and Σ N := ({0, 1} × {0, 1}) B N where the correspondence between an element (ξ, ω) ∈ Σ N and η ∈ Ω N is given as follows: for x ∈ B N , η(x) = 0 ⇐⇒ (1 − ξ(x))(1 − ω(x)) = 1, η(x) = 1 ⇐⇒ ξ(x)(1 − ω(x)) = 1, η(x) = 2 ⇐⇒ (1 − ξ(x))ω(x) = 1, η(x) = 3 ⇐⇒ ξ(x)ω(x) = 1. (2.1) In other words, (ξ(x), ω(x)) = (0, 0) if x is in state 0, (1, 0) if it is in state 1, (0, 1) if it is in state 2 and (1, 1) if it is in state 3. Also, in order to describe the evolution of the density of sites in state 1, resp. 2, resp 3, resp 0, we define for x in B N and a configuration η ∈ Ω N with associated configuration (ξ, ω) ∈ Σ N ,        η 1 (x) := 1 η(x)=1 = ξ(x)(1 − ω(x)), η 2 (x) := 1 η(x)=2 = (1 − ξ(x))ω(x), η 3 (x) := 1 η(x)=3 = ξ(x)ω(x), η 0 (x) := 1 η(x)=0 = (1 − ξ(x))(1 − ω(x)). (2.2) Finally, we also express the correspondence (2.1) by the following application from Σ N to Ω N : η = η(ξ, ω), where, for any x ∈ B N , η(x) = 2ω(x) + ξ(x). (2.3) • Generator for the diffusion mechanism: it corresponds to the usual stirring mechanism where each site has an exponential clock with rate D and independent from all the other clocks. When the clock rings, a neighbouring site is chosen uniformly at random and the states of both sites are exchanged. The action of the generator on functions f : Σ N → R is therefore given by: L N f (ξ, ω) := d k=1 (x,x+e k )∈B N D f (ξ x,x+e k , ω x,x+e k ) − f (ξ, ω) (2.4) where (e 1 , ..., e d ) is the canonical basis of Z d and for ζ ∈ {0, 1} B N and x, y ∈ B N , ζ x,y is the configuration obtained from ζ by exchanging the occupation variables ζ(x) and ζ(y), i.e, ζ x,y (z) =    ζ(x) if z = y, ζ(y) if z = x, ζ(z) otherwise. • Generator for the contact process in the bulk: following the description of the CPRS in the introduction, the birth and death mechanism in the bulk has the following rates: for η ∈ Ω N and x ∈ B N , 0 → 1 at rate λ 1 n 1 (x, η) + λ 2 n 3 (x, η), 1 → 0 at rate 1, 0 → 2 at rate r, 2 → 0 at rate 1, 1 → 3 at rate r, 3 → 1 at rate 1, 2 → 3 at rate λ 1 n 1 (x, η) + λ 2 n 3 (x, η), 3 → 2 at rate 1. (2.5) Therefore, using the correspondence (2.1), the generator L N = L N,λ1,λ2,r of the CPRS acts as follows on functions f : Σ N → R: L N f (ξ, ω) = x∈B N L x B N f (ξ, ω) ,(2.6) where for x ∈ B N , L x B N f (ξ, ω) := r(1 − ω(x)) + ω(x) f (ξ, σ x ω) − f (ξ, ω) + β B N (x, ξ, ω)(1 − ξ(x)) + ξ(x) f (σ x ξ, ω) − f (ξ, ω) , (2.7) β B N (x, η) := λ 1 y∼x y∈B N η 1 (y) + λ 2 y∼x y∈B N η 3 (y) where x ∼ y means that x and y are neighbouring sites in B N , and where for ζ ∈ {0, 1} B N , σ x ζ is the configuration obtained from ζ by flipping the configuration at x, i.e. σ x ζ(z) = 1 − ζ(x) if z = x, ζ(z) otherwise. • Generator for the boundary dynamics: the generator of the dynamics at the boundary is parametrized by θ = (θ , θ r ) with θ , θ r ≥ 0 and a positive function b = (b 1 , b 2 , b 3 ) : Γ → R 3 + satisfying the following conditions: there exists a neighbourhood V of B in R × T d−1 and a smooth function g = (g 1 , g 2 , g 3 ) : V → (0, 1) 3 in C 2 (V, R) such that 0 < c * < min 1≤i≤3 |g i | ≤ max 1≤i≤3 |g i | ≤ C * < 1 (2.8) for two positive constants c * and C * , and such that the restriction of g to Γ is equal to b. The dynamics at the boundary can then be described as follows: a site x ∈ Γ − N , resp. x ∈ Γ + N goes from state i ∈ {0, 1, 2, 3} to state j ∈ {0, 1, 2, 3} \ {i} at rate N −θ b j (x/N ), resp. N −θr b j (x/N ) . In order to express the generator of the boundary dynamics, we make use of η i = η i (ξ, ω) for i ∈ {0, 1, 2, 3} which is the configuration in {0, 1} B N obtained from (ξ, ω) ∈ Σ N according to (2.2). For f : Σ N → R, the boundary generator acts on f as follows: L b, θ,N f (ξ, ω) := x∈Γ N L x b, θ,N f (ξ, ω), where L x b, θ,N f (ξ, ω) := c x ( b(x/N ), ξ, σ x ω) f (ξ, σ x ω) − f (ξ, ω) + c x ( b(x/N ), σ x ξ, ω) f (σ x ξ, ω) − f (ξ, ω) + c x ( b(x/N ), σ x ξ, σ x ω) f (σ x ξ, σ x ω) − f (ξ, ω) , and with c x ( b(x/N ), ξ, ω) := N −θ 3 i=0 b i (x/N )η i (x) if x ∈ Γ − N , N −θr 3 i=0 b i (x/N )η i (x) if x ∈ Γ + N and b 0 (x/N ) := 1 − 3 i=1 b i (x/N ). Note that the following simpler expression holds for L b, θ,N : for f : Σ N → R and (ξ, ω) ∈ Σ N , L b, θ,N f (ξ, ω) = N −θ 3 i=0 x∈Γ − N b i (x/N ) f (σ i,x (ξ, ω)) − f (ξ, ω) + N −θr 3 i=0 x∈Γ + N b i (x/N ) f (σ i,x (ξ, ω)) − f (ξ, ω) (2.9) with σ i,x (ξ, ω) := σ i,x η(ξ, ω), the configuration in Σ N associated to σ i,x η, where σ i,x η(z) := i if z = x, η(ξ, ω)(z) otherwise with η(ξ, ω) as defined in (2.3). Fix a time horizon T > 0 and denote {(ξ N t , ω N t ), t ∈ [0, T ]} the Markov process associated to the generator L N := N 2 L N + N 2 L b, θ,N + L N . (2.10) Let D Σ N ([0, T ]) be the path space of càdlàg trajectories with values in Σ N . Given a measure µ N on Σ N , denote by P µ N the probability measure on D Σ N ([0, T ]) induced by µ N and (ξ t , ω t ) t≥0 and denote E µ N the expectation with respect to P µ N . Invariant measures for the diffusive and boundary dynamics: Consider α = (α 1 , α 2 , α 3 ) : B −→ (0, 1) 3 a smooth function satisfying the following conditions: ∃ c * , C * > 0, 0 < c * < min 1≤i≤3 |α i | ≤ max 1≤i≤3 |α i | ≤ C * < 1 (2.11) and ∀x ∈ Γ, α(x) = b(x). (2.12) Denote ν N α the Bernoulli product measure on B N with parameter α: for (ξ, ω) ∈ Σ N , ν N α (ξ, ω) := 1 Z α,N exp 3 i=1 x∈B N log α i (x/N ) α 0 (x/N ) η i (x) , where Z α,N is the normalizing constant Z α,N = x∈B N 1 + 3 i=1 exp(α i (x/N ) and with α 0 = 1 − α 1 − α 2 − α 3 . Note that ν N α is such that for every 1 ≤ i ≤ 3 and x ∈ B N , E ν N α [η i (x)] = α i (x/N ). The following statements are easy to verify: • Consider α a constant profile. Then ν N α is an invariant measure for the exchange dynamics so for any f : Σ N → R, Σ N L N f (ξ, ω)dν N α (ξ, ω) = 0. (2.13) • Consider α a smooth profile satisfying (2.11) and (2.12). Then ν N α is an invariant measure for the boundary dynamics so for any f : Σ N → R, Σ N L b, θ,N f (ξ, ω)dν N α (ξ, ω) = 0. (2.14) For any θ ∈ (R + ) 3 , at fixed N , the dynamics defined by (2.10) is irreducible and the state space is finite. Therefore, there is a unique invariant measure that in the sequel we denote µ ss N ( θ). Useful (in)equalities: For any A, B > 0, A(B − A) = − 1 2 (B − A) 2 + 1 2 (B 2 − A 2 ). (2.15) For any a, b, A and N ∈ N, 2ab ≤ N A a 2 + A N b 2 . (2.16) The macroscopic equations Let us first introduce a few notations. We will write functions with values in R with normal letters (for instance G) and the ones with values in R 3 with letters with a hat (for instance G) . For n, m ∈ N, denote For θ = (θ , θ r ) in (R + ) 2 , we will use the following notations to denote these functional spaces: C θ :=        C 1,2 0 if θ ∈ [0, 1) 2 , C 1,2 0,− if θ ∈ [0, 1), θ r ≥ 1, C 1,2 0,+ if θ r ∈ [0, 1), θ ≥ 1, C 1,2 if θ r , θ ≥ 1. (2.17) Let < ., . > be the L 2 (B) inner product and < ., . > µ the inner product with respect to a measure µ. For f = (f 1 , f 2 , f 3 ) and g = (g 1 , g 2 , g 3 ) in L 2 (B) 3 , < f , g >= 3 i=1 < f i , g i >. Introduce H 1 (B) the set of functions in L 2 (B) such that for any 1 ≤ k ≤ d, there exists an element ∂ e k g in L 2 (B) such that for any ϕ in C ∞ k (B), < ∂ e k ϕ, g >= − < ϕ, ∂ e k g > . The H 1 (B) norm is then defined as follows: g H 1 (B) = g 2 L 2 (B) + d k=1 ∂ e k g 2 L 2 (B) 1/2 . Denote L 2 [0, T ], H 1 (B) the space of functions f : [0, T ] → H 1 (B) such that T 0 f (t, .) 2 H 1 (B) dt < ∞. In order to define the value of an element G in H 1 (B) at the boundary, we need to introduce the notion of trace of functions on such Sobolev spaces. The trace operator in the Sobolev space H 1 (B) can be defined as a bounded linear operator, Tr : H 1 (B) → L 2 (Γ) such that Tr extends the classical trace, that is Tr(G) = G |Γ , for any G ∈ H 1 (B) ∩ C(B). We refer to [6, Part II Section 5] for a detailed survey of the trace operator. In the sequel, for s, u ∈ R + × Γ and for any f ∈ L 2 ((0, T ); H 1 (B)), f (s, u) stands for T r(f (s, .))(u). To lighten notations, for a function G depending on time and space we will often write G s instead of G(s, .). Finally, for θ ∈ (R + ) 2 , introduce the following linear functional on L 2 [0, T ], H 1 (B) parametrized by a test function G in C θ : for t ∈ [0, T ], I G ( ρ)(t) := < ρ t , G t > − < ρ 0 , G 0 > − t 0 < ρ s , ∂ s G s > ds − D t 0 < ρ s , ∆ G s > ds − t 0 < F ( ρ s ), G s > ds (2.18) where F = (F 1 ( ρ), F 2 ( ρ), F 3 ( ρ)) : [0, 1] 3 → R 3 is defined by    F 1 (ρ 1 , ρ 2 , ρ 3 ) = 2d(λ 1 ρ 1 + λ 2 ρ 3 )ρ 0 + ρ 3 − (r + 1)ρ 1 F 2 (ρ 1 , ρ 2 , ρ 3 ) = rρ 0 + ρ 3 − 2d(λ 1 ρ 1 + λ 2 ρ 3 )ρ 2 − ρ 2 F 3 (ρ 1 , ρ 2 , ρ 3 ) = 2d(λ 1 ρ 1 + λ 2 ρ 3 )ρ 2 + rρ 1 − 2ρ 3 , (2.19) with ρ 0 = 1 − ρ 1 − ρ 2 − ρ 3 . The hydrodynamic equation is a reaction diffusion coupled equation with mixed boundary conditions depending on θ. If θ , resp. θ r in [0, 1), the boundary conditions are of Dirichlet type on Γ − , resp. Γ + . If θ = 1, resp θ r = 1, they are of Robin type on Γ − , resp. Γ + . If θ > 1, resp. θ r > 1, they are of Neumann type on Γ − , resp. Γ + . We will focus on the cases where θ ∈ [0, 1), θ r = 1 resp. θ > 1, θ r = 1 corresponding to a Dirichlet boundary condition on Γ − and a Robin boundary condition on Γ + , resp. a Neumann boundary condition on Γ − and a Robin boundary condition on Γ + . All the other cases can easily be adapted (see the Table 1). Definition 1. Let γ : B → R 3 be a continuous function. • Hydrodynamic equation for θ ∈ [0, 1) and θ r = 1. We say that a bounded function ρ = (ρ 1 , ρ 2 , ρ 3 ) : [0, T ] × B → R 3 is a weak solution of the Dirichlet + Robin mixed boundary problem        ∂ t ρ = D∆ ρ + F ( ρ) in B × (0, T ), ρ(0, .) = γ in B, ρ(t, .) |Γ − = b for 0 < t ≤ T, ∂ e1 ρ(t, .) |Γ + = 1 D ( b − ρ) |Γ + for 0 < t ≤ T (2.20) if, for any 1 ≤ i ≤ 3, ρ i ∈ L 2 ((0, T ), H 1 (B)), (2.21) for any function G ∈ C θ , for any t ∈ [0, T ], I G ( ρ)(t) + D 3 i=1 t 0 Γ − b i (r)(∂ e1 G i,s )(r)n 1 (r).dS(r)ds + D 3 i=1 t 0 Γ + ρ i (s, r)(∂ e1 G i,s )(r)n 1 (r).dS(r)ds − 3 i=1 t 0 Γ + G i (r)(b i (r) − ρ i (s, r))n 1 (r).dS(r)ds = 0, (2.22) where n 1 (r) is the outward unit normal vector to the boundary surface Γ and dS(r) is an element of surface on Γ. And, ρ(0, .) = γ(.) almost surely. (2.23) • Hydrodynamic equation for θ > 1 and θ r = 1. We say that a bounded function ρ = (ρ 1 , ρ 2 , ρ 3 ) : [0, T ] × B → R 3 is a weak solution of the Neumann + Robin mixed boundary problem        ∂ t ρ = D∆ ρ + F ( ρ) in B × (0, T ), ρ(0, .) = γ in B, ∂ e1 ρ(t, .) |Γ − = 0 for 0 < t ≤ T ∂ e1 ρ(t, .) |Γ + = 1 D ( b − ρ) |Γ + for 0 < t ≤ T (2.24) if ρ satisfies conditions (2.21) and (2.23) as well as the following: for any G ∈ C θ , for any t ∈ [0, T ], The letters D, resp. R, resp. N denote a Dirichlet, resp. Robin, resp. Neumann boundary condition. For instance (D ; N) denotes a left hand side Dirichlet boundary condition and a right hand side Neumann boundary condition. I G ( ρ)(t) + D 3 i=1 t 0 Γ − ρ i (s, r)(∂ e1 G i,s )(r)n 1 (r).dS(r)ds + D 3 i=1 t 0 Γ + ρ i (s, r)(∂ e1 G i,s )(r)n 1 (r).dS(r)ds − Γ + G i (r)(b i (r) − ρ i (s, r))n 1 (r).dS(r)ds = 0.• A function ρ = (ρ 1 , ρ 2 , ρ 3 ) in H 1 (B) 3 is said to be a stationary solution of (2.20) if for every function G = (G 1 , G 2 , G 3 ) ∈ C 2 0 (B) 3 , for all 1 ≤ i ≤ 3, D < ρ i , ∆G i > + < F i ( ρ), G i > = D Γ − b i (r)(∂ e1 G i )(r)n 1 (r).dS(r) + D Γ + ρ i (r)(∂ e1 G i )(r)n 1 (r).dS(r) − Γ + G i (r)(b i (r) − ρ i (r))n 1 (r).dS(r). (2.26) • A function ρ = (ρ 1 , ρ 2 , ρ 3 ) in H 1 (B) 3 is said to be a stationary solution of (2.24) if for every function G = (G 1 , G 2 , G 3 ) ∈ C 2 0 (B) 3 , for all 1 ≤ i ≤ 3, D < ρ i , ∆G i > + < F i ( ρ), G i > = D Γ − ρ i (r)(∂ e1 G i )(r)n 1 (r).dS(r) + D Γ + ρ i (r)(∂ e1 G i )(r)n 1 (r).dS(r) − Γ + G i (r)(b i (r) − ρ i (r))n 1 (r).dS(r). (2.27) Hydrodynamic and hydrostatic results Let us state the main results proved in this paper. The first one (Theorem 1) establishes the hydrodynamic limit of the dynamics defined above and the second one (Theorem 2) establishes its hydrostatic limit. Before stating Theorem 1, let us first define the empirical measure ( π N (ξ t , ω t )) t≥0 = ( π N t ) t≥0 associated to (ξ t , ω t ) t≥0 . Recall how in (2.2), we built η i ∈ {0, 1} B N from (ξ, ω) ∈ Σ N for 0 ≤ i ≤ 3. For any t ≥ 0, π N (ξ t , ω t ) := 1 N d x∈B N η 1,t (x)δ x/N , 1 N d x∈B N η 2,t (x)δ x/N , 1 N d x∈B N η 3,t (x)δ x/N =: (π N 1,t (ξ t , ω t ), π N 2,t (ξ t , ω t ), π N 3,t (ξ t , ω t )) where δ x/N is the point mass at x/N . For G in C 1,2 ([0, T ] × B) and t ≥ 0, write < π N t , G t >:= 3 i=1 < π N i,t , G i (t, .) >= 3 i=1 1 N d x∈B N η i,t (x)G i (t, x N ). The empirical measure is therefore the triplet of empirical measures associated to the density of sites in state 1, resp. 2, resp. 3. Denote M the set of positive measures on B with total mass bounded by 1 2 d . The process ( π N t ) t≥0 , is a Markov process with state space M 3 and its trajectories are in D([0, T ], M 3 ), the path space of cadlàg time trajectories with values in M 3 . We endow the path space with the Skorohod topology (we refer to [3] for a detailed survey on the Skorohod topology). For θ ∈ (R + ) 2 and µ N a measure on Σ N , denote Q θ N = P µ N ( π N ) −1 the law of the process ( π N (ξ t , ω t )) t≥0 when (ξ 0 , ω 0 ) ∼ µ N and where (ξ t , ω t ) t≥0 evolves according to the dynamics given by (2.10), with parameter θ for the boundary reservoirs. The hydrodynamic result states as follows: Theorem 1. (Hydrodynamic limit). For any sequence of initial probability measure (µ N ) N ≥1 on Σ N , the sequence of probability measures (Q θ N ) N ≥1 is weakly relatively compact and all its converging subsequences converge to some limit Q θ, * that is concentrated on the set of weak solutions of hydrodynamic equation that are in L 2 (0, T ; H 1 (B)). Furthermore, if there is an initial continuous profile γ : B → [0, 1] 3 such that for any δ > 0 and any G ∈ C ∞ k (B), lim sup N →∞ P µ N < π N , G > − < γ, G > > δ = 0, then, (Q θ N ) N ≥1 converges to the Dirac mass Q θ concentrated on the unique weak solution ρ of the boundary value problem associated to θ and with initial condition γ. Therefore, for any t ∈ [0, T ], δ > 0 and any function G ∈ C 1,2 c ([0, T ] × B), lim sup N →∞ P µ N < π N t , G t > − < ρ t , G t > > δ = 0. We prove Theorem 1 in Section 3. For the hydrostatic limit, we need to introduce the following sets of conditions: (H 1 ) :    D ≥ 1 r + 1 > 2d(λ 1 − λ 2 ) 1 > 2dλ 2 (H 2 ) : Dδ 1 + r + 2 > 2d(λ 1 − λ 2 ) Dδ 1 + 1 > 2dλ 2 (H 3 ) : r + 2 > 2d(λ 1 − λ 2 ) 1 > 2dλ 2 where δ 1 is the smallest eigenvalue of the Laplacian with Dirichlet boundary conditions (see (3.70)). Recall that µ ss N ( θ) denotes the sequence of unique invariant measures for the irreducible dynamics defined by (2.10). The hydrostatic result states as follows. Theorem 2. (Hydrostatic limit). Suppose that conditions (H 1 ) hold. There exists a unique stationary solution of (2.20) that we denote ρ D,R , and a unique stationary solution of (2.24) that we denote ρ N,R . Furthermore, the following statements hold. • Consider θ = (θ , θ r ) with θ ∈ [0, 1) and θ r = 1. For any continuous function G : B → [0, 1] 3 , lim N →∞ E µ ss N ( θ) 3 i=1 1 N d x∈B N η i (x)G i (x/N ) − 3 i=1 B G i (u)ρ D,R i (u)du = 0. (2.28) In other words, the sequence (µ ss N ( θ)) N ≥1 is associated to the unique stationary profile ρ D,R . • Consider θ = (θ , θ r ) with θ > 1 and θ r = 1. For any continuous function G : B → [0, 1] 3 , lim N →∞ E µ ss N ( θ) 3 i=1 1 N d x∈B N η i (x)G i (x/N ) − 3 i=1 B G i (u)ρ N,R i (u)du = 0. (2.29) In other words, the sequence (µ ss N ( θ)) N ≥1 is associated to the unique stationary profile ρ N,R . Proof of the hydrodynamic limit As said before, we focus on the cases where θ ∈ [0, 1), θ r = 1 and θ > 1, θ r = 1. We follow the entropy method introduced by Guo, Papanicolaou and Varadhan in [13] to prove the hydrodynamic limit. First, we prove tightness of the sequence of measures (Q θ N ) N ≥1 . Then, we show that any limit point of (Q θ N ) N ≥1 is a Dirac mass concentrated on a weak solution of (2.20) if θ ∈ [0, 1), θ r = 1, or (2.24) if θ > 1, θ r = 1. Finally, we prove uniqueness of the solution of the hydrodynamic equations at fixed initial data. The martingale property and tightness By Dynkin's formula (see [15, Appendix A.1] ), for 1 ≤ i ≤ 3, t ∈ [0, T ] and G ∈ C 1,2 c ([0, T ] × B), M N i,t ( G) :=< π N i,t , G i,t > − < π N i,0 , G i,0 > − t 0 < π N i,s , ∂ s G i,s > ds − N 2 t 0 L N < π N i,s , G i,s > ds − t 0 L N < π N i,s , G i,s > ds − N 2 t 0 L b, θ,N < π N i,s , G i,s > ds (3.1) is a martingale with respect to the natural filtration F t = σ(η s , s ≤ t) and with quadratic variation given by: N 2 t 0 L N < π N i,s , G i,s > 2 ds + t 0 L N < π N i,s , G i,s > 2 ds + N 2 t 0 L b, θ,N < π N i,s , G i,s > 2 ds − 2N 2 t 0 < π N i,s , G i,s > L N < π N i,s , G i,s > ds − 2 t 0 < π N i,s , G i,s > L N < π N i,s , G i,s > ds − 2N 2 t 0 < π N i,s , G i,s > L b, θ,N < π N i,s , G i,s > ds. (3.2) We then have that M N t ( G) := In order to develop the integral terms in (3.1), introduce the discrete second derivative in the direction e k (for 1 ≤ k ≤ d) in the bulk, the discrete laplacian, and the discrete gradient in the direction e 1 at the boundary: for x ∈ B N \ Γ N , (∂ N e k ) 2 G(x/N ) := d k=1 N 2 G x + e k N + G x − e k N − 2G x N ,∆ N G(x/N ) := d k=1 (∂ N e k ) 2 G(x/N ), (∂ N e1 ) − H(x/N ) := N H x N − H x − e 1 N and (∂ N e1 ) + H(x/N ) := N H x + e 1 N − H x N . Computations yield M N i,t ( G) =< π N i,t , G i,t > − < π N i,0 , G i,0 > − t 0 < π N i,s , ∂ s G i,s > ds − t 0 D N d x∈B N \Γ N ∆ N G i,s (x/N )η i,s (x)ds − t 0 D N d x∈Γ N d k=2 (∂ N e k ) 2 G i,s (x/N )η i,s (x)ds − t 0 D N d−1 x∈Γ + N (∂ N e1 ) − G i,s (x/N )η i,s (x) − D N d−1 x∈Γ − N (∂ N e1 ) + G i,s (x/N )η i,s (x) ds − t 0 1 N d x∈B N G i,s (x/N )τ x f i (η s )ds + N 2 N d+θ t 0 x∈Γ − N G i,s (x/N ) η i,s (x) − b i (x/N ) ds + N 2 N d+θr t 0 x∈Γ + N G i,s (x/N ) η i,s (x) − b i (x/N ) ds,(3.3) where we used that L N < π N i,s , G i,s >= 1 N d x∈B N G i,s (x/N )τ x f i (η s ), (3.4) with L N η 1 (0) = β B N (0, η)η 0 (0) + η 3 (0) − (r + 1)η 1 (0) =: f 1 (ξ, ω), L N η 2 (0) = rη 0 (0) + η 3 (0) − β B N (0, η)η 2 (0) − η 2 (0) =: f 2 (ξ, ω), L N η 3 (0) = β B N (0, η)η 2 (0) + rη 1 (0) − 2η 3 (0) =: f 3 (ξ, ω) . The second and third line in (3.3) correspond to the computation of the time integral associated to N 2 L N , the fourth line in (3.3) corresponds to the time integral associated to L N and the last term, to the integral associated to N 2 L b, θ,N . Proposition 1. The sequence of probability measures (Q θ N ) N ≥1 is tight in the Skorohod topology of D([0, T ], M 3 ) . We refer to [15,Section 4] for details regarding the proof of tightness of a sequence of probability measures. Recall that it is enough to show that for every H in a dense subset of C(B) for the L 2 norm, for every 1 ≤ i ≤ 3, lim sup δ→0 lim sup N →∞ E µ N sup |t−s|≤δ < π i,t , H > − < π i,s , H > = 0. (3.5) By density of C 2 0 (B) in C 2 (B) for the L 1 norm, it is enough to show (3.5) with H in C 2 0 (B). To prove that, we use the martingale and its quadratic variation introduced in (3.1) and (3.2), and show that lim sup δ→0 lim sup N →∞ E µ N sup |t−s|≤δ M N i,t (H) − M N i,s (H) = 0,(3.E µ N sup |t−s|≤δ t s L N < π N i,r , H > dr = 0. (3.7) We get (3.6) using the triangular inequality, Doob's inequality and the fact that the quadratic variation converges to zero. To prove (3.7), one shows that there is a constant C depending only on H such that for every r ∈ [0, T ], L N < π N i,r , H > ≤ C. (3.8) For that, use the decomposition of L N and the fact that H vanishes at the boundary as well as explicit computations and the fact that the f i 's are uniformly bounded in N . Replacement Lemmas In order to characterize the limit points of a sequence (Q θ N ) N ≥1 , we need to close the equation (3.3). That means that we want to show that each term of the martingale converges to a term that appears in the weak formulation of the solution of the hydrodynamic equation, and that the martingale converges to zero. For that, we perform a replacement lemma in the bulk and one at the boundary. The replacement lemma in the bulk (Proposition 2) is exactly the same as in [18,Lemma 4.2] and we refer to that article for a detailed proof. Here we focus on the replacement lemmas at the boundary and more specifically on the left hand side boundary (the same statements hold on the right hand side). There are two replacement lemmas: one for θ ∈ [0, 1) whose formulation coincides with the replacement lemma at the boundary in [18,Proposition 4.3] (corresponding to a Dirichlet condition), and one for θ r ≥ 1, whose formulation involves particle densities over small macroscopic boxes. Dirichlet forms Let us recall the expressions introduced in [18, Section 5] of the Dirichlet forms associated to each dynamics: for f : Σ N → R and µ a measure on Σ N , D N (f, µ) := d k=1 (x,x+e k )∈B 2 N Σ N D f (ξ x,x+e k , ω x,x+e k ) − f (ξ, ω) 2 dµ(ξ, ω), D b, θ,N (f, µ) := N −θ 3 i=0 x∈Γ − N Σ N b i (x/N )(1 − η i (x)) f (σ i,x (ξ, ω)) − f (ξ, ω) 2 dµ(ξ, ω) + N −θr 3 i=0 x∈Γ + N Σ N b i (x/N )(1 − η i (x)) f (σ i,x (ξ, ω)) − f (ξ, ω) 2 dµ(ξ, ω) and D N (f, µ) := x∈B N Σ N r(1 − ω(x)) + ω(x) f (ξ, σ x ω) − f (ξ, ω) 2 dµ(ξ, ω) + x∈B N Σ N β B N (x, ξ, ω)(1 − ξ(x)) + ξ(x) f (σ x ξ, ω) − f (ξ, ω) 2 dµ(ξ, ω). In the proofs of the Replacement lemmas, we will widely make use of the following inequalities. Lemma 1. (i) Consider α a smooth profile which satisfies (2.11) and (2.12). There is a constant C 1 > 0 such that for any density function f : Σ N → R with respect to the measure ν N α , < L N f , f > ν N α ≤ − 1 4 D N (f, ν N α ) + C 1 N d−2 . (3.9) (ii) Consider α a smooth profile which either satisfies (2.11) and (2.12) or is constant. There is a constant C 2 > 0 such that for any density function f : Σ N → R with respect to the measure ν N α , < L N f , f > ν N α ≤ C 2 N d . (3.10) (iii) Consider α a smooth profile which satisfies (2.11) and (2.12), then for any density function f : Σ N → R with respect to the measure ν N α , < L b, θ,N f , f > ν N α = − 1 2 D b, θ,N (f, ν N α ). (3.11) Proof. Both (i) and (ii) are stated and proved in [18, Section 6, Lemma 6.1] and (iii) follows from the fact that ν N α is invariant under the dynamics generated by L b, θ,N when α satisfies (2.11) and (2.12). We recall the proof of (ii). Using the correspondence (2.1) and the expression of the generator for the contact process in the bulk (2.6) we have < L N f , f > ν N α = x∈B N β B N (x, ξ, ω)(1 − ξ(x)) + ξ(x) f (ξ, ω) f (σ x ξ, ω) − f (ξ, ω) dν N α (ξ, ω) + x∈B N r(1 − ω(x)) + ω(x) f (ξ, ω) f (ξ, σ x ω) − f (ξ, ω) dν N α (ξ, ω). As the terms β B N (x, ξ, ω)(1 − ξ(x)) + ξ(x) and r(1 − ω(x)) + ω(x) are uniformly bounded in N by a constant C(λ 1 , λ 2 , r, d) , the above is less than I 1 + I 2 : = x∈B N C(λ 1 , λ 2 , r, d) f (ξ, ω) f (σ x ξ, ω) + f (ξ, ω) dν N α (ξ, ω) + x∈B N C(λ 1 , λ 2 , r, d) f (ξ, ω) f (ξ, σ x ω) + f (ξ, ω) dν N α (ξ, ω). To deal with I 1 (the same method holds for I 2 ). Use (2.16) with A = 2N to get that I 1 ≤ 2C(λ 1 , λ 2 , r, d) x∈B N f (ξ, ω)dν N α (ξ, ω) + C(λ 1 , λ 2 , r, d) x∈B N f (σ x ξ, ω)dν N α (ξ, ω). Since f is a density with respect to ν N α , the first term is less than CN d where C > 0 is a uniform constant in N . Now note that for x ∈ B N , using f (σ x ξ, ω)dν N α (ξ, ω) = 0≤i≤3 η i (x)f (σ x ξ, ω)dν N α (ξ, ω) ≤ 0≤i≤3 j =i α j (x/N ) =0 α i (x/N ) α j (x/N ) η j (x)f (ξ, ω)dν N α (ξ, ω) where we used the change of variable (ξ, ω) → (σ x ξ, ω) and formulas (A.5) in the second inequality. Now as α satisfies (2.11) and (2.12), the set α i (x/N ) α j (x/N ) , α j (x/N ) = 0, 0 ≤ i, j ≤ 3 is uniformly bounded in N . Using again that f is a density we get the desired result. Replacement lemma in the bulk. Let us first introduce a few notations. Given a smooth profile α, and a function φ : Σ N → R, denote ∼ φ( α) the expectation of φ under ν N α . For ∈ N, introduce Λ x = {y ∈ B N , y − x ≤ } where y − x = max{|y i − x i |, 1 ≤ i ≤ d}, and denote η i (x) the average of η in Λ x , that is, η i (x) = 1 |Λ x | y∈Λ x η i (y), for 1 ≤ i ≤ 3. Introduce the vector η (x) = ( η 1 (x), η 2 (x), η 3 (x)) and for ε > 0, V εN (ξ, ω) = 1 |Λ εN x | y∈Λ εN x τ y φ(ξ, ω) − ∼ φ( η εN (0)) . In the sequel, we will write εN instead of εN . The replacement lemma in the bulk stated and proved in [18,Lemma 4.2] is the following: Proposition 2. For any G ∈ C 1,2 0 and for any function φ : Σ N → R , lim sup ε→0 lim sup N →∞ E µ N 1 N d x∈B N T 0 G s (x/N ) τ x V εN (ξ s , ω s )ds = 0. 3.2.3 Replacement lemma at the left hand side boundary for θ ∈ [0, 1). Here we fix θ in [0, 1) and prove the replacement lemma at the left hand side boundary. It essentially states that when performing the macroscopic limit N → ∞, one can replace η i (x) by b i (x/N ). For θ r ∈ [0, 1), the replacement lemma at the right hand side boundary is exactly the same. Recall that this result has been proved for θ = θ r = 0 in [18, Section 6] and we generalize it here to the case where the left hand side (or right hand side) parameter θ is allowed to vary in [0, 1). P µ N t 0 1 N d−1 x∈Γ − N G(s, x/N ) η i,s (x) − b i (x/N ) ds > δ = 0. (3.12) Note that the replacement lemma at the right hand side boundary for θ r ∈ [0, 1) states as above, with the sum in x carrying over Γ + N rather than Γ − N . Proof. Fix an i ∈ {1, 2, 3}. It is enough to show that lim sup N →∞ 1 N d log P µ N t 0 1 N d−1 x∈Γ − N G(s, x/N ) η i,s (x) − b i (x/N ) ds > δ = −∞. Consider α a smooth profile satisfying conditions (2.11) and (2.12). For a > 0, P µ N t 0 1 N d−1 x∈Γ − N G(s, x/N ) η i,s (x) − b i (x/N ) ds > δ ≤ sup (ξ,ω)∈ Σ N dµ N dν N α (ξ, ω) × P ν N α t 0 1 N d−1 x∈Γ − N G(s, x/N ) η i,s (x) − b i (x/N ) ds > δ ≤ exp(K 0 N d )E µ N exp aN d t 0 1 N d−1 x∈Γ − N G(s, x/N ) η i,s (x) − b i (x/N ) ds exp(−aδN d ). We used, in the first inequality, that the Radon-Nikodym derivative of µ N with respect to ν N α is bounded by exp(K 0 N d ) with K 0 a constant, and Tchebychev's inequality in the second line. Therefore, 1 N d log P µ N t 0 1 N d−1 x∈Γ − N G(s, x/N ) η i,s (x) − b i (x/N ) ds > δ ≤ −aδ + K 0 + 1 N d log E ν N α exp aN d t 0 1 N d−1 x∈Γ − N G(s, x/N ) η i,s (x) − b i (x/N ) ds . (3.13) It is enough to show that the last term is uniformly bounded in a and N and then, take a → ∞ with N . Since e |x| ≤ e x + e −x and lim sup N 1 N log(a N + b N ) = max lim sup N 1 N log a N , lim sup N 1 N log b N ,(3.14) we show that the last term in (3.13) without the absolute values, is uniformly bounded in a and N . Apply Feynman-Kac's formula with V (s, (ξ s , ω s )) = aN d N d−1 x∈Γ − N G(s, x/N )(η i,s (x) − b i (x/N )). Since G is bounded, 1 N d log E ν N α exp aN d t 0 1 N d−1 x∈Γ − N G(s, x/N ) η i,s (x) − b i (x/N ) ds ≤ t 0 ds sup f Σ N a N d−1 x∈Γ − N G(s, x/N ) b i (x/N ) − η i (x) f (ξ, ω)dν N α (ξ, ω) + 1 N d < L N f , f > ν N α . Note that for x ∈ Γ − N , b i (x/N ) − η i (x) = j =i (b i (x/N )η j (x) − b j (x/N )η i (x)) and for j = i, η i (x)b j (x/N )f (ξ, ω)dν N α (ξ, ω) = η j (x)f (σ i,x (ξ, ω))b i (x/N )dν N α (ξ, ω). (3.15) Therefore, G(s, x/N ) b i (x/N ) − η i (x) f (ξ, ω)dν N α (ξ, ω) = G(s, x/N ) b i (x/N ) j =i η j (x)f (ξ, ω)dν N α (ξ, ω) − G(s, x/N ) b i (x/N ) j =i η j (x)f (σ i,x (ξ, ω))dν N α (ξ, ω) = −G(s, x/N ) b i (x/N )(1 − η i (x))(f (σ i,x (ξ, ω)) − f (ξ, ω))dν N α (ξ, ω) ≤ A 2 b i (x/N )(1 − η i (x)) f (σ i,x (ξ, ω)) − f (ξ, ω) 2 dν N α (ξ, ω) + 1 2A G(s, x/N ) 2 b i (x/N )(1 − η i (x)) f (σ i,x (ξ, ω)) + f (ξ, ω) 2 dν N α (ξ, ω) (3.16) where we used (2.16) in the last line replacing A by AN , with A > 0. Summing (3.16) over Γ − N and multiplying by a N d−1 yields, η a N d−1 x∈Γ − N (b i (x/N ) − η i (x))f (ξ, ω)dν N α (ξ, ω) ≤ aAN θ 2N d−1 D b, θ,N (f, ν N α ) + a b ∞ G 2 ∞ 2A where the second term comes from Cauchy-Schwarz's inequality, the fact that f is a density, the change of variable formula (3.15) and the fact that b is bounded and that there are of order N d−1 terms in Γ − N . Therefore, taking a = A = N 1−θ 2 , using (3.9), (3.10) and (3.11) to bound < L N √ f , √ f > ν N α and the fact that a Dirichlet form is positive we are left with 1 N d log E ν N α exp aN d t 0 1 N d−1 x∈Γ − N G(s, x/N ) η i,s (x) − b i (x/N ) ds ≤ T sup f N 2 2N d D b, θ,N (f, ν N α ) + G 2 ∞ b ∞ 2 + 1 N d < L N f , f > ν N α ≤ T sup f G 2 ∞ b ∞ 2 − N 2 4N d D N (f, ν N α ) + C 1 + C 2 ≤ T sup f aAN θ 2N d−1 − N 2 2N d D b, θ,N (f, ν N α ) + G 2 ∞ a b ∞ 2A + T C 1 + T C 2 ≤ T b ∞ G 2 ∞ 2 + T C 1 + T C 2 , hence the desired result. 3.2.4 Replacement lemma at the left hand side boundary for θ ≥ 1. For θ ≥ 1, the replacement lemma at the boundary involves particle densities over small macroscopic boxes. Again, the same replacement lemma holds at the right hand side boundary for θ r ≥ 1. In fact, we will see in the proof that the lemma holds for any positive value of θ , resp. θ r regardless of whether θ resp. θ r ≥ 1. Proof. Consider α a smooth profile satisfying conditions (2.11) and (2.12) . By the entropy inequality (see [15,Appendix 1]), for any A > 0, .18) is bounded by K 0 /A. Let us show that the second term tends to zero when N → ∞ and ε → 0 and then take A arbitrarily big. Again, by (3.14), it is enough to show that the second term in (3.18) without the absolute values in the exponential, tends to zero. By Feynman-Kac's inequality, E µ N 1 N d−1 x∈Γ − N t 0 G(s, x/N )(η εN i,s (x) − η i,s (x))ds ≤ 1 AN d H(µ N |ν N α ) + 1 AN d log E ν N α exp AN d 1 N d−1 x∈Γ − N t 0 G(s, x/N ) η εN i,s (x) − η i,s (x) ds . (3.18) As B N is finite, there is a constant K 0 > 0 such that H(µ N |ν N α ) ≤ K 0 N d so the first term in (31 AN d log E ν N α exp AN d 1 N d−1 x∈Γ − N t 0 G(s, x/N ) η εN i,s (x) − η i,s (x) ds ≤ t 0 ds sup f 1 N d−1 x∈Γ − N G(s, x/N ) η εN i (x) − η i (x) f (ξ, ω)dν N α (ξ, ω) + 1 AN d < L N f , f > ν N α (3.19) where the supremum is taken over densities. Write η εN i (x) − η i (x) as a d-dimensional telescopic sum. For that, note that for any y ∈ Λ εN , there is an integer p y ≤ dεN and p y points u 1 = x, ..., u p = x + y such that for every 1 ≤ j ≤ p y − 1, u j+1 − u j = e rj where r j ∈ {1, ..., d}. Therefore, η εN i (x) − η i (x) = 1 (εN ) d y∈Λ εN py−1 j=1 η i (u j + e rj ) − η i (u j ) . (3.20) Performing the change of variable (ξ, ω) → (ξ uj ,uj +er j , ω uj ,uj +er j ) =: (ξ uj , ω uj ) and using (A.2), 1 N d−1 x∈Γ − N G(s, x/N ) η εN i (x) − η i (x) f (ξ, ω)dν N α (ξ, ω) (3.21) = 1 2N d−1 x∈Γ − N G(s, x/N ) 1 (εN ) d y∈Λ εN py−1 j=1 η i (u j + e rj ) − η i (u j ) f (ξ, ω) − f (ξ uj , ω uj ) dν N α (ξ, ω) (3.22) + 1 2N d−1 x∈Γ − N G(s, x/N ) 1 (εN ) d y∈Λ εN py−1 j=1 η i (u j + e rj ) − η i (u j ) f (ξ uj , ω uj ) 1 − ν N α (ξ uj , ω uj ) ν N α (ξ, ω) dν N α (ξ, ω). (3.23) First we deal with (3.22). Using that |η i (u j + e rj ) − η i (u j )| ≤ 1 and inequality (2.16), (3.22) ≤ 1 2N d−1 x∈Γ − N B 2(εN ) d y∈Λ εN py−1 j=1 f (ξ uj , ω uj ) − f (ξ, ω) 2 dν N α (ξ, ω) + 1 2N d−1 x∈Γ − N G 2 ∞ 2B(εN ) d y∈Λ εN py−1 j=1 f (ξ uj , ω uj ) + f (ξ, ω) 2 dν N α (ξ, ω) (3.24) where B > 0 will be chosen later. Now for x ∈ Γ − N , y∈Λ εN py−1 j=1 f (ξ uj , ω uj ) − f (ξ, ω) 2 ≤ dεN d k=1 (z,z+e k )∈x+Λ εN f (ξ z,z+e k , ω z,z+e k ) − f (ξ, ω) 2 and y∈Λ εN py−1 j=1 f (ξ uj , ω uj ) + f (ξ, ω) 2 ≤ dεN d k=1 (z,z+e k )∈x+Λ εN f (ξ z,z+e k , ω z,z+e k ) + f (ξ, ω) 2 . Therefore, x∈Γ − N 1 (εN ) d y∈Λ εN py−1 j=1 f (ξ uj , ω uj ) − f (ξ, ω) 2 dν N α (ξ, ω) ≤ dεN x∈Γ − N 1 (εN ) d d k=1 (z,z+e k )∈x+Λ εN f (ξ z,z+e k , ω z,z+e k ) − f (ξ, ω) 2 dν N α (ξ, ω) ≤ d d k=1 (z,z+e k )∈B 2 N f (ξ z,z+e k , ω z,z+e k ) − f (ξ, ω) 2 dν N α (ξ, ω) = d D D N (f, ν N α ) (3.25) where the last inequality comes from an integration by part formula and the fact that there are of order N d−1 elements in Γ − N . Now using that f is a density, 1 2N d−1 x∈Γ − N 1 (εN ) d y∈Λ εN py−1 j=1 f (ξ uj , ω uj ) + f (ξ, ω) 2 dν N α (ξ, ω) (3.26) ≤ dεN + 1 N d−1 x∈Γ − N dεN (εN ) d (z,z+e k )∈x+Λ εN f (ξ z,z+e k , ω z,z+e k )dν N α (ξ, ω). (3.27) Note that (ξ, ω) → f (ξ z,z+e k , ω z,z+e k ) is not a density. Using the explicit expression of the product measure ν N α , one has that for (x, x + e k ) ∈ B 2 N ν N α (ξ x,x+e k , ω x,x+e k ) ν N α (ξ, ω) = 3 i=1 α i (x/N ) α 0 (x/N ) ηi(x+e k )−ηi(x) α i ((x + e k )/N ) α 0 ((x + e k )/N ) . Now using that α i x+e k N = α i x N + O 1 N we have the following inequality: there is a constant C > 0 such that 1 − ν N α (ξ x,x+e k , ω x,x+e k ) ν N α (ξ, ω) ≤ C N . (3.28) Therefore, for (x, x + e k ) ∈ B 2 N , f (ξ x,x+e k , ω x,x+e k )dν N α (ξ, ω) ≤ 1 + C N (3.29) so the last term in (3.27) is smaller than 1 N d−1 x∈Γ − N dεN (εN ) d z,z+e k ∈x+Λ εN 1 + C N = dεN B + dε B . (3.30) Collecting inequalities (3.24), (3.25) and (3.30), we get that (3.22) ≤ dB 2D D N (f, ν N α ) + G 2 ∞ dεN B + dε B . (3.31) Now we deal with (3.23). By inequalities (3.28) and (3.29) and using that η i ∈ {0, 1}, (3.23) ≤ G ∞ N d−1 x∈Γ − N 1 (εN ) d y∈Λ εN py−1 j=1 f (ξ uj , ω uj ) 1 − ν N α (ξ uj , ω uj ) ν N α (ξ, ω) dν N α (ξ, ω) ≤ C G ∞ N d x∈Γ − N 1 (εN ) d y∈Λ εN py−1 j=1 1 + C N ≤ G ∞ 1 + C N Cdε. (3.32) Therefore, collecting inequalities (3.31) and (3.32), for any density f , 1 N d−1 x∈Γ − N η εN i (x) − η i (x) f (ξ, ω)dν N α (ξ, ω) ≤ dB 2DN d−1 D N (f, ν N α ) + dεN B + dε B + Cdε + C 2 dε N . This, combined with (3.19) as well as Lemma 1 yields: 1 AN d log E ν N α exp AN d 1 N d−1 x∈Γ − N t 0 G(s, x/N ) η εN i,s (x) − η i,s (x) ds ≤ sup f T G ∞ dB 2DN d−1 − N 2 4AN d D N (f, ν N α ) + T G ∞ ε B (N + d) + T G ∞ Cdε 1 + C N + C 4 A (3.33) with C 4 > 0, a constant that only depends on d. Taking B = (N D)/(2dT G ∞ A) and putting together (3.19) and (3.33) yields E µ N 1 N d−1 x∈Γ − N t 0 G(s, x/N )(η εN i,s (x) − η i,s (x))ds ≤ K 0 Aε 1 + 1 N + T G ∞ Cdε 1 + C N + C 4 A with K 0 a constant depending on T, G, d and taking ε → 0 and then A → ∞ we get the desired result. Energy estimates In view of the proof of uniqueness of the limit of the sequence of probability measures (Q θ N ) N ≥1 , we state that any limiting measure Q θ is concentrated on a trajectory belonging to a specific functional space. This allows to define the hydrodynamic limit at the boundary. Proposition 5. Let θ ∈ (R + ) 2 and Q θ be a limit point of the sequence of probability measures (Q θ N ) N ≥1 . Then, the probability measure Q θ is concentrated on paths ρ(t, u)du such that for every 1 ≤ i ≤ 3, ρ i belongs to L 2 ((0, T ); H 1 (B)). This follows from the Lemma below and the Riesz Representation Theorem. A similar proof can be found, for instance in [15, section 5]. Lemma 2. For any θ ∈ (R + ) 2 , there is a constant K θ > 0 such that for every 1 ≤ i ≤ 3, E Q θ sup H T 0 B d k=1 ∂ e k H(s, u)ρ i (s, u)duds − K θ T 0 B H(s, u) 2 duds < ∞ (3.34) where the supremum is carried over functions H ∈ C 0,2 c ([0, T ] × B). For the proof of Lemma 2, one can follow the arguments in [15,Section 5]. First prove (3.34) for a dense and countable set of elements of C 0,2 c ([0, T ] × B) thanks to Feynmann-Kac's inequality. Then, use an integration by part to deal with the spatial derivatives in H, as well as a change of variable. Characterization of the limit point in the (Dirichlet ; Robin) mixed regime In order to show that the limit point of the sequence of probability measures (Q θ N ) N ≥1 lies on the trajectory with density profile the unique solution of the hydrodynamic equation associated to θ and γ, we give a characterization result (see Proposition 6). We will focus on the (Dirichlet ; Robin) mixed regime since the (Neumann ; Robin) mixed regime can be proved following the same lines. Therefore, take θ ∈ [0, 1) and θ r ≥ 1. As mentioned in the introduction, in one dimension, the macroscopic trajectories are continuous in space and their values at the boundaries are defined in the classical sense. This is no longer valid in higher dimension. To deal with this difficulty we use the regularity of the trajectories proved in Proposition 5: the trajectories lie in L 2 ((0, T ); H 1 (B)) so their values at the boundary are defined via the trace operator (see Lemma 3). Proposition 6. If Q θ is a limit point of the sequence of probability measures (Q θ N ) N ≥1 , then Q θ π, I G ( ρ)(t) + D 3 i=1 t 0 Γ − b i (r)(∂ e1 G i,s )(r)n 1 (r).dS(r) + Γ + ρ i (s, r)(∂ e1 G i,s )(r)n 1 (r).dS(r) ds − 3 i=1 t 0 Γ + G i (r)(b i (r) − ρ i (s, r))n 1 (r).dS(r)ds = 0, ∀t ∈ [0, T ], ∀ G ∈ C θ = 1 (3.35) where recall that I G ( ρ) was defined in (2.18). Proof. The fact that any limit point is concentrated on trajectories which are absolutely continuous with respect to the Lebesgue measure comes from Proposition 5. Let Q θ be a a limit point of the sequence of probability measures (Q θ N ) N ≥1 , To prove (3.35), it is enough to show that for any fixed δ > 0 and G ∈ C 1,2 0,− , Q θ π, sup 0≤t≤T I G ( ρ)(t) + D 3 i=1 t 0 Γ − b i (r)(∂ e1 G i,s )(r)n 1 (r).dS(r) + Γ + ρ i (s, r)(∂ e1 G i,s )(r)n 1 (r).dS(r) ds − 3 i=1 t 0 Γ + G i (r)(b i (r) − ρ i (s, r))n 1 (r).dS(r)ds > δ = 0. (3.36) Here, note that for s ∈ [0, T ] and r ∈ Γ, ρ i (s, r) stands for Tr(ρ)(s, r) which is well defined since ρ is in H 1 (B)). By the triangular inequality, it suffices to prove that for any 1 ≤ i ≤ 3, r))n 1 (r).dS(r)ds > δ = 0. L 2 ([0, T ],Q θ π, sup 0≤t≤T I Gi (ρ i )(t) + D t 0 Γ − b i (r)(∂ e1 G i,s )(r)n 1 (r).dS(r) + Γ + ρ i (s, r)(∂ e1 G i,s )(r)n 1 (r).dS(r) ds − t 0 Γ + G i (r)(b i (r) − ρ i (s, (3.37) As usual, we would like to approximate ρ by a convolution of its associated empirical measure with an approximation of the identity. Indeed, that convolution product can then be written in terms of the mean value of the configuration in a microscopic box. This is straightforward in the bulk, however, for the boundary terms, one needs to justify that such an approximation works (see (3.49)).Without loss of generality, let us deal with i = 1. We turn to our martingales (3.1) M N 1,t ( G) and recall that we have proved that its quadratic variation vanishes as N ↑ ∞. For ε > 0, introduce the set B N,ε = {−N (1 − ε), · · · , N (1 − ε)} × T d−1 N . By Proposition 2, M N 1,t ( G) =< π N 1,t , G 1,t > − < π N 1,0 , G 1,0 > − t 0 < π N 1,s , ∂ s G 1,s > ds − t 0 D N d x∈B N \Γ N ∆G 1,s (x/N )η 1,s (x)ds + t 0 D N d−1 x∈Γ − N b 1 (x/N )∂ e1 G 1,s (x/N )ds + t 0 D N d−1 x∈Γ − N ∂ e1 G 1,s (x/N ) η 1,s (x) − b 1 (x/N ) ds − t 0 D N d−1 x∈Γ + N ∂ e1 G 1,s (x/N )η εN 1,s (x)ds + D N d−1 x∈Γ + N ∂ e1 G 1,s (x/N ) η εN 1,s (x) − η 1,s (x) ds + t 0 D N d−1 x∈Γ + N G 1,s (x/N ) η εN 1,s (x) − b(x/N ) ds − t 0 1 N d x∈B N,ε G 1,s x N 2d(λ 1 η εN 1 (x) + λ 2 η εN 3 (x))η εN 0 (x) + η εN 3 (x) − (r + 1)η εN 1 (x) ds + R N, ε, G 1 , (η t ) t∈[0,T ] , (3.38) where R N, ε, G 1 , (η t ) t∈[0,T ] is a random variable satisfying lim ε→0 lim N →∞ E µ N R N, ε, G 1 , (η t ) t∈[0,T ] = 0. From Proposition 3 and Proposition 4, the martingale M N 1,t ( G) can be rewritten as M N 1,t ( G) =< π N 1,t , G 1,t > − < π N 1,0 , G 1,0 > − t 0 < π N 1,s , ∂ s G 1,s > ds − t 0 D N d x∈B N \Γ N ∆G 1,s (x/N )η 1,s (x)ds + t 0 D N d−1 x∈Γ − N b 1 (x/N )∂ e1 G 1,s (x/N )ds − t 0 D N d−1 x∈Γ + N ∂ e1 G 1,s (x/N )η εN 1,s (x)ds + D N d−1 x∈Γ + N G 1,s (x/N ) η εN 1,s (x) − b(x/N ) ds − t 0 1 N d x∈B N,ε G 1,s x N 2d(λ 1 η εN 1 (x) + λ 2 η εN 3 (x))η εN 0 (x) + η εN 3 (x) − (r + 1)η εN 1 (x) ds + R N, ε, G 1 , (η t ) t∈[0,T ] , (3.39) where R N, ε, G 1 , (η t ) t∈[0,T ] is a random variable satisfying lim ε→0 lim N →∞ E µ N R N, ε, G 1 , (η t ) t∈[0,T ] = 0. On the other hand, a computation of the quadratic variation of the martingale M N 1,t ( G) shows that its expectation vanishes as N ↑ ∞. Therefore, by Doob's inequality, for every δ > 0, lim sup N →∞ P µ N sup 0≤t≤T M N 1,t ( G) > δ = 0, (3.40) Now, introduce the following approximations of the identity on B: u ε (x) = 1 (2ε) d 1 [−ε,ε] d (x), (3.41) u right ε (x) = 1 ε(2ε) d−1 1 [0,ε]×[−ε,ε] d−1 (x), and u lef t ε (x) = 1 ε(2ε) d−1 1 [−ε,0]×[−ε,ε] d−1 (x). (3.42) Note that for ε > 0, 1 ≤ i ≤ 3, x ∈ B N,ε , y ∈ Γ + N , and z ∈ Γ − N , η εN i (x) = (2εN ) d (2εN + 1) d π N i * u ε (x/N ), (3.43) η εN i (y) = (2εN ) d−1 (2εN + 1) d−1 π N i * u right ε (y/N ), and η εN i (z) = (2εN ) d−1 (2εN + 1) d−1 π N i * u lef t ε (z/N ). (3.44) Here we will only make used of (3.43) and the first relation in (3.44) since we need to replace elements in the bulk and the right hand side boundary of the system to recover the weak formulation of the equation in the (Dirichlet; Robin) regime. For regimes where a replacement is needed on the left hand side boundary, we use the second relation in (3.44) in the same way. We may thus replace in (3.39) and (3.40), η εN i by π N i * u ε in the bulk and η εN i by π N i * u right ε at the right boundary. Therefore, for any δ > 0. lim sup ε→0 lim sup N →∞ Q θ N sup 0≤t≤T F G,t 1,N, π ≥ δ = 0, where for any trajectory π and for any t ∈ [0, T ], F G,t 1,N, π =< π 1,t , G 1,t > − < π 1,0 , G 1,0 > − t 0 < π 1,s , ∂ s G 1,s > ds − t 0 D π 1,s , ∆G 1,s ds + t 0 D N d−1 x∈Γ − N b 1 (x/N )∂ e1 G 1,s (x/N )ds − t 0 D N d−1 x∈Γ + N ∂ e1 G 1,s (x/N ) π 1,s * u right ε (x)ds + t 0 D N d−1 x∈Γ + N G 1,s (x/N ) π 1,s * u right ε (x) − b(x/N ) ds − t 0 1 N d x∈B N,ε G 1,s (x/N ), F 1 π 1,s * u ε (x/N ), π 2,s * u ε (x/N ), π 3,s * u ε (x/N ) ds ,(3.Q θ N sup 0≤t≤T F G,t 1, π ≥ δ = 0 , where for any trajectory π and for any t ∈ [0, T ], F G,t 1, π =< π 1,t , G 1,t > − < π 1,0 , G 1,0 > − t 0 < π 1,s , ∂ s G 1,s > ds − D t 0 π 1,s , ∆G 1,s ds + D t 0 Γ − b 1 (r)∂ e1 G 1,s (r) drds − D t 0 Γ + ∂ e1 G 1,s (r) π 1,s * u right ε (r) drds + t 0 Γ + G 1,s (r) π 1,s * u right ε (r) − b(r) drds − t 0 B1,ε G 1,s (r), F 1 π 1,s * u ε (r), π 2,s * u ε (r), π 3,s * u ε (r) drds , (3.46) with B 1,ε = [−1 + ε, 1 + ε] × T d−1 . By the continuity of the function π → F G,t 1, π , for each ε > 0, we get for any limit point Q θ of the sequence of probability measures (Q θ N ) N ≥1 , lim sup ε→0 Q θ sup 0≤t≤T F G,t 1, π ≥ δ = 0 . (3.47) To conclude the proof, it remains to prove that we may replace the convolutions appearing in the functional F G,t 1, by the associated density of the trajectory. By Proposition 5, Q θ is concentrated on paths ( π(t, dr)) t∈[0,T ] = ( ρ(t, r)dr) t∈[0,T ] which are absolutely continuous with respect to the Lebesgue measure and such that for every 1 ≤ i ≤ 3, ρ i belongs to L 2 ((0, T ); H 1 (B)). For the replacement of the convolution with the density in the bulk, since u ε is an approximation of the identity in L 1 (B) and the functions F i are Lipschitz, the random variables For the other terms in F G,t 1, , by the dominated convergence Theorem, for almost every trajectory ( π(t, dr)) t∈[0,T ] = ( ρ(t, r)dr) t∈[0,T ] with ρ 1 ∈ L 2 ((0, T ); H 1 (B)), t 0 B1,ε G 1,s (r), F 1 π 1,s * u ε (r), π 2,s * u ε (r), π 3,s * u ε (r) drds converge Q θ almost surely to t 0 B1,ε G 1,s (r), F 1 ρ 1,s (r), ρ 2,s (r), ρ 3,s (r) drds .lim ε→0 D t 0 Γ + ∂ e1 G 1,s (r) π 1,s * u r ε (r) drds − t 0 Γ + G 1,s (r) π 1,s * u r ε (r) − b(r) drds = D t 0 Γ + ∂ e1 G 1,s (r)Tr(ρ 1,s )(r) drds − t 0 Γ + G 1,s (r) Tr(ρ 1,s )(r) − b(r) drds. Uniqueness of the limit points In order to finish the proof of the hydrodynamic limit specific to each regime we are left to show that each boundary valued problem (2.20) and (2.24) with fixed initial data admits a unique solution. For that, we use the standard method which consists in decomposing the difference of two solutions on the orthonormal basis of a well chosen eigenvectors of the Laplacian. The choice of the family of eigenvectors is not necessarily intuitive and depends on the boundary conditions of the mixed regime considered. As we are working in dimension d ≥ 1, we will need to control integral terms on the boundary. Therefore, we will make use of the following result regarding the continuity of the trace operator. We refer to [6, Part II Section 5] for a detailed survey of the trace operator. Theorem 3. (see [6]) Fix 1 ≤ p < ∞ and Ω an open subspace of R d with smooth boundary ∂Ω. There is a constant C tr > 0 depending only on Ω and p such that for any ϕ ∈ C ∞ (Ω), ϕ L p (∂Ω) ≤ C tr ϕ W 1,p where . L p (∂Ω) denotes the L p norm on ∂Ω and . W 1,p the Sobolev norm on Ω given by ϕ W 1,p = ϕ p L p (Ω) + ∇ϕ p L p (Ω) 1/p where ∇ϕ p L p (Ω) = d i=1 ∂ ei ϕ p L p (Ω) . Remark 3. For p = 2 and Ω = B, ϕ 2 L 2 (∂Ω) ≤ ϕ 2 L 2 (Ω) + ∇ϕ 2 L 2 (Ω) (3.51) In particular, C tr = 1. In the sequel we only make use of (3.51) but we stated Theorem 3 by sake of completeness. Proof. By Liouville's Theorem stated for instance in [6], there is a countable system {V n , α n , n ≥ 1} of eingensolutions for the problem −∆φ = αφ ∂ e1 φ |Γ = 0 (3.52) in H 1 (B) and containing all possible eigenvalues. The set {V n , n ≥ 1} forms a complete, orthonormal system in the Hilbert space L 2 (B) and the eigenvalues 0 ≤ α 1 < α 2 < ... < α n −→ n→∞ ∞ have finite multiplicity. Note that for any U, W ∈ H 1 (B), < U, W > 2 = lim n→∞ n k=1 < U, V k >< W, V k >, (3.53) < ∇U, ∇W > 2 = lim n→∞ n k=1 α k < U, V k >< W, V k >, (3.54) U 2 L 2 (Γ + ) = lim n→∞ n k=1 Γ + U (r)V k (r)n 1 (r)dS(r) 2 . (3.55) One can check that since we are working on (−1, 1) × T d−1 , for k = (k 1 , ..., k d ) ∈ N × (N * ) d−1 , V k (x 1 , ...x d ) = 2 d−1 2 cos k 1 πx 1 2 + π 2 d i=2 sin(k i πx i ) and α k = (k 1 π) 2 4 + d i=2 k 2 i π 2 , V k (x 2 , ..., x d ) = 2 d−1 2 d i=2 sin(k i πx i ) andα k = d i=2 k 2 i π 2 . (3.56) Note that by abuse of notations we indexed the family V k by N * instead of N × (N * ) d−1 but this is not a problem because we can give an order to elements of N × (N * ) d−1 . Consider ρ 1 and ρ 2 two solutions of (2.24) associated to the same initial profile and for n ∈ N and t > 0, introduce G n (t) = 3 i=1 n k=1 | < ρ 1 i − ρ 2 i , V k > | 2 . (3.57) Let us show that lim n→∞ G n (t) = ρ 1 − ρ 2 2 2 =: G(t) = 0. For that, apply the weak formulation (2.24) with V k : for any 1 ≤ i ≤ 3 < (ρ 1 i − ρ 2 i )(t, .), V k > = −Dα k t 0 < (ρ 1 i − ρ 2 i )(s, .), V k > ds + t 0 < (F i ( ρ 1 ) − F i ( ρ 2 ))(s, .), V k > ds − t 0 Γ + (ρ 1 i − ρ 2 i )(s, r)V k (r)n 1 (r).dS(r)ds. (3.58) Therefore < ρ 1 i (t, .) − ρ 2 i (t, .), V k > is time differentiable with derivative: ∂ t < ρ 1 i (t, .) − ρ 2 i (t, .), V k > = −Dα k < ρ 1 i (t, .) − ρ 2 i (t, .), V k > + < F i ( ρ 1 (t, .)) − F i ( ρ 2 (t, .)), V k > − Γ + (ρ 1 i − ρ 2 i )(t, r)V k (r)n 1 (r).dS(r) (3.59) and so is G n , with G n (t) = −2D 3 i=1 n k=1 α k < ρ 1 i,t − ρ 2 i,t , V k > 2 + 2 3 i=1 n k=1 < F i ( ρ 1 t ) − F i ( ρ 2 t ), V k >< ρ 1 i,t − ρ 2 i,t , V k > − 2 3 i=1 n k=1 Γ + (ρ 1 i − ρ 2 i )(t, r)V k (r)n 1 (r).dS(r) < ρ 1 i,t − ρ 2 i,t , V k > ≤ −2D 3 i=1 n k=1 α k < ρ 1 i,t − ρ 2 i,t , V k > 2 + 3 i=1 n k=1 < F i ( ρ 1 t ) − F i ( ρ 2 t ), V k > 2 +G n (t) + 1 A 3 i=1 n k=1 Γ + (ρ 1 i − ρ 2 i )(t, r)V k (r)n 1 (r).dS(r) 2 + AG n (t),− 2D ∇( ρ 1 − ρ 2 ) 2 2 + 3 i=1 F i ( ρ 1 ) − F i ( ρ 2 ) 2 2 + (1 + A) ρ 1 − ρ 2 2 2 + 1 A ρ 1 − ρ 2 2 L 2 (Γ) . (3.61) By the trace inequality (3.51), ρ 1 − ρ 2 2 L 2 (Γ) ≤ ρ 1 − ρ 2 2 L 2 (B) + ∇( ρ 1 − ρ 2 ) 2 2 . (3.62) Furthermore, using that ρ 1 and ρ 2 take their values in [0, 1] 3 , there is a constant C := C(λ 1 , λ 2 , r, d) > 0 such that for any ρ a , ρ b ∈ [0, 1] 3 and 1 ≤ i ≤ 3, F i ( ρ a ) − F i ( ρ b ) ≤ C 3 j=1 |ρ a j − ρ b j |. Then, by Cauchy-Schwarz's inequality, there is a constant C > 0 such that for any 1 ≤ i ≤ 3, Proof. The proof follows the same lines as the previous one except that we consider another family of eigenfunctions of the Laplacian. Indeed, consider the following boundary-eigenvalue problem for the Laplacian: F i ( ρ a ) − F i ( ρ b ) 2 2 ≤ C 3 j=1 ρ a i − ρ b i 2 2 .       −∆φ = γφ φ(x) = 0 for x ∈ Γ − × T d−1 ∂ e1 φ(x) = 0 for x ∈ Γ + × T d−1 φ ∈ H 1 (B). (3.65) Again, one can check that the countable system of eigensolutions {W n , γ n , n ≥ 1} given below (in (3.66)) for the problem (3.65) contains all possible eigenvalues and is a complete, orthonormal system in the Hilbert space L 2 (B), that the eigenvalues γ n have finite multiplicity and that 0 < γ 1 ≤ γ 2 ... ≤ γ n → ∞. Furthermore, (3.53), (3.54) and (3.55) stay valid when one replaces V k by W k , where, for k = (k 1 , ..., k d ) ∈ N × (N * ) d−1 , W k (x) = 2 d−1 2 (−1) k1 cos π 4 + k 1 π 2 x + sin π 4 + k 1 π 2 x d i=2 sin(k i πx i ) (3.66) with γ k = π 4 + k 1 π 2 2 + d i=2 k 2 i π 2 . Again, by abuse of notations we have indexed the W k 's by N * instead of (N * ) d . As before, take ρ 1 and ρ 2 two solutions of (2.20) with same initial data and introduce H n (t) = 3 i=1 n k=1 < ρ 1 i (t, .) − ρ 2 i (t, .), W k > 2 (3.67) and H(t) = ( ρ 1 − ρ 2 )(t, .) 2 2 . (3.68) Using the weak formulation (2.22) with W k , we get that for any 1 ≤ i ≤ 3, < (ρ 1 i − ρ 2 i )(t, .), W k > = −Dγ k t 0 < (ρ 1 i − ρ 2 i )(s, .), W k > ds + t 0 < (F i ( ρ 1 ) − F i ( ρ 2 ))(s, .), W k > ds − t 0 Γ + (ρ 1 i − ρ 2 i )(s, r)W k (r)n 1 (r).dS(r)ds (3.69) where theW k =V k are defined in (3.56). Then, one concludes following exactly the same lines as the proof of Theorem 4. Uniqueness of the solution in the other regimes In order to prove uniqueness in the other regimes, one can follow the same classic method used above. The orthonormal basis used to decompose the difference of two solutions as in (3.57) or (3.67) then depends on the boundary conditions. For the (Dirichlet ; Dirichlet) regime, the decomposition is carried out on the eigenvectors of the following boundary-eigenvalue problem for the Laplacian: for which the associated family of eigenvectors is −∆φ = δφ φ ∈ H 1 0 (B).U k (x 1 , ...x d ) = 2 d−1 2 d i=1 sin(k i πx i ), with eigenvalues given by δ k = d i=1 k 2 i π 2 for k = (k 1 , ..., k d ) ∈ (N * ) d . As before, for V, W ∈ L 2 (B), < V, W > 2 = lim n→∞ n k=1 < V, U k > 2 < W, U k > 2 , (3.71) < ∇V, ∇W > 2 = lim n→∞ n k=1 δ k < V, U k >< W, U k >, (3.72) V 2 L 2 (Γ) = lim n→∞ n k=1 Γ V (r)Ǔ k (r)n 1 (r)dS(r) 2 (3.73) where theǓ k =V k are defined in (3.56). Hydrostatic limit In this section, we prove Theorem 2 which states that when the parameters r, λ 1 , λ 2 , d, D satisfy certain conditions, starting from an invariant measure, the system converges to the stationary profile of the corresponding hydrodynamic equation. Precisely, recall that in Section 2, for θ ∈ (R + ) 3 we defined µ ss N ( θ) as the sequence of unique invariant measures for the irreducible dynamics defined by (2.10). The hydrostatic principle states that this sequence is associated to the unique stationary solution of the hydrodynamic equation, if existence and uniqueness of such a solution hold. For the proof, we were inspired by [9] and the key argument relies on the convergence of all the trajectories satisfying the hydrodynamic equation to the unique stationary profile of these equations. In [9], the convergence of trajectories is established thanks to a comparison principle. The difficulty here is that we are dealing with a system of coupled equations and we need to define a specific order for which such a comparison principle holds. Now in [17], it has been proved that at the microscopic level, the interacting particle system is attractive only for the following order: 2 < 0 < 3 < 1. (4.1) That means that given two configurations η ≤ ∼ η, it is possible to build a coupling between (η t ) t≥0 and ( ∼ η t ) t≥0 where both these processes evolve according to the dynamics given by (2.10), such that η 0 ≤ ∼ η 0 and almost surely, for all t ≥ 0, η t ≤ ∼ η t pointwise in the sense of (4.1). Note that using the main result from [4], one can show that the system remains attractive when adding an exchange and reservoir dynamics. It is then natural to think that attractiveness also holds at the macroscopic level through a comparison principle. A comparison principle means that if two profiles are such that at a certain time, one is smaller than the other almost everywhere, then the same is true at any later time. Considering the microscopic order (4.1) it is natural to consider that the largest state at the macroscopic level corresponds to (ρ 1 = 1, ρ 2 = 0, ρ 3 = 0) and the smallest state to (ρ 2 = 1, ρ 1 = ρ 3 = 0). We will work under the following change of coordinates:    ρ 1 T := ρ 1 + ρ 3 R := 1 − (ρ 2 + ρ 3 ) (4.2) which is consistent with the fact that (1, 1, 1) corresponds to the largest profile (ρ 1 = 1, ρ 2 = 0, ρ 3 = 0) and (0, 0, 0) with the lowest one (ρ 2 = 1, ρ 1 = ρ 3 = 0). In the sequel, we will say that given two profiles ρ and φ, ρ ≤ φ if:    ρ 1 ≤ φ 1 ρ 1 + ρ 3 ≤ φ 1 + φ 3 1 − (ρ 2 + ρ 3 ) ≤ 1 − (φ 2 + φ 3 ) (4.3) almost everywhere. We prove a comparison principle under that change of coordinates (see Lemma 3). As previously, since we are working in any dimension d ≥ 1 with mixed boundary conditions, some care must be taken to deal with the integral terms on Γ when proving the comparison principle. For that, we strongly rely on analytical tools stated in [21]. Under the change of coordinates (4.2), the coupled equations in the bulk become, :    ∂ t ρ 1 = D∆ρ 1 + F 1 (ρ 1 , T, R) ∂ t T = D∆T + H(ρ 1 , T, R) ∂ t R = D∆R + J(R) (4.4) with    F 1 (ρ 1 , T, R) = 2d (λ 1 − λ 2 )ρ 1 + λ 2 T (R − ρ 1 ) + T − (r + 2)ρ 1 H(ρ 1 , T, R) = 2d (λ 1 − λ 2 )ρ 1 + λ 2 T (1 − T ) − T J(R) = −(r + 1)R + 1. (4.5) We will see that the comparison principle stated and proved in Lemma 3 yields the following Theorem which is used to prove Theorem 2. Note that this result can be equivalently formulated in the change of coordinates (4.2) and we will prove it in that setting in the next subsection. In order to answer that, we simulated the solutions to the equation in the (Neumann ; Neumann) regime for which the constant profile (ρ 1 = 0, ρ 2 = r r+1 , ρ 3 = 0) is stationary. Indeed, F 1 0, r r + 1 , 0 = F 2 0, r r + 1 , 0 = F 3 0, r r + 1 , 0 = 0 and it corresponds to the extinction regime, that is, there are no more wild insects. We observed (see below in the Appendix B) that in dimensions 1, for parameters λ 1 = 1, λ 2 = 0.75 and D = r = 1, for which conditions (H 1 ) are not satisfied, the solution of the hydrodynamic equation starting from ρ 1 = 1, ρ 2 = ρ 3 = 0 converges to a constant profile which is not (0, r r+1 , 0) so uniqueness does not hold. Simulations confirm that Theorem 6 does not hold in all generality and that conditions on the parameters are necessary, although conditions (H 1 ) might not be the optimal ones. Proof of the hydrostatic principle Let us prove Theorem 2. We prove the first point, the second one follows in the same way. Denote A T ⊂ D([0, T ], M + 3 ) the set of trajectories { ρ(t, u)du, 0 ≤ t ≤ T } whose density ρ = (ρ 1 , ρ 2 , ρ 3 ) satisfies conditions (2.21) and (2.23) of the definition of a weak solution of (2.20) for some initial profile ρ 0 . Consider Q * ss ( θ) a limit point of the sequence (Q N µ ss N ( θ) ) N ≥1 associated to the invariant measures. By Theorem 1, Q * ss ( θ) A T = 1. (4.8) Now consider Q N k ss ( θ) a subconverging sequence of (Q N µ ss N ( θ)) N ≥1 . By stationarity of µ ss N ( θ) E Q N k ss ( θ) < π N , G > − < ρ, G > = E Q N k ss ( θ) < π N T , G > − < ρ, G > (4.9) and lim k→∞ E Q N k ss (θ) < π N T , G > − < ρ, G > = E Q * ss (θ) < π T , G > − < ρ, G > 1 A T ≤ 3 i=1 G i ∞ sup ρ 3 i=1 ρ i (T, .) − ρ i (.) 1 (4.10) where the supremum is taken over trajectories which are solutions to (2.20). Then, one concludes thanks to (4.6) in Theorem 6. Proof of Theorem 6 In order to prove Theorem 6 we first establish a comparison principle (Lemma 3). Then, we show that the difference between the largest solution and the smallest solution vanishes (Lemma 4). Using an integration by parts, it is useful to rewrite the weak formulations (2.22) and (2.22), in the following suitable forms: for any 0 ≤ τ ≤ t ≤ T , for any G ∈ C 2 ([0, T ] × B), < ρ t , G t > − < ρ τ , G τ >= t τ < ρ s , ∂ s G s > ds − D t τ B ∇ ρ s · ∇ G s (r)drds − t τ < F ( ρ s ), G s > ds − D 3 i=1 t τ Γ − b i (r)(∂ e1 G i,s )(r)n 1 (r).dS(r)ds + 3 i=1 t τ Γ + G i (r)(b i (r) − ρ i (s, r) )n 1 (r).dS(r)ds = 0, (4.11) and r))n 1 (r).dS(r)ds = 0. < ρ t , G t > − < ρ τ , G τ >= t τ < ρ s , ∂ s G s > ds − D t τ B ∇ ρ s · ∇ G s (r)drds − t τ < F ( ρ s ), G s > ds − Γ + G i (r)(b i (r) − ρ i (s, (4.12) Lemma 3. Consider ρ 1 0 and ρ 2 0 two initial profiles. • Denote ρ 1 t resp. ρ 2 t the solutions to the (Dirichlet ; Robin) boundary problem (2.20) associated to each of those initial profiles. Assume that there is an s ≥ 0 such that almost surely (in the Lebesgue measure sense), ρ 1 1 (s, u) ≤ ρ 2 1 (s, u), T 1 (s, u) ≤ T 2 (s, u) and R 1 (s, u) ≤ R 2 (s, u). Then, for all s ≥ t, ρ 1 1 (t, u) ≤ ρ 2 1 (t, u), T 1 (t, u) ≤ T 2 (t, u) and R 1 (t, u) ≤ R 2 (t, u) almost surely. • The same result holds when ρ 1 t resp. ρ 2 t are two solutions to the (Neumann ; Robin) boundary problem (2.24). Note that Lemma 3 holds for all parameters r, λ 1 , λ 2 , d and D, regardless of conditions (H 1 ). Proof. We prove the first point and the proof of the second one follows in the same way. Introduce A(t) = B ρ 1 1 − ρ 2 1 2 + (t, u)du + B T 1 − T 2 2 + (t, u)du + B R 1 − R 2 2 + (t, u)du := A 1 (t) + A 2 (t) + A 3 (t) (4.13) where x + denotes max(x, 0), the positive part of x. We show that A(t) = 0 for all t ≥ s. Using the weak formulation (2.22) of the solution of the (Dirichlet ; Robin) boundary problem and using Lemma 7.3 and Remark 7.5 in [21], we get: 1 2 d dt A 1 (t) = 1 2 d dt B ρ 1 1 − ρ 2 1 2 + (t, u)du = −D B ∇(ρ 1 1 − ρ 2 1 )∇ ρ 1 1 − ρ 2 1 + (t, u)du + B F 1 ( ρ 1 ) − F 2 ( ρ 2 ) ρ 1 1 − ρ 2 1 + (t, u)du − Γ + (ρ 1 1 − ρ 2 1 ) 2 + (t, u)du. (4.14) Using that ∇ (ρ 1 1 − ρ 2 1 ) + = 1 (ρ 1 1 −ρ 2 1 )≥0 ∇ ρ 1 1 − ρ 2 1 and that B ρ 1 1 − ρ 2 1 2 + (0, u)du = 0, we are left with: Proceeding in the same way for B T 1 − T 2 2 + (t, u)du and B R 1 − R 2 2 + (t, u)du we get: and 1 2 B ρ 1 1 − ρ 2 1 2 + (t, u)du ≤ − t 0 B D1 (ρ 1 1 −ρ 2 1 )≥0 ∇ ρ 1 1 − ρ 2 1 2 2 (r, u)dudr + t 0 B F 1 ( ρ 1 ) − F 2 ( ρ 2 ) (ρ 1 1 − ρ 2 1 + (r, u)dudr.1 2 B T 1 − T 2 2 + (t, u)du ≤ − t 0 B D1 (T 1 −T 2 )≥0 ∇ T 1 − T 2 2 2 (r, u)dudr + t 0 B H( ρ 1 ) − H( ρ 2 ) (T 1 − T 2 + (r, u)dudr1 2 B R 1 − R 2 2 + (t, u)du ≤ − t 0 B D1 (R 1 −R 2 )≥0 ∇ R 1 − R 2 2 2 (r, u)dudr + t 0 B J(R 1 ) − J(R 2 ) (R 1 − R 2 + (r, u)dudr. (4.17) Therefore, 1 2 A(t) ≤ t 0 B F 1 ( ρ 1 ) − F 2 ( ρ 2 ) ρ 1 1 − ρ 2 1 + (r, u)dudr + t 0 B H( ρ 1 ) − H( ρ 2 ) (T 1 − T 2 + (r, u)dudr + t 0 B J(R 1 ) − J(R 2 ) (R 1 − R 2 + (r, u)dudr. (4.18) Now let us use the explicit expressions of F 1 , J and H. We also use the following inequality: for any C ≥ 0 x, y ∈ R, Cxy + ≤ Cx + y + . In order to avoid confusions, a squared term will always be put between brackets, while, for instance ρ 2 1 refers to the first coordinate of ρ 2 . We will denote C a constant which depends on λ 1 , λ 2 , r, d with values possibly changing from one line to the next. F 1 ( ρ 1 ) − F 2 ( ρ 2 ) ρ 1 1 − ρ 2 1 + = 2d(λ 1 − λ 2 )(R 2 − ρ 1 1 − ρ 2 1 ) − 2dλ 1 T 1 − (r + 2) (ρ 1 1 − ρ 2 1 ) 2 + + 2dλ 1 ρ 1 1 + 2dλ 2 ρ 1 3 (R 1 − R 2 )(ρ 1 1 − ρ 2 1 ) + + 1 + 2dλ 2 (1 − ρ 2 2 − ρ 3 3 − ρ 2 1 ) (T 1 − T 2 )(ρ 1 1 − ρ 2 1 ) + ≤ C(ρ 1 1 − ρ 2 1 ) 2 + + 2dλ 1 ρ 1 1 + 2dλ 2 ρ 1 3 (R 1 − R 2 ) + (ρ 1 1 − ρ 2 1 ) + + 1 + 2dλ 2 (1 − ρ 2 2 − ρ 3 3 − ρ 2 1 ) (T 1 − T 2 ) + (ρ 1 1 − ρ 2 1 ) + ,(4.20) where we used (4.19) and the fact that 2dλ 1 ρ 1 1 + 2dλ 2 ρ 1 3 ≥ 0 and 1 + 2dλ 2 (1 − ρ 2 2 − ρ 3 3 − ρ 2 1 ) ≥ 0 in the last line. H( ρ 1 ) − H( ρ 2 ) (T 1 − T 2 ) + = 2dλ 2 − 2d(λ 1 − λ 2 )ρ 1 1 − 2dλ 2 ((T 2 ) 2 + (T 1 ) 2 ) − 1 (T 1 − T 2 ) 2 + + 2d(λ 1 − λ 2 )(1 − ρ 2 1 − ρ 2 3 ) (ρ 1 1 − ρ 2 1 )(T 1 − T 2 ) + ≤ C(T 1 − T 2 ) 2 + + 2d(λ 1 − λ 2 )(1 − ρ 2 1 − ρ 2 3 ) (ρ 1 1 − ρ 2 1 ) + (T 1 − T 2 ) + (4.21) where again, we used (4.19) in the last line, the fact that λ 1 ≥ λ 2 and that (1 − ρ 2 1 − ρ 2 3 ) ≥ 0. Finally, J(R 1 ) − J(R 2 ) (R 1 − R 2 + = −(r + 1)(R 1 − R 2 2 + . (4.22) Collecting (4.20), (4.21) and (4.22) we are left with 1 2 A(t) ≤ C t 0 B ρ 1 1 − ρ 2 1 2 + (r, u) + T 1 − T 2 2 + (r, u) + R 1 − R 2 2 + (r, u) dudr = C t 0 A(r)dr (4.23) where C is a constant which depends on λ 1 , λ 2 , r, d and by Grönwall's lemma, A(t) = 0. Corollary 1. Denote ρ 0 = (ρ 0 1 , T 0 , R 0 ), resp. ρ 1 = (ρ 1 1 , T 1 , R 1 ) the weak solution of (4.4) with (Dirichlet ; Robin) boundary conditions and initial data ρ 0 1 = T 0 = R 0 = 0, resp. ρ 1 1 = T 1 = R 1 = 1. Then for every t ≥ s, ρ 0 1 (s, .) ≤ ρ 0 1 (t, .), T 0 (s, .) ≤ T 0 (t, .) and R 0 (s, .) ≤ R 1 (t, .), resp. ρ 1 1 (s, .) ≥ ρ 1 1 (t, .), T 1 (s, .) ≥ T 1 (t, .) and R 1 (s, .) ≥ R 1 (t, .) almost surely. Furthermore, any other solution (ρ 1 , T, R) of (4.4) with (Dirichlet ; Robin) boundary conditions satisfies: ρ 0 1 ≤ ρ 1 ≤ ρ 1 1 , T 0 ≤ T ≤ T 1 and R 0 ≤ R ≤ R 1 almost surely. The same result holds for ρ 0 = (ρ 0 1 , T 0 , R 0 ) resp. ρ 1 = (ρ 1 1 , T 1 , R 1 ) the weak solution of (4.4) with (Neumann ; Robin) boundary conditions and initial data ρ 0 1 = T 0 = R 0 = 0, resp. ρ 1 1 = T 1 = R 1 = 1. Proof. We prove the result for the (Dirichlet ; Robin) boundary problem and for ρ 0 . The proof is the same for ρ 1 and for the (Neumann ; Robin) case. Fix s ≥ 0 and consider τ s ρ 0 : (t, u) → ρ 0 (t + s, u). τ s ρ 0 is the solution of (4.4) with initial condition u → ρ 0 (s, u) and almost surely in B, τ s ρ 0 (0, u) ≥ (0, 0, 0) = ρ 0 (0, u). Applying Lemma 3 to τ s ρ 0 and ρ 0 with s = 0 and t = t − s yields ρ 0 (t, u) ≥ ρ 0 (s, u) almost surely. • Denote ρ 0 = (ρ 0 1 , T 0 , R 0 ), resp. ρ 1 = (ρ 1 1 , T 1 , R 1 ) the weak solution of (4.4) with (Dirichlet ; Robin) boundary conditions and with initial data (0, 0, 0), resp. (1, 1, 1). Then, lim t→∞ 3 i=1 B ρ 1 1 (t, u) − ρ 0 1 (t, u)| + |T 1 (t, u) − T 0 (t, u)| + |R 1 (t, u) − R 0 (t, u)| du = 0. (4.24) • Denote ρ 0 = (ρ 0 1 , T 0 , R 0 ), resp. ρ 1 = (ρ 1 1 , T 1 , R 1 ) the weak solution of (4.4) with (Neumann ; Robin) boundary conditions and with initial data (0, 0, 0), resp. (1, 1, 1). Then, lim t→∞ 3 i=1 B ρ 1 1 (t, u) − ρ 0 1 (t, u)| + |T 1 (t, u) − T 0 (t, u)| + |R 1 (t, u) − R 0 (t, u)| du = 0. (4.25) By Corollary 1, R 1 is almost surely decreasing and R 0 increasing therefore R 1 t −R 0 t is almost surely decreasing and the above inequality implies R 1 t − R 0 t 2 2 −→ t→∞ 0. We are now left to show that lim t→∞ lim n→∞ B n (t) + C n (t) = 0. (4.29) We proceed following the same steps as for A n . B n (t) = −2D n k=1 γ k | < ρ 1 1,t − ρ 0 1,t , W k > | 2 + 2 n k=1 < F 1 ( ρ 1 t ) − F 1 ( ρ 0 t ), W k >< ρ 1 1,t − ρ 0 1,t , W k > − 2 n k=1 < ρ 1 1,t − ρ 0 1,t , W k > Γ+ (ρ 1 1,t − ρ 0 1,t )(r)W k (r)n 1 (r).dS(r). (4.30) To lighten notations we will not write the subscript t in the computations. Let us compute the second term. n k=1 < F 1 ( ρ 1 ) − F 1 ( ρ 0 ), W k >< ρ 1 1 − ρ 0 1 , W k >= 2d(λ 1 − λ 2 ) n k=1 < ρ 1 1 (R 1 − R 0 ), W k >< ρ 1 1 − ρ 0 1 , W k > + 2d(λ 1 − λ 2 ) n k=1 < R 0 (ρ 1 1 − ρ 0 1 ), W k >< ρ 1 1 − ρ 0 1 , W k > −2d(λ 1 − λ 2 ) n k=1 < (ρ 1 1 ) 2 + (ρ 0 1 ) 2 , W k >< ρ 1 1 − ρ 0 1 , W k > + 2dλ 2 n k=1 < R 1 (T 1 − T 0 ), W k >< ρ 1 1 − ρ 0 1 , W k > +2dλ 2 n k=1 < T 0 (R 1 − R 0 ), W k >< ρ 1 1 − ρ 0 1 , W k > − 2dλ 2 n k=1 < T 1 (ρ 1 1 − ρ 0 1 ), W k >< ρ 1 1 − ρ 0 1 , W k > −2dλ 2 n k=1 < ρ 0 1 (T 1 − T 0 ), W k >< ρ 1 1 − ρ 0 1 , W k > + n k=1 < T 1 − T 0 , W k >< ρ 1 1 − ρ 0 1 , W k > −(r + 2) n k=1 | < ρ 1 1 − ρ 0 1 , W k > | 2 . (4.31) Using Lemma 3 and the Cauchy-Schwarz inequality, we get: − 1 2 B n (t) ≥ n k=1 Dγ k + r + 2 − 2d(λ 1 − λ 2 ) < ρ 1 1 − ρ 0 1 , W k > 2 − (1 + 2dλ 2 ) n k=1 < T 1 − T 0 , W k >< ρ 1 1 − ρ 0 1 , W k > − 2dλ 1 A n (t) B n (t) − B n (t) n k=1 Γ+ (ρ 1 1 − ρ 0 1 )(r)W k (r)n 1 (r)dS(r) 2 . Integrating this between 0 and T and using the Cauchy-Scwharz inequality we are left with 1 2 B n (0) − B n (T ) ≥ T 0 n k=1 Dγ k + r + 2 − 2d(λ 1 − λ 2 ) < ρ 1 1 − ρ 0 1 , W k > 2 dt − (1 + 2dλ 2 ) T 0 B n (t) C n (t)dt − 2dλ 1 T 0 A n (t) B n (t)dt − T 0 B n (t)dt T 0 n k=1 Γ+ (ρ 1 1 − ρ 0 1 )(r)W k (r)n 1 (r)dS(r) 2 dt. (4.32) Now C n (t) = −2D n k=1 γ k | < T 1 − T 0 , W k > | 2 + 2 n k=1 < H( ρ 1 ) − H( ρ 0 ), W k >< T 1 − T 0 , W k > − 2 n k=1 < T 1 − T 0 , W k > Γ+ (T 1 − T 0 )(r)W k (r)n 1 (r).dS(r). (4.33) Again, we compute the second term using the explicit expression of H: n k=1 < H( ρ 1 ) − H( ρ 0 ), W k >< ρ 1 1 − ρ 0 1 , W k >= 2d(λ 1 − λ 2 ) n k=1 < ρ 1 1 − ρ 0 1 , W k >< T 1 − T 0 , W k > + (2dλ 2 − 1) n k=1 | < T 1 − T 0 , W k > | 2 − 2d(λ 1 − λ 2 ) n k=1 < ρ 1 1 (T 1 − T 0 ), W k >< T 1 − T 0 , W k > − 2d(λ 1 − λ 2 ) n k=1 < T 0 (ρ 1 1 − ρ 0 1 ), W k >< T 1 − T 0 , W k > −2dλ 2 n k=1 < (T 1 ) 2 − (T 0 ) 2 , W k >< T 1 − T 0 , W k > . (4.34) Using Lemma 3 and the Cauchy-Schwarz inequality, we get: − 1 2 C n (t) ≥ n k=1 Dγ k + 1 − 2dλ 2 < T 1 − T 0 , W k > 2 − C n (t) n k=1 Γ+ (T 1 − T 0 )(r)W k (r)n 1 (r)dS(r) 2 − 2d(λ 1 − λ 2 ) C n (t) B n (t). Integrating this between 0 and T and using the Cauchy-Scwharz inequality we are left with: Summing inequalities (4.32) and (4.35), using that B n is uniformly bounded by a constant C 1 and C n by a constant C 2 , we obtain For the proof in the (Neumann ; Robin) regime, one proceeds in the same way, but decomposing the difference between ρ 1 and ρ 0 on the basis (V k ) k≥1 . Now, we are able to prove Theorem 6. 1 2 B n (0) − B n (T ) + C n (0) − C n (T ) ≥ T 0 n k=1 Dγ k + r + 2 − 2d(λ 1 − λ 2 ) < ρ 1 1 − ρ 0 1 , W k > 2 dt − max Proof. Again, we focus on the (Dirichlet ; Robin) regime and the proof is the same for all the others. As said before, it is enough to prove uniqueness of a solution of By Corollary 1, the above sets are almost sure and so is U := ∩ n≥0 (U 0 n ∩ U 1 n ). On U , the sequence of profiles { ρ 1 (n, .), n ≥ 1} (resp.{ ρ 0 (n, .), n ≥ 1}) decreases (resp. increases) to a limit that we denote ρ + (.) = (ρ + 1 (.), T + (.), R + (.)) (resp. ρ − (.) = (ρ − 1 (.), T − (.), R − (.))). By Lemma 4, ρ + = ρ − everywhere on U so almost surely on B . Denote this profile ρ and consider ρ(t, .) the solution to (4.38) with initial condition ρ. Since for all t ≥ 0, ρ 0 (t, .) ≤ ρ(.) ≤ ρ 1 (t, .) almost surely, by Lemma 3 we have that for every s, t ≥ 0, ρ 0 (t + s, .) ≤ ρ(s, .) ≤ ρ 1 (t + s, .) almost surely and letting t → ∞ we get that ρ(s, .) = ρ(.) for all s so ρ is a solution of (4.38). (ii) Uniqueness: Note that by Lemma 3 and Corollary 1, for any profiles ρ a = (ρ a 1 , T a , R a ) and ρ b = (ρ b 1 , T b , R b ) satisfying (4.4) with any initial condition, for every t > 0 B ρ a 1 (t, u) − ρ b 1 (t, u)| + T a (t, u) − T b (t, u)| + R a (t, u) − R b (t, u)| du ≤ B ρ 1 1 (t, u) − ρ 0 1 (t, u)| + T 1 (t, u) − T 0 (t, u)| + R 1 (t, u) − R 0 (t, u)| du. (4.41) Applying (4.41) to two stationary solutions and using Lemma 4, one gets uniqueness. As said before, existence and uniqueness of a solution ρ of (4.38) yields existence and uniqueness of the stationary solution of (2.20). Similarly, the proof of (4.6) comes from the fact that B |ρ 1 (t, u) − ρ 1 (u)| + |T (t, u) − T (u)| + |R(t, u) − R(u)| du ≤ B ρ 1 1 (t, u) − ρ 0 1 (t, u)| + T 1 (t, u) − T 0 (t, u)| + R 1 (t, u) − R 0 (t, u)| du (4.42) where again, we applied (4.41) and the fact that the right hand side term converges to 0. A Change of variable formulas The following change of variable formulas have been established in [18,Section 5.2]. Recall that for i, j ∈ {0, 1, 2, 3} and x, y ∈ B N , v j (x/N ) = log(α j (x/N )), and B Simulations The hydrodynamic equations have been simulated with an Euler explicit scheme, in dimension 1 and spatial domain [0, 1], with Neumann boundary conditions. The fixed parameters chosen for the simulations are: • Time horizon: T=100 • Time subdivision: δ T = 5.10 5 • Space subdivision: δ x = 100 • r = 1 and D = 1. In the first simulation we took λ 1 = 0.75 and λ 2 = 0.25. The conditions (H 1 ) are then satisfied and in Figures 2 and 3 we present the limiting profiles starting from the initial condition (ρ 1 = 1, ρ 2 = 0, ρ 3 = 0), resp. (ρ 1 = 0, ρ 2 = 1, ρ 3 = 0) in the system of coordinates (ρ 1 , ρ 1 + ρ 3 , 1 − ρ 2 − ρ 3 ) and we observe that both limiting profiles coincide. In the second simulation we took λ 1 = 1 and λ 2 = 0.75. The conditions (H 1 ) are not satisfied and in Figures 4 and 5 we present the limiting profiles starting from the initial condition (ρ 1 = 1, ρ 2 = 0, ρ 3 = 0), resp. (ρ 1 = 0, ρ 2 = 1, ρ 3 = 0) and we observe that both limiting profiles do not coincide. 0 if there are no insects in x, 1 if there are only wild insects in x, 2 if there are only sterile insects in x, 3 if there is a combination of wild and sterile insects in x. C n,m ([0, T ] × B) the space of functions that are n times differentiable in time and m times differentiable in space, C n,m 0 , resp. C n,m 0,− , resp. C n,m 0,+ the ones in C n,m ([0, T ] × B) which are zero on Γ, resp. Γ − , resp. Γ + . Denote C ∞ k (B) the space of smooth functions with compact support in B, C m (B) the space of functions that are m times differentiable in space and C(B) when m = 0. (2. 25 ) 25Remark 1. In (2.22), the integral over Γ − corresponds to the Dirichlet boundary condition. In (2.25) the integral over Γ − comes from an integration by part of the terms involved in the bulk. Both in (2.22) and (2.25) the first integral over Γ + comes from an integration by part of the terms involved in the bulk and the second integral over Γ + corresponds to the Robin boundary condition.Definition 2. Stationary solution of the hydrodynamic equation. Figure 1 : 1Mixed boundary conditions depending on the values of θ and θ r . Remark 2 . 2For all the other mixed boundary regimes corresponding to other values of θ, the hydrostatic principle states in the same way, replacing ρ stationary solution of the associated hydrodynamic equation. In the cases where only Dirichlet and Robin boundary conditions are involved, one can slightly weaken the conditions (H 1 ) by using conditions (H 2 ) and (H 3 ) instead. Precisely: in the (D ; D), (D ; R), (R; D) regimes, the hydrostatic principle holds under conditions (H 2 ) and in the (N ; N) regime, it holds under conditions (H 3 ). The proof of Theorem 2 is done in Section 4. It essentially relies on an intermediate result stated in Theorem 6 regarding the convergence of solutions of the hydrodynamic equation towards the unique stationary state. This result is non standard as it involves a system of coupled equations and we prove it in the second section of Section 4. Proposition 3 . 3For any sequence of measures (µ N ) N ≥0 on Σ N , for any G ∈ C 1,2 ([0, T ] × B) and any i ∈ {1, 2, 3}, for any t ∈ [0, T ], for all δ > Proposition 4 .G 4For any sequence of probability measures (µ N ) N ≥0 on Σ N , for any G ∈ C 1,2 ([0, T ] × B), for all i ∈ {1, 2, 3} and any t ∈ [0, (s, x/N )(η εN i,s (x) − η i,s (x))ds = 0. (3.17) replacement of the convolution at the boundary we use the following result which follows from [7, Section 5.3]: for any H ∈ H 1 3.47), (3.48), (3.49) and (3.50), we obtain (3.37) and conclude the proof. 3.5. 1 1Uniqueness of the solution in the (Neumann ; Robin) mixed regime Theorem 4. There exists a unique solution to the Neumann + Robin boundary problem (2.24). A > 0, where we used both the Cauchy-Schwarz and (2.16) inequalities in the last line. By (3.53), (3.54) and (3.55), the right hand side of (3.60) converges to ), taking A > 1 D and applying the dominated convergence theorem, we are left with G (t) ≤ (C + 2 + A)G(t).(3.64) Grönwall's inequality and the fact that G(0) = 0 yields G(t) = 0 at any time. 3.5.2 Uniqueness of the solution in the (Dirichlet ; Robin) mixed regime Theorem 5. There exists a unique solution to the Dirichlet + Robin boundary problem (2.20). Theorem 6 . 6Suppose that conditions (H 1 ) hold. Then, there exists a unique stationary solution ρ D,R , resp. ρ N,R of (2.20), resp. (2.24). Furthermore, for any solution ρ D,R , resp. ρ N,R to the boundary value problem (2.20), resp. Remark 4 . 4One could ask if conditions on the parameters are necessary to establish existence and uniqueness of the stationary solution of the hydrodynamic equation. Could we not generalize the result to all parameters? Lemma 4 . 4Assume conditions (H 1 ) are satisfied. k + 1 − 2dλ 2 < T 1 − T 0 , W k > 2 dt − 2d(λ 1 − λ 2 ) A n (t)dt − (1 + 2dλ 1 )C 2 T 0 B n (t)dt (4.36)Taking n to infinty and using the dominated convergence theorem as well as the trace inequality T 1 are almost surely decreasing and ρ 0 1 and T 0 increasing, therefore ρ 1 1 − ρ 0 1 and T 1 − T 0 are almost surely decreasing and the above inequalities imply 1 + F 1 (ρ 1 , T, R) = 0, ρ 1 |Γ − = b 1 (.), ∂ e1 ρ 1 (t, .) |Γ + = 1 D (b 1 − ρ 1 ) |Γ + D∆T + H(ρ 1 , T, R) = 0, T |Γ − = b 1 (.) + b 3 (.), ∂ e1 T (t, .) |Γ + = 1 D (b 1 + b 3 − ρ 1 − ρ 3 ) |Γ + D∆R + J(R) = 0, R |Γ − = 1 − b 2 (.) − b 3 (.), ∂ e1 R(t, .) |Γ + = 1 D (ρ 2 + ρ 3 − b 1 − b 3 ) |Γ + . Existence: For n ∈ N, define U 0 n = {u ∈ B, ρ 0 1 (n, .) ≤ ρ 0 1 (n + 1, .), T 0 (n, .) ≤ T 0 (n + 1, .), R 0 (n, .) ≤ R 0 (n + 1, .)} (4.39) and U 1 n = {u ∈ B, ρ 1 1 (n, .) ≤ ρ 1 1 (n + 1, .), T 1 (n, .) ≤ T 1 (n + 1, .), R 1 (n, .) ≤ R 1 (n + 1, .)}. (4.40) Figure 2 :Figure 3 :Figure 4 :Figure 5 : 2345j ( α) = exp v j (y/N ) − v j (x/N ) − v i (y/N ) − v i (x/N ) − 1. (A.1) Note that R x,y i,j ( α) = O(N −1 ). Consider f : Σ N → R and x, y ∈ B N . (i) For (i, j) ∈ {0, 1, 2, 3} 2 such that i = j, Σ N η i (x)η j (y)f (ξ x,y , ω x,y )dν N α (ξ, ω) = Σ N η j (x)η i (y)(R x,y i,j ( α) + 1)f (ξ, ω)dν N α (ξ, ω). (x)f (σ x ξ, σ x ω)dν N α (ξ, ω) = Σ N η j (x) exp(v i (x/N ) − v j (x/N ))f (ξ, ω)dν N α (ξ, ω). (x)f (σ x ξ, ω)dν N α (ξ, ω) = η j (x) exp(v i (x/N ) − v j (x/N ))f (ξ, ω)dν N α (ξ, ω). (x)f (ξ, σ x ω)dν N α (ξ, ω) = η j (x) exp(v i (x/N ) − v j (x/N ))f (ξ, ω)dν N α (ξ,ω). λ 1 = 0.75, λ 2 = 0.25, ρ(0, .) λ 1 = 0.75, λ 2 = 0.25, ρ(0, .) λ 1 = 1, λ 2 = 0.75, ρ(0, .) λ 1 = 1, λ 2 = 0.75, ρ(0, .) = (0, 1, 0). i=1 M N i,t ( G)is also a martingale whose quadratic variation is known. Acknowledgements: This work has been conducted within the FP2M federation (CNRS FR 2036). The authors would like to thank Camille Pouchol for the useful advice regarding numerical simulations.Proof. We start with the proof of the (Dirichlet ; Robin) regime. It is enough to show that lim t→∞ ρ 1 1 (t, .) − ρ 0 1 (t, .) 2 2 + T 1 (t, .) − T 0 (t, .) 2 2 + R 1 (t, .) − R 0 (t, .) 2 2 = 0. (4.26)Consider the eigenvalue problem for the Laplacian (3.65) and the countable system {W n , γ n , n ≥ 1} of eigensolutions for that problem. For n ≥ 1 introduceRecall that by (3.53), one hasLet us first prove that lim t→∞ lim n→∞ A n (t) = 0. 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[ "Toward Global Sensing Quality Maximization: A Configuration Optimization Scheme for Camera Networks", "Toward Global Sensing Quality Maximization: A Configuration Optimization Scheme for Camera Networks" ]
[ "Xuechao Zhang ", "Xuda Ding ", "Yi Ren ", "Yu Zheng ", "Chongrong Fang ", "Jianping He " ]
[]
[]
The performance of a camera network monitoring a set of targets depends crucially on the configuration of the cameras. In this paper, we investigate the reconfiguration strategy for the parameterized camera network model, with which the sensing qualities of the multiple targets can be optimized globally and simultaneously. We first propose to use the number of pixels occupied by a unit-length object in image as a metric of the sensing quality of the object, which is determined by the parameters of the camera, such as intrinsic, extrinsic, and distortional coefficients. Then, we form a single quantity that measures the sensing quality of the targets by the camera network. This quantity further serves as the objective function of our optimization problem to obtain the optimal camera configuration. We verify the effectiveness of our approach through extensive simulations and experiments, and the results reveal its improved performance on the AprilTag detection tasks. Codes and related utilities for this work are open-sourced and available at https://github.com/sszxc/MultiCam-Simulation.
10.1109/iros47612.2022.9982246
[ "https://export.arxiv.org/pdf/2211.15166v2.pdf" ]
254,044,638
2211.15166
45672e92f716bb792e6222dfb41d94bf86c1dbc9
Toward Global Sensing Quality Maximization: A Configuration Optimization Scheme for Camera Networks Xuechao Zhang Xuda Ding Yi Ren Yu Zheng Chongrong Fang Jianping He Toward Global Sensing Quality Maximization: A Configuration Optimization Scheme for Camera Networks The performance of a camera network monitoring a set of targets depends crucially on the configuration of the cameras. In this paper, we investigate the reconfiguration strategy for the parameterized camera network model, with which the sensing qualities of the multiple targets can be optimized globally and simultaneously. We first propose to use the number of pixels occupied by a unit-length object in image as a metric of the sensing quality of the object, which is determined by the parameters of the camera, such as intrinsic, extrinsic, and distortional coefficients. Then, we form a single quantity that measures the sensing quality of the targets by the camera network. This quantity further serves as the objective function of our optimization problem to obtain the optimal camera configuration. We verify the effectiveness of our approach through extensive simulations and experiments, and the results reveal its improved performance on the AprilTag detection tasks. Codes and related utilities for this work are open-sourced and available at https://github.com/sszxc/MultiCam-Simulation. I. INTRODUCTION A. Background and motivation Cameras play important roles in society surveillance [1], environmental monitoring [2], and scene perception, localization and mapping in robotics [3], etc. Compared to singlecamera systems, camera networks tend to have a larger field of view and more viewing angles for each object, leading to stronger anti-interference ability and better detection performance [4], [5]. Fig. 1 shows a typical application for camera networks as drones tracking ground vehicles with cameras. A typical task of camera networks is to provide adequate coverage for the targets [6], [7]. However, meeting the qualitative coverage requirement does not necessarily ensure the desirable sensing quality of the targets in the scene. The distortion inevitably introduced in the imaging process of cameras depends on the lens's physical properties, the camera's parameters, viewing angle and the objects' relative position. Different configurations of the camera network may result in different sensing quality of the targets. Configuring a camera network properly to cover multiple targets while ensuring desirable sensing quality is challenging due to the vast amount of camera parameters involved and a lack of widely accepted sensing performance metric [8]. In this paper, we propose a straightforward metric that captures the coverage and sensing quality of objects for a This camera network, based on which we automate the search for the configuration that optimizes the proposed metric. B. Related Work Recent surveys [9] [10] summarize the historical developments in the area of camera network coverage. Globally optimal configuration for such systems, similar to the Art Gallery Problem [11] and Watchmen Tour Problem [12], has been proved to be NP-hard, and using general optimization methods could lead to sub-optimal solutions. Recent literature reports several approaches that have been proposed to solve this problem. In [13] [14], the authors focus on complex indoor environments and present a method for pan-tilt-zoom (PTZ) camera reconfiguration. The proposed method considers the targets, camera distortion, and environment illumination, and a particle swarm optimizer gives the solution. However, the performance of the method relies on complete modeling of the environment, which is generally very difficult to obtain. The authors in [15] take into consideration the realistic constraints of computer vision and aim to achieve a balance between coverage and resolution for multiple cameras. They split the 2D plane into polygons with equal areas and use a greedy algorithm to segment the polygons for a simplified camera model and determine the cameras' configuration. Recently, a class of gradient methods based on Voronoi diagrams has been widely used to solve the problem of sensor allocation. In [16], the authors focus on a 2D convex environment and propose a sensing quality metric using prior knowledge of the imaging process. They successfully use a provably correct greedy algorithm to configure cameras for given event distribution. C. Contributions and Organization In this paper, we build an automatic reconfiguration system for camera networks that significantly improves collective sensing performance. The main contributions are as follows. • We propose a new metric that measures the coverage and sensing quality of the targets for a camera. This metric is easy to compute and is shown to be effective in simulation and experiments. • We propose a novel model to quantitatively describe the effect of the camera distortion on sensing quality, which is suitable for any pre-calibrated cameras and targets in 3D space. • We verify the reliability and usability of the strategy through sufficient experiments both in virtual and physical environments. The rest of the paper is organized as follows. Section II explains the relationship between sensing quality and the number of pixels occupied by the target, and presents the proposed sensing quality model for a multi-camera system. In Section III, we illustrate the effectiveness of the proposed reconfiguration model using simulations and experimental results, respectively. Section IV concludes with a summary of our contributions and a brief discussion of future directions. II. PROBLEM STATEMENT AND FORMULATION A general camera network consists of several heterogeneous cameras that may have distinct properties. Dynamic configuration adjustment of the camera network must be performed collectively to ensure that the entire network of cameras is globally optimized concerning the specified task or performance requirements. The sensing process is cameraspecific and mutually exclusive, but when the cameras' fields of view overlap the fusion results should be calculated jointly. As shown in Fig. 2, we consider a model for a camera C i ∈ C for i ∈ {1, 2, . . . , N c } with a conic field of view, whose configuration is specified by its fixed position o i ∈ R 3 , variable optical-axis direction φ i ∈ R 3 , angle of view 2α i ∈ (0, π) and maximum radial resolution of w i pixels. The position of the target T j ∈ T for j ∈ {1, 2, . . . , N t } in the sensing task is denoted by p j ∈ R 3 . Let o i be the point such that − − → o i p j ⊥ φ i . The plane that is parallel to the camera plane and contains o i is the target plane. Let β i j be the angle between − − → o i p j and φ i , and γ i j be the angle between − − → o i p j and − − → o i x . Ifp j is the estimated position for T j , then the sensing error is defined by j = p j −p j 2 .(1) Counting the number of pixels a target occupies in the image before correcting distortion is an intuitive description of the sensing quality. To illustrate the idea, suppose the target with the length of l occupies n pixels in the image. Moreover, let the actual length represented by each pixel on the target plane be k. Then the measurement l has an error |l − l | < 2k. The reason is that each pixel on complementary metal-oxide-semiconductor (CMOS) measures the average brightness information through integration [17], and the error caused by sampling will make both ends of the target captured as an uncertain pixel. It shows that when the same target occupies more pixels, the sensor has richer information to describe the target, making the downstream object detection more accurate. In other words, a shorter actual length represented by each pixel results in a smaller relative error. The error of positioning j is bounded by j < k. (2) A. Single-Camera Model This section discusses how the number of pixels occupied by the target as a metric of sensing quality is related to two critical parameters, i.e., the camera's perspective factor and the distortion factor. 1) Perspective: Perspective primarily depends on the angle of view and focal length which considerably deteriorates the sensing quality. Objects appear smaller as their distance from the observer increases. In imaging process, the actual distance between the camera and the object has a great influence on the number of pixels that the object occupies, illustrated in Fig. 3. It is shown that the number of pixels occupied by an object with unit length is inversely proportional to the distance between the camera and the object. To improve sensing quality, the camera should be placed closer to the target. Suppose that camera C i performs the task of tracking target T j and assume only perspective factor is considered. According to [18], the sensing quality here can be denote as Fig. 4: The arrows indicate the mapping relationship of the pixels during the distortion correction, and the blue triangles represent the origin areas on the sensor occupied by two triangles of same size after correction. In this example, there is 48.4% less information at the edge of the image than in the center. q p 2| − − → o i p j | cos β i j tan α i w i .(3) 2) Distortion: Distortions can be introduced at the image acquisition and processing stages. With a specific lens and perspective, distortions are inherent in the captured videos which can seriously affect the achievable results [19]. In the pinhole camera model system, a widely-used mathematical formulation to correct camera distortions is the Brown-Conrady model [20] which includes radial and slight tangential distortion. x = xK r + 2s 1 xy + s 2 (r 2 + 2x 2 ), y = yK r + s 1 (r 2 + 2y 2 ) + 2s 2 xy,(4) where r 2 = x 2 + y 2 and K r = 1 + k 1 r 2 + k 2 r 4 + k 3 r 6 1 + k 4 r 2 + k 5 r 4 + k 6 r 6 . In (4), (x, y) and (x , y ) stands for the points before and after the distortion, respectively. k 1 , k 2 , k 3 , k 4 , k 5 , and k 6 are radial distortion coefficients. s 1 and s 2 are tangential distortion coefficients. These coefficients can be easily calculated by using classical calibration algorithms [21]. Distortion correction is necessary for performing detection, positioning, and recognition tasks. The distortion correction process of an image is a non-linear mapping, which does not bring any new information about the targets as the differences in pixel density are compensated by interpolation algorithms. After the stretching, the effective information at the edge and the center of the corrected image has a huge gap, as shown in Fig. 4. Furthermore, the model coefficients for distortion correction can also be used to evaluate the effective pixel density obtained by the sensor during the imaging process. The pixels are stretched in different proportions in the directions of x and y. Denote x t , x t+1 as two adjacent pixels after distortion correction, then |x t − x t+1 | represents the original information provided by the sensor on x direction. According to (4), the influence of distortion factor can be written as the derivative of x , y . The pixel density is the product of the distortion factor in the directions of x and y. Therefore, for camera C i and target T j , we use q d to denote the influence factor of distortion on pixel density. x = tan β i j cos γ i j tan α i , y = tan β i j sin γ i j tan α i , q d ∂x ∂x ∂y ∂y .(6) Thus, the influence of different types of lenses can be described uniformly, no matter how they distort lights. This constraint indicates that the camera should be configured to align its optical axis with the object. Finally, combining (3) and (6) yields a comprehensive description of the sensing quality for a single camera as Q j = q p q d > j .(7) An illustration of this model is shown in Fig. 5. B. Multi-Camera Model To extend the single-camera model to multiple cameras, some modifications need to be made to take into account the information fusion. Now consider two cameras C a , C b observe the same target, for example, and Q a , Q b (let Q a < Q b ) denote the actual length represented by the unit pixel at the target location of the two cameras. Fusing data from the two cameras allows for more precise measurements than sensing with a single camera. More specifically, the fused result is the intersection of the origin range, that is j ∈ [0, Q a ] considering all possibilities of each camera's relative position and focal length. A one-dimensional example for intuitive demonstration is given in Fig. 6. Suppose that the overlap between cameras occurs randomly. The expectation of the error bound can be written as Q j = 1 1/Q a + 1/Q b .(8) Similarly, it can be extended to the camera network of arbitrary scale. For a camera C i and a target j, use V ij to denote the visibility. V i j = 1, β i j ≤ α i 0, β i j > α i .(9) Then, the fusion of every cameras' estimation is written as Q j = 1 Ci∈C V i j /Q i j > j .(10) C. Optimization The tasks of a camera network in the real life vary greatly, resulting in different optimization problems. One of the most common goals is that each target is covered by the field of view of at least one camera and the average sensing quality (b) Distortion factor (c) Complete model certain camera mounted at (0, 0, 3000)mm. The XY axis represents the position, be written (11) ty, different l with other ptimize the (12) camera netm 1 can be 3. ents demoning quality camera netde (cameras to evaluate ild a virtual Algorithm 1: Camera Network Configuration Input: camera angle of view α i , camera distortion coefficients, camera resolution w i pixels, user constrains on o i , φ i . Output: optimal camera configuration, targets' position p j . 1 Initial scan to get targets' position p j ; 2 while true do 3 Solve the optimization problem (11) if target T j gets more then one estimation from the camera network then 10 Update p j to the fusion of estimations; 11 end 12 end 13 end visual fiducial marker AprilTag [24], which facilitates the vision-based six degrees of freedom robot localization. The task of the camera network is to obtain the positions of all of all targets is maximized. The objective function based on the proposed sensing quality metric (10) can be written as Minimize 1 N t Tj ∈T Q j subject to Ci∈C V i j ≥ 1, ∀T j ∈ T.(11) Further, with the formulation of sensing quality, different optimization problems can be constructed to deal with other application requirements. For example, we can optimize the worst-case sensing quality as Minimize max Tj ∈T Q j subject to Ci∈C V i j ≥ 1, ∀T j ∈ T.(12) Based on (11), Algorithm 1 is proposed for camera network configuration systems. Apparently, Algorithm 1 can be used to solve (12) with the modification of Line 3. III. PERFORMANCE EVALUATION A. Simulation In this section, we provide simulation experiments demonstrating the claimed strength of proposed sensing quality model for the reconfiguration of heterogeneous camera networks, e.g., PTZ camera networks or drone mode (cameras carried by unmanned aerial vehicles). In order to evaluate the sensing quality, we build a virtual environment to simulate all the camera views. In the designed experiment, several robots move randomly on the 5m × 3m floor [22], [23]. Each robot is attached with an individual visual fiducial marker AprilTag [24], which facilitates the vision-based six degrees of freedom robot localization. The task of the camera network is to obtain the positions of all the robots as accurately as possible. Specifically, the robots' position p j are randomly given and cameras have fixed mounting position o i and flexible opticalaxis direction φ i in the PTZ camera mode, while flexible position o i (minimum height is limited to 1m) but fixed downward optical-axis direction φ i in the drone mode. All the cameras' coefficients are calibrated from real cameras. Our virtual environment generates images corresponding to the given camera configuration, shown in Fig. 7. AprilTag detection algorithm is used to estimate the position of robots according to the images. The optimization problem (11) is solved by fmincon in MATLAB 2021, which runs on an AMD Ryzen 5 3600 CPU, 3.60GHz processor and 32GB RAM. We generate 12 different simulation scenarios and run 10 experiments in each scenario. The results are shown in Fig. 8 and Table I. First, the average errors of the PTZ camera systems are larger than that of the drone camera systems. This is because the fixed positions of the PTZ cameras lead to more serious distortion of the sight when monitoring robots at a large relative angle, while drones can fly directly above the robots to get a sight with better sensing quality. Besides, we compare the performance of the two algorithms, i.e., Interior Point OPTimizer (IPOPT) and Sequential Quadratic Programming (SQP). The results show that SQP computes faster with larger error. Moreover, the sensing quality model in [16] is used for comparison. It is obvious that our model achieves more accurate localization since our model incorporates the effects of distortion. Lastly, as the numbers of cameras and targets increase, the average computing time increases too due to the increasing complexity of the optimization problem, and it becomes harder to find a global optimal solution. B. Experiment To experimentally validate the proposed model, we set up a 3m (width) × 5m (length) × 2.5m (height) rectangular boxshaped environment with three self-build interchangeablelens PTZ cameras mounted on the ceiling, the intrinsic and extrinsic coefficients of which are pre-calibrated. We set several Omni-directional robots attached with AprilTags moving on the ground as sensing targets. The cameras' positions are fixed, and the optical-axis directions perform as optimization variables, limited by PTZ cameras' physical properties. Since the estimation of the depth depends on the size of the tags in the image, this experiment can examine both the positioning and measurement ability simultaneously. The ground truth of the robots' positions is given by manual measurement with laser rangefinder. Our algorithm will give an optimal camera configuration for sensing quality, as shown in Fig. 9 and video. We first use manual configuration of the cameras to cover all the robots and get the average error. Then, the configuration are set according to Algorithm 1. The average Q is 1.165 and average error decreased by 37.0%, down to 35.3. Finally, we use sensing model proposed by [16] as another comparison. The results are shown in Table II. IV. CONCLUSION In this paper, we considered the balance between the coverage for all targets and sensing quality of each single one for a camera network. The problem has been formulated as an optimization problem which takes into account the perspective, distortion factor and multi-camera fusion results. Optimal configuration for camera network can be achieved without modeling the environment, suitable for any pre-calibrated cameras and targets anywhere in 3D space. The proposed method demonstrates superior performance as compared to previous approaches, increasing the average accuracy in AprilTag detection task by 38.9%. We would like to introduce some potential research directions: i) developing optimization approach for online tracking and large-scale problems, ii) optimizing sensing quality while taking the surveillance of unknown areas into account, and iii) developing a refined model for objects which need to be tracked from a certain direction. Fig. 1 : 1A network of three cameras mounted on the drones for tracking autonomous ground vehicles. Fig. 2 : 2Camera symbol definition. Fig. 3 : 3Perspective comparison: (left) image space and (right) world space. The farther away from the lens, the larger the area described by a single pixel, and vice versa. Fig. 5 : 5The sensing quality on the ground plane by a certain camera mounted at (0, 0, 3000)mm. The XY axis represents the position, and the lighter the color, the higher the quality. Fig. 6 : 6A one-dimensional example for fusion: The task is to estimate the length of the green line, and the fused result is the intersection of two cameras with different Qj. Fig. 7 : 7AprilTag detection task in the virtual environment, (left) image space and (right) world space. The farthest tag was not detected successfully because it occupied few pixels on the image. (a) PTZ, Nc = 3, Nt = 3 (b) PTZ, Nc = 3, Nt = 9 (c) PTZ, Nc = 5, Nt = 10 (d) PTZ, Nc = 5, Nt = 20 (e) Drone, Nc = 3, Nt = 3 (f) Drone, Nc = 3, Nt = 9 (g) Drone, Nc = 5, Nt = 10 (h) Drone, Nc = 5, Nt = 20 Fig. 8 : 8Simulations for different network scales. The XY axis represents the experiment plane. The Z axis represents the sensing quality, which has been normalized for better presentation. The targets' positions are marked with circles. Fig. 9 : 9Experimental demonstration of camera network configuration for optimal sensing quality. (left) Top view of the environment and visualization of a optimal configuration for certain situation. (a) Three PTZ cameras mounted on the ceiling. (b)(c)(d) The camera view with the optimized configuration. work is supported by the CIE-Tencent Robotics X Rhino-Bird Focused Research Program. The authors would like to thank Dr. Xiaoming Duan, Han Wang and Hao Jiang for their helpful advice on technical issues. * : the Dept. of Automation, Shanghai Jiao Tong University,Shanghai, China. email: {zhang.xc, dingxuda, crfang, jphe}@sjtu.edu.cn. † : the Tencent Robotics X Lab, Shenzhen, China. email: {evanyren, petezheng}@tencent.com TABLE I : ISimulation Results on Multi-Robot DetectionNc Nt PTZ Drone Method Computing Time/s Error/mm Method Computing Time/s Error/mm 3 3 IPOPT 0.684 11.97 IPOPT 0.534 5.13 SQP 0.223 11.85 SQP 0.120 5.67 Arslan's - 13.56 Arslan's - 9.71 5 10 IPOPT 1.065 21.96 IPOPT 1.321 14.62 SQP 0.490 25.69 SQP 0.586 17.51 Arslan's - 47.11 Arslan's - 22.18 7 20 IPOPT 1.492 23.55 IPOPT 1.915 17.44 SQP 0.720 27.23 SQP 0.746 21.99 Arslan's - 51.02 Arslan's - 22.93 * Arslan's method [16] did not give comparisons on computing time. 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[ "https://github.com/sszxc/MultiCam-Simulation." ]
[ "P 3 OVD: Fine-grained Visual-Text Prompt-Driven Self-Training for Open-Vocabulary Object Detection", "P 3 OVD: Fine-grained Visual-Text Prompt-Driven Self-Training for Open-Vocabulary Object Detection" ]
[ "Yanxin Long ", "Jianhua Han ", "Runhui Huang ", "Xu Hang ", "Yi Zhu ", "Chunjing Xu ", "Xiaodan Liang " ]
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Inspired by the success of visual-language methods (VLMs) in zero-shot classification, recent works attempt to extend this line of work into object detection by leveraging the localization ability of pre-trained VLMs and generating pseudo labels for unseen classes in a self-training manner. However, since the current VLMs are usually pre-trained with aligning sentence embedding with global image embedding, the direct use of them lacks fine-grained alignment for object instances, which is the core of detection. In this paper, we propose a simple but effective Pretrain-adaPt-Pseudo labeling paradigm for Open-Vocabulary Detection (P 3 OVD) that introduces a fine-grained visual-text prompt adapting stage to enhance the current selftraining paradigm with a more powerful fine-grained alignment. During the adapting stage, we enable VLM to obtain fine-grained alignment by using learnable text prompts to resolve an auxiliary dense pixel-wise prediction task. Furthermore, we propose a visual prompt module to provide the prior task information (i.e., the categories need to be predicted) for the vision branch to better adapt the pre-trained VLM to the downstream tasks. Experiments show that our method achieves the state-of-the-art performance for open-vocabulary object detection, e.g., 31.5% mAP on unseen classes of COCO.
10.48550/arxiv.2211.00849
[ "https://export.arxiv.org/pdf/2211.00849v1.pdf" ]
253,255,271
2211.00849
d8cb2631b393aa227c4b6787464d22c4aa3bda56
P 3 OVD: Fine-grained Visual-Text Prompt-Driven Self-Training for Open-Vocabulary Object Detection Yanxin Long Jianhua Han Runhui Huang Xu Hang Yi Zhu Chunjing Xu Xiaodan Liang P 3 OVD: Fine-grained Visual-Text Prompt-Driven Self-Training for Open-Vocabulary Object Detection 1Index Terms-Open-vocabulary Object DetectionSelf- trainingPrompt LearningVision-language Inspired by the success of visual-language methods (VLMs) in zero-shot classification, recent works attempt to extend this line of work into object detection by leveraging the localization ability of pre-trained VLMs and generating pseudo labels for unseen classes in a self-training manner. However, since the current VLMs are usually pre-trained with aligning sentence embedding with global image embedding, the direct use of them lacks fine-grained alignment for object instances, which is the core of detection. In this paper, we propose a simple but effective Pretrain-adaPt-Pseudo labeling paradigm for Open-Vocabulary Detection (P 3 OVD) that introduces a fine-grained visual-text prompt adapting stage to enhance the current selftraining paradigm with a more powerful fine-grained alignment. During the adapting stage, we enable VLM to obtain fine-grained alignment by using learnable text prompts to resolve an auxiliary dense pixel-wise prediction task. Furthermore, we propose a visual prompt module to provide the prior task information (i.e., the categories need to be predicted) for the vision branch to better adapt the pre-trained VLM to the downstream tasks. Experiments show that our method achieves the state-of-the-art performance for open-vocabulary object detection, e.g., 31.5% mAP on unseen classes of COCO. I. INTRODUCTION T HE dominant object detection paradigm uses supervised learning to predict limited categories of the object. However, the existing object detection datasets usually contain only a few categories due to the time-consuming labeling procedure, e.g., 20 in PASCAL VOC [23], 80 in COCO [17]. Some methods [24] attempt to expand the categories to a larger scale by adding more labeled datasets. However, it is time-consuming and difficult to guarantee sufficient instances in each class due to the naturally long-tailed distribution. On the other hand, previous methods [18], [42]- [44] follow the setting of zeroshot detection and align the visual embeddings with the text embeddings generated from a pre-trained text encoder on base categories, but they still have a significant performance gap compared to the supervised counterpart. Benefiting from the massive scale of pre-trained datasets collected from the web [35], [39], recent vision-language pre-training models (VLM), e.g., CLIP [22], have shown a surprising zero-shot classification capability by aligning the image embeddings with corresponding caption. However, it is a challenging research direction to transfer this zero-shot classification ability to object detection in dense prediction framework, because fine-grained pixel-level alignment, which is essential for dense tasks, is missing in the current visual language pre-training models. There exist some attempts to train an open-vocabulary detector by leveraging the zero-shot classification ability of a pre-trained VLM. The work [20] proposes a basic self-training pipeline, which utilizes the activation map of the noun tokens in the caption from the pre-trained VLM and generates the pseudo bounding-box label. However, existing OVD methods just take advantage of the global text-image alignment ability of VLMs, thus failing to capture dense text-pixel alignment and severely hindering the self-training performance of OVD tasks. Directly using them by activation map cannot fully adapt to downstream detection tasks, which require better dense representations. For example, as shown in the upper part of Fig. 2, directly utilizing the pre-trained VLM can only obtain a low-quality and incomplete dense score map for the dog in the input image, which is harmful for the next pseudo labeling stage. Recent works [4], [7] attempt to build generic object detection frameworks by scaling to larger label spaces, while they are costly required to acquire large-scale annotations from bigger datasets. To achieve fine-grained alignment over dense pixels and avoid extra data cost, we propose P 3 OVD, a Pretrain-adaPt-Pseudo labeling paradigm for Open-Vocabulary Detection (P 3 OVD), to improve the robustness and generalization capability. Inspired by the recent advance in learningbased prompting methods [45], [47] from Natural Language Processing (NLP) community, we design a novel promptdriven self-training framework ( Fig.1) for better adaption of the pre-trained VLM to new detection tasks. In detail, P 3 OVD designs a fine-grained visual-text prompt adapting stage to obtain more powerful dense alignments for better pseudo label generation by introducing an additional dense prediction task. Specifically in adapting stage, to reduce the domain gap of upstream and downstream tasks and obtain the semantic-aware visual embedding, we introduce the visual and text prompt modules into the learnable image encoder and text encoder of the original VLM respectively. The text prompts provide dense alignment task cues to enhance category embedding, and the visual prompt module aligns enhanced categories information to each pixel. Furthermore, with the prompt-enhanced VLM after adapting stage, a better pseudo label generation strategy is proposed for novel classes by leveraging the non-base categories' names as the label input and aligning each connect region of score map for each category. As far as we know, our P 3 OVD is the first work to employ a learnable prompt-driven adapting stage in OVD task to capture fine-grained pixel-wise alignment and therefore to generate better pseudo labels. We evaluate our P 3 OVD on the popular object detection dataset COCO [17] under the well-known zero-shot setting, and further validate its generalization capability on PASCAL VOC [23], Objects365 [68] and LVIS [24] benchmarks. Experiment results show that our P 3 OVD achieves the state-of-the-art performance on detecting novel classes (without annotations), e.g., 31.5% mAP on the COCO dataset. Besides, P 3 OVD also outperforms other open-vocabulary detection (OVD) methods when directly adapting the model trained on COCO to perform open-vocabulary detection on the three other object detection datasets. To demonstrate the effectiveness of the two learnable prompt modules, further experiments are conducted to show that fine-tuned with these two modules, P 3 OVD can generate a better dense score map (4.3% mIoU higher) compared to directing using the pre-trained CLIP [22] model. II. RELATED WORK A. Zero-shot Object Detection. Most zero-shot detection(ZSD) methods [18], [42]- [44] align the visual embeddings to the text embeddings of the corresponding base category generated from a pre-trained text encoder. Inspired by the success of VLMs [21], [22], several methods attempt to perform ZSD by training with image captions or directly leveraging a pre-trained VLM. [16] propose to pre-train the CNN backbone and a Vision to Language Module (V2L) via grounding task on an image caption dataset, and then the whole architecture is fine-tuned with an additional RPN module. However, it still suffers from a large performance gap with the SOTA, and the domain of upstream and downstream datasets must remain similar to maintain performance [20]. Based on a pre-trained VLM, [19] distills the learned image embeddings of the cropped proposal regions to a student detector, therefore requiring huge computation cost because each proposal needs to be fed forward into the image encoder of VLM. [20] proposes a basic selftraining pipeline based on ALBEF [21], which firstly utilizes Grad-CAM [25] to obtain dense activation region of specific words in the caption and then generate the pseudo boundingbox label by selecting the proposal that has the largest overlap with the activation region. However, most VLMs lack the ability to perform pixel-wise classification since it is pre-trained with the correspondence of text embedding and global visual token instead of pixel embeddings. Besides, the reliance on caption data limits the application of this generalization to some datasets without caption. B. Vision-language Pre-training. Inspired by the success of pre-train-and-fine-tune scheme in the domains of natural language processing [30], [31] and computer vision [32], [40], many Vision-and-Language Pretraining (VLP) models have been proposed. The pre-training tasks of VLP models can be divided into two categories: image-text contrastive learning tasks and Language Modeling (LM) based tasks. The first category, e.g., CLIP [22], ALIGN [39] and UNIMO [35], aims to align the visual feature with textual feature in a cross-modal common semantic space. The other category, e.g., VisualBERT [34], UNITER [35], M6 [36], DALL-E [37], ERNIE-ViLG [38], employs LM-like objects, include both autoregressive LM (e.g., image captioning, textgrounded image generation) and masked LM (e.g., Masked Language/Region Modeling). Different from previous works that only focus on global feature alignment, our proposed model can achieve pixel-aware alignments via prompts and perform better in downstream tasks of pixel-wise dense prediction. C. Prompt Tuning. Freezing the pre-trained models with only tuning the soft prompts can benefit from efficient serving and matching the performance of full model tuning. Prompt tuning has been verified as an effective method to mitigate the gap between pre-training and fine-tuning. As a rapidly emerging field in NLP [45], prompt tuning is originally designed for probing Fig. 3: Overview of proposed open-vocabulary detection pipeline P 3 OVD. P 3 OVD consists of three stages: fine-grained visual-text prompt adapting stage to improve the dense pixel-wise alignment via learnable vision-text prompts to obtain better pseudo bounding-box labels; pseudo labeling stage to obtain non-base classes' pseudo labels by utilizing the dense classification ability of prompt-enhanced VLM and the localization ability from a pre-trained RPN; self-training stage to feed the pseudo labels into an open-vocabulary detector for further training. Different line color denotes different stages. knowledge in pre-trained language models [62] and now applied in various NLP tasks, e.g., language understanding [63] and generation [56]. Prompt tuning has now been extended to vision-language models. Instead of constructing hand-crafted prompts in CLIP [22], CoOp [47] proposes tuning soft prompts with unified context and class-specific context in downstream classification task. CPT [60] proposes colorful cross-modal prompt tuning to explicitly align natural language to finegrained image regions. The main differences between our usage of learnable multi-modal prompts with the previous prompt tuning works lie in three aspects: 1) Unaligned upstream and downstream tasks; 2) Multi-modal; 3) Learnable prompts for self-training-based open-vocabulary object detection (refer to Appendix E for more detail). III. METHOD In this section, we first briefly introduce the P 3 OVD framework. Then we describe the details of different stages in P 3 OVD: a) learnable multi-modal prompts in fine-grained alignment stage, which are used to enable VL model to obtain fine-grained pixel-wise alignment ability; b) better pseudo label generation strategy in the pseudo labeling stage and c) the details of the final self-training stage. Basic Notations: We construct a combined categories set (denoted as C) via extracting the categories from several large-scale detection datasets. The base (seen) categories, novel (unseen) categories, and non-base categories are denoted as C B , C N , C B (i.e., C = {C B , C B }). The input image is represented as X. We use the Φ v to represent for the image encoder and Φ t for the text encoder of the original VLM, and regard Φ v and Φ t as the image encoder (with visual prompt) and text encoder (with text prompt) of the VLM after the fine-tuning stage. A. P 3 OVD Framework Fig. 3 illustrates the overall pipeline of the proposed P 3 OVD. Firstly, at the fine-grained visual-text prompt adapting stage , by employing the learnable text prompt module prompt t and visual prompt module prompt v into the text encoder Φ t and image encoder Φ v , we can fine-tune the whole network under dense supervision of pseudo dense score map with other parameters fixed. Secondly, for pseudo labeling stage, we obtain the pseudo labels of non-base classes by leveraging the dense classification ability of the fine-tuned VL model (Φ t , Φ v ) and location ability of a pre-trained (on base classes) Region Proposal Network (RPN) Φ r . Finally, in the final self-training stage, an open-vocabulary detector Φ d is further trained with the combination of the ground truth of base classes and the pseudo labels of non-base classes to fulfill the self-training pipeline. B. Adapting Stage Via Multi-Modal Prompts The crucial part of the self-training pipeline for openvocabulary detection (OVD) lies in the pseudo labeling stage that determines the final detection performance of novel classes. To generate better pseudo labels for the detection task, our method first aligns each pixel with a category and then adopts a pre-trained RPN for bounding-box localization. Since most vision-language methods (VLM) are trained via the alignment between the whole image and the corresponding caption, they lack the dense alignment ability between pixels and categories. Thus in this section, we focus on modifying the pre-trained vision-language model (VLM) to enhance the current selftraining paradigms with fine-grained adapting stage via a newly designed dense alignment loss function and learnable text/visual prompts. Note that to obtain the dense pixel-level visual embeddings instead of the global visual embedding, following [48], we modified the image encoder by removing the query and key embedding layers. 1) Dense Alignment Task: As shown in Fig. 3, we introduce a dense alignment task to fine-tune the VLM under the supervision of pseudo dense score map S ∈ R HW ×|C| , where C denotes the all categories, including multi dataset Fig. 4: Details of the visual prompt. The visual prompt encoder consists of a cross-attention module to obtain the semanticaware visual embeddings. categories, and |C| represents the number of C. The S is generated by calculating the similarity of pixel-wise visual embeddings of input image X ∈ R HW with each category via the original VLM. Specifically, we first compute the dense visual embeddings Φ v (X) ∈ R HW ×D and the text embeddings Φ t (C) ∈ R |C|×D . The dense score map S is calculated by: S = Φ v (X), Φ t (C)(1) where ·, · denotes the inner product operation. To reduce the domain gap of upstream and downstream tasks and obtain the semantic-aware visual embedding, we further introduce the visual and text prompt modules into the learnable image encoder and text encoder of the original VLM respectively. Compared to most previous prompt-based tuning works [45]- [47] are adopted for downstream tasks with different data domains while keeping the same task form, our P 3 OVD utilizes learnable multi-modal prompts to obtain the finegrained feature alignment, which is different from the upstream global alignment in the task domain. The objective of dense alignment stage is: L F T = L CE (S,Ŝ),(2) where L CE represents the cross-entropy loss. Similar to Eq. (1), the model predictionŜ ∈ R HW ×C with the image input X is calculated as:Ŝ = Φ v (X), Φ t (C)(3) where Φ v and Φ t denote the image encoder and text encoder plugged with the prompt module. 2) Text Prompt: Prompt tuning has been verified as an effective method to probe knowledge in pre-trained language models [62] and applied in various NLP tasks [56], [63]. For example, the hand-crafted text prompts, e.g., 'a photo of a {}.', have been adopted to adapt the pre-trained VLM to the different downstream tasks [19], [22]. Inspired by these works, we further introduce a learnable text prompt module to provide dense alignment task cues for enhancing categories embeddings since downstream text-pixel alignment task is quite different from the upstream global text-image alignment task. The text prompt module maps the learnable prompt word q l ∈ R D into the prompt embeddings h l ∈ R D . We believe that the h l should depend on each other since the words in a natural sentence always have a strong contextual relationship. Therefore for the structure of proposed text prompt module, it contains a bidirectional long-short term memory network (LSTM) to get the contextual information followed by a ReLU activated two-layer multilayer perceptron (MLP) [52]. h l = M LP ([LST M (q 0:l ) : LST M (q l:m )])(4) where l denotes the length of prompt tokens added before the category token and m denotes the total number of prompt tokens. 3) Visual Prompt: The text prompts employ dense alignment task cues to enhance category embedding. Therefore, to align the enhanced categories information to each pixel, a visual prompt module is built upon the cross-attention mechanism to generate semantic-aware visual embedding for each pixel to improve dense alignment. Similar to [65], which utilizes positional embeddings as visual prompt to provide additional temporal information, ours is also formed as a post-network prompt module [65] to provide additional semantic cues and we only finetune the visual prompt module while fix the visual encoder. By taking each pixel embedding in visual features F = Φ v (X) ∈ R HW ×D as the input of the query and the text embeddings of all categories Φ t (C) ∈ R |C|×D as the input of key and value, the cross-attention block output the semanticaware visual promptsF ∈ R HW ×D for each pixel. Then they're concatenate with the visual features followed by an 1-layer MLP to obtain the semantic-aware visual embeddings F = Φ v (X) ∈ R HW ×D (see Fig. 4). F = M LP (F ⊕F ) (5) F = sof tmax( F × Φ t (C) T √ D ) · Φ t (C)(6) where ⊕ and D represent the concatenation operation and the feature dimension of Φ t (C). M LP denotes the 1-layer multi-layer perceptron to reduce back the feature dimension. C. Pseudo Label Generation Based on the more precise dense score map generated by the fine-tuned VLM with learnable text and visual prompts, we can obtain the non-base classes' pseudo labels by additionally leveraging the location ability of a pre-trained region proposal network (RPN) Φ r (shown in Fig. 3(b)). Note that Φ r is only trained on base classes, and the experiments in [19] already demonstrate that training only on the base categories can achieve comparable recall to average recall of the novel categories. Previous pseudo labeling generation strategy [20] used the objects of interest in the caption, which not only harms the transfer ability to the detection datasets without the caption but also is limited to the uncompleted description of caption data. For example, such a strategy cannot generate the pseudo labels of classes that are not included in the caption. To address these issues, We directly use non-base classes' names as the input of fine-tuned text encoder Φ t . Besides, to better distinguish the background classes, instead of directly using the word "background", we treat all the base classes as background since we already have the ground-truth annotation of these classes. Then we obtain the pseudo dense score map of non-base classes S B ∈ R HW ×|C B | from the fine-tuned VLM with the procedure same to Eq. (3). For each image x i , we firstly compute the connected regions r j ∈ R HW for j-th non-base category on S B by setting a similarity threshold δ, then we adopt the intersection of union (IoU) of k-th proposal p k i from RPN and r j as confidence score: score i,j,k = IoU (p k i , r j )(7) where p i = Φ r (x i ) and the score i,j,k denotes the confidence score that the proposal p k belongs to the j-th non-base category for image x i . Finally, a hard score threshold γ is adopted on the confidence score to filter out pseudo boxes with less confidence. Detailed pseudo labeling strategy comparison with [20] refer to Appendix D. Self-training stage. After obtaining the pseudo boundingbox labels for non-base classes, together with the groundtruth annotations of base classes, we are able to train a final open-vocabulary detector Φ d to fulfill the self-training pipeline (shown in Fig. 3(c)) via Faster-RCNN [69]. We build the Φ d by replacing the last classification layer with the text embeddings generated by the text encoder on all categories Φ t (C). The objective of self-training stage L T is calculated as: L T = L CLS (Y,Ŷ ) + L REG (Y,Ŷ )(8) Where the L CLS and L REG denote the cross-entropy loss for classification head and L1-loss for regression head,Ŷ and Y represent the model prediction Φ d (X) and annotations combined with pseudo labels of non-base classes and the ground-truth of base classes. The trained detector Φ d can then transfer to other detection datasets by providing the class names. IV. EXPERIMENTS A. Open-Vocabulary Object Detection Setups Benchmark Setting: We benchmark on the widely-used object detection dataset COCO 2017 [17]. Following the wellknown settings [18] adopted by many OVD methods, we divide the categories used in COCO into 48 base (seen) classes and 17 novel (unseen) classes. Each method can only be trained on annotations of the base categories and then predicts the novel categories without seeing any annotations of these categories. We report the detection performance of both the base and novel classes during inference, as the generalized settings used in [16]. We also evaluate the generalization ability of the trained detector by directly transferring it to three other object detection datasets, including PASCAL VOC [23], LVIS [24] and Objects365 [68]. Evaluation Metric: We use mean average precision (mAP) with IoU threshold 0.5 as the evaluation metric. We pay more attention to the novel class performance since we aim to build an open-vocabulary detector and the annotations of base classes are already provided. B. Implementation Details We utilize a pre-trained CLIP (RN50×16) [22] model as the VLM. Note that our method is compatible with conventional VL models. For simplicity, we take CLIP as an example. All the detection models are implemented on the mmdetection [72] codebase and follow the default setting as Mask-RCNN (ResNet50) [73], 1X schedule unless otherwise mentioned. At the adapting stage, the prompt-enhanced CLIP is trained for 5 epochs, with the text prompt learning rate set to 1e-1 and the visual prompt set to 1e-5 separately. For the pseudo labeling stage, the RPN is trained only on the base classes in the 2X schedule. The objectness score threshold for RPN is set to 0.98. The similarity threshold δ and score threshold γ are set to 0.6 and 0.4. For the self-training stage, following the default setting, we adopt a Mask-RCNN (ResNet50) [73] with the last classification layer replaced by class embeddings output by the text encoder. We train the detector for 12 epochs, with the learning rate decreased by a factor of 0.1 at 8 and 11 epochs. The initial learning rate is set to 0.04 with batch size 32, and the weight decay is set to 1e-4. We extract the categories of large-scale object detection dataset (i.e., LVIS [24] and Objects365 [68]) as the combined categories set C, which contains about 1k categories. C. Main Results COCO Dataset. We compare our P 3 OVD with existing open-vocabulary methods on the COCO dataset [17]. As shown in Table I, P 3 OVD achieves the state-of-the-art performance (i.e., 31.5% mAP) on the novel categories of the COCO dataset and 4.5% mAP improvement on overall categories compared to another self-training based method OVD-ALBEF [20]. Besides, by comparing with the P 3 OVD (w/o FT), e.g., 31.5% mAP vs. 29.8% mAP, we further demonstrate the necessity of the fine-grained adapting stage with learnable visual and text prompts for dense alignment tasks. Note that we utilize the basic vision-language model (VLM) CLIP [22] and do not use the caption data, guaranteeing the transfer ability to other VLMs or pre-training datasets without caption information. Clarification should be made that self-training-based methods (e.g., OVD-ALBEF [20] and ours) usually achieve slightly worse on base classes compared to the knowledge-distillation (KD) method (e.g., ViLD [19]). We attribute this to the reason that self-training methods make the model optimize more towards novel classes via generating massive pseudo labels on them, while KD-based methods try to keep the performance of the base class by only distilling the classification ability. Generalization Abilities. To further demonstrate the openvocabulary ability of the detector trained through the P 3 OVD pipeline, we directly transfer the final detection model trained on COCO datasets, to other detection datasets, including PAS-CAL VOC [23], Object365 [68] and LVIS [24]. Experimental results in Table IV show that our P 3 OVD achieves the best generalization ability even adopted to the datasets with much more categories than the pre-trained dataset, i.e., 81 classes in COCO vs. 1203 classes in LVIS. Note that we do not compare with ViLD [19] since it does not provide either this experiment result or the code. I: Open-vocabulary detection performance comparison on COCO datasets. P 3 OVD (w/o FT) denotes the P 3 OVD without fine-grained adapting stage. We can observe that P 3 OVD achieves the state-of-the-art detection performance on the novel classes. Without the fine-tuning stage, the performance on the novel classes suffers a large drop. Note that OVR-CNN pre-trains a self-designed vision-language model with a vision-to-language (V2L) module by itself. ViLD* trains the detector with data augmentations of large-scale jittering (LSJ) [6] and longer training schedule (16x). mAP (%) is reported. D. Ablation Study a) What's the effect of the learnable prompt modules for pseudo label generation?: We introduce two learnable prompt modules to obtain the dense alignment ability, which is important for pseudo bounding-box label generation. To analyze the effect of proposed visual and text prompts, we conduct the quantitative experiments with different combinations of prompt modules to show the quality of pseudo labels (with mAP) and dense score map (with mIoU on novel classes). As shown in Table II, adopting hand-crafted prompts improves the baseline (without prompt) by 1.2% mAP increment on pseudo labels' quality, while adding text and visual prompts separately gain additional 1.7% and 1.4% increment, respectively. Combining visual and text prompts makes better performance (+3.5%) and the best dense score map (35.4% mIoU). After adopting the self-training stage, it reaches the state-of-the-art detection performance (31.5% mAP) on COCO novel classes. b) What's the effect of different text prompt structures?: As shown in Table III, we evaluate the effect of different structures of only the text prompts. Based on the input learned embeddings, separately using LSTM and MLP obtain a relatively small improvement due to the lack of the contextual information of prompt tokens. Adopting both LSTM and MLP for the association of prompt tokens achieves the best performance, i.e., 25.4% mAP for generated pseudo labels on novel classes of COCO. The ablations of the category token position and prompt token number which are decided by l and m in Eq.4 are shown in Table V Fig. 6: Visualization of the learnable prompt modules' effect on dense alignment ability and the generated pseudo labels of novel classes on COCO dataset. The Vision.P and Text.P denote the vision prompt and text prompt modules, respectively. c) How the pseudo labeling robust to hyper parameters?: We conduct the ablation studies on the two most important hyper parameters, including the similarity threshold δ for dense score map to compute connected regions and objectness threshold for RPN to filter the low-confident proposals, in fine-tuning stage. Note that when we ablation one parameter, we keep the other fixed as the default value. Observation can be made that our pseudo labeling strategy is robust to this two parameters and achieves the highest when similarity threshold and objectness threshold set to 0.6 and 0.98 respectively. Low similarity and objectness threshold bring much false positives while high similarity and objectness threshold decrease the overall recall. E. Qualitative Results Qualitative examples. We visualize the final detection performance of P 3 OVD on the novel classes of COCO dataset and the transfer performance on LVIS in Fig. 5. We can observe that our P 3 OVD can obtain high-quality bounding-boxes for novel classes on COCO even without any annotation of these categories. Besides, as shown in the second row of Fig. 5, our P 3 OVD can detect rare classes like fireplug and polar bear, demonstrating its open-vocabulary detection ability. However, we also find our P 3 OVD fails in detecting several novel categories, including skateboard, snowboard, etc, during the pseudo labeling stage and self-training stage. By observing the dense score map and the pseudo labels of skateboard class in Fig. 8, we attribute this phenomenon to the generation of wrong pseudo labels. To explain the reason, we think that some objects are often related to the environment, such as skateboards, skateboarders, and skateboard playground often appear together, thus trained with the alignment based on global image information and text embeddings, current vision-language model usually cannot distinguish these objects from other objects in same scene. Effect of prompt modules. In Fig. 6, we visualize the effect of proposed prompt modules on the generation of the pseudo labels of novel classes. By comparing the dense score maps in the third and sixth column, observation can be made that through learnable visual-text prompts, the VLM can achieve a better dense alignment and improve the quality of pseudo labels. Specifically, adopting the visual prompt tends to fill the dense alignment of the object, while adopting the text prompt usually explores the new classes or objects. Combining these two prompts makes the better performance to generate more precise pseudo bounding-boxes. We also visualize the distribution of the pixel embeddings generated by the vision-language model CLIP (with or without learnable prompt modules) via t-SNE [53] on novel classes of COCO dataset in Fig. 9. Observation can be made that adopting learnable prompt modules helps cluster the pixel embeddings in the same class and separate pixel embeddings from different categories, demonstrating their effects on dense alignments. Fig. 9: Illustration of the visual embeddings generated by the (a) original CLIP and (b) CLIP with the proposed visual prompts via t-SNE [53]. Point in different color denotes the pixel belongs to different novel classes. V. CONCLUSIONS In this paper, we propose a novel open-vocabulary pipeline named P 3 OVD. P 3 OVD introduces a new fine-tuning stage to enhance the self-training paradigm with dense alignments by adopting two learnable visual and text prompt modules. Experimental results show that P 3 OVD achieves the state-ofthe-art performance on the novel classes of COCO datasets and the best transfer performance when directly adapting the model trained on COCO to PASCAL VOC, Object365, and LVIS datasets. Additional experiments also show that after finetuning with two learnable prompt modules, P 3 OVD obtains a more precise dense score map (4.3% higher on mIoU) on novel classes. Nevertheless, P 3 OVD is a general pipeline of adopting the pre-trained vision-image encoder to dense prediction tasks, which can be easily extended to other tasks, e.g., open-vocabulary instance segmentation. We hope P 3 OVD can serve as a strong baseline for future research on different open-vocabulary tasks. Fig. 1 : 1Illustration of our proposed prompt-driven self-training pipeline P 3 OVD compared with the conventional self-training pipeline for open-vocabulary object detection. We introduce an additional adapting stage of VLM to obtain fine-grained alignment via adopting learnable visual and text prompt modules in image encoder and text encoder. Fig. 2 : 2Illustration of the dense score map of our proposed prompt-enhanced VLM compared with the original VLM. Fig. 5 : 5Qualitative detection results on the novel classes of COCO dataset and the transfer performance on LVIS dataset. Note that the detection model is trained without any annotations of these categories. Fig. 7 : 7Ablation studies of pseudo label generation on the (a) similarity threshold for dense score map (b) objectness threshold for RPN. Fig. 8 : 8Illustration of the dense score map and the pseudo labels of class skateboard. Corresponding author: Xiaodan Liang. Yanxin Long, Runhui Huang and Xiaodan Liang are with the School of Intelligent Systems Engineering, Sun Yat-sen University, Shenzhen 518107, China (e-mail: [email protected]; [email protected]; [email protected]). Jianhua Han, Xu Hang, Yi Zhu and Chunjing Xu is with Huawei Noah'ark Lab, Shanghai 201206, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). -grained Visual-text Prompt-driven Self-training PipelineFineAdapting stage Modified Image Encoder Visual Prompt Text Prompt Text Encoder Text Encoder Image Input Text Input Self-training stage Pseudo labeling stage Pre-trained RPN Pre-trained VLM (a) Conventional method (b) Ours Adapting VLM Fine-grained Alignment TABLE .cake, scissors, knife dog, couch bus, airplane, snowboard cat, cup, dog dog, umbrella bus, umbrella Input Original w/ Vision.P w/ Text.P GT Original dog, cat, couch cat, couch, cup, sink, keyboard P 3 OVD P 3 OVD TABLE II : IIAblation Study for different prompt settings on the novel classes of COCO dataset. H.P, T.P, V.P, S.T denote the human pre-defined prompt, text prompt, vision prompt and last training stage. The mAP(%) of pseudo labels and the mIoU(%) of dense score map on novel classes are reported. H.P T.P V.P S.T MAP(%) MIOU (%) +1.2 31.1 +0.2 25.4 +2.9 31.3 +0.4 25.1 +2.6 33.1 +2.2 26.0 +3.5 35.4 +4.5 31.5 +9.0 /22.5 30.9 23.7 TABLE III : IIIAblation study for different structure of text prompt. Embedding, LSTM and MLP denote the learnable embeddings, a bidirectional LSTM and a two-layer multilayer perceptron. The mAP(%) of pseudo labels are reported.Embedding LSTM MLP mAP(%) 22.5 23.6 +1.1 24.3 +1.8 25.4 +2.9 TABLE IV : IVGeneralization performance when adopted to other detection datasets. 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Girshick, "Mask r-cnn," in Proceedings of the IEEE international conference on computer vision, 2017, pp. 2961-2969. He works at the Human Cyber Physical Intelligence Integration Lab under the supervision of Prof. Xiaodan Liang. Before that, He received his Bachelor Degree from the Comunication College, Xidian University in 2020. Sun Yat-Sen UniversityYanxin Long is a third-year master in the School of Intelligent Systems Engineering. His research interests include 2D detection and vision-and-language understandingYanxin Long is a third-year master in the School of Intelligent Systems Engineering, Sun Yat-Sen University. He works at the Human Cyber Physical Intelligence Integration Lab under the supervision of Prof. Xiaodan Liang. Before that, He received his Bachelor Degree from the Comunication College, Xi- dian University in 2020. His research interests include 2D detection and vision-and-language understanding. He is currently a re-searcher with the Noahs Ark Laboratory. Huawei Technologies Co ., LtdChinaJianhua Han received the Bachelor Degree and Master Degree from Shanghai Jiao Tong UniversityHis research interests lie primarily in deep learning and computer visionJianhua Han received the Bachelor Degree and Master Degree from Shanghai Jiao Tong University, China. He is currently a re-searcher with the Noahs Ark Laboratory, Huawei Technologies Co ., Ltd. His research interests lie primarily in deep learning and computer vision. He works at the Human Cyber Physical Intelligence Integration Lab under the supervision of Prof. Xiaodan Liang. Before that, he received the Bachelor Degree from the School of Intelligent Systems Engineering. School of Intelligent Systems Engineering, Sun Yat-sen University ; Sun Yat-sen UniversityRunhui Huang is a two-year master in the. in 2021. His reasearch interests include vision-and-language pre-trainingRunhui Huang is a two-year master in the School of Intelligent Systems Engineering, Sun Yat-sen University. He works at the Human Cyber Physical Intelligence Integration Lab under the supervision of Prof. Xiaodan Liang. Before that, he received the Bachelor Degree from the School of Intelligent Systems Engineering, Sun Yat-sen University in 2021. His reasearch interests include vision-and-language pre-training. Hang Xu is currently a senior researcher in Huawei Noah Ark Lab. He received his BSc in Fudan University and Ph.D in Hong Kong University in Statistics. He has published 50+ papers in top-tier AI conferences such as NeurIPS, CVPR, ICCV, ICLR and some statistics journals. CSDA, Statistical ComputingHang Xu is currently a senior researcher in Huawei Noah Ark Lab. He received his BSc in Fudan University and Ph.D in Hong Kong University in Statistics in 2018. His research Interest includes machine Learning, vision-language models, object detection and AutoML. He has published 50+ papers in top-tier AI conferences such as NeurIPS, CVPR, ICCV, ICLR and some statistics journals, e.g. CSDA, Statistical Computing. Since 2015, she has been a Ph.D student in computer science with the School of Electronic, Electrical, and Communication Engineering. Guangzhou, China; Beijing, ChinaYi Zhu received the B.S. degree in software engineering from Sun Yat-sen University ; University of Chinese Academy of SciencesHer current research interests include object recognition, scene understanding, weakly supervised learning and visual reasoningYi Zhu received the B.S. degree in software engi- neering from Sun Yat-sen University, Guangzhou, China, in 2013. Since 2015, she has been a Ph.D student in computer science with the School of Elec- tronic, Electrical, and Communication Engineering, University of Chinese Academy of Sciences, Beijing, China. Her current research interests include object recognition, scene understanding, weakly supervised learning and visual reasoning. He was Assistant Professor and then Associate Professor at Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences. He became director of the computer vision lab in Noah's Ark lab. Chunjing Xu received his Bachelor degree in Math from Wuhan University ; Master degree in Math from Peking University 2002, and PhD from Chinese University of Hong KongChunjing Xu received his Bachelor degree in Math from Wuhan University 1999, Master degree in Math from Peking University 2002, and PhD from Chinese University of Hong Kong 2009. He was Assistant Professor and then Associate Professor at Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences. He became director of the computer vision lab in Noah's Ark lab, Central research institute in 2017. His main research interests focus on machine learning and computer vision. She received her PhD degree from Sun Yat-sen University in 2016, advised by Liang Lin. She has published several cutting-edge projects on humanrelated analysis, including human parsing, pedestrian detection and instance segmentation. Xiaodan Liang is currently an Associate Professor at Sun Yat-sen UniversityShe was a postdoc researcher in the machine learning department at Carnegie Mellon University, working with Prof. Eric Xing. 2D/3D human pose estimation and activity recognitionXiaodan Liang is currently an Associate Professor at Sun Yat-sen University. She was a postdoc researcher in the machine learning department at Carnegie Mel- lon University, working with Prof. Eric Xing, from 2016 to 2018. She received her PhD degree from Sun Yat-sen University in 2016, advised by Liang Lin. She has published several cutting-edge projects on human- related analysis, including human parsing, pedestrian detection and instance segmentation, 2D/3D human pose estimation and activity recognition.
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[ "Analytical and Numerical Studies of Noise-induced Synchronization of Chaotic Systems", "Analytical and Numerical Studies of Noise-induced Synchronization of Chaotic Systems" ]
[ "Raúl Toral \nInstituto Mediterráneo de Estudios Avanzados\nIMEDEA *\nCSIC-UIB\n\n\nDepartament de Física\nUniversitat de les Illes Balears\n07071-Palma de MallorcaSpain\n", "Claudio R Mirasso \nDepartament de Física\nUniversitat de les Illes Balears\n07071-Palma de MallorcaSpain\n", "Emilio Hernández-García \nInstituto Mediterráneo de Estudios Avanzados\nIMEDEA *\nCSIC-UIB\n\n\nDepartament de Física\nUniversitat de les Illes Balears\n07071-Palma de MallorcaSpain\n", "Oreste Piro \nInstituto Mediterráneo de Estudios Avanzados\nIMEDEA *\nCSIC-UIB\n\n\nDepartament de Física\nUniversitat de les Illes Balears\n07071-Palma de MallorcaSpain\n" ]
[ "Instituto Mediterráneo de Estudios Avanzados\nIMEDEA *\nCSIC-UIB\n", "Departament de Física\nUniversitat de les Illes Balears\n07071-Palma de MallorcaSpain", "Departament de Física\nUniversitat de les Illes Balears\n07071-Palma de MallorcaSpain", "Instituto Mediterráneo de Estudios Avanzados\nIMEDEA *\nCSIC-UIB\n", "Departament de Física\nUniversitat de les Illes Balears\n07071-Palma de MallorcaSpain", "Instituto Mediterráneo de Estudios Avanzados\nIMEDEA *\nCSIC-UIB\n", "Departament de Física\nUniversitat de les Illes Balears\n07071-Palma de MallorcaSpain" ]
[]
We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of 1-d maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon analytically. Finally, we analyze the structural stability of the phenomenon.
10.1063/1.1386397
[ "https://arxiv.org/pdf/nlin/0104044v1.pdf" ]
34,587,077
nlin/0104044
17e28bfa25d3c1a32839bdb9a9b08963a13ff461
Analytical and Numerical Studies of Noise-induced Synchronization of Chaotic Systems 18 Apr 2001 Raúl Toral Instituto Mediterráneo de Estudios Avanzados IMEDEA * CSIC-UIB Departament de Física Universitat de les Illes Balears 07071-Palma de MallorcaSpain Claudio R Mirasso Departament de Física Universitat de les Illes Balears 07071-Palma de MallorcaSpain Emilio Hernández-García Instituto Mediterráneo de Estudios Avanzados IMEDEA * CSIC-UIB Departament de Física Universitat de les Illes Balears 07071-Palma de MallorcaSpain Oreste Piro Instituto Mediterráneo de Estudios Avanzados IMEDEA * CSIC-UIB Departament de Física Universitat de les Illes Balears 07071-Palma de MallorcaSpain Analytical and Numerical Studies of Noise-induced Synchronization of Chaotic Systems 18 Apr 2001(April 5, 2001)arXiv:nlin/0104044v1 [nlin.CD] We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of 1-d maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon analytically. Finally, we analyze the structural stability of the phenomenon. The synchronization of chaotic systems has been the subject of intensive research in the last years. Besides its fundamental interest, the study of the synchronization of chaotic oscillators has a potential application in the field of chaos communications. The main idea resides in the hiding of a message within a chaotic carrier generated by a suitable emitter. The encoded message can be extracted if an appropriate receiver, one which synchronizes to the emitter, is used. One of the conditions to be fulfilled in order to achieve synchronization is that the receiver and the emitter have very similar device parameters, hence making it very difficult to intercept the encoded message. Although the usual way of synchronizing two chaotic systems is by injecting part of the emitted signal into the receiver, the possibility of synchronization using a common random forcing has been also suggested. However, there have been some contradictory results in the literature on whether chaotic systems can indeed be synchronized using such a common source of noise and the issue has began to be clarified only very recently. In this paper we give explicit examples of chaotic systems that become synchronized by the addition of Gaussian white noise of zero mean. We also analyze the structural stability of the phenomenon, namely, the robustness of the synchronization against a small mismatch in the parameters of the chaotic sender and receiver. I. INTRODUCTION One of the most surprising results of the last decades in the field of stochastic processes has been the discovering that fluctuation terms (loosely called noise) can actually induce some degree of order in a large variety of non-linear systems. The first example of such an effect is that of stochastic resonance [1,2] by which a bistable system responds better to an external signal (not nec-essarily periodic) under the presence of fluctuations, either in the intrinsic dynamics or in the external input. This phenomenon has been shown to be relevant for some physical and biological systems described by nonlinear dynamical equations [3][4][5]. Other examples in purely temporal dynamical systems include phenomena such as noise-induced transitions [6], noise-induced transport [7], coherence resonance [8][9][10][11], etc. In extended systems, noise is known to induce a large variety or ordering effects [12], such as pattern formation [13,14], phase transitions [15][16][17][18], phase separation [19,20], spatiotemporal stochastic resonance [21,22], noise-sustained structures [23,24], doubly stochastic resonance [25], amongst many others. All these examples have in common that some sort of order appears only in the presence of the right amount of noise. There has been also some recent interest on the interplay between chaotic and random dynamics. Some counterintuitive effects such as coherence resonance, or the appearance of a quasi-periodic behavior, in a chaotic system in the presence of noise, have been found recently [26]. The role of noise in standard synchronization of chaotic systems has been considered in [27,28], as well as the role of noise in synchronizing non-chaotic systems [29,30]. In this paper we address the different issue of synchronization of chaotic systems by a common random noise source, a topic that has attracted much attention recently. The accepted result is that, for some chaotic systems, the introduction of the same noise in independent copies of the systems could lead (for large enough noise intensity) to a common collapse onto the same trajectory, independently of the initial condition assigned to each of the copies. This synchronization of chaotic systems by the addition of random terms is a remarkable and counterintuitive effect of noise and although some clarifying papers have appeared recently, still some contradictory results exist for the existence of this phenomenon of noise-induced synchronization. It is the purpose of this paper to give further analytical and numerical evidence that chaotic systems can synchronize under such circumstances and to analyze the structural stability of the phenomenon. Moreover, the results presented here clarify the issue, thus opening directions to obtain such a synchronization in electronic circuits, for example for encryption purposes. Common random noise codes have been used in spread spectrum communication since a long time ago [31]. The main idea is to mix a information data within a noisy code. At the receiver, the information is recovered using a synchronized replica of the noise code. More recently, the use of common noise source has been also proposed as a useful technique to improve the encryption of a key in a communication channel [32]. The issue of ordering effect of noise in chaotic systems was considered already at the beginning of the 80's by Matsumoto and Tsuda [33] who concluded that the introduction of noise could actually make a system less chaotic. Later, Yu, Ott and Chen [34] studied the transition from chaos to non-chaos induced by noise. Synchronization induced by noise was considered by Fahy and Hamman [35] who showed that particles in an external potential, when driven by the same random forces, tend to collapse onto the same trajectory, a behavior interpreted as a transition from chaotic to non-chaotic behaviors. The same system has been studied numerically and analytically [36][37][38]. Pikovsky [39] analyzed the statistics of deviations from this noise-induced synchronization. A paper that generated a lot of controversy was that of Maritan and Banavar [40]. These authors analyzed the logistic map in the presence of noise: x n+1 = 4x n (1 − x n ) + ξ n(1) where ξ n is the noise term, considered to be uniformly distributed in a symmetric interval [−W, +W ]. They showed that, if W was large enough (i.e. for a large noise intensity) two different trajectories which started with different initial conditions but used otherwise the same sequence of random numbers, would eventually coincide into the same trajectory. The authors showed a similar result for the Lorenz system (see section III). This result was heavily criticized by Pikovsky [41] who proved that two systems can synchronize only if the largest Lyapunov exponent is negative. He then argued that the largest Lyapunov exponent of the logistic map in the presence of noise is always positive and concluded that the synchronization was, in fact, a numerical effect of lack of precision of the calculation. The analysis of Pikovsky was confirmed by Longa et al. [42] who studied the logistic map with arbitrary numerical precision. The criterion of negative Lyapunov exponent has also been shown to hold for other types of synchronization of chaotic systems and Zhou and Lai [43] noticed that previous results by Shuai, Wong and Cheng [44] showing synchronization with a positive Lyapunov exponent were again an artifact of the limited precision of the calculation. In addition to the above criticisms, Herzel and Freund [45] and Malescio [46] pointed out that the noise used to simulate Eq.(1) and the Lorenz system in [40] is not really symmetric. While the noise in the Lorenz system is non-symmetric by construction, in the case of the map, the non-zero mean arises because the requirement x n ∈ (0, 1), ∀n, actually leads to discard the values of the random number ξ n which would induce a violation of such condition. The average value of the random numbers which have been accepted is different from zero, hence producing an effective biased noise, i.e. one which does not have zero mean. The introduction of a non-zero mean noise means that the authors of [40] were altering essentially the properties of the deterministic map. Furthermore, Gade and Bassu [47] argued that the synchronization observed by Maritan and Banavar is due to the fact that the bias of the noise leads the system to a nonchaotic fixed point. With only this basis, they concluded that a zero-mean noise can never lead to synchronization in the Lorenz system. The same conclusion was reached by Sánchez et al. [48] who studied experimentally a Chua circuit and concluded that synchronization by noise only occurs if the noise does not have a zero mean. The same conclusion is obtained in [49] by studying numerically a single and an array of Lorenz models, and in [50] from experiments in an array of Chua circuits with multiplicative colored noise. Therefore, from these last works, a widespread belief has emerged according to which it is not possible to synchronize two chaotic systems by injecting the same noisy unbiased, zero-mean, signal to both of them. Contrary to these last results (but in agreement with the previously mentioned results [32][33][34][35][36][37][38][39]), Lai and Zhou [51] have shown that some chaotic maps can indeed become synchronized by additive zero-mean noise. A similar result has been obtained by Loreto et al. [52], and by Minai and Anand [32,53,54], in the case where the noise appears parametrically in the map. The implications to secure digital communications have been considered in [32,53,55], and an application to ecological dynamics in fluid flows is presented in [56]. An equivalent result about the synchronization of Lorenz systems using a common additive noise has been shown by the authors of the present paper in [57]. The actual mechanism that leads to synchronization has been explained by Lai and Zhou [51], see also [58]. As Pikovsky [41] required, synchronization can only be achieved if the Lyapunov exponent is negative. The presence of noise allows the system to spend more time in the "convergence region" where the local Lyapunov exponent is negative, hence yielding a global negative Lyapunov exponent. This argument will be developed in more detail in section II, where an explicit calculation in a simple map will confirm the analysis. The results of Lai and Zhou have been extended to the case of coupled map lattices [59] where Pikovsky's criterion has been extended for spatially extended systems. In this paper we give further evidence that it is possible to synchronize two chaotic systems by the addition of a common noise which is Gaussianly distributed and not biased. We analyze specifically some 1-d maps and the Lorenz system, all in the chaotic region. The necessary criterion introduced in Ref. [41] and the general arguments of [51] are fully confirmed and some heuristic arguments are given about the general validity of our results. The organization of the paper is as follows. In section II we present numerical and analytical results for some 1-d maps, while section III studies numerically the Lorenz system. In section IV we analyze the structural stability of the phenomenon, i.e. the dependence of the synchronization time on the parameter mismatch. Finally, in section V we present the conclusions as well as some open questions relating the general validity of our results. II. RESULTS ON MAPS The first example is that of the map: x n+1 = F (x n ) = f (x n ) + ǫξ n (2) where ξ n is a set of uncorrelated Gaussian variables of zero mean and variance 1. As an example, we use explicitly f (x) = exp − x − 0.5 ω 2(3) Studying the convergence or divergence of trajectories of Eq. (2) starting from different initial conditions under the same noise ξ n is equivalent to analyzing the converge or divergence of trajectories from two identical systems of the form (2) driven by the same noise. We plot in Fig.(1) the bifurcation diagram of this map in the noiseless case. We can see the typical windows in which the system behaves chaotically. The associated Lyapunov exponent, λ, is positive in these regions. For instance, for ω = 0.3 (the case we will be considering throughout the paper) it is λ ≈ 0.53. In Fig.(2) we observe that the Lyapunov exponent becomes negative for most values of ω for large enough noise level ǫ. Again for ω = 0.3 and now for ǫ = 0.2 it is λ = −0.17. A positive Lyapunov exponent in the noiseless case implies that trajectories starting with different initial conditions, but using the same sequence of random numbers {ξ n }, remain different for all the iteration steps. In this case, the corresponding synchronization diagram shows a spread distribution of points (see Fig.(3a)). However, when moderate levels of noise (ǫ > ∼ 0.2) are used, λ becomes negative and trajectories starting with different initial conditions, but using the same sequence of random numbers, synchronize perfectly, see the synchronization diagram in Fig.(3b). Obviously, the noise intensity in the cases shown is not large enough such as to be able to neglect completely the deterministic part of the map. Therefore, the synchronization observed does not trivially appear as a consequence of both variables becoming themselves identical to the noise term. 2) and (3) with ω = 0.3. Each realization consists of 10,000 points which have been obtained by iteration of the map starting in each case from a different initial condition (100,000 initial iterations have been discarded and are not shown). In figure (a) there is no noise, ǫ = 0 and the trajectories are independent of each other. In figure (b) we have used a level of noise ǫ = 0.2, producing a perfect synchronization (after discarding some initial iterations). According to [41], convergence of trajectories to the same one, or lack of sensitivity to the initial condition, can be stated as negativity of the Lyapunov exponent. The Lyapunov exponent of the map (2) is defined as λ = lim N →∞ 1 N N i=1 ln |F ′ (x i )|(4) It is the average of (the logarithm of the absolute value of) the successive slopes F ′ found by the trajectory. Slopes in [−1, 1] contribute to λ with negative values, indicating trajectory convergence. Larger or smaller slopes contribute with positive values, indicating trajectory divergence. Since the deterministic and noisy maps satisfy F ′ = f ′ one is tempted to conclude that the Lyapunov exponent is not modified by the presence of noise. However, there is noise-dependence through the trajectory values x i , i = 1, ..., N . In the absence of noise, λ is positive, indicating trajectory separation. When synchronization is observed, the Lyapunov exponent is negative, as required by the argument in [41]. Notice that this definition of the Lyapunov exponent assumes a fixed realization of the noise terms, and it is the relevant one to study the synchronization phenomena addressed in this paper. One could use alternative definitions [52]. For instance, if one considers the coupled system of both the x variable and the noise generator producing ξ, then the largest Lyapunov exponent of the composed system is indeed positive (and very large for a good random number generator). This simply tells us that there is a large sensitivity to the initial condition of the composed system (x, ξ) as shown by the fact that a change of the seed of the random number generator completely changes the sequence of values of both ξ and x. We consider in this paper the situation described by definition (4) with fixed noise realization. By using the definition of the invariant measure on the attractor, or stationary probability distribution P st (x), the Lyapunov exponent can be calculated also as λ = log |F ′ (x)| = log |f ′ (x)| ≡ P st (x) log |f ′ (x)|dx(5) Here we see clearly the two contributions to the Lyapunov exponent: although the derivative f ′ (x) does not change when including noise in the trajectory, the stationary probability does change (see Fig.4), thus producing the observed change in the Lyapunov exponents. Synchronization, then, can be a general feature in maps, such as (3), which have a large region in which the derivative |f ′ (x)| is smaller than one. Noise will be able to explore that region and yield, on the average, a negative Lyapunov exponent. This is, basically, the argument developed in [51]. In order to make some analytical calculation that can obtain in a rigorous way the transition from a positive to negative Lyapunov exponent, let us consider the map given by Eq. (2) and f (x) =          a(1 − exp(1 + x)) if x < −1 −2 − 2x if x ∈ (−1, −.5) 2x if x ∈ (−.5, .5) 2 − 2x if x ∈ (.5, 1) a(−1 + exp(1 − x)) if x > 1 (6) with 0 < a < 1. This particular map, based in the tent map [60], has been chosen just for convenience. The following arguments would apply to any other map that in the absence of noise takes most frequently values in the region with the highest slopes, but which visits regions of smaller slope when noise is introduced. This is the case, for example, of the map (3). In the case of (6), the values given by the deterministic part of the map, after one iteration from arbitrary initial conditions, fall always in the interval (−1, 1). This is the region with the highest slope |F ′ | = 2. In the presence of noise the map can take values outside this interval and, since the slopes encountered are smaller, the Lyapunov exponent can only be reduced from the deterministic value. To formally substantiate this point, it is enough to recall the definition of Lyapunov exponent (4): an upper bound for |F ′ (x)| is 2, so that a bound for λ is immediately obtained: λ ≤ ln 2. Equality is obtained for zero noise. The interesting point about the map (6) and similar ones is that one can demonstrate analytically that λ can be made negative. The intuitive idea is that it is enough to decrease a in order to give arbitrarily small values to the slopes encountered outside (−1, 1), a region accessible only thanks to noise. To begin with, let us note that |F ′ (x)| = 2 if x ∈ (−1, 1), and |F ′ (x)| < a if |x| > 1, so that an upper bound to (4) can be written as λ ≤ lim N →∞ N I N ln 2 + N O N ln a = p I ln 2 + p O ln a = ln 2 − p O ln(2/a).(7) N I /N and N O /N are the proportion of values of the map inside I = (0, 1) and outside this interval, respectively, and we have used that as N → ∞ they converge to p I and p O , the invariant measure associated to I and to the rest of the real line, respectively (p I + p O = 1). A sufficient condition for x n+1 = f (x n ) + ǫξ n to fall outside I is that |ξ n | > 2/ǫ. Thus, p O = Probability(|x n+1 | > 1) > Probability(|ξ n | > 2/ǫ) = erfc( √ 2/ǫ) ≡ T , where we have used the Gaussian character of the noise. In consequence, one finds from (7) λ ≤ ln 2 − T ln(2/a). The important point is that T = erfc( √ 2/ǫ) is independent on the map parameters, in particular on a. Thus, (8) implies that by decreasing a the value of λ can be made as low as desired. By increasing ǫ such that T > ln 2/ ln(2/a), λ will be certainly negative. Thus we have shown analytically that strong enough noise will always make negative the Lyapunov exponent of the map (6) and, accordingly, it will induce yield "noise-induced synchronization" in that map. III. THE LORENZ SYSTEM In this section we give yet another example of noiseinduced synchronization. We consider the well known Lorenz [61] model with additional random terms of the form [40]:ẋ = p(y − x) y = −xz + rx − y + ǫξ (9) z = xy − bz ξ is white noise: a Gaussian random process of mean zero, ξ(t) = 0 and delta-correlated, ξ(t)ξ(t ′ ) = δ(t − t ′ ). We have used p = 10, b = 8/3 and r = 28 which, in the deterministic case, ǫ = 0 are known to lead to a chaotic behavior (the largest Lyapunov exponent is λ ≈ 0.9 > 0). As stated in the introduction, previous results seem to imply that synchronization is only observed for a noise with a non-zero mean. However, our results show otherwise. We have integrated numerically the above equations using the stochastic Euler method [62]. Specifically, the evolution algorithm reads: x(t + ∆t) = x(t) + ∆t [p(y(t) − x(t))] y(t + ∆t) = y(t) + ∆t [−x(t)z(t) + rx(t) − y(t)] (10) + ǫ √ ∆tg(t) z(t + ∆t) = z(t) + ∆t [x(t)y(t) − bz(t)] The values of g(t) are drawn at each time step from an independent Gaussian distribution of zero mean and variance one and they have been generated by a particularly efficient algorithm using a numerical inversion technique [63]. The time step used is ∆t = 0.001 and simulations range typically for a total time of the order of t = 10 4 (in the dimensionless units of the Lorenz system of equations). The largest Lyapunov exponent has been computed using a simultaneous integration of the linearized equations [64]. For the deterministic case, trajectories starting with different initial conditions are completely uncorrelated, see Fig. (5a). This is also the situation for small values of ǫ. However, when using a noise intensity ǫ = 40 the noise is strong enough to induce synchronization of the trajectories. Again, the presence of the noise terms forces the largest Lyapunov exponent to become negative (for ǫ = 40 it is λ ≈ −0.2). As in the examples of the maps, after some transient time, two different evolutions which have started in completely different initial conditions synchronize towards the same value of the three variables (see Fig. (5b) for the z coordinate). Therefore, these results prove that synchronization by common noise in the chaotic Lorenz system does occur for sufficiently large noise intensity. This result contradicts previous ones in the literature [46,48]. The main difference with these papers is in the intensity of the noise: it has to be taken sufficiently large, as here, in order to observe synchronization. Notice that although the noise intensity is large, the basic structure of the "butterfly" Lorenz attractor remains present as shown in Fig. (6). Again, this result shows that, although the noise intensity used could be considered large, the synchronization is rather different from what would be obtained from a trivial common synchronization of both systems to the noise variable by neglecting the deterministic terms. IV. STRUCTURAL STABILITY An important issue concerns the structural stability of this phenomenon, in particular how robust is noise synchronization to small differences between the two systems one is trying to synchronize. Whether or not the synchronization of two trajectories of the same noisy Lorenz system (or of any other chaotic system) observed here, equivalent to the synchronization of two identical systems driven by a common noise, could be observed in the laboratory, depends on whether the phenomenon is robust when allowing the two Lorenz systems to be not exactly equal (as they can not be in a real experiment). If one wants to use this kind of stochastic synchronization in electronic emitters and receivers (for instance, as a means of encryption) one should be able to determine the allowed discrepancy between circuits before the lack of synchronization becomes unacceptable. Additional discussions on this issue may be found in [39,32,65]. We consider the following two maps forced by the same noise: x n+1 = f (x n ) + ξ n (11) y n+1 = g(y n ) + ξ n (12) Linearizing in the trajectory difference u n = y n − x n , assumed to be small, we obtain u n+1 = g ′ (x n )u n + g(x n ) − f (x n ) ≡ g ′ (x n )u n + ∆(x n )(13) We have defined ∆(x) ≡ g(x) − f (x), and we are interested in the situation in which the two systems are just slightly different, for example, because of a small parameter mismatch, so that ∆ will be small in some sense specified below. Iteration of (13) leads to the formal solution: u n = M (n − 1, 0)u 0 + n−1 m=0 M (n − 1, m + 1)∆(x m )(14) We have defined M (j, i) = j k=i g ′ (x k ), and M (i−1, i) ≡ 1. An upper bound on (14) can be obtained: |u n | 2 ≤ |M (n − 1, 0)| 2 |u 0 | 2 + n−1 m=0 |M (n − 1, m + 1)| 2 |∆(x m )| 2(15) The first term in the r.h.s. is what would be obtained for identical dynamical systems. We know that M (n − 1, 0) → e λn as n → ∞, where λ is the largest Lyapunov exponent associated to (12). We are interested in the situation in which λ < 0, for which this term vanishes at long times. Further analysis is done first for the case in which ∆(x) is a bounded function (or x is a bounded trajectory with ∆ continuous). In this situation, there is a real number µ such that |∆(x m )| < µ. We then get: |u n | 2 ≤ µ 2 n−1 m=0 |M (n − 1, m + 1)| 2(16) an unequality valid for large n. Let us now define K = max x |g(x)|, the maximum slope of the function g(x). A trivial bound is now obtained as: |u n | 2 ≤ µ 2 1 − K 2n 1 − K 2(17) This can be further improved in the case K < 1, where we can write: |u n | 2 ≤ µ 2 1 1 − K 2(18) As a consequence, differences in the trajectories remain bounded at all iteration steps n. Since, according to the definition (4), ln K is also an upper bound for the Lyapunov exponent for all values of ǫ and, in particular, for the noiseless map, ǫ = 0, this simply tells us that if the deterministic map is non-chaotic, then the addition of a common noise to two imperfect but close replicas of the map will still keep the trajectory difference within well defined bounds. The situation of interest here, however, concerns the case in which a negative Lyapunov exponent arises only as the influence of a sufficiently large noise term, i.e. the deterministic map is chaotic and K > 1. In this case, the sum in Eq. (16) contains products of slopes which are larger or smaller than 1. It is still true that the terms in the sum for large value of n − m can be approximated by M (n − 1, m + 1) ≈ e (n−m−1)λ and, considering this relation to be valid for all values of n, m, we would get: n−1 m=0 |M (n − 1, m + 1)| 2 ≈ e 2λn n−1 m=0 e −2λ(m+1) = 1 − e 2λn 1 − e 2λ .(19) and, thus, at large n: |u n | 2 < ∼ µ 2 (1 − e 2λ ) −1(20) It can happen, however, that the product defining M (n− 1, m + 1) contains a large sequence of large slopes g ′ (x i ). These terms (statistically rare) will make the values of |u n | to violate the above bound at sporadic times. Analysis of the statistics of deviations from synchronization was carried out in [39]. Although for λ < 0 the most probable deviation is close to zero, power-law distributions with long tails are found, and indeed its characteristics are determined by the distribution of slopes encountered by the system during finite amounts of time, or finite-time Lyapunov exponents, as the arguments above suggest. Therefore, we expect a dynamics dominated by relatively large periods of time during which the difference between trajectories remains bounded by a small quantity, but intermittently interrupted by bursts of large excursions of the difference. This is indeed observed in the numerical simulations of the maps defined above. This general picture is still valid even if |∆(x)| is not explicitly bounded. We have performed a more quantitative study for the case in which two noisy Lorenz systems with different sets of parameters, namely: x 1 = p 1 (y 1 − x 1 ) y 1 = −x 1 z 1 + r 1 x 1 − y 1 + ǫξ (21) z 1 = x 1 y 1 − b 1 z 1 andẋ 2 = p 2 (y 2 − x 2 ) y 2 = −x 2 z 2 + r 2 x 2 − y 2 + ǫξ (22) z 2 = x 2 y 2 − b 2 z 2 are forced by the same noise ξ(t). In order to discern the effect of each parameter separately, we have varied independently each one of the three parameters, (p, b, r), while keeping constant the other two. The results are plotted in Fig. 7. In this figure we plot the percentage of time in which the two Lorenz systems are still synchronized with a tolerance of 10%. This means that trajectories are considered synchronized if the relative difference in the z variable is less than 10%. According to the general discussion for maps, we expect departures from approximate synchronization from time to time. They are in fact observed, but from Fig. 7 we conclude that small variations (of the order of 1%) still yield a synchronization time of more than 85%. In Fig. 8 we show that the loss of synchronization between the two systems appears in the form of bursts of spikes whose amplitude is only limited by the size of the attractor in the phase space. Moreover, it can be clearly seen in the same figure that large (but infrequent) spike amplitudes appear for arbitrarily small mismatch. FIG. 7. Percentage of time that two slightly dissimilar Lorenz systems, subjected to the same noise, remain synchronized (up to a 10% discrepancy in the z variable). For one of the two systems we fix p1 = 10, b1 = 8/3 and r1 = 28 while for the other we vary systematically one of the parameters keeping the other two constant: in panel (a) the parameter b2 varies , in panel (b) the parameter r2 varies and in panel (c) the parameter p2 varies. Notice that the percentage of synchronization time is still higher than 85% if the relative difference between the parameters is less than 1%. In the realm of synchronization of chaotic oscillators, two different types of analogous intermittent behavior have been associated also to the fluctuating character of the finite-time conditional Lyapunov exponents as above. One is on-off intermitency [66] where the synchronization manifold is sligthly unstable on average but the finite time Lyapunov exponent is negative during relatively long periods of time. In the other one, named bubbling [67], the synchronization is stable on average but the local conditional Lyapunov exponent becomes occasionally positive. While in the former case bursting always occurs due to the necessarily imperfect initial synchronization, in the latter it is strictly a consequence of the mismatch of the entraining systems. In this sense, the behavior reported in the preceding paragraph should be considered as a manifestation of bubbling in synchronization by common noise. FIG. 8. Time evolution of the difference between two trajectories z1 and z2 corresponding to two Lorenz systems driven by the same noise but with a small mismatch in the r parameter: panel (a) r1 = 28 and r2 = 0.99999 × r1 and in panel (b) r2 = 0.999 × r1. Notice that although the synchronization time diminishes with increasing parameter mismatch, the maximum absolute difference between the two variables does not depend on the mismatch. V. CONCLUSIONS AND OPEN QUESTIONS In this paper we have addressed the issue of synchronization of chaotic systems by the addition of common random noises. We have considered three explicit examples: two 1-d maps and the Lorenz system under the addition of zero-mean, Gaussian, white noise. While the map examples confirm previous results in similar maps, and we have obtained with them analytical confirmation of the phenomenon, the synchronization observed in the Lorenz system contradicts some previous results in the literature. The reason is that previous works considered noise intensities smaller than the ones we found necessary for noise-synchronization in this system. Finally, we have analyzed the structural stability of the observed synchronization. In the Lorenz system, synchronization times larger than 85% (within an accuracy of 10%) can still be achieved if the parameters of the system are allowed to change in less than 1%. It is important to point out that noise-induced synchronization between identical systems subjected to a common noise is equivalent to noise induced order, in the sense that the Lyapunov exponent defined in (4) becomes negative in a single system subjected to noise. One can ask whether the state with negative Lyapunov exponent induced by noise may be still be called 'chaotic' or not. This is just a matter of definition: if one defines chaos as exponential sensibility to initial conditions, and one considers this for a fixed noise realization, then the definition of Lyapunov exponent implies that trajectories are not longer chaotic in this sense. But one can also consider the extended dynamical system containing the forced one and the noise generator (for example, in numerical computations, it would be the computer random number generator algorithm). For this extended system there is strong sensibility to initial conditions in the sense that small differences in noise generator seed leads to exponential divergence of trajectories. In fact, this divergence is at a rate given by the Lyapunov exponent of the noise generator, which approaches infinity for a true Gaussian white process. Trajectories in the noise-synchronized state are in fact more irregular than in the absence of noise, and attempts to calculate the Lyapunov exponent just from the observation of the time series will lead to a positive and very large value, since it is the extended dynamical system the one which is observed when analyzing the time series [68] (typically such attempts will fail because the high dimensionality of good noise generators, ideally infinity, would put them out of the reach of standard algorithms for Lyapunov exponent calculations). Again, whether or not to call such irregular trajectories with just partial sensibility to initial conditions 'chaotic' is just a matter of definition. More detailed discussion along these lines can be found in [52]. There remain still many open questions in this field. They involve the development of a general theory, probably based in the invariant measure, that could give us a general criterion to determine the range of parameters (including noise levels) for which the Lyapunov exponent becomes negative, thus allowing synchronization. In this work and similar ones, the word synchronization is used in a very restricted sense, namely: the coincidence of asymptotic trajectories. This contrasts with the case of interacting periodic oscillations where a more general theory of synchronization exists to explain the phenomenon of non trivial phase locking between oscillators that individually display very different dynamics. Indications of the existence of analogue non trivial phase locking have been reported for chaotic attractors [69]. There a "phase" with a chaotic trajectory defined in terms of a Hilbert transform is shown to be synchronizable by external perturbations in a similar way as it happens with periodic oscillators. Whether or not this kind of generalized synchronization can be induced by noise is, however, a completely open question. Last, but not least, it would be also interesting to explore whether analogs of the recently reported synchronization of spatio-temporal chaos [70,71] may be induced by noise. FIG. 1 . 1Bifurcation diagram of the map given by Eqs.(2) and (3) in the absence of noise terms. FIG. 2. Lyapunov exponent for the noiseless map (ǫ = 0, continuous line) and the map with a noise intensity ǫ = 0.1 (dotted line) and ǫ = 0.2 (dot-dashed line). FIG. 3. Plot of two realizations x (1) , x (2) of of the map given by Eqs. 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[ "KIRSZBRAUN'S THEOREM VIA AN EXPLICIT FORMULA", "KIRSZBRAUN'S THEOREM VIA AN EXPLICIT FORMULA", "KIRSZBRAUN'S THEOREM VIA AN EXPLICIT FORMULA", "KIRSZBRAUN'S THEOREM VIA AN EXPLICIT FORMULA" ]
[ "Daniel Azagra ", "Erwan Le Gruyer ", "Carlos Mudarra ", "Daniel Azagra ", "Erwan Le Gruyer ", "Carlos Mudarra " ]
[]
[]
Let X, Y be two Hilbert spaces, E a subset of X and G : E → Y a Lipschitz mapping.
10.4153/s0008439520000314
[ "https://arxiv.org/pdf/1810.10288v3.pdf" ]
119,177,132
1810.10288
62fcd4c33c3b2e0694069abfa58c72576caa8294
KIRSZBRAUN'S THEOREM VIA AN EXPLICIT FORMULA 21 Apr 2020 Daniel Azagra Erwan Le Gruyer Carlos Mudarra KIRSZBRAUN'S THEOREM VIA AN EXPLICIT FORMULA 21 Apr 2020arXiv:1810.10288v3 [math.FA] Let X, Y be two Hilbert spaces, E a subset of X and G : E → Y a Lipschitz mapping. An explicit formula for Kirszbraun's theorem In 1934 M.D. Kirszbraun [23] proved that, for every subset E of R n and every Lipschitz function f : E → R m , there exists a Lipschitz extension F : R n → R m of f such that Lip(F ) = Lip(f ). Here Lip(ϕ) denotes the Lipschitz constant of ϕ, that is, Lip(ϕ) = sup x =y ϕ(x) − ϕ(y) x − y . This theorem was generalized for Hilbert spaces X, Y in place of R n and R m by F.A. Valentine [30] in 1945, and the result is often referred to as the Kirszbraun-Valentine theorem. The proof is rather nonconstructive, in the sense that it requires to use Zorn's lemma or transfinite induction at least in the nonseparable case. In the separable case the proof can be made by induction, considering a dense sequence {x k } in X and at each step managing to extend f from E ∪ {x 1 , . . . , x m } to E ∪ {x 1 , . . . , x m+1 } while preserving the Lipschitz constant of the extension by using Helly's theorem or intersection properties of families of balls, but still it is not clear what the extension looks like. Several other proofs and generalizations which are not constructive either have appeared in the literature; see [26,21,15,11,27,5,1]; apart from Zorn's lemma or induction these proofs are based on intersection properties of arbitrary families of balls, or on maximal extensions of non-expansive operators and Fitzpatrick functions. In 2008 H.H. Bauschke and X. Wang [9] gave the first constructive proof of the Kirszbraun-Valentine theorem of which we are aware; they relied on their previous work [8] on extension and representation of monotone operators and the Fitzpatrick function. See also [2], where some of these techniques are used to construct definable versions of Helly's and Kirszbraun's theorems in arbitrary definably complete expansions of ordered fields. Finally, in 2015 E. Le Gruyer and T-V. Phan provided sup-inf explicit extension formulas for Lipschitz mappings between finite dimensional spaces by relying on Le Gruyer's solution to the minimal C 1,1 extension problem for 1-jets; see [25,Theorem 32 and 33] and [24]. In this note we present a short proof of the Kirszbraun-Valentine theorem in which the extension is given by an explicit formula. This proof is based on our previous work concerning C 1,1 extensions of 1-jets with optimal Lipschitz constants of the gradients [4]. See [14] for an alternative construction of such C 1,1 extensions on the Hilbert space, and [12,16,17,18] for the much more difficult question of extending functions (as opposed to jets) to C 1,1 or C m,1 functions on R n . If X is a Hilbert space, E ⊂ X is an arbitrary subset and (f, G) : E → R × X is a 1-jet on E, we will say that (f, G) satisfies condition (W 1,1 ) with constant M > 0 on E provided that (1) f (y) ≤ f (x) + 1 2 G(x) + G(y), y − x + M 4 x − y 2 − 1 4M G(x) − G(y) 2 , for all x, y ∈ E. In [32,24] it was proved that condition (W 1,1 ) with constant M > 0 is a necessary and sufficient condition on f : E → R, G : E → X for the existence of a function F ∈ C 1,1 (X) with Lip(∇F ) ≤ M and such that F = f and ∇F = G on E. Here ∇F (x) denotes the gradient of F at the point x, that is, the unique vector ∇F (x) ∈ X for which DF (x)(v) = ∇F (x), v for every v ∈ X, where DF (x) ∈ X * denotes the Fréchet derivative of F at the point x. More recently, as a consequence of a similar extension theorem for C 1,1 convex functions, we have found an explicit formula for such an extension F. Theorem 3.4] Let E be a subset of a Hilbert space X. Given a 1-jet (f, G) satisfying condition (W 1,1 ) with constant M on E, the formula Theorem 1. [4,F = conv(g) − M 2 · 2 , g(x) = inf y∈E f (y) + G(y), x − y + M 2 x − y 2 + M 2 x 2 , x ∈ X, defines a C 1,1 (X) function with F | E = f , (∇F ) | E = G, and Lip(∇F ) ≤ M . Here conv(g) denotes the convex envelope of g, defined by (2) conv(g)(x) = sup{h(x) : h is convex, proper and lower semicontinuous, h ≤ g}. Another expression for conv(g) is given by (3) conv(g)(x) = inf    k j=1 λ j g(x j ) : λ j ≥ 0, k j=1 λ j = 1, x = k j=1 λ j x j , k ∈ N    , and also by the Fenchel biconjugate of g, that is, (4) conv(g) = g * * , where (5) h * (x) := sup v∈X { v, x − h(v)}; see [13,Proposition 4.4.3] for instance. In the case that X is finite dimensional, say X = R n , the expression (3) can be made simpler: by using Carathéodory's Theorem one can show that it is enough to consider convex combinations of at most n + 1 points. That is to say, if g : R n → R then (6) conv(g)(x) = inf    n+1 j=1 λ j g(x j ) : λ j ≥ 0, n+1 j=1 λ j = 1, x = n+1 j=1 λ j x j    ; see [28,Corollary 17.1.5] for instance. In general, the convex envelope does not preserve smoothness of orders higher than C 1 and C 1,1 . For instance, the function g(x, y) = x 2 + e −y 2 defined on R 2 is real analytic and its convex envelope is conv(g)(x, y) = |x| for every (x, y) ∈ R 2 ; see [10]. In [22], Kirchheim and Kristensen proved that the convex envelope of a differentiable function g on R n is of class C 1 , provided that g is coercive. On the other hand, if g is of class C 1,1 on a Hilbert space X, or even if g only satisfies g(x + h) + g(x − h) − 2g(x) ≤ M h 2 , x, h ∈ X for some M > 0, then conv(g) is of class C 1,1 and Lip(∇ conv(g)) ≤ M ; see [4,Theorem 2.3]. In particular, if g = inf i (g i ) is the infimum of an arbitrary family of parabolas g i , whose second derivatives are uniformly bounded by a constant M > 0, then conv(g) is of class C 1,1 with Lip(∇ conv(g)) ≤ M, provided that g has a convex lower semicontinuous minorant. However, conv(g) is not necessarily of class C 2 even when g is the minimum of two parabolas: if we define g( x) = min{x 2 , (x − 1) 2 } for x ∈ R, then conv(g)(x) = x 2 for x ≤ 0, conv(g) = 0 for 0 ≤ x ≤ 1 and conv(g) = (x − 1) 2 for x ≥ 1; and therefore conv(g) ∈ C 1,1 (R) \ C 2 (R). Theorem 2 (Kirszbraun's theorem via an explicit formula). Let X, Y be two Hilbert spaces, E a subset of X and G : E → Y a Lipschitz mapping. There exists G : X → Y with G = G on E and Lip( G) = Lip(G). In fact, if M = Lip(G), then the function G(x) := ∇ Y (conv(g))(x, 0), x ∈ X, where g(x, y) = inf z∈E G(z), y Y + M 2 x − z 2 X + M 2 x 2 X + M y 2 Y , (x, y) ∈ X × Y, defines such an extension. Here · X and · Y denote the norm on X and Y respectively. Also, the inner products in X and Y are denoted by ·, · X and ·, · Y respectively. For any function F, ∇ Y F will stand for the Y -partial derivatives of F, that is, the canonical projection from X × Y onto Y composed with ∇F . Proof. We consider on X × Y the norm given by (x, y) = x 2 X + y 2 Y for every (x, y) ∈ X × Y. Then X × Y is a Hilbert space whose inner product is (x, y), (x ′ , y ′ ) = x, x ′ X + y, y ′ Y for every (x, y), (x ′ , y ′ ) ∈ X × Y. We define the 1-jet (f * , G * ) on E × {0} ⊂ X × Y by f * (x, 0) = 0 and G * (x, 0) = (0, G(x)). Then the jet (f * , G * ) satisfies condition (W 1,1 ) on E × {0} with constant M (see inequality (1)). Indeed, by the definition of f * and G * we can write, for every (x, 0), (y, 0) ∈ E × {0}, f * (x, 0) − f * (y, 0) + 1 2 G * (x, 0) + G * (y, 0), (y, 0) − (x, 0) + M 4 (x, 0) − (y, 0) 2 − 1 4M G * (x, 0) − G * (y, 0) 2 = 1 2 (0, G(x)) + (0, G(y)), (y, 0) − (x, 0) + M 4 x − y 2 X − 1 4M G(x) − G(y) 2 Y = M 4 x − y 2 X − 1 4M G(x) − G(y) 2 Y , and the last term is nonnegative because G is M -Lipschitz on E. Therefore, Theorem 1 asserts that the function F defined by F = conv(g) − M 2 · 2 , where g(x, y) = inf z∈E f * (z, 0) + G * (z, 0), (x − z, y) + M 2 (x − z, y) 2 + M 2 (x, y) 2 , is of class C 1,1 (X × Y ) with (F, ∇F ) = (f * , G * ) on E × {0} and Lip(∇F ) ≤ M. In particular, the mapping X ∋ x → G(x) := ∇ Y F (x, 0) ∈ Y is M -Lipschitz and extends G from E to X. Finally, the expressions defining G and g can be simplified as G(x) = ∇ Y conv(g) − M 2 · 2 (x, 0) = ∇ Y (conv(g))(x, 0) − ∇ Y M 2 · 2 (x, 0) = ∇ Y (conv(g))(x, 0) and g(x, y) = inf z∈E G(z), y Y + M 2 x − z 2 X + M 2 x 2 X + M y 2 Y . Let X be a Hilbert space with inner product and associated norm denoted by ·, · and · respectively. If E ⊂ X is arbitrary and G : E → X is a mapping, we say that G is firmly non-expansive if G(x) − G(y), x − y ≥ G(x) − G(y) 2 for all x, y ∈ E. Important examples of firmly non-expansive mappings are the metric projections onto closed convex sets of Hilbert spaces and the proximal mappings prox f : X → X of proper lower semicontinuous convex functions f : X → (−∞, +∞]; see [6,Chapter 12]. Firmly non-expansive mappings arise naturally in convex feasibility problems too: given a family C 1 , . . . , C N of closed convex sets of a Hilbert space, find a point x ∈ i C i . Also, these mappings are known to be resolvents J A = (A + I) −1 of monotone or maximally monotone operators A : X ⇒ X, and they play a crucial role in the following basic problem that arises in several branches of applied mathematics: given a maximally monotone operator A : X ⇒ X, find a point x ∈ X with 0 ∈ Ax. For more information about firmly non-expansive mappings and their applications; see [5,6,7,9] and the references therein. It is well-known that a mapping G : E → X is firmly non-expansive if and only 2G − I : E → X is 1-Lipschitz, where I denotes the identity map; see [6, Proposition 4.2] for a proof of this fact. Using this characterization and Theorem 2 we obtain the following corollary. Corollary 3. Let G : E → X be a firmly non-expansive mapping defined on a subset E of a Hilbert space X. Then G can be extended to a firmly non-expansive mapping G : X → X by means of the formula G(x) := 1 2 (P 2 (∇(conv(g))(x, 0)) + x) x ∈ X, where P 2 (x, y) = y, (x, y) ∈ X × X, and g(x, y) = inf z∈E 2 G(z), y + 1 2 z − (x + y) 2 + 1 2 x − y 2 , (x, y) ∈ X × X. Extensions of strongly biLipschitz mappings In this section we consider strongly biLipschitz mappings, which appear naturally as derivatives of strongly convex C 1,1 functions, and we provide an extension result for this class of mappings. Definition 4. Let E be a subset of a Hilbert space X. We say that a mapping G : E → X is strongly biLipschitz provided that SBilip(G) := inf x,y∈E; x =y 2 x − y, G(x) − G(y) x − y 2 + G(x) − G(y) 2 > 0. Strongly biLipschitz mappings may be interesting in regard to some problems in computer vision or image processing where one needs to match points in R n : for instance, given two sets of points in R n with equal cardinality, find a homeomorphism from R n onto itself which does not distort distances too much and takes one set onto the other. Supposing that the data satisfy the strongly biLipschitz condition, our explicit formula for such an extension can be useful. Also, in [7,Corollary 4.5] it is shown that strongly biLipschitz mappings are closely related to contractive mappings: a maximally monotone mapping G is strongly biLipschitz if and only if its reflected resolvent N = 2(G + I) −1 − I is a contractive mapping. See [29,Chapter 12] for more information about resolvent mappings and maximally monotone operators. It should also be noted that it is not generally true that a biLipschitz mapping whose domain and range is a subset of the same Hilbert space X extends to a total one-to-one continuous mapping, as shown by the following example. Example 5. Let | · | denote the euclidean norm on R n . Let C = {x ∈ R n : |x| = 1} ∪ {p}, p be any point with |p| > 1, and g : C → R n be defined by g(x) = x for |x| = 1 and g(p) = 0. Then both g and g −1 are Lipschitz but no continuous extension of g to R n can be one-to-one. However, this is true for the class of strongly biLipschitz mappings, and moreover, the extension can be performed without increasing what seems natural to call the strong biLipschitz constant SBilip(G), as we will show by using the extension formula for Lipschitz mappings given by Theorem 2. Proposition 6. If G : E → X is strongly biLipschitz then G is biLipschitz. Proof. For every x, y ∈ E we have (7) 2 x − y G(x) − G(y) ≥ 2 x − y, G(x) − G(y) ≥ α x − y 2 + G(x) − G(y) 2 , where α = SBilip(G) is as in Definition 4. It follows that G is one-to-one. Note that we can write (7) in the equivalent form (8) G(x) − G(y) − 1 α (x − y) 2 ≤ 1−α 2 α 2 x − y 2 . Setting λ := G(x) − G(y) x − y for x = y, (7) holds if and only if (9) λ 2 − 2 α λ + 1 ≤ 0, which is equivalent to 1 K ≤ λ ≤ K; where K = 1 α + ( 1 α 2 − 1) 1/2 . This means that for any x, y ∈ E with x = y, we have 1 K ≤ G(x) − G(y) x − y ≤ K. Therefore G is a biLipschitz mapping. Remark 7. (i) If X = R, then the strongly biLipschitz functions are exactly the strictly increasing biLipschitz functions. (ii) If G : E → X is such that SBilip(G) = 1, then G is the restriction of a translation. (iii) If G : E → X is an isometry such that α := inf x,y∈E; x =y x − y, G(x) − G(y) x − y 2 > 0, then G is a strongly biLipschitz function with SBilip(G) = α. However the composition of strongly biLipschitz isometries need not be strongly biLipschitz (for instance, if r : R 2 → R 2 is defined by r(z) = e πi/4 z then r is strongly biLipschitz, but r 2 is not (and in fact r 4 = −id is not strongly biLipschitz locally on any disk). Theorem 8. Let G : E → X be a strongly biLipschitz mapping. Then G extends to a strongly biLipschitz mapping on X preserving the strongly bilipchitz constant SBilip(G). Moreover, if α = SBilip(G), the formula G(x) := P 2 (∇(conv(g))(x, 0)) + 1 α x, x ∈ X; where P 2 (x, y) = y, (x, y) ∈ X × X, and g(x, y) = inf z∈E G(z), y − 1 α z, y + √ 1−α 2 2α x − z 2 + √ 1−α 2 α 1 2 x 2 + y 2 , (x, y) ∈ X × X, defines such an extension. Proof. We know from the characterization (8) that G − 1 α I is Lipschitz on E with Lip G − 1 α I ≤ 1−α 2 α 2 . By Theorem 2, the mapping T : X → X defined as T (x) := P 2 (∇(conv(g))(x, 0)) x ∈ X; where P 2 (x, y) = y for every (x, y) ∈ X × X, and g(x, y) = inf z∈E G(z), y − 1 α z, y + √ 1−α 2 2α x − z 2 + √ 1−α 2 α 1 2 x 2 + y 2 , (x, y) ∈ X × X, is an extension of G − 1 α I to all of X such that Lip(T ) = Lip G − 1 α I ≤ 1−α 2 α 2 . Therefore, if we define the function G = T + 1 α I, we have that G(x) − G(y) − 1 α (x − y) 2 = T (x) − T (y) 2 ≤ 1−α 2 α 2 x − y 2 for all x, y ∈ X. We obtain from (8) that G is strongly biLipschitz on X with SBilip( G) = α. Also, since T is an extension of G − 1 α I, it is obvious that G is an extension of G. C 1,1 strongly convex functions In this section we characterize the 1-jets which can be interpolated by strongly convex functions of class C 1,1 in Hilbert spaces. A function F : X → R is strongly convex if F − c · 2 is convex for some c > 0. In Proposition 10 below we will see that the gradient of a C 1,1 strongly convex function is a biLipschitz mapping. These functions arise naturally when studying smooth manifolds of positive curvature as well as in problems involving Monge-Ampére equations. See the papers [19,20,31] for some results and problems involving smooth strongly convex functions. Throughout this section X denotes a Hilbert space with norm and inner product denoted by ·, · and · respectively. Definition 9. Let E ⊆ X be arbitrary, (f, G) : E → R × X be a 1-jet and c ∈ R, M > 0 constants such that M > c. We say that (f, G) satisfies condition (SCW 1,1 ) with constants (c, M ) provided that f (x) ≥ f (y) + G(y), x − y + c M −c G(x) − G(y), y − x + cM 2(M −c) x − y 2 + 1 2(M −c) G(x) − G(y) 2 for every x, y ∈ E. Proposition 10. Assume that (f, G) : E → R × X satisfies condition (SCW 1,1 ) with constants (c, M ). Then, the following properties hold. (i) For every x, y ∈ E we have (c + M ) G(x) − G(y), x − y ≥ cM x − y 2 + G(x) − G(y) 2 . (ii) G is Lipschitz with c ≤ Lip(G) ≤ M. (iii) If c > 0, then G is strongly biLipschitz with SBilip(G) ≥ 2 c+M min{1, cM }. (iv) For c = −M we recover Wells's condition W 1,1 considered in [32,24,4]. For c = 0, (SCW 1,1 ) is just condition (CW 1,1 ) of [3,4]. For c ∈ (0, M ] we have what can be called a C 1,1 strongly convex 1-jet, which in the extreme case c = M becomes the restriction of a quadratic function to E. Proof. (i) Let x, y ∈ E. By summing the inequalities f (x) ≥ f (y) + G(y), x − y + c M −c G(x) − G(y), y − x + cM 2(M −c) x − y 2 + 1 2(M −c) G(x) − G(y) 2 f (y) ≥ f (x) + G(x), y − x + c M −c G(y) − G(x), x − y + cM 2(M −c) x − y 2 + 1 2(M −c) G(x) − G(y) 2 we obtain 0 ≥ 1 + 2c M −c G(x) − G(y), y − x + cM M −c x − y 2 + 1 M −c G(x) − G(y) 2 , which is equivalent to the desired estimation. (ii) Let x, y ∈ E be such that x = y. Writing λ = G(x) − G(y) / x − y , the inequality in (i) yields λ 2 − (c + M )λ + cM ≤ 0, which in turn implies c ≤ λ ≤ M. (iii) It follows immediately from (i) and Definition 4. We say that a 1-jet (f, G) : E → R × X satisfies condition (CW 1,1 ) with constant M > 0 on E provided that 2 for every x, y ∈ E. In [4,Theorem 2.4] it was shown that (CW 1,1 ) is a necessary and sufficient condition on (f, G) for the existence of a C 1,1 (X) convex extension F of f with ∇F = G on E. Proof. Assume first that (f, G) satisfies (SCW 1,1 ) with constants (c, M ) on E. We have f (x) ≥ f (y) + G(y), x − y + 1 2M G(x) − G(y)f (x) − f (y) − G(y), x − y − 1 2(M −c) G(x) − G(y) 2 = f (x) − f (y) − G(y), x − y − c 2 x − y 2 − 1 2(M −c) G(x) − G(y) 2 + c 2 x − y 2 + 2c G(x) − G(y), y − x = f (x) − f (y) − G(y), x − y − c M −c G(x) − G(y), y − x − cM 2(M −c) x − y 2 − 1 2(M −c) G(x) − G(y) 2 ≥ 0. Conversely, if ( f , G) = (f − c 2 · 2 , G − cI) satisfies condition (CW 1,1 ) with constant M − c, we can write f (x) − f (y) − G(y), x − y − c M −c G(x) − G(y), y − x − cM 2(M −c) x − y 2 − 1 2(M −c) G(x) − G(y) 2 = f (x) − f (y) − G(y), x − y + c 2 x − y 2 − c M −c G(x) − G(y), y − x + c 2 M −c x − y 2 − cM 2(M −c) x − y 2 − 1 2(M −c) G(x) − G(y) 2 + c 2 x − y 2 + 2c G(x) − G(y), x − y = f (x) − f (y) − G(y), x − y − 1 2(M −c) G(x) − G(y) 2 ≥ 0. Proposition 12. Let F ∈ C 1,1 (X) be such that Lip(∇F ) ≤ M and g := F − c 2 · 2 is a convex function, where c ∈ R, M > 0 are two constants (if c > 0, we call such a function F a C 1,1 (globally) strongly convex function). Then the following properties hold. Therefore, according to Definition 9, (F, ∇F ) satisfies condition (SCW 1,1 ) on X with constants (c, M ). Proof. (i) and (ii) Given x, h ∈ X, the fact that Lip(∇F ) ≤ M and the convexity of g yield 0 ≤ g(x + h) + g(x − h) − 2g(x) = F (x + h) + F (x − h) − 2F (x) − c 2 x + h 2 + x − h 2 − 2 x 2 ≤ (M − c) h 2 . This shows that c ≤ M and, again by convexity of g, that Lip(∇g) ≤ M − c. Finally observe that if M = c, then g is affine and therefore F is a quadratic function. (iii) By virtue of (ii), the convexity of g implies that (g, ∇g) satisfies condition (CW 1,1 ) with constant M − c on X. Thus Lemma 11 gives the desired inequality. Theorem 13. Let E be an arbitrary subset of a Hilbert space X, f : E → R, G : E → X be two functions, and c ∈ R, M > 0 be two constants. There exists a function F ∈ C 1,1 (X) such that F = f, ∇F = G on E, Lip(∇F ) ≤ M, and F − c 2 · 2 is a convex function if and only if the jet (f, G) satisfies condition (SCW 1,1 ) with constants (c, M ) on E. In fact, F can be defined by means of the formula F = conv(g) + c 2 · 2 , g(x) = inf y∈E f (y) + G(y), x − y + M 2 x − y 2 − c 2 x 2 , x ∈ X. Moreover, if H is another function of class C 1,1 (X) satisfying the above properties, then H ≤ F on X. Proof. The necessity of the condition (SCW 1,1 ) with constants (c, M ) on the jet (f, G) follows immediately from Proposition 12 (iii). Conversely, assume that (f, G) satisfies condition (SCW 1,1 ) with constants (c, M ) on E. By Lemma 11, the jet ( f , G) = (f − c 2 · 2 , G − cI) satisfies condition (CW 1,1 ) with constant M − c and we can apply [4,Theorem 2.4] to obtain that Lemma 11 . 11The 1-jet (f, G) satisfies (SCW 1,1 ) with constants (c, M ) on E if and only if the 1-jet ( f , G) = (f − c 2 · 2 , G − cI) satisfies condition (CW 1,1 ) with constant M − c on E. (i) We have M ≥ c and if M = c, then F is a quadratic function. (ii) g is of class C 1,1 (X) with Lip(∇g) ≤ M − c. (iii) Assume that M > c. For every x, y ∈ X we have F (x) ≥ F (y)+ ∇F (y), x−y + c M −c ∇F (x)−∇F (y), y−x + cM 2(M −c) x−y 2 + 1 2(M −c) ∇F (x)−∇F (y) 2 . f (y) + G(y), x − y + M 2 x − y 2 − c 2 x 2 is convex and of class C 1,1 (X) with Lip(∇ F ) ≤ M − c and ( F , ∇ F ) = ( f , G) on E. If we consider the function F := F + c 2 · 2 , Lemma 11 says that (F, ∇F ) satisfies condition (SCW 1,1 ) with constants (c, M ) on X (because ( F , ∇ F ) satisfies (CW 1,1 ) with constant M − c on X) and (F, ∇F ) = (f, G) on E. It is obvious that F − c 2 · 2 isconvex on X and, by Remark 10 (ii), Lip(∇F ) ≤ M. Finally, if H is a function of class C 1,1 (X) such that (H, ∇H) = (f, G) on E, Lip(∇H) ≤ M and H := H − c 2 · 2 is convex, then it is easy to see (using the same calculations as in the proof of Proposition 12 (ii)) that Lip(∇ H) ≤ M − c, and obviously ( H, ∇ H) = ( f , G) on E. We thus have from [4, Theorem 2.4] that H ≤ F on X, and therefore H ≤ F on X. A constructive proof of Kirszbraun's theorem. 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Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem. Proceedings of the. S Reich, S Simons, American Mathematical Society133S. Reich, S. Simons, Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem. Pro- ceedings of the American Mathematical Society, vol. 133 (2005), 2657-2660. T Rockafellar, Convex Analysis. Princeton, NJPrinceton Univ. PressT. Rockafellar, Convex Analysis. Princeton Univ. Press, Princeton, NJ, 1970. T Rockafellar, R J , .-B Wets, Variational Analysis. New YorkSpringerT. Rockafellar, R.J.-B. Wets, Variational Analysis. Springer, New York (1998) A Lipschitz condition preserving extension for a vector function. F A Valentine, Amer. J. Math. 671F.A. Valentine, A Lipschitz condition preserving extension for a vector function. Amer. J. Math., 67 No. 1 (1945), 83-93. Extension of Convex Function. M Yan, J. Convex Anal. 214M. Yan, Extension of Convex Function. J. Convex Anal. 21 (2014) no. 4, 965-987. Differentiable functions on Banach spaces with Lipschitz derivatives. J C Wells, J. Differential Geometry. 8J.C. Wells, Differentiable functions on Banach spaces with Lipschitz derivatives. J. Differential Geometry 8 (1973), 135-152. DISCLAIMER: The first-named author is affiliated to Universidad Complutense de Madrid, but this does not mean this institution has offered him all the support he expected; on the contrary, the Biblioteca Complutense has hampered his research by restricting his access to many books. E-mail address: [email protected] INSA de Rennes & IRMAR. Icmat (csic-Uam-Uc3-Ucm) , Departamento De Análisis Matemático Y Matemática Aplicada, CS 70839 F-35708address: [email protected] ICMAT (CSIC-UAM-UC3-UCM). Madrid, Spain; Avenue des Buttes de Coësmes; Rennes Cedex 7, France E-mail20Universidad ComplutenseFacultad Ciencias Matemáticas. Calle Nicolás Cabrera 13-15. 28049 Madrid, Spain E-mail address: [email protected] (CSIC-UAM-UC3-UCM), Departamento de Análisis Matemático y Matemática Aplicada, Facultad Ciencias Matemáticas, Universidad Complutense, 28040, Madrid, Spain. DISCLAIMER: The first-named author is affiliated to Universidad Complutense de Madrid, but this does not mean this institution has offered him all the support he expected; on the contrary, the Biblioteca Complutense has hampered his research by restricting his access to many books. E-mail address: [email protected] INSA de Rennes & IRMAR, 20, Avenue des Buttes de Coësmes, CS 70839 F-35708, Rennes Cedex 7, France E-mail address: [email protected] ICMAT (CSIC-UAM-UC3-UCM), Calle Nicolás Cabrera 13-15. 28049 Madrid, Spain E-mail address: [email protected]
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[ "Full analytical solution of finite-length armchair/zigzag nanoribbons", "Full analytical solution of finite-length armchair/zigzag nanoribbons" ]
[ "A García-Fuente ", "D Carrascal ", "G Ross ", "J Ferrer ", "\nDepartamento de Física\nNanomaterials and Nanotechnology Research Center (CINN)\nUniversidad de Oviedo\nE-33007OviedoSpain\n", "\nCSIC-Universidad de Oviedo\nEl EntregoE-33424Spain\n" ]
[ "Departamento de Física\nNanomaterials and Nanotechnology Research Center (CINN)\nUniversidad de Oviedo\nE-33007OviedoSpain", "CSIC-Universidad de Oviedo\nEl EntregoE-33424Spain" ]
[]
Finite-length armchair graphene nanoribbons can behave as one dimensional topological materials, that may show edge states in their zigzag-terminated edges, depending on their width and termination. We show here a full solution of Tight-Binding graphene rectangles of any length and width that can be seen as either finite-length armchair or zigzag ribbons. We find exact analytical expressions for both bulk and edge eigen-states and eigen-energies. We write down exact expressions for the Coulomb interactions among edge states and introduce a Hubbard-dimer model to analyse the emergence and features of different magnetic states at the edges, whose existence depends on the ribbon length. We find ample room for experimental testing of our predictions in N = 5 armchair ribbons. We compare the analytical results with ab initio simulations to benchmark the quality of the dimer model and to set its parameters. A further detailed analysis of the ab initio Hamiltonian allows us to identify those variations of the Tight-Binding parameters that affect the topological properties of the ribbons. arXiv:2303.00325v1 [cond-mat.mes-hall]
10.1103/physrevb.107.115403
[ "https://export.arxiv.org/pdf/2303.00325v1.pdf" ]
257,255,321
2303.00325
3114a2a938f235f2be6fee44ac01d5860c285a89
Full analytical solution of finite-length armchair/zigzag nanoribbons A García-Fuente D Carrascal G Ross J Ferrer Departamento de Física Nanomaterials and Nanotechnology Research Center (CINN) Universidad de Oviedo E-33007OviedoSpain CSIC-Universidad de Oviedo El EntregoE-33424Spain Full analytical solution of finite-length armchair/zigzag nanoribbons (Dated: March 2, 2023) Finite-length armchair graphene nanoribbons can behave as one dimensional topological materials, that may show edge states in their zigzag-terminated edges, depending on their width and termination. We show here a full solution of Tight-Binding graphene rectangles of any length and width that can be seen as either finite-length armchair or zigzag ribbons. We find exact analytical expressions for both bulk and edge eigen-states and eigen-energies. We write down exact expressions for the Coulomb interactions among edge states and introduce a Hubbard-dimer model to analyse the emergence and features of different magnetic states at the edges, whose existence depends on the ribbon length. We find ample room for experimental testing of our predictions in N = 5 armchair ribbons. We compare the analytical results with ab initio simulations to benchmark the quality of the dimer model and to set its parameters. A further detailed analysis of the ab initio Hamiltonian allows us to identify those variations of the Tight-Binding parameters that affect the topological properties of the ribbons. arXiv:2303.00325v1 [cond-mat.mes-hall] I. INTRODUCTION The experimental identification of graphene sheets almost two decades ago 1 lead to the development of a whole new branch of condensed matter physics, that of 2D materials. Since then, several new 2D materials, such as silicene, 2 phosphorene 3 or MoS 2 4 have been fabricated, presenting different and exotic properties. However, the interest in graphene-based structures has not diminished during the years. In particular, graphene nanoribbons (GNRs) keep attracting attention due to their characteristic electronic and magnetic properties, usually related to the presence of topologically protected edge states around their zigzag terminations. Experimentally, bottom-up techniques have enabled the fabrication of long armchair GNRs of different widths and finite length from molecular precursors with atomic precision. [5][6][7][8][9][10] The existence of edge states at the zigzag ends of some of these ribbons has been confirmed by scanning tunneling microscopy, 7 while transport measurements have demonstrated their magnetic character. 11 From the theoretical point of view, the existence of edge states localized at the zigzag edges of GNRs [12][13][14][15][16][17][18] and graphene islands of different shapes 19 was predicted long time ago. But, only after the work of Cao et al in 2017, 20 the topological nature of these edge states has been unveiled. Cao et al made use of a Z 2 topological invariant that depended on the ribbon width and termination and could be computed by determining the Zak phase from the Tight-Binding (TB) wavefunctions. 21,22 Finite-length armchair ribbons could be classified into a Z 2 = 1 topological class, where ribbons host robust edge states, and a Z 2 = 0, topologically trivial class. Furthermore, GNR-based heterostructures were proposed and found, where protected edge states emerge at the boundaries between GNRs of different topology. 20,23 This work led to a renovated interest in finite-length GNRs and the topological states at their ends, with new efforts dedicated to further characterize them both computationally 24 and experimentally. 25,26 We analyse here the emergence and features of edge states in finite-length GNRs, where we map the ribbons to a waveguide of Schrieffer-Heeger-Su (SSH) 27 transverse modes. The ribbons that we discuss here can be viewed as either armchair or zigzag depending on the width/length aspect ratio, or more generally as graphene rectangles or rectangulenes. We present a full analytical solution of a graphene TB Hamiltonian with open boundary conditions in all directions to take into account the ribbons finite width and length. We uncover the bulkboundary condition 28 by relating the ribbon Hamiltonian winding number to the quantization condition for the bulk and edge states. Our analysis goes beyond a topological classification since we are able to characterize fully the edge wave-function spatial distribution, that determines the strength of electron-electron interactions and hence the magnetic properties of the ribbons. We also show how and why topological predictions for edge states fail for short enough ribbons. The analytical solution of infinite-length ribbons with armchair and zigzag or arbitrary orientation has been known for a long time now, 29,30 where a band of edge states associated to zigzag-like terminations appears at the lateral edges of the ribbons. Akhmerov and coworkers 31 analysed the nature of edge states in finitesize graphene dots. Little effort has been done however in obtaining the analytical solution of finite-length ribbons, where a small set of edge states appears at the ribbon ends rather than along the ribbon. In addition, previous solutions usually required the definition of a one dimensional unit cell having several (more than two) basis states to generate the ribbon, while our analysis shows that two orbitals suffice just as in bulk graphene if one chooses the adequate boundary conditions. Hence the connection to bulk graphene and to the SSH model is made transparent. We also deduce a double-site Hubbard model that accounts for the magnetic states of small-width armchair GNR (AGNR), and show how different magnetic states emerge as the ribbon length increases. We find that the length windows between transitions is large enough for N = 5 AGNR to leave ample room for experimental testing. We complement our analytical TB approach with Density Functional Theory (DFT) simulations to deliver a complete theoretical characterization of the ribbons, with the possibility of getting in closer contact to current-day experiments. We are therefore able to characterize completely the TB parameters, where we discuss how needed second-and third-neighbour hopping elements affect the topology of a given ribbon. The outline of this article is as follows. Section II introduces the ribbon TB Hamiltonian, our handling of open boundary conditions and explains the full analytical solution, together with a complete analysis of the exact bulk and edge states. Section III introduces an effective Hubbard dimer model that accounts for the electron-electron interactions between edge states, whose parameters are fully determined thanks to the knowledge of the exact wave-functions. The section includes a detailed analysis of the magnetic mean-field solutions of the model, where their existence is found to depend on the ribbon length. Section IV compares the analytical results to ab initio DFT simulations of the ribbons. A close inspection and handling of the DFT Hamiltonian allows us to map it to a third-nearest neighbour TB Hamiltonian. We discuss how the extra neighbour terms affect the robustness of the edge states. Section V summarizes our main conclusions. Appendix A delivers a pedagogical description of the analytical solution of finite-length one dimensional chains. Appendix B shows our DFT results for N = 7 and 9 AGNR. II. ANALYTICAL SOLUTION OF FINITE-LENGTH GNRS A. Hamiltonian, eigen-states and eigen-functions of an infinite graphene sheet We discuss here shortly the solution of an infinite graphene sheet to introduce notation that will help us to discuss the finite-length case. We consider the primitive unit cell depicted in Fig. 1 (a), where we consider a single p z orbital per carbon atom as usual. Lattice vectors R are spanned in terms of the primitive vectors a 1 and a 2 . Distances along the X and Y axes are measured in units of the primitive vector components lengths a x 2.13Å and a y 1.23Å. This choice simplifies the algebraic expressions below, rendering our results independent of uniform distortions of the lattice from the hexagonal structure (however notice that lattice distortions affect the value of the hopping integrals, as we will discuss later). Then, the Hamiltonian of the system can be written as follows: H = − t R δ=0,a1,a2 â † RbR−δ +b † RâR+δ (1) whereâ † R (b † R ) andâ R (b R ) are the creation and annihilation operators acting on site A (B) of the unit cell defined by the lattice vector R. We are considering only nearest-neighbors hopping integrals −t ( Fig. 1 (a)) and set all on-site energies to zero. We gather the basis states centered at sites A or B of each R unit cell into a vector |R = |R, A |R, B(2) Then, any eigen-state wave-function can be written as the linear combination |Ψ = R C R |R(3) where translational symmetry dictates that the Bloch coefficients must be decomposed as C R = c A R c B R = e ik R C k = e ik R c A k c B k(4) The wave-vectors k label the Bloch eigen-states. They must be real to guarantee that the wave-function is normalizable, and are determined by imposing suitable (periodic) boundary conditions. The 2×2 Hamiltonian can be written as: H = 0 −f * (k) −f (k) 0 = − |f (k)| 0 e −iθ k e iθ k 0(5) where f (k) = t 1 + e ika1 + e ika2 = t 1 + ∆ y e ikx (6) |f (k)| = t 1 + ∆ 2 y + 2 ∆ y cos (k x ) and θ k is the polar angle of f (k). We have dumped all the k y dependence into the function ∆ y = ∆(k y ) = 2 cos (k y ) that depends only on the modulus of k y . It is now straightforward to see that the eigen-values and eigen-function coefficients can be written as follows: ε k τ = −τ |f (k)| C k τ = 1 √ 2 1 τ e iθ k(8) where τ = ± labels graphene's valence and conduction bands. B. Open boundary conditions in a finite armchair ribbon We consider now armchair nanoribbons of finite length, defined by their width N (e.g.: the number of atomic rows) and their length M (e.g.: the number of hexagons along the length of the ribbon) as shown in Fig. 1 (b). We focus on odd values of N because those are the kind of ribbons that can be obtained experimentally. However, most of our analytical results are also valid for an even value of N , and we also comment briefly those cases in the following sections. In contrast with the infinite sheet, translational symmetry is broken now because edge atoms exist that have a coordination number of two instead of three. We can however restore translational symmetry by inserting fake atoms at the edges as drawn in Fig. 1 (b), so that edge atoms recover a coordination number of three. The cost for doing so consists of inserting extra equations that ensure that the wave-function is exactly zero at the fake-atom positions. These extra equations are the finite-length boundary conditions that replace the periodic boundary conditions of the infinite sheet. We note now that the Bloch coefficients C R in Eq. (4) are non-zero for all R, so that they cannot meet the boundary condition equations. We can however take advantage of the fact that any linear combination of sameenergy bulk coefficients C k is also an eigen-state of the system with the same energy. We therefore search for those linear combinations that fulfill the boundary conditions. Graphene bulk eigen-states have large degeneracies at most energies. This is illustrated in Fig. 1 (c), where isoenergy curves within graphene's bulk Brillouin zone are drawn. The set of possible linear combinations can however be restricted by noticing that it is the edges that mix waves as we illustrate in Figs. 1 (d) and (e). Indeed, any wave with wave-vector k i that impinges on an edge must bounce back with a momentum k b whose components satisfy k b = k i and k b ⊥ = −k i ⊥ . Fig. 1 (d) shows a wave inpinging on an edge that has irregular shape, typical of a chaotic cavity. This edge gives rise to many outgoing waves, and all of them must be included in the linear combination. In contrast, Fig. 1 (e) shows equalenergy waves inside one of the ribbons that we study in this article. Then the edges' symmetries restrict the possible linear combinations to just four waves for each incident wave-vector k i . We denote the set of four waves by k σ,σ = (σk x , σ k y ), where σ, σ = ±. These considerations imply that the wave-function coefficients consist of the summation of four Bloch coefficients: C R = σ,σ =± A σ,σ e ik σ,σ R C k σ,σ(9) with boundary conditions c A (Rx,Ry=0) = c A (Rx,Ry=N +1) = 0; R x = 2, 4...2M c B (Rx,Ry=0) = c B (Rx,Ry=N +1) = 0; R x = 0, 2, ...2M − 2 c A (Rx=0,Ry) = c B (Rx=2M,Ry) = 0; R y = 2, 4...N − 1 (10) The eigen-functions of the system are characterized by a single wave-vector k that lies inside a region within the first quadrant of the Brillouin Zone region. We draw in Fig. 1 (f) several possible choices for the region. We have chosen the region enclosed by the dashed red lines (k x ∈ [0, π] , k y ∈ 0, π 2 ) because the function ∆ y ≥ 0 inside it. To proceed, we notice that the Bloch coefficients C k in Eq. (8) depend only on the modulus of k y , e.g.: C (kx,ky) = C (kx,−ky) and we denote these by C kx below. This means that we can factorize the wave-function coefficients as follows: C R = D Rx (k x , k y ) E Ry (k y )(11)D Rx (k x , k y ) = D + e ikx Rx C kx + D − e −ikx Rx C −kx (12) E Ry (k y ) = E + e iky Ry + E − e −iky Ry(13) Notice that D Rx (k x , k y ) is a vector of components d A Rx , d B Rx , while E Ry (k y ) is just a scalar. Similarly, the boundary conditions in Eq. (10) can be written in a factorized form as follows: d A Rx=0 = d B Rx=2 M = 0 (14) E Ry=0 = E Ry=N +1 = 0(15) Eqs. (11) through (15) are the first central result of this article. We can infer from them that a finite-length armchair GNR system is a separable problem in the sense that it can be decomposed into two much simpler finitelength one-dimensional models as follows. Eqs. (13) and (15) correspond to a simple N -site mono-atomic chain that lies along the Y-direction. As also shown in Appendix A, the boundary condition of Eq. (15) quantizes the k y wave-vectors as follows: sin ((N + 1)k y ) = 0 ⇒ k y = k α = π α N + 1 (16) where we have labeled the allowed wave-vectors by the integer number α, with α = 1, ..., N +1 2 . These k α wavevectors lie all inside the ribbon Brillouin zone shown in red lines in Fig. 1 (f). The quantized k α wave-vectors enter Eqs. (12) and (14) as a parameter through the function ∆(k y ), and we call ∆ α = ∆(k α ) = 2 cos (k α ) to simplify the notation below. Then, these two equations correspond to a set of dimerized chains lying along the X-axis that have 2 M cells. Each of the chains correspond to a different ∆ α . The dimerized TB chain is solved in detail in Appendix A. The boundary conditions of Eq. (14) fix the k x, α allowed values for each k y, α via the equation sin (2M k x, α + θ k ) = 0 ⇒ 2M k x, α + θ k = βπ(17) with the additional condition that the wave-vectors must lie within the ribbon Brillouin Zone, k x, α ∈ [0, π]. For each given value of α, the integer value β univocally defines the value of k x , so we label k x, α, β = k α β . We define a critical ∆ c α = 1, that corresponds to a critical wave-vector k c α = π/3. Then, the above equation has 2 M real solutions so that β = 1, ..., 2 M if (a) ∆ α > ∆ c α (k α < π/3), or (b) if ∆ α < ∆ c α (k α > π/3 ) and the chain length M < M c = ∆α 2(1−∆α) . However, the above equation has only 2 M − 1 real solutions if ∆ α < ∆ c α (k α > π/3) and the chain M > M c , so that β = 1, ..., 2 M − 1 in this case. The missing solution can be found by setting k x, α = π − i q α where q α is determined by the equation sinh 2M q α + θ q = 0 → 2M q α + θ q = 0(18) where we have introduced θ q in analogy to θ k as: e 2θq = f (π − iq, k α ) f (π + iq, k α )(19) The case of k α = π is especial. In that case ∆ α = 0, but E Ry = 0 for all even values of R y . Therefore, there is no condition over k x , instead we obtain M degenerated states of ε k = −τ t. However, we can still use condition (17) to obtain these M bulk states in the range k x ∈ [0, π 2 ]. The central panels in Fig. 2 show the resulting grid of real (k x, α, β , k y, α ) = (k α β , k α ) solutions within the ribbon Brillouin Zone for ribbons of two selected widths. In these central panels, the quantization condition (16) is represented by red horizontal lines, while blue lines represent the quantization condition (17). The curvature of the latter represents the dependence of quantized k x values in k y . The last blue curve hits k x = π at k y > k c α , where complex values of k x arise. Considering only one of the quantization conditions we recover the band structure of armchair (left) or zigzag (right) ribbons. Considering both conditions we obtain a grid of points that represent the actual (k αβ , k α ) states of the ribbon. band structure. The number of edge states of the ribbon is given by the number of allowed k α ∈ (π/3, π/2), which gives floor( N +1 6 ) for odd values of N . Each putative edge state must also fulfill the extra condition M > M c . The bulk eigen-states have wave-functions and eigenenergies given by |Ψ α β τ = 8 (N + 1) A α β R sin (k α R y ) sin (k α β R x ) τ (−1) β+1 sin (k α β (2M − R x )) | R (20) A α β = M − sin (M k α β ) sin (k α β ) (21) ε α β τ t = −τ 1 + ∆ 2 α + 2 ∆ α cos (k α β )(22) where τ = ±, and we have used the shorthand M = 4 M + 1, while the edge eigen-states wave-functions and eigen-energies are |Ψ α τ = 8 (N + 1) B α R sin (k α R y ) (−1) Rx sinh (q α R x ) τ (−1) Rx+1 sinh (q α (2M − R x )) | R (23) B α = sinh (M q α ) sinh (q α ) − M (24) ε α τ t = −τ 1 + ∆ 2 α − 2∆ α cosh (q α ) (25) = −τ sinh (q α ) sinh ((2M + 1) q α ) All these results are valid for both odd and even values of N . The only noticeable difference is that the especial case of k α = π only appears for ribbons with odd N , and that in this case the number of edge states of the ribbon is given by floor( N +4 6 ). Eqs. (16) through (25) give the full solution of the TB finite length nanoribbon and are the second central result of this article. C. Number of edge states and topology We note the well-known fact that SSH chains can be classified according to two topological categories depending on the ratio between their hopping integrals. In the correspondence between the GNR along the X direction and the dimerized chain, this ratio is just ∆ c α . SSH chains with ∆ α > ∆ c α are topologically trivial in the sense that they host only bulk states. SSH chains with ∆ α < ∆ c α are topological, they host topologically protected edge states (beyond a certain length). Appendix A shows in detail the content of the bulk/boundary principle for SSH chains. We can separate armchair ribbons into 3 groups, corresponding to N = 3p, 3p + 1 or 3p + 2. For long enough ribbons with an odd value of N , N = 3p contains p−1 2 edge states for each edge, while N = 3p + 1 contains p 2 . Within this approach N = 3p + 2 infinite ribbons are found to be metallic, as the k α = π 3 band passes through the Dirac point K. However, DFT results show that these ribbons have a small gap. 15 A modification of our TB model that has different hopping integrals −t and −t in the longitudinal and transverse directions of the ribbon (see Fig. 3) reproduces this behavior. We redefine ∆ α = 2 t t cos (k α ), so that the rest of the problem remains the same. Then, if t > t, the region of reciprocal space that represents ∆ α < 1 is reduced and the N = 3p + 2 ribbons can only present p−1 2 edge states. If t < t, the same region is increased and these ribbons can present p 2 edge states. This is shown in Fig. 3. cating Z 2 = 1+(−1) N 3 + N +1 2 2 . Z 2 = 1 (0) is equivalent to a topologically protected odd (even) number of edge states. This is consistent with our results if t < t. Cao et al also reported Z 2 values for ribbons with open edges, reporting for them Z 2 = 1−(−1) N 3 + N +1 2 2 , that corresponds to the opposite value of Z 2 from that of ribbons of the same width N and closed edges, as those analyzed here. The analytical solution of these new ribbons is very similar to that presented here, but in this case condition (14) must be satisfied for odd values of R y , and that is not immediately satisfied in k α = π as for the ribbons with closed edges. This leads to an extra couple of edge states in the limit ∆ α = 0, fully localized on the edge atoms and with ε ατ = 0. For ribbons with an even value of N , N = 3p contains Z 2 = 1−(−1) N 3 + N +1 2 2 for these ribbons. This is again consistent with our calculations if t < t, where we obtain p+2 2 edge states for N = 3p + 2 ribbons. III. HUBBARD DIMER MODEL FOR INTER-EDGE COULOMB INTERACTIONS Some of the most relevant features of graphene nanoribbons such as their magnetic, electrical or optical properties originate from the strong electron-electron interactions existing among edge states, that go beyond the single-electron picture described above. We drop in this section the bulk states and set up a model of interacting edge electrons for the case where we have a single edge-state solution q α . A. Left and Right edge states The above |Ψ α τ edge eigen-states are delocalized over both edges and both A and B sub-lattices as we show in Fig. 4. But we can define alternatively orthogonal zero-energy states that are located at either the left/B or right/A edge/sublattice (but are not eigen-states) as follows: |Ψ α (L,R) = 1 √ 2 (|Ψ α − ± |Ψ α + )(26) Alternatively, |Ψ α τ can be viewed as the bonding and antibonding states formed by the interaction between the single-edge states |Ψ α L and |Ψ α R via an effective hopping integral t α : t α t = |ε α τ | t = sinh (q α ) sinh ((2M + 1) q α ) −−−−−→ M qα 1 (1−∆ 2 α ) ∆ 2M α(27) B. Hubbard Dimer model for inter-edge Coulomb interactions We assume now that electrons in a graphene ribbon obey the Hubbard model to a good approximation. One can then show that two electrons in the same single-edge state q α have opposite spins. We find that their dynamics can be described to a good approximation by the following Hubbard dimer model H α = t α σ ĉ † Lσĉ Rσ +ĉ † Rσĉ Lσ +U α (n L↑nL↓ +n R↑nR↓ )(28)U α = Ψ αi↑ | ⊗ Ψ αi↓ |Û |Ψ αi↓ ⊗ |Ψ αi↑ (29) = 3 sinh (2Mqα) sinh (2qα) − 4 sinh (Mqα) sinh (qα) + 3M (N + 1) sinh (Mqα) sinh (qα) − M 2 (30) −−−−−→ M qα 1 3 1 − ∆ 2 α (N + 1) (1 + ∆ 2 α ) U = U 0 α(31) where U is the local interaction within one atom. We show in Fig. 6 (a) and (b) the dependence of t α /t and U α /U with the ribbon length M for a N = 5 ribbon and different possible values of ∆ α . We find that t α /t decays exponentially to zero with M . In contrast, the Hubbard U α /U parameter decreases strongly for short ribbons, but then levels off and converges to a constant value U 0 α as each edge state adquires its maximum delocalization. C. Mean field analysis at half-filling We perform a mean field treatment of the Hamiltonian, where we denote n iσ = n iσ . We also denote by m i = n i↑ − n i↓ the magnetic moment in units of µ B at either the i = L or the i = R ribbon edge. We shall restrict the analysis to the half-filled case so that n L↑ + n R↑ + n L↓ + n R↓ = 2. We find always a non-magnetic (NM) solution to the mean-field equations. In addition, an antiferromagnetic (AFM) solution exists if U α /t α > 2, that is always more stable than the NM solution whenever it exists. A ferro-magnetic (FM) solution also exists if U α /t α > 4. The FM solution is less stable than the AFM one, but more stable than the NM solution. Fig. 5 is a graphical summary of these three solutions, where we draw the one-electron eigen-states, and write down the total energies and local magnetic moments. We analyse now whether the three magnetic states can be realized in short-width ribbons that host a single edge state. Although at this point we do not know the exact values of the parameters that define the ribbon, we can make an educated guess that may shed some light on the expected behavior of the ribbons. We consider ribbons of N = 5, 7, and 9, and we estimate t = U . Then, for each ribbon we can calculate M c , M AF M and M F M as a function only of ∆ α (that, for each value of N , only needs t t to be defined). We show our results in Fig. 7, where we focus especially in the ∆ α region where t t ∈ [0.9, 1.1]. In all cases we find the 4 types of behavior, but both N = 7 and N = 9 ribbons reach M F M already for ribbons with a few unit cells. More interesting is what happens with N = 5 ribbons. In this case, M c , M AF M and M F M all become much larger, and we can expect to be able to distinguish a quite wide range of integer M values within each regime. IV. DFT SIMULATIONS OF FINITE-LENGTH GNRS The goal of this section is two-fold. We want to check in the first place whether our results and predictions above using a simple TB model agree with more realistic DFT simulation. Second, we wish to determine the U , t and t parameters of our model that reproduce the DFT simulations. We have performed DFT simulations of finite graphene nanoribbons of widths N = 5, 7, and 9 and different lengths from M = 2 to M = 10 or 30, depending on the width. We have used for this task the code SIESTA. 32,33 The choice is based on the fact that the SIESTA code expands wave-functions into a variational basis of atomiclike functions. Therefore the SIESTA Hamiltonian is already written in the TB language. Difficulties arise however because (a) SIESTA's atomic-like functions are not orthogonal to each other; (b) SIESTA's basis includes usually multiple-ζ atomic functions at each atom, that have the same angular symmetry (e.g.: two or three swave-functions, etc.); (c) atomic-like functions have a radius larger than several times the inter-atomic distance, so that hopping integrals exist to several neighbor shells. We shall explain below our procedure to handle these difficulties and achieve an accurate mapping. A. Simulation details We have chosen the generalized gradient approximation (GGA) parametrized by Perdew, Burke and Ernzerhof (PBE) 34 for the exchange and correlation potential. The code SIESTA uses the pseudopotential method as implemented by Troullier and Martins, 35 where core electrons are integrated out and valence electrons feels semi-local potentials. We have employed standard pseudopotential parameters for both carbon and hydrogen atoms. We have employed a double ζ polarized (DZP) basis set for the carbon atoms, that includes 2 pseudoatomic orbitals for each 2s and 2p atomic state, and a p-polarized (e.g.: a d) function; we have used a simpler double ζ basis set for H with 2 orbitals for its 1s states. We have used a real-space grid defined by a mesh cut-off of 250 Ry. We have also relaxed all atom positions in the nanoribbons simulated until all forces were smaller than 0.001 eV/Å. We have employed our own MATLAB scripts to post-process the SIESTA Hamiltonian. B. Tight-Binding model accuracy and parameters We have searched for NM, AFM and FM DFT selfconsistent solutions for each of the ribbons that we have simulated. We have found that all those ribbons have a NM solution while AFM and FM solutions only exist for ribbons larger than given critical lengths. These facts fully agree with the TB and Hubbard dimer model predictions. We have taken advantage of the fact that DFT is in effect a mean-field method. This means that we can use the Kohn-Sham (KS) eigen-energies to perform estimates and make comparisons with the eigen-energies of both the TB and the Hubbard dimer models, by using the equations in Fig. 5. First, we note that the eigen-energy of any bulk/edge state must lie inside the band/gap of the corresponding infinite-length ribbon. We can therefore simply look into the NM DFT solutions to establish the critical length M DF T c as the length in which in-gap states nucleate for the first time. Second, we can extract the effective hopping between DFT edge states t DF T α from the NM edge states KS eigen-energies (see the top panel in Fig. 5): t DF T α = ε N M LU M O − ε N M HOM O 2(32) Third, we can extract the Hubbard-U interaction between DFT edge states U DF T α from the AFM edge states KS eigen-energies: U DF T α = ε AF M LU M O − ε AF M HOM O(33) We can then extract the TB parameters t, t and U by fitting t α in Eq. (27) to t DF T α and U α in Eq. (30) to U DF T α . We show the results of this fitting procedure for t α and U α , for N = 5 ribbons, in the top two panels of Fig. 8. We then write down in Table I the fitted values of t, t and U . We estimate now ∆ α , M c , M AF M and M F M from these fitted parameters, and compare them with the DFT values, that are also shown in Table I. We stress that the two panels and the values of the critical lengths show that both model and DFT simulations agree truly well. The high quality of the mapping can be further tested by looking into more complex magnitudes. We have chosen here the energy differences between different magnetic solutions E N M − E AF M and E F M − E AF M , as well as the magnetic moment of the AFM solution. The bottom panels in Fig. 8 shed more weight on the quality of the mapping. We have chosen N = 5 ribbons for the present discussion because they have the highest potential for experimental testing of our predictions. The results for N = 7 and 9 ribbons is qualitatively similar and therefore relegated to Appendix B. Table I indicates a possible significant trouble for the validity of our results, since the fitted t value of about 4 to 5 eV is much larger than the universally accepted value for bulk graphene of about 2.7 eV. 36 This discrepancy has prompted us to perform a deeper analysis of the DFT Hamiltonian. C. DFT Hamiltonian downsizing We devote this section to trim the SIESTA DFT Hamiltonian gradually from the initial full-basis form H f ull down to the simple TB expression given in Eq. (1). Our first step is to reduce the basis set and leave only the 2p z carbon orbitals. This is equivalent to picking the Hamiltonian box containing only matrix elements among 2p z orbitals. We call the resulting Hamiltonian H DZ because each atom contains two p z orbitals. The drastic reduction of the Hamiltonian is justified by the fact that the lowest-lying valence and conduction bands of graphene have 2p z flavor to a very large extent. The second step consists of reducing the basis from two 2p z orbitals per carbon atom to a single one. This is accomplished by making use of the variational principle and integrating out the unwanted high-energy degrees of freedom. The single remaining p z orbital is defined by the linear combination of the 2 original p z orbitals that minimizes the energy of the HOMO and LUMO states. We denote the resulting Hamiltonian H SZ SIESTA orbitals are non-orthogonal to each other, and so are the orbitals of the single-ζ basis defined in the previous paragraph. We therefore compute the overlap matrix S SZ and orthogonalize the basis. The resulting Hamiltonian H SZ,orth is already rather similar to the Hamiltonian in Eq. (1). There remain however three differences: first, H SZ,orth has non-zero hopping integrals to first, second and third nearest neighbors, that we denote by −t 1 , t 2 and −t 3 , respectively; second, non-zero on-site energies ε 0 appear; third, both on-site energies and hopping integrals are non-uniform across the ribbon. We define t 1 and t 3 with a negative sign in front of them so that all numbers are real positive. We show in Fig. 9 the spatial distribution of on-site energies and hopping integrals for a N = 5, M = 10 ribbon to achieve further insight on their non-uniformities. The figure shows that all values of t 1 fall in the range (2.6, 2.9) eV in agreement with the accepted values of nearest neighbor hopping integrals in graphene. 36 We find that ε 0 ∼ t 2 ∼ t 3 , and that the three are one order of magnitude smaller than t 1 . This later fact has prompted us Fig. 11 (a). t1N (meV) t2N (meV) t3N (meV) t a 1N 2713 t a 2N 259 t a 3N 149 t b 1N 2753 t b 2N 252 t b 3N 154 t c 1N 2694 t c 2N 312 t c 3N 161 t d 1N 2772 t d 3N 208 t e 1N 2899 to undertake two further trimmings on the Hamiltonian. The first consists of setting all on-site energies to zero, the resulting Hamiltonian being called H 3N . A second trimming consists of picking H 3N and chopping off all t 2 and t 3 hopping integral, whereby the resulting Hamiltonian H 1N indeed conforms to Eq. (1). We assess now the impact of each the above Hamiltonian reductions for a N = 5 ribbon. We show first t α computed from the different Hamiltonians as a function of the ribbon length in Fig. 10. We find that all of them deliver estimates for t α in close agreement to the full DFT Hamiltonian. The single exception is H 1N , the one Hamiltonian that looks like Eq. (1). We then reach the conclusion that the simplest DFT-based Hamiltonian that reproduces the simulations is H 3N . D. Parameter mapping Fig. 9 shows that the hopping integrals t i are mainly affected by their proximity to the edges, so that we should assess whether those changes modify the topological protection and existence of edge states defined by the full Hamiltonian. To do so, we define a new TB Hamiltonian for infinite-length N = 5 ribbons H T B whose hopping integrals are defined graphically in Fig. 11, and are written down in Table II. The hopping integrals t 1a , t 1c and t 1e correspond to TB model t, while t 1b and t 1d correspond to t . The table displays some apparent paradoxes because t 1a , t 1c and t 1e are not equal, and furthermore they are not really larger than t 1b and t 1d , which is a requisite for the appearance of an edge state for the N = 5 ribbon within the TB model. E. Z2 invariant We have computed the Z 2 invariant using H T B , and have found that Z 2 = 1 as expected, hence confirming the presence of topologically protected edge states. We modify now each of the different hopping integrals in the model at a time to identify which of them affect most the Z 2 value. Our results, shown in Fig. 11 (b), demonstrate that changes in any t 2 , t 3 , or t 1a , t 1b do not modify Z 2 , while small variations in t 1c , t 1d or t 1e do, and kill the edge states. For a moment, let's just focus on a first neighbor Hamiltonian. In this case, our model indicates that for N = 5, inside the relevant region of the reciprocal space to find edge states k α = π 3 , and the coefficients E y (k y ) defined in equation (13) vanish in the central row of the ribbon R y = 3. This condition, fixed by the boundary conditions in the Y direction, is maintained even if we change the different values of t 1N , as far as the axial symmetry around the axis defined by the central row of C atoms is conserved. Then, any interaction with the central C-atoms of the ribbon, that is, t a 1N and t b 1N , becomes irrelevant for the properties of the edge states; and the edge states of the ribbon are exactly those of a SSHlike chain with t and t hopping integrals, formed by the 2 upper or 2 lower C chains of the ribbon structure (as it is clear from the value of f (k) = t + t e ikx ). We show in Fig. 11 (c) the value of Z 2 of the ribbon as we modify t c 1N , t d 1N and t e 1N in pairs, considering only the t 1N interactions (upper panels), or all the interactions shown in (a) (lower panels). With only the t 1N interactions, we can make a correspondence t ↔ t d 1N , t ↔ t c 1N t e 1N . Similar relations between our simplified t, t parameters and the t 1N values of the real ribbon are expected for ribbons of other widths. Then, we obtain that the transition between Z 2 = 0 and Z 2 = 1 occurs exactly at t = t , as expected in our model. Cao et al 20 indicate that a distortion at the edges leading to a stronger hopping between the edge atoms (t e 1N in our calculations) is enough to open a GAP in the band struc-ture of these ribbons and obtain Z 2 = 1 for N = 5. This agrees with our results, where the strongest value of t 1N is indeed t e 1N , and is crucial to fulfill the t < t condition. However, we go beyond this edge-distorted model, as we consider the effect of changing any of the t 1N parameters. With the values of Table II, Z 2 = 1 but ∆ y = 0.997 and M c 187, that explains why no edge states are shown in Fig. 10 for H 1N . Including t 2N and t 3N interactions changes the results, increasing the region where Z 2 = 1. Although several factors affect to this change, the most important is the inclusion t b 3N and t d 3N that modify the SSH-like chain formed by 2 C chains. If we include in our model an average t 3N = t b 3N t d 3N , f (k) in the SSH-like chain becomes: f (k) =t + t e ikx + t 3N e i2kx = =(t − t 3N ) + (t + 2t 3N cos (k x ))e ikx(34) In this case, when k x → π, θ k → 0 if t − t 3N < t, and the condition to obtain edge states becomes less restrictive, in agreement to what is shown in Fig. 11 (c). We can make a rough estimation of the equivalent ∆ α = t −t 3N t = 0.927, in good agreement with the result of ∆ α = 0.933 obtained from our fitting of t and t . Therefore, we can assume that the obtained value of ∆ y in our fitted TB model, that is the main responsible of the behavior of the edge states in the ribbon, is correct, but it is obtained at the cost of getting unrealistic values of t and t that take care of the effects of interactions between other neighbors and of the differences in the hopping integrals as we move closer to the edges. V. CONCLUSIONS We have presented a full analytical solution of the TB model of finite-length AGNRs, that we have also called rectangulenes. We have indeed shown that the above problem can be separated as the product of a one-dimensional finite-length mono-atomic chain times a one-dimensional finite-length dimerized chain. We have written down the explicit expressions for the quantum numbers, the eigen-functions and the eigen-energies. We have found that finite-length armchair ribbons witness a cascade of magnetic transitions as a function of the ribbons length. We have found ample room for experimental testing of the prediction in N = 5 AGNRs. We have also performed DFT simulations of N = 5, 7 and 9 ribbons where the above TB-based estimates are confirmed. We have then performed a mapping between the TB and the DFT Hamiltonian to check the robustness of the predictions and determine the model parameters. VI. ACKNOWLEDGMENTS The research carried out in this article was funded by project PGC2018-094783 (MCIU/AEI/FEDER, EU) and by Asturias FICYT under grant AYUD/2021/51185 with the support of FEDER funds. G. R. received a GEFES scholarship. Appendix A: Open boundary conditions in TB chains In this appendix we show the analytical solution of the TB Hamiltonian of a monoatomic chain (Fig. 12 (a)) and of a dimerized chain (Fig. 12 (b)), also known as the SSH model, 27 with open boundary conditions. The solution of the monoatomic chain is quite straightforward. We consider a chain of n sites (where we use lower case letters to avoid confusion with the definition of the graphene ribbon structure in the main text), with all on-site energies shifted to zero and first neighbors interaction of value −t. In the basis of the orbitals located on each cell l, labeled |l , any wave-function can be described from a set of coefficients C l as: |Ψ = l C l |l (A1) In particular, a Block wave-function of the system |u k can be written as: −2t cos (k), leads to a degeneracy ε k = ε −k . Therefore, we write the following trial wave-function, of energy ε k : |u k = l e ikl |l(|Ψ = A 1 |u k + A 2 |u −k (A3) to try to fulfill the open boundary conditions, consisting in: C 0 = C n+1 = 0 (A4) We obtain the following solution: C l = A sin (kl); k = απ n + 1 ; α = 1...n (A5) where A is just a normalization constant. We define the dimerized chain ( Fig. 12 (b)) as follows. Each unit cell l contains 2 orbitals a and b, so we write |l, a , |l, b to identify our basis. Those can be gathered in a single vector for each cell as: |l = |l, a |l, b(A6) All on-site energies are shifted to zero, and each orbital of type a (b) interacts only with its neighbors of type b (a) with an interaction labeled −t i or −t o depending on if it occurs within the same unit cell of between neighboring cells. In this basis, any wave-function can be written as: |Ψ = l C l |l (A7) while Bloch wave-functions |u k verify: C l = c a l c b l = e ik l C k = e ik l c a k c b k (A8) where the coefficients c a k and c b k have to be obtained from the diagonalization of a 2 × 2 effective Hamiltonian: H = 0 −t i − t o e −ik −t i − t o e ik 0 = = 0 −t i 1 + ∆e −ik −t i 1 + ∆e ik 0 = = 0 −f * (k) −f (k) 0 = − |f (k)| 0 e −iθ k e iθ k 0 (A9) where we defined ∆ = to ti , f (k) = t i + t o e ik and θ k as the polar angle of the complex number f (k). The Bloch wave-functions are then described by: c a k = 1; c b k = τ e iθ k ; (τ = ±) (A10) with energy: ε k = −τ |f (k)| = −τ t i 1 + ∆ 2 + 2∆ cos (k) (A11) We now focus on the open boundary conditions for a SSH chain of 2m atoms. Like for the monoatomic chain, ε k = ε −k and therefore we use the same linear combination of Bloch wave-funtions of equation (A3) as trial wave-functions. Two different cases can be considered ( Fig. 12 (b)). If the chain contains only complete unit cells, we call this chain a closed-cell SSH chain. If the cells at the edges contain only one atom belonging to the chain, we call this an open-cell SSH chain. It is clear that we can transform one system into the other by exchanging the labels t i and t o . Therefore, we solve explicitly the closed-cell case, and at the end we do the needed transformations to obtain the solution of the open-cell case, which is relevant in the context of graphene ribbons. The open boundary conditions at one edge define the general shape of the wave-function: This relation allows us to rewrite the coefficients c a l as: c a l = A(−1) p+1 sin (k (m + 1 − l)) (A15) Equation (A14) must be solved numerically, under the restriction that k ∈ (0, π), as both k = 0 and k = π lead to c a l = c b l = 0 for any l. All these real values of k lead to states delocalized over all the chain, that is, bulk states. However, unlike what happens for an infinite chain or for a chain with periodic boundary conditions, in the finite chain the loss of translational symmetry opens the door to the existence of states located close to the limits of the chain, that is, edge states. These states can also be described with a wave-vector k, but with an imaginary part. Our objective now is to determine wether these states exist in the chain or not. c b 0 = 0 ⇒c a l = A sin (kl − θ k ) c b l = τ A sin (kl) (A12) where A is The problem can be faced from the perspective of topology. The bulk-boundary correspondence establishes that we can define a topological invariant from the bulk wave-functions, whose value determines the existence or not of edge states at the boundaries. 28 This correspondence supposes a closed-cell structure at the edges. In the case of a one dimensional system that can be described with a 2 × 2 Hamiltonian H(k) in terms of the Pauli matrices σ x and σ y from a two dimensional vector d(k) = (d x (k), d y (k)) as: H(k) = d x (k)σ x + d y (k)σ y = d(k) σ (A16) the relevant topological invariant is the winding number ν. ν is just the number of loops that d performs around the origin when k goes through the first Brillouin zone. Topology states that if ν = 0, all k values are real and no edge states appear, while if ν = 1 there is a k with an imaginary part that leads to a couple of edge states. Notice that, besides a global sign, d is just f (k) in the XY instead of the complex plane. Therefore, we can analyze ν by analyzing the evolution of θ k as k goes from −π to π. Fig. 13 (a) shows the evolution of θ k through the first Brillouin zone, as well as the evolution of f (k) in the polar plane. It is clear that ν = 1 (ν = 0) if ∆ > 1 (∆ < 1). Our chain of m cells must contain m values of k, whether real or complex. Looking into equation (A14), if we had θ k = 0 for all values of k, g(k) would be a straight line and the valid values of k would be just those of a monoatomic chain of n = m atoms, as shown in equation (A5). As θ k is a continuous function of k, the values of k deviate from those of the monoatomic chain, but we know that each time g(k) crosses an integer value times π in the range k ∈ (0, π), a new real solution of k arises. If ∆ < 1, θ k (0) = θ k (π) = 0, the values of g(0) and g(π) do not change from those of the monoatomic chain, and therefore the existence of m real values of k is guaranteed by the continuity of g(k). If ∆ > 1, however, θ k (π) = π and g(π) decreases a π-step from the monoatomic case. Therefore, continuity of g(k) only guarantees the existence of m − 1 real values of k. This is exactly the result obtained from ν. In other words, the winding number is just a measurement of the change of θ k through the first Brillouin zone that reduces the number of bulk states that can be guaranteed by continuity. However, this is not the whole story, as continuity of g(k) only fixes a lower bound to the number of bulk states, but it can not guarantee the existence of edge states. Looking at the behavior of θ k as a function of k for ∆ > 1 (Fig. 13 (a)), θ k is a monotonous function of k that increases first slowly, but finally fast as k is close to π. Then, g(k) can become a decreasing function around k = π. In this case, an extra real value of k appears and the system has no edge states, even although ν = 1. This condition translates to: dg (k) dk k=π = m + 1 − ∆ ∆ − 1 < 0 ⇒ m < m c = 1 ∆ − 1 (A17) If the length of the chain m is below a certain threshold m c , we still have m bulk values of k. Alternatively, for a fixed value of m, if ∆ is below a critical value ∆ c = m+1 m , we also have m bulk states. If this is not the case, we must find a complex value of k. We show an example of the different possible behaviors of g(k) in Fig. 13 (b). We search for complex values of k by analytical continuation of k in the limits of its validity range, k = 0 − iq or k = π − iq. It can be demonstrated that only the second case leads to a valid solution. The Hamiltonian of equation (A9) then becomes: H = 0 −t i (1 − ∆e −q ) −t i (1 − ∆e q ) 0 = = f (q) 0 e −θq e θq 0 (A18) where f (q) is the geometric mean of the off-diagonal terms of the Hamiltonian (that is positive as ∆ > 1), and θ q = 1 2 log 1−∆e q 1−∆e −q is introduced to mimic θ k in equation (A9). We require q ∈ (−q lim , q lim ) to guarantee that θ q is real, with q lim = | log (∆)|. The solutions of the Hamiltonian are then: c a q = 1; c b q = τ e θq ; (τ = ±) (A19) with energy: ε q = τ f (q) = τ t i 1 + ∆ 2 − 2∆ cosh (q) (A20) Once again, we have to apply the open boundary conditions, with the first one defining the general shape of the wave-function: Condition (A23) is always satisfied for q = 0, but this leads to the invalid, real solution k = π. Other possible values of q must be obtained numerically, but we can determine if these solutions exist by analyzing the behavior of the functiong(q) (see Fig. 13 (c)). For ∆ < 1 we only findg(0) = 0. For ∆ > 1 this function is continuous inside the defined range of q, odd, andg (q → ±q lim ) = ∓∞. Then, there are other two solutions ofg(q) = 0, of value ±q, if: dg (q) dq q=0 = m + 1 − ∆ ∆ − 1 > 0 ⇒ m > m c (∆ > ∆ c ) (A25) Notice that solutions of value of ±q lead to the same coefficients of the wave-function in equations (A21) and (A24), up to a sign. Therefore, it is enough to consider the solution with q > 0. Results of equations (A17) and (A25) are consistent. For a given chain defined by ∆ and m, if ∆ < 1, or ∆ > 1 but m < m c (equivalent to ∆ < ∆ c ), the chain presents m real values of k leading to 2m bulk solutions. If ∆ > 1 and m > m c (equivalent to ∆ > ∆ c ), the chain contains m − 1 real values of k to define 2m − 2 bulk states, but also a complex value of k = π − iq, leading to 2 localized edge states. The value of q indicates the level of localization of the edge states, as q −1 is a measurement of the penetration depth of the state in units of d. The exact value of q for a given value of ∆ and m must be obtained numerically solving equation (A23), or any of the following, equivalent equations: tanh (qm) = sinh (q) ∆ − cosh (q) (A26) ∆ sinh (qm) = sinh (q (m + 1)) (A27) We can obtain an approximated value of q if it is close to q lim with the following expression: q = q lim − ∆ 2 − 1 1 + ∆ 2m+2 − 2m (∆ 2 − 1) (A28) Alternatively, we propose the following iterative solution that, starting at q 0 =q lim , converges quickly to the exact value of q: C i = tanh (mq i−1 ) (A29) q i = log   ∆C i 1 + C i + ∆C i 1 + C i 2 + 1 − C i 1 + C i   (A30) Fig. 14 shows the evolution of q/q lim with m for several values of ∆. Notice that in all cases q evolves asymptotically to q lim , reaching q lim faster the larger the value of ∆. The value of q lim decreases as ∆ decreases, leading to more delocalized edge states for ∆ closer to one. Edge states given by equations (A21) and (A24), that we can label |Ψ e τ , are non-zero eigen-states distributed over both edges and both sublattices. We can define zero-energy states, that are not eigen-states, but that are located only over the left (|Ψ e L ) or right (|Ψ e R ) edge, by: |Ψ e L = 1 √ 2 |Ψ e + + |Ψ e − |Ψ e R = 1 √ 2 |Ψ e − − |Ψ e +(A31) These states are not only localized over different edges, but also over different sublattices of the chain. We can then see the eigen-states |Ψ e τ as the result of the interaction of two zero-energy states, located at different edges, interacting via an effective hopping integral of value f (q). Finally, we look at the open-cell case. We can solve again the SSH chain, now with the following open boundary conditions: c a 0 = c b m = 0 (A32) However, we can also obtain this new solution making the following transformations to the closed-cell solution. First, we exchange the role of t i and t o . This changes the role of ∆ to ∆ −1 . This leads, for example, to the following changes in f (k) and θ k f (k) = t o + t i e ik = t o 1 + ∆ −1 e ik = |f (k)|e iθ k (A33) FIG. 1 . 1(a) Representation of an infinite graphene sheet, where the A− and B−sublattice carbon atoms are shown as light and dark red spheres, respectively. The unit cell is defined by the lattice vectors a1 and a2. Hopping integrals between first-neighbour pz orbitals are shown in green. ax and ay are lengths used to scale all distances along the X and Y axes, respectively. (b) Scheme of a finite graphene ribbon characterized by a width N and a length M . Blue spheres represent f ake atoms that define the boundary conditions. A dotted grid and values of Rx and Ry are depicted to identify the lattice cells and XY-components of their corresponding lattice vectors. (c) Iso-energy curves of an infinite graphene sheet in the reciprocal space. Different colors are used to identify different energies. (d) An incident wave with wave-vector k i (green) impinging in a rough edge bounces back in several directions indicated by the orange wave-vectors k b , which leads to a chaotic cavity. (e) Incident waves that bounce back in the straight edges of a finite ribbon only change the sign of one of their wave-vector components at a time. (f) Representation of the reciprocal space of graphene. The boundaries of the First Brillouin zone are drawn with blue dashed lines. The area marked in shaded blue represents the region of the First Brillouin zone where all the states of the finite ribbon can be folded. Green and red dashed lines show other equivalent regions of the reciprocal space. We use the red dashed region in this article because ∆y ≥ 0 inside it. Fig. 2 (a)and(b) show how the number of edge states (represented by green dots) of a ribbon of a given width depends on its length. These edge states are part of the zigzag band structure, but fall inside the energy gap of the armchairFIG. 2. Grid of quantized solutions (k x, α β , ky, α) of a graphene rectangulene with (a) N = 9, M = 2; (b) N = 9, M = 8; (c) N = 45, M = 12 ribbon. Each central panel shows the grid where black and green dots represent the real k x, α β (bulk states) and complex π − i qα (edge states) solutions. Red and blue lines are an eye guide to visualized correlations. The quantized eigen-energies can be visualized from the band structure of an infinite armchair (left panel) or zigzag (right panel) ribbon. FIG. 3 . 3(a) Sketch of the modified TB approach with 2 different hopping integrals t and t . (b) Having t = t impacts on the size of the allowed region of the ribbon BZ where edge states appear, because changing t shifts the Dirac point up and down. Cao et al reported the values of the Z 2 topological invariant of these infinite ribbons with closed edges, 20 indi- p 2 2edge states, N = 3p + 1 contains p+1 2 , while N = 3p + 2 are again metallic if t = t . Cao et al 20 also predicted whereĉ † iσ andĉ iσ are the spin-σ creation and annihilation operators acting on the edge states at the i = L, R edge, andn iσ are their corresponding number operators. The Hubbard-U parameter is given byFIG. 4. Spatial representation of the |Ψα + , |Ψα − , |Ψα L and |Ψα R edge states, for a N = 7, M = 8 ribbon. Black crosses indicate the carbon atom positions. Red (blue) circles at each of those sites indicate positive (negative) values of the wave-function coefficients, where the circle radii are proportional to the magnitude of the coefficient. FIG. 5 . 5Graphical summary of the different mean field magnetic solutions of the Hubbard dimer model at half filling of the states. Orange and green lines in the central panel show the one-electron eigen-energies for each solution, where occupied states are called HOMO states and are indicated by a dot, while empty states are called LUMO and do not have a dot. The left and right panels show the wave-function shapes for a N = 7, M = 5 ribbon. Mean field total energies and magnetic moments in units of µB are written at the far right side of the figure. FIG. 6 . 6(a) tα/t, (b) Uα/U and (c) Uα/tα (in units of t/U ) as a function of the length M for an AGNR with N = 5. The different curves show the results for different values of of ∆α. The inset in figure (a) shows tα/t in logarithmic scale to highlight its exponential decay. Figure 6 ( 6c) shows that the ratio U α /t α increases monotonically as the ribbon length M grows. We then define M AF M and M F M as the critical edge state lengths for which U α /t α = 2 and U α /t α = 4, respectively. Theexpected behavior of a given edge state p as a function of the ribbon length M can be summarized as follows, where we assume that M c < M AF M . For very short ribbons M < M c , no edge state exists. Once M > M c , a NM edge state emerges. If M grows beyond M AF M , the edge state becomes AFM. And if M > M F M , both AFM and FM solutions can be found for the edge state, with the AFM solution being more stable in all cases. FIG. 7 . 7Mc, MAF M and MF M as a function of ∆α and M for ribbons with N = 5, 7, and 9. Regions with no edge states (NES), and non-magnetic (NM), antiferromagnetic (AFM) and ferromagnetic (FM) solutions are marked by different colors. The scale in each of the three figures is different to help focusing on the different relevant ranges of ∆α and M for each case. Dashed black lines indicate different reference values of t . FIG. 8 . 8(a) Effective hopping tα, (b) effective Hubbard-U Uα, (c) total energy difference between the NM and AFM solutions ENM − EAF M , (d) total energy difference between the FM and AFM solutions EF M − EAF M and (e) magnetization m as a function of the ribbon length M for N = 5 AGNR. The red dots correspond to our DFT simulations, while the blue dashed lines are the results of our TB model where we have fitted them in the upper two panels to obtain the optimal values of U and t. The results in the lower three figures are parameter-free. FIG. 9 . 9Spatial representation of the different on-site energies ε0 (referred to their average value) and first-, second-and thid-neighbor hopping integrals, −t1, t2 and −t3 respectively, for a N = 5, M = 10 AGNR. Black crosses indicate the carbon atom positions. FIG. 10 . 10tα as a function of M for a N = 5 AGNR. Different curves refer to different versions of the DFT Hamiltonian: H f ull , H DZ , H SZ , H 3N and H 1N . FIG. 11 . 11(a) Graphical definition of the different hopping integrals t1i, t2i and t3i in infinite-length N = 5 ribbons. The color palette is consistent with the color scale shown in Fig. 9. (b) Values of Z2 obtained from the bulk Hamiltonian defined in (a) and Table II, when each of the hopping integrals is individually changed by a percentage of ± 50 %. (c) Values of Z2 when t c 1N , t d 1N and t e 1N are changed by pairs, considering only t1N interactions (upper panels) or t1N , t2N and t3N interactions (lower panels). The dashed lines indicate the values t d 1N = t c 1N t e 1N (t = t) The crosses indicate the reference values of each couple of t1N . A2) where k is measured in units of the inverse of the lattice constant, d −1 . The expression of the energy for |u k , ε k = FIG. 12. Structure of (a) a monoatomic chain of n atoms and (b) a dimerized chain (SSH) of 2m atoms, with open boundary conditions. Unit cells of lattice constant d are separated by dotted lines and labeled in black, with a central, l cell shaded in blue. Red spheres represent atoms belonging to the chain, with a light and dark red used in the dimerized chain to differentiate between the two types of site, labeled a and b. Blue spheres represent f ake atoms used to define the open boundary conditions. Different hopping integrals are shown in different shades of green. The closed-cell and open-cell structures at the edges of the SSH chain are also represented. a normalization constant. The conditions at the other edge determine the possible values of k: c a m+1 = 0 ⇒ sin (k (m + 1) − θ k ) = 0; (A13) g(k) := k(m + 1) − θ k = βπ; β = 1...m (A14) FIG. 13 . 13(a) Representation of the evolution of θ k and f (k) as k evolves through the first Brillouin zone for SSH chains with different values of ∆. Dotted lines are used in the range k ∈ (−π, 0), while continuous lines are used in the range k ∈ (0, π), relevant for equation (A14). (b) Representation of g(k) for a SSH chain of m = 6, with different representative values of ∆. Different symbols are used to identify the different real solutions of k in equation (A14). (c) Representation of g(k) for the same chain and ∆ values of (b), with blue circles showing the imaginary part of the complex solutions for ∆ > ∆c. a l = A(−1) l sinh ql − θ q c b l = τ A(−1) l sinh (ql) (A21)where A is a normalization constant. The conditions at the other edge determines the possible values of q:c a m+1 = 0 ⇒ sinh q (m + 1) − θ q = 0; (A22) g(q) := q(m + 1) − θ q = 0 (A23)This relation allows us to rewrite the coefficients c a l as: c a l = A(−1) l+1 sinh (q (m + 1 − l)) (A24) FIG. 14 . 14Values of q normalized by q lim = | log (∆)| as a function of the SSH chain length m for different values of ∆ (red lines). Red dots indicate the solutions for integer values of m. The approximated solutions for q → q lim given by equation (A28) are shown with blue dotted lines. TABLE I . IHubbard dimer model parameters t, t and U , obtained by fitting the TB estimates for tα and Uα to the corresponding DFT results. TheTable also includes themodel ∆α, Mc, MAF M and MF M , and their DFT counterparts (M DF Tc , M DF T AF M and M DF T F M ). N t (meV) t (meV) U (meV) ∆α Mc MAF M MF M M DF T c M DF T AF M M DF T F M 5 4027.4 3758.7 5348.1 0.933 6.99 11.46 17.14 9 12 16 7 3881.7 3422.8 3872.6 0.675 1.04 2.51 3.46 2 3 3 9 5322.7 4089.3 3480.7 0.452 0.45 1.68 2.15 2 2 2 TABLE II . IIFirst, second and third nearest neighboring hopping integrals in a infinite-length N = 5 AGNR, as defined in K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grig-orieva, and A. A. 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[]
[ "The collapse of atomically-cooled primordial haloes. I. High Lyman-Werner backgrounds", "The collapse of atomically-cooled primordial haloes. I. High Lyman-Werner backgrounds" ]
[ "Samuel J Patrick \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n", "Daniel J Whalen \nInstitute of Cosmology and Gravitation\nUniversity of Portsmouth\nPO1 3FXPortsmouthUK\n", "Muhammed A Latif \nPhysics Department\nCollege of Science\nEmirates University\nPO Box 15551Al-AinUnited Arab, UAE\n", "Jacob S Elford \nSchool of Physics and Astronomy\nCardiff University\nThe ParadeCF24 3AACardiffQueen's BuildingsUK\n" ]
[ "Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK", "Institute of Cosmology and Gravitation\nUniversity of Portsmouth\nPO1 3FXPortsmouthUK", "Physics Department\nCollege of Science\nEmirates University\nPO Box 15551Al-AinUnited Arab, UAE", "School of Physics and Astronomy\nCardiff University\nThe ParadeCF24 3AACardiffQueen's BuildingsUK" ]
[ "MNRAS" ]
Pristine, atomically-cooled haloes may be the sites of primordial quasar formation because atomic cooling triggers rapid baryon collapse that can create 10 4 -10 5 M black hole seeds. However, no numerical simulation has ever followed the collapse of these haloes for the times required to form supermassive stars and direct-collapse black holes (DCBHs). We have now modeled baryon collapse in atomically-cooled haloes with a wide range of spin parameters and assembly histories for times that are sufficient for DCBH formation. Fragmentation of accretion disks after ∼ 500 kyr is nearly ubiquitous in these haloes and in most cases leads to the formation of binary or multiple supermassive stellar systems. They also confirm that rapid baryon collapse proceeds for the times required for these stars to form DCBHs. Our simulations suggest that binary or even multiple DCBH formation was the rule rather than the exception in the primordial Universe.
10.1093/mnras/stad1179
[ "https://export.arxiv.org/pdf/2012.11612v3.pdf" ]
229,348,945
2012.11612
b664edd32d536efd78de19210844246a118c4f98
The collapse of atomically-cooled primordial haloes. I. High Lyman-Werner backgrounds 2020 Samuel J Patrick Institute for Astronomy University of Edinburgh Royal Observatory Blackford HillEH9 3HJEdinburghUK Daniel J Whalen Institute of Cosmology and Gravitation University of Portsmouth PO1 3FXPortsmouthUK Muhammed A Latif Physics Department College of Science Emirates University PO Box 15551Al-AinUnited Arab, UAE Jacob S Elford School of Physics and Astronomy Cardiff University The ParadeCF24 3AACardiffQueen's BuildingsUK The collapse of atomically-cooled primordial haloes. I. High Lyman-Werner backgrounds MNRAS 0002020Accepted XXX. Received YYY; in original form ZZZPreprint 17 April 2023 Compiled using MNRAS L A T E X style file v3.0quasars: general -black hole physics -early universe -dark agesreionizationfirst stars -galaxies: formation -galaxies: high-redshift Pristine, atomically-cooled haloes may be the sites of primordial quasar formation because atomic cooling triggers rapid baryon collapse that can create 10 4 -10 5 M black hole seeds. However, no numerical simulation has ever followed the collapse of these haloes for the times required to form supermassive stars and direct-collapse black holes (DCBHs). We have now modeled baryon collapse in atomically-cooled haloes with a wide range of spin parameters and assembly histories for times that are sufficient for DCBH formation. Fragmentation of accretion disks after ∼ 500 kyr is nearly ubiquitous in these haloes and in most cases leads to the formation of binary or multiple supermassive stellar systems. They also confirm that rapid baryon collapse proceeds for the times required for these stars to form DCBHs. Our simulations suggest that binary or even multiple DCBH formation was the rule rather than the exception in the primordial Universe. INTRODUCTION Hot, atomically-cooled primordial haloes at z ∼ 15 -20 may have been the birthplaces of the earliest quasars in the universe, more than 200 of which have now been discovered at z > 6 (e.g., Fan et al. 2003), including nine at z > 7 ( Mortlock et al. 2011;Bañados et al. 2018;Matsuoka et al. 2019;Yang et al. 2020;Wang et al. 2021). In this picture, primordial haloes grow to masses of 10 7 -10 8 M and reach virial temperatures of ∼ 10 4 K without ever having formed a primordial (Pop III) star, either by being immersed in strong Lyman-Werner (LW) UV backgrounds that destroy all their H2 (e.g., Agarwal et al. 2012;Yue et al. 2014;Dijkstra et al. 2014;Johnson et al. 2014;Schauer et al. 2015Schauer et al. , 2017a or in highly supersonic baryon streaming motions that delay the collapse of the halo even if H2 is present (Tseliakhovich & Hirata 2010;Greif et al. 2011;Stacy et al. 2011;Schauer et al. 2017b;Hirano et al. 2017). Virial temperatures of 10 4 K trigger atomic cooling that leads to catastrophic baryon collapse at infall rates of ∼ 0.1 -1 M yr −1 . Stellar evolution models predict that such flows, if they persist, would E-mail: [email protected] build up cool, red supermassive stars (SMSs) before dying as 100,000 -300,000 M direct-collapse black holes (DCBHs; e.g., Hosokawa et al. 2013;Umeda et al. 2016;Woods et al. 2017;Haemmerlé et al. 2018a;Woods et al. 2021b;Herrington et al. 2023). However, it has now been shown that the rare, turbulent haloes that have been shown to form quasars by z 6 in large-scale cosmological simulations (Di Matteo et al. 2012Lupi et al. 2019;Valentini et al. 2021;Lupi et al. 2021) produced DCBHs without UV backgrounds, supersonic baryon streaming motions, or even atomic cooling (Latif et al. 2022). DCBHs are currently the leading candidates for the seeds of the first quasars because ordinary Pop III star BHs are only a few tens to hundreds of solar masses at birth (e.g., Hirano et al. 2014Hirano et al. , 2015 and form in low densities that preclude rapid initial growth (Whalen et al. 2004;Kitayama et al. 2004;Alvarez et al. 2009;Whalen & Fryer 2012;Smith et al. 2018). In contrast, DCBHs can grow much more rapidly because they are born in dense environments in massive host haloes capable of retaining their fuel supply, even when it is heated by X-rays (Johnson et al. 2013). Runaway stellar collisions in dense, marginally-enriched clusters can create BHs of up to a few thousand solar masses (Devec-chi & Volonteri 2009;Latif et al. 2016;Sakurai et al. 2017;Reinoso et al. 2018;) but even these objects may not be massive enough to become quasars by z > 6 ( Smidt et al. 2018;Zhu et al. 2020 -see also Woods et al. 2019;Maio et al. 2019). Although numerical simulations of rapid baryon collapse in primordial atomically-cooling haloes have steadily improved over the past decade, they remain a trade-off between resolution and evolution time. The first simulations either resolved sub-AU scales that could capture the formation of the protostar but not the accretion disk around it on parsec (pc) scales (Wise et al. 2008) or larger scales that could follow the evolution of the disk for a few dynamical times but not the SMS at its center (Regan & Haehnelt 2009a,b). These studies found large central infall rates at the onset of collapse but could not determine how long they lasted. Later work at high resolution and somewhat longer evolution times found that these infall rates continued down to scales approaching those of the star but did not run for nearly enough times to follow its evolution (Latif et al. 2013a;Regan et al. 2014). The implementation of pressure floors (Machacek et al. 2001) and sink particles at high resolution confirmed that accretion rates could remain large at the smallest scales and persist for a few tens of thousands of years (Latif et al. 2013c;Shlosman et al. 2016;Chon et al. 2016). Chon et al. (2018) and Regan & Downes (2018b) used sink particles with radiative feedback from the star and a simple prescription for its evolution to follow collapse for 100 kyr and 250 kyr, respectively. They found that radiation from the star was unable to suppress accretion, as did Ardaneh et al. (2018) and Luo et al. (2018), and reported some small-scale fragmentation in the disk at later times (see also Becerra et al. 2015Becerra et al. , 2018. Chon et al. (2018) found that almost half of the fragments in one of their haloes form in binaries that they suspected could later become SMSs, corroborating idealised simulations by Bromm & Loeb (2003). However, neither study evolved the disks for long enough times to determine if any of the fragments became stars or were simply subsumed into the central object later on. Suazo et al. (2019) examined the collapse of atomically cooled haloes at intermediate resolutions in high LW backgrounds for ∼ 600 kyr, longer than previous studies but still well short of the formation of a DCBH. The large inflow rates lasting for millions of years required to form SMSs and DCBHs have only recently been confirmed to occur in numerical simulations Regan et al. 2020). These models sacrificed the extreme resolution of earlier studies in order to capture the evolution of the gas at the center of the halo over many dynamical times. found that the disk can fragment into binary and multiple SMS systems, but only in a few haloes with high spin parameters that were special cases. Here, we follow the collapse of an ensemble of atomically-cooling haloes over the full range of spin parameters expected for these objects and for a variety of merger histories. We evolve them for up to 3 Myr, enough time for SMS formation and collapse at their centers. For simplicity, we approximate high LW backgrounds by deactivating H2 chemistry in our runs and defer intermediate LW backgrounds with H2 to later studies. This should be considered to be a strong but useful upper limit to actual LW backgrounds in the primordial Universe because H2 could self-shield in the cores of massive haloes and survive even intense UV fluxes from nearby Pop III stars (Shang et al. 2010;Latif et al. 2014b. Even small amounts of H2 could cool gas in the core and result in much less massive stars. In Section 2 we describe our numerical simulations and discuss halo evolution, disk dynamics and central accretion rates in Section 3. We conclude in Section 4. NUMERICAL METHOD We first performed a series of low-resolution dark-matter (DM) only runs to identify haloes with a range of assembly histories and spin parameters. These haloes are then resimulated with full baryonic physics at much higher resolution to follow their collapse and the dynamics of the disks at their centers. Enzo We use the Enzo adaptive mesh refinement (AMR) cosmology code to model the haloes in our study (Bryan et al. 2014). Enzo has an N −body adaptive particle-mesh scheme for evolving DM (Efstathiou et al. 1985;Couchman 1991;Bryan & Norman 1997) that is self-consistently coupled to hydrodynamics and nonequilibrium primordial gas chemistry (Anninos et al. 1997;Glover & Abel 2008). We employ the piecewise parabolic method for gas dynamics (PPM; Woodward & Colella 1984;Bryan et al. 1995) and apply the HLLC scheme for enhanced stability with strong shocks and rarefaction waves (Toro et al. 1994). Our simulations include six species (H, He, e − , H + , He + , He 2+ ) with no H2 in order to ensure isothermal cooling and collapse. Our chemistry model includes collisional excitational and ionizational cooling by H and He, recombination cooling, bremsstrahlung cooling, and inverse Compton cooling by the cosmic microwave background (CMB). Simulation Setup Haloes of interest are identified from a series of unigrid DMonly simulations in 1.5 h −1 Mpc cosmological boxes with a resolution of 256 3 and initial conditions generated at z = 200 with MUSIC (Hahn & Abel 2011). We use the second-year Planck cosmological parameters: ΩM = 0.308, ΩΛ = 0.691, Ω b h 2 = 0.0223, σ8 = 0.816, h = 0.677 and n = 0.968 (Planck Collaboration et al. 2016). The Rockstar halo finder (Behroozi et al. 2013) is used to find haloes with masses of a few times 10 7 M in these boxes. We restart our simulations at z = 200 with a top grid resolution of 256 3 and three additional nested grids centered on the halo (each with resolution of 256 3 ) which yield effective DM and baryon mass resolutions of 28 M and 34 M , respectively. Up to 15 levels of refinement are allowed for a maximum physical resolution of 0.014 pc. A grid is flagged for refinement when baryon or DM overdensities exceed 8.0 times the mean density (δρ/ρ > 8.0). We resolve the Jeans length by 32 cells to capture the formation of turbulent structures and prevent artificial fragmentation, and we activate refinement on Jeans length at z = 30 (Truelove et al. 1997;Latif et al. 2013a). DM particles are smoothened at the tenth refinement level, which corresponds to a comoving resolution of 5.72 pc, to avoid spurious effects caused by the discrete sampling of the DM potential. Below pc scales, collapse is dominated by baryons, and if there is an insufficient number of DM particles at higher resolutions they can unphysically accelerate and heat gas in their vicinity. Smoothing mitigates such effects. We also turn on a pressure floor at the maximum level of refinement in our simulations that prevents collapse on scales below our maximum resolution by ensuring that the cells on that level are Jeans stable. The pressure floor in our runs smoothens the gravitational potential over 4∆x, where ∆x is the length of a cell at the highest level of refinement (Machacek et al. 2001), so clumps can be smeared out on those scales. Pressure floors never reverse collapse in our models, just prevent it at the highest level of refinement, and they never result in local expansion of the gas that drives shocks, especially at temperatures of 8000 K. These points are corroborated by previous simulations of collapse in atomically-cooled haloes with pressure floors and with sink particles without floors that produced essentially identical results (Latif et al. 2013c(Latif et al. , 2022. We output the data every 10 kyr during the run and follow the gravitational collapse for about 3 Myr. Halo Selection Criteria We initially selected 20 haloes from our DM-only runs that reached masses of at least 10 7 M by z ∼ 14 -20. From this set we chose only eight haloes for resimulation because of their merger histories and spin parameters at the onset of atomic cooling. They primarily grew through accretion, major mergers or some combination of the two, where a major merger is defined to be the collision of two haloes with a mass ratio of 1/5 or more. They were also chosen so that they spanned the likely range of spin parameters, λ, for atomically-cooled haloes, where λ is (Peebles 1969;Bullock et al. 2001 ) λ = |L| |E| GM 5 2 = |L| √ 2GRM 3 .(1) Here, L, R, and M are the angular momentum, virial radius and virial mass of the halo, respectively. Values for λ for all eight haloes are listed in Table 1. The remainder of the 20 haloes were discarded for having merger histories similar to those in our final set. We save outputs at redshift intervals ∆z = 0.5 from z = 25 -15, use Rockstar to locate positions, masses and velocities of the haloes in each redshift bin, and then port them to ytree (Smith & Lang 2019) to build merger trees. Assembly histories for these haloes, designated 1, 2, 8, 10, 12, 16, 19 and 20, are shown in Figure 1. The trees extend from z = 25 at the bottom to z = 15 at the top in increments ∆z = 0.5. Each circle represents an ancestor of the final halo and has a radius proportional to its logarithmically-scaled mass. This proportionality varies between haloes in Figure 1 so the same circle radius in different trees does not imply the same mass. The main progenitor line is marked in red. Halo 12 is an example of a halo that has grown primarily by accretion while halo 16 has three major mergers. The others lie somewhere between these two extremes. Most of the haloes have lines of low-mass haloes in their ancestry, visible as the long vertical black tails with small black dots along them. Halo 1 grows slowly at early times but then suddenly rises in mass at z = 17.5 because of a surge in accretion and a merger. Halo 2 has the highest accretion rate of the sample but does not begin to cool before reaching a mass of 8.47 × 10 7 M at z = 14.5, likely because of turbulence driven by rapid infall. Halo 8 begins to cool at the earliest times at a mass of 1.15 × 10 7 M at z = 20.4. Halo 10 has a major merger early in its history and has the highest spin parameter at the time it begins to cool at z = 17.3 at a mass of 2.6 × 10 7 M . Halo 12 has the next highest spin when it begins to cool at z = 16.8 at a mass of 1.93 × 10 7 M , which is interesting because it is almost entirely the product of accretion, not mergers that could have spun it up by tidal torquing. Halo 16 undergoes three major mergers at z = 18.5, 17.5, and 16.5 and minor mergers at z=18.5 and 15. It has the lowest accretion rates in our sample. Halo 19 begins to cool at the lowest redshift, z = 13.9 at a mass of 2.12 × 10 7 M , and has the lowest spin at the onset of cooling. Like halo 12, which has the second highest spin parameter, it too is mostly the result of accretion, not mergers. Halo 20, which has the second lowest spin parameter, is also mostly built up by accretion but has a number of minor mergers in its history. We summarise these results in Table 1. In sum, haloes 1, 10 and 16 have 10 -2 10 -1 10 0 10 1 10 2 10 3 radius (pc) undergone major mergers during their formation, haloes 2, 12 and 19 have grown mainly via accretion, and haloes 8 and 20 are products of both minor mergers and accretion. Redshifts at the onset of atomic cooling are determined in our runs with both gas and DM for accuracy. RESULTS Rapid baryon collapse triggered by atomic cooling leads to the formation of large, rotationally-supported disks at the centers of all eight haloes. We now discuss the evolution of these disks, their accretion rates, and their stability over time. Initial Baryon Collapse We show spherically-averaged density and temperature profiles for our haloes at the onset of cooling and collapse in Figure 2. The densities all closely approximate the Larson-Penston solution, ρ ∝ r −2.2 , which slightly deviates from the classic r −2 density profile expected for self-gravitating isothermal spheres because of the presence of DM in the halo. The virial shock of the halo is visible as the bump in density and temperature at 200 pc -800 pc in the profiles. Densities range from 10 6 -10 7 cm −3 in the core to 0.1 -1 cm −3 at the virial shock. They level off at small radii because of the absence of torques within the Jeans sphere (not because of pressure floors, which are imposed just after this stage of collapse). The large bump in density at ∼ 200 pc in halo 2 is due to another halo that is about to merge with it. The temperature profiles show that collapse due to atomic cooling is nearly isothermal, with flat ∼ 8000 K temperatures extending out to ∼ 100 pc. Gas falling onto the halo is first heated to 10,000 K as it passes through the virial shock and then cools to ∼ 8000 K as it settles deeper into the gravitational potential well of the halo. Because of our resolution limits, gas never reaches densities where it becomes optically thick to Lyα photons (e.g., Smith et al. 2017;Smith & Bromm 2019). At higher levels of refinement these photons could become partly trapped by frequent resonant scatterings, reducing cooling efficiencies and increasing temperatures deep in the core. We also note that at number densities n 10 8 cm −3 H − cooling becomes important and would reduce gas temperatures to ∼ 5000 K before the gas becomes optically thick to Lyα, further increasing densities and potentially trapping more cooling photons at later times. Disk Formation / Evolution Rapid baryon collapse leads first to the formation of a dense core and then a rotationally-supported, self-gravitating disk with an initial radius of ∼ 0.2 pc by ∼ 200 kyr. A bar instability then appears and creates two spiral arms. The disk thereafter can exhibit a range of morphologies, as we discuss below, but continues to grow in mass and radius and generally becomes more turbulent over time. This turbulence increases the effective alpha parameter of the disk and it grows to over 1 pc in radius in all eight haloes. Density projections of the disks in haloes 10, 12 and 1 are shown in Figure 3. They represent the range of disk evolution in our runs: relatively quiescent ones, as in halo 10; more turbulent disks that undergo multiple episodes of fragmentation, as in halo 12; and the prompt formation of interacting binary disks at early times, as in halo 1. In the initial stages of disk formation the collapse of low-angular momentum gas creates the small, dense cores like the one at 51 kyr in halo 12. They are soon followed by the appearance of barred spiral arms as gas at higher angular momentum rains down onto the disk, as shown at 226 kyr in halo 10. Since radiation from the star cannot prevent gas from con- tinuing to fall onto the disk at high rates, it grows rapidly in radius, in some cases to ∼ 3 pc by the end of the runs. Neither the bars nor viscous processes can transport angular momentum out from the center of the disk as fast as gas rains down on it so its spiral arms cannot maintain a single angular velocity and they begin to fracture into clumps, generally by 500 -700 kyr but as early as ∼ 250 kyr in halo 1. Disk fragmentation thus proceeds here via the spiral arm instability (SAI) mechanism discussed in Inoue & Yoshida (2020) in which spiral arms form first and then become locally unstable, overcoming the stabilizing effects such as gas pressure and Coriolis force. Most fragments spiral into the center of the disk but some persist for times that are sufficient to form satellite disks, as we discuss below. Tidal torquing between fragments and the disk often cause the center of the disk to eject tidal tails when the clumps crash into it. Turbulence in the disks also breaks up spiral arms to some degree. The disk in halo 10 is the most stable one and does not begin to fragment until 1.14 Myr, when it has grown to ∼ 2 pc in radius and become more turbulent. It thereafter forms a number of clumps, but they are quickly torn apart by gravitational torques and taken back up into the spiral arms except for one that forms at 1.26 Myr about 1.2 pc from the center. This clump grows in mass over time, collapsing into a second, smaller disk that is in a highly elliptical orbit with the first. This second disk exchanges mass with the first in a series of close encounters by the end of the run at 2.01 Myr. Because the second disk appears well after the first one, either a DCBH -SMS binary system or a binary DCBH could form after the second star collapses. There is more turbulence in the disk in halo 12 at early times and it produces two episodes of violent fragmentation, shown at 1.17 Myr and 1.85 Myr in Figure 3. Between episodes, most of the short-lived clumps are torn apart by gravitational torques and ejected in the tidal streams visible at 1.43 Myr, but one clump flattens into a secondary disk that persists from 889 kyr -1.39 Myr. Consequently, it is likely that a second SMS will form in this system, as discussed below. The star at the center of the main disk likely collapses to a DCBH before the second episode of clumping, so some of the clumps would rapidly fuel the growth of the BH when they crash into the center of the disk. The disk in halo 1 exhibits morphologies similar to the others at first but begins to fragment at early times, 289 kyr after formation. This episode is short-lived, however, because of the formation of a second, satellite disk at 366 kyr at an initial distance of ∼ 1 pc from its more massive companion. It orbits its companion until being torn apart and subsumed into the main disk at 947 kyr, which results in the ejection of the multiple tidal tails shown in Figure 3 at 995 kyr. The orbit is highly eccentric and precesses around the main disk, with disk-disk separations varying from ∼ 0.5 -1.2 pc. Mass exchange between the two disks occurs a number of times during close encounters, as shown at 879 kyr. Tidal torques between the two disks suppress subsequent fragmentation in both, with only two short-lived clumps forming after the appearance of the second disk (fragmentation can also be pre-empted as the satellite disk sweeps up gas along its orbit). The survival of the second disk for ∼ 600 kyr leads to the likely formation of a second SMS that is later drawn into a close binary with the star in the main disk, with final separations of less than 0.1 pc at the time they collapse to DCBHs. Large accretion rates in both disks corroborate the likely eventual formation of a binary DCBH, as discussed below. Binary and Multiple Disk Systems Although we noted the appearance of binary accretion disks in haloes 1, 10 and 12 in Section 3.2, they form in all the haloes and in some cases multiple disks appear. For example, three additional disks appear in halo 2: from 666 kyr -1.04 Myr, 937 kyr -1.48 Myr and 1.18 Myr -1.32 Myr (right panel of Figure 4). There is almost no overlap in time between the first satellite disk and the other two, which coevolve and even briefly exchange mass with each other at one point. In halo 8 just one binary disk appears in the run from 685 kyr -1.0 Myr. In halo 16, which hosts a relatively stable disk like that in halo 10, a fragment forms at 1.75 Myr, is flung into a highly elliptical orbit by three-body interactions with the main disk and another clump at 2.01 Myr, and then flattens into a satellite accretion disk by 2.7 Myr that is still orbiting the original disk at the end of the simulation at 3.05 Myr. Two additional disks form in halo 19, from 0.675 kyr -907 kyr and 907 kyr -1.41 Myr. Finally, in halo 20, two disk fragments collide at 639 kyr and form a dense clump that flattens into a disk by 774 kyr. It grows and soon rivals the original disk in mass (see the left panel of Figure 4) but is then almost destroyed in close encounters with this disk twice, at 1.36 Myr and at 1.61 Myr. In both cases tidal streams are ejected from the site of the interaction and the satellite disk retains only a small fraction of its mass after 1.61 Myr. We show center-to-center separations between the main disk and its satellite over time in haloes 8 and 20 in falls somewhere in between these two extremes. Separation distances vary from ∼ 0.3 pc -2 pc, and the satellite disks typically make 5 -10 orbits before either crashing into the center of the main disk or being torn apart by gravitational torques. The binary disks in our models live for shorter times than those in . Their disks survive for longer times because their simulations probed the upper extremes in spin parameter for the host haloes. Their large angular momenta allowed the satellite disks to orbit the main disks for longer times before crashing into them. In contrast, the haloes in our study have a range of spin parameters that do not reach their upper or lower extremes so satellite disks are taken up into the main disks in less time. Accretion Rates We plot accretion rates for all the disks in our haloes in Figure 6. They are calculated by dividing the change in mass in a 0.135 pc sphere centered on the densest cell at the center of the disk over 10 kyr intervals by 10 kyr, the time between data dumps. We use this radius to ensure that the tally sphere is always resolved by at least 10 zones while excluding the spiral arms of the disk, which could produce spurious contributions to the infall rates. We also found that the accretion rates converged at this radius as we decreased the radius. The initial time t = 0.0 is when the densest cell first reaches the maximum refinement level and catastrophic collapse has begun. Main Disk In each case there is an initial jump in accretion of 0.3 -1 M yr −1 for 200 -300 kyr that is due to the formation of the dense core of the disk, like the one shown at 51.5 kyr in halo 12 in Figure 3. Rates then fall as conservation of angular momentum of the gas increases its rotational velocity and it flattens into a self-gravitating, rotationally-supported disk, which hinders further collapse. Over the next few hundred kyr, accretion can be relatively smooth as in haloes 10 and 16, highly turbulent as in haloes 1, 8 and 19, or clumpy as in haloes 2, 12 and 20 because of fragmentation of the disk and collision of the fragments with its center. The larger jumps in accretion rate at 300 -600 kyr are from turbulence (200 -400 kyr in halo 8), mass exchange with a smaller, satellite disk (500 -600 kyr in halo 1), or clumps crashing into its center (500 -600 kyr in halo 12). The largest peaks, like those at 1.0 Myr in halo 1 and 1.4 Myr in halo 19, are from the collision of a satellite disk with the main disk. After the initial jump in accretion due to the formation of the dense core of the disk, the rates in some of the disks fall to lower values and flatten out as gas flows circling the center of the disk become Keplerian. This effect is most prominent in the disks in haloes 10 and 16. Other disks are too turbulent for the flows to ever become Keplerian and tidal interactions with satellite disks or collisions of fragments with the center of the disk disrupt circularised flows at later times, even in disks 10 and 16. At times, the accretion rate can become negative if turbulence drives gas out of the center of the disk (and thus out of the tally sphere) or during mass exchange with another disk in a close encounter. This process occurs on much larger scales than infall onto the star itself, which would not be expected to lose mass during these episodes. During such times accretion onto the star would at most be temporarily halted. Average accretion rates after the initial surge vary from ∼ 0.1 -0.5 M yr −1 but can peak at up to 2 M yr −1 for up to 100 kyr. Our rates are consistent with those reported at earlier times in previous studies ( 0.1 M yr −1 ; e.g., Latif & Volonteri 2015). Note that these average infall rates would produce SMSs that collapse via the general-relativistic instability rather than depletion of their hydrogen fuel ( 0.04 M yr −1 ; Woods et al. 2017). We show the mass accumulated at the center of the main disk in each halo in the left panel of Figure 7. The final masses at the end of the runs vary from 2.4 × 10 5 M to over 5 × 10 5 M . The occasional dips in mass correspond to the episodes of negative accretion rate visible in Figure 6. Comparison of these masses to the final SMS masses plotted as a function of accretion rate in Figure 4 of Woods et al. (2017) indicates that all the disks have been evolved for sufficient times for SMSs at their centers to have collapsed to DCBHs prior to the end of the run. Satellite Disks Accretion rates for the longest-lived binary disks in our haloes are shown in red in Figure 3. Although they only live for a fraction of the time that the main disk evolves, their infall rates can rival those of the main disk. Peaks in accretion in the satellite disks often coincide with dips in the main disk and peaks in the main disk often coincide with dips in the satellite. These correlations arise from mass exchange between the disks during close encounters, like the ones at 540 kyr and 879 kyr in halo 1 in Figure 3. Other peaks in accretion rate are due to collisions between the satellite disk and other clumps orbiting the main disk. Both mass exchange and collisions can perturb the orbits of binary disks. Accretion in the binary companion disks ends when they collide with the center of the main disk or are destroyed by tidal forces. The masses accumulated at the centers of the binary disks range from approximately 75,000 M to nearly 200,000 M and are shown in the right panel of Figure 7. At our resolution (and without a detailed stellar evolution model) it is not clear how much of this mass is taken up into a star at the center of the disk, but it is likely that at least some of these disks host SMSs. As noted earlier, in some cases such as halo 19 the second SMS would form early and coevolve with the star at the center of the main disk, possibly producing an SMS binary when the second disk later crashes into the first. In other cases, such as halo 16, the second SMS would likely form after the first collapses to a DCBH and an SMS -DCBH binary would form. Angular Momentum The evolution of the angular momentum, L, in the central 0.136 pc of the main disk in all eight haloes is shown in Figure 8. It mostly tracks the enclosed mass (compare with Figure 7), implying that no large external torques are exerted on the disk that could change the angular momentum without mass transfer. The large jumps in L coincide with the collisions of massive clumps or satellite disk with the main disk, which also coincide with the large jumps in accretion rates in the disks. The subsequent drops in L are due to the ejection of tidal streams after the collision that transport L out of the center of the disk. There is no apparent correlation between halo spin at the onset of atomic cooling and large jumps in L because they are present in halo 19, which has the lowest of the spin parameters, and halo 12, which has the second-highest spin. Likewise, there is no obvious connection between previous mergers and sudden increases in L because haloes 1 and 16 both exhibit mergers during assembly but not jumps in L. Note that the angular momenta shown in Figure 8 are likely not taken up by the SMS at the center of the disk, which we do not resolve in our models. Previous studies going to higher resolution for shorter times report that barswithin-bars instabilities arise on scales that are intermediate to those of the star and center of the disk that rapidly transport angular momentum outward (e.g., Wise et al. 2008). If such mechamisms did not arise SMSs could not form at the center of the disk because they can only accrete at most a few percent of the Keplerian angular momentum without being destroyed by the ΩΓ radiative instability (Haemmerlé et al. 2018b). Disk Stability The gravitational stability of disks is often parametrised by the radially-dependent Toomre Q parameter (Toomre 1964), Q(r) = c 2 s + v 2 turb κ πGσ , where cs is the sound speed, v turb is the turbulent speed, σ is the surface density, and κ is the epicyclic frequency (Binney & Tremaine 1987;Oh & Haiman 2002), κ = 2 V φ r v φ r + dv φ dr . If the rotation of the disk is Keplerian, κ reduces to the rotational frequency. The Q parameter represents the ability of restorative forces in the disk (rotation and pressure gradients) to counteract gravitational perturbations. If Q falls below unity, perturbations in the disk grow rather than dampen, and it is prone to fragmentation at that radius. Radial profiles of the Toomre parameter for the disks in haloes 10, 12, and 1 are shown in Figure 9 for the times in Figure 3. At 0.226 kyr in halo 10, Q is greater than 1 at all radii and, as shown in Figure 3, the disk is stable. At 1.029 Myr Q dips below 1 at ∼ 0.25 pc and 0.75 pc and the disk becomes prone to gravitiational instabilities that distort the spiral arms near their base and create turbulence but do not break them up or produce long-lived clumps at this time. However, the instabilities persist and later cause the disk to fragment at 1.14 Myr and produce the clumps visible at 1.261 Myr. Q falls below 1 over a greater range of radii by this time because of the growth of the disk in mass and surface area but it again exceeds 1 everywhere by 2.006 Myr, when the fragments have coalesced into a satellite disk whose orbit around the main disk has stabilised it against fragmentation. In contrast to halo 10, the disk in halo 12 exhibits multiple episodes of fragmentation. Q falls below 1 at several radii at 1.174 Myr and again at 1.851 Myr in halo 12 and clumps are visible at those radii in Figure 3. Outward transport of angular momentum due to the destruction of clumps by tidal torques from the disk stabilise the disk somewhat at 1.435 Myr, when tidal tails are visible instead of fragments, but Q has again fallen below 1 at 0.4 pc and beyond 0.9 pc at this time, portending the next round of fragmentation at 1.851 Myr. As in halo 10, as the disk in halo 12 grows in mass more of it becomes subject to gravitational instabilities. The Toomre parameter falls below 1 by 0.289 Myr at radii of 0.5 pc -1.1 pc in halo 1, when the disk has indeed produced the two massive clumps in Figure 3. The disk continues to be unstable at these radii at 0.540 Myr but then becomes stable at 1.261 Myr after the clumps have merged to produce a satellite disk who mass rivals that of the main disk. Their mutual orbit suppresses fragmentation in both disks. The satellite then merges with the main disk by 0.995 Myr, which ejects the tidal tails in Figure 3, and it again becomes prone to fragmentation as Q again falls below 1 at ∼ 0.4 pc and beyond 1.1 pc. Disk Rotation / Turbulence Rotational velocities for the main disks in all eight haloes at the end of the simulations are shown in the left panel of Figure 10. After multiple dynamical times the disks have all settled into rotationally supported structures with mostly Keplerian velocities. Rotational velocities are greatest at 0.1 -0.2 pc and reach 75 -110 km s −1 . The highest rotation speeds occur in the disks with the largest accretion rates, such as halo 8, while disks exhibiting the the lowest rotation rates have the lowest accretion rates, such as haloes 10 and 16. These velocities are consistent with those in Regan & Haehnelt (2009b), who report rotation rates of up to 60 km s −1 , with higher velocities corresponding to more massive disks. Their velocities are somewhat lower than ours because their disks are evolved for much shorter times, when they have only grown to 0.3 -0.6 pc in radius. The peaks and dips in velocity visible in some of the haloes at the end of the run are due to satellite disks that are orbiting the main disk at faster or slower rates. The absence of these features in the central 0.3 -0.5 pc, along with our Toomre profiles, indicate that fragmentation does not occur near the centers of the disks at later times. The bump at ∼ 200 pc in halo 2 is due to an interloper that later merges with the halo. The widths of the peaks in velocity mark the outer boundaries of the disks, as gas at larger radii falling onto the disks has much lower rotation rates. We calculate turbulent velocities in the disks by subtracting the infall and rotational velocities from the absolute velocities in quadrature: v 2 RMS = v 2 abs − v 2 φ − v 2 r . These profiles are shown for all eight disks at the end of the simulation in the right panel of Figure 10. They peak at slightly larger radii than the rotational velocities, 0.2 -0.3 pc, as gas from the spiral arms crashes into the center of the disk and the kinetic energy of bulk inflow builds up turbulent cascades and the gas becomes chaotic. Turbulent motions transport angular momentum outward through shocks that dissipate kinetic energy and they help support the disks against further collapse. Disk Evolution vs. Halo Assembly History Although we find a variety of disk evolution ranging from relatively stable disks to violently fragmenting ones, for the most part there is no apparent correlation between their evolution and the assembly histories or spin parameters of our eight haloes. For example, we find stable disks in haloes with high and low spin parameters and with as few as one and as many as three major mergers in their assembly histories (haloes 10 and 16, respectively). Likewise, disks in haloes 1 and 20 both fragment at early times but halo 1 has twice the spin parameter of halo 20. Furthermore, there is no obvious connection between longevity of binary disks and halo spin. Halo 1 has a much longer-lived satellite disk than halo 2 but a similar spin parameter. Baryons become gravitationally decoupled from DM in atomically-cooled haloes on scales of 5 -10 pc so the evolution of the disk is governed by local gas flows and turbulence rather than DM dynamics on larger scales. DM dynamics does appear to affect the evolution of the disks in haloes 10 and 16. From Figure 7 it can be seen that they are only half as massive as those in the other haloes and have correspondingly lower central accretion rates. These haloes begin to atomically cool at z = 17.2 and 16.5, just after major mergers at z = 18 and 17.5 that give them the highest spin parameters of all the haloes. The gas inherits some of this angular momentum as it collapses to form the disks, which have more rotational support due to the larger centrifugal forces on the gas. The disks consequently grow more slowly and accumulate less mass at their centers. DISCUSSION AND CONCLUSION Suazo et al. (2019) studied fragmentation in three primordial haloes that grew primarily by accretion, major mergers, or some mix of the two in a variety of LW backgrounds. They found no fragmentation in the highest backgrounds, which approximated atomic cooling, but only evolved these haloes for a few hundred kyr. Our results are consistent with their models because fragmentation generally happened after these times in our disks. Binary and multiple SMSs have previously been shown to form in atomically-cooled disks at high redshift, but only in the special case of haloes with high spin parameters ). Our new simulations prove that multiple SMS formation was the rule rather than the exception in atomically-cooled haloes because it occurred over their full range of spin parameters and for a variety of assembly histories. Fragmentation in the vicinity of the star on AU scales has also recently been used to invoke the formation of multiple massive stars instead of a single SMS (e.g., Regan & Downes 2018a). However, they could only follow the evolution of the clumps for at most a few hundred kyr, so it is not clear if they grow to large enough masses to become stars or are just taken up into the central object, as are most of the fragments in our models on larger scales. Regardless of the final fates of fragments on small scales, our simulations show that binaries or small multiples will eventually form in the halo on larger scales. Because DCBHs in our models typically end up in close proximity to each other after formation (0.1 -0.2 pc) and in high ambient densities ( 10 7 cm −3 ), radiative drag forces would rapidly decay their orbits and produce mergers whose GW emission could be detected by future observatories such as LISA (Hartwig et al. 2018) in addition to NIR or radio from the BHs themselves (Pacucci et al. 2015Natarajan et al. 2017;Barrow et al. 2018;Whalen et al. 2020aWhalen et al. ,b, 2021Vikaeus et al. 2022) or their progenitor stars (Surace et al. , 2019. If a SMS assumes a highly elliptical or-bit around a DCBH companion it could be destroyed in a tidal disruption event (TDE). Luminosities for TDEs of 10 -40 M Pop III stars with DCBHs have been calculated (Kashiyama & Inayoshi 2016) but not those for an SMS, which could produce an extended afterglow because of its large mass, so it is not known if such events would produce enough NIR flux to be detected today. Both types of TDEs may be possible in a given halo because X-rays from the DCBH are known to trigger lower-mass Pop III star formation in its vicinity at later times (Machacek et al. 2003;Aykutalp et al. 2014Aykutalp et al. , 2020. Using the accretion rates in Figure 6 in the Kepler stellar evolution code (Weaver et al. 1978;Woosley et al. 2002), we have determined that the SMSs in our disks would all evolve as cool, red supergiants with little ionizing UV that could affect their growth rates (Woods et al. 2021a,b). Consequently, radiation transport was never required in our Enzo models to obtain their true final masses. We do not include magnetic fields in our simulations, which are thought to suppress fragmentation on small scales and result in fewer, more massive objects (Turk et al. 2012;Latif et al. 2013bLatif et al. , 2014aSharda et al. 2020). However, magnetic dynamos typically affect gas flows on much smaller spatial scales than those of clump formation in our models so their absence in our simulations probably did not affect fragmentation. Woods et al. (2021a) have now shown that SMS -SMS, SMS -DCBH, and DCBH -DCBH binaries could form in our haloes. Supermassive X-ray binaries with unique spectra could result from SMS -DCBH systems if they are in close enough proximity. Because we do not resolve flows close to the surface of the star, our accretion rates should be taken to be upper limits. Nevertheless, as noted in the Introduction, numerical simulations that follow the collapse of atomicallycooled flows down to protostellar scales for short times find that accretion proceeds at rates similar to ours even down to these radii because of efficient angular momentum transport by bar instabilities. We only considered isothermal collapse, in which no H2 is present. In reality, it is difficult to destroy all H2 in the cores of massive haloes because of self shielding, even with very high LW backgrounds, and low to intermediate LW backgrounds were far more common in the early Universe. Studies have shown that even small amounts of H2 can dramatically reduce central infall rates that only produce 3000 -30,000 M stars, not 100,000 M stars (see Latif et al. 2021). There may thus have been two or three tiers of DCBH mass that led to populations of less massive quasars at high redshifts, not just the 10 9 M SMBHs at z > 6. H2 cooling also promotes fragmentation of the gas that could create even less massive stars. Given that primordial haloes fragment even in the absence of H2 cooling, it was likely a common occurence in all primordial haloes. We are investigating longterm disk evolution in a grid of LW backgrounds in all eight of our haloes now. Yue B., Ferrara A., Salvaterra R., Xu Y., Chen X., 2014, MNRAS, 440, 1263 Zhu Q., Li Y., Li Y., Maji M., Yajima H., Schneider R., Hernquist L., 2020, arXiv e-prints, p. arXiv:2012.01458 This paper has been typeset from a T E X/L A T E X file prepared by the author. Figure 1 . 1Merger histories. Left to right: haloes 1, 2, 8, 10, 12, 16, 19 and 20. Figure 2 . 2Spherically-averaged density (left) and temperature (right) profiles of the eight haloes at the onset of atomic cooling and rapid baryon collapse. Figure 3 . 3Projections of gas density in 3 of the 8 disks in our simulations. Top row: the relatively stable disk in halo 10. Left to right: 226 kyr, 1.02 Myr, 1.26 Myr, and 2.01 Myr. Center row: the more turbulent, rapidly fragmenting disk in halo 12. Left to right: 51 kyr, 1.17 Myr, 1.43 Myr and 1.85 Myr. Bottom row: early binary disk formation in halo 1. Left to right: 289 kyr, 540 kyr, 879 kyr and 995 kyr. Figure 4 . 4Binary and multiple disk systems in our runs. Left: the main disk and its satellite in halo 20 at 1.19 Myr. Right: Two smaller disks orbiting the main disk at 1.29 Myr in halo 2. Figure 5 . 5. Orbits vary from fairly regular, as in halo 8, to highly irregular, as in halo 20. Binary motion in the Center-to-center separations between the main disk and satellite disk in haloes 8 and 20. Figure 6 .Figure 7 . 67Accretion rates at the centers of the main and binary disks in all eight haloes. Total mass accreted at the center of the disks over time. Left: main disks. Right: satellite disks. Figure 8 . 8The evolution of angular momentum in the central 0.136 pc of the disks over time. Figure 9 . 9Toomre Q parameter. Top: halo 10 at 0.226 Myr, 1.029 Myr, 1.261 Myr and 2.006 Myr. Center: halo 12 at 0.051 Myr, 1.174 Myr, 1.435 Myr and 1.851 Myr. Bottom: halo 1 at 0.289 Myr, 0.540 Myr, 0.879 Myr and 0.995 Myr. All these times correspond to those inFigure 3. Figure 10 . 10Left: final rotational velocities as a function of radius. Right: final turbulent velocities. Table 1. Halo properties at the onset of atomic cooling. From left to right: halo, collapse redshift, mass at the onset of atomic cooling, spin parameter and number of major mergers.halo z col mass (M ) spin mergers 1 16.7 3.68e7 0.0389 1 2 14.5 8.47e7 0.0388 0 8 20.4 1.15e7 0.0321 0 10 17.3 2.60e7 0.0500 1 12 16.8 1.93e7 0.0471 0 16 16.5 2.91e7 0.0258 3 19 13.9 2.12e7 0.0072 0 20 17.7 3.56e7 0.0199 0 Patrick et al. MNRAS 000, 1-14 (2020) ACKNOWLEDGEMENTSThe authors thank the anonymous referee, whose critique improved the quality of this paper, and Britton Smith for valuable advice on the use of ytree. S. J. P. was supported by STFC grant ST/N504245/1. D. J. W. was supported by the Ida Pfeiffer Professorship at the Institute of Astrophysics at the University of Vienna and by STFC New Applicant Grant ST/P000509/1. M. A. L. thanks the UAEU for funding via UPAR grant No. 31S390 and startup grant No 31S372. 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[ "Qimera: Data-free Quantization with Synthetic Boundary Supporting Samples", "Qimera: Data-free Quantization with Synthetic Boundary Supporting Samples" ]
[ "Kanghyun Choi [email protected] \nDepartment of Computer Science\nYonsei University\n\n", "Deokki Hong [email protected] \nDepartment of Artificial Intelligence\nYonsei University\n\n", "Noseong Park [email protected] \nDepartment of Computer Science\nYonsei University\n\n\nDepartment of Artificial Intelligence\nYonsei University\n\n", "Youngsok Kim [email protected] \nDepartment of Computer Science\nYonsei University\n\n\nDepartment of Artificial Intelligence\nYonsei University\n\n", "Jinho Lee [email protected] \nDepartment of Computer Science\nYonsei University\n\n\nDepartment of Artificial Intelligence\nYonsei University\n\n" ]
[ "Department of Computer Science\nYonsei University\n", "Department of Artificial Intelligence\nYonsei University\n", "Department of Computer Science\nYonsei University\n", "Department of Artificial Intelligence\nYonsei University\n", "Department of Computer Science\nYonsei University\n", "Department of Artificial Intelligence\nYonsei University\n", "Department of Computer Science\nYonsei University\n", "Department of Artificial Intelligence\nYonsei University\n" ]
[]
Model quantization is known as a promising method to compress deep neural networks, especially for inferences on lightweight mobile or edge devices. However, model quantization usually requires access to the original training data to maintain the accuracy of the full-precision models, which is often infeasible in real-world scenarios for security and privacy issues. A popular approach to perform quantization without access to the original data is to use synthetically generated samples, based on batch-normalization statistics or adversarial learning. However, the drawback of such approaches is that they primarily rely on random noise input to the generator to attain diversity of the synthetic samples. We find that this is often insufficient to capture the distribution of the original data, especially around the decision boundaries. To this end, we propose Qimera, a method that uses superposed latent embeddings to generate synthetic boundary supporting samples. For the superposed embeddings to better reflect the original distribution, we also propose using an additional disentanglement mapping layer and extracting information from the full-precision model. The experimental results show that Qimera achieves state-of-the-art performances for various settings on data-free quantization. Code is available at https://github.com/iamkanghyunchoi/qimera.
null
[ "https://arxiv.org/pdf/2111.02625v1.pdf" ]
242,757,321
2111.02625
b72f590d0992539748b5e106aef447ce470cd535
Qimera: Data-free Quantization with Synthetic Boundary Supporting Samples Kanghyun Choi [email protected] Department of Computer Science Yonsei University Deokki Hong [email protected] Department of Artificial Intelligence Yonsei University Noseong Park [email protected] Department of Computer Science Yonsei University Department of Artificial Intelligence Yonsei University Youngsok Kim [email protected] Department of Computer Science Yonsei University Department of Artificial Intelligence Yonsei University Jinho Lee [email protected] Department of Computer Science Yonsei University Department of Artificial Intelligence Yonsei University Qimera: Data-free Quantization with Synthetic Boundary Supporting Samples Accepted to NeurIPS 2021. Author's copy Model quantization is known as a promising method to compress deep neural networks, especially for inferences on lightweight mobile or edge devices. However, model quantization usually requires access to the original training data to maintain the accuracy of the full-precision models, which is often infeasible in real-world scenarios for security and privacy issues. A popular approach to perform quantization without access to the original data is to use synthetically generated samples, based on batch-normalization statistics or adversarial learning. However, the drawback of such approaches is that they primarily rely on random noise input to the generator to attain diversity of the synthetic samples. We find that this is often insufficient to capture the distribution of the original data, especially around the decision boundaries. To this end, we propose Qimera, a method that uses superposed latent embeddings to generate synthetic boundary supporting samples. For the superposed embeddings to better reflect the original distribution, we also propose using an additional disentanglement mapping layer and extracting information from the full-precision model. The experimental results show that Qimera achieves state-of-the-art performances for various settings on data-free quantization. Code is available at https://github.com/iamkanghyunchoi/qimera. Introduction Among many neural network compression methodologies, quantization is considered a promising direction because it can be easily supported by accelerator hardwares [1] than pruning [2] and is more lightweight than knowledge distillation [3]. However, quantization generally requires some form of adjustment (e.g., fine-tuning) using the original training data [4,5,6,7,8] to restore the accuracy drop due to the quantization errors. Unfortunately, access to the original training data is not always possible, especially for deployment in the field, for many reasons such as privacy and security. For example, the data could be medical images of patients, photos of confidential products, or pictures of military assets. Therefore, data-free quantization is a natural direction to achieve a highly accurate quantized model without accessing any training data. Among many excellent prior studies [9,10,11,12], generative methods [13,14,15] have recently been drawing much attention due to their superior performance. Generative methods successfully generate synthetic samples that resemble the distribution of the original dataset and achieve high accuracy using information from the pretrained full-precision network, such as batch-normalization statistics [15,13] or intermediate features [14]. However, a significant gap still exists between data-free quantized models and quantized models finetuned with original data. What is missing from the current generative data-free quantization schemes? We hypothesize that the synthetic samples of conventional methods lack boundary supporting samples [16], which lie on or near the decision boundary of the full-precision model and directly affect the model performance. The generator designs are often based on conditional generative adversarial networks (CGANs) [17,18] that take class embeddings representing class-specific latent features. Based on these embeddings as the centroid of each class distribution, generators rely on the input of random Gaussian noise vectors to gain diverse samples. However, one can easily deduce that random noises have difficulty reflecting the complex class boundaries. In addition, the weights and embeddings of the generators are trained with cross-entropy (CE) loss, further ensuring that these samples are well-separated from each other. In this work, we propose Qimera, a method for data-free quantization employing superposed latent embeddings to create boundary supporting samples. First, we conduct a motivational experiment to confirm our hypothesis that samples near the boundary can improve the quantized model performance. Then, we propose a novel method based on inputting superposed latent embeddings into the generator to produce synthetic boundary supporting samples. In addition, we provide two auxiliary schemes for flattening the latent embedding space so that superposed embeddings could contain adequate features. Qimera achieves significant performance improvement over the existing techniques. The experimental results indicate that Qimera sets new state-of-the-art performance for various datasets and model settings for the data-free quantization problem. Our contributions are summarized as the following: • We identify that boundary supporting samples form an important missing piece of the current state-of-the-art data-free compression. • We propose using superposed latent embeddings, which enables a generator to synthesize boundary supporting samples of the full-precision model. • We propose disentanglement mapping and extracted embedding initialization that help train a better set of embeddings for the generator. • We conduct an extensive set of experiments, showing that the proposed scheme outperforms the existing methods. 2 Related Work Data-free Compression Early work on data-free compression has been led by knowledge distillation [3], which usually involves pseudo-data created from teacher network statistics [19,20]. Lopes et al. [19] suggested generating pseudo-data from metadata collected from the teacher in the form of activation records. Nayak et al. [20] proposed a similar scheme but with a zero-shot approach by modeling the output space of the teacher model as a Dirichlet distribution, which is taken from model weights. More recent studies have employed generator architectures similar to GAN [21] to generate synthetic samples replacing the original data [22,23,24,25]. In the absence of the original data for training, DAFL [22] used the teacher model to replace the discriminator by encouraging the outputs to be close to a one-hot distribution and by maximizing the activation counts. KegNet [26] adopted a similar idea and used a low-rank decomposition to aid the compression. Adversarial belief matching [23] and data-free adversarial distillation [24] methods suggested adversarially training the generator, such that the generated samples become harder to train. One other variant is to modify samples directly using logit maximization as in DeepInversion [25]. While this approach can generate images that appear natural to a human, it has the drawback of skyrocketing computational costs because each image must be modified using backpropagation. Data-free quantization is similar to data-free knowledge distillation but is a more complex problem because quantization errors must be recovered. The quantized model has the same architecture as the full-precision model; thus, the early methods of post-training quantization were focused on how to convert the full-precision weights into quantized weights by limiting the range of activations [9,12], correcting biases [9,10], and equalizing the weights [10,12]. ZeroQ [15] [29]. To the extent of our knowledge, this work is the first to propose using superposed latent embeddings to generate boundary supporting samples explicitly for data-free quantizations. Boundary Supporting Samples In the context of knowledge distillation, boundary supporting samples [16] are defined as samples that lie near the decision boundary of the teacher models. As these samples contain classification information about the teacher models, they can help the student model correctly mimic the teacher's behavior. Heo et al. [16] applied an adversarial attack [30] to generate boundary supporting samples and successfully demonstrated that they improve knowledge distillation performance. In AMKD [31], triplet loss was used to aid the student in drawing a better boundary. Later, DeepDig [32] devised a refined method for generating boundary supporting samples and analyzed their characteristics by defining new metrics. Although boundary supporting samples have been successful for many problems, such as domain adaptation [33] and open-set recognition [34], they have not yet been considered for data-free compression. Motivational Experiment To explain the discrepancy between the accuracy of the model fine-tuned with the original training data and the data-free quantized models, we hypothesized that the synthetic samples from generative data-free quantization methods lack samples near the decision boundary. To validate this hypothesis, we designed an experiment using the CIFAR-100 [35] dataset with the ResNet-20 network [36]. First, we forwarded images in the dataset into the pre-trained full-precision ResNet-20. Among these, we selected 1500 samples (3% of the dataset, 15 samples per class) from samples where the highest confidence value was lower than 0.25, forming a group of 'confusing' samples. Then, we combined the confusing samples with synthetic samples generated from a previous study [13] to fine-tune the quantized model with 4-bit weights and 4-bit activations. We also selected as a control group an equal number of random samples from the images classified as unconfusing and fine-tuned the quantized model using the same method. The results are presented in Table 1. The quantized model with synthetic + unconfusing real samples exhibited only 0.52%p increase in the accuracy from the baseline. In contrast, adding confusing samples provided 2.36%p improvement, filling almost half the gap towards the quantized model fine-tuned with the original data. These results indirectly validate that the synthetic samples suffer from a lack of confusing samples (i.e., boundary supporting samples). We aim to address the issue in this paper. As we indicate in Section 5, Qimera achieves a 1.71%p performance gain for the same model in a data-free setting, close to that of the addition of confusing synthetic samples. Generating Boundary Supporting Samples with Superposed Latent Embeddings Baseline Generative Data-free Quantization Recent generative data-free quantization schemes [13,14] employ a GAN-like generator to create synthetic samples. In the absence of the original training samples, the generator G attempts to generate synthetic samples so that the quantized model Q can mimic the behavior of the full-precision model P . For example, in GDFQ [13], the loss function L GDF Q is L GDF Q (G) = L P CE (G) + αL P BN S (G),(1) where the first term L CE guides the generator to output clearly classifiable samples, and the second term L BN S aligns the batch-normalization statistics of the synthetic samples with those of the batch-normalization layers in the full-precision model. In another previous work ZAQ [14], L ZAQ (G) = L P,Q o (G) + βL P,Q f (G) + γL P a (G),(2) where the first term L o separates the prediction outputs of P and Q, and the second term L f separates the feature maps produced by P and Q. These two losses let the generator be adversarially trained and allow it to determine samples where P cannot mimic Q adequately. Lastly, the third term L a maximizes the activation map values of P so that the created samples do not drift too far away from the original dataset. The quantized model Q is usually jointly trained with G, such that L GDF Q (Q) = L Q CE (Q) + δL P KD (Q),(3) for GDFQ, and L ZAQ (Q) = −L P,Q o (G) − βL P,Q f (G),(4) for ZAQ, respectively, where L KD from Eq. 3 is the usual KD loss with Kullback-Leibler divergence. While these two methods exhibit great performance, they both model the distribution of per-class embeddings in the latent input space as a Gaussian distribution, and generate diverse samples using random Gaussian noise inputs. However, based on the Gaussian distribution, it is difficult to correctly reflect the boundary between the two different classes, especially when all samples have one-hot labels. Figure 1 visualizes such problems in a simplified two-dimensional space. With samples generated from gaussian noise (Figure 1a), the two per-class distributions are far away, and a void exists between the two classes (see Figure 3b for plots from experimental data). This can cause the decision boundaries to be formed at nonideal regions. One alternative is to use noise with higher variance, but the samples would overlap on a large region, resulting in too many samples with incorrect labels. Furthermore, in the above approaches, the loss terms of Eqs. 1 and 2 such as L CE and L o encourage the class distributions to be separated from each other. While this is beneficial for generating clean, well-classified samples, it also prevents generating samples near the decision boundary, which is necessary for training a good model [16,31]. Therefore, in this paper, we focus on methods to generate synthetic samples near the boundary from the full-precision model P ( Figure 1b). Superposed Latent Embeddings The overview of Qimera is presented in Figure 2. Often, generators use learned embeddings to create samples of multiple classes [18,37]. Given an input one-hot label y representing one of C classes and a random noise vector z, a generator G uses an embedding layer E ∈ R D×C to create a synthetic samplex:x = G(z + E y ), z ∼ N (0, 1).(5) To create boundary supporting samples, we superpose the class embeddings so that the generated samples have features lying near the decision boundaries of P . With two embeddings superposed, new synthetic samplex becomeŝ x = G(z + (λE y1 + (1 − λ)E y2 )), z ∼ N (0, 1), 0 ≤ λ ≤ 1.(6) To avoid too many confusing samples from complicating the feature space, we also apply soft labels in the same manner as a regularizer: y = λy 1 + (1 − λ)y 2 , 0 ≤ λ ≤ 1.(7) Generalizing to K embeddings, Eqs. 6 and 7 become (x ,ŷ ) = G S(e) , K k λ k y k , S(e) = z + K k λ k e k , z ∼ N (0, 1),(8)λ i = Sof tmax(p i ) = exp(pi) / K k (exp(p k )), p i ∼ N (0, 1).(9) where e ∈ R D×K has K embeddings from E as the column vectors (i.e., e = [E y0 , . . . , E y K−1 ]), and S : R D×K → R D is a superposer function. Applying this to existing methods is straightforward and incurs only a small amount of computational overhead. Similar to knowledge distillation with boundary supporting samples [16,31], the superposed embeddings are supposed to help transfer the decision boundary of the full precision (teacher) model to the quantized (student) model. Although the superposed embedding scheme alone produces a substantial amount of performance gains, the generator embedding space is often not flat enough [38]; therefore, linearly interpolating them can result in unnatural samples [39]. To mitigate this, the embedding space used in Qimera should possess two characteristics. First, the embedding space should be as flat as possible so that the samples generated from Eq. 8 reflect the intermediate points in the feature space. Second, the individual embeddings should still be sufficiently distinct from each other, correctly representing the distance between each class distribution. In the remaining two subsections, we describe our strategies for enforcing the embeddings to contain the above characteristics. Disentanglement Mapping To perform superposing in a flatter manifold, we added a learnable mapping function M : R D → R d before the embeddings are superposed, where D is the embedding dimension and d is the dimension of the target space. Thus,x from Eq. 8 becomesx = G S(m) , where m k = M (e k ). Although we do not add any specific loss that guides the output space of M to be flat, training to match the output of the full-precision model (i.e., L CE ) using the superposedŷ encourages M to map the input to a flatter space. In practice, we modeled M as a single-layer perceptron, which we call the disentanglement mapping layer. The experimental results from Section 5 demonstrate that the disentanglement mapping provides a considerable amount of performance gain. Extracted Embedding Initialization For the embeddings to be flat, we want the distributions of the embeddings fused with noise to be similar to the feature space. For this purpose, we utilize the fully connected layer of the full precision model P . Given f , the output features of the full-precision model before the last fully connected layer, we minimize C y dist P (f |y), P (g|y) , where C is the number of classes, g is the input of the generator, and dist is some distance metric. We do not have knowledge of the distribution of f ; thus, we modeled it as a Gaussian distribution, such that f |y ∼ N (µ y , 1). Therefore, solving it against Eq. 5 simply yields E(y) = µ y . In practice, we use the corresponding column from the weight of the last fully connected layer of the full-precision model because its weights represent the centroids of the activations. If the weights of the last fully connected layer W = [w 1 , w 2 , ..., w C ], we set µ y = w y . Our experiments reveal that extracting the weights from the full-precision model and freezing them already works well (see Appendix B.2). However, using them as initializations and jointly training them empirically works better. We believe this is because fully connected layers have bias parameters in addition to weight parameters. Because we do not extract these biases into the embeddings, a slight tuning is needed by training them. This outcome aligns with the findings from the class prototype scheme used in self-supervised learning approaches [40,33]. Experimental Results Experiment Implementation Our method is evaluated on CIFAR-10, CIFAR-100 [35] and ImageNet (ILSVRC2012 [41]) datasets, which are well-known datasets for evaluating the performance of a model on the image classification task. CIFAR10/100 datasets consist of 50k training sets and 10k evaluation sets with 10 classes and 100 classes, respectively, and is used for small-scale experiments in our evaluation. ImageNet dataset is used for large-scale experiments. It has 1.2 million training sets and 50k evaluation sets. To keep the data-free environment, only the evaluation sets were used for test purposes in all experiments. For the experiments, we chose ResNet-20 [36] for CIFAR-10/100, and ResNet-18, ResNet-50, and MobileNetV2 [42] for ImageNet. We implemented all our techniques using PyTorch [43] and ran the experiments using RTX3090 GPUs. All the model implementations and pre-trained weights before quantization are from pytorchcv library [44]. For quantization, we quantized all the layers and activations using n-bit linear quantization, described by [7], as below: θ = θ − θ min interval(n) − 2 n−1(10) where θ is the full-precision value, θ is the quantized value, interval(n) is calculated as θmax−θmin 2 n −1 . θ min and θ max are per-channel minimum and maximum value of θ. To generate synthetic samples, we built a generator using the architecture of ACGAN [18] and added a disentanglement mapping layer after class embeddings followed by a superposing layer. Among all batches, the superposing layer chooses between superposed embeddings and regular embeddings in p : 1 − p ratio. The dimension of latent embedding and random noise was set to be the same with the channel of the last fully connected layer of the target network. For CIFAR, the intermediate embedding dimension after the disentanglement mapping layer was set as 64. The generator was trained using Adam [45] with a learning rate of 0.001. For ImageNet, the intermediate embedding dimension was set to be 100. To maintain label information among all layers of the generator, we apply conditional batch normalization [46] rather than regular batch normalization layer, following SN-GAN [47]. The optimizer and learning rate were the same as that of CIFAR. To fine-tune the quantized model Q, we used SGD with Nesterov [48] as an optimizer for Q with a learning rate of 0.0001 while momentum and weight decay terms as 0.9 and 0.0001 respectively. The generator G and the quantized model Q were jointly trained with 200 iterations for 400 epochs while decaying the learning rate by 0.1 per every 100 epochs. The batch size was 64 and 16 for CIFAR and ImageNet respectively. While Qimera can be applied almost all generator-based approaches, we chose to adopt baseline loss functions for training from GDFQ [13], because it was stable and showed better results for large scale experiments. Thus, loss functions L(G) and L(Q) are equal to Eq. 1 and Eq. 3 with α = 0.1 and δ = 1.0, following the baseline. Visualizations of Qimera-generated Samples on Feature Space To ensure that the superposed latent embedding creates boundary supporting samples, we conducted an experiment to compare generated synthetic samples on feature space visually. The experimental results are based on the generators trained with ResNet-20 for CIFAR-10 dataset. The features were extracted from the intermediate activation before the last fully connected layer of the full-precision model. For the Qimera-based generator, we set K=2 during the sample generations for clarity. After extracting the features, we projected the features into a two-dimensional space using Principal Component Analysis (PCA) [49]. The results are presented in Figure 3. Compared with the original training set data (Figure 3a), the samples from the GDFQ-based generator ( Figure 3b) show a lack of boundary supporting samples and the class distributions are confined to small regions around the centroid of each class. A generator trained with Qimera, however, exhibits different characteristics (Figure 3c). Samples that are generated from superposed latent embeddings are displayed as black dots. The black dots are mostly located on the sparse regions between the class clusters. This experimental result shows that our method, Qimera, successfully generates samples near the decision boundaries. In other words, superposed latent embeddings are not only superposed on embedding space but also in the feature space, serving as synthetic boundary supporting samples. Table 2 displays the classification accuracy on various datasets, target models, and bit-width settings. Note that nwma means n-bit quantization for weights and m-bit quantization for activations. As baselines, we selected ZeroQ [15], ZAQ [14], and GDFQ [13] as the important previous works on generative data-free quantization. In addition, we implemented Mixup [50] and Cutmix [51] on top of GDFQ, which are data augmentation schemes that mix input images. To implement these schemes, we created synthetic images from GDFQ, and applied the augmentations to build training images and labels. All baseline results are from official code of the authors, where a small amount of The results demonstrate that Qimera outperforms the baselines at almost all settings. Its performance improvement is especially large for low-bitwidth (4w4a) cases. For 5w5a setting, the gain is still significant, and its performance reaches near that of the full-precision model, which represents the upper bound. In addition, Qimera is not limited to small datasets having a low spatial dimension. The ImageNet results prove that Qimera performs beyond other baselines with considerable gaps, on a large-scale dataset with many classes. Interestingly, the result of GDFQ+Mixup and GDFQ+Cutmix did not produce much noticeable improvement except for GDFQ+Mixup in 4w4a ResNet -18 and -50. This implies that a mixture of generated synthetic samples of different classes in the sample space is not sufficient to represent boundary supporting samples. In summary, Qimera achieves superior accuracy on various environments regardless of dataset or model scale. Quantization Results Comparison Qimera on Top of Various Algorithms In this paper, we have applied GDFQ [13] as a baseline. However, our design does not particularly depend on a certain method, and can be adopted by many schemes. To demonstrate this, we implemented Qimera on top of ZAQ [14] and AutoReCon [27]. The results are shown in Table 3. AutoReCon [27] strengthens the generator architecture using a neural architecture search, and therefore applying this is no different from that of the original Qimera implemented on top of GDFQ. However, original ZAQ does not use per-class sample generation. To attach the techniques from Qimera, we extended the generator with an embedding layer, and added a cross-entropy loss into the loss function as the following: L(G) = L P,Q o (G) + βL P,Q f (G) + γL P a (G) + ρL P CE (G),(11)L ( Q) = −L P,Q o (G) − βL P,Q f (G) + ρL Q CE (G),(12) where ρ was set to 0.1, and L P CE (G), L Q CE (G) are calculated based on Eq. 8. For all cases, applying the techniques of Qimera improves the performance by a significant amount. Especially for Qimera + ZAQ, the performances obtained was better than those of our primary implementation Qimera + GDFQ. Unfortunately, it did not converge with ImageNet, and thus we did not consider this implementation as the primary version of Qimera. Further Experiments We discuss some aspects of our method in this section, which are effect of each scheme introduced in Section 4 on the accuracy of Q, a sensitivity study upon various p and K settings, and a closer investigation into the effect of disentanglement mapping (DM) and extracted embedding initialization (EEI) on the embedding space. All experiments in this subsection are under 4w4a setting. Ablation study. We conducted an ablation study by adding the proposed components one by one on the GDFQ baseline. The results are presented in Table 4. As we can see, the superposed embedding (SE) alone brings a substantial amount of performance improvement for both Cifar-100 (1.16%p) and ImageNet (11.98%p). On Cifar-100, addition of EEI provides much additional gain of 0.68%p, while that of DM is marginal. ImageNet, on the other hand, DM provides 1.97%p additional gain, much larger than that of EEI. When both of DM and EEI are used together with SE, the additional improvement over the best among (+SE, DM) and (+SE, EEI) is relatively small. We believe this is because DM and EEI serve for the similar purposes. However, applying both techniques can achieve near-best performance regardless of the dataset. Sensitivity Analysis. Table 4 also provides a sensitivity analysis for K (number of classes for superposed embeddings) and p (ratio of synthetic boundary supporting samples within the dataset). As shown in the table, both parameter clearly have an impact on the performance. For Cifar-100, the sweet spot values for K and p are both small, while those of the ImageNet were both large. We believe is because ImageNet has more classes. Because the number of class pairs (and the decision boundaries) is a quadratic function of number of classes, there needs more boundary supporting samples with more embeddings superposed to draw the correct border. Investigation into the Embedding Space. To investigate how DM and EEI help shaping the embedding space friendly to SE, we conducted two additional experiments with ResNet-20 in Table 5. First, setting K=2 (between two classes), we measured the ratio of the perceptual distance and Euclidean distance between the two embeddings in the classifier's feature space. The perceptual distance is measured by sweeping λ from Eqs. 6 and 7 from 0 to 1 by 0.01 and adding all the piecewise distances between points in the feature space (See Figure 5 in Appendix). We want the distance ratio to be close to 1.0 for the embedding space distribution to be similar to that of the feature space. Second, we defined 'intrusion score', the sum of logits that are outside the chosen pair of classes. If the score is large, that means the synthetic boundary samples are regarded as samples of non-related classes. Therefore, lower scores are desired. For comparison, we have also included the Mixup as a naive method and measured both metrics. It shows that the DM and EEI are effective for in terms of both the distance ratio and the intrusion score, explaining how DM and EEI achieves better performances. Discussion Does it Cause Invasion of Privacy? The motivation for data-free quantization is that the private original data might not be available. Using a generator to reconstruct samples that follow the distribution of original data might indicate the invasion of privacy. However, as displayed in Figure 4, at least with current technologies, there is no sign of privacy invasion. The generated data are far from being human interpretable, which was also the case for previous works [15,14,13]. See Appendix ( Figure 6) for more generated images. Geirhos et al. [52] reveal an interesting property that CNNs are heavily biased by local textures, not global shapes. BagNet [53] supports this claim by confirming that images with distorted shapes but preserved textures can still mostly be correctly classified by CNNs. In such regard, it is no wonder that the generated samples are non-interpretable, because there is only a few combinations that maintains the global shape out of all possibilities that preserve the textures. Nonetheless, not being observed does not guarantee the privacy protection. We believe it is a subject for further investigations. Limitations Even though Qimera achieves a superior performance, one drawback of this approach is the fact that it utilizes embeddings of multiple classes to generate the boundary supporting samples. This restricts its application to classification tasks and its variants. For example, generation of datasets such as image segmentation or object detection do not take class embeddings as the input. However, generators for such data are not extensively studied yet, and we envision adding diversity to them would require inputting some form of labels or embeddings similar to paint-to-image [54], which would allow Qimera be easily applied. Conclusion In this paper, we have proposed Qimera, a simple yet powerful approach for data-free quantization by generating synthetic boundary supporting samples with superposed latent embeddings. We show in our motivational experiment that current state-of-the-art generative data-free quantization can be greatly improved by a small set of boundary supporting samples. Then, we show that superposing latent embeddings can close much of the gap. Extensive experimental results on various environments shows that Qimera achieves state-of-the-art performance for many networks and datasets, especially for large-scale datasets. A Code The whole code is available at https://github.com/iamkanghyunchoi/qimera, including the training, evaluation, and visualization for all settings. This project code is licensed under the terms of the GNU General Public License v3.0. B Additional Experimental Results B.1 Baseline with Different Noise Level Instead of the superposed embeddings, we tried the alternative of adjusting the variance of the noise inputs, as discussed in Section 4.1 of the main body. We have tested five values as standard deviation σ z : 0.25, 0.5, 1.0, 1.5, and 2.0. The results in Table 6 show that as we hypothesized, just by increasing the noise level did not provide much improvement on the performance of the baseline. Instead, we have noticed a slight improvement on GDFQ when the noise level is decreased, and we believe this is due to the generation of clearer samples. Nonetheless, the accuracy is far below that of the proposed Qimera. To show that the weight from the last fully connected layer of the full-precision model is a good candidate for the initial embeddings, we have performed an experiment where the embeddings are frozen right after initialization. The results are presented in Table 7. Qimera with frozen embeddings are not better than the primary Qimera method with trained embeddings. However, compared to the two baselines (ZAQ and GDFQ), they provides a comparable accuracy on Cifar-10 and better accuracies on Cifar-100 and ImageNet. Furthermore, the accuracy on Cifar-10 dataset is close to the upper bound for all techniques under comparison, and thus the differences are minimal. B.3 Sensitivity Study on Number of DM layers Num. DM Layers Accuracy Table 9: Further Sensitivity analysis In addition to our choice of hyperparameters presented in the main body, we have performed a further extensive sensitivity study on those parameters, which is displayed in Table 9. All experiments are against 4w4a configuration, equal to the Table 3 (Section 5.4) in the main body. Regardless of the choice in p and K, the results are all better than the two baselines ZAQ and GDFQ. Furthermore, while they all provide a meaningfully good performance, the results show a clear trend: lower p, K for Cifar-10/100 and higher p, K for ImageNet as sweet spots. This result supports the use of Qimera in that these parameters are easily tunable, not something that must be exhaustively seAutoReConhed for optimal values. B.5 Comparison with DSG Qimera is conceptually similar to DSG [28] which tries to diversify the sample generation by relaxing the batch-norm stat alignments. However, Qimera is different from DSG because we explicitly try to Table 10 shows the comparison of Qimera with DSG. We use the reported numbers for DSG, and perform a new set of experiments for Qimera to match the settings. We use the lowest-bit settings for each network evaluated in DSG. As displayed in the table, Qimera outperforms DSG in all settings, especially for 4w4a cases. C Class-Pairwise Visualization To look closely onto the visualization of the samples from Section 5.3 ( Figure 3) in the main body, we have plotted them in a pair-wise manner. Even though 10 classes in total gives 45 possible pairs, we chose nine symbolically adjacent pairs in the figure. Although being symbolically adjacent does not have much meaning, we believe having nine pairs is enough for our purpose rather than showing all 45 possible pairs. The colors match that of the Figure 3, where the lightgreen dots represent the synthetic boundary supporting samples. Also, we have plotted the path between the centroids of the two clusters in black, by varying λ (the ratio of superposition) from 0 to 1 by 0.01 without any noise. Each 10th percentile is denoted as larger black dots. The results show that the samples and the path lie relatively in the middle of the two clusters. Please note that we have performed PCA plot for each pair to best show the distribution, so the position and orientation of the clusters do not exactly match those from Figure 3. D More Generated Images Lastly, Figure 6 shows more samples generated from Qimera. Figure 6a displays the synthetic boundary supporting samples generated from Cifar-10 dataset, with K = 2 and λ = 0.5. Each row and column represents a class from Cifar-10. For example, the image at row 0 (airplane) and column 2 (bird) represents a sample generated from superposed embeddings of airplane and bird. Although still not very human-recognizable, we find that each sample in Figure 6a has some features adopted from each of the source classes in Figure 4d. Figure 6c shows the sample images created from ImageNet. Because there are too many classes within ImageNet (1000), we chose 10 classes from them, which are {0: 'tench, Tinca tinca', 100: 'black swan, Cygnus atratus', 200: 'Tibetan terrier, chrysanthemum dog', 300: 'tiger beetle', 400: 'academic gown, academic robe, judge's robe", 500: 'cliff dwelling', 600: 'hook, claw', 700: 'paper towel', 800: 'slot, one-armed bandit', 900: 'water tower'}, and the original samples from those classes are shown in Figure 6b. As in Cifar-10, the generated samples are far from human-recognizable, but each row is clearly distinguishable from the others. In addition, Figure 6d contains the synthetic boundary supporting samples from ImageNet, following the same rules from Figure 6a. Again, we see that each position in the sample matrix adopts features from the rows of the corresponding class pair in Figure 6c. (c) Synthetic samples from ImageNet. (d) Synthetic boundary supporting samples from ImageNet. Figure 6: Additional synthetic samples. Synthetic boundary supporting samples with superposed latent embeddings (proposed). Figure 1 : 1Diagram of synthetic samples in the feature space of the full-precision network. The black curves represent the decision boundary of the full-precision model, which are considered ideal for the quantized model (orange dotted curves) to mimic. When synthetic images are generated with per-class embeddings and noises as in (a), their features do not support the decision boundary, whereas the proposed approach in (b) generates the boundary supporting samples, helping the quantized model to set the near-ideal decision boundary. Figure 2 : 2An overview of the proposed method. Proposed components are denoted as colored shapes. Figure 3 : 3PCA plots of the features before the last layer of the full-precision model. In the plots of Qimera-generated samples (c), the black dots represent synthetic boundary supporting samples generated with the superposed latent embeddings, which fill the space between the clusters. GDFQgenerated samples (b) form clusters that are smaller than that of the original data (a) and lack samples in the mid-ground. Best viewed in color. Figure 4 : 4Synthetic samples generated for Cifar-10 dataset. Each row represents one of the 10 classes, except for ZAQ which generates samples without labels. Figure 5 : 5Visualization of the generated samples in the feature space. The lightgreen cloud represents the synthetic boundary supporting samples. The black dots represents the path between the two embeddings without any noise, where every 10th percentile is denoted as larger dots. The colors match that of theFigure 3of the main body, but the PCA dimension has been adjusted to best show each chosen class pair.(a) Synthetic boundary supporting samples from Cifar-10. (b) Original ImageNet samples from the selected 10 classes. Highest among the baselines † Did not convergeDataset Model (FP32 Acc.) Bits ZeroQ ZAQ GDFQ GDFQ +Cutmix GDFQ +Mixup Qimera (%p improvement) Cifar-10 ResNet-20 4w4a 79.30 92.13 * 90.25 89.58 88.69 (-3.44) 91.26 ±0.49 (-0.87) (93.89) 5w5a 91.34 93.36 93.38 * 92.75 92.79 (-0.59) 93.46 ±0.03 (+0.08) Cifar-100 ResNet-20 4w4a 47.45 60.42 63.39 * 62.74 62.99 (-0.40) 65.10 ±0.33 (+1.71) (70.33) 5w5a 65.61 68.70 * 66.12 67.51 67.78 (-0.92) 69.02 ±0.22 (+0.32) ImageNet ResNet-18 4w4a 22.58 52.64 60.60 * 58.90 61.72 (+1.12) 63.84 ±0.30 (+3.24) (71.47) 5w5a 59.26 64.54 68.40 * 68.05 68.67 (+0.27) 69.29 ±0.16 (+0.89) ResNet-50 4w4a 08.38 53.02 * 52.12 51.80 59.25 (+6.24) 66.25 ±0.90 (+13.23) (77.73) 5w5a 48.12 73.38 * 71.89 70.99 71.57 (-1.81) 75.32 ±0.09 (+1.94) MobileNetV2 4w4a 10.96 00.10 † 59.43 * 57.23 59.99 (+0.56) 61.62 ±0.39 (+2.19) (73.03) 5w5a 59.88 62.35 68.11 * 67.61 68.83 (+0.72) 70.45 ±0.07 (+2.34) * Table 2 : 2Comparison on data-free quantization schemesDataset Model (FP32 Acc.) Bits GDFQ Qimera + GDFQ ZAQ Qimera + ZAQ AutoReCon Qimera + AutoReCon Cifar-10 ResNet-20 4w4a 90.25 91.26 (+1.01) 92.13 93.91 (+1.78) 88.55 91.16 (+2.61) (93.89) 5w5a 93.38 93.46 (+0.08) 93.36 93.84 (+0.48) 92.88 93.42 (+0.54) Cifar-100 ResNet-20 4w4a 63.39 65.10 (+1.71) 60.42 69.30 (+8.88) 62.76 65.33 (+2.57) (70.33) 5w5a 66.12 69.02 (+2.90) 68.70 69.58 (+0.88) 68.40 68.80 (+0.40) Table 3 : 3Performance of Qimera implemented on top of GDFQ, ZAQ and AutoReCon modifications have been made on the GDFQ baseline for applying Mixup and Cutmix. We report top-1 accuracy for each experiment. The numbers inside the parentheses of Qimera results are improvements over the highest baseline. Table 4 : 4Ablation study and Sensitivity analysisMethod Cifar-10 Cifar-100 Dist. Ratio Intrusion Dist. Ratio Intrusion Mixup 2.44 0.800 3.14 0.400 SE Only 1.58 0.00260 1.64 0.053 SE+DM 1.67 0.00073 1.59 0.044 SE+DM+EEI 1.57 0.00013 1.52 0.029 Table 5 : 5Embedding Distance Ratio and Intrusion Score Table 6 : 6Experimental results on noise variance test B.2 Extracted Embedding Initialization without Training Highest among the baselines † Did not convergeDataset Model (FP32 Acc.) Bits ZeroQ ZAQ GDFQ Qimera Extracted Init + Freeze Cifar-10 ResNet-20 4w4a 79.30 92.13 * 90.25 91.26 90.37 (93.89) 5w5a 91.34 93.36 93.38 * 93.46 93.25 Cifar-100 ResNet-20 4w4a 47.45 60.42 63.39 * 65.10 63.83 (70.33) 5w5a 65.61 68.70 * 66.12 69.02 68.76 ImageNet ResNet-18 4w4a 22.58 52.64 60.60 * 63.84 63.67 (71.47) 5w5a 59.26 64.54 68.40 * 69.29 69.23 ResNet-50 4w4a 08.38 53.02 * 52.12 66.25 63.20 (77.73) 5w5a 48.12 73.38 * 71.89 75.32 74.84 MobileNetV2 4w4a 10.96 00.10 † 59.43 * 61.62 60.46 (73.03) 5w5a 59.88 62.35 68.11 * 70.45 68.82 * Table 7 : 7Extracted embedding initialization without Training Table 8 : 8Sensitivity Study on Number of DM LayersTo have a deeper look into the DM layers, we have conducted a sensitivity study on the number of DM layers inTable 8. In the table, all results are from 4w4a setting with p=0.4, K=2 for Cifar-10 and K=10 for Cifar-100. As displayed, we found that there are sometimes small improvements from using more DM layers above one, but a severe drop in performance has been observed for using too many layers(Cifar-100, 8 layers). B.4 More Sensitivity Study on Hyperparameters 64.18 64.62 64.90 64.95 64.76 64.89 10 64.85 64.63 65.10 64.76 64.52 63.86 25 64.53 64.91 64.72 64.66 64.40 64.01 100 64.37 64.66 64.64 64.27 64.79 63.65 ImageNet (ResNet-50) 100 58.74 60.64 61.43 61.47 63.87 65.73 250 61.30 61.28 62.16 64.03 64.50 65.23 500 58.96 60.11 58.69 63.05 66.25 66.19 1000 58.65 59.62 61.20 58.86 65.12 64.24Dataset K p 0.10 0.25 0.40 0.55 0.70 0.85 Cifar-100 (ResNet-20) 2 Table 10 : 10Comparison with DSG generate boundary supporting samples, instead of relying on diversification. This would led to better performance as demonstrated in the motivational experiment of Section 3. Acknowledgments and Disclosure of Funding In-datacenter performance analysis of a tensor processing unit. 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[ "https://github.com/iamkanghyunchoi/qimera.", "https://github.com/iamkanghyunchoi/qimera," ]
[ "Models and Datasets for Cross-Lingual Summarisation", "Models and Datasets for Cross-Lingual Summarisation" ]
[ "Laura Perez-Beltrachini \nInstitute for Language, Cognition and Computation School of Informatics\nUniversity of Edinburgh\n10 Crichton StreetEH8 9ABEdinburgh\n", "Mirella Lapata \nInstitute for Language, Cognition and Computation School of Informatics\nUniversity of Edinburgh\n10 Crichton StreetEH8 9ABEdinburgh\n" ]
[ "Institute for Language, Cognition and Computation School of Informatics\nUniversity of Edinburgh\n10 Crichton StreetEH8 9ABEdinburgh", "Institute for Language, Cognition and Computation School of Informatics\nUniversity of Edinburgh\n10 Crichton StreetEH8 9ABEdinburgh" ]
[ "Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing" ]
We present a cross-lingual summarisation corpus with long documents in a source language associated with multi-sentence summaries in a target language. The corpus covers twelve language pairs and directions for four European languages, namely Czech, English, French and German, and the methodology for its creation can be applied to several other languages. We derive cross-lingual document-summary instances from Wikipedia by combining lead paragraphs and articles' bodies from language aligned Wikipedia titles. We analyse the proposed cross-lingual summarisation task with automatic metrics and validate it with a human study. To illustrate the utility of our dataset we report experiments with multi-lingual pretrained models in supervised, zero-and fewshot, and out-of-domain scenarios.
10.18653/v1/2021.emnlp-main.742
[ "https://www.aclanthology.org/2021.emnlp-main.742.pdf" ]
243,865,371
2202.09583
936f5bca0e3c3f68513e00acd335cdbb7be28c4e
Models and Datasets for Cross-Lingual Summarisation November 7-11, 2021 Laura Perez-Beltrachini Institute for Language, Cognition and Computation School of Informatics University of Edinburgh 10 Crichton StreetEH8 9ABEdinburgh Mirella Lapata Institute for Language, Cognition and Computation School of Informatics University of Edinburgh 10 Crichton StreetEH8 9ABEdinburgh Models and Datasets for Cross-Lingual Summarisation Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing the 2021 Conference on Empirical Methods in Natural Language ProcessingNovember 7-11, 20219408 We present a cross-lingual summarisation corpus with long documents in a source language associated with multi-sentence summaries in a target language. The corpus covers twelve language pairs and directions for four European languages, namely Czech, English, French and German, and the methodology for its creation can be applied to several other languages. We derive cross-lingual document-summary instances from Wikipedia by combining lead paragraphs and articles' bodies from language aligned Wikipedia titles. We analyse the proposed cross-lingual summarisation task with automatic metrics and validate it with a human study. To illustrate the utility of our dataset we report experiments with multi-lingual pretrained models in supervised, zero-and fewshot, and out-of-domain scenarios. Introduction Given a document in a source language (e.g., French), cross-lingual summarisation aims to produce a summary in a different target language (e.g., English). The practical benefits of this task are twofold: it not only provides rapid access to salient content, but also enables the dissemination of relevant content across speakers of other languages. For instance, providing summaries of articles from French or German newspapers to non-French or non-German speakers; or enabling access to summary descriptions of goods, services, or knowledge available online in foreign languages. Figure 1 shows an example of an input document in French (left) and its summary in English and other languages (right). Recent years have witnessed increased interest in abstractive summarisation (Rush et al., 2015;Zhang et al., 2020) thanks to the popularity of neural network models and the availability of datasets (Sandhaus, 2008;Hermann et al., 2015;Grusky et al., 2018) containing hundreds of thousands of document-summary pairs. Although initial efforts have overwhelmingly focused on English, more recently, with the advent of cross-lingual representations (Ruder et al., 2019) and large pre-trained models (Devlin et al., 2019;Liu et al., 2020), research on multi-lingual summarisation (i.e., building monolingual summarisation systems for different languages) has been gaining momentum (Chi et al., 2020b;Scialom et al., 2020). While creating large-scale multi-lingual summarisation datasets has proven feasible (Straka et al., 2018;Scialom et al., 2020), at least for the news domain, cross-lingual datasets are more difficult to obtain. In contrast to monolingual summarisation, naturally occurring documents in a source language paired with summaries in different target languages are rare. For this reason, existing approaches either create large-scale synthetic data using back-translation (Zhu et al., 2019;Cao et al., 2020), translate the input documents (Ouyang et al., 2019), or build document-summary pairs from social media annotations and crowd-sourcing (Nguyen and Daumé III, 2019). Recent efforts (Ladhak et al., 2020) have been directed at the creation of a large-scale cross-lingual dataset in the domain of how-to guides. Despite being a valuable resource, how-to guides are by nature relatively short documents (391 tokens on average) and their summaries limited to brief instructional sentences (mostly commands). To further push research on cross-lingual summarisation, we propose a large dataset with document-summary pairs in four languages: Czech, English, French, and German. 1 Inspired by past research on monolingual descriptive summarisation (Sauper and Barzilay, 2009;Zopf, 2018;Liu et al., 2018;Liu and Lapata, 2019a;Perez-Beltrachini et al., 2019;Hayashi et al., 2021), we derive cross- lingual datasets from Wikipedia 2 , which we collectively refer to as XWikis. We exploit Wikipedia's Interlanguage links and assume that given any two related Wikipedia titles, e.g., Huile d'Olive (French) and Olive Oil (English), we can pair the the lead paragraph from one title with the body of the other. We assume that the lead paragraph can stand as the summary of the article (see Figure 1). Our dataset covers different language pairs and enables different summarisation scenarios with respect to: degree of supervision (supervised, zeroand few-shot), combination of languages (crosslingual and multi-lingual), and language resources (high-and low-resource). To illustrate the utility of our dataset we report experiments on supervised, zero-shot, few-shot, and out-of-domain cross-lingual summarisation. For the out-of-domain setting, we introduce Voxeurop, a cross-lingual news dataset. 3 In experiments, following recent work (Ladhak et al., 2020), we focus on All-to-English summarisation. In addition to assessing supervised and zero-shot performance of multilingual pre-trained models (Liu et al., 2020;Tang et al., 2020), we also provide a training mechanism for few-shot cross-lingual summarisation. 4 The XWikis Corpus Wikipedia articles are organised into two main parts, a lead section and a body. For a given Wikipedia title, the lead section provides an overview conveying salient information, while the body provides detailed information. Indeed, the body is a long multi-paragraph text generally structured into sections discussing different aspects of the Wikipedia title. We can thus consider the body and lead paragraph as a document-summary pair. Furthermore, a Wikipedia title can be associated with Wikipedia articles in various languages also composed by a lead section and body. Based on this insight, we propose the cross-lingual abstractive document summarisation task of generating an overview summary in a target language Y from a long structured input document in a source language X. Figure 1 illustrates this with an example. For the Wikipedia title Huile d'Olive (Olive Oil), it shows the French document on the left and overview summaries in German, French, Czech, and English on the right. Below, we describe how our dataset was created, analyse its main features (Section 2.1), and present a human validation study (Section 2.2). Cross-Lingual Summarisation Pairs From a set of Wikipedia titles with articles (i.e., lead paragraph and body) in N languages, we can create N ! (N −2)! cross-lingual summarisation sets D X →Y , considering all possible language pairs and directions. Data points (Doc X , Sum Y ) in D X →Y are created, as discussed in the previous section, by combining the body of articles for titles t X in language X with the lead paragraph of articles for corresponding titles t Y in language Y . In this work, we focus on four languages, namely English (en), German (de), French (fr), and Czech (cs). To create such summarisation sets D X →Y , we first use Wikipedia Interlanguage Links to align titles across languages, i.e., align title t X ∈ X with t Y ∈ Y . 5 Then, from the aligned titles t X − t Y , we retain those whose articles permit creating a data point (Doc X , Sum Y ). In other words, t X 's article body and t Y 's lead section should obey the following length restrictions: a) the body should be between 250 and 5,000 tokens long and b) and the lead between 20 and 400 tokens. Table 1 shows the number of instances in each set D X →Y that we created following this procedure. Wikipedia titles exist in different language subsets, thus, language sets D X →Y will include different sets of titles. For better comparison in the evaluation of our models, we would like to have exactly the same set of titles. To achieve this, we take 7,000 titles in the intersection across all language sets. We call this subset XWikis-parallel and the sets with remaining instances XWikis-comparable. For further details about the data collection process, see the Appendix A. Monolingual Summarisation Data A byproduct of our data extraction process is the creation of multi-lingual summarisation data. Each D X →Y set has its corresponding monolingual D X →X version. Data points (Doc X , Sum X ) in D X →X are created by combining the body of articles for titles t X in language X with the lead paragraph of articles in the same language X. Features of XWikis Dataset Comparison with Existing Datasets Our dataset departs from existing datasets in terms of size, summarisation task, and potential for extension to additional languages. statistics for our XWikis corpus and existing datasets. Our dataset is larger in terms of number of document-summary pairs. WikiLingua (Ladhak et al., 2020) is also larger than previous datasets, in terms of number of instances, however, the summarisation task is different. In XWikis, the input documents are more than twice as long (average number of tokens). As for the number of languages, although in this work we focus on four European ones, the proposed data creation approach allows to extend the dataset to a large number of languages including more distant pairs (e.g., English-Chinese), as well as low-resource and understudied languages (e.g., Gujarati and Quechua). Summarisation Task We carry out a detailed analysis of our XWikis corpus to characterise the summarisation task it represents and assess the validity of the created summarisation data points (Doc X , Sum Y ). In the first instance, we do this through automatic metrics. Since metrics that are based on token overlap (Grusky et al., 2018;Narayan et al., 2018) cannot be directly applied to our cross-lingual data, we carry out some automatic analysis on the monolingual version of the corpus instead, i.e., (Doc X , Sum X ) instances. We first validate the assumption that the lead paragraph can serve as a summary for the article body. Table 3 provides statistics per language pair, for XWikiscomparable 6 , and averaged over all language pairs for XWikis-parallel. Size. The top part of Table 3 provides an overview of the summarisation task in terms of size. The documents are long, with an overall average of 952 tokens, 40 sentences (note that sentence length is thus~23 tokens) and 6 sections. , 2021). We analyse the average number of sections per document as a proxy for the complexity of the content selection sub-task. A summariser will need to learn which aspects are summary-worthy and extract content from different sections in the input document. Summaries are also long with 60 tokens and 3 sentences on average. Content Diversity. To assess the diversity of content in the corpus, we report the number of distinct top level section titles as an approximation (without doing any normalisation) of aspects discussed (Hayashi et al., 2021). These high numbers, together with the average number of sections per document, confirm that our dataset represents multi-topic content. Level of Abstraction. To characterise the summarisation task in terms of level of abstraction, we analyse content overlap of document-summary pairs using automatic metrics (Grusky et al., 2018;Narayan et al., 2018) and then evaluate the performance of two extractive summarisation ap-proaches. 7 When the summarisation task is extractive in nature (i.e., the summaries copy text spans from the input document), extractive methods ought to perform well. The set of automatic metrics proposed in Grusky et al. (2018), indicates the extent to which a summary is composed by textual fragments from the input document, i.e., extractive fragments. Coverage, measures the average number of tokens in the summary that are part of an extractive fragment; Density, indicates the average length of the set of extractive fragments. As shown in Table 3, Coverage is high, specially for de and fr sets, while Density is quite low. This indicates that the summaries overlap in content with the input documents but not with the same phrases. Although summaries are not short, the compression ratio is high given the size of the input documents. This highlights the rather extreme content selection and aggregation imposed by the summarisation task. The second set of metrics proposed in Narayan et al. (2018), measures the percentage of new n-grams appearing in the summary (i.e., not seen in the input document), and shows a similar trend. The percentage of novel unigrams is low but increases sharply for higher ngrams. The last two rows in Table 3 report ROUGE-L for two extractive methods. LEAD creates a summary by copying the first K tokens of the input document, where K is the size of the reference and performs well when the summarisation task is biased to content appearing in the first sentences of the document. EXT-ORACLE selects a subset of sentences that maximize ROUGE-2 (Lin, 2004) with respect to the reference summaries (Nallapati et al., 2017;Narayan et al., 2018) and performs well when the summarisation task is mostly extractive. As we can see, LEAD is well below EXT-ORACLE (~4 ROUGE-L points on average), indicating no lead bias (i.e., summary-worthy content is not in the beginning of the document). EXT-ORACLE performs better, however, considering the high levels of Coverage, it does not seem to cover all salient content. This indicates that important content is scattered across the document in different sentences (not all of which are selected by EXT-ORACLE) and that phrasing is different (see jump from % of novel unigrams to bigrams). The French subset, has the highest Coverage (conversely the lower % of novel unigrams), and thus is more amenable to the extractive methods. Validation through Human Evaluation To further complement automatic evaluation, we carried out a human evaluation study to assess the quality of cross-lingual data instances (Doc X , Sum Y ). In other words, we validate the assumption that given a pair of aligned titles t X − t Y , the lead paragraph in language Y is a valid overview summary of the document body in language X. As this evaluation requires bilingual judges, we selected three language pairs, namely D de→en , D f r→en and D cs→en and recruited three judges per pair, i.e., bilingual in German-English, French-English, and Czech-English. We selected 20 data instances from each set and asked participants to give an overall judgement of summary adequacy. Specifically, they were asked to provide a yes/no answer to the question Does the summary provide a general overview of the Wikipedia title?. In addition, we elicited more fine-grained judgments by asking participants to ascertain for each sentence in the summary whether it was supported by the document. We elicited yes/no answers to the question Does the sentence contain facts that are supported by the document?. We expect judges to answer no when the content of a sentence is not discussed in the document and yes otherwise. Table 4 shows the proportion of yes answers given by our judges for the three language pairs. Overall, judges view the summary as an acceptable overview of the Wikipedia title and its document. The same picture emerges when considering the more fine-grained sentence-based judgments. 66.2% of the summary sentences are supported by the document in the German-English pair, 77.4% for French-English, and 60.5% for Czech-English. We also used Fleiss's Kappa to establish inter-annotator agreement between our judges. This was 0.48 for German-English speakers, 0.55 for French-English, and 0.59 for Czech-English. All-to-English Summarisation Task Following previous work (Ladhak et al., 2020), the specific cross-lingual task that we address is generating English summaries from input documents in different (source) languages. In the context of cross-lingual summarisation, we assume that a) we have enough data to train a monolingual summarizer in a source language; b) we want to port this summarizer to a different target language without additional data (zero-shot) or a handful of training examples (few-shot); and c) the representations learnt by the monolingual summarizer to carry out the task, i.e., select relevant content and organise it in a short coherent text, should transfer or adapt to the cross-lingual summarisation task. The main challenges in this setting are understanding the input documents in a new language which may have new relevance clues and translating them into the target language. Specifically, we assume we have access to monolingual English data (Doc en , Sum en ) to learn an English summariser, and we study the zero-and few-shot cross-lingual scenarios when the input to this model is in a language other than English (i.e., German, French, and Czech). We further exploit the fact that our XWikis corpus allows us to learn cross-lingual summarisation models in a fully supervised setting, and establish comparisons against models with weaker supervision signals. Our fully supervised models follow state-of-the-art approaches based on Transformers and pre-training (Liu and Lapata, 2019b; Lewis et al., 2020). We simulate zero-and few-shot scenarios by considering subsets of the available data instances. Approach We formalise cross-lingual abstractive summarisation as follows. Given input document Doc X in language X represented as a sequence of tokens x = (x 1 · · · x |x| ), our task is to generate Sum Y in language Y . The target summary is also represented as sequence of tokens y = (y 1 · · · y |y| ) and generated token-by-token conditioning on x by a summarisation model p θ as |y| t=1 p θ (y t |y 1..t−1 , x). Our summarisation model is based on mBART50 (Tang et al., 2020), a pre-trained multi-lingual sequence-to-sequence model. mBART50 (Tang et al., 2020) is the result of fine-tuning mBART (Liu et al., 2020) with a multi-lingual machine translation objective (i.e., fine-tuning with several lan-guage pairs at the same time). The fine-tuning process extends the number of languages from 25 to 50. BART (Liu et al., 2020) follows a Transformer encoder-decoder architecture (Vaswani et al., 2017). It was trained on a collection of monolingual documents in 25 different languages to reconstruct noised input sequences which were obtained by replacing spans of text with a mask token or permuting the order of sentences in the input. Although pre-trained models like mBART50 provide multi-lingual representations for language understanding and generation, they require adjustments in order to be useful for abstractive summarisation. Given a training dataset D with document-summary instances {x n , y n } |D| n=1 starting from a model with parameters θ given by mMBART50, we fine-tune to minimise the negative log likelihood on the training dataset, L N LL = − 1 |D| |D| n=1 log p θ (y n |x n ). If D is in- stantiated by a cross-lingual dataset (i.e., D X→Y ) we directly fine-tune on the target cross-lingual task. However, in our zero and few-shot settings we only have monolingual summarisation data available. We therefore assume D to be an English monolingual set (i.e., D en→en ). In the zero-shot scenario, a monolingual summariser English summariser is used for crosslingual summarisation and we assume that the parameters of the English model will be shared to a certain extent across languages (Chi et al., 2020a). In the few-shot scenario, we assume that in addition to monolingual summarisation data, we also have access to a small dataset S X→en with cross-lingual summarisation examples. Although it might be possible to curate cross-lingual summaries for a small number of examples, using these in practice for additional model adaptation can be challenging. In this work propose an approach reminiscent of the few-shot Model Agnostic Meta-Learning (MAML) algorithm (Finn et al., 2017). MAML is an optimisation-based learning-tolearn algorithm which involves meta-training and meta-testing phases. Meta-training encourages learning representations which are useful across a set of different tasks and can be easily adapted, i.e., with a few data instances and a few parameter updates, to an unseen task during meta-testing. More concretely, meta-training consists of nested optimisation iterations: inner iterations take the (meta) model parameters θ meta and compute for each task T i a new set of parameters θ i . In the outer iteration, the (meta) model parameters are updated according to the sum of each task T i loss on task-specific parameters θ i . 8 At test time, the (meta) model parameters can be adapted to a new task with one learning step using the small dataset associated with the new task. We assume that the multi-lingual and MT pretraining of mBART50 (and mBART) are a form of meta-training involving several language tasks which learn shared representations across different languages. We then adapt the English monolingual summariser to the cross-lingual task T X→en with a small set of instances S X→en . We perform a single outer loop iteration and instead of taking a copy of the (meta) parameters and updating them after the inner loop, we combine the support set with a monolingual sample of similar size. We call this method light-weight First Order MAML (LF-MAML). We also observe that in a real-world scenario, in addition to the small set with cross-lingual examples S X→en , there may exist documents in the source language Doc X without corresponding summaries in English. To further train the model with additional unlabelled data, we apply a Cross-View Training technique (CVT; Clark et al. 2018). We exploit the fact that our fine-tuning does not start from scratch but rather from a pre-trained model which already generates output sequences of at least minimal quality. We augment the set of document summary pairs x, y in S X→en with instanceŝ x,ŷ whereŷ is generated by the current model and x is a different view of x. We cheaply create different views from input x by taking different layers from the encoder. Experimental Setup Datasets and Splits We work with the D de→en , D f r→en , and D cs→en directions of our XWikis corpus (i.e., first column in Table 1) and evaluate model performance on the XWikis-parallel set. We split XWikis-comparable into training (95%) and validation (5%) sets. To train an English monolingual summariser, we created a monolingual dataset D en→en following the procedure described in Section 2 (lead paragraph and body of Wikipedia articles). We selected a set of Wikipedia titles disjoint from those Table 5: ROUGE-L recall for source document against reference monolingual summary computed against all input tokens (All), the first 800 tokens and the 600 tokens extracted with paragraph-based LEXRANK. in our XWikis corpus. This dataset has 300,000 instances with 90/5/5 percent of instances in training/validation/test subsets. It follows similar characteristics to the data in our XWikis corpus with an average document and summary length of 884 and 70 tokens, respectively. Paragraph Extraction To deal with very long documents, we carry out an initial extractive step (Liu et al., 2018;Liu and Lapata, 2019a). Specifically, we rank document paragraphs (represented as vectors of their tf-idf values) using LEXRANK (Erkan and Radev, 2004) and then select the top ranked paragraphs up to a budget of 600 tokens. Table 5 reports ROUGE-L recall of the input against the reference summary (note that to measure this we take the monolingual summary associated with the document rather than the cross-lingual one). As can be seen, the extractive step reduces the document to a manageable size without sacrificing too much content. Note that after ranking, selected paragraphs are kept in their original position to avoid creating a bias towards important information coming at the beginning of the input sequence. Out of Domain Data To evaluate the robustness of cross-lingual models on non-Wikipedia text, we created an out of domain dataset from the European news site Voxeurop. This site contains news articles composed of a summary section (with multisentence summaries) and a body written and translated into several languages by professional journalists and translators. We extracted from this site 2,666 summary-article pairs in German, French, Czech, and English. The average document length in tokens is 842 and the summary length 42. We used 2,000 instances for evaluation and reserved the rest for model adaptation. Models We evaluated a range of extractive and abstractive summarisation models detailed below. In cases where translation is required we used the Google API. 9 Extractive We applied extractive approaches on the source documents. Extracted sentences were the translated into English to create a summary in the target language. 1. EXT-ORACLE This extractive approach builds summaries by greedily selecting sentences from the input that together maximize ROUGE-2 against the reference summary. We implemented this upper bound following Nallapati et al. (2017)'s procedure. For datasets D X→en , we take the monolingual summary associated to the input document as a proxy for ROUGE-based selection. 2. LEAD The first K tokens from the input document are selected where K is the length of the reference summary. 3. LEXRANK This approach uses tf-idf graphbased sentence ranking (Erkan and Radev, 2004) to select sentences from the input and then takes first K tokens (where K is the length of the reference summary). Supervised We fine-tuned three separate models based on mBART (Liu et al., 2020) and mBART50 (Tang et al., 2020) in a supervised fashion on the three cross-lingual datasets (D de→en , D f r→en , and D cs→en ). This provides an upper-bound on achievable performance. Additionally, we trained an English summariser on the separate English dataset D en→en (described in the previous section) for our zero and few-shot scenarios. Translated This is a translate and summarise pipeline approach. We first translate the input documents Doc de , Doc f r , and Doc cs into English and then apply a monolingual English summariser. Zero-Shot A monolingual English summariser is directly applied to summarise Doc de , Doc f r , and Doc cs documents into English. We fine-tune the entire network except the embedding layer. We report experiments with mBART50 (and mBART). Few-Shot These models are based on fine-tuned monolingual English summarisers subsequently adapted to the cross-lingual task with a small set of examples S X→en . We present experiments with mBART and mBART50 pre-trained models. We evaluate three few-shot variants (see Section 3.2). LF-MAML is the light-weight First Order MAML version, FT is a fine-tuned version where only cross-attention and layer normalisation layers are fine-tuned, and CVT incorporates additional unlabelled instances into the adaptation step. We also consider two settings with |S X→en | being 300 and 1,000 few instances. Note that in each case we take 1/3 for validation, and the rest for training. For CVT, we generate two views,x m andx u , for each input document x in S X→en by taking a middle encoder representation (x m the hidden states at layer 6) and another by taking an upper encoder representation (x u the hidden states at layer 11). Intuitively, these provide different levels of abstraction from the input document. Results and Analysis In this section we discuss our cross-lingual summarisation results (Table 6 and Table 7 Table 6). This gap is highest when summarizing from Czech to English. Can we Beat Machine Translation? In agreement with previous work (Ladhak et al., 2020), we find that Supervised models are better than Translated ones. Zero versions with mBART50 perform slightly below Translated, except for German-to-English (this is more surprising for mBART which has not seen any cross-language links during pretraining). Interestingly, Few with mBART50 and 300 training instances achieves comparable performance, which indicates that the summariser can improve on the new cross-lingual task by seeing only a few examples. We observe a similar trend for mBART even though it never sees any crosslingual examples during pre-training. Which Few-Shot Model is Better? FL-MAML performs well across languages both in the 300 and 1K settings. Indeed, in this last configuration it beats Translated and gets closer to Supervised using a relatively small training set (~600 instances -the rest is used for validation). The performance of FT and CVT variants varies depending on the language. FT (which only fine-tunes crossattention) helps when summarizing from French whereas CVT helps when summarizing from Czech. The latter model benefits from potentially noisier unlabelled instances. Is Out-of-Domain Summarisation Feasible? Table 7 shows the performance of a monolingual English summariser (trained on XWikis) and tested on the Voxeurop dataset. There is indeed a penalty for domain shift by approximately 10 ROUGE points (compare row Zero in Table 7 with rows Supervised/Zero in Table 6). Overall, Few-shot manages to improve upon Zero-shot, even though the few training instances come from a more distant distribution than the one used to pre-train the monolingual summariser (i.e., different genres). Which Pre-trained Model? Our experiments identify mBART as the weakest pre-trained model, reporting lower ROUGE scores across languages, domains, and training settings (e.g., supervised, zero-and few-shot). mBART50 benefits from finetuning on machine translation and this knowledge is useful to our summarisation task. Are there Differences between Languages? In the XWikis corpus (and mostly with mBART) Czech-to-English has the lowest performance. However, this gap disappears when applying Few-shot variants to the summarisation task. In Voxeurop, there are no discernible differences amongst language pairs; this is probably due to the fact that document-summary pairs are translations across languages. How Hard is Cross-lingual Summarisation? The task is very challenging! XWikis documents are long, and summarisation models must be able to represent multi-paragraph text adequately and isolate important content which is interspersed through the document. This difficulty is further compounded by the translation of content in different languages and the need for models to abstract, rephrase, and aggregate information. Our results in Tables 6 and 7 show that there is plenty of room for improvement. Conclusion We presented a new summarisation dataset in four languages (German, French, Czech, and English) which we hope will be a valuable resource for crosslingual and monolingual summarisation. We evaluated a wide range of models on the cross-lingual summarisation task, including zero-and few-shot variants some of which show promising results. Future work directions are many and varied. We would like to further investigate MAML variants for few-shot summarisation, and expand on document views for CVT (e.g., by looking at semantic roles and discourse relations). A The XWikis Corpus Dataset Creation Our corpus was created with English, German, French and Czech Wikipedia dumps from June 2020. 10 We adapted Wikiextractor (Attardi, 2015) to obtain the lead section and body of Wikipedia articles. We preserved the structure of the input document, and section mark-ups were kept (e.g., <h2>). We used a dump of the same date for the table containing the Wikipedia Interlanguage Links. 11 We performed text normalisation (a variant of NFKC normalization) with sentence-piece (Kudo and Richardson, 2018). B Experiments All our models were built on top of the fairseq library (Ott et al., 2019) code base. Text Processing For sentence splitting and tokenisation in German, French and English, we used the Stanza Python NLP Package (Qi et al., 2020). For Czech, we used the MorphoDiTa package (Straka and Straková, 2016). Training Details For mBART50 (Tang et al., 2020), we used the checkpoint provided as mMBART 50 finetuned many-to-many and for mBART the mBART.cc25 checkpoint, both available in the fairseq library (Ott et al., 2019). We reused mBART's 250K sentencepiece (Kudo and Richardson, 2018) model which was trained using monolingual data for 100 languages. However, to reduce the size of the model to fit our GPU availability we carried out the following modifications. We trimmed the vocabulary to 135K. We first applied the sentencepiece encoder to the language sets in our XWikis corpus (Table 1) and the English data (used to train the monolingual summariser D en→en ) to generate a reduced dictionary. Then, we trimmed the dictionary and the models' embeddings (taking care to map indices from the original dictionary to the reduced one). We further slimmed-down the position embeddings layer from 1,024 to 600. Supervised fine-tuning of mBART and mBART50 was carried out for 20K updates with a batch size of 80 instances, following previous work (Lewis et al., 2020;Liu et al., 2020). We used the Adam optimizer ( =1e-6 and β 2 =0.98) with linear learning rate decay scheduling. We set dropout rate to 0.3 and attention-dropout to 0.1. We used half precision (fp16) and additionally set the weight decay to 0.01 and clipped gradients norm to 0.1. We fine-tuned with label smoothing and α=0.2. When fine-tuning on English mono-lingual summarisation, we freeze the embedding layer for mBART50 as it showed better zero-shot results (but not for mBART as zero shot results were not improved). We used 4 GPUs with 12GB of memory, fine-tuning took 2 days of training. For the few-shot adaptation, we kept similar hyperparameters, except that we used a much smaller batch size, i.e., 8 instances, and ran 1K updates (300 few-shot) and 5k (1k few-shot). We monitored validation perplexity and obtained checkpoints with best perplexity. All few-shot variants used 1 GPU with 12GB of memory (and needed less than 10 hours of training). For the Few approaches, we sampled a subset of English S en→en instances of size similar to the support set S X→en of the adaptation task T X→en and doubled its size when in addition applying CVT. The sample of unlabelled CVT instances had also size similar to the task support set. Adding more unlabelled data for CVT hurts performance. We combined data from the three tasks, English monolingual, Few cross-lingual instances (task support set) and unlabelled cross-lingual instances. We computed a weighted loss with weights 0.5, 1 and 0.1 respectively (note that variants with no CVT have 0 in the third weight). We followed the same instance formatting as used in Liu et al. (2020). We use special language ID tokens <LID>, postpend sentences with the </S>, and prepend <S> at the beginning of each sequence. Tables 8 and 9 are the extended versions of Tables 6 and 7 in the paper. Here, we report ROUGE-1/2/L F1 metrics. C Results Full Set of Metrics and Results Example Outputs Tables 10, 11, and 12 show example outputs by mBART50 model variants for the three language pairs German-English, French-English, and Czech-English, respectively. Table 13 shows example outputs for the different mBART50 model variants on the Voxeurop dataset. Les Italiens, à qui il était demandé si le retour au nucléaire était une voie praticable compte tenu des coûts, du facteur temps et des risques, ont dans leur grande majorité, définitivement exclu cette éventualité, pour la seconde fois en un quart de siècle. Ce deuxième "non" au nucléaire impose une vaste réflexion qui ne se limite pas à gérer les problèmes immédiats que le referendum a imposés. [...] On notera que le choix de l'Allemagne n'est pas dicté seulement par la peur du présent ou par une angoisse intellectuelle qui a ses racines dans sa propre histoire : c'est un pays qui, avant de dire adieu au nucléaire, investit depuis au moins une vingtaine d'années dans les énergies renouvelables et qui, ces huit dernières années, a vu doubler les emplois dans ce secteur. En termes d'expérience et de stratégies industrielles, il peut être utile d'en tenir compte. En ce qui concerne la France, en dépit de ses 58 réacteurs et de ses projets de centrales de nouvelle génération, il faut avoir présent à l'esprit qu'après Fukushima et après la décision allemande, un fort pourcentage de Français s'est déclaré favorable à une révision de la politique de l'atome. Le président Sarkozy, tout en réaffirmant, après le désastre japonais, le choix historique du général De Gaulle, avait créé au début de son mandat un grand ministère de l'Ecologie, en lui donnant pour mission d'élargir le champ des énergies renouvelables et de diminuer la dépendance envers le nucléaire. Conservateur comme Angela Merkel, Nicolas Sarkozy, a compris que le "renouvelable" est aussi un marché et que les partis traditionnels risquent gros face aux mouvements écologique et antinucléaire. Les Verts français ont inséré la question de l'énergie nucléaire dans leur programme d'alliance pour 2012 avec les socialistes (en majorité pro-nucléaires). L'Europe dénucléarisée reste une utopie En matière énergétique, les choix stratégiques nationaux sont et seront prédominants dans une vision d'ensemble européenne, mais si deux puissances in dustrielles telles que l'Italie et l'Allemagne, membres du G8 et pays fondateurs de l'Europe, abandonnent le nucléaire, il n'est pas illusoire de considérer que ce choix va exercer une forte incitation au changement et aura une grande influence sur les opinions publiques des autres pays. Le nouvel axe anti-atomique Abandon de l'atome en Allemagne, puis rejet du retour au nucléaire en Italie : le volte-face de deux membres fondateurs de l'UE pourrait pousser les autres Etats membres à tourner la page du nucléaire et à miser sur les énergies renouvelables. Erst der Atomausstieg Deutschlands, dann die Ablehnung einer Rückkehr zur Atomenergie in Italien: Dieser Sinneswandel zweier EU-Gründungsmitglieder könnte die übrigen Mitgliedsstaaten dazu bewegen, sich endgültig von der Kernkraft zu verabschieden und künftig auf erneuerbare Energien zu setzen. De Germany is phasing out nuclear power and Italy has rejected its reintroduction. This about-face by two founding members of the European Union could encourage other member states to turn the nuclear page and to develop renewable energies. Německo a Itálie rozhodly vzdát se jaderné energie -radikální obrat v pozicích dvou zakládajících členů EU by mohl přimět další členské státy k odklonu od jádra a zaměřit se na obnovitelné zdroje. Reference "Never Smile at a Crocodile" is a comic song with music by Frank Churchill and lyrics by Jack Lawrence. The music, without the lyrics, was first heard in the Walt Disney animated film "Peter Pan". Following the film's release in 1953, the sung version with Lawrence's lyrics went on to become a children's song classic. En Supervised "Never Smile at a Crocodile" is a song by American singer-songwriter Charles Churchill. It was first released on the Disney soundtrack to the 1937 Disney film "Blanche-Neige et les Sept Nains" in 1939, when "Peter Pan" was already in the planning stages. However, Disney decided to discontinue the recording until 1949. Translated "Never Smile at a Crocodile" is a song from the 1937 Disney film "Snow White and the Seven Dwarfs". It was composed by Charles Churchill, who had composed most of the soundtrack for the 1937 film, "Peter Pan". However, the original lyrics were later recorded by several singers, including Jerry Lewis and Rolf Harris. The song became one of the top ten best-selling children's songs that year. Zero "Never Smile at a Crocodile" is a song by American singer-songwriter John Churchill. It was first released in 1953 as the soundtrack to the 1937 Disney film, "Blanche-Neige and the Seven Nains". The song was later re-released as a CD in 1997. FT "Never Smile at a Crocodile" is a song by American singer-songwriter John Churchill. It was first released in 1939 as the soundtrack to the 1937 Disney film "Blanche-Neige and the Seven Nains". The song was later re-released as the lead single from the 1953 film "Peter Pan". Table 11: Example with mBART50 based models outputs from the validation set for French-to-English. Gold One in every five young Europeans is out of a job, and even one in two in some countries. Numbers like these were enough to have the young generation rebel against governments in the Arab world, remarks a Polish columnist. What will happen if our social model deprives young people of all hope? en ORACLE For many international education experts, a university education -bachelor or master's degree, doctorate -is the measure of all things. And it is true that the time-frame may not be ideal, as the German system is strongly dependent on the economy. en LEAD More than 5.5m young Europeans are without jobs. In the crisis countries in southern Europe, a generation is coming of age with few prospects: one in two Spaniards and Greeks under 25 are unemployed, and it's one in three in Italy and Portugal . To them, Germany must en LEXRANK As do young southern Europeans who are leaving home to come to Germany to find a job or receive vocational training. They not only lack companies willing to create apprenticeship positions, and patient "masters" happy to pass on their know-how to "their" apprentices, but also the institutions, and en Zero Youth unemployment in Europe has risen to 52% in Spain and Greece. In countries such as the United Kingdom, the jobs that are on offer are invariably short-term contracts. Precarious work is now the only option for a generation threatened by employment and poverty. However, in Europe, we may not have dictators to depose, but Monti's remarks are an indirect admission of the capitulation of democracy in response to the crisis. de-en Zero This article is a list of the events that have taken place in Greece, Italy, Spain, and the United Kingdom in the last decades of the twentieth century. The events that took place in Italy, Greece, Spain and Italy in the first decade of the twenty-first century have been described as "the most important events in the history of the European Union". Few In Europe, youth unemployment is on the rise. In Spain and Greece, it is rising to 52 per cent. But what will happen if the governments of Greece, Spain and Italy stop cutting their pensions? fr-en Zero This is a list of events that have taken place in the last decades of the twentieth century in Europe. The most recent events in the history of the European Union have been the events in Greece, Spain, Spain and the United Kingdom. Few A message of hope for young people in Europe has been delivered by Italian Prime Minister Mario Monti, who has deplored the fact that the unemployment rate of 20 years old is now a lost generation. But what will happen when they are no longer in the world or when the governments of Greece, Spain and Italy reduce the level of pensions? cs-en Zero This is a list of events that have taken place in the European Union in the past two decades. This list includes the events that occurred in the last decade of the twentieth century, including the events of the Arab revolutions, the collapse of the European social model, and the fall in the living standards of young people Few Whatever leaders do this week, they are not going to bridge the gap between unemployment in Europe and poverty in the Middle East. Instead, young people should take to the streets in Brussels to express their support for Europe, argues Mario Monti. Figure 1 : 1Example source document in French and target summaries in German, French, Czech and English. [...] Le ministre français de L'Industrie et de l'Energie Eric Besson réclame des négociations européennes sur les conséquences de cette décision nationale. [...] Table 2 shows 5 25https://en.wikipedia.org/wiki/Help: Interlanguage_linksDataset Lang Pairs SumL DocL MultiLing'13 40 30 185 4,111 MultiLing'15 38 30 233 4,946 Global Voices 15 229 51 359 WikiLingua 18 45,783 39 391 XWikis (comp.) 4 213,911 77 945 XWikis (para.) 4 7,000 76 972 Table 2 : 2Number of languages (Lang), average number of document-summary pairs (Pairs), average summary (SumL) and document (DocL) length in terms of num- ber of tokens. Table 3 : 3XWikis statistics (number of words and sen- tences per document (/Doc) and summary (/Sum)) and task characterisation metrics. Such lengthy documents are challenging for cur- rent neural summarisation models which struggle to represent multi-paragraph text; most approaches rely on an initial separate extractive step (Liu et al., 2018; Liu and Lapata, 2019a; Perez-Beltrachini et al., 2019). Each section describes a different aspect of its related Wikipedia title (Hayashi et al. Table 4 : 4Proportion of yes answers given to questionsof Overall summary and Sentence adequacy. Judg- ments elicited for cross-lingual document-summary pairs in three languages. Table 6 : 6ROUGE-L F1 X → en XWikis test sets. Table 7 : 7ROUGE-L F1 X → en Voxeurop test sets.Does Zero-Shot Work? Zero-shot (with a mono- lingual English summariser) grasps the gist of the document, and some representations are in- deed transferred. Despite the summariser being learnt on monolingual English data, when pre- sented with documents in other languages (i.e., German/French/Czech) it manages to produce a summary which, according to ROUGE, is better than extractive baselines (including EXT-ORACLE). However, across languages, zero-shot results are be- low the Supervised upper-bound (see second block in Cs Fr CsFigure 2: Example from Voxeurop dataset: source document in French and target summaries in German, French, English, and Czech. ORACLE 36.40/14.21/31.33 27.78/ 5.51/23.75 29.37/ 6.73/25.01 29.33/ 5.99/25.09 LEAD 29.99/ 6.07/25.45 29.25/ 5.58/24.95 28.97/ 5.57/24.74 28.58/ 5.10/24.35 LEXRANK 30.06/ 6.43/25.23 28.71/ 5.57/24.22 28.82/ 5.64/24.33 27.88/ 5.04/23.68 mBART Supervised 35.57/13.23/31.62 35.88/13.14/32.37 35.76/12.86/32.18 36.43/13.58/32.84 Translated -34.64/11.71/30.69 34.56/11.46/30.63 34.22/11.23/30.39 Zero -33.45/11.22/30.10 33.15/10.46/29.78 31.70/10.10/28.64 49/12.46/31.53 35.30/12.33/31.35 35.15/12.07/31.25 Zero -35.37/12.32/31.70 34.66/11.49/30.97 34.73/11.83/31.14 Few 300 LF-MAML -35.61/12.74/31.96 34.82/11.86/31.17 35.35/12.32/31.73 300 FT -35.45/12.45/31.77 35.01/12.04/31.39 35.43/12.30/31.67 300 CVT -35.45/12.41/31.77 34.77/11.53/31.08 35.53/12.52/31.91 1K LF-MAML -35.69/12.73/32.01 35.09/12.08/31.46 35.63/12.65/32.00en de-en fr-en cs-en EXT-Few 300 LF-MAML - 34.32/11.53/30.84 33.87/10.96/30.44 33.58/11.04/30.15 300 FT - 34.50/11.78/31.06 33.82/11.04/30.39 33.77/11.07/30.36 300 CVT - 33.81/11.36/30.40 33.57/10.82/30.12 32.63/10.61/29.39 1K LF-MAML - 34.76/11.93/31.19 34.25/11.27/30.77 34.50/11.91/31.02 mBART50 Supervised 36.60/13.73/32.53 36.64/13.96/32.95 35.59/12.70/31.84 37.56/14.57/33.72 Translated - 35. Table 8 : 8ROUGE-1/2/L F1 X → en XWikis test sets.en de-en fr-en cs-en EXT-ORACLE 29.16/9.94/20.83 25.12/5.07/17.81 25.41/5.52/17.90 24.93/4.69/17.63 LEAD 24.62/3.98/17.17 24.26/3.58/17.13 23.60/3.51/16.61 24.27/3.62/17.07 LEXRANK 24.22/3.59/16.65 23.20/3.09/16.32 23.32/3.21/16.32 23.32/3.16/16.48 mBART Zero 26.72/5.13/21.68 23.16/3.77/19.54 23.30/3.75/19.49 22.43/3.41/18.92 Few 300 LF-MAML - 27.55/4.76/22.32 27.68/4.71/22.42 27.42/4.64/22.26 300 FT - 26.62/4.60/21.86 26.78/4.48/21.74 26.42/4.57/21.72 mBART50 Zero 26.07/5.00/21.28 25.32/4.71/21.04 25.02/4.60/20.66 25.81/4.68/21.30 Few 300 LF-MAML - 27.44/4.87/21.87 27.36/4.83/21.90 27.41/4.82/22.11 300 FT - 27.13/4.76/21.79 27.01/4.61/21.53 27.35/4.79/21.95 Table 9 : 9ROUGE-1/2/L F1 X → en Voxeurop test sets. The Maria Canals International Music Competition (, ) is a music competition held yearly in the Palau de la Música Catalana, Barcelona. It was founded as a piano competition in 1954, but in 1964 it was expanded so other modalities could be held occasionally.SupervisedThe "Maria Canals International Music Competition" () is an international music competition in Spain. It was founded in 1954 by Maria Canals i Cendrós and her husband Rossend Llates i Serrat. In 1964, the competition was expanded to include more than 120 competitions, with over 8.000 participants from 100 countries.TranslatedTheMaria Canals International Music Competition is an international music competition organised by the Spanish Ministry of the Interior. It is a member of the World Federation of International Music competitions. The competition is held every year from 1954 to 2019. The Maria Canals International Music Competition is a multi-disciplinary music competition organised by the Spanish Ministry of the Interior. The competition is held annually in Spain. It was founded in 1954 by the Catalan pianist Maria canals i Cendrós and her husband, Rossend Llates i Serrat. 300 LF-MAML The Maria Canals International Music Competition is a music competition held annually in Catalonia, Spain. The competition was founded in 1954 by the Catalan pianist Maria canals i Cendrós and her husband, the composer Rossend Llates i Serrat (1899-1973).Reference Zero Table 10 : 10Example with mBART50 based models outputs from the validation set for German-to-English. Table 13 : 13Examples from Voxeurop datasets. We show Gold summary together with three extractive baselines (EXT-ORACLE, LEAD and LEXRANK) on the input English document for comparison. For each cross lingual task (de-en, fr-en, and cs-en), we report BART50 Zero and Few Shot FL-MAML variants. Although we focus on this language subset in this paper, we plan to release further languages in the future. https://www.wikipedia.org/ 3 http://voxeurop.eu. We were given authorisation by Voxeurop SCE publishers https://voxeurop.eu/ en/legal-notice-privacy/ 4 Code and data are available at https://github. com/lauhaide/clads. We compute statistics and metrics on the monolingual subsets from de-en, fr-en, and cs-en. Extractive methods were run on validation splits. A simplified version, First-Order MAML, updates the (meta) model parameters directly with the derivative of the last inner loop gradient update(Finn et al., 2017). Translation was supported by Google Cloud Platform credits. https://dumps.wikimedia.org 11 https://en.wikipedia.org/wiki/Help: Interlanguage_links Acknowledgments We thank the anonymous reviewers for their feedback. We also thank Yumo Xu for useful discussions about the models. We are extremely grateful to our bilingual annotators and to Voxeurop SCE publishers. We gratefully acknowledge the support of the European Research Council (award number 681760). . Giusepppe Attardi, Giusepppe Attardi. 2015. Wikiextractor. https:// github.com/attardi/wikiextractor. Reference Privacy International (PI) is a UK-based registered charity that defends and promotes the right to privacy across the world. First formed in 1990, registered as a non-profit company in 2002 and as a charity in 2012, PI is based in London. Its current executive director. since 2012, is Dr Gus HoseinReference Privacy International (PI) is a UK-based registered charity that defends and promotes the right to privacy across the world. First formed in 1990, registered as a non-profit company in 2002 and as a charity in 2012, PI is based in London. Its current executive director, since 2012, is Dr Gus Hosein. Supervised Privacy International is an international non-governmental organization dedicated to protecting the privacy of individuals, companies, and organizations. It is the world's largest private security organization. Washington, D.Cand operates under the auspices of the Worldwide Interbank Financial Telecommunication (SWIFTSupervised Privacy International is an international non-governmental organization dedicated to protecting the privacy of individuals, companies, and organizations. It is the world's largest private security organization. The organization is headquartered in Washington, D.C., and operates under the auspices of the Worldwide Interbank Financial Telecommunication (SWIFT). Zero Privacy International is a non-profit organization dedicated to protecting the privacy and privacy of financial institutions. It was founded in 1997 and has been active since 2003. It is the largest private security organization in the United States. The organization is a member of the World Bank and the International Monetary Fund. 1k LF-MAML Privacy International is a non-profit organization dedicated to protecting and protecting the privacy of individuals, companies and corporations. It was founded in 1997 and is one of the largest private sector organizations in the United States. The organization's mission is to protect and protect the privacy and data of individuals. Translated The Society for Worldwide Interbank Financial Telecommunication (SWIFT) is a non-profit organization dedicated to providing financial information to customers of financial institutions under SWIFT. The organization was founded in 1997 and is currently headquartered in. Los Angeles, Californiaand companies Table 12: Example with mBART50 based models outputs from the validation set for Czech-to-EnglishTranslated The Society for Worldwide Interbank Financial Telecommunication (SWIFT) is a non-profit organization dedicated to providing financial information to customers of financial institutions under SWIFT. The organization was founded in 1997 and is currently headquartered in Los Angeles, California. Zero Privacy International is a non-profit organization dedicated to protecting the privacy and privacy of financial institutions. It was founded in 1997 and has been active since 2003. It is the largest private security organization in the United States. The organization is a member of the World Bank and the International Monetary Fund. 1k LF-MAML Privacy International is a non-profit organization dedicated to protecting and protecting the privacy of individuals, companies and corporations. It was founded in 1997 and is one of the largest private sector organizations in the United States. The organization's mission is to protect and protect the privacy and data of individuals and companies Table 12: Example with mBART50 based models outputs from the validation set for Czech-to-English.
[]
[ "Attend, Memorize and Generate: Towards Faithful Table-to-Text Generation in Few Shots", "Attend, Memorize and Generate: Towards Faithful Table-to-Text Generation in Few Shots" ]
[ "Wenting Zhao \nDepartment of Computer Science\nUniversity of Illinois at Chicago\nILUSA\n", "Ye Liu \nDepartment of Computer Science\nUniversity of Illinois at Chicago\nILUSA\n", "Yao Wan [email protected] \nSchool of Computer Sci. & Tech\nHuazhong University of Science and Technology\nChina\n", "Philip S Yu [email protected] \nDepartment of Computer Science\nUniversity of Illinois at Chicago\nILUSA\n" ]
[ "Department of Computer Science\nUniversity of Illinois at Chicago\nILUSA", "Department of Computer Science\nUniversity of Illinois at Chicago\nILUSA", "School of Computer Sci. & Tech\nHuazhong University of Science and Technology\nChina", "Department of Computer Science\nUniversity of Illinois at Chicago\nILUSA" ]
[]
Few-shot table-to-text generation is a task of composing fluent and faithful sentences to convey table content using limited data. Despite many efforts having been made towards generating impressive fluent sentences by finetuning powerful pre-trained language models, the faithfulness of generated content still needs to be improved. To this end, this paper proposes a novel approach Attend, Memorize and Generate (called AMG), inspired by the text generation process of humans. In particular, AMG (1) attends over the multi-granularity of context using a novel strategy based on table slot level and traditional token-by-token level attention to exploit both the table structure and natural linguistic information; (2) dynamically memorizes the table slot allocation states; and (3) generates faithful sentences according to both the context and memory allocation states. Comprehensive experiments with human evaluation on three domains (i.e., humans, songs, and books) of the Wiki dataset show that our model can generate higher qualified texts when compared with several state-ofthe-art baselines, in both fluency and faithfulness. 1
10.18653/v1/2021.findings-emnlp.347
[ "https://arxiv.org/pdf/2203.00732v1.pdf" ]
244,119,648
2203.00732
e18c8ca90b61494ad9d30fd2f3221b095df6d302
Attend, Memorize and Generate: Towards Faithful Table-to-Text Generation in Few Shots Wenting Zhao Department of Computer Science University of Illinois at Chicago ILUSA Ye Liu Department of Computer Science University of Illinois at Chicago ILUSA Yao Wan [email protected] School of Computer Sci. & Tech Huazhong University of Science and Technology China Philip S Yu [email protected] Department of Computer Science University of Illinois at Chicago ILUSA Attend, Memorize and Generate: Towards Faithful Table-to-Text Generation in Few Shots Few-shot table-to-text generation is a task of composing fluent and faithful sentences to convey table content using limited data. Despite many efforts having been made towards generating impressive fluent sentences by finetuning powerful pre-trained language models, the faithfulness of generated content still needs to be improved. To this end, this paper proposes a novel approach Attend, Memorize and Generate (called AMG), inspired by the text generation process of humans. In particular, AMG (1) attends over the multi-granularity of context using a novel strategy based on table slot level and traditional token-by-token level attention to exploit both the table structure and natural linguistic information; (2) dynamically memorizes the table slot allocation states; and (3) generates faithful sentences according to both the context and memory allocation states. Comprehensive experiments with human evaluation on three domains (i.e., humans, songs, and books) of the Wiki dataset show that our model can generate higher qualified texts when compared with several state-ofthe-art baselines, in both fluency and faithfulness. 1 Introduction Table- to-text generation, which aims to translate a semi-structured table into natural language descriptions while preserving the conveyed table information, are drawing increasing interest over the past few years. It has been widely applied in many real-world scenarios, such as automatically generating weather forecasting reports (Liang et al., 2009), biographies (Lebret et al., 2016;, restaurant descriptions (Novikova et al., 2017), task-oriented conversations (Budzianowski et al., 2018;Williams et al., 2013) as well as healthcare descriptions (DiMarco et al., 2007; Hasan and 1 All the source code and experimental dataset are available at https://github.com/wentinghome/AMG. Farri, 2019). Despite such significant gains, current approaches are driven by large-scale well-labeled training data, hindering the generalization to other scenarios with limited labeled data. In addition, the faithfulness of generated contents is still not well explored. Few-shot natural language generation (Brown et al., 2020;Schick and Schütze, 2021;Xia et al., 2020a) has been in increasing demand since sufficient labeled data are always unavailable in many scenarios. To improve the table-to-text generation in few-shot scenarios, many existing works (Chen et al., 2020c;Gong et al., 2020;Peng et al., 2020) resort to the pre-training techniques which have been widely adopted in NLP, that is, pre-training a model first on large-scale unlabeled data, and then transfer the learned knowledge in pre-trained model to the few-shot scenario of table-to-text generation. Although these pre-trained models have achieved promising performance on generating fluent descriptions, from our investigation, they are still suffering from three major limitations: (1) The structure of table has not been well preserved. On table representation, existing methods (Chen et al., 2020c;Gong et al., 2020;Chen et al., 2020a) used to flatten the table into sequential sentences, ignoring the structured features (e.g., correlation between words within each table slot) among tables, which is also critical for table-to-text generation. (2) Generation bias. Current approaches that directly fine-tune the model on target data make the model in favor of the knowledge learned from pretraining rather than specific target task knowledge, hurting the faithfulness because extra information irrelevant to the input table is introduced. For example, as shown in Figure 1, given a table in the top box, the aim is to generate a coherent and faithful sentence with high coverage of table slots, as well as less out-of-table information. From this table, we can observe that current stateof-the-art models tend to generate sentences with hallucinated contents. For example, GPT-2 introduces wrong middle name "kelly" and the nationality "american". In addition, the table coverage of contents generated by current approaches is low. For example, BART does not mention the event "marathon". These observation motivate us to design a model that can generate faithful texts from tables while keeping the fluency. To tackle the aforementioned limitations, this paper proposes a novel approach Attend, Memorize and Generate (called AMG) for faithful table-totext generation in few-shots. Inspired by the human generation process which copies a consecutive slot span to compose a sentence using the context, we propose a table slot attention mechanism to empower the model generalization ability in inference by strengthening the dependency between the generated sentence with the input table. In addition, to avoid generating hallucinated contents, we design a memory unit to monitor the visits of each table slot. Particularly, the memory unit is initialized as all the meta-data of table slots, and then updated by checking the generated words as well as the current memory state. Looking back to Figure 1, we can also observe several advantages of AMG. First of all, we can see AMG allows the to-be-predicted word "1998" from "birth_date" table slot to attend on the table as well as the previously generated sentence "robert . . . born", while the attention on within table slot words are prohibited. Thus, the model is enforced to capture the table span structure and rely on the table span value to generate. To this end, the model learns to capture the slot level table representation. Furthermore, as shown in Figure 1, "M 0 " is the memory initial state where all the slot are available to be chosen (marked by green). After predicting the last word of table slot "name", "M 1 " will be updated since it detects that the table slot "name" is present in the generated sentence, thus making the state of "name" unavailable (marked by red). In addition, the generation of word "1998" takes the context and table slot allocation into account, therefore "1998" is selected by locating the value of table span "birth_date" as well as the activated signal of table slot "birth_date" (marked by blue) from memory allocation status. To summarize, the primary contributions of this paper are as follows: (1) To better preserve the structure of table, we design a multi-grain attention that can attend over the table word as well as table slots level. (2) It is the first time that we introduce a memory mechanism to improve the faithfulness of generated texts by tracking the allocation of table slots. (3) We have conducted comprehensive experiments on three domains (i.e., Humans, Books and Songs) of the Wiki dataset to validate the effectiveness of our proposed approach. Preliminaries Problem Definition Given a table T of m attribute-value pairs {(a i , v i )} m i=1 , where a i and v i refer to the attribute name and value of i-th table slot, respectively, the table-to-text generation task aims at producing a coherent text Y = (y 1 , · · · , y L ) that can describe the table information with fluency and faithfulness, where L denotes the length of generated text. UniLM To alleviate the under-fitting issue caused by insufficient training examples in few shot learning, AMG adopts the state-of-art pre-trained language model UniLM (Dong et al., 2019) structure to integrate the external knowledge. UniLM is a multilayer Transformer network which can be applied into both tasks of natural language understanding (NLU) and natural language generation (NLG). In this paper, we configure UniLM using Seq2Seq self-attention mask to aggregate the context of the masked i-th to-be-predicted word y [M ASK] i that are source sequence words from table T , and the previously generated target words y <i . The proposed model computes the conditional probability for the to-be-predicted word using the masked language model objective function, as follows: P (Y |T ; θ) = L i=1 P ( AMG Approach 3.1 Overview Figure 2 illustrates the overall architecture of our model, which is composed of three components, i.e., attend, memorize, and generate. (1) Attend. We propose a multi-granularity attention mechanism which attends over both token level and the table slot level to capture the linguistic knowledge as well as table structure information. We think that these knowledge can improve the faithfulness of generated texts. (2) Memory. We develop a memory to store and keep track of the table slot allocation status. (3) Generate. We take both the context representation and the table slot allocation states into account while making predictions. The above three building blocks interweave and lead the model to generate descriptions from tables faithfully. Table Representation Table Linearization Table-to-text generation receives semi-structured table as input. However, our proposed model AMG is built upon the UniLM architecture which requires natural sentence as input. Therefore, the first step we need to do is to translate the table into a natural sentence by linearization (Chen et al., 2020c). For the table example shown in Figure 1, the attribute value pair "name: robert kiprono cheruiyot" can be linearized Representing the History of Table Slot Allocation AMG makes prediction on the to-bepredicted token by taking the memory allocation status into account. The memory at different time step is updated by the previously generated table slots. Thus, we need to prepare the previously generated table slot representation his t at time step t by using the static UniLM model. For example, in Figure 2, when making prediction for "[MASK]", the representation of table slot allocation history is computed by feeding "robert kiprono cheruiyot" to the static UniLM model and obtain the average of hidden states. Multi-Granularity Attention AMG introduces the multi-granularity attention (MA) which is the combination of two granularity of attention, i.e., token level and Figure 2, the memory augmented attention A is the average of token level attention A ta and table slot level attention A sa , as following: A = (A ta + A sa )/2 ,(2) where the token level self-attention mechanism learns a unique series of query matrix W l Qta , key matrix W l Kta , and value matrix W l Vta at the l-th Transformer layer for each attention head. Then, AMG maps the (l − 1)-th Transformer layer output T l−1 to three matrices: query Q ta , key K ta , and value V ta . The output of a self-attention head A ta is computed as Eq. (3), where M ask ta ∈ R N ×N is the seq2seq attention mask, allowing the to-bepredicted token to attend to table tokens as well as the previously generated tokens. N refers to the total token length of table, previously generated tokens and the current to-be-predicted token. A l ta = softmax( Q ta K T ta √ d k + M ask ta ) · V ta ,(3) Table Slot Attention Table slot attention works in a similar way with the self attention, while the major difference is to learn new key and value mapping matrices W l Ksa and W l Vsa and project memory M l−1 using W l Ksa and W l Vsa to obtain K sa and V sa . The query Q sa is computed by the projection of UniLM hidden state h l−1 using mapping matrix W l Qsa . Memory M in AMG is defined as a R d h ×slotn matrix where slot n is the maximum number of table slots. The j-th column of memory at time step t is denoted as M t j , and the initial state of memory M 0 j is the average embedding of the j-th table slot value computed using static UniLM model. The output of slot level attention head A l sa is as follows: Q sa = h l−1 W l Qsa K sa = M l−1 W l Ksa V sa = M l−1 W l Vsa A l sa = softmax( Q sa K T sa √ d k + M ask slot ) · V sa .(4) Instead of applying the original seq2seq attention from UniLM to the input, a table slot attention mask M ask slot ∈ R N ×N is introduced to decide which word should be attended. In our case, we prohibit the to-be-predicted token to attend the previously generated words within the same table slots, while allow to attend the rest of generated words and the table. As shown in Figure 2, "1998" from the descriptive sentence can attend to both the table " name is . . . , birth_date is . . . " and previously generated words "robert kiprono cheruiyot ( born", while is not allowed to attend to words within the same table slot "august 10 ,". into the reference. Memory is updated using the gated mechanism, following (Henaff et al., 2016): M t j = tanh(W a M t−1 j + W b his t−1 ) z t j = δ(W c M t−1 j + W d his t−1 ) M t j = (1 − z t j )M t−1 j + z t jM t j .(5) In Eq. (5), W a , W b , W c and W d are trainable parameters. First,M t j is the new candidate memory to be combined with the existing memory M t−1 j . Then, the gate function z t j employs a sigmoid function δ to determine how much memory M t j will be influenced. At last, we retain M t j by using gate function to control how much each cell in memory is updated by considering the history of table slot appearance in the target sentence, as well as the last memory. Text Generation When predicting the next token at each time step, AMG considers both the context representation and the table slot allocation status from memory shown in Eq. (6) where tb refers to the table representation, tk t denotes the token predicted at time t by AMG, and tk 0...t−1 denote the tokens previously generated from time 0 to t − 1. (his t , M t , tk t ) = AMG(tb, his t−1 , M t−1 , tk 0...t−1 ) . (6) Task-Adaptive Pre-Training AMG is built upon the pre-trained UniLM and introduces additional weight. The memory updater depends on W a , W b , W c and W d to project memory and history values, as shown in Eq.(5). Besides, the newly added special token [E_CLS] and [E_SEP] is supposed to learn appropriate embedding weight from scratch. It is challenging to expect the newly introduced weight can be learned properly if we directly fine-tune AMG under the few shot scenario. Inspired by the pre-trained language models and the task adaptive pre-training (Gururangan et al., 2020), we collect the unlabelled table side data to do a second phase task adaptive pre-training. We first linearize the input During pre-training, AMG modifies the UniLM model architecture by designing a novel slot attention mask as well as slot memory mechanism which introduces additional weights. There are two goals for pre-training: 1) tune UniLM weights to incorporate slot attention mask , and 2) learn proper weights for slot memory block. We divide the pretraining stage into two phases: slot attention based pre-training and slot memory based pre-training. We incrementally incorporate the slot attention and slot memory elements to the UniLM model along the two pre-training phases. First, the model structure of slot attention based pre-training is to add the slot attention mask to the last 6 layers of UniLM. We also learn the embedding of two special tokens [E_CLS] and [E_SEP] by adding them into the UniLM vocabulary. We load the UniLM checkpoint model weight as the initial weight for slot attention based pre-training. The second slot memory based pre-training phase adopts the full AMG model, and is loaded with the checkpoint obtained after the slot attention mask based pre-training. Fine-Tuning and Inference In fine-tuning stage, AMG first loads the model weight after the further pre-training stage which exploits valuable information from plenty of unlabelled task relevant data. The input for our proposed model is the concatenation of the linearized table and the reference sentence. The model is trained end to end in masked language model fashion. Around 70% words in the reference are masked, and the cross entropy loss is used to min-imize the discrepancy between the masked token and the groundtruth. For inference, table side data is present while the reference sentence is missing. Our approach generates sentence auto-regressively. When making prediction on the t-th word, we need to inform the model previously generated table slots through table slot history representation his t . Experiment In this section, we explore the following experimental questions: (1) Can the proposed model generate fluent sentences?; and (2) Is the generated sentence faithful to the fact given by input table? We also perform ablation analysis to investigate the two main components of AMG, namely the slot attention and slot memory mechanism. Dataset Task Adaptive Dataset for Pre-training To pretrain AMG, we collect additional unlabelled data from WikiBio (Lebret et al., 2016) (Chen et al., 2020c) respectively as the pre-training data. Dataset for Fine-Tuning Inspired by the experimental settings of few-shot natural language generation in (Chen et al., 2020c), we conduct experiments on three domains, i.e., humans, songs and books of Wiki dataset denoted as Wiki-Humans, Wiki-Songs and Wiki-Books. For each domain, we fine tune AMG to inspect the model performance on various few shot settings by sampling different amount of training examples (e.g. 500, 200, 100, 50). The validation set for each domain includes 1000 instances, and test sets of humans, songs and books domain have 13587, 11879 and 5252 examples. We set the maximum length of the linearized table and the generated sentence as 300 and 64 respectively. BLEU-4 METEOR ROUGE-L PARENT(P/R/F) PARENT-T(P/R/F) Implementation Details The base model for AMG is UniLM-base model with 12 Transformer layers, 768 hidden state dimensions, and 110M parameters in total. The implementation of AMG is divided into two stages in total: 1) two-phase task adpative pre-training, and 2) fine-tuning on the target wiki dataset. We run the program on a single 1080Ti GPU with 12GB memory. Due to the memory constraint, the batch size on all stages is set as 4 and gradient is accumulated every 11 steps which results in a comparable 44 batch size. The learning rate is 5e-5. The Adam (Kingma and Ba, 2015) optimizer is used and the weight decay is set as 0.01. For fine-tuning, we fine-tune the AMG on target dataset by setting the maximum number of epoch as 50. For inference, we decode on the test set using the best checkpoints according to the validation set result. During inference, we use beam search with beam size 3 and length penalty 1. Baselines We compare the proposed model with strong pretrained language models. UniLM (Dong et al., 2019) is a pre-trained language model for both natural language understanding and generation using three types of language modeling tasks. BART (Lewis et al., 2020) introduces a denoising autoencoder for pre-training sequence-tosequence models. GPT-2 (Radford et al., 2019) is a powerful unidirectional model pre-trained on millions of webpages in auto-regressive fashion. GPT2+copy (Chen et al., 2020c) designed for fewshot table-to-text generation learns how to alternate between copying from table and generating functional words using GPT-2. TableGPT (Gong et al., 2020) is a followup work of (Chen et al., 2020c) while considers to minimize the contradicting part of the generated sentence give the table information. Automatic Evaluation Following other generation tasks, we choose three automatic evaluation metrics BLEU-4 (Papineni et al., 2002), ROUGE-L (Lin, 2004) and ME-TEOR (Banerjee and Lavie, 2005) to evaluate the overlapping between the generated sentence and the reference sentence. Besides, to evaluate the faithfulness of generated sentence with the source table, we adopt PARENT (Dhingra et al., 2019) reference, but also takes how much table slot information is reflected in the generated sentence into account. In addition, to further evaluate the faithfulness of the generated text, PARENT-T (Wang et al., 2020) which only measures the matching between the generated text and the corresponding table is also included. Results We first compare AMG with state-ofthe-art models mentioned in section 4.3. Table 1 shows the performance of AMG and baseline models on three domains of Wiki dataset using 500 training examples. For (Chen et al., 2020c), we copy the code that the author released on GitHub and replicate the result denoted as GPT2+copy (our replication). Regarding the conventional overlapping based metrics BLEU-4, METEOR, ROUGE-L, We can see that AMG provides the best overall performance under various domains and evaluation metrics. AMG outperforms the base model UniLM 3.71%/3.32%/2.46% on BLEU-4 under Humans/Books/Songs domains, and AMG gains 0.73%/0.53%/0.16% more than the second best model BART on METEOR. AMG outperforms the second best model BART 1.07%/0.48%/0.90% on the F score of PARENT which is a strong indication that AMG can achieve the strongest balance between the fluency and faithfulness. Regarding the overlapping between the generated sentence with table content, F scores of PARTENT-T metric shows that AMG provides the most informative results on Humans and Songs domains while still very competitive with the best model BART on Books domain. Besides, to verify the stability of AMG when the amount of training data varies to 50, 100, 200 and 500, we show PARENT score for the proposed and other baseline models in Table 2 3.39 GPT2 (Radford et al., 2019) 3.73 1.69 3.61 BART (Lewis et al., 2020) 4.017 1.53 3.24 UniLM (Dong et al., 2019) 3.92 1.65 3.52 AMG 4.023 1.75 3.22 And human domains achieves the most gain since we collect most pre-training data for the task adaptive pre-training, thus it would be beneficial for the further work to collect more task adaptive pre-training data for Books and Songs domains to further boost the model performance. Analysis We further analysis the faithfulness and the overall quality of the generated descriptions by conducting human evaluation. Then, we design ablation studies to investigate the importance of two building blocks of AMG: span attention and memory mechanism. In addition, we sample a specific input table and compare sentence generated by AMG with the state-of-the-art models shown in Figure 3. BART AMG 50 shots rating 3.87 4.11 p = 0.002 500 shots rating 4.46 4.55 p = 0.24 Human Evaluation Following (Wang et al., 2020;Chen et al., 2020c), we recruit three human annotators who pass the College English Test (CET-6) English test 2 to judge the quality of the generated sentence. We sample 100 test tables and collect corresponding outputs from AMG, and baseline models. The sentences are randomly shuffled to reduce human variance. We provide instructions for human annotators to evaluate the sentence quality from two aspects: faithfulness and overall quality. First, for faithfulness, they are supposed to identify the number of entities mentioned in the sentence. Then, they need to compare the entities with ones from source table. Finally, they are supposed to report the number of fact supported and contradicted from the table respectively. Subsequently, we compute the average number of supported and unsupported entities denoted by #sup and #con in Table 3. The second study evaluates the overall quality of the generated sentence from their fluency, grammatical correctness, and the information consistency with the table. To compare the overall quality of various models, annotators rank the sentences generated using different models from 1 (best) to 6 (worst) by comparing the sentence. The "overall" column refers to the average ranking of the model. Table 3 shows that AMG generates better quality sentences compared with other models. Specifically, the outputs generated by AMG contains the most information supported by the table and the overall quality is ranked the first place. Although it shows the number unsupported by the table is higher than other models, the overall quality still outperforms other models. The overall ranking in Table 3 between BART and AMG is quite close, thus we ask 3 human evaluators to rate the generated sentences from 3 criteria, and then calculate the statistical significance of the overall rating between BART and AMG. We randomly sample 50 sentences for 50 and 100 training examples in few-shot cases respectively. Three annotators are instructed to re-evaluate the overall sentence quality by rating them from 1 (worst) to 5 (best) by considering the following 3 criteria: (1) #sup, (2) #con (see Table 3), (3) naturalness and grammar correctness. The results are listed as follows. As shown in Table 4, comparing BART with AMG, the p-value p 0.002 of Wilcoxon signedrank tests shows at 95% confidence level, AMG is 2 A national English as a foreign language test in China. Figure 3 provides a sample input table from test set along with various model outputs. Case Study The top box contains an input table while the bottom box includes model generations. In the bottom box, we leave the content supported by table as black, unsupported as light brown, and blue for the remaining words. We find that the output of pre-trained baseline models suffer from the following problems: (1) repetition, e.g., BART fails to generate person name "wayne" correctly while repeats the last two letters as "waynene", (2) hallucination, e.g., GPT-2 generates a middle name "wayne" which is out of table, and GPT2+copy attempts to copy the "office" slot but fail to copy the entire information by introducing unsupported information "the oak house" and "2003 ... brotherwayne.". By contrast, AMG provides the highest table coverage while keeping the sentence fluent which demonstrates the table slot span attention and memory mechanism enables the model to copy from the table slot level correctly and enhance the generation faithfulness. (Lebret et al., 2016;Liu et al., 2018;Wiseman et al., 2018;Ma et al., 2019;Liu et al., 2019a). Ma et al. (2019) extend the table-to-text generation to low-resource scenario and put forward a Transformer-based model. Of late, as the pretraining language model (e.g, BERT and GPT) has achieved significant successes in NLP, many works also propose to pre-train a model for table understanding. Yin et al. (2020) pre-train a model for jointly understanding of tabular data around textual descriptions on large-scale paired data. Herzig et al. (2020) extend the architecture of BERT to encode tables as input, and propose a weakly supervised pre-training model for question answering over tables. Kale (2020) investigate the performance of pre-trained T5 (Raffel et al., 2019) on multiple table-to-text tasks and provide a benchmark for the future research. To keep the faithfulness of table on generation, one related work to ours is (Wang et al., 2020), which introduces a new table-text optimaltransport matching loss and a table-text embedding similarity loss based on the Transformer model to enforce the faithfulness during text generation. Related Work Pre-Trained Language Model Our work is also related to model pre-training for NLP, which has brought dramatic improvements on natural language understanding (Devlin et al., 2019;Liu et al., 2019c;Clark et al., 2020; and generation (Song et al., 2019;Dong et al., 2019;Liu et al., 2020bLiu et al., , 2019b. The widely used pretrained models (PTMs) for table-to-text generation can be categorized into two classes: text-to-text PTMs (Radford et al., 2018;Devlin et al., 2019;Dong et al., 2019;Lewis et al., 2020;Joshi et al., 2020) and structured data-to-text PTMs (Chen et al., 2020b;Herzig et al., 2020;Xing and Wan, 2021). Recently, many pre-training models (Liu et al., 2021(Liu et al., , 2020aYao et al., 2019) start to incorporated the structured information from knowledge bases (KBs) or other structured semantic annotations into pre-training, which is also related to our work. Few-shot text generation Few-shot text generation learns with minimal data while maintaining decent generation capacity. Few-shot text generation can be used to augment the scarce training data to better assist the down-stream task, e.g., (Xia et al., 2020a,b) for spoken language intent detection, (Bražinskas et al., 2020) for opinion summary generation. In addition, to better utilize the available resources, Chang et al. (2021) investigates the training instance selection on unlabelled data, and (Schick and Schütze, 2020) adapts patternexploiting training strategy to fine-tune a PTM. Conclusion In this paper, we have proposed a novel approach AMG for faithful table-to-text generation in few shots. We first attend over the multi-granularity of context using a novel span level and traditional token-by-token level attention strategy to exploit both the table structural and natural linguistic information. Then, we design a memory unit to memorize the table slot allocation states dynamically. Extensive experiments on three domains of Wiki dataset verify the effectiveness of our proposed model on generating fluent and faithful descriptions from tables. Figure 1 : 1robert kiprono cheruiyot ( born august 10 , [MASK] … name : robert kiprono cheruiyot birth_date : 10 august 1988 nationality : kenyan sport : running event : marathon birth_place : bomet , rift valley province , kenya [BART]: robert kiprono cheruiyot ( born 10 august 1988 ) is a kenyan runner . [GPT-2]: robert kelly cheruuyot ( born august 10 , 1988 ) is an american runner . [Ours]: robert kiprono cheruiyot ( born august 10 , 1988 ) is a kenyan marathon runner . [Ref]: robert kiprono cheruiyot ( born august 10 , 1988 ) is a kenyan marathon runner . A motivating example. y[M ASK] i |y <i , T ; θ) . (1) linear robert kiprono cheruiyot ( born august 10 , [MASK] ) … … name birth_date birth_place … M 0 … … … … M 1 M 2 …… M t b) Memorize … M i name birth_date birth_place … … robert kiprono cheruiyot … K ta Q ta V ta V sa K sa A sa A ta a) Attend h i his t M t-1 linear linear tanh (1-g) . g . M t name is …, birth_date is 10 august 1988, … … Memory Update august 10 , 1988 c) Generate his t M t Static UniLM his t+1 Linearized table Descriptive sentence AMG AVG A Figure 2: An overview of AMG. The input to AMG is the concatenation of linearized table (marked in grey) and the descriptive sentence(marked in orange). The bottom box shows the memory update process. The top three boxes show the building blocks of AMG, designed to attend, memorize and generate descriptions from tables. as "name is [E_CLS] robert kiprono cheruiyot [E_SEP];", where [E_CLS] and [E_SEP] are two special tokens to indicate the beginning and the end of table slot value. Table Slot Memory SlotUpdate AMG updates the memory matrix multiple times dynamically depending on how many times the generated sentence finishes generating one entire table slot value. To give a clear signal for the model to detect the beginning and the end of the table slot value, we introduce two additional special tokens [E_CLS] and [E_SEP] table and add special token [E_CLS] and [E_SEP] to indicate the beginning and the end of the table slot value respectively. Then, around 20% tokens are masked and the cross entropy loss is employed as the objective function. One corrupted example for further pre-training stage is "[CLS] name is [E_CLS] [MASK] kiprono [MASK] [E_SEP]; birth_date is [E_CLS] 10 august [MASK] [E_SEP]; . . . [SEP]". and Wiki dataset. First, Wiki-Humans is a subset of WikiBio dataset which contains massive training examples collected from Wikipedia, a cleaned-up version of original WikiBio dataset by setting a vocabulary bound and removing those include out-of-vocabulary words that are not in the given table. Since pre-training only requires the table side data and focuses on reconstructing the corrupted text, we collect the rest of table side data (around 500K from WikiBio by removing all the train/valid/test data used in Wiki-Humans heuristically. Second, for songs and books domain, we collect around 26K and 17K filtered out table data from Table 1: Test results on three domains Humans/Books/Songs of Wiki dataset using 500 training data. "P/R/F" denotes the precision/recall/F score.Humans 1 GPT2+copy (Chen et al., 2020c) 41.7 - - - - 2 GPT2+copy (our replication) 42.05 33.36 63.90 68.47/37.28/45.59 47.90/40.18/41.58 3 TableGPT2 (Gong et al., 2020) 45.6 - - - - 4 GPT2 (Radford et al., 2019) 24.26 25.20 53.90 59.45/18.51/25.89 41.60/27.93/31.57 5 BART (Lewis et al., 2020) 48.31 37.24 68.24 74.04/41.46/50.79 51.50/41.98/44.20 6 UniLM (Dong et al., 2019) 45.31 37.10 68.36 72.90/40.24/49.61 50.06/41.67/43.46 7 AMG 49.02 37.97 69.37 74.14/42.74/51.86 51.20/43.03/44.70 Books 1 GPT2+copy (Chen et al., 2020c) 40.30 - - - - 2 GPT2+copy (our replication) 40.39 34.48 67.59 69.68/35.10/44.87 51.34/35.34/40.45 3 TableGPT2 (Gong et al., 2020) 41.6 - - - - 4 GPT2 (Radford et al., 2019) 19.12 24.99 54.83 55.22/17.72/24.94 40.41/28.21/32.14 5 BART (Lewis et al., 2020) 43.53 36.45 68.93 72.86/37.84/48.11 54.35/37.51/42.97 6 UniLM (Dong et al., 2019) 40.56 35.71 68.85 71.90/35.60/45.87 53.07/35.58/41.15 7 AMG 43.88 36.98 70.57 73.26/38.18/48.59 53.89/37.29/42.69 Songs 1 GPT2+copy (Chen et al., 2020c) 42.20 - - - - 2 GPT2+copy (our replication) 42.41 33.43 65.18 66.34/35.72/44.75 42.05/33.99/36.27 3 TableGPT2 (Gong et al., 2020) 42.30 - - - - 4 GPT2 (Radford et al., 2019) 22.48 24.09 55.92 55.05/17.90/25.65 30.96/21.53/24.42 5 BART (Lewis et al., 2020) 43.88 34.69 67.22 69.22/36.31/46.00 43.48/34.55/37.26 6 UniLM (Dong et al., 2019) 42.63 34.79 67.92 68.19/34.74/44.55 41.32/32.64/35.24 7 AMG 45.09 35.55 67.38 67.60/37.63/46.90 42.78/35.21/37.36 as our main metric. PARENT not only considers the matching between the generated sentence with theDomain Humans Books Songs # of training examples 50 100 200 500 50 100 200 500 50 100 200 500 GPT2+copy (our replication) 30.59 34.59 40.54 45.59 42.67 42.79 43.44 44.87 40.18 41.72 43.97 44.75 GPT2 (Radford et al., 2019) 0.17 12.90 19.02 25.89 0.71 20.82 24.18 24.94 0.85 17.08 24.72 25.65 BART (Lewis et al., 2020) 37.73 41.37 47.41 45.45 41.68 43.43 43.65 48.11 41.74 42.44 44.12 46.00 UniLM (Dong et al., 2019) 35.80 41.83 46.08 49.61 38.28 41.39 44.06 45.87 40.17 41.95 42.45 44.55 AMG 43.55 47.72 50.13 51.86 43.42 46.03 47.45 48.59 42.03 43.30 45.93 46.90 Table 2 : 2PARENT F score on three domains using 50/100/200/500 training examples. Table 3 : 3Results of human evaluation.mans, UniLM by 3.39% on Books, and BART by 1.81% on Songs. The results demonstrate that lever- aging the table slot attention as well as the memory mechanism provide a stable and competitive per- formance of faithful generation. On the other hand, on the Humans/Books/Songs domain with 50 train- ing examples, AMG gains 5.82%/1.74%/0.29% improvements than the second best model BART respectively which shows that our model has pow- erful generative ability even only 50 examples are present. Table 4 : 4Statistical significance on human evaluation. UniLM]: wayne r . parry( born may 15 , 1963 ) is an american politician in the state of maine .[GPT2+copy]: wayne r. parry( born may 15 , 1963 ) is an american politician from oak portthouse , who has been a republican member of the oak house of representatives from 2003 parry to 2004 , when he was succeeded by his brother brother wayne .# [Ours]: wayne r. parry ( born may 15 , 1963 ) is an american politician from maine , who has been a republican member of the maine house of representatives from the 140th district .Figure 3: A case study of a specifictable input for qualitative analysis of table-to-text generation. statistically significant with BART when training examples are as scarce as 50. While at 75% confidence level, AMG is statistically significant with BART when training examples increase to 500.name : wayne r. parry office : member of the maine house of representatives for the 140th district ( arundel ) term_start : december 2010 party : republican birth_date : 15 may 1963 birth_place : portland , maine alma_mater : windham high school residence : arundel , maine article_title : wayne parry [Ref]: wayne r. parry is an american politician from maine . [BART]: waynene r. parry ( born 15 may 1963 ) is a maine politician . [GPT-2]: wayne`` wayne '' parry ( born may 15 , 1963 ) is a former republican politician from windham . [Model BLEU METEOR PARENT PARENT-T AMG 49.02 37.97 51.86 44.70 AMG w/o span 47.28 37.10 50.24 43.36 AMG w/o mem 48.92 38.14 51.38 43.76 AMG w/o extra 46.78 36.99 49.83 44.00 Table 5 : 5Ablation study of the proposed model. Study We also conduct ablation studies to understand each component of the proposed model, including slot attention and slot memory mechanism.Table 5provides the ablation results under different evaluation metrics. It shows that AMG can still outperform all these two variants overall, certifying the effectiveness of each designed component in our model and we demonstrate that incorporating table slot attention and memory mechanism with the pre-trained model UniLM can boost the model performance.Ablation Table - -to-Text Generation Recent years have witnessed much success on representing the semistructured tabular data and generating text to describe the table. From our investigation, most existing methods for table-to-text generation are based on the RNN-based encoder-decoder framework AcknowledgementsWe would like to thank all the anonymous reviewers for their helpful comments. This work is supported by NSF under grants III-1763325, III-1909323, III-2106758, and SaTC-1930941. 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[ "https://github.com/wentinghome/AMG." ]
[ "ASPECTS OF THE CATEGORY SKB OF SKEW BRACES", "ASPECTS OF THE CATEGORY SKB OF SKEW BRACES" ]
[ "Dominique Bourn ", "ANDAlberto Facchini ", "Mara Pompili " ]
[]
[]
We examine the pointed protomodular category SKB of left skew braces. We study the notion of commutator of ideals in a left skew brace. Notice that in the literature, "product" of ideals of skew braces is often considered. We show that Huq=Smith for left skew braces. Finally, we give a set of generators for the commutator of two ideals, and prove that every ideal of a left skew brace has a centralizer.for every a, b, c ∈ A. Here a − * denotes the inverse of a in the group (A, * ). The inverse of a in the group (A, •) will be denoted by a −• .
10.1080/00927872.2022.2151609
[ "https://arxiv.org/pdf/2205.04171v1.pdf" ]
248,572,252
2205.04171
5500dc9046af3baa8a560c84e8e9d31326523b6e
ASPECTS OF THE CATEGORY SKB OF SKEW BRACES Dominique Bourn ANDAlberto Facchini Mara Pompili ASPECTS OF THE CATEGORY SKB OF SKEW BRACES arXiv:2205.04171v1 [math.CT] 9 May 2022 We examine the pointed protomodular category SKB of left skew braces. We study the notion of commutator of ideals in a left skew brace. Notice that in the literature, "product" of ideals of skew braces is often considered. We show that Huq=Smith for left skew braces. Finally, we give a set of generators for the commutator of two ideals, and prove that every ideal of a left skew brace has a centralizer.for every a, b, c ∈ A. Here a − * denotes the inverse of a in the group (A, * ). The inverse of a in the group (A, •) will be denoted by a −• . Introduction Braces appear in connections to the study of set-theoretic solutions of the Yang-Baxter equation. A set-theoretic solution of the Yang-Baxter equation is a pair (X, r), where X is a set, r : X×X → X×X is a bijection, and (r×id)(id×r)(r×id) = (id × r)(r × id)(id × r) [13]. Set-theoretic solutions of the Yang-Baxter equation appear, for instance, in the study of representations of braid groups, and form a category SYBE, whose objects are these pairs (X, r), and morphisms f : (X, r) → (X ′ , r ′ ) are the mappings f : X → X ′ that make the diagram X × X f ×f / / r X ′ × X ′ r X × X f ×f / / X ′ × X ′ commute. One way to produce set-theoretic solutions of the Yang-Baxter equation is using left skew braces. Definition [15] A (left) skew brace is a triple (A, * , •), where (A, * ) and (A, •) are groups (not necessarily abelian) such that (B) a • (b * c) = (a • b) * a − * * (a • c) A brace is sometimes seen as an algebraic structure similar to that of a ring, with distributivity warped in some sense. But a better description of a brace is probably that of an algebraic structure with two group structures out of phase with each other. For every left skew brace (A, * , •), the mapping r : A × A → A × A, r(x, y) = (x − * * (x • y), (x − * * (x • y)) −• • x • y), is a non-degenerate set-theoretic solution of the Yang-Baxter equation ([15, Theorem 3.1] and [19, p. 96]). Here "non-degenerate" means that the mappings π 1 r(x 0 , −) : A → A and π 2 r(−, y 0 ) : A → A are bijections for every x 0 ∈ A and every y 0 ∈ A. The simplest examples of left skew braces are: (1) For any associative ring (R, +, ·), the Jacobson radical (J(R), +, •), where • is the operation on J(R) defined by x • y = xy + x + y for every x, y ∈ J(R). (2) For any group (G, * ), the left skew braces (G, * , * ) and (G, * , * op ). Several non-trivial examples of skew braces can be found in [23]. A complete classification of braces of low cardinality has been obtained via computer [19]. A homomorphism of skew braces is a mapping which is a group homomorphism for both the operations. This defines the category SKB of skew braces. From [15], we know that in a skew brace the units of the two groups coincide. So, SKB appears as a fully faithful subcategory SKB ֒→ DiGp of the category DiGp of digroups, where a digroup is a triple (G, * , •) of a set G endowed with two group structures with same unit. This notion was introduced in [8] and devised during discussions between the first author and G. Janelidze. There are two forgetful functors U i : DiGp → Gp, i ∈ {0, 1}, associating respectively the first and the second group structures. They both reflect isomorphisms. Since U 0 is left exact and reflects isomorphisms, it naturally allows the lifting of the protomodular aspects of the category Gp of groups to the category DiGp. In turn, the left exact fully faithful embedding SKB ֒→ DiGp makes SKB a pointed protomodular category. The protomodular axiom was introduced in [5] in order to extract the essence of the homological constructions and in particular to induce an intrinsic notion of exact sequence. In this paper, after recalling the basic facts about protomodular categories, we study the "protomodular aspects" of left skew braces, in particular in relation to the category of digroups. We study the notion of commutator of ideals in a left skew brace (in the literature, "product" of ideals of skew braces is often considered). We show that Huq=Smith for left skew braces. Notice that Huq = Smith for digroups and near-rings [18]. We give a set of generators for the commutator of two ideals, and prove that every ideal of a left skew brace has a centralizer. Basic recalls on protomodular categories In this work, any category E will be supposed finitely complete, which implies that it has a terminal object 1. The terminal map from X is denoted τ X : X → 1. Given any map f : X → Y , the equivalence relation R[f ] on X is produced by the pullback of f along itself. The map f is said to be a regular epimorphism in E when f is the quotient of R[f ]. When it is the case, we denote it by a double head arrow X ։ Y . 1.1. Pointed protomodular categories. The category E is said to be pointed when the terminal object 1 is initial as well. Let us recall that a pointed category A is additive if and only if, given any split epimorphism f : X ⇄ Y, f s = 1 Y , the following downward pullback: Kerf / / k f / / X f 1 / / 0Y / / O O 0K O O Y O O s O O is an upward pushout, namely if and only if X is the direct sum (= coproduct) of Y and Kerf . Let us recall the following: Definition 1.1. [5] A pointed category E is said to be protomodular when, given any split epimorphism as above, the pair (k f , s) of monomorphisms is jointly strongly epic. This means that the only subobject u : U X containing the pair (k f , s) of subobjects is, up to isomorphism, 1 X . It implies that, given any pair (f, g) : X ⇒ Z of arrows which are equalized by k f and s, they are necessarily equal (take the equalizer of this pair). Pulling back the split epimorphisms along the initial map 0 Y : 1 Y being a left exact process, the previous definition is equivalent to saying that this process reflects isomorphisms. The category Gp of groups is clearly pointed protomodular. This is the case of the category Rng of rings as well, and more generally, given a commutative ring R, of any category R-Alg of any given kind of R-algebras without unit, possibly nonassociative. This is in particular the case of the category R-Lie of Lie R-algebras. Even for R a non-commutative ring, in which case R-algebras have a more complex behaviour (they are usually called R-rings, see [2, p. 36] or [14, p. 52]), one has that the category R-Rng of R-rings is pointed protomodular, as can be seen from the fact that the forgetul functor R-Rng → Ab reflects isomorphisms and Ab is protomodular. The pointed protomodular axiom implies that the category E shares with the category Gp of groups the following well-known Five Principles: (1) a morphism f is a monomorphism if and only if its kernel Kerf is trivial [5]; (2) any regular epimorphism is the cokernel of its kernel, in other words any regular epimorphism produces an exact sequence, which determines an intrinsic notion of exact sequences in E [5]; (3) there a specific class of monomorphisms u : U X, the normal monomorphisms [7], see next section ; (4) there is an intrinsic notion of abelian object [7], see section 3.1.1; (5) any reflexive relation in E is an equivalence relation, i.e. the category E is a Mal'tsev one [6]. So, according to Principle (1), a pointed protomodular category is characterized by the validity of the split short five lemma. Generally, Principle (5) is not directly exploited in Gp; we shall show in Section 3.4.2 how importantly it works out inside a pointed protomodular category E. Pointed protomodular varieties of universal algebras are characterized in [12]. 1.2. Normal monomorphisms. Definition 1.2. [7] In any category E, given a pair (u, R) of a monomorphism u : U X and an equivalence relation R on X, the monomorphism u is said to be normal to R when the equivalence relation u −1 (R) is the indiscrete equivalence relation ∇ X = R[τ X ] on X and, moreover, any commutative square in the following induced diagram is a pullback: U × U d U 0 / /ǔ / / d U 1 R d R 0 d R 1 U / / u / / s U 0 O O X s R 0 O O In the category Set, provided that U = ∅, these two properties characterize the equivalence classes of R. By the Yoneda embedding, this implies the following: Proposition 1.3. Given any equivalence relation R on an object X in a category E, for any map x, the following upper monomorphismx = d R 1 . x is normal to R: I x R ; / /x / / R d R 0 d R 1 / / X 1 / / x / / X In a pointed category E, taking the initial map 0 X : 1 X gives rise to a monomorhism ι R : I R X which is normal to R. This construction produces a preorder mapping ι X : Equ X E → Mon X E from the preorder of the equivalence relations on X to the preorder of subobjects of X which preserves intersections. Starting with any map f : X → Y , we get I R[f ] = Kerf which says that any kernel map k f is normal to R[f ]. Principle (3) above is a consequence of the fact [7] that in a protomodular category a monomorphism is normal to at most one equivalence relation (up to isomorphism). So that being normal, for a monomorphism u, becomes a property in this kind of categories. This is equivalent to saying that the preorder homomorphism ι X : Equ X E → Mon X E reflects inclusions; so, the preorder Norm X of normal subobjects of X is just the image ι X (Equ X ) ⊂ Mon X . Then, in the regular context, given any map f : X → Y , the following canonical factorization m is necessarily a monomorphism: Im f ! ! m ! ! X q f = = = = ④ ④ ④ ④ ④ ④ ④ f / / Y This produces a canonical decomposition of the map f in a monomorphism and a regular epimorphism which is stable under pullbacks. Now, given any regular epimorhism f : X ։ Y and any subobject u : U X, the direct image f (u) : f (U ) Y of u along the regular epimorphism f is given by f (U ) = Im f.u Y . Any variety in the sense of Universal Algebra is exact and regular epimorphisms coincide with surjective homomorphisms. 1.4. Homological categories. The significance of pointed protomodular categories grows up in the regular context since, in this context, the split short five lemma can be extended to any exact sequence. Furthermore, the 3 × 3 lemma, Noether isomorphisms and snake lemma hold; they are all collected in [3]. This is the reason why a regular pointed protomodular category E is called homological. (1) i : (G, * ) (K, * ) is normal in Gp, (2) i : (G, •) (K, •) is normal in Gp, (3) for all (x, y) ∈ K × K, x − * * y ∈ G if and only if x −• • y ∈ G. Skew braces. The following observation is very important: Proposition 2.2. Let (G, * , •) be any skew brace. Consider the mapping λ : G × G → G defined by λ(a, u) = a − * * (a • u). Then: (1) λ a = λ(a, −) is underlying a group homomorphism (G, •) → Aut(G, * ) , and this condition is equivalent to (B); (2) we have (1) λ(a −• , u) = (a −• ) − * * (a −• • u) = a −• • (a * u). Proof. For (1), see [15]. For (2), we have (a −• • a) * (a −• ) − * * (a −• • u) = (a −• ) − * * (a −• * u) = λ(a −• , u). 2.3. First properties of skew braces. The following observation is straightforward: Proposition 2.3. SKB is a Birkhoff subcategory of DiGp. This means that any subobject of a skew brace in DiGp is a skew brace and that, given any surjective homomorphism f : X ։ Y in DiGp, the digroup Y is a skew brace as soon as so is X. In this way, any equivalence relation R in DiGp on a skew brace X actually lies in SKB since it determines a subobject R ⊂ X × X in DiGp and, moreover, its quotient in SKB is its quotient in DiGp. The first part of this last sentence implies that any normal subobject u : U X in DiGp with X ∈ SKB is normal in SKB. We are now going to show that the normal subobjects in SKB coincide with the ideals of [15]. (1) i : (G, * ) (K, * ) is normal in Gp, (2) i : (G, •) (K, •) is normal in Gp, (3 ′ ) λ x (G) = G for all x ∈ K. Proof. Suppose (1) and (2). We are going to show (3) ⇐⇒ (3 ′ ), with (3) given in Proposition 2.1. (i) x −• • y ∈ G ⇒ x − * * y ∈ G if and only if λ x (G) ⊂ G, setting y = x • u, u ∈ G. (ii) from (1): x − * * y ∈ G ⇒ x −• • y ∈ G if and only if λ x −• (G) ⊂ G, setting y = x * u, u ∈ G. Finally λ x (G) ⊂ G for all x is equivalent to λ x (G) = G. Corollary 2.5. A subobject i : (G, * , •) (K, * , •) is normal in the category SKB if and only if it is an ideal in the sense of [15], namely is such that: 1) i : (G, •) (K, •) is normal, 2) G * a = a * G for all a ∈ K, 3) λ a (G) ⊂ G for all a ∈ K. Proof. Straightforward. Being a variety in the sense of Universal Algebra, SKB is finitely cocomplete; accordingly it has binary sums (called coproducts as well). So, SKB is a semiabelian category according to the definition introduced in [17]: Definition 2.6. A pointed category E is said to be semi-abelian when it is protomodular, exact and has finite sums. From the same [17], let us recall the following observation which explains the choice of the terminology: a pointed category E is abelian if and only if both E and E op are semi-abelian. Similarly the category SKBT op is homological as well, without being semi-abelian. As any category of topological semi-abelian algebras, both DiGpT op and SKBT op are finitely cocomplete, see [4]. Skew braces and their commutators 3.1. Protomodular aspects. 3.1.1. Commutative pairs of subobjects, abelian objects. Given any pointed category E, the protomodular axiom applies to the following specific downward pullback: X / / rX / / X × Y pY 1 / / 0Y / / O O 0K O O Y O O lY O O where the monomorphisms are the canonical inclusions. This is the definition of a unital category [6]. In this kind of categories there is an intrisic notion of commutative pair of subobjects: Definition 3.1. Let E be a unital category. Given a pair (u, v) of subobjects of X, we say that the subobjects u and v cooperate (or commute) when there is a (necessarily unique) map ϕ, called the cooperator of the pair (u, v), making the following diagram commute: U { { lU { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ u ❅ ❅ ❅ ❅ ❅ ❅ U × V ϕ / / X V c c rV c c • • • • • • • > > v > > ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ We denote this situation by [u, v] = 0. A subobject u : U Y is central when [u, 1 X ] = 0. An object X is commutative when [1 X , 1 X ] = 0. Clearly [1 X , 1 X ] = 0 gives X a structure of internal unitary magma, which, E being unital, is necessarily underlying an internal commutative monoid structure. When E is protomodular, this is actually an internal abelian group structure, so that we call X an abelian object [7]. This gives rise to a fully faithful subcategory Ab(E) ֒→ E, which is additive and stable under finite limits in E. From that we can derive: Proposition 3.2. [7] A pointed protomodular category E is additive if and only if any monomorphism is normal. 3.1.2. Connected pairs (R, S) of equivalence relations. Since a protomodular category is necessarily a Mal'tsev one, we can transfer the following notions. Given any pair (R, S) of equivalence relations on the object X in E, take the following rightward and downward pullback: R − → × X S pS / / pR S d S 0 o o rS o o R d R 1 / / O O lR O O X o o s R 0 o o O O s S 0 O O where l R and r S are the sections induced by the maps s R 0 and s S 0 . Let us recall the following definition from [10]: Definition 3.3. In a Mal'tsev category E, the pair (R, S) is said to be connected when there is a (necessarily unique) morphism p : R − → × X S → X, xRySz → p(xRySz) such that pr S = d S 1 and pl R = d R 0 , namely such that the following identities hold: p(xRySy) = x and p(yRySz) = z. This morphism p is then called the connector of the pair, and we denote the situation by [R, S] = 0. From [11], let us recall that: R − → × X Sf / / p (R,S) R − → × YS p (R,S) X f / / Y wheref is the natural factorization induced by f −1 (R) and S ⊂ f −1 (S). A pointed Mal'tsev category is necessarily unital. From [10], in any pointed Mal'sev category E, we have necessarily (2) [R, S] = 0 ⇒ [I R , I S ] = 0 In this way, the "Smith commutation" [22] implies the "Huq commutation" [16]. 3.2. Huq=Smith. The converse is not necessarily true, even if E is pointed protomodular, see Proposition 3.6 below. When this is the case, we say that E satisfies the (Huq=Smith) condition. Any pointed strongly protomodular category satisfies (Huq=Smith), see [10]. (Huq=Smith) is true for Gp by the following straighforward: Proof. We can use the counterexample introduced in [8] for another purpose. Start with an abelian group (A, +) and an object a such that −a = a. Then define θ : A × A → A × A as the involutive bijection which leaves fixed any object (x, y) except (a, a) which is exchanged with (−a, a). Then defined the group structure (A × A, •) on A × A as the transformed along θ of (A × A, +). So, we get: (x, z) • (x ′ , z ′ ) = θ(θ(x, z) + θ(x ′ , z ′ )) Clearly we have (a, a) −• = (a, −a). Since the second projection π : A × A → A is such that πθ = π, we get a digroup homomorphism π : (A × A, +, •) → (A, +, +) whose kernel map is, up to isomorphism, ι A : (A, +, +) (A × A, +, •) defined by ι(a) = (a, 0 Let T be any finitary algebraic theory, and denote by T(E) the category of internal T-algebras in E. Let us recall that, given any variety of algebras V(T), we have a Yoneda embedding for the internal T-algebras, namely a left exact fully faithful factorization of the Yoneda embedding for E: (x, y) − (x ′ , y) + (x", y) = (x, y) • (x ′ , y) −• • (x", y) namely (x, y) • (x ′ , y) −• • (x", y) = (x − x ′ + x",T(E)Ȳ T / / ❴ ❴ ❴ ❴ ❴ ❴ U T F (E op , V(T)) F (E op ,U ) E Y / / F (E op , Set) where U : V(T) → Set is the canonical forgetful functor. Proof. Straightforward, setting ϕ(u, v) = u + v and using an Eckmann-Hilton argument. Proposition 3.10 (SKB does satisfy (Huq=Smith)). Let R and S be two equivalence relations on an object X ∈ SKB. The following conditions are equivalent: (1) [I R , I S ] = 0; (2) [U 0 (U ), U 0 (V )] = 0, [U 1 (U ), U 1 (V )] = 0 and x * y − * * z = x • y −• • z for all xRySz; (3) [R, S] = 0. Proof. The identity x * y − * * z = x • y −• • z is equivalent to y −• • z = x −• • (x * y − * * z) = (x −• • x) * (x −• ) − * * (x −• • (y − * * z)) = = (x −• ) − * * (x −• • (y − * * z)), which, in turn, is equivalent to λ x −• (y − * * z) = y −• • z. Suppose xRySy. Setting z = y * v, v ∈ I S , this is equivalent to λ x −• (v) = y −• • (y * v) = λ y −• (v) by (1). This in turn is equivalent to λ y • λ x −• (v) = λ y•x −• (v) = v, v ∈ I S . Setting y = u • x, u ∈ I R , this is equivalent to λ u (v) = v, (u, v) ∈ I R × I S . Now, by Proposition 3.9, [I R , I S ] = 0 is equivalent to: for all (u, v) ∈ I R × I S , we get λ u (v) = v, [U 0 (U ), U 0 (V )] = 0 and [U 1 (U ), U 1 (V )] = 0. So we get [1) ⇐⇒ 2)]. Suppose (2). From [U 0 (U ), U 0 (V )] = 0, we know by Proposition 3.9 that p(x, y, z) = x * y − * * z is a group homomorphism (R − → × X S, * ), → (X, * ), and from [U 1 (U ), According to Theorem 3.8, we get the following: Corollary 3.11. Given any category E, the category SKBE satisfies (Huq= Smith). This is the case in particular for the category SKBT op of topological skew braces. U 1 (V )] = 0 that q(x, y, z) = x • y −• • z is a group homomorphism (R − → × X S, •) → (X, •). If p = q, Homological aspects of commutators. 3.4.1. Abstract Huq commutator. Suppose now that E is any finitely cocomplete regular unital category. In this setting, we gave in [9], for any pair u : U X, v : V X of subobjects, the construction of a regular epimorphism ψ (u,v) which universally makes their direct images cooperate. Indeed consider the following diagram, where Q[u, v] is the limit of the plain arrows: U z z lU z z t t t t t t t t t t " " v " " ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ U × Vψ (u,v) / / Q[u, v] X ψ (u,v) o o V d d rV d d ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ < < v < < ① ① ① ① ① ① ① ① ① The map ψ (u,v) is necessarily a regular epimorphism and the mapψ (u,v) induces the cooperator of the direct images of the pair (u, v) along ψ (u,v) . This regular epimorphism ψ (u,v) measures the lack of cooperation of the pair (u, v) in the sense that the map ψ (u,v) is an isomorphism if and only if [u, v] = 0. We then get a symmetric tensor product: I R[ψ (−,−) ] : Mon X × Mon X → Mon X of preordered sets. Since the map ψ (u,v) is a regular epimorphism, its distance from being an isomorphism is its distance from being a monomorphism, which is measured by the kernel equivalence relation R[ψ (u,v) ]. Accordingly, in the homological context, it is meaningful to introduce the following definition, see also [20]: Definition 3.12. Given any finitely cocomplete homological category E and any pair (u, v) of subobjects of X, their abstract Huq commutator [u, v] is defined as I R[ψ (u,v) ] or equivalently as the kernel map k ψ (u,v) . By this universal definition, in the category Gp, this [u, v] coincides with the usual [U, V ]. Abstract Smith commutator. Suppose E is a regular category. Then, given any regular epimorphism f : X ։ Y and any equivalence relation R on X, the direct image f (R) Y × Y of R X × X along the regular epimorphism f × f : X × X ։ Y × Y is reflexive and symmetric, but generally not transitive. Now, when E is a regular Malt'sev category, this direct image f (R), being a reflexive relation, is an equivalence relation. Suppose moreover that E is finitely cocomplete. Let (R, S) be a pair of equivalence relations on X, and consider the following diagram, where Q[R, S] is the colimit of the plain arrows: Since the map χ (R,S) is a regular epimorphism, its distance from being an isomorphism is its distance from being a monomorphism, which is exactly measured by its kernel equivalence relation R[χ (R,S) ]. Accordingly, we give the following definition: From that, we can assert the following: Skew braces and their commutators. Since the categories SKB and SKBT op are finitely cocomplete homological categories, all the results of the previous section concerning commutators do apply and, in particular, thanks to the (Huq=Smith) condition, the two notions of commutator are equivalent. It remains now to make explicit the description of the Huq commutator. We will determine a set of generators for the Huq commutator of two ideals in a skew brace: Proof. Assume that the mapping µ : I × J → A/K, µ(i, j) = i * j * K is a skew brace morphism for some ideal K of A. Then (i • j) • K = (i • K) • (j • K) = (i * K) • (j * K) = = µ(i, 1) • µ(1, j) = µ((i, 1) • (1, j)) = µ(i, j) = µ((1, j) • (i, 1)) = = µ((1, j) • µ(i, 1)) = (j * K) • (i * K) = (j • K) • (i • K) = (j • i) • K. This proves that the set (1) is contained in K. Similarly, (i * j) * K = (i * K) * (j * K) = µ(i, 1) * µ(1, j) = µ((i, 1) * (1, j)) = µ(i, j) = = µ((1, j) * (i, 1)) = µ((1, j) * µ(i, 1)) = (j * K) * (i * K) = (j * i) * K. Thus the set (2) is contained in K. Also, (i • j) * K = (i • j) • K = (i • K) • (j • K) = (i * K) • (j * K) = = µ(i, 1)) • µ(1, j) = µ((i, 1) • (1, j)) = µ(i, j) = µ((i, 1) * (1, j)) = = µ(i, 1) * µ(1, j) = (i * K) * (j * K) = (i * j) * K. Hence the set (3) is also contained in K. Conversely, let K be the ideal of A generated by the union of the three sets. It is then very easy to check that he mapping µ : I × J → A/K, µ(i, j) = i * j * K is a skew brace morphism. Hence if both J 1 and J 2 centralize I, then J 1 * J 2 ⊆ C (A, * ) (I), and J 1 • J 2 ⊆ C (A,•) (I) ∩ ker λ| I . Therefore J 1 * J 2 = J 1 • J 2 centralizes I. It follows that the set of all the ideals of A that centralize I is a lattice. Hence the maximal element in the set of all the ideals of A that centralize I is the largest element in that set. In particular, the centralizer of the improper ideal of a left skew brace A is the center of A. A description of the free left skew brace over a set X is available, in a language very different from ours, in [21]. 1. 3 . 3Regular context. Let us recall from [1] the following: Definition 1.4. A category E is regular when it satisfies the two first conditions, and exact when it satisfies all the three conditions: (1) regular epimorphisms are stable under pullbacks; (2) any kernel equivalence relation R[f ] has a quotient q f ; (3) any equivalence relation R is a kernel equivalence relation. 2 . 2Protomodular aspects of skew braces 2.1. Digroups. From [8], we get the characterization of normal monomorphisms in DiGp: Proposition 2.1. A suboject i : (G, * , •) (K, * , •) is normal in the category DiGp if and only if the three following conditions hold: Proposition 2.4. A subobject i : (G, * , •) (K, * , •) is normal in the category SKB if and only if the three following conditions hold: 2. 4 . 4Internal skew braces. Given any category E, the notion of internal group, digroup and skew brace is straightforward, determining the categories GpE, DiGpE and SKBE. Since GpE is protomodular, so are the two others. An important case is produced with E = T op the category of topological spaces. Although T op is not a regular category, so is the category GpT op, the regular epimorphisms being the open surjective homomorphisms. So GpT op is homological but not semi-abelian. Now let f : X → Y be any map in DiGpT op. Let us show that R[f ] has a quotient in DiGpT op. Take its quotient q R[f ] : X ։ Q f in DiGp, then endow Q f with the quotient topology with respect to R[f ]; then q R[f ] is an open surjective homomorphism since so is U 0 (q R[f ] ). Accordingly, a regular epimorphism in DiGpT op is again an open surjective homomorphism. Moreover this same functor U 0 : DiGpT op → GpT op being left exact and reflecting the homeomorphic isomorphisms, it reflects the regular epimorphisms; so, these regular epimorphisms in DiGpT op are stable under pullbacks. Accordingly the category DiGpT op is regular. Lemma 3 . 4 . 34Let E be a Mal'tsev category, f : X → Y any map, (R, S) any pair of equivalence relations on X, (R,S) any pair of equivalence relations on Y such that R ⊂ f −1 (R) and S ⊂ f −1 (S). Suppose moreover that [R, S] = 0 and [R,S] = 0. Then the following diagram necessarily commutes: Proposition 3. 5 . 5Let (R, S) be a pair of equivalence relations in Gp on the group (G, * ). The following conditions are equivalent: (1) [I R , I S ] = 0; (2) p(x, y, z) = x * y −1 * z defines a group homomorphism p : G × G × G → G; (3) [R, S] = 0.Proposition 3.6. The category DiGp of digroups does not satisfy (Huq=Smith). y). Now take y = a = x ′ and a = x = −a. Then we get:(x, a) • (a, a) −• • (x", a) = (x, a) • (a, −a) • (x", a) = (x + a, 0) • (x", a) = (x+a+x", a), if moreover a = x" = −a. Now, clearly we get x+a+x" = x−a+x" since a = −a.However we have the following very general observation:Proposition 3.7. Let E be any pointed Mal'tsev satisfying (Huq=Smith). So is any functor category F (C, E). Proof. Let (R, S) be a pair of equivalence relation on an object F ∈ F (C, E). We have [R, S] = 0 if and only if for each object C ∈ C we have [R(C), S(C)] = 0 since, by Lemma 3.4, the naturality follows. In the same way, if (u, v) is a pair of subfunctors of F , we have [u, v] = 0 if and only if for each object C ∈ C we have [u(C), v(C)] = 0. Suppose now that E satisfies (Huq=Smith), and that [I R , I S ] = 0. So, for each object C ∈ C we have [I R (C), I S (C)] = 0, which implies [R(C), S(C)] = 0. Accordingly [R, S] = 0. Theorem 3. 8 . 8Let T be any finitary algebraic theory such that the associated variety of algebras V(T) is pointed protomodular. If V(T) satisfies (Huq=Smith), so does any category T(E).Proof. If V(T) satisfies (Huq=Smith), so does F (E op , V(T)) by the previous proposition. Accordingly,Ȳ T being left exact and fully faithful, so does T(E). 3. 3 . 3Any category SKBE does satisfy (Huq=Smith). Proposition 3 . 9 . 39Given any pair (U, V ) of subobjects of X in SKB, the following conditions are equivalent:(1) [U, V ] = 0; (2) for all (u, v) ∈ U × V , we get u • v = u * vand this restriction is commutative; (3) for all (u, v) ∈ U × V, λ u (v) = v, [U 0 (U ), U 0 (V )] = 0 and [U 1 (U ), U 1 (V )] = 0. Accordingly, an abelian object in SKB is necessarily of the form (A, +, +) with (A, +) abelian. this produces the desired R − → × X S → X in SKB showing that [R, S] = 0. Whence [(2) ⇒ (3)]. We have already noticed that the last implication [(3) ⇒ (1)] holds in any pointed category. Notice that, here, in consideration of the pullback defining R − → × X S, the role of the projections d 0 and d 1 have been interchanged. This map χ (R,S) measures the lack of connection between R and S, see[9]:Theorem 3.13. Let E be a finitely cocomplete regular Mal'tsev category. Then the map χ (R,S) is a regular epimorphism and is the universal one which makes the direct images χ (R,S) (R) and χ (R,S) (S) connected. The equivalence relations R and S are connected (i.e. [R, S] = 0) if and only if χ (R,S) is an isomorphism. Definition 3 . 14 . 314Let E be any finitely cocomplete regular Mal'tsev category. Given any pair (R, S) of equivalence relations on X, their abstract Smith commutator [R, S] is defined as the kernel equivalence relation R[χ (R,S) ] of the map χ (R,S) . In this way, we define a symmetric tensor product [−, −] = R[χ (−,−) ] : Equ X × Equ X → Equ X of preorered sets. It is clear that, with this definition, we get [R, S] = 0 in the sense of connected pairs if and only if [R, S] = ∆ X (the identity equivalence relation on X) in the sense of this new definition. This is coherent since ∆ X is effectively the 0 of the preorder Equ X . Let us recall the following: Proposition 3.15. Let E be a pointed regular Mal'tsev category. Let f : X ։ Y be a regular epimorphism and R an equivalence relation on X. Then the direct image f (I R ) of the normal subjobject I R along f is I f (R) . Proposition 3 . 16 . 316Let E be a finitely cocomplete homological category. Given any pair (R, S) of equivalence relations on X, we have [I R , I S ] ⊂ I [R,S] .Proof. From (2), we get[χ (R,S) (R), χ (R,S) (S)] = 0 ⇒ [I χ (R,S) (R) , I χ (R,S) (S) ] = 0By the previous proposition we have:0 = [I χ (R,S) (R) , I χ (R,S) (S) ] = [χ (R,S) (I R ), χ (R,S) (I S )].Accordingly, by the universal property of the regular epimorphism ψ(IR,IS ) I R , I S ] Q[R, S] which shows that [I R , I S ] ⊂ I [R,S] . Theorem 3.17. In a finitely cocomplete homological category E the following conditions are equivalent: (1) E satisfies (Huq=Smith); (2) [I R , I S ] = I [R,S] for any pair (R, S) of equivalence relations on X. Under any of these conditions, the regular epimorphisms χ (R,S) and ψ (IR,IS ) do coincide. Proof. Suppose (2). Then [I R , I S ] = 0 means that ψ (IR,IS ) is an isomorphism, so that 0 = [I R , I S ] = I [R,S] . In a homological category I [R,S] = 0 is equivalent to [R, S] = 0. Conversely, suppose (1). We have to find a factorization: X ψ (I R ,I S ) / / / / show that [ψ (IR,IS ) (R), ψ (IR,IS ) (S)] = 0. By (1) this is equivalent to 0 = [I ψ (I R ,I S ) (R) , I ψ (I R ,I S ) (S) ], namely to 0 = [ψ (IR,IS ) (I R ), ψ (IR,IS) (I S )] by Proposition 3.15. This is true by the universal property of the regular epimorphism ψ (IR,IS ) . Proposition 3 . 18 . 318If I and J are two ideals of a left skew brace (A, * , •), their Huq commutator [I, J] is the ideal of A generated by the union of the following three sets:(1) the set { i • j • (j • i) −• | i ∈ I, j ∈ J }, (which generates the commutator [I, J] (A,•) of the normal subgroups I and J of the group (A, •)); (2) the set { i * j * (j * i) − * | i ∈ I, j ∈ J }, (which generates the commutator [I, J] (A, * ) of the normal subgroups I and J of the group (A, * )); and (3) the set { (i • j) * (i * j) − * | i ∈ I, j ∈ J }. Proposition 3 . 19 . 319For an ideal I of a left skew brace A, there is a largest ideal of A that centralizes I (the centralizer of I).Proof. The zero ideal centralizes I and the union of a chain of ideals that centralize I centralizes I. Hence there is a maximal element in the set of all the ideals of A that centralize I. Now if J 1 and J 2 are two ideals of A, then J 1 * J 2 = J 1 • J 2 is the join of{J 1 , J 2 } in the lattice of all ideals of A. Now J 1 centralizes I if and only if (1) J 1 ⊆ C (A, * ) (I), the centralizer of the normal subgroup I in the group (A, * ); (2) J 1 ⊆ C (A,•) (I), the centralizer of the normal subgroup I in the group (A, •); and (3) J 1 is contained in the kernel of the group morphism λ| I : (A, •) → Aut(I, * ), which is a normal subgroup of (A, •). Similarly for J 2 . ). The commutativity of the law + makes [ι A , ι A ] = 0 insideDiGp. We are going to show that, however we do not have [R[π], R[π]] = 0. If it was the case, according to the previous proposition and considering the images by U 0 and U 1 of the desired ternary operation, we should have, for any triple (x, y)R[π](x ′ , y)R[π](x", y): This product is with respect to the product · in the brace A defined, for every x, y ∈ A by x · y = y − * * λ x (y). Then, for every i ∈ I and j ∈ J, i · j = j − * * λ i (j) = j − * * i − * * (i • j) = (. study of product I · J of two ideals I, J of a (left skew) brace (A, * , •). i * j) − * * (i • jIt the literature, great attention has been posed in the study of product I · J of two ideals I, J of a (left skew) brace (A, * , •). This product is with respect to the product · in the brace A defined, for every x, y ∈ A by x · y = y − * * λ x (y). Then, for every i ∈ I and j ∈ J, i · j = j − * * λ i (j) = j − * * i − * * (i • j) = (i * j) − * * (i • j). . Clearly, for a left skew brace A, the Huq commutator [I, J] is equal to the Huq commutator [J, IClearly, for a left skew brace A, the Huq commutator [I, J] is equal to the Huq commutator [J, I]. so that the left annihilator of I in (A, ·) is equal to the right annihilator of I in (A, ·). Moreover, the condition "I · J = 0" can be equivalently expressed as. I · J = (j · I) − * Also, the kernel of the group homomorphism λ| I : (A, •) → Aut(I, * ). Also, I · J = (J · I) − * , so that the left annihilator of I in (A, ·) is equal to the right annihilator of I in (A, ·). Moreover, the condition "I · J = 0" can be equivalently expressed as "J is contained in the kernel of the group homomorphism λ| I : (A, •) → Aut(I, * ). Exact categories. M Barr, Lecture Notes in Math. 236SpringerM. Barr, Exact categories, Lecture Notes in Math. 236 (1971), Springer, Berlin, pp. 1-120. Modules over coproducts of rings. G M Bergman, Trans. Amer. Math. Soc. 200G. M. Bergman, Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 1-32. Mal'cev, Protomodular, Homological and Semi-Abelian Categories. F Borceux, D Bourn, Math. and Its Applications. 566KluwerF. Borceux and D. Bourn, Mal'cev, Protomodular, Homological and Semi-Abelian Categories, Math. and Its Applications 566, Kluwer, 2004. Topological semi-Abelian algebras. F Borceux, M M Clementino, Advances in Math. 190F. Borceux and M. M. Clementino, Topological semi-Abelian algebras, Advances in Math. 190 (2005), 425-453. Normalization equivalence, kernel equivalence and affine categories. D Bourn, Lecture Notes in Math. 1488SpringerD. Bourn, Normalization equivalence, kernel equivalence and affine categories, Lecture Notes in Math. 1488 (1991), Springer, Berlin, pp. 43-62. Mal'tsevCategories and fibration of pointed objects. D Bourn, Applied categorical structures. 4D. Bourn, Mal'tsevCategories and fibration of pointed objects, Applied categorical structures 4 (1996), 302-327. Normal subobjects and abelian objects in protomodular categories. D Bourn, J. Algebra. 228D. Bourn, Normal subobjects and abelian objects in protomodular categories, J. Algebra 228 (2000), 143-164. Normal functors and strong protomodularity. D Bourn, Theory Appl. Categ. 79D. Bourn, Normal functors and strong protomodularity, Theory Appl. Categ. 7(9) (2000), 206-218. Galois theory, Hopf algebras, and Semiabelian categories. D Bourn, Commutator theory in regular Mal'tsev categories. B. Pareigis and W. Tholen EdsAmer. Math. Soc43D. Bourn, Commutator theory in regular Mal'tsev categories, in: "Galois theory, Hopf al- gebras, and Semiabelian categories", G. Janelidze, B. Pareigis and W. Tholen Eds, Fields Institute Communications 43, Amer. Math. Soc. (2004), pp. 61-75. Centrality and normality in protomodular categories. D Bourn, M Gran, Theory and Applications of Categories. 9D. Bourn and M. Gran, Centrality and normality in protomodular categories, Theory and Applications of Categories 9 (2002), 151-165. Centrality and connectors in Maltsev categories. D Bourn, M Gran, Algebra Universalis. 48D. Bourn and M. Gran, Centrality and connectors in Maltsev categories, Algebra Universalis 48 (2002), 309-331. Characterization of protomodular varieties of universal algebras. D Bourn, G Janelidze, Theory Appl. Categ. 116D. Bourn and G. Janelidze, Characterization of protomodular varieties of universal algebras, Theory Appl. Categ. 11(6) (2003), 143-147. On some unsolved problems in quantum group theory. V G P Drinfel&apos;d ; P, Kulish Ed, Lecture Notes in Math. 1510. Leningrad; BerlinSpringerQuantum groupsV. G. Drinfel'd, On some unsolved problems in quantum group theory, in "Quantum groups (Leningrad, 1990)", P. P. Kulish Ed., Lecture Notes in Math. 1510, Springer, Berlin, 1992, pp. 1-8. Semilocal categories and modules with semilocal endomorphism rings. A Facchini, Progress in Math. 331Birkhäuser/SpringerA. Facchini, "Semilocal categories and modules with semilocal endomorphism rings", Progress in Math. 331, Birkhäuser/Springer, Cham, 2019. Skew braces and the Yang-Baxter equation. L Guarnieri, L Vendramin, Math. Comp. 86307L. Guarnieri and L. Vendramin, Skew braces and the Yang-Baxter equation, Math. Comp. 86(307) (2017), 2519-2534. Commutator, nilpotency and solvability in categories. S A Huq, Quart. J. Oxford. 19S.A. Huq, Commutator, nilpotency and solvability in categories, Quart. J. Oxford 19, 1968, 363-389. Semi-abelian categories. G Janelidze, L Márki, W Tholen, J. Pure Appl. Alg. 168G. Janelidze, L. Márki and W. Tholen, Semi-abelian categories, J. Pure Appl. Alg. 168 (2002), 367-386. Commutators for near-rings: Huq = Smith. G Janelidze, L Márki, S Veldsman, Algebra Universalis. 762G. Janelidze, L. Márki and S. Veldsman, Commutators for near-rings: Huq = Smith, Algebra Universalis 76(2) (2016), 223-229. On skew braces and their ideals. A Konovalov, A Smoktunowicz, L Vendramin, Exp. Math. 301A. Konovalov, A. Smoktunowicz and L. Vendramin, On skew braces and their ideals, Exp. Math. 30 (2021), no. 1, 95-104. Normalities and commutators. S Mantovani, G Metere, J. Algebra. 324S. Mantovani and G. Metere, Normalities and commutators, J. Algebra 324 (2010), 2568- 2588. A construction of the free skew brace. J Orza, J. Orza, A construction of the free skew brace, https://arxiv.org /abs/2002.12131. Mal'tsevVarieties. J D H Smith, Lecture Notes in Math. 554SpringerJ. D. H. Smith, "Mal'tsevVarieties", Lecture Notes in Math. 554, Springer, Berlin-New York, 1976. On skew braces (with an appendix by N. Byott and L. Vendramin). A Smoktunowicz, L Vendramin, J. Comb. Algebra. 21A. Smoktunowicz and L. Vendramin, On skew braces (with an appendix by N. Byott and L. Vendramin), J. Comb. Algebra 2(1) (2018), 47-86. Università di Padova, 35121 Padova. Matematica Dipartimento Di, Tullio Levi-Civita. Italy Email address: [email protected] di Matematica "Tullio Levi-Civita", Università di Padova, 35121 Padova, Italy Email address: [email protected] Università di Padova, 35121 Padova. Matematica Dipartimento Di, Tullio Levi-Civita. Italy Email address: [email protected] di Matematica "Tullio Levi-Civita", Università di Padova, 35121 Padova, Italy Email address: [email protected]
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[ "Topological Generality and Spectral Dimensionality in the Earth Mineral Dust Source Investigation (EMIT) using Joint Characterization and the Spectral Mixture Residual", "Topological Generality and Spectral Dimensionality in the Earth Mineral Dust Source Investigation (EMIT) using Joint Characterization and the Spectral Mixture Residual" ]
[ "D Sousa [email protected] \nDepartment of Geography\n\n", "C Small [email protected] \nLamont Doherty Earth Observatory San Diego State University\nColumbia University San Diego\n92182, 10984PalisadesCA, NY\n" ]
[ "Department of Geography\n", "Lamont Doherty Earth Observatory San Diego State University\nColumbia University San Diego\n92182, 10984PalisadesCA, NY" ]
[]
NASA's Earth Surface Mineral Dust Source Investigation (EMIT) mission seeks to use spaceborne imaging spectroscopy (hyperspectral imaging) to map the mineralogy of arid dust source regions. Here we apply recent developments in Joint Characterization (JC) and the spectral Mixture Residual (MR) to explore the information content of data from this novel mission. Specifically, for a mosaic of 20 spectrally diverse scenes we find: 1) a generalized three-endmember (Substrate, Vegetation, Dark; SVD) spectral mixture model is capable of capturing the preponderance (99% in 3 dimensions) of spectral variance with low misfit (99% pixels with <3.7% RMSE); 2) manifold learning (UMAP) is capable of identifying spatially coherent, physically interpretable clustering relationships in the spectral feature space; 3) UMAP yields results that are at least as informative when applied to the MR as when applied to raw reflectance; 4) SVD fraction information usefully contextualizes UMAP clustering relationships, and vice-versa (JC); and 5) when EMIT data are convolved to spectral response functions of multispectral instruments (Sentinel-2, Landsat 8/9, Planet SuperDove), SVD fractions correlate strongly across sensors but UMAP clustering relationships for the EMIT hyperspectral feature space are far more informative than for simulated multispectral sensors. Implications are discussed for both the utility of EMIT data in the near-term, and for the potential of high SNR spaceborne imaging spectroscopy more generally, to transform the future of optical remote sensing in the years and decades to come.This document is a non-peer reviewed preprint.
10.3390/rs15092295
[ "https://export.arxiv.org/pdf/2303.04876v1.pdf" ]
257,427,301
2303.04876
cf65895c4371c4db6d94239f92de3da5594d1bc0
Topological Generality and Spectral Dimensionality in the Earth Mineral Dust Source Investigation (EMIT) using Joint Characterization and the Spectral Mixture Residual D Sousa [email protected] Department of Geography C Small [email protected] Lamont Doherty Earth Observatory San Diego State University Columbia University San Diego 92182, 10984PalisadesCA, NY Topological Generality and Spectral Dimensionality in the Earth Mineral Dust Source Investigation (EMIT) using Joint Characterization and the Spectral Mixture Residual This document is a non-peer reviewed preprint. 1 of 19 2 of 19EMITjoint characterizationspectral mixture residualhyperspectraldimensionalitySVD model NASA's Earth Surface Mineral Dust Source Investigation (EMIT) mission seeks to use spaceborne imaging spectroscopy (hyperspectral imaging) to map the mineralogy of arid dust source regions. Here we apply recent developments in Joint Characterization (JC) and the spectral Mixture Residual (MR) to explore the information content of data from this novel mission. Specifically, for a mosaic of 20 spectrally diverse scenes we find: 1) a generalized three-endmember (Substrate, Vegetation, Dark; SVD) spectral mixture model is capable of capturing the preponderance (99% in 3 dimensions) of spectral variance with low misfit (99% pixels with <3.7% RMSE); 2) manifold learning (UMAP) is capable of identifying spatially coherent, physically interpretable clustering relationships in the spectral feature space; 3) UMAP yields results that are at least as informative when applied to the MR as when applied to raw reflectance; 4) SVD fraction information usefully contextualizes UMAP clustering relationships, and vice-versa (JC); and 5) when EMIT data are convolved to spectral response functions of multispectral instruments (Sentinel-2, Landsat 8/9, Planet SuperDove), SVD fractions correlate strongly across sensors but UMAP clustering relationships for the EMIT hyperspectral feature space are far more informative than for simulated multispectral sensors. Implications are discussed for both the utility of EMIT data in the near-term, and for the potential of high SNR spaceborne imaging spectroscopy more generally, to transform the future of optical remote sensing in the years and decades to come.This document is a non-peer reviewed preprint. Introduction NASA's Earth Mineral Dust Source Investigation (EMIT) mission is designed to study the mineralogy of Earth's dust-forming regions using spaceborne imaging spectroscopy [1]. The EMIT instrument is a Dyson imaging spectrometer with an 11° cross-track field of view, with a fast (F1/8) and wide-swath (1240 samples) optical system achieving roughly 7.4 nm spectral sampling across the 380 -2500 nm spectral range at high signal-to-noise [2]. EMIT was launched on July 14, 2022 via SpaceX Dragon and successfully autonomously docked to the forward-facing port of the International Space Station (ISS) [3]. EMIT data and algorithms are freely available for public use. While the stated purpose of the EMIT mission is to measure surface mineralogy and mineral dust in Earth's dust forming regions, these data also provide an unprecedented opportunity to advance our fundamental understanding of the spectral properties of the Earth surface more generally. Sensors like Landsat have collected multispectral satellite imagery for decades [4], but spaceborne hyperspectral (imaging spectroscopy) observations have been much more limited. Early missions like Hyperion [5] and HICO [6] were characterized by nontrivial noise limitations. High quality airborne data from sensors like AVIRIS [7] are available, but with spatial and temporal coverage limitations inherent to airborne platforms. A new generation of spaceborne imaging spectrometers is now starting to come online, with significant involvement from multiple space agencies. Such missions include the Italian Space Agency's CHIME [8] and PRISMA [9], DLR's DESIS [10] and EnMAP [11], JAXA's HISUI [12], and more. EMIT contributes an exciting new aspect to this international constellation and marks an important step towards a global hyperspectral monitoring system. EMIT began collecting high quality data shortly after launch, and some scenes are already available for download. While geographic coverage of the EMIT mission is inherently limited by the orbital parameters of the ISS, the scenes that have been acquired to date do include broad spectral diversity spanning a wide range of biogeophysical settings. These novel data offer an unprecedented opportunity to test the utility of recent developments in spectral image analysis, including both characterization and modeling. Here, we use a compilation of 20 spectrally diverse EMIT scenes to investigate the differences between high SNR spaceborne hyperspectral data and simulated multispectral data from common multispectral sensors like Sentinel-2, Landsat, and SuperDove. Specifically, we apply two novel approaches to hyperspectral image analysis: joint characterization [13] and the spectral mixture residual [14]. Joint characterization assumes that important spectral signals may be distributed across multiple scales of variance and provides a way to characterize these signals in a physically interpretable way. The mixture residual uses spectral mixture analysis to isolate lowvariance spectral signals (e.g., narrow mineral absorptions) from high-variance signals (e.g., land cover modulated variability in continuum shape and amplitude). Synthesizing these two novel approaches using a novel dataset, we address the following questions: 1) To what extent are EMIT reflectance spectra well characterized by a generalized threeendmember Substrate, Vegetation, Dark (SVD) model, such as has been shown effective for analysis of multispectral satellite imagery? 2) What quantitative and qualitative differences in spectral dimensionality and feature space topology are observed between EMIT reflectance and simulated multispectral data? 3) Does the spectral mixture model residual from EMIT data contain substantially more information than the mixture residual computed from multispectral data? If so, is this effectively captured by traditional dimensionality metrics like variance partition? Does the spectral feature space also manifest as substantially different manifold structure? 4) To what extent can joint characterization be used to reveal subtle but physically meaningful spectral signals in EMIT data? Are these signals spatially coherent? Data 20 EMIT scenes were selected on the basis of geographic and spectral diversity ( Figure 1). Scenes span 4 continents, sampling important global deserts (Sahara, Arabia, Atacama, Taklamakan, Gobi, Great Basin, Caspian), geologic structure (Zagros, Jabal Tuwaiq, Bushveld, Atacama, Hindu Kush) agriculture (San Joaquin, South African Cape, Hindu Kush), natural floristic diversity (Mata Atlântica, South African Cape, Okavango, Sierra Nevada), as well as some cryospheric targets (Patagonia, Tian Shan) and human settlements (Los Angeles). While this compilation does not achieve comprehensive global sampling, at least some representation is included from most major biomes. Data were downloaded in netCDF format from the USGS Land Processes Distributed Active Archive Center (LPDAAC) through the web portal: https://search.earthdata.nasa.gov/. Both reflectance and mask files were acquired. All 20 scenes were compiled into a single image mosaic ( Figure 2). For subsequent analysis, reflectance data were masked using the "Aggregate Flag" included in Band 7. SceneIDs are provided in Supplementary Table S1. This document is a non-peer reviewed preprint. of 19 Analysis Variance-Based Spectral Feature Space Figure 3 shows the low-order variance-based spectral feature space of the reflectance mosaic. The first three dimensions of these data are bounded by snow/ice (I), soil and rock substrates (S), illuminated photosynthetic vegetation (V), and dark targets like shadow and water (D). This loworder topology is consistent with previous regional compilations of AVIRIS imaging spectroscopy [14,15], as well as global compilations of Landsat [16][17][18], MODIS [19], and Sentinel-2 [20,21]. Significant spectral diversity in the S endmember is observed, associated with geologic variability in sand, bedrock, and soil of the sparsely vegetated scenes in the mosaic. Reflectance spectra for generalized endmembers are shown in the lower right. The image mosaic was then unmixed using the S, V and D endmembers and wavelength-specific mixture residual was retained, following [14]. The SVD model was found to yield a good fit, with average root-mean-square error (RMSE) approximately 3.1%, and 99% of pixels showing RMSE < 3.7%. The low-order feature space of the mixture residual mosaic is shown in Figure 4. The mixture residual effectively accentuates substrate EM variability by removing the highvariance component of spectral variability which is modeled by a simple linear mixing model. Multiple additional substrate EMs are clearly identifiable in the variance-based mixture residual feature space. Importantly, this endmember variability demonstrates correspondence to VNIR spectral curvature and narrow SWIR absorptions, rather than simple differences in albedo. In order to investigate EMIT's information content relative to multispectral imagery, the reflectance mosaic was convolved using the spectral response functions of the Sentinel-2A, Landsat 8/9 OLI, and Planet SuperDove sensors. Variance-based characterization was then repeated, including computation of the mixture residual. The difference in information content was then quantified using the partition of variance captured by eigenvalues of the low-order PC dimensions of the reflectance and residual spectra from each sensor ( Figure 5). Cumulative variance for the reflectance data (left) shows surprisingly little difference for EMIT data relative to Landsat and Sentinel-2, with minor differences persisting at Dimension 2 but near convergence by Dimension 3. SuperDove shows considerably lower dimensionality, presumably due to the absence of SWIR bands. In contrast, partition of variance from mixture residual spectra (right) shows EMIT dimensionality to consistently exceed all multispectral sensors, and Landsat/Sentinel consistently exceeding SuperDove. The multispectral feature spaces are effectively 2D and 3D while EMIT's hyperspectral feature space is at least 14D to 99.9% of variance. Manifold-Based Feature Space -UMAP Low-variance spectral feature spaces were further examined using manifold learning. Here, we use the Uniform Manifold Approximation and Projection (UMAP) algorithm [22], implemented using the Python-based 'umap-learn' package. Briefly, UMAP is a nonlinear dimensionality reduction algorithm that assumes the data are uniformly distributed on a locally connected Riemannian manifold with (approximately) locally constant metric. Following these assumptions, the data are modeled using a fuzzy topological structure, and then an embedding is found which maximally approximates (preserves) this topological structure. Both 2D and 3D UMAP embeddings were computed for the mosaic and individual EMIT scenes. UMAP results for the EMIT mosaic, as well as convolved Sentinel, Landsat, and SuperDove mosaics, are shown in Figure 6. When UMAP is applied to reflectance spectra (top row), the greater information content of the EMIT mosaic is manifest as a more complex topology characterized by numerous tendrils with varying degrees of differentiation from the main body of the manifold. In contrast, the multispectral sensors demonstrate diminished complexity with fewer identifiable tendrils and a more continuous structure. The difference in manifold structure between EMIT and multispectral spectra is further accentuated when UMAP is applied to the mixture residual mosaic (bottom row). Here, spectral differences within and among tiles result in clearly separated, well-defined clusters for EMIT. The multispectral sensors are not characterized by such spectral separability. For these data, manifolds are visibly well-connected, without such clearly separable gaps. The implications of this difference in manifold structure for both discrete and continuous image analysis are discussed below. Figure 6. Effect of spectral resolution on manifold structure. Pixels from each EMIT scene are visualized using distinct colors. Joint Characterization It has been noted previously that important information may exist at multiple scales of spectral variance in the same dataset, and that such information may be usefully examined using Joint Characterization (JC) in which bivariate distributions are used to simultaneously visualize both global and local spectral features [13]. Figure 7 illustrates JC as implemented for the EMIT reflectance mosaic. Here, the S, V, D endmembers are used as the global variance metric (x axis), and UMAP dimensions are used as the local variance metric (y axis). Tendrils at similar values of each mixture model fraction (similar x values, but distinct y values) correspond to statistically distinct clusters with broadly similar overall spectral continua but distinct absorption features (e.g., endmember variability). These tendrils frequently correspond to spatially contiguous clusters of pixels in individual EMIT scenes. This is examined in greater detail below for example EMIT scenes. Single-Scene Examples Joint characterization of individual EMIT scenes illustrates additional spectral feature space structure not apparent in the 20 scene mosaic. Figure 8 shows joint characterization as applied to vegetation spectra from the single San Joaquin scene. Red, yellow, and cyan regions of interest are identified and labeled (different colors) as clearly separable clusters from the JC scatterplot (upper right), then projected onto the SVD fraction space (upper left) for context. Average spectra from pixels in all three labeled regions of interest (bottom row) clearly correspond to photosynthetic vegetation. Differences in red edge slope, mesophyll reflectance and liquid water absorptions are present, as well as subtle differences in pigment absorption at visible wavelengths (lower right). Figure 9 shows joint characterization as applied to substrate spectra from the single Atacama scene. Differently colored regions of interest are identified and labeled as clearly separable clusters from the JC scatterplot (upper right), then projected onto the SVD fraction space (upper left) for context. Average spectra from pixels in all regions of interest (bottom row) clearly correspond to exposed geologic substrates. All regions are substantially more absorptive than the global sand endmember (lower left, thick red). Differences in albedo and VNIR curvature are present, as well as specific absorption features in the 2.0 to 2.5 micron region (lower right). Figure 8. Joint characterization for discrimination of vegetation spectra in the San Joaquin Valley. Red, cyan, and yellow regions of interest are clearly separable in the joint space (upper right), but not in the variance-based SVD fraction space (upper left). Mean spectra from each region of interest (bottom row) show differences in mesophyll reflectance, red edge slope, liquid water absorption, and cellulose/lignin absorption features. All regions are more absorptive throughout visible wavelengths than the mosaic V endmember (green, lower right). Clusters identified from JC are geographically coherent at the field scale in map space (right). Figure 10. Joint characterization for discrimination of dark spectra in Bahia Union coastal ocean. Regions of interest are clearly separable in the joint space (upper right), but not in the variance-based SVD fraction space (upper left). Mean spectra from each region (bottom row) show differences in amplitude and curvature throughout the spectrum, including significant differences at visible wavelengths (lower right). Clusters identified from JC are geographically coherent in map space (right). Discussion This analysis was guided by four major questions. We discuss lessons learned for each question below: Generality of the SVD Model It has been acknowledged for decades that, for most of the Earth's land surface, the variance in decameter-scale multispectral satellite imagery can be contained in three dimensions. Early observations of the "brightness", "greenness", and "third" dimensions conceptualized by the Tasseled Cap [23,24] were subsequently extended to the domain of spectral mixture analysis with generalized global endmembers (EMs) corresponding to soil and rock Substrate, illuminated photosynthetic Vegetation, and Dark targets like shadow and water (S, V, and D) [16]. The SVD model has been repeatedly confirmed for larger compilations of Landsat [17,18], as well as decameter multispectral data with more spectral bands from Sentinel-2 [20]. Decameter to meter spatial scaling has been characterized using Landsat:WorldView-2 image pairs [25], as well as decameter to hectometer spatial scaling using coincident Landsat:MODIS observations [19]. Spectral unmixing was largely developed in the context of imaging spectroscopy [26][27][28], and models using soil, photosynthetic vegetation, and shadow have been applied to imaging spectroscopy data for decades, often with the addition of a non-photosynthetic vegetation (NPV) endmember (e.g., [26,29]). As noted above, the majority of such studies used airborne imaging spectroscopy, and so largely operated at local to regional scales. Studies of compilations of AVIRIS flight lines have also been performed [15,[30][31][32]. Such studies largely focus on the related but distinct concept of intrinsic dimensionality, e.g., [33][34][35]. Those that do focus on SVD model generality are limited in spatial scope by data availability to North America, primarily California [14,15,36]. Evaluation of the generality of the SVD model with geographically and spectrally diverse EMIT data was a primary objective of this study. To our knowledge, this is the most comprehensive study to date to demonstrating generality of the SVD model for imaging spectroscopy data, and the first to do so with decameter spaceborne data. Feature Space Dimensionality and Topology: Hyperspectral vs Multispectral The cross-sensor generality of the SVD model is intrinsically related to the similarity (or lack thereof) in spectral feature space dimensionality (variance partition) and topology. As noted above, intrinsic dimensionality of imaging spectroscopy data has been studied previously, but studies have been limited by both data coverage and line-to-line differences in sensor calibration and atmospheric correction. The EMIT reflectance product used for this study may exhibit substantially enhanced image-to-image radiometric stability relative to compilations of multiple flight lines from airborne sensors. The similarity in dimensionality between EMIT and simulated multispectral sensors when quantified using eigenvalues computed from reflectance -and dissimilarity when quantified using variance partition computed from the mixture residualaligns with and extends previous results of [14,37] in clearly demonstrating that the greater spectral information content in hyperspectral image data can be effectively conceptualized as greater departure from a simple 3 endmember linear mixing model. The fundamental differences in topology between the UMAP(MR) results for EMIT versus all other sensors, discussed below, also significantly strengthen and extend this finding. Mixture Residual Efficacy: Hyperspectral vs Multispectral UMAP results from EMIT spectra at full spectral resolution indicate a demonstrably distinct manifold structure from all simulated multispectral sensors ( Figure 6). While this distinction is observed when examining reflectance spectra, differences are much more apparent with the mixture residual. The separability among clusters of MR spectra both within and across EMIT scenes is unambiguous. This result strongly suggests that the spectral signatures captured by EMIT can differentiate biogeophysically distinct Earth surface materials which are not resolved by multispectral sensors like Landsat and Sentinel. The further loss of the distinct tendrils observed in Landsat and Sentinel in the SuperDove MR manifolds suggests that SWIR bands are especially important for differentiation of these land cover types. This is particularly true for mineral absorptions in substrates, consistent with expectations of important information at SWIR wavelengths. (local) axis of the JC, UMAP scores are shown to provide useful information. Clusters identified from the JC space are consistently found to be statistically distinct and geographically coherent. Cluster position is not generally interpretable in UMAP space, but introducing S, V, D fractions effectively provides physical order. Efficacy of Joint Characterization with EMIT Data Limitations and Future work While the results of this study are promising, we do note some significant limitations: First, sampling is not truly global so spectral diversity is underrepresented. While a wide range of geologic and floristic landscapes are sampled, several important areas are not yet included. Notably: a) no scenes are included from Europe or Australia, b) only one urban area (Los Angeles) is included, c) only one major agricultural basin is sampled (San Joaquin), d) no boreal (e.g. tundra) environments are included and e) cryospheric diversity is underrepresented. Future studies with greater data coverage may significantly extend these results, particularly in the form of more extreme endmember spectra. Second, while the generalized SVD model is effective at modeling a wide range of terrestrial environments, it is intentionally exclusive of some materials. Such materials not well-fit by the SVD model include natural materials like evaporites, cyrosphere (snow & ice), and shallow water substrates (e.g. reefs); as well as anthropogenic materials like roofing materials, plastics, and paint. While we recommend analyses to include both global and local EMs, it is likely that analyses which include significant areal coverage of evaporite pans, cryosphere, and/or urban areas may especially benefit from local EM selection and (potentially) mixture models with more than three endmembers. Third, the results of this approach are inherently statistical, data-driven characterizations. Physical meaning, particularly of differences in cluster spectra identified from JC, does require user knowledge of reflectance spectroscopy. This approach is capable of identifying statistically distinct spectral signatures -but interpretation of the physical meaning of those features is likely to benefit from models with a different purpose. In particular, synergy with models with direct physical interpretation like Tetracorder [38] is likely to be particularly profitable. Conclusions We analyze a spectrally and geographically diverse mosaic of 20 scenes from NASA's novel Earth Mineral Dust Source Investigation (EMIT) mission. We evaluate the applicability of the generalized Substrate, Vegetation, Dark (SVD) linear mixture model from previous studies to these data, and find the model to successfully fit the broad, high variance signatures in EMIT reflectance (average RMSE of non-masked pixels 3.1% ; 99% of pixels < 3.7%). EMIT data are convolved to the spectral response functions of three common multispectral sensors. We find the partition of variance of EMIT reflectance spectra to be comparable to modeled Landsat and Sentinel reflectance spectra, but significant and consistent differences to be present in partition of variance among sensors for the spectral mixture residual. Similarly, UMAP-estimated manifold structure for EMIT mixture residual is topologically distinct (more clustered) from manifold structure of the multispectral mixture residual. Joint characterization is found to effectively synergize the physical interpretability of the SVD mixture model with the statistical strengths of UMAP to effectively render additional potentially useful information. These results synthesize recent developments in hyperspectral high dimensional characterization, highlight the superb data quality from the novel EMIT mission, and demonstrate the quantitative and qualitative added value of spaceborne imaging spectroscopy over traditional multispectral satellite imaging. Supplement Figure 1 . 1Index Map. Each of the 20 scenes used for this analysis is shown as a red dot. While the sample is not global, it does include a wide range of land cover including globally significant hotspots for agriculture (San Joaquin), cryosphere (Patagonia, Tian Shan), desert (Taklamakan, Arabian, Saharan, Gobi) and floristic diversity (South African Cape, Mata Atlântica). Figure 2 . 2Mosaic of 20 spectrally diverse EMIT scenes. Figure 3 . 33D spectral feature space and spectral endmembers. Scatterplots of orthogonal principal components reveal the straight edges and well-defined apexes of the spectral feature space. Density clustering along the substrate limb between dark and sand results from geologic diversity of arid environments. However, no clusters are geographically specific. All are represented in multiple sample locations. Figure 4 . 43D mixture residual feature space and example spectra. Scatterplots of orthogonal principal components show some geographically distinct clusters (labeled) on periphery but the body of the distribution comingles residuals from almost all sample locations. Tuwaiq 1 and 2 correspond to bedrock and sand respectively. The Zagros spectrum corresponds to shallow water in evaporite pans. The composite sand endmember used to compute the mixture residual is shown in red for comparison. Figure 5 . 5Partition of variance. Figure 7 . 7Joint characterization of the 20 scene mosaic spectral feature space. 2D UMAP manifold (upper left) shows distinct 2D continua for vegetation and substrates with distinct tendril continua for spectrally distinctive sands, bedrock lithology, and snow. Combining individual dimensions of 2D UMAP manifold with individual endmember fractions shows physical properties of distinct spectra. Note geographically specific lithologic endmembers in Substrate+UMAP1 space in contrast to geographically comingled vegetation and nonphotosynthetic vegetation (NPV) endmembers in Vegetation+UMAP2 space. Figure 9 . 9Joint characterization for discrimination of substrate spectra in the Atacama Desert. Red, cyan, and yellow regions of interest are clearly separable in the joint space (upper right), but not in the variance-based SVD fraction space (upper left). Mean spectra from each region (bottom row) show differences in amplitude and curvature throughout the spectrum, including minor but perceptible differences in SWIR wavelengths (lower right). All regions are darker throughout VSWIR wavelengths than the mosaic S endmember (thick red, lower left). Clusters identified from JC are geographically coherent in map space (right). Figure 10 10shows joint characterization as applied to dark spectra from the single Bahia Union scene. Differently colored regions of interest are identified and labeled as clearly separable clusters from the JC scatterplot (upper right), then projected onto the SVD fraction space (upper left) for context. Average spectra from pixels in all regions of interest (bottom row) clearly correspond to different shallow and suspended sediment. Differences in overall brightness and VNIR curvature are present, likely corresponding to factors like bathymetry and turbidity (lower right). Figures 7 - 710 clearly indicate that Joint Characterization (JC) has significant potential to assist with exploratory analysis of high SNR decameter spaceborne imaging spectroscopy data. SVD mixture fractions provide natural quantities for the variance-based (global) axis of the JC. For this purpose, mixture model fractions have important advantages (e.g., physically interpretable) which are not generally true for other global metrics like PC dimensions. For the topology-based When computed from surface reflectance, all 4 sensors show >99% of spectral variance contained in the first 3 dimensions. After computing and removing the generalized (SVD) spectral mixture model, partition of variance much more clearly reflects spectral differences among sensors. EMIT data show highest dimensionality, with 6 additional dimensions required to capture 99% of the remaining variance. Sentinel-2 and Landsat 8 are comparable, each reaching 99% of variance with 2 additional dimensions. SuperDove dimensionality is demonstrably lower, presumably as a result of the lack of SWIR bands. Note differences in both x and y axis scaling between plots. Table S1 : S1Emit scenes used in this study. Latitude and longitude refer to the northwest corner of the scene.Title Short Name Latitude Longitude EMIT_L2A_RFL_001_20220909T145335_2225209_006 Horqueta -41.53 -68.60 EMIT_L2A_RFL_001_20220903T163129_2224611_012 Bahia Union -39.24 -62.09 EMIT_L2A_RFL_001_20220903T101734_2224607_026 Eastern Cape -33.01 23.50 EMIT_L2A_RFL_001_20220830T065605_2224205_022 Tuwaiq 24.74 46.30 EMIT_L2A_RFL_001_20220828T174405_2224012_007 Los Angeles 34.99 -118.51 EMIT_L2A_RFL_001_20220817T140711_2222909_021 Murzuq 26.30 12.39 EMIT_L2A_RFL_001_20220815T042838_2222703_003 Caspian 40.12 54.22 EMIT_L2A_RFL_001_20220815T025827_2222702_016 Gurbantunggut 45.68 88.96 EMIT_L2A_RFL_001_20220814T223520_2222615_004 Black Rock 41.36 -119.54 EMIT_L2A_RFL_001_20220814T160517_2222611_005 Sierra Nevada 38.45 -119.69 EMIT_L2A_RFL_001_20220909T131308_2225208_011 Atacama -21.95 -69.18 EMIT_L2A_RFL_001_20220905T083937_2224806_033 Bushveld -24.46 26.61 EMIT_L2A_RFL_001_20220827T043253_2223903_002 Tian Shan 41.95 77.10 EMIT_L2A_RFL_001_20220814T160505_2222611_004 San Joaquin 37.97 -120.41 EMIT_L2A_RFL_001_20220901T034405_2224403_006 Hindu Kush 36.73 68.68 EMIT_L2A_RFL_001_20220909T114035_2225207_003 Mata Atlântica -22.75 -44.88 EMIT_L2A_RFL_001_20220909T070044_2225204_005 Okavango -18.83 22.51 EMIT_L2A_RFL_001_20220912T154138_2225510_002 Patagonia -49.58 -74.14 EMIT_L2A_RFL_001_20220816T070436_2222805_008 Gobi 41.72 104.40 EMIT_L2A_RFL_001_20220901T052019_2224404_013 Zagros 27.70 55.64 EMIT Team NASA's Earth Surface Mineral Dust Source Investigation: An Earth Venture Imaging Spectrometer Science Mission. 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[]
[ "Measuring the Inflaton Coupling in the CMB", "Measuring the Inflaton Coupling in the CMB" ]
[ "Marco Drewes \nCentre for Cosmology\nParticle Physics and Phenomenology\nUniversité catholique de Lou-vain\nB-1348Louvain-la-NeuveBelgium\n" ]
[ "Centre for Cosmology\nParticle Physics and Phenomenology\nUniversité catholique de Lou-vain\nB-1348Louvain-la-NeuveBelgium" ]
[]
We study the perspectives to extract information about the microphysical parameters that governed the reheating process after cosmic inflation from CMB data. We identify conditions under which the inflaton coupling to other fields can be constrained for a given model of inflation without having to specify the details of the particle physics theory within which this model is realised. This is possible when the effective potential during reheating is approximately parabolic, and when the coupling constants are smaller than an upper bound that is determined by the ratios between the inflaton mass and the Planck mass or the scale of inflation. We consider scalar, Yukawa, and axion-like interactions and estimate that these conditions can be fulfilled if the inflaton coupling is comparable to the electron Yukawa coupling or smaller, and if the inflaton mass is larger than 10 5 GeV. Constraining the order of magnitude of the coupling constant requires measuring the scalar-to-tensor ratio at the level of 10 −3 , which is possible with future CMB observatories. Such a measurement would provide an important clue to understand how a given model of inflation may be embedded into a more fundamental theory of nature.
10.1088/1475-7516/2022/09/069
[ "https://export.arxiv.org/pdf/1903.09599v4.pdf" ]
85,459,540
1903.09599
cb5d8e35f3cf89301dfc1d01d4206cd51d91536d
Measuring the Inflaton Coupling in the CMB Sep 2022 Marco Drewes Centre for Cosmology Particle Physics and Phenomenology Université catholique de Lou-vain B-1348Louvain-la-NeuveBelgium Measuring the Inflaton Coupling in the CMB Sep 2022Prepared for submission to JCAP We study the perspectives to extract information about the microphysical parameters that governed the reheating process after cosmic inflation from CMB data. We identify conditions under which the inflaton coupling to other fields can be constrained for a given model of inflation without having to specify the details of the particle physics theory within which this model is realised. This is possible when the effective potential during reheating is approximately parabolic, and when the coupling constants are smaller than an upper bound that is determined by the ratios between the inflaton mass and the Planck mass or the scale of inflation. We consider scalar, Yukawa, and axion-like interactions and estimate that these conditions can be fulfilled if the inflaton coupling is comparable to the electron Yukawa coupling or smaller, and if the inflaton mass is larger than 10 5 GeV. Constraining the order of magnitude of the coupling constant requires measuring the scalar-to-tensor ratio at the level of 10 −3 , which is possible with future CMB observatories. Such a measurement would provide an important clue to understand how a given model of inflation may be embedded into a more fundamental theory of nature. Introduction Cosmic inflation [1][2][3] is currently the most popular explanation for both, the overall homogeneity and isotropy of the observable universe, and the observed correlations amongst the small cosmological perturbations that are visible in the Cosmic Microwave Background (CMB) and formed the seeds for galaxy formation. While the general idea that he universe underwent a phase of accelerated expansion before it entered the radiation dominated epoch provides an excellent explanation for the observed data, it is unclear what mechanism drove this expansion, 1 and how it should be embedded into a more fundamental theory of nature. Given the high energy scales that most inflationary models invoke, it seems unlikely that the mechanism of inflation and its connection to particle physics theories can be directly probed in laboratory experiments in the foreseeable future. 2 An indirect probe of this connection can, however, be obtained by studying the impact that the production of particles after inflation, known as cosmic reheating [6][7][8][9][10][11][12][13], has on cosmological observables. It is well-known that this process leaves an imprint in CMB observables through its effect on the expansion history [14][15][16], 3 which is primarily governed by the equation of state parameter w re during reheating, and by the duration of the reheating process in terms of e-folds N re . In [20] it was pointed out that this effect can be used to constrain "fundamental" (microphysical) parameters, in particular the inflaton coupling(s) g to other fields. This connection has been explored quantitatively for α-attractors in [21][22][23]. However, translating a constraint on N re into a constraint on g relies on the assumption that a relation between g and CMB observables can be established without fixing other details of the underlying particle physics model. In the present work we outline general conditions under which this is justified. We focus on the simplest scenarios, in which the energy density during inflation is dominated by the potential energy density V(ϕ) of the condensate ϕ = Φ of a single scalar field Φ, but it is straightforward to generalise the considerations to multi-field inflation. To phrase the goal of this work more precisely, we distinguish between the following sets of microphysical parameters. • Model of inflation: A model on inflation is defined by the choice of the effective potential V(ϕ). V(ϕ) is specified by the complete set of the inflaton's self-interactions (i.e., operators that are constructed from Φ only) with coupling constants {v i }. • Inflaton couplings: We refer to all terms in the action that contain both, Φ and other fields, as inflaton couplings. Their strength is determined by a set of coupling constants {g i }. • Full particle physics model: A given model of particle physics is typically defined by its field content and its symmetries. In general it contains a much larger set of parameters than the combined sets {v i } and {g i }, including the masses of the particles produced during reheating as well as their interactions amongst each other and with all other particles. We refer to the set of all parameters other than {v i } and {g i } as {a i }. This set e.g. includes the parameters of the SM. We shall in the following assume that a model of inflation has been chosen, i.e., V(ϕ) has been specified in terms of the {v i }. We further assume that the numerical values of the {v i } are either fixed by theoretical considerations or can be extracted from data. Since the number of cosmologically observable parameters is quite limited (here we consider the spectral index n s of scalar CMB perturbations, their amplitude A s , and the scalar to tensor ratio r), meaningful constraints on any individual microphysical parameter can only be derived if N re depends on just a small number of these parameters. In the present work we address the following question. Suppose the inflaton reheats the universe through a particular type of interaction that is governed by a dimensionless coupling constant g ∈ {g i }, what is the range of values for {v i } and g for which these parameters can be constrained from CMB data independently of the underlying particle physics model? 2 Connecting inflation to particle physics For the present discussion we assume that the time evolution of ϕ can be described by an equation of the form 4φ + (3H + Γ ϕ )φ + ∂ ϕ V(ϕ) = 0 (2.1) The impact of reheating on the CMB The the primordial spectrum of perturbations before reheating is determined by V(ϕ) and therefore depends only on the {v i }. If the transition from inflation to the radiation dominated epoch were instantaneous, then this would entirely fix the CMB power spectrum (leaving aside foreground effects, and assuming a standard thermal history after inflation). However, reheating takes a number N re of e-folds and should be regarded as a separate reheating epoch in cosmic history. This epoch begins when the universe stops accelerating (the equation of state exceeds w > −1/3) and ends when the energy density of radiation 5 ρ R exceeds the energy density ρ ϕ ≃φ 2 /2 + V(ϕ) stored in ϕ, i.e., the universe becomes radiation dominated (w = 1/3). The modified equation of state parameter w re during reheating affects the redshifting of cosmological perturbations and thereby leaves an imprint in the CMB [14][15][16]. Since we can only observe the time-integrated effect, the quantities that the CMB is directly sensitive to are the duration of the reheating era in terms of e-folds N re and the averaged equation of statē w re during reheating, withw re = 1 N re Nre 0 w(N )dN. (2.2) Within a given model of inflation, all parameters that affect the CMB power spectrum except N re are fixed by the {v i } alone, as V(ϕ) permits to compute the initial spectrum of cosmological perturbations, fixes the moment when inflation ends (∂ ϕ V(ϕ) > 3Hφ) and in good approximation determines w re (since the energy density is by definition dominated by ϕ during reheating). The impact of N re can be expressed in terms of the energy density at the end of reheating ρ re . In practice ρ re is often parameterised in terms of an effective temperature of the plasma formed by the decay products, the reheating temperature π 2 g * 30 T 4 re ≡ ρ re = ρ end exp(−3N re (1 +w re )), (2.3) with ρ end the energy density at the end of inflation. At first sight the dependence on N re is quite inconvenient, as it poses a fundamental uncertainty in the fitting of the {v i } from CMB data. It is, however, possible to turn the tables and use this dependence to constrain N re and T re from CMB data for a given model V(ϕ) by fitting N re to CMB data. Such kind of constraints have been derived for a large number of models, including Starobinski inflation [24], α-attractor inflation [21][22][23][25][26][27][28][29][30], natural inflation [24,[31][32][33], power law [34][35][36][37][38] and polynomial potentials [24,[39][40][41], Higgs inflation [24,34], curvaton models [42], hilltop type inflation [24,34], axion inflation [34,43], inflection point inflation [44], fiber inflation [45], tachyon inflation [46], Kähler moduli inflation [47,48], and other SUSY models [34,49]. In most of these N re (or equivalently T re ) is practically treated as a free fit parameter, without relating it to microphysics. In the present article we investigate the general conditions under which information about N re obtained from the CMB can be translated into constraints on individual microphysical parameters (specifically g). The dependence of CMB observables on microphysical parameters Reheating ends and ρ ϕ is rapidly converted into radiation once Γ ϕ exceeds the Hubble rate H. As far as the effect on the expansion history is concerned this occurs almost instantaneously [14]. We work in this approximation in the following, which permits to establish the well-known relation between T re or ρ re and Γ ϕ , π 2 g * 30 T 4 re = ρ R Γϕ=H = ρ re . as T re = Γ ϕ M pl 90 π 2 g * 1/4 Γϕ=H , (2.4) with g * the effective number of relativistic degrees of freedom and M pl = 2.435 × 10 18 GeV the reduced Planck mass. Via the Friedmann equation H 2 = ρ 3M 2 pl with ρ end ≃ 4 3 V end ≡ 4 3 V(ϕ end ) (2.5) (with ϕ end the value of ϕ at the end of inflation), the condition Γ ϕ = H can be used to translate a bound on N re into a constraint on the damping rate Γ ϕ , Γ ϕ | Γϕ=H = 1 M pl ρ end 3 1/2 e −3(1+wre)Nre/2 (2.6) Within a given model of inflation, all parameters on the right hand side of eq. (2.6) can be constrained with CMB data. The details of this connection are not crucial for the present work, a sketch of the derivation and an explicit expression for N re in terms of observables are given appendix A. Our task here is to evaluate the conditions under which (2.6) can be translated into a measurement of g. In the simplest case Γ ϕ depends only on the {v i } and g (e.g. if reheating is driven by elementary decays of inflaton quanta into light particles in vacuum, cf. table 1), meaning that a constraint on g can directly be read off (2.6). However, in practice the relation between g and cosmological observables is usually complicated and depends on the underlying particle physics model (and therefore the {a i }) in several ways. • Feedback effects on Γ ϕ . At the end of inflation the universe is practically empty, but as soon as reheating commences, it becomes filled with particles. These particles affect Γ ϕ even if the energy density is still dominated by ϕ, e.g. by triggering induced transitions or by scattering with the quanta that form the condensate ϕ. The feedback not only poses a practical challenge by making the reheating process highly non-linear and difficult to simulate [50], it also makes it impossible to constrain g independently of the particle physics model, as feedback necessarily depends on the properties of the produced particles and their occupation numbers. • Number of degrees of freedom. The relation between N re and the CMB parameters (A s , n s , r) depends logarithmically on the effective numbers of degrees of freedom g * and g s * related to the plasma's energy and entropy densities at the time of reheating, respectively, cf. (A.4). These parameters can only be fixed when the complete field content of the particle physics model is specified. At first sight this makes it impossible to constrain g without specifying this model. However, the resulting dependency is quite weak. Plugging (A.4) into (2.6) yields Γ ϕ ∝ (g 1/3 s * /g 1/4 * ) 6(1+wre)/(1−3wre) , For g s * = g * and w re = 0 this reduces to Γ ϕ ∝ √ g * . Assuming Γ ϕ ∝ g 2 this only leads to a very mild dependence g ∝ g 1/4 * , which leads to an uncertainty that is well below the sensitivity of near future CMB missions unless g * changed by several orders of magnitude. • Plasma equilibration. T re can be interpreted as a physical temperature only under the assumption that the bath reaches thermal equilibrium instantaneously. The equilibration affects the present discussion insofar as that deviations of the produced particles' phase space distributions from equilibrium can lead to deviations from w = 1/3, and that it affects the relation between the right-hand side of (2.6) and the CMB parameters, cf. equation (15) and following in [21]. A recent discussion of the equilibriation can e.g. be found in [51][52][53][54]. In typical UV-completions of the SM that contain coupling constants of order one the resulting error is small, hence the assumption of fast thermalisation only imposes a comparably weak condition on the {a i }. • Standard expansion history. Throughout this work we assume a standard expansion history, i.e., radiation domination (w = 1/3) between the end of reheating and big bang nucleosynthesis (BBN). 6 If e.g. entropy was released into the plasma after reheating, then back-extrapolation of the present CMB temperature leads to an overestimate of the temperature before the moment of the release. Hence, if an unknown amount of entropy is injected after reheating, one can only obtain an upper bound on g from the CMB. • Gravitational waves. Tensor perturbations can also be caused by gravitational waves generated after inflation, and this can affect the observed value of r. 7 If the gravitational waves are caused by the inflationary sector (e.g. they were generated during reheating), their effect on r can at least in principle be computed with knowledge of the {v i }, otherwise this brings in dependencies on the {a i }. In models where r is dominated by contributions that were generated after inflation and that are not calculable from knowledge of the {v i }, one cannot use r for the approach proposed here. One may, however, still utilise n s (and potentially future measurements of its running). • Foreground and late time effects. Assuming that all astrophysical foregrounds have been identified and removed from the CMB data, there can be cosmological effects that are not part of the concordance model. These can e.g. be caused by comparably light new particles at later times, which can modify CMB observables in different ways. On one hand, they can affect the propagation of photons after their decoupling, e.g. through cosmic birefringence caused by a modification of Maxwell's equation (cf. [57] and reference therein), On the other hand, they can modify the interactions of particles before photon decoupling [58]. A well-studied example are neutrino non-standard interactions [59][60][61]. The existence of light new particles is strongly constrained by cosmology and experiments. If they are light during photon decoupling they introduce deviations from the SM prediction for the effective number of relativistic species N SM eff = 3.0440 ± 0.0002 [62] (cf. also [63][64][65]) which are strongly constrained by observations [66,67]. If they are heavier, there are constraints from experiment and astrophysics, cf. [68] for a review. The approach proposed here is not applicable to specific models that can avoid all of these constraints. • Instantaneous decay. We assume that the conversion of the remaining inflaton energy density ρ ϕ into ρ R occurs instantaneously when Γ ϕ = H. While this is usually a good approximation, relaxing it introduces a mild dependence on g * during the transitional phase, which we ignore here. • Radiative corrections. We define the {v i } as the microphysical parameters that characterise the effective potential during reheating. The CMB anisotropies are generated during the slow roll phase (which is sensitive to a different part of V(ϕ)) and do not directly probe the shape of the potential near its minimum. Since inflation and reheating can occur at different energy scales, a fully consistent determination of the {v i } may require solving renormalisation group equations that in principle bring in a dependence on the {a i }. Radiative corrections can further affect the relation between the parameters in the Lagrangian and the properties of the perturbations generated during inflation in models where the field space trajectories during inflation can be sensitive to small corrections (e.g. inflection point inflation). In section 4 we find that a determination of g from the CMB is only possible for comparably small values of the inflaton coupling constants, which suggests that these effects are subdominant in the regime where the methods discussed here can be applied [69,70]. • Multifield effects. Many single field models of inflation are motivated by particle physics theories that predict the existence of multiple scalar fields. During inflation one can justify the single field description by identifying the inflaton with the relevant direction in field space. It is in general not clear whether this can be justified during reheating, cf. e.g. [71]. There are at least two qualitatively different effects that can arise from this even if we assume that all other scalar fields have relaxed to their minimal before or during inflation. Firstly, quantum fluctuations [72,73] of light spectator fields can build up on super-horizon scales (cf. [74] and references therein for a recent discussion) and locally displace those fields away from the minima of their potential. They may either get trapped in false vacua that subsequently decay or oscillate around the true minima. This can affect the thermal history during reheating [75]. These effects strongly depend on the field content of the theory in which a given model. Based on the considerations in appendix B it seems that they are unlikely to affect N re unless the energy stored in these degrees of freedom dominates the universe. Secondly, in models that feature kinetic mixing between the fields, a non-trivial metric in field space that can introduce instabilities [76] that lead to violent particle production. This is, for instance, wellknown to occur in α-attractor models [77][78][79] and in models with non-minimal coupling to gravity [80,81] (including Higgs inflation [82]), where the non-trivial field space metric is introduced by the mapping between the Einstein and Jordan frame. 8 In the present work we focus on phenomena that can be described within the single field framework, additional conditions that arise from specific embeddings in multifield models may exist, but have to be studied in each model separately. To summarise, within the set of models where our approach is in principle applicable, the main complication arises from feedback effects of the produced particles on the dissipation rate Γ ϕ . All other effects are either subdominant in the regimes identified in section 4, or only lead to mild restrictions on the properties of the underlying particle physics theories. We shall therefore focus on the conditions that can be derived from the requirement to avoid feedback. Conditions to measure the inflaton coupling Feedback typically becomes relevant when the occupation numbers in the plasma approach unity. One can roughly distinguish two effects. 1) Quantum statistical effects. When the occupation numbers of individual modes in which particles are being produced reach unity, quantum statistical effects considerably modify the particle production rate for these modes. Depending on the spin of the final state particles, this either enhances Γ ϕ by induced transitions or suppresses the rate due to Pauli blocking. 2) Plasma effects. When the overall density of particles in the plasma becomes too large, then scatterings between the (quasi)particles that make up the plasma and those that form ϕ become frequent and modify Γ ϕ . In the following we restrict the discussion to scenarios where the cosmos is reheated during oscillations of ϕ around its potential minimum. Other forms of reheating, such as tachyonic instabilities, are generally driven by non-perturbative processes that introduce a complicated dependence of Γ ϕ on time and the {a i }. In this situation, one can broadly distinguish the following mechanisms to reheat the universe, (i) elementary decays of inflaton quanta, (ii) annihilations of inflaton quanta, (iii) scatterings of inflaton quanta with other particles. (iv) non-perturbative production of particles from the time-dependent background provided by ϕ, (v) gravitational effects. We shall neglect (v) in the following, as it tends to be sub-dominant when other channels exist. 9 The mechanisms (i), (ii) and (iv) all can potentially generate large occupation numbers in specific momentum modes well before the overall density of the produced particles becomes large. We shall therefore discuss feedback of type 1) first. It is convenient to introduce the small parameters ε pl ≡ m ϕ /M pl , ε ϕ ≡ m ϕ /ϕ end . (3.1) Restrictions on the potential: qualitative discussion We will see in section 4.1 that the requirement to avoid a so-called broad resonance due to the non-perturbative particle production (iv) practically restricts us to scenarios where ϕ end is small enough that V(ϕ) can be approximated by a parabola, i.e., the mildly non-linear regime where reheating can be modelled by a harmonic oscillator with time dependent frequency and damping. We can intuitively understand this point already without diving too deeply into computations by using the adiabaticity condition on the dispersion relations W φ for screened φ-particles ∂ t W φ W 2 φ ≪ 1,(3.2) which we use as a rough estimator for the presence of highly efficient non-perturbative particle production in a so-called broad resonance. The full dispersion relation W φ is defined as a pole of the resummed propagator, 10 in a simple scalar field theory it can be approximated as W φ ≃ Ω 2 φ = k 2 + M 2 φ , with M φ an effective mass that in general depends on ϕ and the properties of the screening plasma. For vanishing field value it reduces to Ω 2 φ = k 2 + M 2 φ = Ω 2 φ | ϕ=0 , with M φ a thermal mass that includes corrections to the tree-level pole mass m φ at the potential minimum in vacuum due to screening by the plasma. 11 The key point is that the same types of Feynman diagrams from which Γ ϕ and V are computed also appear in the Φ-self-energy and therefore determine the pole of the Φ-propagator, which defines Ω φ . Hence, oscillations with an amplitude that is large enough that non-linear terms in the equation of motion for ϕ dominate automatically imply large ϕ-dependent corrections to the pole of the φ-propagator that violate the adiabaticity condition (3.2). To illustrate this, let us consider a symmetric tree level potential with a ground state at vanishing field value, which we expand as V (Φ) = 1 2 m 2 φ Φ 2 + 1 4! λ φ Φ 4 + O[Φ 6 ]. If we (for simplicity) place Φ in a thermal bath of temperature T , the dispersion relation reads W 2 φ ≃ Ω 2 φ = k 2 + M 2 φ , while we find ∂ 2 ϕ V(ϕ) = M 2 φ + O[ϕ 6 ] , with the same correction M 2 φ = m 2 φ + λ φ T 2 /4! + λ φ 2 ϕ 2 = M 2 φ + λ φ 2 ϕ 2 . 12 The quartic term in V(ϕ) dominates for ϕ 2 > ϕ 2 c , with ϕ c ≡ 12M 2 φ /λ φ . (3.3) When the amplitude of the oscillations greatly exceeds ϕ c we can approximate V(ϕ) ≃ λ φ ϕ 4 /4!. In this regime the ϕ-dependent term λ φ 2 ϕ 2 in M 2 φ is larger than the ϕ-independent term M 2 φ . Hence, M 2 φ is strongly time dependent, leading to a violation of the condition (3.2). Neglecting dissipation, (2.1) can be solved by 13 ϕ(t) ≃ ϕ end cn(1/ √ 2, λ φ /6ϕ end t) ϕ end ≫ ϕ c ϕ end cos(M φ t) ϕ end ≪ ϕ c (3.4) where ϕ end is the field value at the end of inflation. Expanding the elliptic cosine 14 one finds that ϕ(t) ≃ ϕ end cos(ωt) (3.5) in both, the quadratic regime (amplitude ≪ ϕ c ) and the quartic regime (amplitude ≫ ϕ c ), with ω ≃ λ φ /6 ϕ end for ϕ end ≫ ϕ c , ω ≃ M φ for ϕ end ≪ ϕ c ,(3.6) which can be inserted into Ω φ . Demanding that (3.2) holds at all times imposes an upper bound on λ φ ϕ 2 end that can be written as λ φ ϕ 2 end ≪ ω 2 f (Ω φ /ω) (3.7) The most general expression for f (x) is lengthy and not illuminating. For the present discussion it is sufficient to know that f (x) ≃ 27 2 x 4 x > 1 4x 3 x ≪ 1 ,(3.8) and that f (1) ≃ 14.45. Together with (3.6) and the knowledge that ϕ c ≫ M φ this shows that (3.2) can never be fulfilled for ϕ end > ϕ c . In the quadratic regime ϕ end < ϕ c , on the other hand, the adiabaticity condition translates into λ φ ϕ 2 end ≪ M 2 φ f (1) ≃ 27 2 Ω 4 φ /M 2 φ , which is always fulfilled for ϕ end ≪ ϕ c and mildly saturated for ϕ end = ϕ c . Hence, (3.2) is always 12 Here we have taken the high temperature limit and only considered the leading term, more accurate expressions can e.g. be found in [86,87] and references therein. 13 This approximation is conservative because the oscillations usually do not immediately start after the end of inflation, but there is a brief period of "fast roll". Also neglecting this is conservative in the present context because ϕ would decrease during this period. 14 To be explicit, cn(1/ √ 2, x) = A ∞ n=0 cn cos(bnx) with A = 4.79257, cn = e −π(n+1/2) 1+e −2π(n+1/2) , and bn = π 3.70815 (2n + 1). violated in the quartic regime and generally fulfilled in the quadratic regime. As stated before, the reason lies in the fact that the same types of terms appear in the equation of motion for ϕ and the Φ-propagator. 15 Hence, a determination of g independent of the underlying particle physics model is only possible if the inflaton oscillations occur in the mildly non-linear regime with ϕ end < ϕ c , M φ ≃ M φ , ω ≃ M φ , V end ≃ 1 2 M 2 φ ϕ 2 end . (3.9) This conclusion will be backed up by a more detailed computation in section 4.1. Avoiding large occupation numbers in individual modes Large occupation numbers can be avoided in two ways: Either the produced particles decay into other particles within one oscillation, or the expansion of the universe redshifts the particles' momenta out of the resonance bands. Since the decay rates of the produced particles necessarily depend on the {a i }, Γ ϕ in such scenarios necessarily depends on the {a i }. 16 The redshifting, on the other hand, depends on the equation of state, which is dominated by ϕ, and therefore in good approximation independent of the {a i }. Hence, redshifting is the only way to reduce the particle numbers in a a i -independent way. If the range ∆k of momenta k = |k| over which particles are produced is narrow enough, large occupation numbers can be avoided by redshifting alone, and Γ ϕ is a function of g only (in addition to the {v i }). This is much more difficult if the particles are produced efficiently over a broad range of momenta. We will therefore, to be conservative, impose two conditions, I) that efficient particle production occurs only for a narrow range ∆k ≪ k of momenta k and II) that the universe expands fast enough to redshift these momenta before the occupation numbers in mode k reach unity. The second condition can be quantified by estimating how many particles are being produced during the time ∆t that it takes to redshift a momentum k out of the band ∆k. For ∆k/k ≪ 1 we can Taylor expand in ∆t to estimate ∆k/k = 1−a(t)/a(t+∆t) ≃ H(t)∆t, so ∆t ≃ ∆k/(kH). Denoting the rate at which the occupation numbers in mode k grow by Γ k , we arrive at the simple condition Γ k H ∆k k ≪ 1. (3.10) Non-perturbative particle production. The non-perturbative particle production (iv) tends to be very efficient, leading to O[1] occupation numbers, strong feedback, and resonant particle production within a few oscillations [9]. We shall therefore deal with the nonperturbative production first. Using (3.5), the equations for the mode functions X k of fields 15 This connection is of course absent when it comes to the adiabaticity condition for the modes of fields X other that Φ, whose effective mass MX is not directly connected to V(ϕ) and may be smaller or larger than M φ . Hence, ΩX may be smaller or larger than ω in both, the quadratic or quartic regimes (and even for larger elongations), depending on the vacuum mass of X , its interactions, and T , all of which enter MX and are part of the {ai}. 16 In the adiabatic regime with MX ≃ MX the decay is generally slower that the oscillations because the leading processes typically require MX < ω, cf. table 1. At the same time one can only speak of particles for ΓX < MX , with ΓX the inverse lifetime. Hence, it is impossible to achieve a decay within one oscillation, which would require ΓX > ω. However, the time dependence of MX also induces a time-dependence of ΓX . Assuming ΓX ∼ a 2 MX and MX ∼ aϕ, cf. appendix C, it is straightforward to see that ΓX can easily exceed ω when the elongation is maximal (ϕ ∼ ϕ end ). This is e.g. occurs for Higgs inflation [88]. that couple to ϕ can be brought into the form of the usual Mathieu equation, X ′′ k (z) + A k − 2q cos(2z) X k (z) = 0. (3.11) with z ∼ tω (for the moment ignoring prefactors and phase shifts). The Mathieu equation shows instabilities that indicate resonant particle production. The conditions under which resonant production occurs have been studied in detail in ref. [13]. Here we use the results of this work to phrase our conditions I) and II) in terms of q and A k , I) The width of the resonance bands of (3.11) scales as ∼ q, so the condition to avoid a broad resonance translates into q ≪ 1. 17 II) In the regime q < 1 the rate of resonant particle production can be estimated as ∼ qω. The effective width of the first band is roughly ∼ qω/2, while the momenta of the produced particles can be estimated as ω/2. Hence, (3.10) requires q 2 ω ≪ H. For simplicity and to be conservative we impose the condition at the beginning of reheating, which is sufficient to ensure that the condition holds during reheating in the scenarios that we consider here because q 2 decreases quicker with time than H. Elementary decays and scatterings. The conditions q ≪ 1 and q 2 ω ≪ H ensure that the non-perturbative particle production (iv) does not trigger a parametric resonance. Elementary 1 → 2 decays (i) and 2 → 2 annihilations (ii) of particles at rest also lead to high occupation numbers because they exclusively populate narrow ranges of modes k ∼ M φ /2 and k ∼ M φ , respectively. 18 Let us consider 1 → 2 decays of quanta that compose the condensate into X -quasiparticles of mass M X , which dominate at small field values. The range ∆k over which the produced particles' momenta k are spread is affected by two factors: The "natural line width" for fixed masses M φ and M X , and the spread caused by the time dependence of the effective masses M φ and M X . We can approximate the former by Γ ϕ→X X and the latter by qω, 19 where Γ ϕ→X X is the rate of elementary decays. Here we have used the fact that the discussion following (3.2) restricts us to the mildly non-linear regime (3.9) to set ω ≃ M φ and M φ ≃ M φ . Further recalling that condition I) enforces M 2 X ≪ M 2 φ , and that the contribution to Γ k from narrow resonance in the first band is also ∼ qω, we can conclude that ∆k can be estimated by the natural line width Γ k whenever perturbative decay dominates over the narrow resonance. Hence, condition II) as expressed in Eq. (3.10) applied to perturbative decay reads Γ 2 k /ω ≪ H with Γ k ≃ Γ ϕ→X X . Since Γ k ≤ Γ ϕ and Γ k ≪ ω, this condition is always fulfilled before the moment when H = Γ ϕ and reheating ends, and the perturbative decay does not trigger a parametric resonance. Hence, the conditions q < 1 and q 2 ω ≪ H are sufficient to avoid {a i }-dependencies of the CMB observables due to resonant particle production triggered by quantum statistical effects. Avoiding plasma effects For the discussion of plasma effects we assume that the plasma of decay products can be described by a thermal bath with an effective temperature T . 20 The presence of the plasma can 17 This is a conservative criterion for which we assumed that A k can be small for small momenta. If the produced particles are heavier than ω, then the range of A k may be restricted to be larger than 1, in which case the width of the bands is smaller. 18 In principle higher harmonics in the expansion of ϕ(t) beyond (3.5) and sub-leading terms in the expansion in the couplings both give rise to processes with energies given by integer multiples of ω in the initial state, cf. e.g. Sec. 6 in [89], leading to an extended set of possible momenta for the produced particles. For the present discussion it is sufficient to consider the leading terms because they would reach large occupation numbers first. 19 The estimate is consistent with (4.10). 20 This is well-justified if ω < ΓX , with ΓX the typical relaxation rate for the most relevant modes in the plasma of decay products X . In this regime elementary decays are, however, typically forbidden because they affect Γ ϕ in several ways, cf. e.g. [84,[90][91][92][93][94][95][96][97][98][99][100][101][102][103][104][105][106] and references therein. Firstly, quantum statistical effects (induced transitions and Pauli blocking) can enhance or suppress the elementary decays (i) and annihilations (ii) that are already present at T = 0 in a way that depends on the time evolution of T , and therefore on the {a i }. This roughly occurs when the temperature T reaches m φ . 21 Secondly, the presence of the plasma opens up new channels of dissipation via elementary processes, such as scatterings (iii) of the quanta in the condensate ϕ with quasiparticles from the bath. These typically also become relevant for T ∼ m φ . Finally, thermal corrections to the quasiparticle dispersion relations (typically of order aT , with a ∈ {a i } the relevant coupling) strongly affect the kinematics for T > M φ /a. Since the thermal correction in M φ is always smaller than T , this practically amounts to T > m φ /a. We give some explicit examples to illustrate this behaviour in table 1. Hence, we can neglect thermal feedback on Γ ϕ if T remains below m φ . 22 One may therefore be tempted to impose the condition T max < m φ , with T max the maximal temperature during the reheating epoch. This condition would, however, be too conservative: While thermal feedback on Γ ϕ generally modifies the thermal history during reheating (which can e.g. have an impact on the abundance of thermal relics) [84,85,107], it does not necessarily mean that the CMB observables are affected by the {a i } because the CMB is not directly sensitive to the thermal history during reheating [20]. The thermal corrections only affect the observables (n s , A s , r) if they shift the moment Γ ϕ = H and thereby modify N re , i.e., change the expansion history. However, even this overly conservative criterion turns out to be weaker than the bounds derived from the requirement q 2 ω ≪ H derived from condition II), as we will confirm in Sec. 4. We denote by Γ 0 the rate of perturbative reheating at T = 0 and by T n the temperature at which thermal corrections become sizeable, i.e, the solution of the equation |Γ ϕ − Γ 0 |/Γ 0 = c in T , with c some number of order unity. For the sake of definiteness we shall use c = 1. From the explicit expressions in table 1 one can see that T n ∼ m φ , neglecting numerical prefactors of order unity that depend on the process under consideration. For the following discussion it is useful to introduce the would-be reheating temperature T re0 = Γ 0 M pl ( 90 π 2 g * ) 1/4 and the would-be maximal temperature T max0 ≃ (Γ 0 M pl /g * ) 1/4 V 1/8 end that one obtains by setting Γ ϕ = Γ 0 in (2.4) and (B.1), respectively [108]. Thermal corrections affect the thermal history of the universe (i.e., the time evolution of T ) if T max0 > T n [84]. For T n ∼ m φ and V end ∼ 1 2 m 2 φ ϕ 2 end this roughly happens when Γ 0 /m φ > g * m φ M pl m φ ϕ end . For the 1 → 2 decays in table 1 this translates into an upper bound on the coupling constant g which parametrically reads as g ≪ (g * ε pl ε ϕ ) 1/2 , where the prefactor depends on the interaction. This constraint on g is weaker than the constraints on g that we derive from the absence of parametric resonances in the next sections. We can therefore conclude that thermal effects do not modify N re in require ω > MX , cf. table 1, while one generally expects ΓX < MX . For ω > ΓX the use of an effective temperature T to characterise the occupation numbers in the plasma can still be justified if the ϕ elongation is small enough that deviations from (3.9) can be treated perturbatively, and provided that ΓX ≫ Γϕ. Since the conditions I) and II) restrict us to scenarios with g ≪ 1 (while the decay products can have gauge interactions with a ∼ O [1]) this is a reasonable assumption. A more detailed discussion can e.g. be found in [51][52][53][54]. 21 In principle thermalised particles are less dangerous in this regard than the narrow resonance bands, as a given energy density ρR results in lower occupation numbers in each mode when it is spread over a wider range of momenta. On the other hand this also makes it more difficult to reduce the occupation numbers by redshifting once they reach unity. This is the reason why the conditions I) and II) previously formulated are not necessarily sufficient to avoid a thermally induced resonance. 22 One may wonder to what degree this argument can be generalised beyond the specific interactions given in table 1. In fact this generalisation is straightforward because it only relies on kinematic considerations. Neglecting subleading corrections (cf. footnote 18) and still restricting ourselves to the mildly non-linear regime, 1 → 2 decays and annihilations of the quanta that constitute ϕ amongst each other primarily produce particles with energies ∼ M φ /2 and ∼ M φ , respectively. interaction process contribution to Γ ϕ gΦχ 2 ϕ → χχ g 2 8πM φ 1 − 2M χ /M φ 1/2 1 + 2f B (M φ /2) θ(M φ − 2M χ ) [109] h 4 Φ 2 χ 2 ϕϕ → χχ h 2 ϕ 2 256πM φ 1 − M χ /M φ 1/2 1 + 2f B (M φ ) θ(M φ − M χ ) [102] α Λ ΦF µνF µν ϕ → γγ α 2 4π M 3 φ Λ 2 1 − 2M γ /M φ 1/2 1 + 2f B (M φ /2) θ(M φ − 2M γ ) [110] yΦψψ ϕ → ψψ, M ψ ≃ m ψ y 2 8π M φ 1 − 2m ψ /M φ 3/2 1 − 2f F (M φ /2) θ(M φ − 2m ψ ) [101] ϕ → ψψ, M ψ ≫ m ψ y 2 8π M φ 1 − 2M ψ /M φ 1/2 1 − 2f F (M φ /2) θ(M φ − 2M ψ ) [101] . Table 1: Contributions to Γϕ from different decay and annihilation processes in a thermal plasma involving scalars Φ and χ, fermions ψ with gauge interactions, and U(1) gauge bosons γ with field strength tensor Fµν . Small letters denote vacuum masses and capital letters denote effective masses in a thermal bath (including the vacuum mass and thermal corrections). fB(ω) = (e ω/T − 1) −1 and fF (ω) = (e ω/T + 1) −1 are the Bose-Einstein and Fermi-Dirac distributions, respectively, and Λ is some heavy mass scale. Some of the rates given here have been computed for particle decays (rather than condensate decay), but we can take advantage of the fact that these rates in the regime (3.9) are at leading order identical for 1 → 2 decays, cf. appendix C and footnote 18. the regime where conditions I) and II) are fulfilled. We present a slightly more quantitative discussion of this point in appendix B. The range of measurable inflaton couplings In the following we study the conditions introduced in section 3 quantitatively for specific types of interactions. The parametric dependence of the restrictions on g turns out to be rather generic, indicating that these results hold for a broad range of interaction terms. Large occupation numbers can be avoided if g is sufficiently small, so that the transfer of energy from ϕ to radiation is rather slow and the occupation numbers are reduced due to redshift before they can reach unity. 23 On the other hand, if g is too small, the universe fails to reheat the universe efficiently. A model-independent condition can be phrased as III) T re must be larger than the temperature T BBN needed to explain the observed abundances of light elements with BBN. The earliest process that directly affects BBN is the freeze-out of the SM neutrinos, which occurs at temperatures around 1 MeV for neutrino momenta of order of the temperature. We shall take T BBN = 10 MeV as a conservative estimate to avoid any impact on BBN. Most scenarios of baryogenesis or dark matter production require much higher temperatures, but the precise value is strongly model-dependent. The condition T re > T BBN is both conservative and robust in the sense that is it model-independent and based on observation. Note that condition III) is qualitatively different from conditions I) and II) because it is a hard observational bound; failure to fulfil it implies that a given scenario is ruled out. Conditions I) and II), on the other hand, only restrict the range of values for which g can be constrained from CMB data independently of the {a i }, and may be violated in nature. In [22] it was found that the constraint (2.6) on Γ ϕ can, roughly speaking, be converted into a constraint on g if g < 10 −5 . The reheating temperature is high enough to be consistent 23 Since the relation (2.6) constrains Γϕ at the moment when reheating ends, it is strictly speaking not necessary to demand that the occupation numbers remain below unity at all times as long as the feedback does not enhance Γϕ so much that it exceeds the rate of Hubble expansion H. An enhanced Γϕ that remains below H modifies the thermal history, but not Nre [20,84]. More precisely, our results still apply as long as Γϕ is dominated by elementary processes in the moment when Γϕ = H. with the observational constraint from BBN if g > 10 −15 . This leaves about ten orders of magnitude for which the inflaton coupling can be "measured" from the CMB. These results reported in ref. [22] are rather independent of the specific interaction that couples the inflaton to radiation (i.e., whether g is a coupling to bosons or fermions). In the present work we use simple well-known relations to investigate the origin of this general behaviour, which we find is not restricted to α-attractor models. Self-interactions and the effective potential Splitting Φ = ϕ + φ into the expectation value ϕ and fluctuations φ, we shall first consider the production of inflaton particles φ due to self-interactions. For later use, we consider general couplings of the form L ⊃ c (n,m) Φ n χ m Λ 4−m−n (4.1) between Φ and another scalar field χ, with dimensionless Wilson coefficients c (n,m) and some mass scale Λ. For terms with mass dimension D > 4 (with D = n + m here) we shall interpret (4.1) as an effective field theory with a cutoff Λ that is larger than all other physical scales. In terms with n+m < 4, on the other hand, one should not interpret Λ as the cutoff in an effective field theory, but rather as a conveniently chosen scale to make the coupling dimensionless. We may therefore assume Λ = m φ for n + m < 4 , Λ ≫ m φ , T, ϕ for n + m > 4 ,(4.2) where the former choice is arbitrary and for convenience. The terms in (4.1) generate terms n l c (n,m) ϕ l φ n−l χ m Λ 4−m−n with l ≤ n. ϕ(t) l = [ϕ end cos(m φ t)] l , can be decomposed into a series of terms ∝ ϕ l cos(ωt), with ω given by integer multiples of m φ up to lm φ . These operators generate time dependent masses for φ and χ, leading to particle production. At tree level, only the terms with (m, l) = (0, n − 2) and (m, l) = (2, n) contribute. 24 The terms with m = 0 and n−l = 2 in (4.1) lead to tree level contributions to the effective φ masses M φ and can therefore induce self-resonances. The criterion (3.9) is a necessary, but not a sufficient one to avoid a self-resonance. The mode equation for φ then receives contributions from all terms in (4.1) with n − l = 2 and m = 0, φ k (t) + k 2 + m 2 φ + n n(n − 1)c (n,0) ϕ n−2 Λ 4−n φ k (t) = 0 (4.3) Here we consider only renormalisable interactions, as terms with n + m > 4 are suppressed by negative powers of Λ. This leaves us with the terms for n = 3 and n = 4. Assuming that only one or the other is important at a given time, we can use the following relations A k = (2ω φ ) 2 ω 2 , q = 2g φ ϕ end ω 2 , z = (tω + π)/2 [for the g φ 3! Φ 3 interaction] (4.4) A k = ω 2 φ ω 2 + 2q , q = λ φ 8 ϕ 2 end ω 2 , z = tω + π/2 [for the λ φ 4! Φ 4 interaction],(4.5) where we have defined c (4,0) = λ φ /4!, c (3,0) = g φ /(3!Λ) and ω φ = k 2 + m 2 φ . Using the standard relations (2.5) this can be translated into λ φ ≪ 8 ω 2 ϕ 2 end , λ φ ≪ 8 2 3 V 1/4 end ϕ end ω ϕ end ω M pl , g φ ≪ 6ω 2 ϕ end , g φ ≪ √ 24 V 1/4 end ϕ end ω ω M pl . (4.6) 24 A systematic way to include quantum corrections has e.g. been developed in [111], cf. appendix C. Using (3.9) with m φ ∼ M φ at the end of reheating (cf. Sec. 3.3), this simplifies to λ φ ≪ 8ε 2 ϕ , λ φ ≪ 5.5 ε 3/2 ϕ ε 1/2 pl , g φ /m φ ≪ 6ε ϕ , g φ /m φ ≪ 33 ε 1/2 ϕ ε 1/2 pl . (4.7) in the notation (3.1). Ignoring all numerical factors, one can therefore roughly summarise (4.7) as c (4,0) ≪ min(ε 2 ϕ , ε 3/2 ϕ ε 1/2 pl ) , c (3,0) ≪ min(ε ϕ , ε 1/2 ϕ ε 1/2 pl ) (4.8) At this point we shall question our restriction to values n ≤ 4, which we justified with the suppression of higher order terms in (4.3) by powers of ϕ/Λ, assuming that Λ must be larger than ϕ in (4.2). This assumption is motivated by an interpretation of (4.1) as an effective field theory during reheating [112] with a cutoff Λ that exceeds all other relevant scales. Terms with Λ ∼ M pl can certainly be expected from gravitational effects. An effective field theory approach with a lower cutoff can also be justified after integrating out fields with masses that exceed m φ and the energies of particles produced during reheating, in which case Λ should be identified with the mass of the lightest field that was integrated out. In principle there is no reason why ϕ should remain smaller than the cutoff, as ϕ does not correspond to the mass of any physical particle. However, the non-perturbative mechanism (iv) can produce particles with masses much larger than m φ [113]. This is evident from the fact that the Mathieu equation (3.11) has resonance bands for A k ≫ 1. 25 Hence, when using a cutoff Λ < M pl one has to check explicitly that particle production from terms with n > 4 does not lead to feedback effects. In summary, the previous considerations show that the upper bound on the inflaton selfcouplings from the requirement that Γ ϕ is independent of the {a i } is given by the ratios (3.1) of m φ and other physical scales, and that the power at which these ratios appear is determined by the power n at which Φ appears in the respective term. More precisely, the power of the small parameters is given by the power at which ϕ appears in Ω 2 k . These conditions appear to be consistent with what was found in an analysis for the specific case of α-attractor models in [114]. Before moving on, we shall briefly come back to the point that g can only be independent of the {a i } if ϕ end is small enough the oscillations happen in the mildly non-linear regime in which V(ϕ) is approximately parabolic, which we already mentioned after (3.2). It is straightforward to see that it is impossible to fulfil condition I) by plugging (3.6) into (4.6). 26 Interactions with other scalars The mode equation for χ receives contributions from all terms in (4.1) with n = l and m = 2, χ k (t) + k 2 + m 2 χ + n 2c (n,2) ϕ n Λ 2−n χ k (t) = 0. (4.9) 25 The heaviest particles produced gravitationally roughly have a mass mX corresponding to H at the end of inflation. Using (2.5), (3.9) and M φ ≃ m φ we can estimate the maximal mX as ∼ m φ ϕ end /M pl . For non-perturbative production from ϕ-oscillations we can use (3.2) and footnote 26. With (4.12) and (4.16) one parametrically finds that the maximal mX scales as ∼ (gϕ end m φ ) 1/3 and ∼ ϕ end h/8, indicating that (4.2) may be too conservative, and using a cutoff Λ < ϕ end can be justified even during preheating. 26 In this context it is instructive to compare the requirement q ≪ 1 from condition I) to the adiabaticity condition (3.2) that has to be violated to efficiently produce heavy particles. For the c (3,0) Φ 3 interaction the adiabaticity condition (3.2) translates into 2q ≪ A 3/2 k , for quartic interaction λ φ Φ 4 /4! one finds q ≪ (6A k + 9A 2 k − 2 3 ((1 + 6A k ) 3/2 − 1)) 1/2 /6. For A k ≪ 1 this coincides with the condition 2q ≪ A 3/2 k , and for A k ≫ 1 is reproduces the commonly used condition 2q ≪ A k . Note that (ω φ /ω) 2 ≥ 1 in the quadratic regime while (ω φ /ω) 2 < 1 in the quartic regime, so that condition I) tends to be stronger than (3.2) in the quadratic regime while (3.2) is stronger in the quartic regime. Comparing to (3.11), we can infer that the parameter which q quantifies the amplitude of the time-dependent mass and has a contribution that parametrically scales as q ∼ c (n,2) ϕ n Λ 2−n /ω 2 . (4.10) Both conditions I) and II) impose upper bounds on q and therefore c (n,2) . From (4.10) it is clear that the power n at which Φ appears in (4.1) is decisive, and that the upper bound on the coupling c (n,2) is stronger for larger n. We therefore classify the different interactions according to n, which turns out to be more relevant than e.g. the spin of the produced particles. Scalar two body decays. Consider a coupling of the form gΦχ 2 to another scalar χ, where g = c (1,2) Λ in the notation of (4.1). The equation for χ-modes in Minkowski space reads χ k (t) + k 2 + m 2 χ + 2gϕ(t) χ k (t) = 0 (4.11) We can bring the mode equation (4.11) into the form (3.11) with A k = (2ω χ ) 2 ω 2 , q = 4 gϕ end m 2 φ , z = (tω + π)/2 [for the gΦχ 2 interaction], (4.12) with ω χ = k 2 + m 2 χ . With the T → 0 limit of the rate for ϕ → χχ decays given in table 1 and using (3.9) as well as (4.2) the conditions I)-III) translate into ϕ end g m φ < m φ 4 , ϕ end g m φ < m φ 24M pl V 1/4 end , g m φ > T BBN m φ M pl π g * 32 45 1/4 . (4.13) We can simplify this to g m φ ≪ 1 4 ε ϕ , g m φ ≪ 0.2 √ ε ϕ ε pl , g m φ ≫ T BBN m φ 3π √ ε pl g * 106.75 1/4 , (4.14) where 106.75 is the value of g * in the symmetric phase of the SM and we have rounded all numerical prefactors to ratios of integers. If we set T BBN ≃ 10 MeV, ϕ end ∼ M pl , 27 and ignore factors of order one, we obtain the very simple condition 10 −10 g * 106.75 1/4 GeV m φ < g m φ < 10 −19 m φ GeV . (4.15) The set of inequalities (4.15) marks the range of inflaton couplings for which reheating is driven by elementary processes. Since medium effects are negligible in this region (cf. Sec. 3.3), we can use the T → 0 limit of the expressions given in table 1 for Γ ϕ to translate the relation (2.6) into a constraint on the inflaton coupling. If the produced particles are massless, this in the present case yields the very simple well-known expression Γ ϕ→χχ = g 2 /(8πm φ ). Inflaton annihilations. We now consider the operator c (2,2) Φ 2 χ 2 ≡ h 4 Φ 2 χ 2 . Comparing (4.9) and (4.3) it is straightforward to obtain the expressions for A k and q by replacing λ φ → h in (4.5), In order to evaluate condition III) we need the annihilation rate during adiabatic harmonic oscillations from table 1 for which we take Γ ϕϕ→χχ ≃ h 2 ϕ 2 256πmϕ from table 1. This rate can intuitively be interpreted as an annihilation of two Φ-quanta into two χ-quanta. Determining T R is a bit more tricky because Γ ϕ ∝ ϕ 2 , while H ∝ |ϕ|. As a result, the ratio Γ ϕ /H (averaged over a few oscillations) decreases with time. For ϕ end > 256 √ 2πm 2 ϕ /(3h 2 M pl ) this ratio is larger than one already when the oscillations commence, for smaller ϕ end it remains below one at all times, and the universe is not reheated. The requirement to reheat the universe through inflaton annihilations can be translated into the condition on h > 16 × 2 1/4 π/3 √ ε ϕ ε pl . A k = ω 2 χ ω 2 + 2q , q = h 8 ϕ 2 end ω 2 , z = tω + π/2 [for the h 4 Φ 2 χ 2 interaction], ,(4. Comparing this to (4.17) shows that it is not possible to reheat the universe with elementary annihilations (as it was e.g. assumed in [115]): one either encounters a resonance, or the universe is never reheated. Before moving on, we should add that several details have been wiped under the carpet here. We implicitly assumed an averaging over a few oscillations when comparing Γ ϕ to H without specifying how exactly this averaging should be done. This may be important because the ϕ-dependence of Γ ϕ introduces a non-linearity in (2.1), so that the motion of ϕ is in general not that of a simple damped harmonic oscillator even if we set V = 1 2 m 2 φ ϕ 2 . As a result the relation between the rates of elementary processes and the damping of ϕ can be more complicated than in the linear regime [89]. Two body decays into fermions It is usually assumed that reheating primarily produces bosons because quantum statistical effects lead to an enhancement of the rate, while they suppress the production of fermions due to Pauli's principle. However, fermions can also be produced, and they can also experience resonant enhancement [116]. 28 For the present discussion the fermion production is only relevant if it dominates the reheating process and sets N re . This requires that the coupling to fermions is considerably larger than that to light bosons. We here consider the production of a Dirac fermion ψ of mass m ψ through a Yukawa coupling yΦψψ. A resonance parameter can be defined as q = y 2 ϕ 2 ω 2 , which yields q ∼ y 2 ϕ 2 m 2 φ if the potential is approximately quadratic [116,118]. The underlying mechanism for the resonant production of fermions is quite different from bosons. Hence, we cannot use condition II) here. In spite of the differences in the underlying microphysics, it turns out that the condition q ≪ 1 still provides a good criterion to determine the largest coupling where reheating is perturbative [116]. 29 The lower bound on y can again be obtained from condition III). Comparing the perturbative decay rate Γ ϕ→ψψ = y 2 8π m φ to Γ ϕ→χχ reveals that we can literally apply the lower bound in conditions (4.13)-(4.15) with the replacement g m φ → y. We finally arrive at the range T BBN m φ 3π √ ε pl g * 106.75 1/4 ≪ y ≪ 1 4 ε ϕ (4.18) 28 This can have interesting implications even if the fermion production is sub-dominant compared to that of bosons, e.g. for leptogenesis or gravitino production [117,118]. 29 It should be noted that the treatment used in [116] underestimates the fermion production in the presence of large bosonic occupation numbers [119]. This is, however, not relevant in the present discussion because in that case the coupling to bosons would set Nre. Gauge boson production from axion-like coupling Let us now consider axion-like couplings αΦΛ −1 F µνF µν to the field strength tensor F µν of vector bosons. 30 We assume that the gauge bosons are massless for ϕ = 0, which is justified in the symmetric phase of gauge theories, and consider an Abelian gauge theory. This situation is qualitatively different from the scalar interaction because the vertex is momentum dependent, and the time dependent mass term of the gauge fields have a non-trial Lorentz structure. The Mathieu equation for the circular polarisations of the photon field reads [110] ∂ 2 t + k 2 ∓ 4α k m φ Λ ϕ a k± = 0 (4.19) Comparison to (4.11) gives A k = (2k/ω) 2 with k = |k| and q = 8α k ω ϕ Λ m φ ω . It is immediately clear that condition I) is always violated for large momenta. On the other hand this does not necessarily imply that non-perturbative particle production dominates because for sufficiently small α almost all modes remain adiabatic. 31 Here we restrict ourselves to a simple estimate. Both, the perturbative decay and a narrow resonance produce particles with energy m φ /2, so that we can fix the momentum to this value and use (4.4) with the replacements ω φ → m φ /2 and g → αm 2 φ /Λ in A k and q. We obtain the upper bounds on α from conditions I) and II) by making this replacement in (4.13)-(4.15). To obtain the lower bound from BBN, we consider the elementary decay rate into vector bosons from table 1, Γ ϕ→γγ = α 2 4π m 3 φ Λ 2 . Comparison with Γ ϕ→χχ = g 2 8πm φ reveals that the lower bound can be obtained by making the same replacement g → αm 2 φ /Λ in Γ ϕ and multiplying the total rate with a factor 2 to account for the two photon polarisations in the final state, leading to α m φ Λ ≪ 1 4 ε ϕ , α m φ Λ ≪ 0.2 √ ε ϕ ε pl , α m φ Λ ≫ 2 T BBN m φ √ ε pl g * 106.75 1/4 . (4.20) Discussion For the interactions in table 1 that we studied we find that the upper bounds (4.7), (4.14), (4.17), and (4.20) on the dimensionless inflaton coupling g can be summarised in terms of the ratios (3.1) as g ≪ ε n−1/2 ϕ min( √ ε pl , √ ε ϕ ) m φ Λ 4−D , v i ≪ ε n−5/2 ϕ min( √ ε pl , √ ε ϕ ) m φ Λ 4−D , (5.1) with D the mass dimension of the operator and n the power at which Φ appears. In scalar interactions of the form (4.1) tree-level contributions to the mass that can trigger a parametric resonance come from operators with D = n for self-interactions v i and D = n + 2 for couplings g to other scalars. The most strongly constrained interactions are the self-interactions and couplings to other fields with n = 2, as for larger values of n the negative powers of the small quantity m φ /Λ on the RHS of (5.1) soften the constraint. For a Yukawa coupling to fermions we find that only the first of the two conditions in (5.1) applies, cf. (4.18). Based on the observation that the time-dependent contribution to the squared effective fermion mass scales as ∼ (gϕ n ) 2 one can expect constraints of the form g ≪ ε n ϕ (m φ /Λ) 1−n for fermions. This scaling is consistent with (4.18). 30 Though axions may not be needed to solve the strong CP problem [120], axion-like couplings appear in many theories beyond the Standard Model. 31 In addition the photon dispersion relations would also be modified by finite temperature/density effects, and the plasmon mass can help avoiding a resonance. However, this effect of course depends on the underlying particle physics model and the {ai}. There is also a lower bound on g from condition III), i.e., the requirement to reheat the universe, which for the terms with n = 1 in (4.14), (4.18) and (4.20) roughly reads g ≫ T BBN m φ √ ε pl m φ Λ 4−D #g 1/4 * , (5.2) where we have parameterised Γ ϕ = g 2 m φ /# with # a numerical factor, cf. table 1. Ignoring numerical prefactors, this leaves a window of values for √ ε pl T BBN /m φ ≪ g(m φ /Λ) D−4 ≪ min(ε n ϕ , √ ε pl ε n−1/2 ϕ ) that can span several orders of magnitude if m φ is large enough, and closes when m φ violates the lower bound m φ > ϕ end T BBN ϕ end max 1, ϕ end /M pl 1/(n+1/2) , (5.3) which roughly reads m φ > 10 5 GeV for n = 1 and ϕ end ∼ M pl , and approaches M pl for larger n. The constraints (5.1)-(5.3) mark the range of coupling constants and inflaton masses for which it is in principle possible to constrain the inflaton coupling g in a model-independent way. They do not say anything about how accurately one can measure g in practice. To make a simple estimate, we assume a model of inflation where V(ϕ) has a flat plateau shape, as this class of models is amongst those presently preferred by data [66,121]. Let M be the scale of inflation, i.e., V end ∼ M 4 . Then (3.9) implies that m φ ∼ M 2 /ϕ end . Here we have assumed that V end ≃ V(ϕ k ) for a plateau-like potential, with ϕ k the value of ϕ at the Hubble crossing of the pivot mode k used in the CMB data analysis. Using (A.7) and assuming ϕ end ∼ M pl we can estimate m φ ∼ M pl 3π 2 2 rA s , M ∼ M pl 3π 2 2 A s r 1/4 (5.4) and express (5.1) in terms of observable quantities. For n = 1 and renormalisable interactions this yields g ≪ 3π 2 2 rA s . (5.5) If we assume that reheating is driven by elementary decays in vacuum (consistent with what we found in section 3), we can use standard formula (2.4) for the reheating temperature to obtain g > g 1/4 * T BBN m φ M pl # ≃ T BBN M pl g * A s r 1/4 #. (5.6) Assuming that the mild dependence on g * can be neglected, the only unknown in (5.5) and (5.6) is the scalar-to-tensor ratio r. Plugging in the upper bound r = 0.06 from [66] into (5.5) yields g ≪ 4 × 10 −5 . This is about an order of magnitude larger than the electron Yukawa coupling in the SM. From (5.6) we find g > 10 −17 for # = 8π and the SM value of g * . This implies that the current uncertainty in g extends over 12 orders of magnitude. For an inflaton mass near the bound (5.3) feedback effects can only be avoided if T re ∼ T BBN , which requires a very small value of g at the lower end of this window. For larger values of m φ the maximal reheating temperature (2.4) that can be achieved while respecting (5.5) can be estimated as T re < g −1/4 * m 3 φ /(ϕ end #) ∼ m φ √ ε ϕ g −1/4 * . We can now use (5.5) and (5.6) to estimate the accuracy δr with which r will have to be measured in order to determine the order of magnitude of the inflaton coupling g. Let us for a moment suppose that we have fixed all parameters {v i } in the potential V(ϕ). From (2.6) and table 1 we can see that g 2 ∝ e − 3 2 Nre . Further using (A.4), (A.5) 32 and (A.9) we can express the RHS in terms of observables. The resulting expression for log g is in principle very complicated, but within the observationally allowed range, one may linearly expand log g as a function of r around some valuer that is typical for the given model, i.e., log g ∝ r −r within some interval aroundr. For r > 10 −7 the upper and lower bounds (5.5) and (5.6) leave a range of at least ten orders of magnitude for g. At the same time r can at most change by an amount that is smaller than the current upper bound on its value, i.e., ∆r < 6 × 10 −2 [66] or less [67]. 33 Hence, the observational resolution δr must be less than a tenth of ∆r if one wants to constrain the order of magnitude of g. This very rough estimate leads to the requirement 34 δr ∼ 10 −3 . (5.7) An accuracy of a few times 10 −3 can be achieved by several future CMB observatories, including the Simmons Observatory [123], South Pole Observatory [124], LiteBIRD [125], CMB S4 [126], and PICO [127]. Hence, we can conclude that next generation CMB experiments may be able to pin down the order of magnitude of the inflaton coupling. We emphasise that the estimate (5.7) assumes that all other uncertainties can be neglected, which is possible if all parameters in V(ϕ) are either predicted by theory or determined from other observables. While M can be constrained well from A s , the current error bar on n s is too large to simultaneously fit g and additional parameters in V(ϕ) in the family of models considered in [22,122]. This situation may change in the future. For instance, combining CMB observations with data from the EUCLID satellite can reduce the error bar on n s to σ ns = 0.00085, with potential further improvement when adding data from the Square Kilometre Array [128]. The running of the scalar spectral index n s is another potential probe [15] that can provide information on reheating [129], it can be particularly important if r is small [130]. Future CMB observations will be sensitive to the running of n s [131], combining this with data from optical, infrared and 21cm surveys can improve the sensitivity considerably [131][132][133]. Another potentially interesting observable are non-Gaussianities, which can be targeted by a number of probes including CMB observations, galaxy surveys and 21cm observations [134][135][136]. A quantitative study of the perspectives to constrain the inflaton coupling from these and other potential observables goes beyond the scope of the current work, which focuses on the parameters (A s , n s , r) that are already constrained by data. In view of the expected progress on the observational side in the next decade [57,136] our results strongly motivate follow-up work to identify the most promising observables. 32 Note that the integral in (A.5) is dominated by the upper limit ϕ k , where the integrand can be written as M pl 8/r. 33 While it has been shown in [22] for the case of α-attractor models that an approximately linear relationship between log g and the deviation in ns holds across a range of values for ns that is comparable to the current observational error bar on ns, the range of validity for the linear approximation in r −r is typically smaller than the observationally allowed range of values for r (note that r and ns are not independent when all {vi} are fixed, and within the observational limits, one can often establish a unique relation between them). Hence, in specific models there can be considerable deviations from the rough estimate (5.7), necessitating more detailed studies in specific models [122]. 34 Note that the dependence of r on g is stronger for larger values of r, cf. e.g. [22] for an explicit example, which would facilitate the measurement of g. Conclusions We studied the conditions under which a model-independent relation between the inflaton coupling g and the CMB parameters (A s , n s , r) can be established, so that the microphysical parameter g can be "measured" in the CMB. Model-independence here refers to the details of a specific particle physics model in which a given model of inflation can be embedded. We work under the assumptions that an effective single-field description holds during both the inflationary and reheating epochs, that reheating occurs during oscillations of the inflaton condensate ϕ around the minimum of its effective potential V(ϕ), and that radiative corrections to V(ϕ) either remain small or are dominated by inflaton self-interactions during both inflation and reheating. We further assume a standard thermal history after reheating, and that the tensor modes observed in the CMB are either dominated by perturbations generated during inflation or secondary sources can be identified and subtracted. Finally we assume that the mild dependencies on other particle physics parameters summarised in section 2.2 are negligible. This e.g. means that the number of new particles should be small enough that the dependencies ∼ g 1/4 * can be neglected. Under these assumptions the most severe restriction comes from the necessity to avoid feedback effects during reheating, which in general introduce a dependence of (A s , n s , r) on a potentially large number of microphysical parameters {a i } in the underlying particle physics model, making it practically impossible to constrain any individual parameter. Feedback effects and dependencies on additional parameters {a i } can be avoided if conditions I) and II) outlined in section 3 are fulfilled. These conditions can be translated into restrictions on the inflaton's self-couplings {v i } as well as its coupling g to other fields X , which can generally be expressed in terms of the ratios ε ϕ = m φ /ϕ end and ε pl = m φ /M pl defined in (3.1). They necessarily restrict the possibility to measure g to the mildly non-linear regime where deviations from (3.9) can be treated perturbatively, and impose upper bounds (5.1) on the coupling constants. For renormalisable interactions in table 1 that are linear in ϕ and in plateau-like inflation models we can use (5.5) to estimate the numerical value of the upper bound to be about an order of magnitude larger than the electron Yukawa coupling in the SM. At the same time the requirement to successfully reheat the universe leads to the lower bound (5.2) on g. For values of r near the current upper bound from observation the range of g allowed by (5.2) and (5.5) spans about 12 orders of magnitude. This window becomes smaller for lower m φ and closes for m φ ∼ 10 5 GeV for the interactions considered here. This unfortunately suggests that it will not be possible in the foreseeable future to measure g independently in collider experiments and in the CMB, though it should be kept in mind that this is a rough estimate. These conditions permit to identify the range of values for the inflaton coupling for which it in principle can be constrained by CMB observations within a given model of inflation. The practical feasibility of this measurement depends on the model of inflation. The rough estimate (5.7) suggests that next generation CMB experiments will be able to constrain the order of magnitude of the coupling constant by measuring the scalar-to-tensor ratio r. Further improvement will be possible when adding data from cosmological surveys, in particular with EUCLID and SKA. Even a rough measurement would mark a significant achievement from the viewpoint of both cosmology and particle physics. It provides an indirect probe of a microphysical parameter that most likely can never be measured directly in the laboratory, but shaped the evolution of the cosmos by setting the stage for the hot big bang. Further, it can provide one of the very few observational hints on how a given model of inflation can be embedded in a more fundamental theory of nature. -20 -I would like to thank Gilles Buldgen, Drazen Glavan, Jin U Kang, Lei Ming, and Vincent Vennin for their helpful comments on the draft and Christophe Ringeval for helpful discussions. I also thank my grandmother Erika Drewes and my mother Roswitha Drewes for hosting me in their homes and helping us with childcare for a few weeks during the pandemic while I was working on this article. A Relation between N re and observables In this appendix we briefly review the relationship between the RHS of (2.6) and observables. We consider single field inflation models in which the universe is reheated by coherent oscillations of ϕ around he minimum of its effective potential V(ϕ). In this case the evolution of ϕ is governed by the equation of motion (2.1), c.f. appendix C.2. Inflation ends when the universe stops accelerating, which happens when the equation of state exceeds w > −1/3. This roughly corresponds to the moment when ǫ exceeds unity, with ǫ = M 2 pl 2 ∂ ϕ V V 2 , η = M 2 pl ∂ 2 ϕ V V , (A.1) More precisely, we define the reheating epoch as the time between the moment when ǫ| ϕ end = 1 and the moment when the energy density of radiation ρ R exceeds the energy density ρ ϕ ≃ ϕ 2 /2 + V of the condensate ϕ, i.e., the universe becomes radiation dominated (w = 1/3). The latter moment roughly coincides with the moment when Γ ϕ = H. We refer to the duration of the reheating epoch in terms of e-folds as N re = ln(a re /a end ). and the reheating temperature as T re = exp − 3(1 +w re ) 4 N re 40V end g * π 2 1/4 , (A.3) where V end is the potential at the end of inflation, and we have used (2.5). Since the energy density during reheating is (by definition) dominated by V(ϕ),w re only depends on the {v i }. In particular, for a power law potential V(ϕ) ∝ (ϕ/M pl ) p one findsw re = (p − 2)/(p + 2) [137]. In the regime (3.9) permitted by conditions I) and II), this practically impliesw re = 0. Hence, leaving aside the mild dependence on g * (which could be avoided if we were to consider ρ re instead of T re ), N re is the only entity on the RHS of (A.3) that is not determined by fixing the parameters {v i } in V. Hence, determining T re boils down to establishing a relation between N re and (n s , A s , r). A detailed derivation can be found in [21] and has been adapted to our notation in [22], the result reads N re = 4 3w re − 1   N k + ln k a 0 T 0 + 1 4 ln 40 π 2 g * + 1 3 ln 11g s * 43 − 1 2 ln π 2 M 2 pl rA s 2 √ V end   . (A.4) Here N k is the number of e-folds between the horizon crossing of a perturbation with wave number k and the end of inflation, N k = ln a end a k = ϕ end ϕ k Hdφ ϕ ≈ 1 M 2 pl ϕ k ϕ end dϕ V ∂ ϕ V . (A.5) Here a 0 and T 0 = 2.725 K are the scale factor and the temperature of the CMB at the present time, respectively. The subscript notation H k , ϕ k , ǫ k , ϕ k indicates the value of the quantities H, ϕ, ǫ, ϕ at the moment when the pivot-scale k crosses the horizon. ϕ k can be expressed in terms of the spectral index n s and the tensor-to-scalar ratio r by the relation n s = 1 − 6ǫ k + 2η k , r = 16ǫ k . (A.6) In the slow roll approximation, we find H 2 k = V(ϕ k ) 3M 2 pl = π 2 M 2 pl rA s 2 (A.7) with A s = 10 −10 e 3.043 [66] the amplitude of the scalar perturbations from the CMB. T re can be expressed in terms of the observables (n s , A s , r) by plugging (A.4) with (A.5) into (A.3). ϕ k is fixed by solving (A.6) for ϕ k , and V end and ϕ end can be determined by solving ǫ = 1 for ϕ. From (A.6) we obtain ǫ k = r 16 , η k = n s − 1 + 3r/8 2 , (A.8) from which we find ∂ ϕ V(ϕ) V(ϕ) ϕ k = r 8M 2 pl , ∂ 2 ϕ V(ϕ) V(ϕ) ϕ k = n s − 1 + 3r/8 2M 2 pl , (A.9) using (A.1). Together with (A.7) this yields three equations that can be used to relate the effective potential and its derivatives to the observables (n s , A s , r). This is sufficient to express w re and N re in (A.4) in terms of observables, which is all that is needed to determine the RHS of (2.6). B A slightly more detailed discussion of thermal feedback In this appendix we briefly return to the question under what circumstances the modifications of the rate for elementary processes caused by the presence of a plasma of inflaton decay products can modify N re . We confirm the conclusion drawn in section 3.3 that such effects are negligible in the regimes where conditions I) and II) are fulfilled, and therefore do not impose any additional constraints on g. Assuming that the inflaton performs many oscillations before decaying, we can solve the coupled equationsρ ϕ + 3Hρ ϕ + Γ ϕ ρ ϕ = 0 andρ R + 4Hρ R − Γ ϕ ρ ϕ = 0 for the time averaged inflaton energy density ρ ϕ and the radiation density ρ R defined in (2.4). It is convenient to introduce the dimensionless quantities Φ ≡ ρ ϕ a 3 /m φ , R ≡ ρ R a 4 and x ≡ am φ = a/a end and rewrite the equations as dΦ dx = − Γ ϕ Hx Φ , dR dx = Γ ϕ H Φ (B.1) with H = 3 −1/2 m 2 φ M pl R x 4 + Φ x 3 1/2 , T = m φ x 30 π 2 g * R 1/4 . (B.2) Let us first consider the case where thermal effects suppress Γ ϕ , as e.g. for the decay into fermions in table 1. We can then use T n > T re0 as a conservative criterion to avoid feedback. This translates into the bound Γ 0 /m φ < √ g * m φ /M pl . Assuming Γ 0 ∝ g 2 m φ this yields g ≪ ε 1/2 pl g 1/4 * , which is much weaker than the constraints on g that we derive from the absence of parametric resonances in (4.18). To discuss the case where thermal effects enhance Γ ϕ we consider the Taylor expansion Γ ϕ = i Γ i (T /m φ ) i . If Γ ϕ as a function of T is a monomial, the set of equations (B.1)-(B.2) can be solved analytically [20,84,85,107]. We can use this fact to obtain analytic solutions for an arbitrary dependence of Γ ϕ on T by locally approximating the functional dependence by a monomial power law and matching the solutions [84], provided that one term in the Taylor expansion dominates at any given temperature. For the present purpose it is sufficient to consider the parameterisation Γ ϕ = Γ 0 + Γ n (T /m φ ) n , (B.3) which yields T n = m φ (Γ 0 /Γ n ) 1/n . For positive n ≤ 2, Γ ϕ cannot grow faster with temperature than the contribution to H from ρ R , cf. (B.2). If T n > T re0 , then the temperature-dependent correction to Γ ϕ remains smaller than H at all higher temperatures T > T re0 . Hence, for n ≤ 2 the thermal corrections to Γ ϕ can only affect N re if T n < T re0 , which translates into the condition Γ n /Γ 0 < (π 2 g * /90) n/4 ( m φ Γ 0 m φ M pl ) n/2 to avoid a modification of N re . If Γ 0 and Γ n originate from the same interaction, then they tend to have a similar dependence Γ 0 , Γ n ∼ g 2 m φ on g and m φ , cf. table 1, and this requirement translates into g ≪ ε 1/2 pl g 1/4 * , which is much weaker than the bound (5.1) from the absence of a parametric resonance. We can therefore ignore the impact of thermal feedback on N re for n ≤ 2, consistent with what was found in [20]. The cases n = 1 and n = 2 cover several important interactions, cf. table 1: The case n = 1 e.g. describes a Bose enhanced rate for decays 1 → 2 or 2 → 2 annihilations, the case n = 2 can e.g. describe scatterings amongst scalars. 35 For n > 2 the above argument does not hold. We shall instead use the criterion that Γ n (T max /m φ ) n < H, where T max is the maximal temperature during the reheating epoch. Using the explicit solutions for (B.1) found in [84,85] for R = 0 , Φ = ρ end /m 4 φ at x = 1 (B.4) and assuming T max0 > T n (otherwise thermal effects are certainly negligible), we can approximate T max ≃ m φ 30 π 2 g * 1/4 A n 4 1/(4−n) 8 − 2n 3 3 (n−4)(5−2n) . (B.5) with A n = A 0 Γ n Γ 0 30 π 2 g * n/4 , A 0 = √ 2 Γ 0 m φ M pl m φ ϕ end m φ (B.6) This maximal temperature is reached at x max ≃ 8−2n 3 2 5−2n . We can assume that Φ is roughly constant until the moment when Γ ϕ = H and hence set Φ ≃ ρ end /m 4 φ ≃ 2 3 (ϕ end /m φ ) 2 . This gives H ≃ √ 2 3 m φ ϕ end M pl x −3/2 . (B.7) Using the above expressions we can translate the condition that Γ ϕ < H at x = x max into Γ n m φ This imposes an upper bound on Γ n from the requirement that thermal effects do not modify the reheating temperature. 36 If we assume Γ n ∝ g 2 m φ , it turns out that (B.8) is weaker than the conditions I) and II) for n = 1, 2, 3. We have so far assumed that the inflaton is the only field with a non-zero expectation value during reheating. However, in many particle physics models that can accommodate single field inflation the inflaton is not the only scalar field, but simply represents the relevant direction in field space. In fact, leaving aside the specific case of Higgs inflation [138], we know for sure that there are at least four real scalar degrees of freedom, the components of the SM Higgs doublet [75,139]. Even if the dynamics during inflation can effectively be described by a single field model (with Φ simply being the relevant direction in field space), a certain fraction ε of the total energy density may be stored in these other degrees of freedom. If the time that it takes for this energy to be transferred into ρ R is much shorter than the value of N re that one would obtain by neglecting this effect, then one can obtain a conservative bound on the impact that this has on the actual value of N re by replacing the initial conditions (B.4) at x = 1 with Φ = (1 − ε)ρ end /m 4 φ , R = ερ end /m 4 φ and using the analytic solutions presented in [84,85,107]. If the fraction ε is subdominant this will not change our conclusions. If either ε is dominant or its lifetime is longer than that of ϕ, a detailed model-dependent study is necessary. C On the consistent description of reheating A cautious reader may be concerned about the fact that the arguments presented in this work are not derived within a consistent formal description of the reheating process. As a result, it may e.g. not be obvious how the particle production obtained from the classical Mathieu equation (3.11) is related to the rates in table 1 that were computed in thermal quantum field theory. Moreover, the use of an Markovian equation of the form (2.1) during reheating is in general questionable because the effective action for ϕ after tracing out other degrees of freedom is fundamentally non-local. In this appendix we briefly comment on these points and illustrate how the expressions that we used throughout the main text can be derived consistently from nonequilibrium quantum field theory. For a more complete and pedagogical introduction to the topic we e.g. refer the interested reader to [140]. All properties of a system in quantum statistical mechanics are in principle encoded in the von Neumann density operator ̺. Instead of directly dealing with the infinite-dimensional matrix ̺, it is often more convenient to consider a finite set of correlation functions. The most important properties of a scalar field Φ can be encoded in its expectation value and the two independent real-valued two-point functions ϕ(x) ≡ Φ(x) , ∆ − φ (x, y) ≡ i [Φ(x), Φ(y)] , ∆ + φ (x, y) ≡ 1 2 {Φ(x), Φ(y)} − ϕ(x)ϕ(y), (C.1) where {·, ·} and [·, ·] are the anticommutator and commutator operators respectively. All other two-point functions of Φ (such as the Feynman propagator) can be expressed in terms of those defined in (C.1). The expectation value is to be taken with respect to both, quantum and thermal fluctuations encoded in the density operator: . . . = Tr(̺ . . .). The one-point function ϕ describes the scalar condensate or mean field, while the two-point functions ∆ − and ∆ + describe the fluctuations. The poles of the spectral function ∆ − encode information about spectrum of (quasi)particles, while the statistical propagator ∆ + contains information about the occupation numbers of each momentum mode. For illustrative purposes we restrict the discussion to Minkoswki space, the generalisation to curved space is e.g. discussed in [141]. Our starting point to find the equation of motion for ϕ and ∆ ± in the closed-time-path (CTP) formalism [142][143][144] is the two-particle irreducible effective action [145] with arguments on a suitably defined time path C, Γ ϕ [ϕ, ∆] = S[ϕ] + Γ ϕ loop [ϕ, ∆[ϕ]], Γ ϕ loop [ϕ, ∆[ϕ]] = i 2 Tr ln ∆ −1 + i 2 Tr G −1 [ϕ] ∆ + Γ ϕ 2 [ϕ, ∆], (C.2) with S[ϕ] the classical action and iG −1 [ϕ](x 1 .x 2 ) = δ 2 S[Φ] δΦ(x 1 )δΦ(x 2 ) | Φ→ϕ . The trace-terms in (C.2) correspond to the one-loop correction to the effective action, which e.g. gives rise to the Coleman-Weinberg potential [146] in the static limit, while Γ ϕ 2 [ϕ, ∆] contains terms with two or more loops. The equations of motion for the correlation functions defined in (C.1) are found by functional differentiation, δΓ ϕ [ϕ, ∆] δϕ(x) = 0 , δΓ ϕ [ϕ, ∆] δ∆(x, y) = 0 . (C.3) C.1 Equations of motion for propagators and particle numbers The equations of motion for the propagators ∆ ± are obtained from the second equation in (C.3). In the case of an homogeneous and isotropic system, after a Fourier transformation in the spacial relative coordinate x 1 − x 2 , they can be brought into the form ∂ 2 t 1 + Ω 2 φ (t 1 ; k) ∆ − φ (t 1 , t 2 ; k) = − t 1 t 2 dt ′ Π − φ (t 1 , t ′ ; k)∆ − φ (t ′ , t 2 ; k) , (C.4) ∂ 2 t 1 + Ω 2 φ (t 1 ; k) ∆ + φ (t 1 , t 2 ; k) = − t 1 t i dt ′ Π − φ (t 1 , t ′ ; k)∆ + φ (t ′ , t 2 ; k) + t 2 t i dt ′ Π + φ (t 1 , t ′ ; k)∆ − φ (t ′ , t 2 ; k) , (C.5) with t i the initial time. Here Π loc φ , Π − φ and Π + φ are the local, spectral and statistical selfenergies, which can be obtained by decomposing the self-energy on the contour, C Π φ (x, y) = 2iδΓ ϕ 2 [ϕ, ∆]/δ∆, into Π φ (x, y) = −iΠ loc φ (x, x) δ(x − y) C + Π + φ (x, y) − i 2 Π − φ (x, y) sign C (x 0 − y 0 ), (C.6) and Ω 2 φ (t; k) = k 2 + M tree φ (t) 2 + d 3 q (2π) 3 Π loc φ (t, t; k) ≡ k 2 + M φ (t) 2 (C.7) with M tree φ the effective tree level mass defined by iG −1 φ [ϕ](x 1 , x 2 ) = −( x 1 + M tree φ (x) 2 )δ C (x 1 − x 2 ). (C.8) The tree level mass M tree φ differs from m φ due to the contribution from the coupling of φ to the time dependent background ϕ. Together with the local contribution Π loc φ to the selfenergy from forward-scattering M tree φ determines the effective mass M φ that appears in the approximation W φ ≃ Ω φ in (C.7) to the full dispersion relation W 2 φ of screened φ-particles. Ω 2 φ describes a change in the quasiparticle dispersion relation that depends on ϕ and the occupation numbers in the medium, but not on momentum, and therefore can be parameterised by a timedependent effective mass M φ . The full dispersion relation W 2 φ can be obtained by including Π − φ and can be quite complicated (cf. e.g. [147]); it is determined by the poles of the solutions ∆ − to (C.4) in Wigner space, hence the name spectral function for ∆ − . The interpretation of the statistical propagator ∆ + can be illustrated by noticing that there is a contribution ρ φ to the total energy density that can be interpreted as the energy in fluctuations φ (i.e. particles) and to leading order reads 37 ρ φ ≃ 1 2 ∂ tt ∂ t 2 + Ω 2 φ (t 1 ) ∆ + (t 1 , t 2 )|) t 1 =t 2 . (C.9) For the derivation of conditions I) and II) in section 3.2 it is sufficient to consider the leading approximation Ω φ in (3.2) and its equivalents for other particle species, i.e., neglecting the RHS of (C.4). This can be justified by an expansion in the couplings {v i , g i } and in . By making the same approximation in (C.5) we obtain an equation of the form 10) i.e., a Klein-Gordon equation with a time dependent mass M φ . This equation for ∆ + φ has the same form as the equation (4.3) for the mode functions and can be brought into the form (3.11) by making the replacements such as (4.4) and (4.5). Hence, at leading order in the solutions of the Mathieu equation (3.11) describe the behaviour of the statistical propagator ∆ + φ , and, through (C.9), the energy density stored in φ-particles. The same holds for the χ-particles, cf. (4.11) with (4.12) and (4.16). This in particular justifies using the Mathieu equation as a tool to study the non-perturbative particle production (iv) because (3.11) represents the leading term in the -expansion of the full quantum equation (C.5). (∂ 2 t 1 + k 2 + M 2 φ )∆ + φ = 0, (C. Elementary processes are not included at this level; they are encoded in the self-energies on the RHS of (C.5) and contribute at the next order in , cf. e.g. [111]. More precisely, one can define Γ φ = − ImΠ − φ (t, Ω φ ) Ω φ ,Π − φ (t, Ω φ ) = ∞ t i dzΠ − (t, t − z). (C.11) The physical interpretation of Γ φ and ∆ + φ becomes evident in the adiabatic regime, where one can approximate [148] ∆ + φ (t 1 , t 2 ; k) ≃ cos t 1 t 2 dt ′ Ω φ (t ′ ) exp − 1 2 t 1 t 2 dt ′ Γ φ (t ′ ) 2 Ω φ (t 1 )Ω φ (t 2 ) (1 + 2f φ (t)) (C. 12) with t = min(t 1 , t 2 ). Hence, Γ φ describes the damping of the statistical propagator in time [148,149]; it determines the rate at which the quasiparticle phase space distribution f φ and the energy density (C.9) approach equilibrium. In the weak coupling and dilute gas limit f φ fulfils a standard Boltzmann-equation in which the collision term is computed from (C.11). If the fields that Φ interacts with are in thermal equilibrium [150], (C.11) reduces to the well-known statement that the damping is given by the imaginary part (more precisely: discontinuity) of the retarded self-energy Π R φ [151], i.e., Γ φ ≃ ImΠ − (Ω φ )/(2iΩ φ ) = −ImΠ R (Ω φ )/Ω φ . Of course, the same considerations hold for the spectral and statistical propagators for all other fields X . C.2 Equation of motion for ϕ The equation of motion for ϕ can be obtained from (C.2) via the first equation in (C.3). Assuming that we have formally solved the equations of motion (C.4) and (C.5) for the propagators in the background provided by ϕ, we can use the solutions ∆[ϕ] to compute the loops in Γ ϕ loop [ϕ, ∆[ϕ]]. Since we are interested in oscillations near the minimum, we can perform a Taylor expansion in functional space around the ground state configurationφ as ϕ =φ + δφ. For simplicity we shall assume that the ground state corresponds to the field configuration ϕ = 0 and write [152] ∂Γ ϕ loop [ϕ, ∆[ϕ]] ∂ϕ(x) φ+δφ = ∂Γ ϕ loop [ϕ, ∆[ϕ]] ∂ϕ(x) φ + ∞ n=1 1 n! n i=1 C dx 0 i d 3 x i δφ(x i )    δ n δϕ(x 1 ) · · · δϕ(x n )   ∂Γ ϕ loop [ϕ, ∆[ϕ]] ∂ϕ(x)   φ . (C. 13) where the partial functional derivatives ∂/∂ϕ only apply to the explicit dependence of Γ ϕ loop . A Markovian (local) equation can be derived in the regimeφ/ϕ < Γ X [102,[152][153][154] and in the mildly non-linear regime (3.9) [89,92,94,102]. Being interested in reheating, we focus on the latter case. In this mildly nonlinear regime (3.9) we may neglect the ϕ-dependence in the propagators ∆[ϕ] inside the loops, which would typically break the adiabaticity condition (3.2) outside the regime (3.9). We can further locally neglect the effects of nonlinearities inside memory kernels in (C.13) and approximate [94] ϕ(t ′ ) = ϕ t−(t−t ′ ) ≈ ϕ(t) cos M φ (t−t ′ ) −φ (t) M φ sin M φ (t−t ′ ) . (C.14) One then obtains an equation of the form K(ϕ)φ + 1 2 K ′ (ϕ)φ 2 + Γ ϕ (ϕ)φ + V ′ (ϕ) = 0 . (C. 15) We note that there is no noise-term on the RHS of (C.15) and (2.1), as one might expect from a generalised fluctuation-dissipation theorem, because our definition of ϕ includes an average over thermal fluctuations. Near the potential minimum, where reheating occurs, these average to zero. The equation of motion for Φ itself would contain a noise term, see e.g. [91,109,150,[154][155][156][157]. In [89] it has been shown explicitly that the use of Markovian equations of motion of the form (C.15) based on the approximation (C.14) is justified in the mildly non-linear regime, were deviations from (3.9) can be treated as perturbations. The coefficient K(ϕ) does not lead to dissipative behaviour, but causes a perturbative correction to the effective frequency M φ . While the impact of this on the phase of the oscillations can build up after many oscillations, it modifies Γ ϕ only by shifting the mass shell of the quasiparticles that compose the condensate ϕ away from m φ to M φ . While this kinematic effect in principle can be significant [84,[95][96][97][98][99][100][101]103], it is always a small correction in the regime allowed by conditions I) and II), cf. section 3.3. 38 Since we are not interested in the phase, but only in the moment when Γ ϕ = H, we can neglect the correction K(ϕ) and conclude that the use of the equation (2.1) is consistent in the mildly non-linear regime of elongations permitted by the considerations in section 4.1. This still leaves the question how Γ ϕ is computed practically. The operators considered in sections 4.4 and 4.3 as well as the case n = 1 in section 4.2 are linear in Φ. In this case Γ ϕ is (at leading order) independent of ϕ, meaning that the only non-linearities in (C.15) are due to deviations from (3.9) in V(ϕ), which we know must be small. In this case, it is well-known that the leading contribution to Γ ϕ can be identified with the thermal width Γ φ of the propagator in (C.11) because the frequency at which ϕ oscillates coincides with the pole mass of φ-particles, as illustrated before (3.3). Higher order corrections are suppressed by additional small parameters, cf. footnote 18. In this situation the inflaton damping rate Γ ϕ and the rate of particle production are both identical to Γ φ in (C.11) and given by table 1. 39 When Γ ϕ depends on ϕ (as e.g. for n > 1 in section 4.2) the connection between Γ ϕ and the rate of perturbative particle production is more subtle (cf. section 5 in [89]), but the expression in table 1 can still be used. The redshifting relation ρ(N ) = ρ end exp[−3 N 0 (1+w(N ′ ))dN ′ ] yields the second equality in (2.3) and permits to express the Friedmann equation (2. We comment on the justification to use an equation of the form (2.1) in Appendix C. We collectively refer to all relativistic particles as "radiation", irrespectively of their spin and charges. A critical discussion of this assumption can be found in[55].7 Observationally they may e.g. be spotted due to deviations from the relation r + 8nt = 0 between r and the spectral index of tensor perturbations nt, as observations are in principle able to distinguish different sources of tensor modes[56]. If the mixing occurs only with other fields in the (multi-field) inflaton sector, then Γϕ may formally still be independent of the {ai} (because the parameters in the extended inflaton sector belong to the set {vi}), though the determination of its functional form would be practically difficult due to highly non-linear and the non-perturbative behaviour. In practice the explosive particle production will almost always introduce {ai}-dependencies through subsequent particle decays and scatterings. For g within the range permitted by the considerations in section 4, a pure coupling to gravity typically leads to fragmentation[83] before the end of the reheating epoch (cf.[84,85] for analytic estimates). However, as long as this mainly transforms the coherent condensate ϕ into a plasma of non-relativistic Φ-particles that decay perturbatively this should not change our conclusions. The reason is that the perturbative decay rate of the latter equals the decay rate Γϕ of the condensate at leading order when non-linear effects can be neglected.10 To be precised, we define W φ as the real part of the pole of the spectral function, i.e., the Fourier transform of the spectral propagator ∆ − φ defined in (C.1) with respect to x1 − x2.11 Here we for simplicity assume that the potential minimum is at ϕ = 0, which is not true in many inflationary models. However, all subsequent arguments hold if we replace ϕ by the deviation of the field from its value at the minimum. This is conservative because in reality ϕ is typically sub-Planckian when the oscillations commence. For the specific examples given in table 1 this conclusion could also have reached without the expansion (B.3) by noticing that fB(M φ ) becomes of order unity when T ∼ m φ . Surprisingly it does not depend on Γ0. However, note that all of this is only relevant if Tmax0 > Tn. This bring ins an additional conditions that depends on Γ0. Complete expressions for the energy-momentum tensor derived from (C.2) by varying with respect to the metric can e.g. be found in[69]. More precisely: In the mildly non-linear regime, where the field elongation is restricted by (3.9) and occupation numbers in the primordial plasma are restricted by the considerations in section 3.3, corrections of higher order in the loop expansion are suppressed by additional powers of the coupling constants, which are restricted to small values by the conditions (5.1). 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Using (3.11) based on the approximation (C.10) leads to a conservative estimate of the parameter range where feedback due to non-perturbative effects at tree-level can be avoided (because elementary decays encoded in one-loop corrections on the RHS of (C.5) can weaken and possibly avoid resonant particle production by reducing the occupation numbers between two zero-crossings of ϕ). Within that parameter range one can then be confident that nonperturbative effects at tree-level are sub-dominant, and therefore the leading contribution to Γϕ comes from elementary processes encoded in one-loop diagrams. Hence, it is fully consistent to neglect the loop corrections when formulating conditions I) and II). 39 A careful reader may be concerned about the fact that the conditions I) and II) in section 3.2 are formulated in terms of the tree-level Mathieu equation (3.11). neglecting contributions to Γϕ from elementary decays that appear as radiative corrections to the equations of motion in the CTP formalism, while one-loop contributions are later used to compute the coefficients in table 1. This is, however, not inconsistent. and then neglect the tree-level effects when computing Γϕ within the range of parameters where these conditions are fulfilled39 A careful reader may be concerned about the fact that the conditions I) and II) in section 3.2 are formulated in terms of the tree-level Mathieu equation (3.11), neglecting contributions to Γϕ from elementary decays that appear as radiative corrections to the equations of motion in the CTP formalism, while one-loop contributions are later used to compute the coefficients in table 1. This is, however, not inconsistent. Using (3.11) based on the approximation (C.10) leads to a conservative estimate of the parameter range where feedback due to non-perturbative effects at tree-level can be avoided (because elementary decays encoded in one-loop corrections on the RHS of (C.5) can weaken and possibly avoid resonant particle production by reducing the occupation numbers between two zero-crossings of ϕ). 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[ "Can Bad Teaching Induce Forgetting? Unlearning in Deep Networks Using an Incompetent Teacher", "Can Bad Teaching Induce Forgetting? Unlearning in Deep Networks Using an Incompetent Teacher" ]
[ "Vikram S Chundawat \nMavvex Labs\nIndia\n", "Ayush K Tarun [email protected] \nMavvex Labs\nIndia\n", "Murari Mandal [email protected] \nSchool of Computer Engineering\nKalinga Institute of Industrial Technology\nBhubaneswar\n", "† ‡ ", "Mohan Kankanhalli \nSchool of Computing\nNational University of Singapore\n\n" ]
[ "Mavvex Labs\nIndia", "Mavvex Labs\nIndia", "School of Computer Engineering\nKalinga Institute of Industrial Technology\nBhubaneswar", "School of Computing\nNational University of Singapore\n" ]
[]
Machine unlearning has become an important area of research due to an increasing need for machine learning (ML) applications to comply with the emerging data privacy regulations. It facilitates the provision for removal of certain set or class of data from an already trained ML model without requiring retraining from scratch. Recently, several efforts have been put in to make unlearning to be effective and efficient. We propose a novel machine unlearning method by exploring the utility of competent and incompetent teachers in a student-teacher framework to induce forgetfulness. The knowledge from the competent and incompetent teachers is selectively transferred to the student to obtain a model that doesn't contain any information about the forget data. We experimentally show that this method generalizes well, is fast and effective. Furthermore, we introduce the zero retrain forgetting (ZRF) metric to evaluate any unlearning method. Unlike the existing unlearning metrics, the ZRF score does not depend on the availability of the expensive retrained model. This makes it useful for analysis of the unlearned model after deployment as well. We present results of experiments conducted for random subset forgetting and class forgetting on various deep networks and across different application domains. Source code is at: https://github.com/vikram2000b/ bad-teaching-unlearning
10.48550/arxiv.2205.08096
[ "https://export.arxiv.org/pdf/2205.08096v2.pdf" ]
248,834,527
2205.08096
425669c368004dc43bebaa1d4acdd46a6bcca171
Can Bad Teaching Induce Forgetting? Unlearning in Deep Networks Using an Incompetent Teacher Vikram S Chundawat Mavvex Labs India Ayush K Tarun [email protected] Mavvex Labs India Murari Mandal [email protected] School of Computer Engineering Kalinga Institute of Industrial Technology Bhubaneswar † ‡ Mohan Kankanhalli School of Computing National University of Singapore Can Bad Teaching Induce Forgetting? Unlearning in Deep Networks Using an Incompetent Teacher Machine unlearning has become an important area of research due to an increasing need for machine learning (ML) applications to comply with the emerging data privacy regulations. It facilitates the provision for removal of certain set or class of data from an already trained ML model without requiring retraining from scratch. Recently, several efforts have been put in to make unlearning to be effective and efficient. We propose a novel machine unlearning method by exploring the utility of competent and incompetent teachers in a student-teacher framework to induce forgetfulness. The knowledge from the competent and incompetent teachers is selectively transferred to the student to obtain a model that doesn't contain any information about the forget data. We experimentally show that this method generalizes well, is fast and effective. Furthermore, we introduce the zero retrain forgetting (ZRF) metric to evaluate any unlearning method. Unlike the existing unlearning metrics, the ZRF score does not depend on the availability of the expensive retrained model. This makes it useful for analysis of the unlearned model after deployment as well. We present results of experiments conducted for random subset forgetting and class forgetting on various deep networks and across different application domains. Source code is at: https://github.com/vikram2000b/ bad-teaching-unlearning Introduction Machine learning (ML) models are being widely deployed for various applications across different organizations. These models are often trained with large-scale user data. Modern data regulatory frameworks such as European Union GDPR (Voigt and Von dem Bussche 2017), and California Consumer Privacy Act (CCPA) (Goldman 2020) provide for citizens the right to be forgotten. It mandates deletion-upon-request of user data. The regulations also require that user consent must be obtained prior to data collection. This consent for the use of an individual's data in these ML models may be withdrawn at any point of time. Thus, a request for data deletion can be made to the ML model owner. The owner company (of the ML model) is legally obligated to remove the models/algorithms derived from using that particular data. As the ML models usually memorize the training samples (Feldman 2020;Carlini et al. 2019), the company either needs to retrain the model from scratch by excluding the requested data or somehow erase the user's information completely from the ML model parameters. The algorithms supporting such information removal are known as machine unlearning methods. Machine unlearning also offers a framework to prove data removal from the updated ML model. The unlearning methods can be practically applied in the following ways: (i) forgetting single-class or multiple classes of data (Tarun et al. 2021), (ii) forgetting a cohort of data from a single class (Golatkar, Achille, and Soatto 2020a,b), (iii) forgetting a random subset of data from multiple classes (Golatkar et al. 2021). In this paper, we investigate the utility of teacher-student framework with knowledge distillation to develop a robust unlearning method that can support all the three modes, i.e. single/multiple classlevel, sub-class level and random subset-level unlearning. Another important question we raise is how well the unlearned model has generalized the forgetting? Recent studies suggest that the unlearning methods may lead to privacy leakage in the models (Chen et al. 2021). Therefore, it is important to validate whether the unlearned models are susceptible to privacy attacks such as membership inference attacks. Moreover, the trade-off between the amount of unlearning and privacy exposure also should be investigated for better decision-making on the part of the model owner. We propose a new metric to evaluate the generalization ability of the unlearning method. The existing unlearning methods for deep networks put several constraints over the training procedure. For example, (Golatkar et al. 2021) train an additional mixed-linear model along with the actual model which is used in their unlearning method. Similarly, (Golatkar, Achille, and Soatto 2020a,b) strictly require SGD to be used in optimization during model training. These restrictions and the need for other prior information make these methods less practical for realworld applications. We present a method that does not require any prior information about the training procedure. We do not train any extra models to assist in the unlearning. Furthermore, we aim to keep the unlearning process efficient and fast in comparison to the high computational costs of the existing methods. We make the following key contributions: 1. We present a teacher-student framework, consisting of competent and incompetent teachers. The selective knowledge transfer to the student results in the unlearned model. The method works for both single-class and multiple class unlearning. It also works effectively for multiple class random-subset forgetting. 2. We propose a new retrained model-free evaluation metric called zero retrain forgetting (ZRF) metric to robustly evaluate the unlearning method. This also helps in assessing the generalization in the unlearned model on the forget data. 3. Our method works on different modalities of deep networks such as CNN, Vision transformers, and LSTM. Unlike the existing methods, our method doesn't put any constraints over the training procedure. We also demonstrate the wide applicability of our method by conducting experiments in different domains of multimedia applications including image classification, human activity recognition, and epileptic seizure detection. Related Work Machine Unlearning. Bourtoule et al. (Bourtoule et al. 2021) proposed to partition the training dataset into nonoverlapping shards and create multiple models for the disjoint sets. They store the weakly learned models to deal with multiple data removal requests. Ginart et al. (Ginart et al. 2019) adopted the definition of differential privacy to introduce the probabilistic notion of unlearning. It expects high similarity between the output distributions of the unlearned model and the retrained model without using the deletion data. Several subsequent works (Mirzasoleiman, Karbasi, and Krause 2017;Izzo et al. 2021;Ullah et al. 2021) follow this approach in presenting theoretical guarantees in their respective problem settings. We also follow this definition of unlearning in our work. Guo et al. (Guo et al. 2020) give a certified data removal framework to enable data deletion in linear and logistic regression. Neel et al. (Neel, Roth, and Sharifi-Malvajerdi 2021) apply gradient descent to achieve unlearning in convex models. The difference between differential privacy and machine unlearning is studied in (Sekhari et al. 2021). Unlearning in random forests (Brophy and Lowd 2021) and Bayesian setting (Nguyen, Low, and Jaillet 2020) are also studied. These methods are designed specifically for convex problems and are unlikely to work in deep learning models. Our work is aimed at performing unlearning in deep networks. Unlearning in Deep Networks. Golatkar et al. (Golatkar, Achille, and Soatto 2020a) presented one of the early works in deep machine unlearning. They introduced a scrubbing method to remove the information from the network weights. The method impose a condition of SGD based optimization during training. The subsequent work (Golatkar, Achille, and Soatto 2020b) proposed a neural tangent kernel (NTK) based method to approximate the training process. The additional approximated model is used to estimate the network weights for the unlearned model. (Golatkar et al. 2021) train a mixed-linear model along with the original model. The linearized model is specific to different deep networks and requires fine-tuning to work properly. Moreover, all these methods suffer from high computational costs, constraints on the training process, and limitations of the approximation methods. Tarun et al. (Tarun et al. 2021) proposed an efficient class-level machine unlearning method. However, it does not support random subset forgetting. In our work, we do not need to train any additional model to support unlearning. Our method does not demand the use of any specific optimization technique during training or any other prior information about the training process. (Chundawat et al. 2023;Graves, Nagisetty, and Ganesh 2021;Tarun et al. 2022) are some other notable works. Preliminaries Let the complete (multimedia) dataset be D c = {(x i , y i )} n i=1 with n number of samples, where x i is the i th sample, and y i is the corresponding class label. The set of samples to forget is denoted as D f . In class-level unlearning, D f corresponds to all the data samples present in a single or multiple classes. In random-subset unlearning, D f may either consist of a random subset of data samples from a single class or multiple classes. The information exclusive to these data points need to be removed from the model. The set of remaining samples to be retained is denoted by D r . The information about these samples are to be kept unchanged in the model. D f and D r together represent the whole training set and are mutually exclusive, i.e. D r ∪D f = D c and D r ∩D f = ϕ. Each data point is assigned an unlearning label, l u , which is 1 if the sample belongs to D f and 0 if it belongs to D r . The subset used for unlearning is {(x i , l ui )} p i=1 , p is total number of samples, and l ui is unlearning label corresponding to each sample x i . The model trained from scratch without observing the forget samples is called the retrained model or the gold model in this paper. In the proposed teacher-student framework, the competent teacher is the fully trained model or the original model. The competent teacher has observed and learned from the complete data D c . Let T s (x; θ) denote the competent/smart teacher with parameters θ. It takes x as input and outputs the probabilities t s . The incompetent teacher is a randomly initialized model. Let T d (x; ϕ) be the incompetent/dumb teacher with parameters ϕ and output probabilities t d . The student S(x; θ) is a model initialized with parameters θ i.e., the same as the competent teacher. It returns the output probabilities s. It is to be noted that the student is initialized with all the information present in the original model (θ). The incompetent teacher is used to remove the requested information (about the forget data D f ) from this model. The Kullback-Leibler (KL) divergence (Kullback and Leibler 1951) is used as a measure of similarity between two probability distributions. For two distributions p(x) and q(x), the KL-divergence is defined by Proposed Method Unlearning with Competent/Incompetent Teachers We aim to remove the information about the requested datapoints by using two teachers (competent and incompetent) and one student. The student is initialized with knowledge about the complete data i.e., the parameters of the fully trained model. The idea is to selectively remove the information about the forget samples from this model. At the same time, the information pertaining to the retain set should not to be disturbed. Thus, the unlearning objective is to remove the information about D f while retaining the information about D r . We achieve this by using a pair of (competent/smart (T s ) and incompetent/dumb (T d )) teachers to manipulate the student (S) as depicted in Figure 1. The bad knowledge about D f from the incompetent teacher T d is passed on to the student which helps the student to forget D f samples. Such an approach consequently induces random knowledge about the forget set in the student instead of completely making their prediction accuracy zero. This serves as a protection against the risk of information exposure about the samples to forget. The bad (random) inputs from T d may invariably corrupt some of the information about the retain set D r in the student. Therefore, we selectively borrow correct knowledge related to D r from the competent teacher T s as well. In this manner, both the incompetent and competent teachers help the student forget and retain the corresponding information, respectively. For a student S, incompetent/dumb teacher T d , and competent/smart teacher T s , we define the KL-Divergence between T d and S in Eq. 1. KL(T d (x)||S(x)) = i t (i) d log(t (i) d /s (i) )(1) where i corresponds to the data class. Similarly, the KL-Divergence between the fully trained competent teacher T s and student S is given in Eq. 2. KL(T s (x)||S(x)) = i t (i) s log(t (i) s /s (i) )(2) The unlearning objective can be formulated as in Eq. 3. (3) L(x, l u ) = (1 − l u ) * KL(T s (x)||S(x)) + l u * (KL(T d (x)||S(x))) where l u is the unlearning label and x is a data sample. The data samples used by the proposed unlearning method consists of all the samples from D f and a small subset of samples of D r . The student is then trained to optimize the loss function L for all these samples. The intuition behind optimizing over L is that we selectively transfer bad knowledge about forget data D f from T d by minimizing KL-Divergence between S and T d and the accurate knowledge corresponding to D r is fed from T s by minimizing KL-Divergence between S and T s . The student learns to mimic T d for D f , thus removing information exclusively pertaining to those samples while retaining all the generic information which can be obtained by other samples of same class. (Chundawat et al. 2023), and layerwise distance (Tarun et al. 2021). Excluding the forget and retain set accuracy, all of the remaining metrics in the literature require a retrained model i.e., training a model from scratch without using the forget set. These metrics can only be interpreted with reference to such a retrained model. Such dependency on the retrained model for unlearning evaluation would lead to higher time and computational costs. Simply measuring the performance on D f and D r does not reveal whether the information is actually removed from the network weights. Thus it is not a comprehensive measure of unlearning. We propose a novel 'Zero Retrain Forgetting Metric' (ZRF) to enable evaluation of unlearning methods free from dependence on the retrained model. It measures the randomness in the model's prediction by comparing them with the incompetent teacher T d . We calculate the Jensen-Shannon (JS) divergence (Lin 1991) between an unlearned model M and the incompetent teacher T d as below. (4) J S(M (x), T d (x)) = 0.5 * KL(M (x)∥m) + 0.5 * KL(T d (x)∥m) where m = M (x)+T d (x) 2 . The ZRF metric is defined as ZRF = 1 − 1 n f n f i=0 J S(M (x i ), T d (x i ))(5) where x i is i th sample from D f with a total of n f samples. The ZRF compares the output distribution for the forget set in the unlearned model with the output of a randomly initialized model, which is our incompetent teacher in most of the cases. The ZRF score lies between 0 and 1. The score will be close to 1 if the model behaviour is completely random for the forget samples and it will be close to 0 if the model shows some specific pattern. What is an ideal ZRF score? Suppose there is a class aeroplanes that contains images of Boeing aircraft along with other aircraft models in the training set. If we unlearn Boeing aircraft, we don't expect the model to now classify them as animals, vegetables or any other totally unrelated class. We still expect most of these unlearned images to be classified as aeroplanes. This comes from the intuition that the model must have been designed and trained with generalization in mind. An unlearning method that makes the performance much worse than the generalization error for aeroplanes is not actually unlearning. It is just teaching the model to be consistently incorrect when it sees a Boeing aeroplane. The ZRF score will be 0 when the model almost always classifies a Boeing aircraft as an animal or some other totally different class. The ZRF will be 1 if the model always classifies all classes with same random probability for Boeing aircraft. Both of these (∼ 0 or ∼ 1) are not the desirable outcomes. We expect the unlearned model to have a generalization performance similar to that of a model trained without the Boeing aircraft. It will have some random predicted logits since the Boeing aircraft class was not overfitted during training. An ideal value of ZRF score depends on the model, dataset and the forget set. Ideally, the optimal ZRF value is what a model trained without the forget set would have. But in practical scenarios we do not have access to the retrained model. So, a good proxy for the ideal ZRF value could be the ZRF value obtained on a test set. The test set by definition is a set about which the model has never learned anything specifically. It is equivalent to saying, a set that the model has unlearned perfectly. Experiments Datasets used. We evaluate our proposed method on image classification: CIFAR10 (Krizhevsky 2009), CI-FAR100 (Krizhevsky 2009), epileptic seizure recognition (Andrzejak et al. 2001), and activity recognition (Anguita et al. 2013) datasets. Models used. We use ResNet18, ResNet34, Mo-bileNetv2, Vision Transformer, and AllCNN models for learning and unlearning in image classification tasks. We use a 3-layer DNN model for unlearning in epileptic seizure recognition. We use an LSTM model for unlearning in activity recognition task. All the experiments were performed on NVIDIA Tesla V100 (32 GB) with Intel Xeon processors. The experiments are implemented in PyTorch 1.5.0. The KL temperature is set to 1 for all the experiments. Evaluation Measures. We use the following metrics for our analysis of the proposed unlearning method. 1) Accuracy on forget & retain set: The accuracy of the unlearned model on D f and D r sets should be similar to the retrained model. 2) Membership inference attack: A membership inference attack is performed to check if any information about the forget samples is still remaining in the model. The attack probabilities should be lower on the forget set in the unlearned model. 3) Activation distance: This is an average of the L2distance between the unlearned model and retrained model's predicted probabilities on the forget set. A lesser activation distance represents better unlearning. 4) JS-Divergence: JS-Divergence between the predictions of the unlearned and retrained model when coupled with activation distance gives a more complete picture on unlearning. Lesser the divergence, better the unlearning. 5) ZRF score: We introduce this metric to remove the dependence on the retrained model for evaluating the machine unlearning method. Baseline Models. We use the unlearning method from (Graves, Nagisetty, and Ganesh 2021) for a comparative analysis. This best fits our problem statement i.e., unlike most other methods, this method achieves unlearning in an already trained model without putting any constraints on the training procedure. We also use the retrained model for comparison. We perform two types of unlearning: (i) sample unlearning, and (ii) class unlearning. We present the experiments and analysis for each of them below. Forget Acc. Vs Information Exposure Trade-off Machine unlearning of a specific class or cohort often leads to a decrease in accuracy or performance on forget set. Although, it is an expected result when the forget set is orthogonal to the retain set i.e., there are no samples in the retain set similar to the ones in the forget set. But this may not be true when retain set contains data points similar to the forget set samples. The accuracy may drop slightly, may not drop at all, or even increase in some cases. An unlearning method should bring the forget set performance closer to the gold (retrained) model instead of simply reducing it. If the performance of the unlearned model deviates a lot from the gold model, it could lead to Streisand effect. This effect refers to unexpected behaviour of the model on forget samples which may leak information about that data. The leak could be in the form of being consistently & maximally incorrect about only the forget samples, signalling that a deliberate effort was made to forget a selected set of samples. The aim should be to avoid this in order to ensure that the information about the forget set has been properly erased. For example, as mentioned earlier in the aeroplanes example, when method unlearns Boeing aircraft, if it is maximally wrong whenever it sees any Boeing aircraft image and classifies it as sea, animals, mushroom, etc., it will be suspicious. Other unseen aeroplanes will not be classified incorrectly so consistently. That means it has not really erased the information of Boeing aircraft. That information still exists which the model uses to be deliberately incorrect about the forget set. Sample Unlearning CIFARSuper20. The CIFAR100 is made up of 20 super classes i.e., there are different variants for each of these 20 We merge all classes of the CIFAR100 into their super classes and convert it into a 20 class set named CIFAR-Super20. Each class in CIFARSuper20 have 5 sub-classes, which are actual classes of CIFAR100. We conduct experiments on CIFARSuper20 by forgetting one sub-class from each super class. This setup makes unlearning more difficult than a regular scenario as we need to unlearn a sample/class without damaging the information of another sample/class that looks quite similar to it (for example, forget baby from people super class consisting of baby, boy, girl, man, woman). We present unlearning results on ResNet18, ResNet34, and Vision Transformer. We use pretrained models to train/fine-tune for 5 epochs using Adam optimizer with a batch size of 256. The learning rate is 0.001 for final layer and 0.0001 for pretrained weight layers. A learning plateau with patience of 3 and reduce factor 0.5 is used. We conduct multiple runs (5 times) of our algorithm which didn't show any significant variation in performance (refer Table 1). Therefore, we report the results of single run for all the models in this paper. We unlearn various sub-classes from a super-class. We use 30% of retain data and a single epoch of unlearning for all models. The learning rate of 0.0001 is used for unlearning. Table 1 shows unlearning results on ResNet18 and Vision Transformer (Dosovitskiy et al. 2021). The evaluation is performed on all the metrics discussed earlier. It can be observed in Table 1 that performance of our method is very close to that of the retrained model. There is very low probability of membership inference attack on our unlearned model. The accuracy of our method on the forget and retain set when forgetting rocket images from vehicles is almost same as the retrained models. The membership inference attack probability on the model for samples from rocket class drops to 0.002 from 0.982 after unlearning. The JS-Divergence between predictions of the retrained model and our model is 0.04 in the forget set. This implies the output distribution of unlearned model is very close to the retrained model. The ZRF score of our model becomes 0.99 from 0.87 after unlearning, thus indicating effective forgetting. Furthermore, Table 4 shows the unlearning results in ResNet34 with different types of teachers. Epileptic Seizure Detection. The dataset consists of the status of seizure in medical patients. There are a total of 178 predictor variables and 5 classes. A 3-layer DNN is trained for classification. The model is trained for 50 epochs using Adam Optimizer with learning rate of 0.01 and plateau with patience 10 and reduce factor 0.1. We unlearn 50 and 100 randomly selected data points. The results are presented in Figure 2a and Figure 2b. We observe that the proposed method performance is close to the retrained model. The accuracy on the forget set indicates that we have indeed effectively unlearned the forget set as the forget accuracy is reduced from around 100% to a generalized performance. For example, in case of forgetting 100 samples, the accuracy on the forget set drops from 90% to 74% in our method which is close the 70% accuracy of the retrained model. Human Activity Recognition. This is a task of classify- ing the activity of a person using the readings collected from smartphone sensors that an individual is carrying with her. The observation were taken from 30 different persons. The dataset contains 6 different types of activities which can be classified using time-series data with sensors giving 9 readings at each time-step. An LSTM Model with 2 dense layers after each LSTM is trained to predict the activity. The model is trained for 50 epochs using Adam Optimizer with learning rate of 0.01 and plateau with patience 10 and reduce factor 0.1. Table 2 contains the results of forgetting person 1 and person 3. Detailed results and effects of various parameters on unlearning are present in the supplementary material. Comparison with Amnesiac learning (Graves, Nagisetty, and Ganesh 2021). We compare our result with Amnesiac learning which fine-tunes the model with random labels on forget samples. Table 2 shows the comparison between both the methods. We compare the activation distance and accuracy on the forget and retain set. A lower activation distance indicates closeness to the retrained model. This subsequently indicates better unlearning and an accuracy closer to the retrained model is desired on forget and retain set. The activation distance for our method is very low compared to Amnesiac method in most of the cases (refer Table 2). Amnesiac method causes too much damage in the forget set of epileptic seizure and human activity recognition dataset, indicating Streisand effect. The accuracy in epileptic seizure dataset (forget set of 50 samples) is 98% for the original model, 70% for retrained model, 74% for our method, and 30% for amnesiac method. The Amnesiac method damages the performance on forget set by a huge margin. It reduces the forget set accuracy to 30% which otherwise should be close to 70%. It should also be noted that activation distance from retrained model is 0.47 for our method and 0.77 for Amnesiac method. Amnesiac method is causing undesired effects and the generated model is very different from the retrained model. Our method, besides being more effective and robust, requires access to only a subset of retain data. We use only 30% of retain data to obtain the results in our method. Our method is ∼ 2× faster than Amnesiac method, more effective even when limited data is available for use. Class Unlearning We also demonstrate full-class (single and multiple classes) unlearning capability of our method. We show results on CIFAR10 and CIFAR100 with ResNet18, AllCNN, and MobileNetv2 models. Class-level unlearning results are compared with an existing method with configuration as in (Tarun et al. 2021). The model update in our method is performed for 1 epoch using 30% of the retain data. The learning rate at the time of unlearning is set to 0.001. Table 3 gives a performance comparison between the proposed and the existing methods. The accuracy on the retain set in CIFAR10 single-class forgetting is 71.06% for UNSIR, 78.32% for the retrained model, and 78.46% for our method. The results are quite similar in all three methods. The accuracy on forget set is zero in the retrained model and UN-SIR but our method retains some accuracy on the forget set. This is because the method learns from a randomly initialized teacher which does random predictions and predicts each class with 10% probability and forget model learns the Using a Simpler Model as an Incompetent Teacher Our method does not place any constraints on the architecture of the incompetent teacher. Preferably the architecture should be kept same as the student for proper transfer of information. But it can be replaced with smaller models without significantly affecting the results. As the teacher is initialized with random weights, such behaviour can be obtained by a significantly smaller model, or even hard coded algorithms to generate random predictions. A cheaper teacher can make the unlearning process faster without compromising in the quality of unlearning. We replace the incompetent teacher with (i) a small randomly initialized Neural Network, and (ii) a random prediction generator. The random prediction generator first assigns equal probability to all classes and then a adds Gaussian noise to the predictions. The performance with these teachers is shown in Table 4. With ResNet34 as teacher, the performance on D f (forget class: Rocket) is 1% while using the same model as teacher, 15% while using ResNet18, and 5% while using a random predictor as a teacher. Similarly, the performance on retain set while using ResNet34 as teacher is 85.8%, 85.86% while using ResNet18, and 86.04% while using a random predictor as a teacher. There is a negligible change in performance when we use simpler models as teachers. Thus, it can be used to reduce the computational costs without much loss in the performance. Using Partially Retrained Model as an Incompetent Teacher A partially trained (PT) model on a subset of retain data can be used as an incompetent teacher in the proposed framework. Similarly, smaller models trained on a small subset of the retain data can also serve as an incompetent teacher. We show the results of using PT models as incompetent teachers to induce forgetting. Similarly, we further investigate the effectiveness of PT teacher on CIFARSuper20 in Table 5 Efficiency Analysis We compare the run-time comparison of the retrained model, the existing methods, and the proposed methods. The random weights based setup is ∼ 70× faster than retraining and more than 2× faster than Amnesiac learning (Graves, Nagisetty, and Ganesh 2021). The method is faster when cheaper unlearning teachers are used. The proxy model based setup is about 20× faster than retraining method. The ideal trade-off between efficiency and performance can be obtained by using smaller models partially trained on retain data but they come with the expense of additional training. This (partial training) further comes with a trade-off between computational cost and closeness of the model to the retrained model. The right amount of partial training should be decided. We observed that the cheap randomly initialized models are more efficient and generally perform well in most cases. Conclusion We present a novel and general teacher-student framework for machine unlearning. A pair of competent and incompetent teachers is used to selectively transfer knowledge into the student network to obtain the unlearned model. Our work supports single & multiple classes forgetting, sub-class forgetting and random samples forgetting. The effectiveness is evaluated in various application domains and modality of networks. We also introduce a new evaluation metric ZRF that is free from the need of having a retrained model for reference. This metric would be useful in real world scenarios where retrained models are not available or very expensive to obtain. Several possible efficient teachers are also explored to reduce the computational complexity. Future work could focus at the intersection of efficiency and privacy guarantees which may be in the form of either developing better evaluation measures or developing new class of unlearning techniques. Additional Ablation Studies We study the effect of various hyper parameters in the unlearning method for the Human Activity Recognition dataset. We discuss the results achieved at different epochs of unlearning and at different learning rates below. Effect of Different Number of Epochs Without loss of generality, we unlearn the data related to user-1 from the activity dataset. Table 6 shows how varying the size of the data used for retraining (% of D r ) and the number of epochs for fine-tuning effects unlearning performance. The learning rate is fixed at 0.001. From Table 6 we observe that using more data samples from D r result in better randomization in the outputs for D f in the unlearned model. For example, when only a single epoch is run and 100% of D r is used, the ZRF score is 0.79. This goes down to 0.69 if only 50% of D r is used. The ZRF score further declines to 0.67 and 0.62 if only 30% and 10% of D r is used, respectively. Increasing the number of epochs has a similar effect to that of increasing the number of samples from D r . For example, with 10% of D r , the score ZRF is 0.62 after 1 epoch, 0.65 after 2 epochs and 0.75 after 5 epochs, respectively. The decision of selecting the number of epochs and % of D r for retraining would depend on the degree of desired efficiency and the amount of D r available for use. Table 7 shows the effect of using different learning rate during the brief retraining period. The number of epochs and the subset of D r are fixed at 2 and 30%, respectively. Increasing the learning rate increases the amount of randomization in the updated model. The ZRF score steadily goes down from 0.96 (learning rate 0.1) to 0.59 (learning rate 0.0001). Evidently, our method is highly flexible with respect to the amount of randomization desired by the model owner or the customer. Increasing the number of epochs, portion of the retain data or the learning rate increases the ZRF score. As explained in Section 4 in the main paper, we want the ZRF score to be as close to the retrained model as possible. The ZRF score closest to the retrained model is obtained with 30% retain data, 2 epochs and a learning rate of 0.001. The JS-Divergence in this case between the retrained model and the model obtained by our method is 0.03. This indicates that the output distributions of the retrained model and the model obtained using our method are very similar. Figure 3: Sequential Unlearning Requests: Request 1 (R1), Request 2 (R2), Request 3 (R3) are the subsequent unlearning requests for subclasses Rocket, Mushroom and Lamp. We observe that the performance achieved by our method is very close to that of the model trained from scratch (retrained model) on both retain and forget set. Effect of Different Learning Rates Sequential Unlearning We conduct experiments to simulate a real life scenario where unlearning may be requested repeatedly. Figure 3 showcases the performance of our method in the following scenario: we forget sub-classes Rocket, Mushrooms, and Lamp one after the other. We observe that our method performs really well here. Even after multiple requests of unlearning, there is hardly any degradation in performance which indicates the robustness of our method. Partially Retrained Model as a Competent Teacher: Additional Experiments A partially trained (PT) model on a subset of retain data can be used as an incompetent teacher in the proposed framework. Similarly, smaller models trained on a small subset of the retain data can also serve as the incompetent teacher. We show the results of using PT models as incompetent teachers to induce forgetting. Table 8 shows results of class unlearning using PT teacher. The teacher is trained for 2 epochs on 50% of the retain data. The accuracy on forget set with ResNet18 is 0% for retrained model, UNSIR, and our method with PT teacher. Whereas, it is 4.22% in our method with randomly initialized (RI) teacher. The accuracy on retain set for ResNet18 is 78.32% for retrained model, 78.6% for our method with PT teacher, 78.46% for our method with RI teacher, and 71.06% for UNSIR. The performance is closer to the retrained model when PT teacher is used instead of RI teacher. Efficiency Analysis: More Details Figure 4 shows the total run-time comparison between the retrained model, the existing and the proposed methods. Our methods takes substantially less amount of time for unlearning in comparison to both the retrained method and the Amnesiac learning method (Graves, Nagisetty, and Ganesh 2021). Our method is only behind UNSIR (Tarun et al. 2021) in run time efficiency. However, the UNSIR only supports class-level unlearning whereas, the proposed method supports both sample-level and classlevel unlearning. Figure 1 : 1The proposed competent and incompetent teachers based framework for unlearningZero Retrain Forgetting MetricThe effectiveness of an unlearning method is evaluated employing several metrics in the literature. Some frequently used metrics are 'accuracy on forget set and retain set' (Golatkar, Achille, and Soatto 2020a;Tarun et al. 2021;Golatkar et al. 2021;Chundawat et al. 2023), relearn time(Tarun et al. 2021), membership inference attacks(Golatkar et al. 2021;Graves, Nagisetty, and Ganesh 2021), activation distance(Golatkar, Achille, and Soatto 2020a;Golatkar et al. 2021), Anamnesis Index Figure 2 : 2Unlearning random samples (50 and 100 samples, respectively) from Epileptic Seizure Data. Table 1: Unlearning on CIFARSuper20. We show the results for forgetting a sub-class from a super class. The Original Model is trained on complete dataset. The Retrained Model is trained on retain dataset. We use a randomly initialized model as incompetent teacher and the original model as competent teacher. The ZRF score should increase on forget set after unlearning. The JS-Div: Jensen-Shannon Divergence, MR: Mushrooms, Acc.: Accuracy, Orig.: Original Model, Veh2: Vehicles2, Veg: Vegetables, ED: Electrical Devices, NS: Natural Scenes classes.Super Sub Accuracy (D f ↓, D r ↑) ZRF JS- Mem. Attack Prob Class Class Acc. Orig. Retrain Our Orig. Retrain Our Div Orig. Retrain Our ResNet18 Veh2 Rocket D r 85.78 85.79 85.05±0.61 0.87 0.93 0.99 0.04 0.98 0.52 0.00 D f 82 3 2±0.40 Veg MR D r 85.82 85.38 84.79±0.51 0.88 0.93 0.99 0.04 0.99 0.41 0.00 D f 78 4 1±0.36 People Baby D r 85.67 85.73 85.18±0.54 0.84 0.87 0.98 0.05 1.0 0.84 0.58 D f 93 82 77±0.34 ED Lamp D r 85.83 86.28 84.74±0.32 0.88 0.94 0.98 0.03 0.98 0.43 0.01 D f 77 14 5±0.29 NS Sea D r 85.63 85.46 84.58±0.22 0.84 0.87 0.98 0.07 0.99 0.88 0.42 D f 97 83 84±0.68 ViT Veh2 Rocket D r 94.89 95.35 94.84±0.71 0.91 0.96 0.99 0.03 0.99 0.47 0.01 D f 98 9 17±0.2 Veg MR D r 94.94 94.8 94.59±0.65 0.93 0.98 0.99 0.02 0.99 0.25 0.01 D f 93 4 17±0.54 People Baby D r 94.91 95.26 94.45±0.69 0.90 0.92 0.99 0.06 1.0 0.91 0.14 D f 96 92 77±0.33 ED Lamp D r 94.93 94.86 94.990.85 0.91 0.96 0.99 0.03 1.0 0.57 0.02 D f 94 13 21±0.29 NS Sea D r 94.91 94.93 94.31±0.46 0.90 0.92 0.99 0.06 1.0 0.97 0.12 D f 96 85 79±.54 Table 2 : 2Comparison of our method with the Amnesiac learning (Graves, Nagisetty, and Ganesh 2021) RN18: ResNet18, C20: CIFAR20, HAR: Human Action Recognition # Y f Acc. Orig. Retrain UNSIR Our RN18+ 1 D r ↑ 77.86 78.32 71.06 78.46 C10 D f ↓ 81.01 0 0 4.22 Dataset 2 D r ↑ 78.00 79.15 73.61 79.22 D f ↓ 78.65 0 0 9.94 RN18+ 1 D r ↑ 78.68 78.37 75.36 77.00 Pre+C100 D f ↓ 83.00 0 0 0 Dataset 20 D r ↑ 77.84 79.73 75.38 77.78 D f ↓ 82.84 0 0 3.90 AllCNN+ 1 D r ↑ 82.64 85.90 73.90 81.74 C10 D f ↓ 91.02 0 0 9.16 Dataset 2 D r ↑ 84.27 85.21 80.76 77.68 D f ↓ 79.74 0 0 5.64 MNv2+ 1 D r ↑ 77.43 78 75.76 78.22 Pre+C100 D f ↓ 90 0 0 0 Dataset 20 D r ↑ 76.47 77 76.27 76.65 D f ↓ 81.70 0 0 13.65 Table 3 : 3Class-level unlearning on CIFAR10 and CI- FAR100. The results are compared with UNSIR (Tarun et al. 2021). C10: CIFAR10, C100: CIFAR100, RN18: ResNet18, MNv2: MobileNetv2, Pre: Pretrained Table 4 : 4Forgetting sub-class from a super class on CIFAR- Super20+ResNet34 using different types of cheaper incom- petent teacher (T d ) Super-Acc. Orig. Ret. (RI) (P) JS-Div Sub (RI) (P) Veh2-D r ↑ 85.8 85.8 85.1 85.1 0.04 0.02 Rocket D f ↓ 82 3 2 2 Veg- D r ↑ 85.8 85.4 84.8 85.4 0.04 0.08 MR D f ↓ 78 4 1 3 Table 5 : 5Forgetting sub-class from a super class in CI- FARSuper20. Our(RI): Using randomly initialized teacher, Our(P): Using a partially trained model (1 epoch on 50% of retain data) as an incompetent teacher. Ret.: Retrain same. This in turn leads to better protection against the risk of privacy exposure. Table 6 : 6Forgetting the data (activity) of a single person in Human Activity Recognition Task. The results are presented for different number of epochs of unlearning with different size of subsets of retain data. The results are presented for an LSTM model. The JS-Divergence is computed between the retrained and our methodNo. of % Accuracy Original Retrained Our ZRF JS-Div epochs of D r Model Model Method Original Retrained Ours 1 100% D r ↑ 90.46 84.01 88.43 0.57 0.72 0.79 0.03 D f ↓ 100 99.13 94.52 50% D r ↑ 90.46 84.01 88.7 0.58 0.72 0.69 0.02 D f ↓ 100 99.13 97.69 30% D r ↑ 90.46 84.01 88.77 0.58 0.72 0.67 0.02 D f ↓ 100 99.13 98.27 10% D r ↑ 90.46 84.01 90.19 0.58 0.72 0.62 0.01 D f ↓ 100 99.13 100 2 100% D r ↑ 90.46 84.01 88.22 0.57 0.71 0.79 0.02 D f ↓ 100 99.13 94.81 50% D r ↑ 90.46 84.01 86.7 0.58 0.72 0.75 0.02 D f ↓ 100 99.13 96.83 30% D r ↑ 90.46 84.01 87.75 0.58 0.72 0.72 0.03 D f ↓ 100 99.13 93.95 10% D r ↑ 90.46 84.01 89.07 0.57 0.71 0.65 0.02 D f ↓ 100 99.13 99.71 5 100% D r ↑ 90.46 84.01 88.67 0.57 0.71 0.87 0.04 D f ↓ 100 99.13 89.91 50% D r ↑ 90.46 84.01 86.90 0.56 0.71 0.84 0.04 D f ↓ 100 99.13 90.49 30% D r ↑ 90.46 84.01 84.29 0.57 0.71 0.83 0.05 D f ↓ 100 99.13 91.07 10% D r ↑ 90.46 84.01 85.71 0.57 0.71 0.75 0.04 D f ↓ 100 99.13 92.22 Table 7 : 7Learning rate ablation on Human Activity Recognition task. The unlearning is done for 2 epochs with 30% of the retain data.Learning Accuracy Original Retrained Our ZRF JS-Div Rate Model Model Method Original Retrained Ours 0.1 D r ↑ 90.46 84.01 72 0.57 0.71 0.96 0.07 D f ↓ 100 99.13 82.13 0.01 D r ↑ 90.46 84.01 86.19 0.57 0.72 0.93 0.07 D f ↓ 100 99.13 88.76 0.001 D r ↑ 90.46 84.01 87.75 0.58 0.72 0.72 0.03 D f ↓ 100 99.13 93.95 0.0001 D r ↑ 90.46 84.01 90.4 0.57 0.71 0.59 0.01 D f ↓ 100 99.13 100 Table 8 : 8Class-level unlearning for a single class in CIFAR10. Our(R): Using randomly initialized teacher, Our(P): Using partially trained model trained for 2 epochs on 50% of retain data as incompetent teacher.Figure 4: Efficiency comparison of various unlearning methods. We conduct the experiments for class unlearning in ResNet18 over CIFAR10.Model Accuracy Original Retrained UNSIR Ours(R) Ours(P) Model Model (Tarun et al. 2021) ResNet18 D r ↑ 77.86 78.32 71.06 78.46 78.6 D f ↓ 81.01 0 0 4.22 0 AllCNN D r ↑ 82.64 85.90 73.90 81.74 83.76 D f ↓ 91.02 0 0 9.16 0.5 AcknowledgementsThis research is supported by the National Research Foundation, Singapore under its Strategic Capability Research Centres Funding Initiative. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore. Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state. R G Andrzejak, K Lehnertz, F Mormann, C Rieke, P David, C E Elger, Physical Review E. 64661907Andrzejak, R. G.; Lehnertz, K.; Mormann, F.; Rieke, C.; David, P.; and Elger, C. E. 2001. 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[ "https://github.com/vikram2000b/" ]
[ "Caveats in Eliciting Mobile App Requirements", "Caveats in Eliciting Mobile App Requirements" ]
[ "Nitish Patkar [email protected] \nUniversity of Bern\nUniversity of Bern\nUniversity of Bern\nUniversity of Zurich\n\n", "Mohammad Ghafari [email protected] \nUniversity of Bern\nUniversity of Bern\nUniversity of Bern\nUniversity of Zurich\n\n", "Oscar Nierstrasz [email protected] \nUniversity of Bern\nUniversity of Bern\nUniversity of Bern\nUniversity of Zurich\n\n", "Sofija Hotomski \nUniversity of Bern\nUniversity of Bern\nUniversity of Bern\nUniversity of Zurich\n\n" ]
[ "University of Bern\nUniversity of Bern\nUniversity of Bern\nUniversity of Zurich\n", "University of Bern\nUniversity of Bern\nUniversity of Bern\nUniversity of Zurich\n", "University of Bern\nUniversity of Bern\nUniversity of Bern\nUniversity of Zurich\n", "University of Bern\nUniversity of Bern\nUniversity of Bern\nUniversity of Zurich\n" ]
[]
Factors such as app stores or platform choices heavily affect functional and non-functional mobile app requirements. We surveyed 45 companies and interviewed ten experts to explore how factors that impact mobile app requirements are understood by requirements engineers in the mobile app industry.We observed the lack of knowledge in several areas. For instance, we observed that all practitioners were aware of data privacy concerns, however, they did not know that certain third-party libraries, usage aggregators, or advertising libraries also occasionally leak sensitive user data. Similarly, certain functional requirements may not be implementable in the absence of a third-party library that is either banned from an app store for policy violations or lacks features, for instance, missing desired features in ARKit library for iOS made practitioners turn to Android.We conclude that requirements engineers should have adequate technical experience with mobile app development as well as sufficient knowledge in areas such as privacy, security and law, in order to make informed decisions during requirements elicitation.CCS CONCEPTS• Software and its engineering → Requirements analysis.
10.1145/3383219.3383238
[ "https://arxiv.org/pdf/2002.08458v1.pdf" ]
211,205,167
2002.08458
3e971064036f6468997a42f6cbff3d5e99b62f8f
Caveats in Eliciting Mobile App Requirements 19 Feb 2020 Nitish Patkar [email protected] University of Bern University of Bern University of Bern University of Zurich Mohammad Ghafari [email protected] University of Bern University of Bern University of Bern University of Zurich Oscar Nierstrasz [email protected] University of Bern University of Bern University of Bern University of Zurich Sofija Hotomski University of Bern University of Bern University of Bern University of Zurich Caveats in Eliciting Mobile App Requirements 19 Feb 2020ACM Reference Format: Nitish Patkar, Mohammad Ghafari, Oscar Nierstrasz, and Sofija Hotom-ski. 2020. Caveats in Eliciting Mobile App Requirements. In Proceedings of Evaluation and Assessment in Software Engineering (EASE 2020). ACM, New York, NY, USA, 11 pages.Mobile app developmentrequirements engineeringrequirements elicitation Factors such as app stores or platform choices heavily affect functional and non-functional mobile app requirements. We surveyed 45 companies and interviewed ten experts to explore how factors that impact mobile app requirements are understood by requirements engineers in the mobile app industry.We observed the lack of knowledge in several areas. For instance, we observed that all practitioners were aware of data privacy concerns, however, they did not know that certain third-party libraries, usage aggregators, or advertising libraries also occasionally leak sensitive user data. Similarly, certain functional requirements may not be implementable in the absence of a third-party library that is either banned from an app store for policy violations or lacks features, for instance, missing desired features in ARKit library for iOS made practitioners turn to Android.We conclude that requirements engineers should have adequate technical experience with mobile app development as well as sufficient knowledge in areas such as privacy, security and law, in order to make informed decisions during requirements elicitation.CCS CONCEPTS• Software and its engineering → Requirements analysis. INTRODUCTION The number of available mobile apps is constantly increasing ever since app hosting platforms came into existence ten years ago. For example, the Apple App Store now hosts about 2.2 million mobile apps. About 28% of the installed apps are however uninstalled within 30 days, showing that such apps failed to satisfy users. EASE 2020, April 15-17, 2020, Trondheim, Norway 2020. Although there are several similarities between mobile and desktop apps, such as the availability of app stores and multiple available platforms, research has also shown several differences between them [33,49]. Additionally, the availability and possibility of interpreting data obtained from various onboard sensors on mobile phones allow apps to know much more about their users than the users might willingly wish to share [7,51,52], thus jeopardizing the privacy of millions of users. This study does not assume that requirements elicitation process is different for mobile apps, instead we simply reflect on the factors that are more prevalent in the mobile domain, and investigate how they are perceived in industry. We conducted an industry survey in which 45 mobile app development companies from Switzerland, Germany, and the Czech Republic participated. In addition, we interviewed ten industry experts from Europe and India working in the mobile app industry and responsible for requirements elicitation. Consequently, we aim to answer the research question: "Which factors are essential for mobile app requirements, and how well are they understood by practitioners?" Most participants who claimed to be responsible for requirements elicitation in fact shared roles such as CEO, business analyst, or marketing head, and lacked knowledge in several areas. For instance, most of them had no direct technical experience with mobile app development. Consequently, they fail to understand how, for instance, non-functional requirements affected by usability and performance, are also compromised by third-party libraries including app analytics that have been reported to leak sensitive user data. In several cases, we observed that the practitioners lacked additional knowledge of app security, privacy and law. Consequently, they relied heavily on developers' knowledge of security and on external lawyers for privacy matters, failing to communicate the implications of certain requirements choices to their clients. We believe that discussing the implications of the observed factors i.e., platform choices, app store policies, third-party libraries, NFRs, and app type for the mobile app requirements will help RE practitioners to make informed decisions. The remainder of the paper is structured as follows: In section 2, we explain the protocol we followed for the survey and the interview. We then present and discuss our discoveries in section 3. Related work is discussed in section 4, and finally section 6 concludes our findings. RESEARCH METHOD To answer the research question we conducted a qualitative exploratory study and followed the procedure described by Patton et al. [40]. We first performed a survey that ultimately led to the interviews to obtain more detailed information from industry representatives. The survey and interview instruments, and supporting graphs are available online. 1 Survey The online survey was performed from October 2018 to late January 2019. The invitations for participation were sent manually to 276 mobile app companies; it was also advertised on LinkedIn and Twitter. We specifically looked for people who claimed to be responsible for requirements elicitation in mobile app companies. Preparation. The survey questions were selected based on the themes evident from a literature review of 60 publications where the authors identified 24 elicitation techniques that target mobile apps, out of which only half were empirically evaluated [39]. The questions in the survey have been validated and improved over several iterations to accurately answer the proposed research question. It consists of five sections: The first section characterizes the participants, while the remaining four sections focus on various aspects of mobile app development, such as the application domain, elicitation parameters, communication, and the developers. Participants. In total 45 people with various roles, such as requirements engineer, developer, or even CEO, participated in the survey. They all claimed to be responsible for the requirements elicitation in their respective companies. About 60% of them (i.e., 27) had more than five years of experience in this area. About ten participants shared multiple roles within their company. Companies, products, and services. We received responses from both product-based and service-oriented companies. Their apps mainly belonged to but were not limited to categories such as travel, transport, logistics/marketing, business, and productivity. Interview We designed an interview with 52 questions to thoroughly investigate the commonly emerging themes from the survey responses. Preparation. The questions in the interview have been validated and improved in two major iterations: First, they were reviewed and discussed by the collaborators of this paper, and second, they were validated in a pilot interview with an external expert in mobile app development. The pilot interview is not included in the main study. The interview consists of four sections: The first two sections characterize the interviewee and the companies, the third section explores the state of the art requirements elicitation in the mobile app industry, and finally, the fourth section explores the factors that might affect requirements gathering for mobile apps. Participants. Survey respondents who were interested in participating in an interview were invited by email. We also invited other practitioners through our contacts, who we know were responsible for requirements elicitation and additionally advertised the interview on LinkedIn and Twitter. Subsequently we interviewed ten practitioners who agreed to participate in an in-depth interview. We observed that in most small companies in our study, the role 1 https://figshare.com/s/87a07b7c47f37952eef8 has very little to do with actual expertise in RE. People with different roles ranging from business analysts to CEO were responsible for requirements elicitation. This explains why the implications of requirements elicitation are not well understood in practice. An overview of the interviewees can be found in Table 1. For instance, participant P1 works in company C1 as a CEO. This Service-based company has developed 15 apps, and it usually spends 15-20% of its budget on RE. The company has 35 employees who are based in Switzerland and target the Swiss market. Most of the interviewees had a computer engineering background and they all could speak English fluently. They had varying degrees of experience with requirements elicitation ranging from 2 to 20 years, and most of them learned the corresponding elicitation techniques through experience at the workplace. Importantly, a few of them who have an academic background mentioned that they no longer have free access to academic publications, which deprives them of the latest advances in academia. Companies, products, and services. An overview of the interviewees' companies can also be found in Table 1 as well. The companies were based mainly in Switzerland. The companies were of different sizes in terms of number of employees ranging from seven up to 150 employees. Similarly, the numbers of mobile apps they have delivered varied considerably, ranging from one up to twenty, which shows their varied experience in this field. Except for one company, all were service-oriented, which indicates a risk associated with developing an innovative app in the absence of financial funding and through market research, especially for small-sized companies. Data collection and analysis. All ten interviews were carried out between February and March 2019 and were conducted by the first author. Although P10 preferred to fill in an interview template due to time constraints, other interviews were conducted face-to-face. The duration of the interviews was between 60 and 120 minutes. The voices of the interviewees in all face-to-face interviews were recorded, as was the video signal in all video call interviews, with permission from the interviewees. In order to analyze the data, all interviews were digitally transcribed and coded. We created an initial list of codes based on the survey results and complemented it with codes that emerged while reviewing interview transcripts. The codes were used to group related answers and to compare them to each other. DISCOVERIES This section discusses the factors that are known to the research community, but that are not reviewed in prior work from the requirements perspective. In particular, we discuss how (i) platform and third-party libraries, (ii) app stores and their policies, (iii) RE practitioner's experience and expertise regarding NFRs, (iv) learning from app usage preferences, and (v) app type are affecting app requirements, and how practitioners perceive these factors for the success of the app. These factors emerged essentially from the survey responses, and were later discussed in depth during the interviews. For each factor, we report our own empirical observations from the survey and the interviews, provide a discussion referring to the existing literature wherever relevant, finally followed Platform and third-party libraries Platform selection depends on several obvious factors such as cost, budget, and market research. Besides these factors, a few other non-obvious factors such as the role of app stores, or third-party libraries are equally influential, and can directly impact functional requirements. These factors are inter-related and inter-dependant, and their relevance to app requirements should be made explicit. 3.1.1 Empirical observations. Survey. According to the survey participants (e.g., total nine), only few customers have a clear idea about their choice of platform, and in spite of the limited funds, without any exception they would like to be present "everywhere. " Interview. P7 mentioned that apart from platform features, customer bases in different geographical markets also influence the platform selection decision. According to him, Indian customers prefer Android, whereas European and U.S. customers prefer iOS as their first platform for app releases. One of the experts (i.e., P8) further said that they usually choose the platform that best supports the customer's requirements, which means that the same app designed for different platforms might use different features to fully utilize the underlying platform capabilities. For instance, company C3 chose the Android platform for one of their augmented reality projects as the bundled library ARCore provided features unavailable in iOS' ARKit. All the interviewees further agreed that when adopting the Android platform, additional properties such as the minimum OS version supported or numerous screen sizes must also be considered to optimize an app. Interestingly, various challenges posed by platform ecosystems have led interviewees to consider hybrid apps 2 over native Android and iOS apps. According to P1, in most cases customers choose hybrid apps over native ones mainly for economic reasons because they are cheaper to build as they do not require dedicated iOS or Android developers. Interviewees, however, had different opinions about hybrid apps in the mobile app market. P5, for instance said that hybrid platforms add "yet another dependency" in addition to native layer and hence, he fears that developers tend to lose control over security. 3.1.2 Discussion. One should select a mobile platform that brings justice to the end-user requirements. We identified three key themes from our observations: • Platform selection depends on several factors such as cost, budget, market research and platform ecosystem. • End users, customers and developers can have different concerns regarding platform selection. • Functional requirements are affected while selecting a particular platform, and the opposite is true as well, sometimes resulting in favouring hybrid apps. From the customer's point of view, platform selection is based on market research and the number of potential users, e.g., in Switzerland iOS is prevalent, while in India it is Android. Customers also have stringent budget concerns. However, they may not be experts when a decision must be made. Kumar et al. have shown how several mobile app projects suffer from low funding, eventually resulting in low-quality apps [28]. They mention that it is primarily because mobile app development is perceived to be cheap by customers, which in fact, is incorrect. On the other hand, app developers and RE practitioners seem to be more concerned about device fragmentation as it leads to increased cost and efforts. The aforementioned issues have been discussed in previous publications, notably stressing the direct impact on the initial development and post-release maintenance cost [6,47]. The effect of fragmentation on testing and eventually on platform selection is also evident from literature. Miranda et al. report that the variety of compatible devices and OS versions for Android makes Android app testing much easier than iOS apps [34]. Furthermore, they also point out that Java as a programming language is more widespread than Objective-C, both at universities and private companies, making it easier to find Android developers. To the best of our knowledge, there is no evidence of potential solutions in the literature to the conflicting concerns regarding platform selection. Beyond these commonly known factors, the platform ecosystem, including app stores and third-party libraries, affect app requirements and contribute to the platform selection decision. Certain functional requirements may not be implementable on a specific platform due to, for instance, a missing third-party library. Both app stores and third-party libraries impose additional challenges to requirements elicitation. Third-party libraries are heavily used in developing mobile apps, for instance, Minelli et al. from their analysis of a corpus of Android apps found that external calls represented more than 75% of the total number of method invocations [33]. The recent work by Derr et al. discusses an imminent risk of misuse by third-party libraries in Android as they inherit the access rights of their host apps [13]. Similarly, Grace et al. have reported how advertising libraries used by developers were collecting user's private data; a few even fetch and run code from the internet [19]. Only during development when developers discover that a certain third-party library that is vital in fulfilling a functional requirement is unavailable, or is banned from the app store for policy violations, requirements have to be renegotiated with their clients. We discuss further implications of app stores and third-party libraries on requirements in subsection 3.2 and subsection 3.4 respectively. Fragmentation and app store related issues have made practitioners consider hybrid frameworks instead of a specific native platform although with some skepticism. The role and potential of hybrid apps has been discussed in great detail in the literature, and we also explore this later in subsection 3.5. In conclusion, we found that all practitioners were aware about platform fragmentation and the involved cost, however, they failed to provide deeper insights into issues that might arise due to choosing compromised third-party libraries, such as data leaks. Although, all of them were aware of hybrid frameworks, most of them were skeptical about switching from native to hybrid apps even in cases where it would make sense. RE practitioners have to be aware about technical alternatives and their feasibility to be able to proactively discuss the shortcomings of the requirements up front. Such discussions and decisions can save app companies a lot of time and efforts, and money for customers. App stores and their policies App stores are important for both end users and developers as they simplify the app discovery and distribution process. Although the importance of the app stores for app developers has been discussed extensively, their impact on requirements is ill understood. 3.2.1 Empirical observations. Survey. There were no dedicated questions on the role of app stores in our survey. The theme emerged from the survey responses and it was subsequently added to the interview instrument for further discussion with practitioners. Interview. App stores and their time-consuming review processes have affected the company's workflow in the past, said P5, so they had to be careful while accepting customer requirements to avoid any potential conflicts. Especially hot fixes that must be released immediately present a severe problem, as all the interviewees agreed. The complex app release processes introduce additional work for the development teams and the requirements engineers. They must properly plan their release cycles and consider a tradeoff between fast deployment and low costs, or as C3 puts it: "more releases mean more money as they demand more time. " Sudden guideline and policy changes of app stores also become problematic. Interestingly, P4 said that their contracts clearly state that app rejection from app stores shall not be their responsibility. Concerning technical aspects, companies such as C3 have faced app store restrictions like the maximum allowed number of methods per app, or the maximum allowed app size. Hence, they nowadays reconsider every requirement about the inclusion of external libraries and media content. Nevertheless, P1 and P4 mentioned that the current state is improving with newer policies. 3.2.2 Discussion. Although app stores exist for desktop apps, they play a far more important role in the case of mobile apps due to the huge volume of downloaded apps. When Nokia still held the major app market share, Holzer et al. had predicted that mobile app marketplaces from Google and Apple were going to change the app development trends, and consequently, they indicated that developers must familiarize themselves with platform strategies [24]. Ten years since then, we see an explosion of mobile apps, and the app market is by far dominated by only two major platforms i.e., Android and iOS. We drew three conclusions from our observations: • Customers have no concerns regarding app stores, but they are important for both end users and developers as they simplify the app distribution and discovery process • Different app stores have different business models and policies • Policy changes by app stores are affecting app development companies by causing unintentional rework or in worse cases app denial. App stores are usually not a concern for customers, but they play an important role for end users and developers for different reasons. For end users, it is a one-stop marketplace to search for their desired apps, enabling easy download and installation, and additionally providing them with useful data such as user ratings and comments to make informed decisions. User reviews help developers as well to correct their apps by identifying new requirements. User review analysis has gained much traction in academia, especially in the RE community as later discussed in subsection 3.4. For developers, app stores matter in several ways: (i) they allow developers to distribute their app easily and increase their app's discoverability, which is also evident from a study of questions posted on StackOverflow [42]; (ii) different app stores charge differently; (iii) different app stores have different policies regarding app design, usage of third-party libraries etc.; and (iv) policies change, so developers have to keep themselves informed and align their workflows accordingly. Some of these concerns are reflected in the literature. The difference in business models is highlighted by Holzar et al. as they point out that besides distribution, app stores take care of billing and advertising for a certain commission (usually about 30%), which helps developers to increase the visibility of their apps on the platform [24]. However, the cost of publishing and distributing apps through app stores also differs significantly for Android and iOS platforms. It costs developers a one-time payment of about 25$ for Google Play, whereas the Apple App Store charges annually, which can range anywhere from 99$ to 299$ [34]. To the best of our knowledge very few studies discuss the role of app store requirements and policies on the app requirements and workflow. Notably, in their study, Joorabchi et al. highlight the need for testing APIs by app stores so that app developers can check their code for any guideline violations [27]. Similarly, Holzar et al. studied the app distribution process and subsequently outlined the implications of app store centralization. In particular, they claim that such centralization limits the freedom of developers. For instance, Apple's App Store decides which apps will be promoted over others [24]. In a recent exploratory study Al Subaihin et al. conducted interviews with mobile development team managers followed by a questionnaire to gather qualitative data on app engineering practices. They report that 54% of their survey respondents claimed to adopt a release strategy that is influenced by the app store's regulations i.e., by review and approval period [1]. Shortly after Google introduced revised policy changes to maintain the quality of the published apps in 2015, several apps have been removed from Google Play. The effects of policy changes are reflected in literature as well. Wang et al. studied 790K removed apps from Google Play to explore and understand the factors responsible for policy violations [48]. For instance, they discovered that the apps targeting children should never violate the COPPA policy. 3 Interestingly, most of the removed apps were published by spamming developers. Among the important reasons they identified for app removal were apps being classified as either malicious, privacy-risk, spamming, or fake by Google. The exact parameters that Google uses to flag an app as malicious are unclear. The parameters for flagging spamming apps are more explicit, such as those using excessive or inappropriate metadata, especially with misleading references to other apps or products. In conclusion, we observed that all practitioners were aware of the impact app store policies have on their development workflow. However, they never discussed the problematic app store policies with their clients during requirements negotiation. Hence, it has become necessary not just for app developers, but also for RE practitioners, to fully understand what different app stores offer them, and align their development, publishing, and marketing strategies with app store policies. Customers usually aren't aware of involved complications. Hence, RE practitioners need to be aware of the latest policies (on top of ecosystem offerings such as third-party libraries) from different app stores to be able to discuss those with customers. Certain functional requirements may not be implementable because the third-party library is banned from a specific App Store for policy violations. Such issues must be raised with the customer up front, otherwise, they may lead to considerable rework for developers, and in the worst case renegotiation. Non-functional requirements Non-functional requirements are important in general, but for mobile apps they are even more vital as they directly influence enduser acceptance. The General Data Protection Regulation (GDPR) 4 has fostered awareness regarding data privacy, however, eliciting other non-functional requirements is still ill-supported and can only benefit from practitioner expertise. 3.3.1 Empirical observations. Survey. There were no dedicated questions on the role of app stores in our survey. This theme emerged from the survey responses and it was subsequently added to the interview for further discussion with practitioners. Interview. All the interviewees mentioned that the GDPR improved the privacy awareness of customers who are now becoming sensitive to data privacy requirements. Most app developers maintain a GDPR-related checklist that they consult during app development. Other non-functional requirements like security, however, are only considered for basic measures such as maintaining secure data flows during communication. Except for C2 and C6, which are rather big companies, all companies outsource the legal work to the experts, e.g., preparation of contracts and data policy assistance. Sometimes inter-personal skills alone are not enough to elicit certain requirements, particularly non-functional ones. Eliciting such requirements requires experience. For instance, experience is needed to elicit requirements for unpredictable situations such as handling network drops in tunnels to make app usage a pleasant experience. Reflecting on this, P1 told us of an experience: An enterprise app they built for a company suddenly stopped working for a few employees as it could no longer connect to the server, since those employees were connected through a hotel's wireless network that was protected by a captive gate. 5 He said such situations are extremely hard to foresee and can only be tackled through experience. Similarly, P9 mentioned that for mobile apps, users need to manually install the updates, a process that the developers do not control. This can cause the back end to get out of sync if not handled properly. 3.3.2 Discussion. Customers are not good at dictating NFRs, RE practitioners must formulate those themselves. We identified three recurring themes from our observations: • eliciting NFRs requires experience and expertise; • in the case of mobile apps, NFRs such as data privacy can be unintentionally compromised by selecting a certain thirdparty library; and • additionally, device capabilities are an important factor for app's performance and acceptance. RE practitioners require multifaceted expertise to elicit NFRs: (i) Awareness about security and data privacy risks. People we interviewed had no idea about security whatsoever; they relied heavily on the developer's knowledge, and developers in turn relied on platform documentation. Recent research has shown that security issues are prevalent in mobile apps, jeopardizing the privacy and security of millions of users worldwide [17,18]. Interestingly, they discovered that old IDE versions had limited support for identifying security violations resulting in countless security issues to be missed by the developers. Similarly, Jain et al. discuss the complications of using personal mobile phones in a corporate environment; specifically, as it poses a challenge to enforce corporate policies causing an intentional or unintentional corporate data leak. They also briefly discuss several security risks for mobile apps such as using custom cryptographic algorithms instead of standard algorithms that can compromise data confidentiality, or failing to disable insecure OS features in mobile apps can result in sensitive data ending up in the web caches or global OS logs [26]. Privacy has become a serious concern for many. We found a lack of awareness among practitioners about potential factors that can compromise user privacy. For example, the use of third-party libraries and any unintentional use of trackers can compromise the security and data privacy of users. Third-party trackers are commonly used by app developers to gather user information, which helps them eventually to build a detailed user profile. Such a user profile is helpful to draw inferences about shopping habits, socioeconomic class, or political opinions etc. of the user. Large data aggregators who operate as data brokers (accumulating and selling user information) such as Acxiom and BlueKai collect data from a variety of sources, in most cases implementing user tracking technologies such as cookies on the host device with the consent of app developers [35]. End users are often unaware of such data leaks. (ii) Awareness about legal complications. Companies will need a much deeper understanding of problems arising due to diverse legal regulations across the world when the apps must be released globally. Not only that, when apps are using third-party libraries or usage trackers that fall outside their jurisdiction, they need to understand the consequences. Although outsourcing legal work might be affordable for big companies, how a large number of individual app developers and small companies will cope with this is an open question. Third-party trackers mentioned earlier have been revealed to be a highly transnational problem as many of them are operating outside the jurisdiction of the EU. The study by Binns et al. outlines some significant legal compliance challenges such as cross-jurisdictional data flow, profiling, and rights and obligations regarding children, arising from the use of trackers [3]. (iii) Awareness about mobile device capabilities. Device fragmentation affects performance and perceived app quality. End users who eventually accept or reject mobile apps place high value on the quality of the apps. App quality mainly depends on the device capabilities and the app's usability. Different devices mean an app might behave differently for different users; certain features may not be available for those with older OS versions or the apps might run slower. Noei et al. in their exploratory study found that the perception of app quality strongly relates to powerful CPUs and other device attributes, and it is not limited to app attributes such as code size [37]. Similarly, it is also hard to quantify the implications of security measures on user experience. For example, how much extra power will be needed for the added encryption and decryption on variety of devices while choosing HTTPS over HTTP is still unknown [45]. Usability and issues with usability testing of mobile apps have been studied in several publications [23,53]. In conclusion, we observed that all practitioners were aware of data privacy concerns, however, they did not know that certain third-party libraries, usage aggregators, or advertising libraries also occasionally leak sensitive user data. They all agreed that app security is a major concern, and that there is a lack of security experts in the industry, especially in many small mobile app companies. Requirements must be elicited by experts in the field, and, therefore, the app development companies tend to outsource their legal work to legal experts. Similarly, requirements engineers will proactively need to consult the latest literature and experts in the area of security and privacy to raise awareness about identified breaches in existing tools or libraries. Learning about the end users Market research is important for RE in general, however RE practitioners usually have little to do with it. Hence, app development companies rely heavily on other means such as app analytics to learn more about their users. 3.4.1 Empirical observations. Survey. All but four of the participants said that end users are anonymous to them. The locality of the end users, however, appears to be important: 29 said it matters. The ethnicity of the end users, on the other hand, is not a concern for many, i.e., 31 said it has never mattered before. About 33% of the participants said that they consider only young people and adults as their end users. Most of the participants (34, i.e., 75%) agreed that user feedback posted on app stores matters to their customers, but they process it almost without exception manually, as it is usually of low volume. In our survey 18 participants (i.e., 40%) said they never evaluate user feedback on social media platforms such as Twitter or Facebook. Interview. When asked about their end users, all interviewees said that the end users were always anonymous; the reason we believe is most of the corresponding companies were service-oriented. They also mentioned that their customers often did not have any access to the end users either. Worse, market research was entirely the customer's responsibility. According to all of them, when apps are released globally it is hard for requirements engineers to make assumptions about end user preferences. Ethnicity or locality were not concerns for any of them. Acceptance testing is vital to uncover missing requirements and improve usability. P9 told us that it is hard to get real end users for beta testing, and they end up testing the app with the people they know, which were mostly their own colleagues. This poses a great risk of product failure. All interviewees agreed that it is common practice to start making assumptions about the users whenever market data is missing. Another way to eliciting anonymous end user preferences is through app analytics. Post-release app usage statistics, e.g., provided by Google Analytics, are a very powerful tool to improve existing requirements. Interestingly, although none of the interviewees carry out pre-release user and market research themselves, all of them stated that post release analytics services can support app developers in decision making of the global audience's preferences. Interviewees P2, P3, and P5 said that there is usually no extra budget planned for app analytics, instead another (additional) contract is usually set up. New requirements can also be discovered through app reviews. Importantly, similar to the survey participants, all the interviewees were hardly aware of research in the area of opinion mining. Discussion. There are several ways to learn about end-user requirements, and each has its own complications. We identified four important themes from our observations: • market research is the client's responsibility, but it is rarely reliable; • developers use other means to learn about the user requirements such as through beta testing, trackers, analytics, and app reviews; • beta testing suffers from a lack of access to the actual users; trackers and analytics have privacy breaches; and • app review analysis is more popular in academia than in the industry. We see that app development companies rely on their customers to get insights about the potential end users. The quality of market research depends heavily on the skills and the reach of the customers. We learned that customers rarely have access to enough end users for beta testing. We suspect that they make far too many assumptions, specifically because their app users were located in the same country. Such assumptions fail if the app is to be released globally. The issues with beta-testers are previously reported by Joorabchi et al. in 2013, where they mention that "beta-testers are in the order of dozens and not thousands" [27]. It appears that after 6 years the problem is still relevant in the industry. Besides market research, app analytics are widely used in the industry, without developers adequately understanding the consequences. Similar to the implications of third-party trackers, research shows the implications of using popular third-party analytics services. For instance, the study of Chen et al. exposes the vulnerabilities of analytics services. They manipulated user-profiles constructed by such services to influence the ads shown to the users [5]. Analytics services also pose data privacy challenges. Privacy policies of analytics services are often difficult to read and demand too much time to comprehend. As a consequence, Bhardwaj et al. mention that developers are usually not aware of what or how the data is collected by analytics tools [2]. Nevertheless, using app analytics for improving app usability has also been demonstrated by Ferre et al. They extended Google Analytics to store specific low-level user interactions of interest to further test the usability [14]. Another approach to understanding app usage patterns is suggested by Tongaonkar et al. who use in-app advertisements instead of app analytics [46]. End-user review analysis has also gained much traction in academia in recent years. Different approaches suggested by researchers aim to help app developers improve their app's functionality directly from end-user suggestions [9,16,21,29,38]. We observed that user review analysis is practiced in the industry but on a limited scale. The interviewees rely on app stores such as Google Play, which provides automated user review summaries and sentiment analyses. The reason we believe, as also mentioned by P4, could be the paid access to the academic publications, which underlines the need to publicize the work beyond the academic context e.g., writing tweets, online blogs, articles etc. In conclusion, all practitioners agreed that there is a lack of beta testers to gain confidence in the app they develop. Nevertheless, they were all unaware of opinion mining techniques recently proposed in academic publications. They similarly did not know the implications of using app analytics and usage trackers as mentioned earlier. App type To fulfil end-user requirements, one does not always need a native mobile app, but hybrid or Progressive Web Apps 6 (PWAs) can be a viable option too. 6 https://developers.google.com/web/progressive-web-apps/ 3.5.1 Empirical observations. Survey. There were no dedicated questions on the role of app stores in our survey. The theme emerged from the survey responses and it was subsequently added to the interview instrument for further discussion with practitioners. Interview. Interviewees had different opinions regarding hybrid apps. According to P1, performance issues are actually not a concern for the end users as they are not so sophisticated that they can easily identify such details. Additionally, platform-specific customizations generate in most scenarios up to 30% additional code, which is acceptable for him and justifies their purpose. The interviewee believes that discussions regarding performance of hybrid apps are "usually held by technical people who do not reflect the true audience. " PWAs have recently gained some traction, so we asked the interviewees their opinions on PWAs. They mention that PWAs are easy deployed and thus provide potential to kick-start useful discussions while gathering requirements. Most of the interviewees agreed on the potential of PWAs and according to them industry understands their potential, but they also mentioned that it will take some time until they become mature. They said that PWAs are especially helpful as they break the dependency on the app stores, and let developers release changes as quickly as possible, giving them advantages of native apps such as push notifications. Discussion. Cost has been one of the main determining factors for both end users and developers for platform selection. As we mentioned earlier, the cost of mobile devices affects the user base for a specific platform. On the other hand, from the developers' perspective, the budget can affect the platform selection. We identified three key themes from our observations: • hybrid apps and PWAs are seen as alternatives to native apps but with some reservations as to their quality; • previous arguments regarding hybrid apps are not necessarily true due to technological advancements; and • PWAs have the potential to replace native and hybrid apps in certain cases for few obvious advantages. RE practitioners must also familiarize themselves with hybrid frameworks as they can potentially replace native apps, and can be built at a lower price in terms of effort and number of required expert developers. Several studies advocate cross-platform development due to extra effort and time to market caused by platform fragmentation [6,41]. The main complaint about hybrid apps has been their inferior performance [8]. Performance issues seem to be a bit overrated as they are far from obvious for average users. It is also evident from the literature as Malavolta et al. from their survey of 11,917 apps from Google Play conclude that the perceived performance difference between native and hybrid apps is negligible [32]. Another major concern about hybrid apps is platform-specific customization [8], about which our interviewees disagreed. Such observations differ from previous publications in this area; the reason, we believe, is merely due to technological advancements in the past few years [44,49]. PWAs, on the other hand, are a recent phenomenon, and they are expected to occupy the gap between hybrid apps and plain web sites. They even could potentially replace hybrid apps in certain cases where the app's purpose is just about displaying information. PWAs are especially helpful when the native features such as camera access are not important for the app. Additionally, they also simplify the app publishing workflow by breaking the dependency on app stores. In response to a lack of academic involvement in the area of PWAs, the work of Biørn-Hansen et al. provides a performance and feature comparison between cross-platform mobile and progressive web apps; notably they demonstrated that the performance of PWAs was far superior in terms of launch time and time from app icon tap to toolbar rendering [4]. In a similar comparison study, Cardieri et al. explored the aspects of user experience on three different platforms i.e., native, web mobile and PWA, and they reported overall positive user experience despite there being a few interaction issues [11]. Luntovskyy et al. also compare native, web-based and hybrid apps, and report that PWAs can represent an efficient alternative to native mobile apps due to several advantages such as a simplified installation process, and reduced data volume consumption [30]. In conclusion, hybrid apps and PWAs have evolved in recent years into strong alternatives to native app development. Most practitioners seemed to be aware of hybrid frameworks but they mostly had negative impressions about them, regardless of the existence of studies proving that performance differences are, in fact, negligible. We observed that a practitioner had at least heard of PWAs but had never considered it an an option to mainstream mobile app development, despite certain advantages they offer such as breaking the dependency on app stores. To fulfill end-user requirements, a native app may not always be an appropriate fit; alternatives such as hybrid apps may solve the problem more elegantly. RELATED WORK Numerous papers outline differences between mobile and generalpurpose software development [15,27]. The study of Frances et al. investigates aspects related to the development and management of mobile apps. They interviewed four IT managers with experience in mobile app development, and surveyed 510 mobile app developers [15]. Only a small part of the study explores requirements issues regarding mobile apps, and unlike our study, it is limited to the discussion about usability aspects and subsequent GUI optimizations. It explains app maintenance issues and reports that an additional business contract is required with the customer. They present similar challenges with testing as we found in our work, e.g., "beta testers are not available readily, and when at all they are in dozens and not in thousands, " and "manual testing is prevalent. " Finally, they motivate the need for mobilization i.e., using custom mobile apps to fulfil business needs of communication or data processing of organizational processes, which in our opinion is becoming more relevant. Joorabchi et al. interviewed 12 senior mobile app developers and surveyed 188 mobile developers to gain an understanding of the challenges faced by developers [27]. Their findings, unlike ours, are mainly related to the challenges regarding app development for different platforms (at that time) and challenges regarding testing and analysis. They argued that device fragmentation is not only Learning about end users * Market research is the clientâĂŹs responsibility, but it is rarely reliable * Developers use other means to learn about the user requirements such as through beta testing, trackers, analytics, and app reviews * Beta testing suffers from a lack of access to the actual users; trackers and analytics have privacy breaches * App review analysis is more popular in academia than in the industry App type * Hybrid apps and PWAs are seen as alternatives to native apps but with some reservations as to their quality * Previous arguments regarding hybrid apps are not necessarily true due to technological advancements * PWAs have the potential to replace native and hybrid apps in certain cases for few obvious advantages a challenge for development, but also for testing. Besides these aspects, they also pointed out app store requirements such as "changing app store policies" play an important role in mobile app development. We discuss this issue from the app's requirements point of view. Nagappan et al. shed light on the current and future research trends for various stages in the mobile app development life-cycle, e.g., requirements, design and development, testing, and maintenance [36]. They extensively discuss the role of app stores in the mobile app industry, and also list few limitations such as "only a subset of user reviews available", and "no access to source code of the apps". However, they neither explore these issues from the requirements point of view, nor do they discuss how developers tackle such issues. This work also extensively discusses the battery usage of the apps, but interestingly none of the interviewees in our study has such concerns. Wasserman et al. discuss software engineering issues specific to mobile app development, specifically citing performance, reliability, quality, and security among the most important NFRs for mobile apps. However, their future direction regarding NFRs is limited to discovering differences between responsiveness, and data integrity in case of low battery, to name a few [49]. Dehlinger et al. outline and discuss four challenges that mobile application engineering faces, among which are the necessity of creating user interfaces for differently abled users, and platform fragmentation [12]. As for the requirements, they signify the need for self-adapting apps based on the context e.g., providing limited functionality for location-based services while having low battery rather than providing no service at all. There are several studies that are relevant to this paper, but none of them exclusively target mobile development. For instance, recent research has discussed the challenges with current RE practices [20,25,31,36,43]. Groen et al. proposed a CrowdRE approach i.e., performing requirements engineering with the crowd of stakeholders [20]. They mention that data privacy issues become prevalent while using different user feedback channels as the chance of exposing sensitive information increases. Schon et al. reported four relevant key challenges in their Agile RE approach: dependency issues arising between multiple teams due to coordination effort, customers not willing to let developers make independent decisions, inadequate access to the end users, and stakeholder participation throughout the development process [43]. Inayat et al. explain additional issues such as budget and time estimation, negligence regarding NFRs, and fixed-priced contracts [25]. Maalej et al. discuss data-driven RE, which involves user feedback analysis to identify potential requirements [31]. They report that managing huge user input requires substantial human resources for manual processing. Moreover, scalability and the sophisticated tool support are also questionable. There exist several relatively old studies which have outlined the practices and challenges with contemporary RE practices [10,22,54]. The study from Davis et al. questions the effectiveness of the RE techniques [10]. On the other hand, the study of Zowghi et al. highlights the need to overcome the gap between research and industry, and also between novices and experts, as a potential challenge to effective RE [54]. The empirical study from Hickey et al. reports on when to use which elicitation technique [22]. THREATS TO VALIDITY The validity of the findings of a qualitative study is invariably subject to several threats [50]. Construct validity is threatened if the answers from the participants do not accurately reflect the real practice. This could be due to the interviewees not feeling comfortable talking about certain topics or to the interviewer influencing the discussion. To reduce these risks, we avoided judgment and evaluations during the discussions by assuring the interviewees anonymity in the study, and we abstained from communicating our assumptions to the participants. In order to collect reliable data, we only selected participants who are knowledgeable about the companies' practices in gathering requirements. When we interviewed multiple participants from the same company, we asked them not to talk about the interview to others in order to avoid any bias. Unclear questions and misunderstandings between the interviewee and interviewers are also possible threats that cannot be completely ruled out. We mitigated these threats by discussing the interview questions together with other experienced researchers and by conducting a pilot interview. We also tried to explain the questions differently to the interviewees whenever we believed that the participant was suffering from misunderstandings. Reliability threats that relate to researcher bias, however, cannot be completely ruled out, because the interviews and analyses were conducted by a single researcher. External validity issues are related to the inability to generalize the results of the study beyond the studied companies. In order to achieve reasonable generalization during the interview sampling, we selected companies that showed different characteristics in terms of size, the domain of operation, and to some extent location. However, our study suffers from selection bias as eight out of ten companies were from a single country i.e., Switzerland. Similarly, we could interview only ten practitioners due to several reasons such as contacted experts not being available for an interview, or not being interested. Additionally, most of the interviewees were also survey participants, and hence they were already exposed to the topic which adds additional selection bias to our study. Nevertheless, only few interviewees were from our personal contact list. Internal validity is threatened if a causal relationship between treatment and outcome is wrongly established. A possible factor that could negatively impact the internal validity in our case is the interview change. To reduce the effect, we evaluated the interview questions through an internal validation and a pilot interview and revised them before starting the real data collection. CONCLUSION We surveyed 45 companies and interviewed ten experts in the area of mobile app development to understand the challenges of requirements gathering. Notwithstanding mobile and (traditional) software development being different in general, we observed that they are quite similar from the requirements-gathering perspective. Nevertheless, there exist several factors that are more specific to the mobile domain. For instance, RE practitioners need a deep understanding of platform ecosystems, including app store policies and available third-party libraries to make good decisions, e.g., platform or feature selection, that fulfil the functional requirements. Also, dealing with non-functional requirements such as security and usability requires delicate expertise and experience as mobile devices offer customization both in the hardware and the operating system. Furthermore, we realized that practitioners in our study are mostly unaware of several techniques for requirements elicitation such as the use of mobile apps and mobile devices, whereas they are extensively discussed in academia. Similarly, opinion-mining techniques are prevalent in the state-of-the-art literature, but practitioners rather rely on user review summaries provided by app stores. The reason we identified is that practitioners who are responsible for requirements collection have no direct technical experience with mobile app development, instead, they come from business or marketing background. ACKNOWLEDGMENTS We gratefully acknowledge the financial support of the Swiss National Science Foundation for the project "Agile Software Assistance" (SNSF project No. 200020-181973, Feb. 1, 2019 -April 30, 2022). We also thank CHOOSE, the Swiss Group for Original and Outside-the-box Software Engineering of the Swiss Informatics Society, for its financial contribution to the presentation of this paper. Table 1 : 1Interviewees, their roles, and details regarding their employersParticipant ID Company ID Participant Role Experience in years with RE Company Type Number of Published Apps % RE budget Spent Number of Employees Which Markets They Target Country of Origin P1 C1 CEO 5 Service 15 15-20 35 Switzerland Switzerland P2 C2 Team Lead 6 Service 13 10-20 29 Switzerland Switzerland P3 C2 Project manager 2.5 Service 13 10-20 29 Switzerland Switzerland P4 C3 Team Lead 6 Service 9 5-10 7 Switzerland Switzerland P5 C3 CEO 6 Service 9 5-10 7 Switzerland Switzerland P6 C4 Project manager 20 Product 1 10 8 Switzerland Switzerland P7 C5 Director 7 Service 10 40-50 31 Europe, India, USA India P8 C6 Agile consultant 15 Service 10+ ∼20 NA Europe, Asia Switzerland P9 C7 Business developer 10 Service 20 15-20 150 Switzerland Switzerland P10 C8 Product Owner 7 Both 1 ∼1 100 Netherlands Netherlands by conclusions. The summary of all the important findings can be seen in Table 2. Table 2 : 2Main findings Eliciting NFRs requires experience and expertise * In the case of mobile apps, NFRs such as data privacy can be unintentionally compromised by selecting a certain third-party library * Device capabilities are an important factor fora ppâĂŹs performance and acceptanceFactor Findings Platform and third-party libraries * Platform selection depends on several factors such as cost, budget, market research and platform ecosystem * End users, customers and developers can have different concerns regarding platform selection * Functional requirements are affected while selecting a particular platform, and the opposite is true as well, sometimes resulting in favouring hybrid apps App stores and their policies * Customers have no concerns regarding app stores, but they are important for both end-users and developers as they simplify the app distribution and discovery process * Different app stores have different business models and policies * Policy changes by app stores are affecting app development companies by causing unintentional rework or in worse cases app denial Non-functional requirements * A hybrid app is developed using browser supported language e.g., HTML, CSS, JavaScript that runs only in a browser shell and has access to the native platform layer. http://www.coppa.org/ 4 https://eugdpr.org/ A captive gate forwards all requests to the login page instead of the application server until the user has been authenticated. 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[]
[ "Bosonic D-brane Effective Action in Linear Dilaton Background", "Bosonic D-brane Effective Action in Linear Dilaton Background" ]
[ "J Klusoň [email protected] \nDepartment of Theoretical Physics and Astrophysics Faculty of Science\nMasaryk University\nKotlářská 2611 37BrnoCzech Republic\n" ]
[ "Department of Theoretical Physics and Astrophysics Faculty of Science\nMasaryk University\nKotlářská 2611 37BrnoCzech Republic" ]
[]
In this paper we will study tachyon effective action for Dp-brane in bosonic string theory in the linear dilaton background. We obtain the tachyon effective Lagrangian from boundary state coeficient of Dp-brane in the linear dilaton background and compare it with tachyon effective Lagrangians that were proposed in previous papers.
10.1088/1126-6708/2004/02/024
[ "https://export.arxiv.org/pdf/hep-th/0401236v3.pdf" ]
7,568,362
hep-th/0401236
f76f6496d29c268272207aeb8c4b76f16d64e8cf
Bosonic D-brane Effective Action in Linear Dilaton Background 11 Mar 2004 J Klusoň [email protected] Department of Theoretical Physics and Astrophysics Faculty of Science Masaryk University Kotlářská 2611 37BrnoCzech Republic Bosonic D-brane Effective Action in Linear Dilaton Background 11 Mar 2004Preprint typeset in JHEP style. -HYPER VERSIOND-branes In this paper we will study tachyon effective action for Dp-brane in bosonic string theory in the linear dilaton background. We obtain the tachyon effective Lagrangian from boundary state coeficient of Dp-brane in the linear dilaton background and compare it with tachyon effective Lagrangians that were proposed in previous papers. Introduction Study of the tachyon condensation is one of the central themes in the string theory research in last few years [1]. Many powerful methods were developed in order to describe this process, for example it was realised that open string field theory is very useful tool for analysis of the tachyon condensation 1 . In some situations one can also analyse the tachyon condensation using worldsheet conformal field theory description [1,5]. It was also shown that the tachyon condensation could be described in terms of the non-commutative geometry [6]. One of the most remarkable results given in last year is the fact that in some situation the tachyon condensation could be successfully described in terms of the effective field theory including the tachyon field and massless modes living on unstable D-brane 2 . Very nice discussion considering the tachyon effective field theory was given recently in [27]. According to this paper it seems that there is no much sense to find tachyon effective action only since the scale of the masses of an infinite set of massive string modes is the same at the tachyon mass and thus to keeping the tachyon while integrate out all other string modes my appear to be not well defined. On the other hand one may hope that some aspects of string dynamics can be captured by an effective field theory action invoking only the tachyon fields and massless modes, where all other massive modes decouple at a vicinity of certain conformal points. One such an example of an exact conformal point that was studied recently is a time-dependent background which should represent an exact boundary conformal theory [15,16] T = f 0 e µx 0 +f 0 e −µx 0 , (1.1) where in the superstring case µ 2 super = 1 2 and in case of bosonic string theory µ 2 bose = 1. Its special case is the "rolling tachyon" background T = f 0 e µx 0 . (1. 2) The disk partition function in this background was recently studied in [30] suggesting that the corresponding potential term should look like V = 1 1 + T 2 2 . (1.3) Then it was shown in [19,29] 3 that demanding that a generic first-order Lagrangian should have (1.1) (for µ super ) as its exact solution fixes its time-derivative part to be L = − 1 1 + T 2 2 1 + T 2 2 − (∂ 0 T ) 2 . (1.4) If we now assume as in [19] that (1.4) has a direct Lorentz-covariant generalisation we obtain the tachyon effective action for unstable Dp-brane in supersymmetric theory S = − d p+1 x √ −gL , L = e −Φ 1 + T 2 2 1 + T 2 2 + g µν ∂ µ T ∂ ν T . (1.5) The Lagrangian given in (1.5) after field redefinition T √ 2 = sinhT √ 2 becomes "tachyon DBI" Lagrangian L T DBI = −e −Φ V (T ) − det(g µν + ∂ µT ∂ νT ) , V (T ) = 1 coshT √ 2 . (1.6) The arguments based on having (1.1) as an exact solution fixes only the timederivative dependence of the action (1.5). On the other hand it is possible that the covariantisation of (1.4) to (1.5) could be rather subtle [27]. In fact, the exact description for the string partition function evaluated on more general background T (x 0 , x i ) = f (x 0 , x i )e µx 0 (1.7) should be Lorentz-covariant, but that need not apply to its first-derivative part only. Exactly such a behaviour was observed in [27] where the string partition function in bosonic string theory was calculated on the tachyon profile T = f (x i )e x 0 which results into following tachyon effective Lagrangian L = − 1 1 + T 1 − ln(1 + T ) T (1 + T ) (∂ i T ) 2 − s 1 2(1 + T ) ∂ 2 i T + . . . . (1.8) In supersymmetric case, where the tachyon profile is T = f (x i )e x 0 √ 2 (1.9) we obtain tachyon effective Lagrangian L = − 1 1 + T 2 2 1 + 1 1 + T 2 2 1 − ln 1 + T 2 2 + 1 2 s 1 1 − 3 2 T 2 1 + T 2 2 (∂ i T ) 2 + . . . , (1.10) where s 1 can be changed by field redefinition. As was shown in [27] this Lagrangian differs from (1.5) evaluated on the profile (1.9). The discussion given above implies that the form of the tachyon effective action strongly depends on the point in the field theory space where this action is calculated from the string partition function. For example, it was shown recently in [29] that the Lagrangian (1.5) correctly reproduces string S-matrix tree level amplitudes near the conformal point T = e x 0 √ 2 on condition that spatial momenta of initial a final states are small. This fact gives strong support for the validity of (1.5) at least for description of tachyon dynamics close to the rolling tachyon solution (1.2). On the other hand in our recent paper [25] we have shown that there are problems when we try to apply the action (1.5) to the case of linear dilaton background. The lesson from these arguments is that we should be very careful when we try to use tachyon effective actions in regions of field theory space where its validity is not established. In this paper we will continue the study of the tachyon effective actions in the linear dilaton background. The starting point of our analysis will be results given in very nice paper [31] where the string theory on D-branes in linear dilaton background was studied. We use the expression for the boundary state coefficientB(x) obtained there and following the general strategy [32] we will interpret this one point function as a tachyon effective Lagrangian evaluated on the marginal tachyon profile T = πλe βt . Then in order to gain some information about the tachyon effective action we will expand this one point function for case of small T and also for case of "weak" dilaton background V 0 ≪ 1. Then we compare resulting effective Lagrangian with the tachyon effective Lagrangians obtained in previous papers [19,27]. We find that the tachyon effective action obtained here has the similar form as the tachyon effective action for D-brane in bosonic theory (1.8). Then using the results given in [31] we will consider the case when the tachyon field is still marginal however now also depends on the spatial coordinates. We obtain an exact form of the string partition function that is manifestly Lorentz-covariant which implies that the tachyon effective Lagrangian evaluated on marginal tachyon profile is Lorentz-covariant as well. Since the form of this effective Lagrangian is rather complicated we restrict ourselves to the case of weak linear dilaton background. Once again we obtain action that coincides with the action given in [27]. This paper is organised as follows. In section (2) we review results given in [31] that are needed for our calculation. In section (3) we will consider the situation when the time-like component of dilaton vector is small V 0 ≪ 1 and we can perform an expansion of the one point function with respect to the small parameter V 0 . In section (4) we generalise the calculation given in [31] to the tachyon boundary perturbation that depends on the spatial coordinates as well. In conclusion (5) we outline our results and suggest possible extension of our work. Review of the tachyon condensation on D-brane in the linear dilaton background In this section we briefly review the description of the tachyon condensation in the linear dilaton background how was presented in [31]. We begin with the fact that tachyon condensate on unstable D-brane that describes its time-dependent decay is T = πλe βX 0 with β > 0. We will study this process in arbitrary spacelike dilaton background 4 Φ = V µ X µ , V µ V µ > 0 . (2.1) The weight of the worldsheet boundary interaction operator is β(β − V 0 ). Requiring the weight to be equal to 1 we obtain V 0 = β − 1 β . (2.2) As was shown in [31] the time-like part of the worldsheet CFT is the Timelike Boundary Liouville theory [33] − 1 2π Σ ∂X 0 ∂X 0 − V 0 X 0 R 4 + 1 2π ∂Σ (πλe βX 0 + V 0 X 0 K) , (2.3) where K is the extrinsic curvature which integrates to 2π around the boundary of the disc. The action (2.3) is related to the standard boundary Liouville theory with Q = b + 1 b 1 2π Σ ∂φ∂φ + QφR 4 + 1 2π ∂Σ πλe bφ + QφK , (2.4) by analytic continuation X 0 → iφ , β → −ib , V 0 → −iQ. The boundary Liouville theory (2.4) was studied in [34,35,36,37,38,39]. Among many interesting results that were calculated in [31] the most important for our calculation is the boundary state coefficientB(x) that is given by a disc worldsheet one point functioñ B(x) = e V x d D k (2π) D e −ikx e −ikX+V X , (2.5) where D means the dimension of target spacetime. Recall that there is following relation between D and V µ in critical string theory V µ V µ = 26 − D 6 . (2.6) As in [31] we require the dilaton gradient to point along the unstable D-brane. Now we will argue how (2.5) is related to the tachyon effective Lagrangian evaluated on the profile T = πλe βx 0 . The general definition of the stress energy tensor is T IJ = − 2 √ −g δS δg IJ . (2.7) If we now presume that the tachyon effective action has the form S = − d p+1 x √ −gL (2.8) we get T IJ (x) = −g IJ (x)L(x) + 2 δL(x) δg IJ (x) . (2.9) On the other hand it was shown in [31] that the stress energy tensor of D-brane in the linear dilaton background can be expressed in terms of the boundary states coefficientsà µν (x) ,B(x) as T µν (x) = e −2V x à µν (x) −B(x)η µν ,(2.10) where the explicit form ofà µν (x) is not important for us. Comparing (2.9) with (2.10) we can anticipate that the Lagrangian evaluated on the tachyon profile T = πλe βt is equal to L(x) = e −2V xB (x) . (2.11) The expression (2.5) was calculated in [33] with the result B(x 0 ) = e Φ β d D−p k (2π) D−p e ikxλik 0 /β Γ(−ik 0 /β)Γ(1 + iβk 0 ) , (2.12) whereλ = πλ Γ(1 + β 2 ) . (2.13) Following [31] we rewrite (2.12) into the form that will be suitable for our analysis B = e Φ β d D−p−1 ke i D−1 i=p+1 k i x i (2π) D−p−1 dk 0 2π e ik 0 x 0λ ik 0 /β Γ(−ik 0 /β)Γ(1 + ik 0 β) = = δ(x T ) β dk 0 (2π) e ik 0 x 0λ ik 0 /β ∞ 0 dqe −q q −ik 0 /β−1 ∞ 0 dse −s s ik 0 β = = δ(x T ) ∞ 0 ds dx ′0 dk 0 (2π) e ik 0 (x 0 + 1 β lnλ−x ′0 +β ln s) e −e βx ′0 −s = δ(x T ) ∞ 0 ds dx ′0 δ(x 0 + 1 β lnλ − x ′0 + β ln s)e −e βx ′0 −s = = δ(x T ) ∞ 0 ds exp −s − T Γ(1 + β 2 ) s β 2 , (2.14) where we have introduced the tachyon field T = πλe βx 0 ,(2.15) and where δ(x T ) means the transverse delta function projecting onto the Dp-brane. In what follows we suppress this delta function. According to the discussion given above (2.14) implies that the tachyon effective Lagrangian evaluated on the marginal profile (2.15) in the linear dilaton background is equal to L = e −Φ ∞ 0 ds exp −s − T Γ(1 + β 2 ) s β 2 . (2.16) We present various checks that this Lagrangian reduces to the more familiar one in various limits of tachyon field theory space. First of all, for t → −∞ we can write L = e −Φ ∞ 0 dse −s 1 − T Γ(1 + β 2 ) s β 2 = e −Φ (1 − T ) ,(2. Weak linear dilaton background As we said above the formulation weak linear dilaton background means that V 0 ≪ 1. Since we demand that β > 0 in order to describe tachyon condensation where the tachyon field rolls from T = 0 to T = ∞ the condition of marginality β 2 − βV 0 = 1 implies β = V 0 + 4 + V 2 0 2 ≈ 1 + V 0 2 , β 2 ≈ 1 + V 0 . (3.1) Then for small V 0 we get L = e −Φ ∞ 0 ds exp −s − T s(1 + (C − 1)V 0 )e V 0 ln s = = e −Φ ∞ 0 ds exp [−s(1 + T )] 1 + sV 0 T ln 1 x + (1 − C)T V 0 s = = e −Φ 1 1 + T + V 0 T (1 + T ) 2 ln(1 + T ) = = e −Φ 1 + T 1 + (β 2 − 1)T (1 + T ) ln(1 + T ) ,(3.2) where we have used Γ(1 + β 2 ) = Γ(2 + V 0 ) ≈= 1 − (C − 1)V 0 , C = 0.577 , ∞ 0 dxxe −ax ln 1 x = 1 a 2 (ln a − 1 + C) . (3.3) Let us say few comments about (3.2). First of all we must explain why we have replaced V 0 with β 2 − 1 using (3.1). We can argue as follows. The presence of the expression V 0 T suggests that the Lagrangian could contain terms as ∂ µ Φ∂ ν T g µν that are rather unusual and lead to some problems. For example, the variation of the action containing such terms with respect to g µν will produce expressions proportional to ∂ µ Φ∂ ν T = ∂ µ Φ∂ 0 T . These expressions would be presented in the components T i0 of the stress energy tensor. On the other hand we know from [31] that the stress energy tensor for the rolling tachyon solution in the linear dilaton background is diagonal so that T 0i = 0. Since the Lagrangian (3.2) is evaluated on the marginal tachyon profile where (3.1) holds we mean that it is appropriate to replace V 0 with β 2 −1. The next issue is how to interpret the expression (β 2 −1)T . It can be written eitherṪ 2 −T 2 T or asT − T . The first possibility is non-analytic around the point T = 0 but together with the factor ln(1 + T ) ∼ T for T ≪ 1 we get analytic expression. On the other hand we mean that it is reasonable to replace (β 2 − 1)T withT −T in terms where the choiceṪ 2 −T 2 T would lead to the non-analytic behaviour around the point T = 0. In fact, the importance of higher derivative terms in the tachyon effective Lagrangian was discussed recently in [27]. On the other hand it is clear that the expression (β 2 −1)T is in principle ambiguous and that the requirement of the analycity of the tachyon effective Lagrangian around the point T = 0 need not be fundamental. Moreover it is possible that non-analytic terms could play significant role in some situations. We mean that in order to resolve this issue we should calculate the string partition function on more general tachyon background. However calculation such a partition function on general tachyon background is very complicated task which is certainly beyond the scope of this paper. For that reason we will follow the arguments given above so that we write (3.2) as L = e −Φ 1 + T 1 + ln(1 + T )(Ṫ 2 − T 2 ) T (1 + T ) + . . . . (3.4) We see that (3.4) has similar form as the Lagrangian (1.8). We must also mention that in some sense the similarity between (3.4) and (1.8) should not come as a big surprise since both Lagrangians are calculated from the string partition function even if closed string background and hence worldsheet conformal field theories are different. The more interesting question is whether there is a relation between (3.4) and the bosonic version of (1.4) given in [25] 5 S = − d p+1 x e −Φ 1 + T 1 + T − (∂ 0 T ) 2 T , (3.5) where we now presume that (3.5) is valid in the nontrivial dilaton background as well. Since for the weak linear dilaton the factor T − (∂ 0 T ) 2 T = T (1 − β 2 ) = −V 0 T is small we can write 6 L = e −Φ 1 + T 1 + T 2 − (∂ 0 T ) 2 2T (3.6) that is clearly different from (3.4). We mean that this fact is in agreement with the claims given in [27] that the tachyon effective Lagrangians (1.4),(3.5) do not follow directly from the string partition function. We return to this issue in the conclusion. Spatial dependent tachyon profile In this section we will consider the tachyon boundary interaction that also depends on spatial coordinates. More precisely, we presume that the marginal interaction inserted on the boundary of the worldsheet is T = πλe βµx µ , −η µν β µ β ν + η µν β µ V ν = 1 . (4.1) 5 For precise discussion of the bosonic D-brane tachyon effective action, see also [26]. 6 It is important that (3.5) also implies 1 − (β 2 − 1)T > 1 ⇒ T < 1 β 2 −1 . On the other hand for exponential form of T ≈ e βx 0 there will be certainly time x 0 * for which T (x 0 * ) = 1 β 2 −1 . For this time event the Lagrangian is equal to zero and then becomes imaginary. This fact again suggests problems with the generalisation of (3.5) to the linear dilaton background. We can also observe that the same problem arises in the supersymmetric version (1.4) as well. In this case the action for the worldsheet CFT has the form S = 1 2π Σ η µν ∂X µ ∂X ν + V µ X µ 4 R + 1 2π ∂Σ πλe βµX µ + V µ X µ K . (4.2) From this explicit form of the string worldsheet action it is clear that the exact form of the string partition function is manifestly Lorentz covariant. On the other hand it is possible that the expansion of the partition function in powers of ∂T will lead to the Lagrangian where the manifest Lorentz covariance is lost. This is an important issue that was extensively discussed recently in [27]. One can hope that the study the tachyon effective Lagrangian in the linear dilaton background could be helpful for addressing this problem. In fact, we will see that results given in this section suggest possible covariant extension of (1.8). First of all, using the manifest Lorentz covariance we can bring β µ in (4.1) into the frame where T = exp(β µ X µ ) = exp(β ′ 0 X 0 ′ ). The the worldsheet action (4.2) is S = 1 2π Σ η µν ∂X ′µ ∂X ′ν + V ′ µ X ′µ 4 R + 1 2π ∂Σ πλe β ′ 0 X ′0 + V ′ µ X ′ µ K (4.3) that has the same form as the worldsheet action given in section (2). Hence the coefficientB will be the samẽ B(x ′ ) = e V ′ µ x ′ µ ∞ 0 ds exp −s − πλe β ′ x ′0 Γ(1 + β ′2 ) s β ′2 = = e Vµx µ ∞ 0 ds exp −s − T Γ(1 − η µν β µ β ν ) s −η µν βµβν ,(4.4) where we have introduced the tachyon field T = πλe βµx µ (4.5) and we have also used explicit Lorentz covariance ofB. Using the relation betweeñ B and L we obtain the tachyon effective Lagrangian evaluated on the tachyon profile (4.5) in the form L = e −Vµx µ ∞ 0 ds exp −s − T Γ(1 − η µν β µ β ν ) s −η µν βµβν . (4.6) In order to obtain more transparent form of the Lagrangian let us again consider the case when the dilaton gradient is small. Now the condition of marginality implies −η µν β µ β ν = 1 − η µν V µ β ν = 1 + ǫ , ǫ ≪ 1 (4.7) on condition of the weak dilaton background V µ ≪ 1 , V µ V µ > 0. With this assumption we immediately get L = e −Vµx µ ∞ 0 ds exp −s − T Γ(1 − η µν β µ β ν ) s −η µν βµβν = = e −Φ dse −s(1+T ) (1 − T ǫs ln s + (1 − C)T sǫ) = = e −Φ 1 + T 1 + ǫT 1 + T ln(1 + T ) = = e −Φ 1 + T 1 − (η µν β µ β ν + 1)T 1 + T ln(1 + T )] . (4.8) Following discussion given in the previous section we perform the substitution (η µν β µ β ν + 1)T ⇒ η µν ∂µT ∂ν T +T 2 T in the term proportional to ln(1+T ) 1+T so that we get L = e −Φ 1 + T 1 − ln(1 + T ) T (1 + T ) (η µν ∂ µ T ∂ ν T + T 2 ) + . . . . (4.9) The Lagrangian given above is the main result of this paper. Let us now presume that (4.9) is also valid for general tachyon not obeying the condition of marginality. Then we see that (4.9) could be considered as covariant version of (1.8). In fact, when we insert the tachyon field T = f (x i )e x 0 into the Lagrangian (4.9) we get L = e −Φ 1 + T 1 − ln(1 + T ) T (1 + T ) δ ij ∂ i T ∂ j T + . . . . (4.10) that same as the first order derivative part of the Lagrangian (1.8) 7 . Note also that for small T (4.9) has the form L = e −Φ [1 − T − η µν ∂ µ T ∂ ν T ] . (4.11) We see that this Lagrangian is different from the standard bosonic tachyon Lagrangian describing the tachyon dynamics around the perturbative vacuum T = 0 L = e −Φ − 1 2 η µν ∂ µ T ∂ ν T + 1 2 T 2 . (4.12) Note that the same thing was recently observed in [27] in case of the Lagrangian (1.8). Then it is not surprising that the tachyon profile T = e βµx µ is not solution of the equation of motion that arises from (4.11) while is solution of the equation of motion arising from (4.12) ∂ µ e −Φ η µν ∂ ν T + T = 0 (4.13) for η µν β µ β ν −η µν V µ β ν +1 = 0. We can explain this puzzle as follows. In order to derive the correct equation of motion one needs to compute first the partition function for the general tachyon field. In our case we have calculated the partition function on the exact marginal perturbation so that it is possible that in the resulting Lagrangian some terms are missing which would be nonzero for general off shell tachyon profile. We also have ambiguity when we have replaced β 2 with eitherT T or withṪ 2 T 2 . In order to obtain more general effective Lagrangian we should consider arbitrary boundary tachyon perturbation on the string worldsheet and study this problem using the boundary string field theory [40]. Conclusion The main goal of this paper was to study the tachyon effective action for Dp-brane in the bosonic string theory in the linear dilaton background. We have based this analysis on the known form of the boundary state coefficient given in [31] and its relation to the tachyon effective Lagrangian evaluated on the tachyon marginal profile. As we could see in section (2) this Lagrangian which is valid for general spacelike dilaton vector V µ has rather unfamiliar form. However we have shown that in case of vanishing dilaton field V µ = 0 the Lagrangian is the same as the Lagrangians calculated for exactly time dependent tachyon profile in [19,27]. This coincidence also holds in the far past for general V µ . Much more informations we have got when we considered the case when the time-like component of the dilaton vector V 0 is small. Then we could introduce small parameter proportional to V 0 and perform an expansion in the tachyon Lagrangian with respect to it. Using the condition of the marginality of the tachyon profile we have got the tachyon effective Lagrangian that has similar form as the Lagrangian (1.8). This coincidence was much sharper when we have generalised the pure time dependent tachyon condensation to the case when the tachyon profile depends on spatial coordinates as well. We have argued that this can be done very easily using manifest Lorentz-covariance of the boundary state coefficientB(x). Then we have found that the resulting effective tachyon Lagrangian has the same form as the effective tachyon Lagrangian evaluated in [27]. On the other hand we have also shown that this Lagrangian is different from the Lagrangian (3.5) generalised to the case of the linear dilaton background. Then we have argued that this result is in agreement with the claim given in [27] where it was said that (1.4) and (3.5) do not directly follow from the string partition function. Of course this fact does not mean that the Lagrangian (1.4) is not suitable for descriptions of the tachyon condensation. Moreover, it was shown recently in [29] that (1.4) gives the excellent description of the tachyon dynamics around the point in the tachyon field space that corresponds to the rolling tachyon background. We must also mention very interesting paper [42] where it was shown that "tachyon DBI" Lagrangian (1.6) gives excellent description of the moduli space of unstable D-branes on a circle of critical radius. On the other hand we have shown recently in [25] that the application of (3.5) to the description of the tachyon condensation in the case of linear dilaton background leads to results which are different from results given in [31] 8 . The extension of our calculations is obvious. We mean that it would be very interesting to perform the same analysis in the supersymmetric case as well. In order to do this we should firstly understand the time-like form of the N = 1 super Liouville theory [43,44,45], following [33]. We hope to return to this problem in future. dse −s s β 2 = Γ(β 2 + 1) . (2. 18 ) 218One can see that(2.17) is equal to the bosonic version of the Lagrangian (1.4) given in(3.5) and to (1.8) evaluated on time-dependent tachyon profile in the limit of small T . Moreover, for V µ = 0 which corresponds to β = 1, the effective Lagrangian (equal to the Lagrangians (3.5) and (1.8) evaluated on time dependent tachyon profile T = e t . To gain further information about (2.16) we will consider the case when time-like component of the dilaton vector V 0 is small. Let us name this background as "weak linear dilaton background". For reviews, see[2,3,4].2 Some papers, where the effective field theory descriptions of the tachyon dynamics can be found, are[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,23,22,24,25,26,27,28,29]. For similar discussion in case of bosonic string theory see[26]. Our convention is η µν = (−1, 1, . . . , p) , µ, ν = 0, . . . , p , i, j = 1, . . . , p. The indices in target spacetime are labelled with I, J = 0, . . . , D − 1. 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[ "Diagonals-parameter symmetry model and its property for square contingency tables with ordinal categories", "Diagonals-parameter symmetry model and its property for square contingency tables with ordinal categories" ]
[ "Kouji Tahata \nDepartment of Information Sciences\nTokyo University of Science\n\n", "Kohei Matsuda \nDepartment of Information Sciences\nTokyo University of Science\n\n" ]
[ "Department of Information Sciences\nTokyo University of Science\n", "Department of Information Sciences\nTokyo University of Science\n" ]
[]
Previously, the diagonals-parameter symmetry model based on f -divergence (denoted by DPS[f ]) was reported to be equivalent to the diagonals-parameter symmetry model regardless of the function f , but the proof was omitted.Here, we derive the DPS[f ] model and the proof of the relation between the two models. We can obtain various interpretations of the diagonalsparameter symmetry model from the result. Additionally, the necessary and sufficient conditions for symmetry and property between test statistics for goodness of fit are discussed.
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[ "https://export.arxiv.org/pdf/2305.08136v1.pdf" ]
258,686,388
2305.08136
b916b3959574d78a34db59f1078ae42c0266e583
Diagonals-parameter symmetry model and its property for square contingency tables with ordinal categories 14 May 2023 Kouji Tahata Department of Information Sciences Tokyo University of Science Kohei Matsuda Department of Information Sciences Tokyo University of Science Diagonals-parameter symmetry model and its property for square contingency tables with ordinal categories 14 May 2023Last update: May 16, 2023Conditional symmetryf -divergenceGlobal symmetryPartial global symmetry Mathematics Subject Classification: 62H17 Previously, the diagonals-parameter symmetry model based on f -divergence (denoted by DPS[f ]) was reported to be equivalent to the diagonals-parameter symmetry model regardless of the function f , but the proof was omitted.Here, we derive the DPS[f ] model and the proof of the relation between the two models. We can obtain various interpretations of the diagonalsparameter symmetry model from the result. Additionally, the necessary and sufficient conditions for symmetry and property between test statistics for goodness of fit are discussed. Introduction A square contingency table with the same ordinal row and column categories may arise when a categorical variable is measured repeatedly. In such a table, observations tend to concentrate on the main diagonal cells. Our research aims to apply symmetry rather than the independence between row and column categories. Many studies have treated symmetry issues, for example, Bowker (1948), Kateri and Papaioannou (1997), Kateri and Agresti (2007), Tahata andTomizawa (2011), andTahata (2020). Let X and Y respectively denote the row and column variables for an r × r contingency table with ordinal categories. Also, let π ij denote the probability that an observation falls in (i, j)th cell (i = 1, . . . , r; j = 1, . . . , r). Goodman (1979) proposed the diagonals-parameter symmetry (DPS) model, which is defined by π ij = d k ψ ij (i < j), ψ ij (i ≥ j),(1) where ψ ij = ψ ji and k = j − i. The parameter d k in the DPS model is simply the odds that an observation will fall in one of the cells (i, j) where j − i = k; i < j, rather than in one of the cells (j, i) where j − i = k; i < j for k = 1, . . . , r − 1. Additionally, for j − i = k; i < j, the ratio between π ij and π ji can be expressed the constant d k . That is, the ratio depends only on the distance from the main diagonal cells. When equation (1) with d 1 = · · · = d r−1 = 1, the DPS model is reduced to the symmetry (S) model proposed by Bowker (1948). When d k does not depend on i or j in equation (1) with d 1 = · · · = d r−1 , the DPS model is reduced to the conditional symmetry (CS) model proposed by McCullagh (1978). Kateri and Papaioannou (1997) described the DPS model based on the fdivergence (DPS[f ]) model, which is defined as π ij = π S ij F −1 (∆ k + ζ ij ) (i = 1, . . . , r; j = 1, . . . , r), where k = i−j, π S ij = (π ij + π ji )/2, ζ ij = ζ ji and ∆ k + ∆ −k = 0. It should be noted that the function f is a twice-differential and strictly convex, F (t) = f ′ (t), f (1) = 0, f (0) = lim t→0 f (t), 0 · f (0/0) = 0, and 0 · f (a/0) = a lim t→∞ [f (t)/t]. However, the derivation of this model is omitted in Kateri and Papaioannou (1997 The rest of this paper is organized as follows. Section 2 derives equation (2) and interprets the model from an information theory viewpoint. Additionally, the proof that the DPS[f ] model is equivalent to the DPS model regardless of the function f is given. Section 3 discusses the necessary and sufficient condition for the S model and highlights the relationships between the goodness-of-fit test statistics for the S model and the partitioned models. Section 4 gives a numerical example. Section 5 summarizes this paper. Kateri and Papaioannou (1997) noted that the DPS[f ] model is the closest model to the S model in terms of the f -divergence under the conditions where j−i=k π ij (and i−j=k π ij ) for k=1, . . . , r − 1 as well as the sums π ij + π ji for i = 1, . . . , r; j = 1, . . . r, are given. For example, Ireland et al. (1969), Kateri and Agresti (2007), and Tahata (2020) mentioned a similar property for the symmetry (or asymmetry) model. This section derives the DPS[f ] model and describes its properties. Properties of the DPS[f ] model We can obtain the following theorem, although the proof of Theorem 1 is given in Appendix. Theorem 1. In the class of models with given i−j=k π ij , k = 0, and π ij + π ji (i = 1, . . . , r; j = 1, . . . , r), the model π ij = π S ij F −1 (∆ k + ζ ij ) (i = 1, . . . , r; j = 1, . . . , r) with k = i − j, ζ ij = ζ ji and ∆ k + ∆ −k = 0, is the model closest to the complete symmetry model in terms of the f -divergence. The DPS[f ] model can be expressed as F (2π c ij ) =    γ ij + a k (i < j), γ ij (i ≥ j),(3) where k = j − i, γ ij = γ ji and π c ij = π ij /(π ij + π ji ). Note that π c ij is the conditional probability that an observation falls in the (i, j)th cell for a condition where the observation falls in the (i, j)th cell or the (j, i)th cell. Namely, the DPS[f ] model indicates that F (2π c ij ) − F (2π c ji ) = a k (i < j).(4) When a 1 = · · · = a r−1 = 0, the DPS[f ] model is reduced to the S model. If f (x) = x log(x), x > 0, then the f -divergence is reduced to the KL divergence. When we set f (x) = x log(x), equation (3) is reduced to π ij =    π S ij exp(γ ij + a k − 1) (i < j), π S ij exp(γ ij − 1) (i ≥ j), where k = j − i and γ ij = γ ji . We shall refer to this model as the DPS KL model. Under the DPS KL model, the ratios of π ij and π ji for i < j are expressed as π ij π ji = d KL k (i < j),(5) where d KL k = exp(a k ) and k = j − i. Since equation (5) indicates that the ratio of π ij and π ji depends on the distance of k = j − i, the DPS KL model is equivalent to the DPS model proposed by Goodman (1979). Namely, the DPS model is the closest model to the S model in terms of the KL divergence under the conditions where i−j=k π ij , k = 0, and the sums π ij + π ji for i = 1, . . . , r; j = 1, . . . , r are given. This is a special case of Theorem 1. If f (x) = − log(x), x > 0, then the f -divergence is reduced to the reverse KL divergence. Then, the DPS[f ] model is reduced to π ij =        π S ij − 1 γ ij + a k (i < j), π S ij − 1 γ ij (i ≥ j), where k = j − i and γ ij = γ ji . We shall refer to this model as the DPS RKL model. This model is the closest to the S model when the divergence is measured by the reverse KL divergence and can be expressed as 1 π c ij − 1 π c ji = d RKL k (i < j), where d RKL k = −2a k and k = j − i. This model indicates that the difference between inverse probabilities 1/π c ij and 1/π c ji depends on the distance of k = j − i. If f (x) = (1 − x) 2 , then the f -divergence is reduced to the χ 2 -divergence (Pearsonian distance). Then, the DPS[f ] model is reduced to π ij =      π S ij γ ij + a k 2 + 1 (i < j), π S ij γ ij 2 + 1 (i ≥ j), where k = j − i and γ ij = γ ji . We shall refer to this model as the DPS P model. This model is the closest to the S model when the divergence is measured by the χ 2 -divergence and can be expressed as π c ij − π c ji = d P k (i < j), where d P k = a k /4 and k = j − i. This model indicates that the difference between π c ij and π c ji depends on the distance of k = j − i. Moreover, if f (x) = (λ(λ + 1)) −1 (x λ+1 − x), x > 0, where λ is a real-valued pa- rameter, then the f -divergence is reduced to the power-divergence (Read and Cressie 1988). Then, the DPS[f ] model is reduced to π ij =          π S ij λ (γ ij + a k ) + 1 λ + 1 1 λ (i < j), π S ij λγ ij + 1 λ + 1 1 λ (i ≥ j), where k = j − i and γ ij = γ ji . We shall refer to this model as the DPS PD(λ) model. This model is the closest to the S model when the divergence is measured by the power-divergence and can be expressed as (π c ij ) λ − (π c ji ) λ = d PD(λ) k (i < j), where d PD(λ) k = (λa k )/2 λ and k = j − i. This model indicates that the difference between the symmetric conditional probabilities to the power of λ depends on the distance of k = j − i. When we apply the DPS PD(λ) model, we should set the value of λ. Kateri and Papaioannou (1997) reported that the DPS[f ] model is equivalent to the DPS model regardless of f . That is, the all models described above (i.e., DPS KL , DPS RKL , DPS P , and DPS PD(λ) ) are equivalent to the DPS model surprisingly. However, the proof was not given. We prove the following theorem. Read (1977) and Tahata et al. (2016). We consider the distance global symmetry (DGS) model defined as δ U k = δ L k (k = 1, . . . , r − 1),(6) where Next, we consider the global symmetry (GS) model, which is defined as δ U k = j−i=k π ij , δ L k = i−j=k π ij . For k = 1, . . . , r − 1,i<j π ij = i<j π ji . It should be noted that the DGS model implies the GS model. Read (1977) noted that the S model holds if and only if both the CS model and the GS model hold. Fujisawa and Tahata (2020) G 2 (M) = 2 r i=1 r j=1 n ij log n iĵ m ij , with the corresponding degree of freedom (df). It is supposed that model M 3 holds if and only if both models M 1 and M 2 hold. For these three models, Aitchison (1962) discussed the properties of the Wald test statistics, and Darroch and Silvey (1963) described the properties of the likelihood ratio chi-square statistics. Assume that the following equivalence holds: T (M 3 ) = T (M 1 ) + T (M 2 ),(7) where T is the goodness of fit test statistic and the number of df for M 3 is equal to the sum of numbers of df for M 1 and M 2 . If both M 1 and M 2 are accepted with a high probability (at the α significance level), then M 3 is accepted. However, when               m ij = n U k n U k + n L k (n ij + n ji ) (i < j), m ij = n ij (i = j), m ij = n L k n U k + n L k (n ij + n ji ) (i > j),(8) where k = |j − i|, n U k = k=j−i n ij , and n L k = k=j−i n ji (Goodman 1979). Next, we consider the MLEs under the DGS model using the Lagrange function. Since the kernel of the log likelihood is r i=1 r j=1 n ij log π ij , Lagrange function L is written as L = r i=1 r j=1 n ij log π ij + λ r i=1 r j=1 π ij − 1 + r−1 k=1 λ k k=j−i (π ij − π ji ) . Equating the derivation of L to 0 with respect to π ij , λ, and λ k gives               m ij = (n U k + n L k )n ij 2n U k (i < j), m ij = n ij (i = j), m ij = (n U k + n L k )n ij 2n L k (i > j),(9) where k = |j − i|. We obtain the following equivalence from equations (8) and (9). Religious Medical Leaders Leaders Great (1) Fair (2) Little (3) None (4) Table 2 gives the values of the likelihood ratio chi-square statistics G 2 and p value for the models applied to these data. Table 2 indicates that the sum of the test statistics DPS (i.e., DPS[f ]) model and DGS model is equal to that of the S model. The S model fits the data very poorly. We can infer that the marginal distribution for religious leaders is not equal to that for medical leaders. On the other hand, the DPS model fits the data very well. Additionally, the DGS model fits the data poorly. From Theorem 3, the reason of the poor fit of S model is caused by the poor fit of DGS model rather than the DPS model. Let (i, j) denote the pair that the amount of influence religious leaders is ith level and that of medical leaders is jth level. When k = j − i (k = 1, 2, 3), a pair (i, j) isd k times as likely as a pair (j, i) on condition that a pair is (i, j) or (j, i). Fromd k < 1 (k = 1, 2, 3), the probability distribution for religious leaders is stochastically higher than the probability distribution of medical readers. That is, the medical leaders rather than the religious leaders should have influence in government funding for decisions on stem cell research. Moreover, from Theorem 2, we can obtain various interpretations. Since the DPS model holds, the DPS RKL , DPS P , and DPS PD(λ) models also hold. For example, we obtain When k = j − i (k = 1, 2, 3), we can infer that (i) the difference between the reciprocal of conditional probability that a pair is (i, j) and the reciprocal of conditional probability that a pair is (j, i) isd RKL k on condition that the pair is (i, j) or (j, i) from the DPS RKL model, (ii) the difference between the conditional probability that a pair is (i, j) and the conditional probability that a pair is (j, i) isd P k under the same condition from the DPS P model, and (iii) the difference between the conditional probability that a pair is (i, j) to the third power and the conditional probability that a pair is (j, i) to the third power isd Concluding remarks This paper provides the proof that the DPS[f ] model is equivalent to the DPS model proposed by Goodman (1979). This result provides the various interpretations of the DPS model. Additionally, the separation of the test statistic for the S model is considered. The DPS[f ] model and the DGS model is separable and exhibit independence. Kateri and Papaioannou (1997), Kateri and Agresti (2007), Tahata (2020) and Fujisawa and Tahata (2020) considered models based on the f -divergence for the analysis of square contingency tables with ordinal categories. In the future, whether the model based on the f -divergence is equivalent to the conventional model should be studied. Appendix This section provides the proofs of theorems. Proof of Theorem 1 We note that (π S ij ) satisfies the symmetry structure for cell probabilities and is given under the conditions where π ij + π ji = t ij = t ji (i = 1, . . . , r; j = 1, . . . , r) are given. Let I C (π : π S ) denote the f -divergence between (π ij ) and (π S ij ). That is I C (π : π S ) = r i=1 r j=1 π S ij f π ij π S ij ,(11) where f satisfies the conditions described in Section 1. Now minimize (11) under the conditions where the restraints (10) in addition to δ U −k = i−j=−k π ij and δ L k = i−j=k π ij (k = 1, . . . , r − 1)(12) are given. Note that (i) I C (π : π S ) is strictly convex and (ii) the restrictions are linear equations. The Lagrange function is written as L = I C (π : π S ) + r i=1 r j=1 λ ij (π ij + π ji − t ij ) + r−1 k=1 ∆ −k i−j=−k π ij − δ U −k +∆ k i−j=k π ij − δ L k . Equating derivation L to 0 with respect to π ij gives                    f ′ π ij π S ij +∆ −k + λ ij + λ ji = 0 (i < j), f ′ π ij π S ij + λ ij + λ ji = 0 (i = j), f ′ π ij π S ij +∆ k + λ ij + λ ji = 0 (i > j).(13) Let f ′ denote F , and let π * ij denote the solution satisfying (10), (12), and (13). Since f is a strictly convex function, it follows that F ′ (x) = f ′′ (x) > 0 for all x. Hence, F is strictly monotone and ensures that F −1 exists. Let −(λ ij + λ ji ) and −∆ l denote ζ ij and ∆ l , respectively. From equation (13), we obtain        π * ij = π S ij F −1 (∆ −k + ζ ij ) (i < j), π * ij = π S ij F −1 (ζ ij ) (i = j), π * ij = π S ij F −1 (∆ k + ζ ij ) (i > j), where ζ ij = ζ ji and ∆ k + ∆ −k = 0. The minimum value of I C (π : π S ) is attained for π * ij where ζ ij and ∆ l are determined so that π * ij satisfies restraints (10) and (12). Therefore, the DPS[f ] model is the closest model to the S model in terms of the f -divergence under these conditions. Proof of Theorem 2 Let function G be defined as G(x) = F 2x 1 + x − F 2 1 + x (x > 0), where F = f ′ . Then, the derivative of G is G ′ (x) = 2 (1 + x) 2 F ′ 2x 1 + x + F ′ 2 1 + x . Since the function f is twice-differential and strictly convex that G ′ (x) > 0 for x > 0. Hence, G is a strictly increasing function, and G −1 exists. If the DPS model holds, π ij /π ji = d k holds for i < j from equation (1), where k = j − i. Then we can see that for i < j, G(d k ) = F 2d k 1 + d k − F 2 1 + d k , = F 2π c ij − F 2π c ji . This is equivalent to equation (4). Namely, the DPS[f ] model holds. On the other hand, if the DPS[f ] model holds, equation (4) holds. We can see that for i < j, G π ij π ji = a k . Since G −1 exists, we obtain π ij π ji = G −1 (a k ) . Namely, the DPS model holds. The proof is complete. Proof of Theorem 3 It (d k − 1) π ji = 0 (k = 1, . . . , r − 1). Since π ji > 0, we get d k = 1 (k = 1, . . . , r − 1). Namely, the S model holds. Proof of Theorem 4 Theorem 2 shows that the DPS[f ] model is equivalent to the DPS model. Let π = (π 11 , . . . , π 1r , π 21 , . . . , π 2r , . . . , π r1 , . . . , π rr ) t , β = (ρ 1 , . . . , ρ r−1 , ε) t , where ε = (ε 11 , . . . , ε 1r , ε 22 , . . . , ε 2r , . . . , ε rr ). Then, from equation (1), the DPS model is expressed as log π = Xβ = (x 1 , . . . , x r−1 , x 11 , . . . , x 1r , x 22 , . . . , x 2r , . . . , x rr )β,(14) where x l = (w l+1 , . . . , w r , 0, . . . , 0) t is a r 2 × 1 vector (l = 1, . . . , r − 1). Here, w h (1 × r vector) is 1 for the hth element and 0 otherwise. For example, when r = 4, x 1 = (w 2 , w 3 , w 4 , 0, . . . , 0) t = (0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0) t . Additionally, x ij (i ≤ j) is the r 2 × 1 vector shouldering ε ij . Note that the r 2 × K matrix X is a full column rank where K = (r − 1) + r(r + 1)/2. We denote the linear space spanned by the column of matrix X by S(X) with dimension K. S(X) is a subspace of R r 2 . Let U be an r 2 × d 1 full column rank matrix such that the linear space S(U ) spanned by the column of U is the orthogonal complement of the space S(X). Note that d 1 = r 2 − ((r − 1) + r(r + 1)/2) = (r − 1)(r − 2)/2. Since U t X = O d 1 ,K where O d 1 ,K is the d 1 × K zero matrix, the DPS model can be expressed as h 1 (π) = U t log π = 0 d 1 , where 0 s is the s × 1 zero vector. Additionally, the DGS model can be expressed as h 2 (π) = M π = 0 d 2 where M = (g 1 , . . . , g r−1 ) t , and d 2 = r − 1. Here, g l = 2x l − j−i=l x ij . Note that M t belongs to the space S(X). That is, S(M t ) ⊂ S(X). Let p denote π with π ij replaced by p ij , where p ij = n ij /n with n = n ij . From Theorem 3, the S model is equivalent to h 3 (π) = 0 d 3 , where h 3 = (h t 1 , h t 2 ) t and d 3 = d 1 + d 2 = r(r − 1)/2. In an analogous manner to Tahata (2020), we obtain that √ n(h 3 (p) − h 3 (π)) has an asymptotically normal distribution with mean 0 d 3 and covariance matrix H 3 (π)Σ(π)H t 3 (π) = H 1 (π)Σ(π)H t 1 (π) O d 1 ,d 2 O d 2 ,d 1 H 2 (π)Σ(π)H t 2 (π) , where H s (π) = ∂h s (π)/π t and Σ(π) = diag(π) − ππ t . Here, diag(π) denotes a diagonal matrix with the ith component of π as the ith diagonal component. Therefore, W 3 = W 1 + W 2 holds, where W s = nh t s (p)(H s (p)Σ(p)H t s (p)) −1 h s (p). The Wald statistic for the DPS[f ] model (i.e., W (DP S[f ])) is W 1 , that for the DGS model (i.e., W (DGS)) is W 2 , and that for the S model (i.e., W (S)) is W 3 . The proof is complete. this model indicates that the sum of probabilities which are apart distance k = j − i from main diagonal cells is equal to the sum of probabilities which are apart distance k = i − j from main diagonal cells. We obtain the following theorem. (The proof is given in Appendix.) Theorem 3. The S model holds if and only if both the DPS[f ] model and the DGS model hold. proved that the S model holds if and only if the CS[f ] model and the GS model hold. These statements are the same as those from Corollary 1. This section proves the separation of the test statistics for the S model into those for the DPS[f ] model and the DGS model. Let n ij denote the observed frequency in the (i, j)th cell in the r × r square contingency table. Assume that a multinomial distribution applies to the r × r contingency table. Let m ij andm ij denote the expected frequency in the (i, j)th cell and the corresponding maximum likelihood estimate under a model, respectively. Each model can be tested for the goodness of fit by, for example, the likelihood ratio chi-square statistic of model M, which is given as G 2 ( 2S) = G 2 (DP S[f ]) + G 2 (DGS), because the MLEs under the S model arem ij = (n ij + n ji )/2. Therefore, the DPS[f ] model and the DGS model are separable and exhibit independence. Let W (M) denote the Wald statistic for model M. We obtain the following theorem and prove it in Appendix. Theorem 4 . 4W (S) is equal to the sum of W (DP S[f ]) and W (DGS). 1, which is taken fromSmith et al. (2006), describes the amount of influence religious leaders and medical leaders should have in government funding for decisions on stem cell research when surveying 871 people. The influence levels are divided into four categories: from the DPS PD(3) model. gorical data. Japanese Journal of Statistics and Data Science 3, 469-484. Tahata, K., Naganawa, M. and Tomizawa, S. (2016). Extended linear asymmetry model and separation of symmetry for square contingency tables. Journal of the Japan Statistical Society 46, 189-202. Tahata, K. and Tomizawa, S. (2011). Generalized linear asymmetry model and decomposition of symmetry for multiway contingency tables. Journal of Biometrics and Biostatistics 2, 1-6. ). They also noted that (i) the DPS model is the closest model to symmetry in terms of the Kullback-Leibler (KL) distance and (ii) the DPS[f ] model is equivalent to the DPS model. In this study, we derive the DPS[f ] model and the proof about the relation between the two models. We can obtain various interpretations of the DPS model from the result. Additionally, the necessary and sufficient condition for the S model and a property between test statistics for goodness of fit are discussed. Theorem 2 . 2The DPS[f ] model is equivalent to the DPS model regardless of f . poof is given in Appendix. Theorem 2 states that the DPS model holds if and only if the DPS[f ] model holds. That is, if the DPS model fits the given dataset, then we obtain various interpretations for the data.When a 1 = · · · = a r−1 , the DPS[f ] model is reduced to the conditional symme-try model based on the f -divergence (CS[f ]) model. The CS[f ] model is described previouslyKateri and Papaioannou (1997). Additionally,Fujisawa and Tahata (2020) proposed the generalization of CS[f ] model. Similarly, when d 1 = · · · = d r−1 , the DPS model is reduced to the conditional symmetry (CS) model proposed by McCullagh (1978). The CS[f ] model is equivalent to the CS model regardless of f (Kateri and Papaioannou 1997). Hence, Theorem 2 leads to the following corollary. Corollary 1. The CS[f ] model is equivalent to the CS model regardless of f . Here, the equivalence conditions of the S model are discussed. If the S model holds, then the DPS[f ] model with a 1 = · · · = a r−1 = 0 holds. Conversely, if the DPS[f ] model holds, then the S model does not hold generally. Therefore, we are interested in considering an additional condition to obtain the S model when the DPS[f ] model holds. Other studies have discussed such conditionsThe 3 Equivalence conditions for symmetry Table 1 : 1How much influence should religious leaders and medical leaders have in government funding for decisions on stem cell research?(Smith et al. 2006). MLEs under the DPS model b MLEs under the DGS modelTotal Great(1) 36 16 7 7 66 (36.00) a (11.96) (6.22) (7.00) (36.00) b (60.19) (70.95) (67.00) Fair(2) 74 96 22 4 196 (78.04) (96.00) (26.05) (4.78) (42.67) (96.00) (82.76) (40.55) Little(3) 119 174 48 4 345 (119.78) (169.95) (48.00) (3.99) (62.59) (100.34) (48.00) (15.05) None(4) 127 93 26 18 264 (127.00) (92.22) (26.01) (18.00) (67.00) (48.91) (14.99) (18.00) Total 356 379 103 33 871 a Table 2 : 2Likelihood ratio chi-square values G 2 for the models applied toTable 1.Models df G 2 p-value S 6 545.15 < 0.0001 DPS 3 2.45 0.4847 DGS 3 542.70 < 0.0001 The values of MLEs of (d 1 , d 2 , d 3 ) in equation (1) are (0.15, 0.05, 0.06). It should be noted that (d 1 , d 2 , d 3 ) is equal to (d KL 1 , d KL 2 , d KL 3 ) in the DPS KL model. is obvious that if the S model holds, the DPS[f ] model and the DGS model simultaneously hold. Assuming that both the DPS[f ] model and the DGS model hold, we show that the S model holds. From Theorem 2, the DPS[f ] model isequivalent to π ij /π ji = d k for i < j with k = j − i.Since the DGS model holds,we obtain j−i=k AcknowledgmentsThis work was supported by JSPS KAKENHI (Grant Number 20K03756). Large-sample restricted parametric tests. J Aitchison, Journal of the Royal Statistical Society Series B-Statistical Methodology. 24Aitchison, J. (1962). Large-sample restricted parametric tests. Journal of the Royal Statistical Society Series B-Statistical Methodology 24, 234-250. A test for symmetry in contingency tables. A H Bowker, Journal of the American Statistical Association. 43Bowker, A. H. (1948). A test for symmetry in contingency tables. Journal of the American Statistical Association 43, 572-574. On testing more than one hypothesis. J N Darroch, S D Silvey, The Annals of Mathematical Statistics. 34Darroch, J. N. and Silvey, S. D. (1963). On testing more than one hypothesis. The Annals of Mathematical Statistics 34, 555-567. Asymmetry model based on f -divergence and orthogonal decomposition of symmetry for square contingency tables with ordinal categories. K Fujisawa, K Tahata, SUT Journal of Mathematics. 56Fujisawa, K. and Tahata, K. (2020). Asymmetry model based on f -divergence and orthogonal decomposition of symmetry for square contingency tables with ordinal categories. SUT Journal of Mathematics 56, 39-53. Multiplicative models for square contingency tables with ordered categories. L A Goodman, Biometrika. 66Goodman, L. A. (1979). Multiplicative models for square contingency tables with ordered categories. Biometrika 66, 413-418. Symmetry and marginal homogeneity of an r × r contingency table. C T Ireland, H H Ku, S Kullback, Journal of the American Statistical Association. 64Ireland, C. T., Ku, H. H. and Kullback, S. (1969). Symmetry and marginal homogeneity of an r × r contingency table. Journal of the American Statistical Association 64, 1323-1341. A class of ordinal quasi-symmetry models for square contingency tables. M Kateri, A Agresti, Statistics and Probability Letters. 77Kateri, M. and Agresti, A. (2007). A class of ordinal quasi-symmetry models for square contingency tables. Statistics and Probability Letters 77, 598-603. Asymmetry models for contingency tables. M Kateri, T Papaioannou, Journal of the American Statistical Association. 92Kateri, M. and Papaioannou, T. (1997). Asymmetry models for contingency tables. Journal of the American Statistical Association 92, 1124-1131. A class of parametric models for the analysis of square contingency tables with ordered categories. P Mccullagh, Biometrika. 65McCullagh, P. (1978). A class of parametric models for the analysis of square contingency tables with ordered categories. Biometrika 65, 413-418. Partitioning chi-squape in contingency tables: A teaching approach. C B Read, Communications in Statistics -Theory and Methods. 6Read, C. B. (1977). Partitioning chi-squape in contingency tables: A teaching approach. Communications in Statistics -Theory and Methods 6, 553-562. Goodness-of-Fit Statistics for Discrete Multivariate Data. C B Read, N Cressie, Springer-VerlagNew YorkRead, C. B. and Cressie, N. (1988). Goodness-of-Fit Statistics for Discrete Multi- variate Data. Springer-Verlag, New York. General Social Surveys, 1972-2014 [machine-readable data file] /Principal Investigator, Tom W. Smith; Co-Principal Investigator. T W Smith, P Marsden, M Hout, J Kim, Michael Hout; Sponsored by National Science Foundation. -NORC. Peter V. MarsdenCo-Principal InvestigatorNORC at the University of Chicagoproducer and distributorSmith, T. W., Marsden, P., Hout, M. and Kim, J. (2006). General Social Sur- veys, 1972-2014 [machine-readable data file] /Principal Investigator, Tom W. Smith; Co-Principal Investigator, Peter V. Marsden; Co-Principal Investigator, Michael Hout; Sponsored by National Science Foundation. -NORC ed.-Chicago: NORC at the University of Chicago [producer and distributor].
[]
[ "Quantum learning machines", "Quantum learning machines" ]
[ "Gerard Milburn \nSchool of Mathematics and Physics\nARC Centre of Excellence for Engineered Quantum Systems\nThe University of Queensland\n4072BrisbaneAustralia\n" ]
[ "School of Mathematics and Physics\nARC Centre of Excellence for Engineered Quantum Systems\nThe University of Queensland\n4072BrisbaneAustralia" ]
[]
Physical learning machines, be they classical or quantum, are necessarily dissipative systems.The rate of energy dissipation decreases as the learning error rate decreases linking thermodynamic efficiency and learning efficiency. In the classical case the energy is dissipated as heat. We give an example based on a quantum optical perceptron where the energy is dissipated as spontaneous emission. At optical frequencies the temperature is effectively zero so this perceptron is as efficient as it is possible to get. The example illustrates a general point: In a classical learning machine, measurement is taken to reveal objective facts about the world. In quantum learning machines what is learned is defined by the nature of the measurement itself.
null
[ "https://export.arxiv.org/pdf/2305.07801v1.pdf" ]
258,686,412
2305.07801
d7abbcb46ec9bc80b962b71525ed1b1445f0aab6
Quantum learning machines 12 May 2023 Gerard Milburn School of Mathematics and Physics ARC Centre of Excellence for Engineered Quantum Systems The University of Queensland 4072BrisbaneAustralia Quantum learning machines 12 May 2023(Dated: May 16, 2023)1 Physical learning machines, be they classical or quantum, are necessarily dissipative systems.The rate of energy dissipation decreases as the learning error rate decreases linking thermodynamic efficiency and learning efficiency. In the classical case the energy is dissipated as heat. We give an example based on a quantum optical perceptron where the energy is dissipated as spontaneous emission. At optical frequencies the temperature is effectively zero so this perceptron is as efficient as it is possible to get. The example illustrates a general point: In a classical learning machine, measurement is taken to reveal objective facts about the world. In quantum learning machines what is learned is defined by the nature of the measurement itself. I. INTRODUCTION Biology offers abundant evidence that physical systems can learn, that is to say, physical systems can exhibit stable behaviour, conditioned on prior interactions with an external environment, in order to achieve a goal. The existence of a goal does not imply teleology. A goal can arise spontaneously: In biological systems the goal is to survive long enough to reproduce. We are entering an era in which learning machines can be engineered. In which case, what are the physical principles in play? I want to make a distinction between learning algorithms and learning machines. Modern machine learning refers to a class of algorithms running on digital CMOS chips. On the other hand, a learning machine could be instantiated in any physical system and not necessarily digital. Biological learning is not based on algorithms running on digital computers [1], even if it can be simulated that way. In this this paper I discuss the physical principles required for machines to learn. A learning machine is an open, dissipative physical system driven far from thermal equilibrium by access to a low entropy source of energy, for example, a battery. I will focus on simple classification in supervised learning. Here the objective is to learn a binary valued function, f (x) of the input data x by giving the machine a list of examples (x, f (x) and adjusting the parameters of the machine through feedback so that the actual outputf (x) has a low probability of error:f (x) = f (x). The goal is do this while making efficient use * [email protected] 2 of the available thermodynamic resources. I am interested in quantum machines operating at very low temperature, in which case the goal is to learn in the presence of unavoidable quantum noise. As will become clear, noise is essential for learning, so we need to ask how quantum noise might be harnessed for efficient learning. II. PHYSICAL PERCEPTRONS. Neural network machine learning algorithms are based on the concatenation of elementary non linear functions called the activation functions. The preceptron function is an example [2]. In the simplest examples, the input to the function is a linear combination of data elements and the coefficients are called weights. The output is a single real number, y. The learning algorithm consists in evaluating the function over many examples of input data, labelled with a true output value y. The actual outputŷ is compared to the true value and the weights adjusted for the next iteration in such a way (back propagation) as to cause the actual value to equal the true value most of the time. The key point is that as the network learns, the distribution over weights change such that it becomes concentrated on particular points in weight-space with small fluctuations over different training episodes. The algorithm is typically implemented on a standard CMOS processor. In a physical perceptron, the activation function is replaced by a physical device that implements a noisy switch and the concatenation of functions is replaced by a physical network of activation switches using feed-forward signals from the output of one layer of switches to the next. Learning consists in an actual feedback of signals structured to emulate back propagation. In this picture learning is best regarded as cooling in weight space [3]. The physical perceptron is a dissipative non linear dynamical system driven far from thermal equilibrium and subject to noise. It is widely acknowledged that neural network algorithms running on standard CMOS devices consume large amounts of power both for learning and for inference [4]. It is hoped that physical neural networks, based on physical perceptrons, could be more thermodynamically efficient. After all, the human brain is some kind of physical learning machine and consumes a tiny amount of power by comparison with deep learning technology. The noise in an activation switch is due to thermal fluctuations that necessarily accompany the dissipation required for the switch to function (the fluctuation-dissipation theorem [5]). In an elementary switching event the energy dissipated must be proportional to k B T where k B is Boltzmann's constant and T , the ambient temperature. The power dissipated will depend on the product of the switching rate and the local temperature. To see this consider a simple a particle moving in a symmetric double well potential with high friction, see Fig. (1). Applying a time varying bias force will make the double well asymmetric such that one well is deeper than the other. As the bias is slowly increased from zero, work is done on the system. At some value of the bias, and the work done, the thermal noise will cause the particle to overcome the potential barrier between the wells and jump to the lower well. As this is a stochastic process, the work done in that trail, W , is also a random variable. Using the Jarzynski equality [6] we can relate the average of W over many trials to the change in free energy of the equilibrium steady state distributions between the starting and stopping times of the bias variation, e −W/k B T = e −∆F/k B T(1) where ∆F is the change in the free energy between the initial and final equilibrium states. The work done is dissipated as heat after the transition as the particle moves to the new local equilibrium state. Heat δQ flows into the environment due to friction as the particle moves to bottom of the deeper well. The energy lost due to heat is proportional to k B T and one can reduce this cost by reducing the temperature. The problem is that, at low temperatures, the switching rate also goes to zero. This is a consequence of Kramer's theory of first passage times [5]. This would result in the learning rate of a perceptron network going to zero. If we coarse-grain the output observable to be simply the sign of the output, the stochastic dynamics can be approximated by a two-state markov process [3]. Let the state be specified by n = +1 if the particle is localised on the right and n = −1 if the particle is localised on the left. A simple learning model based on a single double well perceptron can now be specified. Suppose we want to train a perceptron to implement a one bit Boolean function. There are four such functions: two produce a constant output regardless of the input, one simply copies the input to output and the other inverts the input to produce the output. This last one is the NOT gate: y out = y in ⊕ 1 where the binary operation is addition mod-2. The input y ∈ 0, 1 is encoded in a linear bias potential with slope x = 2y − 1. The weights correspond to another a linear bias potential of arbitrary slope w a real number. The total bias acting on the particle in the double well is then given by A(w, x) = x.w. We set up the double well such that if A > 0 the particle has a higher probability to transition to the right well, while if A < 0 the particle has a higher probability to transition to the left well. Work is done and heat dissipated each time the bias potential changes, but each of these is a random variable as sometimes the transition does not take place. Typically the conditional probability to make the transition is given by thermal activation so that p(w) = (1 + e −βA(w) ) −1(2) where β = (k B T ) −1 with k B Boltzmann's constant and T the temperature of the environment. This is a sigmoidal function of the weight. The training set is a large number of ordered pairs of the form x, n T drawn at random from {(−1, 1), (1, −1)}. Initially the weights w are drawn from a Gaussian distribution with mean and variance σ. In each single trial we compare the actual output of the switch n to the label n T for that data point and compute the error = (n − n T ) 2 /4. We now define an epoch as a large number of runs using the same training data point and compute the average error¯ and average outputn over the epoch. This is equivalent to sampling the switching probability. It is easy to see that¯ (w) = 1 2 (1 − n Tn (w)) (3) n(w) = 2p(w) − 1.(4) We now apply a feedback to change the weights for the next epoch. This is done in such a way as to decrease the average error and is known as gradient descent. The feedback rule is [3] ∆w = βn T x(1 −n 2 (w))/4. for input x, n T at that epoch. The corresponding change in the average error is ∆¯ (w) = −β 2 (1 −n 2 (w)) 2 /16.(6) In the case of the NOT gate n T x < 0 for every training datum so that the weights always decrease. In this simple example this means if the weight w is initially chosen as negative, the learning is very fast. If we focus on the stochastic dynamics of the weight w we see that it is highly nonlinear process in which the initial broad Gaussian distribution, centered on w = 0, converges to a much more narrow distribution centered on w < 0. In physical terms this is cooling via feedback and is a general feature of all physical learning machines. The process of learning decreases the entropy of the machine while increasing the entropy of the environment by heat dissipation. It can only occur if the learning machine is driven to a non-thermal equilibrium steady state through access to a low entropy source of energy. This has very important consequences for our understanding of how learning machines arise in biological systems through evolution [3]. The evolution of biological systems that learn is favoured by the laws of physics when access to thermodynamic resources determines survival. At very low temperatures the Kramer's rate formula fails, but dissipative quantum tunnelling, also called quantum activation [7], can drive the transitions in an activation switch in a bistable system. Learning machines will need to use quantum switching if they are to make the most of thermodynamic resources. We will discuss a different example drawn from quantum optics in which quantum noise in the form of spontaneous emission drives the transition. III. A QUANTUM OPTICAL PERCEPTRON AT ZERO TEMPERATURE. As an example of a quantum activation switch, and a quantum perceptron, we will consider Raman single-photon sources (RSPS) and Raman single-photon detectors (RSPD) [8]. The ideal RSPS produces a single photon pulse with a controllable temporal mode shape. The ideal RSPD is a single-photon detector which projects a single-photon pulse onto a re-configurable temporal mode. A classical write pulse (RSPS) and a classical read pulse (RSPD) are used to control temporal mode shapes. This level of control can be used to encode information in a single photon and change the weights for a single photon detection. In normal operation, the output of the RSPD is a single binary number corresponding to the absorption of the single photon or reflection of the single photon from the detector. The latter event signals an error and can be detected using a standard single photon counter and the record is a classical random variable, as is the case of a classical activation switch. However it would be better if the output of a quantum perception was the same kind of quantum object as the input, i.e. a temporally encoded single photon. In that way an all-optical feed forward neural network could be constructed. This is easily achieved. Once the write field has effectively stored a photon of an unknown temporal shape, the electronic state of the RSPD can be swapped turning it into a RSPS. Now the control field will produce a single-photon pulse with the same shape as the learned shape, with high probability. The scheme is indicated in Fig. (2). This is fine for a single perceptron, but in a neural network we need to consider the case of multi-photon inputs to the Raman memory. That is beyond the scope of this paper. A single atom in an optical cavity, with a time dependent pump tuned to a Raman resonance, can be configured to realise a single-photon source or detector depending on the initial state of the atom [9]. Let the frequency of the optical cavity be ω a and the frequency of the lower transition be ω b . We then take the carrier frequency of the classical control field as ω c = ω a − ω b as the Raman resonance condition. This enables a single photon source or detector depending on how the Raman system is prepared, see Fig. (3). We will assume that the cavity mode is heavily damped so that the single-photon emission/absorption is fast. This means we can adiabatically eliminate the cavity field dynamics. In the case of a RSPS we prepare the atom in the ground state g. The output mean photon emission rate from the cavity in the adiabatic limit is [10] a † out a out = |E w (t)| 2 |E w (t)| 2 has units of s −1 , so that this is the intensity of the external control field. The cavity emits a single photon excitation of the vacuum of the from |ν(t) = ∞ −∞ dων(ω)a † (ω)|0(8) where a(ω), a † (ω) are the usual bosonic annihilation and creation operators. This is a coherent superposition of a single excitation over many frequency modes weighted with the complex amplitude functionν(ω). In an ideal RSPS, ν(t) = E w (t) is the Fourier transform ofν(ω). Thus we can control the temporal state of the emitted photon by an appropriation modulation of the control field. In the case of a RSPD, we prepare the atom in the ground state g and an itinerant single-photon pulse is absorbed by the cavity if the control field, the read-field, has the right temporal shape. If the photon is not absorbed, it is reflected from the cavity. We design the read-field to optimally absorb a single photon pulse into the atom-cavity system from the external field. If the input field of the cavity is prepared in a single-photon pure state with temporal amplitude ν(t), and the carrier frequency equal to the cavity frequency, the probability for absorption is simply the probability for the atom to make a transition g → e. This is [10] p A (t) = t −∞ dt E * r (t)ν(t )e −(τ (t)−τ (t )/2 2(9) where we define a dimensionless time variable, τ through the change of variable τ (t) = t −∞ dt |E r (t )| 2(10) and E r (t) is the read-field amplitude. This clearly depends on the time over which the readfield E r (t) is non zero. This is the detection time. In the long-time limit p 1 (t) tends to a constant less than unity. As an example we will encode the input y in a pulse-code single-photon state. With respect to a reference time t = 0, if y = 0, the pulse intensity is a maximum for t < 0 while for y = 1 the pulse intensity is a maximum for t > 0 see Fig.(4). The NOT gate corresponds to reversing the times of the peak intensity. ' The modulation required in the single-photon source is defined in terms of a superposition of Hermite Gaussian modes [11], ν y (t) = 1 √ 2 (u 0 (t) + xu 1 (t))(11) where we define x = 2y − 1. The probability for the perceptron to switch is given by Eq. (9) with the read-field at the detector defined by E r (w, t) = 1 √ 2 (u 0 (t) + e iπA(w)/2 u 1 (t))(12) where w ∈ R and the activation is A(w) = xw − 1. As we approach w = −1 the device will approximate the NOT function. In that case, if x = −1 then A(w) = 0 and the output photon has mode function ν 1 (t). If x = +1 then A(w) = 1 the output photon state is ν 0 (t). IV. CONCLUSION Both classical and quantum learning machines are irreversible dissipative devices. In the classical case the energy dissipated is called heat. In the quantum case it is simply spontaneous emission from a bosonic oscillator. In both cases, as the machine learns, the energy dissipated per trial decreases as the error decreases. In the photonic example, the energy is dissipated by photons lost from the system. The loss of energy is one quanta of photonic energy and does not depend on temperature. As the system learns, fewer and fewer trials lead to the relfection of the photon from the detector and the average energy lost per trial falls to a minimum. This is probably as energy efficient as it is possible to get at room temperature. In a full quantum photonic learning machine, there are as many photons as their are input channels however the photons become redistributed across the network in a coherent way. This is reminiscent of boson sampling where the Raman sources and detectors are simply replaced by beam-splitters. The optical scheme we have described is a generalisation of boson sampling to a photonic quantum neural network. More work is needed to determine if this will provide a quantum advantage beyond the improvements in thermodynamic efficiency. The nature of the Raman detection scheme determines what is measured and plays an essential role in what is learned. Suppose we had used a conventional photon counter [12]. In any trial, one and only one photon is counted and the detection times are distributed according to |ν(t)| 2 . What we really want to measure is the temporal mode shape of the photon ν(t), in general this is a complex valued function of time. This is precisely what Eq. (9) depends on. The Raman scheme enables one to sample the probability distribution that a single photon is in a chosen temporal mode function. In effect, what we require is a projection valued measure onto a class of single-photon states P µ (t)dt = |µ(t) µ(t)|dt such that the probability that a single photon in state |ν(t) will be found in state |µ(t) is P (µ|ν) = ∞ −∞ dt µ(t)|ν(t) 2(13) This is what the Raman detection scheme approximates. By variation over the class |µ(t) we can learn the temporal mode shape of a photon. In a classical learning machine measurement simply reveals objective facts about the world. In quantum learning machines what is learned is defined by the nature of the measurement itself. This is a restatement of Bohr's complementarity for quantum learning machines. 11 FIG. 1 . 1A physical activation switch modelled as a particle moving in a double well potential with very high friction. On the left the switch is in the off configuration. As a linear potential is applied, a bias force does work on W the particle until, at some random value for W , it undergoes a thermally activated transition to the other well and moves under friction to the new local equilibrium. It is now in the on configuration. FIG. 2 . 2Data is encoded into the temporal shape of a Raman single-photon source (RSPS) using data to amplitude modulate the write-field. The photon is injected into a Raman single-photon detector (RSPD) using a read-field that is controlled by feedback amplitude modulation. This determines the weights used by the preceptron. When the read-field matches the write-field the photon is absorbed with high probability in the RSPD, otherwise it is reflected and lost. Absorption is by photo counting. This resulting classical signal is used to control the weights in the write field until the error is a minimum. Then it is used to switch the RSPD to a RSPS where upon the write field generates a photon closely matching the data encoded photon. FIG. 3 . 3A representation of the Raman transition for a single-photon source (left) and singlephoton detector (right). An atomic transition is coupled to an optical cavity field (frequency ω a ) by a Raman process in which a classical control field is tuned to the frequency difference of the atomic system and the cavity, ω c = ω a − ω b . The classical control field amplitude is E j (t) where j = 1 corresponds to the write-field, and j = 2 corresponds to the read-field. In the case of the source (left), the atom is prepared in the excited state. In the case of a detector, (right) the atom is prepared in the ground state. The atomic state labelled p is a probe state to enable a dispersive read-out of the occupation of states g, e. FIG. 4 . 4Pulse-code modulation encoding of a single binary variable in a single-photon state. The perceptron learning a NOT gate is trained to reverse the temporal order of the pulses. D Purves, Why Brains Don't Compute. SpringerD. Purves, Why Brains Don't Compute (Springer, 2021). S Russell, P Norvig, Artificial Intelligence: A Modern Approach. PearsonS. Russell and P. Norvig., Artificial Intelligence: A Modern Approach (Pearson, 2010). . G J Milburn, S Basiri-Esfahani, Contemporary Physics. 6334G. J. Milburn and S. Basiri-Esfahani, Contemporary Physics 63, 34 (2022). R Chojnacka, Towards Data Science. R. Chojnacka, Towards Data Science (2021). C Gardiner, Stochastic Methods. Berlin HeidelbergSpringerC. Gardiner, Stochastic Methods (Springer Berlin Heidelberg, 2009). . C Jarzynski, 10.1103/PhysRevLett.78.2690Phys. Rev. Lett. 782690C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). . M I Dykman, Phys. Rev. E. 7511101M. I. Dykman, Phys. Rev. E 75, 011101 (2007). . M Kewming, S Shrapnel, G Milburn, Phys. Rev. A. 10332411M. Kewming, S. Shrapnel, and G. Milburn, Phys. Rev. A 103, 032411 (2021). . D F V James, P G Kwiat, 10.1103/PhysRevLett.89.183601Phys. Rev. Lett. 89183601D. F. V. James and P. G. Kwiat, Phys. Rev. Lett. 89, 183601 (2002). . R Marshman, M Ghadimi, G J Milburn, S Shrapnel, R. Marshman, M. Ghadimi, G. J. Milburn, and S. Shrapnel, xxx xx, xxx (2023). . M Raymer, I Walmsley, Physica Scripta. 9564002M. Raymer and I. Walmsley, Physica Scripta 95, 064002 (2020). D Walls, G Milburn, Quantum Optics. Berlin; HeidelbergSpringerD. Walls and G. Milburn, Quantum Optics (Springer, Springer Berlin, Heidelberg, 2008).
[]
[ "Semantic Communication of Learnable Concepts", "Semantic Communication of Learnable Concepts" ]
[ "Francesco Pase [email protected] ", "Szymon Kobus [email protected] \nImperial College London\nLondonUK\n", "Deniz Gündüz [email protected] \nImperial College London\nLondonUK\n", "Michele Zorzi [email protected] ", "\nUniversity of Padova\nItaly\n" ]
[ "Imperial College London\nLondonUK", "Imperial College London\nLondonUK", "University of Padova\nItaly" ]
[]
We consider the problem of communicating a sequence of concepts, i.e., unknown and potentially stochastic maps, which can be observed only through examples, i.e., the mapping rules are unknown. The transmitter applies a learning algorithm to the available examples, and extracts knowledge from the data by optimizing a probability distribution over a set of models, i.e., known functions, which can better describe the observed data, and so potentially the underlying concepts. The transmitter then needs to communicate the learned models to a remote receiver through a rate-limited channel, to allow the receiver to decode the models that can describe the underlying sampled concepts as accurately as possible in their semantic space. After motivating our analysis, we propose the formal problem of communicating concepts, and provide its rate-distortion characterization, pointing out its connection with the concepts of empirical and strong coordination in a network. We also provide a bound for the distortion-rate function.
10.48550/arxiv.2305.08126
[ "https://export.arxiv.org/pdf/2305.08126v1.pdf" ]
258,686,418
2305.08126
292e46b6db09761ecc31465ea5d67e5c1855dea8
Semantic Communication of Learnable Concepts Francesco Pase [email protected] Szymon Kobus [email protected] Imperial College London LondonUK Deniz Gündüz [email protected] Imperial College London LondonUK Michele Zorzi [email protected] University of Padova Italy Semantic Communication of Learnable Concepts This paper has been accepted for presentation at the We consider the problem of communicating a sequence of concepts, i.e., unknown and potentially stochastic maps, which can be observed only through examples, i.e., the mapping rules are unknown. The transmitter applies a learning algorithm to the available examples, and extracts knowledge from the data by optimizing a probability distribution over a set of models, i.e., known functions, which can better describe the observed data, and so potentially the underlying concepts. The transmitter then needs to communicate the learned models to a remote receiver through a rate-limited channel, to allow the receiver to decode the models that can describe the underlying sampled concepts as accurately as possible in their semantic space. After motivating our analysis, we propose the formal problem of communicating concepts, and provide its rate-distortion characterization, pointing out its connection with the concepts of empirical and strong coordination in a network. We also provide a bound for the distortion-rate function. I. INTRODUCTION AND MOTIVATION With the growing number of mobile devices and sensors, massive amounts of data are collected today at the edge of communication networks. On the one hand, this data is the fuel for training large learning models like deep neural networks (DNNs); on the other hand, these models need to be stored, compressed, and communicated over bandwidth limited channels to the cloud, and protected against security and privacy risks [1]. These issues are increasingly limiting the application of typical centralized training approaches. Various federated/distributed learning paradigms have emerged as potential solutions to mitigate these limitations, which allow the models to be locally trained, and then aggregated in a cloud or edge server without moving local private data [2]. The main paradigm shift in distributed learning is to move the models, rather than the data, throughout the network, providing better privacy guarantees and reducing the communication load. However, today even the sizes of such learned models are becoming a concern, as transmitting huge models back and forth for training or inference purposes can easily congest wireless networks, specifically when considering that the edge devices like mobile phones, cars, robots etc., are usually wirelessly connected to the network, and thus have limited bandwidth [3]. Consequently, it is time to investigate, with the proper information-theoretic models and tools, the fundamental limits This work received funding from the UKRI (EP/X030806/1) for the project AIR (ERC-CoG). For the purpose of open access, the authors have applied a Creative Commons Attribution (CCBY) license to any Author Accepted Manuscript version arising from this submission. of communicating models over rate-limited channels, and not just raw data. To this end semantic communications, which concerns with the semantic aspect of the message, maps naturally to the transmission of these learning models [4]. The communication fidelity of these models can be judged by how close the behavior of the reconstructed model at the receiver is to the desired one, rather than by the accuracy of the reconstruction in the parameter space [3]. C 1 , . . . , C n ∼ P C S i = {z j } m j=1 P (h|S i ) = A(S i ) n i=1 m = {1, . . . , 2 nR } {Q i (h)} n i=1 p h (Y |X)h ∼ Q 1 (h), X ∼ C 1 . . . p h (Y |X)h ∼ Q n (h), X ∼ C n II. RELATED WORK The goal of this paper is to introduce the problem of communicating concepts, which is translated to that of conveying functions that better approximate them by learning their semantic aspects from data. The closest reference to this work is [5], in which the goal is to compress neural networks by applying bits back coding [6] to represent probability distributions over models with the minimum number of bits. The problem studied in [5] is the single-shot version of our problem, focusing on the design of a practical coding scheme, which is called MIRACLE, to efficiently compress neural networks. In [7], the authors study the connections between compressibility and learnability in the context of probably approximately correct (PAC) learning, and show that the two concepts are equivalent when zero/one loss is considered, but not in the case of general loss functions. Another line of research investigates the connections between the generalization capabilities of learning algorithms, and the mutual information between the data and the model [8]- [10]. The logic behind these results is to provide a bound on the generalization gap, i.e., the difference between the expected error and the training one, given some information-theoretic properties of the learning algorithm. However, if the environment imposes a constraint on such quantities, e.g., mutual information between the input and output of the learning rule, for example by introducing a rate-limited communication channel between the data and the final model, this influences not only the generalization gap, but also the training error itself (the output of the learning rule is constrained by the environment now), and so it is not clear how the gap between the best achievable test error changes as a function of the mutual information. This is the scenario studied in this work, where the constraint on the mutual information is not a property of the learning rule, but rather a physical limit imposed by the system. In [11], a similar study is performed on the specific case of contextual multi-armed bandits, where the fundamental quantity is the mutual information I(S; A) between the system states and the action taken by the agents, which is a property of the specific policy adopted. This work generalizes that idea to the supervised learning framework, and considers the effect of the communication rate R on the final performance. It is also interesting to highlight the connections between this work and the study in [12], where the authors quantify the complexity of a learning algorithm output Q with its Kullback-Leibler divergence from a prior model distribution P , which, in our system model, represents the minimum achievable rate to convey Q, when P is set as the prior distribution. To conclude, this work is partially built on top of the results in [13]- [17], which generalize the concept of rate-distortion theory [18] for standard data communication to probability distributions, where the fidelity requirement at the receiver is not to exactly reconstruct the input data, but rather to generate samples according to some input distribution. Indeed, here the semantic aspect of communication is captured by the fact that there is no need to convey the exact dataset sampled by the transmitter, but rather to represent with high fidelity the belief on the underlying concepts acquired after observing the data, which is the post-data probability distribution over the class of feasible models. III. SYSTEM MODEL Let E denote the environment, i.e., the source, that generates a sequence of n concepts, e.g., tasks, {c i } n i=1 , c i ∈ C, sampled with probability P C in an independent and identically distributed (i.i.d.) fashion. While P C is known by both Alice and Bob, neither of them can observe the sampled concepts directly. Alice has access to a sequence of m samples {z i,j } m j=1 , where z i,j = (x i,j , y i,j ) ∈ Z, sampled according to each of the concept distributions p ci (Y |X)p ci (X), ∀i = 1, . . . , n. Alice and Bob agree on a hypothesis class, i.e., the model class H, and on a pre-data coding probability distribution P h , ∀h ∈ H. We call the sequence {z i,j } m j=1 of samples the dataset s i . Alice applies a learning algorithm A : Z m → Φ(H) on s i , which is a possibly stochastic function mapping a dataset to a probability distribution Q h|si = A(s i ) over the set of models H, and so, over the subset of all possible probability mappings h : X → Φ(Y), representing Alice's concept belief. With Φ(X ) we denote the set of all possible probability distributions over the set X . Consequently, the models are functions used by Alice and Bob to represent (or, more precisely, to approximate) the concepts' relation among data. To measure how well a model h approximates a concept c, a per-sample loss c (h, z) : H×Z → [0, 1] is defined, which compares the discrepancy between p c (y|x) and h(x). In this work, we assume a bounded loss within [0, 1] for the sake of clarity of exposition. Then, c (Q, z) : Φ(H) × Z → [0, 1] is the performance of the model belief Q, which is defined as c (Q, z) = E h∼Q [ c (h, z)]. Upon observing the data, Alice can compute her empirical performance by using the empirical loss on her dataset S as L C (Q, S) = 1 |S| z∈S C (Q, z),(1) where Q = Q H|S = A(S) is the post-data distribution inferred by Alice, given the data. We assume that, for any sequences of datasets s n , s n , Alice's distribution can be factorized as Q n h n |s n = n i=1 Q hi|si such that s i = s j ⇒ Q hi|si = Q hj |s j . However, to assess how well the belief Q represents the concept C, in machine learning we are usually interested in the true loss L C (Q) = E Z∼C [ C (Q, Z)],(2) i.e., the expected performance on a new unseen sample. Given the realization of the datasets {s i } n i=1 , the problem for Alice is then to convey a message to Bob through a constrained communication channel, which limits the maximum number of bits she can convey per model, so that Bob can use the received information to reconstruct models {ĥ i } n i=1 that can approximate the concepts {c i } n i=1 by minimizing the loss on random samples {z i } n i=1 distributed according to the sequence of concepts, i.e., the true loss in Equation (2). We can observe that the task for Bob is not to exactly reconstruct the sequence {h i } n i=1 sampled by Alice, but rather to obtain samples {ĥ i } n i=1 whose probability distributions are close to the target ones, i.e., {Q hi|si } n i=1 . Remark. We now briefly discuss why we are interested in learning rules A(S) that output model distributions, rather than single-point solutions: • First of all, the case in which Alice finds a point-wise estimate of the best model h * is included as a special case Q h|S = δ h * . • Alice may want to express her uncertainty around the best choice h * , which may be intrinsic in the learning algorithm A, through the distribution Q h|S . • Usually, optimization algorithms used to train DNNs, like stochastic gradient descent (SGD), are stochastic algorithms. • When H is the set of all DNNs h ω with a specific architecture parameterized by the parameter vector ω, there exist many vectors ω performing in the same way. Moreover, small perturbations to the parameters usually does not reduce the final performance. This means that it is not required for Bob to reconstruct the exact value of the network parameters, but rather a nearby or an equivalent solution, and this variability is represented by Q hω|S . More importantly, Q hω|S can be exploited to reduce the rate needed to convey the models, thus saving network resources [5]. This is the semantic aspect of communication captured by our framework, as the meaning of a concept c, i.e., the real unknown mapping, is conveyed through the model belief Q h|S , whose loss expressed in Equation (2) quantifies its fidelity with respect to the real concept c. IV. THE RATE-DISTORTION CHARACTERIZATION In this section, we first characterize the limit of the problem when n = 1, i.e., one-shot concept communication, and then generalize the problem to the n-sequence formulation. For the latter, two kinds of performance metrics are defined: the first one provides average performance guarantees, while the second one ensures the same performance guarantee for each sampleĥ. We will show that the minimum achievable communication rate that can guarantee a certain distortion level is the same in both cases, as long as sufficient common randomness between Alice and Bob is available. A. Single-Shot Problem The single-shot version of the problem has been studied in [5], where the authors propose MIRACLE, a neural network compression framework based on bits back coding [6], providing an efficient single-shot model compression scheme showing empirically that, with enough common randomness, it is possible to convey the model with an average of K D KL (Q||P ) bits with very good performance, where P is the pre-data coding model distribution, and Q is the optimized post-data distribution, providing a belief over well-performing neural networks, or, equivalently, over a set of parameter vectors, as explained in Section III. However, from Lemma 1.5 in [15], the average number of bits E C,S [K] needed to exactly code Q with P , when sufficient common randomness is available, can be bounded by R ≤ E C,S [K] ≤ R+2 log(R + 1) + O(1),(3) where R = E C,S [D KL (Q||P )], while using exactly R bits may lead to samples which are distributed according toQ, slightly different from the target Q [15]. More recent results [16] allow to find even stronger guarantees (Corollary 3.4 in [17]) for this relationship: E C,S [K] ≤ R + log(R + 1) + 4.(4) B. n-Length Formulation We now study the problem depicted in Figure 1, when we let Alice code a sequence of n concept realizations, i.e., datasets, and study the information-theoretic limit of the system as n → ∞. Specifically, we are interested in the tradeoff between the rate R, which is defined as the average number of bits consumed per model by Alice to convey the concept process to Bob, and the performance, which is the true loss that can be obtained by Bob (see Eq. (2)). We start by defining the proper quantities involved. Definition IV.1 (Rate-Distortion Coding Scheme). A (2 nR , n) coding scheme consists of an alphabet X , a reconstruction alphabetX , an encoding function f n : X n → {1, 2, . . . , 2 nR }, a decoding function g n : {1, 2, . . . , 2 nR } → X n , and a distortion measure d : X n ×X n → R + , comparing the fidelity between x n andx n . Specifically, we are interested in the expected distortion E[d(x n ,x n )] = x n ∈X n p(x n )d(x n , g n (f n (x n ))). Definition IV.2 (Rate-Distortion). A rate-distortion pair (R, ) is said to be achievable for a source p(x) and a distortion measure d, if there exists a sequence of (2 nR , n) rate-distortion coding schemes with lim n→∞ E[d(x n , g n (f n (x n )))] < .(5) However, as specified in the remark in Section III, we are interested in conveying beliefs Q ∈ Φ(H), i.e., samples drawn according to the probability Q, obtained from the datasets s n ∈ S n , where S n = Z m . In our case, the coding function f n : S n → {1, 2, . . . , 2 nR } maps the sequence s n = {s i } n i=1 to a message f n (s n ) from which Bob can obtain the modelŝ h n = g n (f n (s n )). Our distortion then considers the difference between theĥ n 's distributionQ n , and the one achievable by Alice Q n = {A(s i )} n i=1 by comparing their samplesĥ n and h n . Consequently, we define withQ S n ,Ĥ n (or simplyQ n ) the joint distribution between the datasets and models induced by a (2 nR , n) coding scheme, whose marginals areQ Si,Ĥi for i = 1, . . . , n. Definition IV.3 (Concept Distortion). For the problem of communicating concepts, we define the following distortion on the model beliefs Q andQ: d sem (Q,Q) = E C,S L C (Q) − L C (Q) .(6) The rationale behind this definition is that Q, which is the target distribution at the transmitter, is optimized on a given dataset S without any constraint. Therefore, it is reasonable to assume L C (Q) − L C (Q) to be always non-negative. We notice that d sem quantifies the gap between the concept reconstruction at the receiver and the one at the transmitter, which is a semantic measure on the unknown true loss L(Q). Definition IV.4 (n-Sequence Concept Distortion). For the problem of communicating concepts, we define the following distortion on the sequence of the model beliefs Q n andQ n : d avg (Q n ,Q n ) = 1 n n i=1 d sem (Q i ,Q i ),(7) and d max (Q n ,Q n ) = max i={1,...,n} d sem (Q i ,Q i ).(8) In practice, d avg defines a constraint on the true loss achievable by Bob averaged over the performance of the sequenceĥ n ∼Q n , whereas d max imposes a constraint on the loss of every marginalĥ i ∼Q n i , i ∈ {1, . . . , n}. Remark. The distortion, easily defined for one-shot communications, can be generalized to a sequence of n concepts in multiple ways. Specifically, if one is interested in a systemlevel loss, then satisfying the constraint on d avg could be enough. However, to provide a per-model guarantee on the performance, then d max is the distortion to use. Definition IV.5 (Rate-Distortion Region). The rate-distortion region for a source is the closure of the set of achievable rate-distortion pairs (R, ). Definition IV.6 (Rate-Distortion Function). The ratedistortion function R( ) is the infimum of rates R such that (R, ) is in the rate-distortion region. C. Average Distortion d avg We first analyze the problem with the average distortion d avg , as defined in Equation (7). where I(S; H) is the mutual information between the data S and the model H [18]. Proof. See Appendix A D. Maximum Distortion d max In this case the distortion function implies a constraint on the performance of each symbol, i.e., model realization. First of all, we just provide a simple scheme in which lim n→∞ d avg (Q n ,Q n ) = 0 does not imply lim n→∞ d max (Q n ,Q n ) = 0, meaning that in general a code that achieves 0 distortion on average, may not achieve 0 distortion model-wise, i.e., we cannot guarantee a singlemodel performance. Example IV.1. Let H = {h 0 , h 1 }, and performance c (h 0 , z) = 0, c (h 1 , z) = 1, ∀z ∈ Z, ∀ c ∈ C. Let ∀ c n ∈ C n , Alice's distribution Q i (h j |S) = 1 2 , where j ∈ {0, 1}, and Q n = n i=1 Q i , while Bob's distribution is deterministiĉ Q 2i (h 0 |S) = 1,Q 2i+1 (h 1 |S) = 1. Then d avg (Q n ,Q n ) = 1 n n 2 i=1 1 − 1 2 + 0 − 1 2 = 0, if n is even 1 n [ n−1 2 i=1 1 − 1 2 + 0 − 1 2 + 1 − 1 2 ] = 1 2n , otherwise Thus, lim n→∞ d avg (Q n ,Q n ) = 0 but d max (Q n ,Q n ) = max 1 − 1 2 , 0 − 1 2 = 1 2 =⇒ lim n→∞ d max (Q n ,Q n ) = 0. The question now is what is needed to ensure the same distortion to the single-model performance, i.e., to ensure d max < , which is of particular interest in the semantic communication of concepts like the one considered here. We now distinguish between the two ways in whichQ n can converge to a target Q n -empirical and strong. These two notions, introduced later, map precisely onto the difference between convergence of d avg and d max . The focus is changed from determining which distortion is achievable to what joint distributions of H and S are feasible under some rate constraint. First, we extend the encoding and decoding functions f n and g n in Definition IV.1 to accept an additional common input ω ∈ Ω, which is generated by a source of common randomness p(ω). We define a (2 nR , 2 nR0 , n) stochastic code consisting of functions f n : S n × {1, . . . , 2 nR0 } → {1, 2, . . . , 2 nR } and g n : {1, 2, . . . , 2 nR } × {1, . . . , 2 nR0 } →Ĥ n , which consumes on average R 0 bits of common randomness per sample. Definition IV.7. A desired distribution Q S,H is achievable for empirical coordination with rate pair (R, R 0 ) if there exists a sequence of (2 nR , 2 nR0 , n) codes and a choice of common randomness distribution p(ω) such that TV Q s nĥn , Q S,H → 0,(10) where TV(Q, Q) indicates the total variation between distri-butionsQ and Q,Q s nĥn (s,ĥ) = 1 n n i=1 1 (si,ĥi)=(s,ĥ) , and 1 I = 1 if the condition I is true, and 0 otherwise. In other words, the empirical coordination property requires that the joint empirical distribution of the pairs (s i , g n (f n (s n )) i ) converges, in total variation, to the desired distribution. Notice how, in Example IV.1, the joint distributions Q n andQ n converge in their empirical distributions, while differing letter-wise. As already pointed out in [13], introducing common randomness does not improve the performance of empirical coordination schemes, meaning that any distribution achievable for empirical coordination by a (2 nR , 2 nR0 , n) coding scheme is also achievable with R 0 = 0. Moreover, empirical coordination schemes can be used to construct rate-distortion schemes for d avg [13]. However, as observed in Example IV.1, they do not equate to the same per-symbol performance requirements. Definition IV.8. A desired distribution Q D,H is achievable for strong coordination with rate pair (R, R 0 ) if there exists a sequence of (2 nR , 2 nR0 , n) coordination codes and a choice of common randomness distribution p(ω) such that TV Q s nĥn , n i=1 Q si,hi → 0,(11) whereQ s nĥn is the joint distribution induced by the stochastic coding scheme. Lemma IV.1 (d max Achievability). With sufficient common randomness, the rate-distortion region (R, ) for d max is the same as the one for d avg . Proof. See Appendix B. Given a constraint on the distortion for d avg achievable with minimum rate of R , if Alice and Bob can use common randomness, then it is possible to satisfy, at the same rate R , the same level of distortion for d max . To translate it into machine learning parlance, we showed that the communication rate needed to provide some performance guarantees on the expected average system test error, and on the expected singlemodel performance, is the same, as long as sufficient common randomness is available. In both cases, the characterization is over the expected performance, thus for any one realization the k-th model might have higher than desired loss. E. Coding Without the Marginal Q H We notice that all the previous achievability results assume knowledge of the exact marginal Q H = c∈C,s∈Z m Q H|s P s|c P c to be used as pre-data coding model distribution (see Section III), which is usually not known and difficult to obtain. Consequently, we are interested in studying the minimum achievable rate, when a generic coding distribution P H is used to code Q H|S . Theorem IV.2 (Achievability with General P H ). For the problem of communicating concepts, the minimum achievable rate for both d avg and d max with pre-data coding distribution P H is R( ) = miñ Q H|S : dsem(Q,Q)≤ E C,S D KL Q H|S P H ,(12) assuming sufficient common randomness. Proof. See Appendix C. It is known that when P H = Q H , i.e., using the marginal, Equation (12) is minimized, and so when the marginal is not known we pay an additional penalty given by V. COMMUNICATING THE DATA VS COMMUNICATING THE MODEL To motivate our research problem, we comment on the advantages for Alice of first compressing a trained model, and then sending it to Bob (scheme 1), versus a second framework (scheme 2), in which Alice communicates a compressed version of the datasetŜ 2 = ρ(S), using which Bob trains his models. Given a quantity X, we indicate with X i the same quantity in the i-th scheme. First of all, we see that in scheme 1, the corresponding Markov chain is C → S R − →Ĥ 1 , where the R above the arrow indicates the information bottleneck between the two random variables. However, the same chain for the second scheme reads C → S R − →Ŝ 2 →Ĥ 2 . By the data processing inequality [18], the rate constraint imposes I(S;Ĥ) ≤ R in both cases, limiting the set of all feasible beliefs Q ∈ Φ(Ĥ). Now, we reasonably assume that the optimal solution Q * (S, H) constrained to I(S; H) ≤ R lies on the boundary of the constraint, i.e., I Q * (S; H) = R, where I Q * (S; H) indicates that the mutual information is computed using Q * . In this case we can see that for scheme 1, I(S;Ĥ 1 ) = R, whereas for scheme 2, I(S; (Ŝ 2 ,Ĥ 2 )) = R. Indeed, in the former scheme Alice conveys just the random variableĤ 1 , i.e., the model, to Bob, whereas in the latter, the pair (Ŝ 2 ,Ĥ 2 ) is communicated. However, the information bottleneck is the same. Consequently, we obtain I(S;Ĥ 1 ) = I(S;Ĥ 2 ) + I(S;Ŝ 2 |Ĥ 2 ). By non-negativity of the mutual information, we always have I(S;Ĥ 1 ) ≥ I(S;Ĥ 2 ), meaning that the rate constraint on the communication channel translates differently into a model constraint for the two schemes, being stricter for the second one. In particular, the gap between the two schemes is exactly I(S;Ŝ 2 |Ĥ 2 ). Consequently, for scheme 2 the optimal compression function ρ(S) must achieve I(S;Ŝ 2 |Ĥ 2 ) = 0, and so the only way to match the performance of scheme 1 is to account for the optimal distribution QĤ |S when computingŜ. A. Distortion Rate Bound In this section, we bound the distortion-rate function for d max , which is useful to translate the rate constraint into a performance gap. We now define ∆ R = R * − R to be the difference between the rate R * of the optimal distribution and the rate R of the channel imposed by the problem. In general, we assume R * ≥ R and E C,S [L(Q * )] ≤ E C,S [L(Q)], where Q * is achievable with rate R * , and Q with rate R. Lemma V.1. Assuming that C (z, h) is upper bounded by L max ∀h ∈ H, ∀z ∈ Z, ∀C ∈ C, and that distortion d max is considered, the distortion-rate function for the problem of communicating concepts when using scheme 1 can be upper bounded by 1 (∆ R ) ≤ L max · min 1 2 ∆ R , 1 − e −∆ R ,(14) and when using scheme 2 by 2 (∆ R ) ≤ L max · min 1 2 ∆ R + I(S;Ŝ 2 |Ĥ 2 ) , 1 − e −(∆ R +I(S;Ŝ 2 |Ĥ 2 )) .(15) Proof. See Appendix D. VI. CONCLUSION We introduced the problem of conveying concepts, where concepts naturally appear as sequences of samples and can be approximated by learnable models, as in standard statistical learning. We study the framework by applying informationtheoretic tools to the problem of communicating many models jointly. We characterized its rate-distortion function for two different notions of system-level distortion, provided a bound for the distortion-rate function, and argued why jointly learning, compressing, and communicating models should be preferred over compressing and sending the datasets. For future investigations, the plan is to study in detail the relations between model compression and test accuracy, and to design practical model communications schemes. APPENDIX A PROOF OF THEOREM IV.1 To prove Theorem IV.1, we need to show that we can translate the average distortion d avg requirement into a constraint on the empirical distributionQ s nĥn (d, h) = 1 n n i=1 1 (si,hi)=(s,h) , where 1 I = 1 if the condition I is true, and 0 otherwise. We can write d avg (Q n ,Q n ) = 1 n n i=1 d(Q i ,Q i ) = E C,S 1 n n i=1 L C (Q i ) − L C (Q i ) (a) = E C,S 1 n n i=1 h∈H L C (h)(1 (S,ĥi)=(S,h) − Q h|S ) (b) = E C,S h∈H L C (h) 1 n n i=1 1 (S,ĥi)=(S,h) − Q h|S = E C,S h∈H L C (h) Q s n ĥn (d, h) − Q h|S = E C,S L(Q s n h n ) − L(Q) = d sem (Q,Q s n ĥn ), where the indicator function in (a) depends on the chosen (2 nR , n) coding scheme, and (b) comes from the fact that Alice's sampling scheme does not depend on the model index i. We then showed that the average distortion requirement translates into a distortion between the empirical distribution Q s nĥn , and the target Q. Given this, Theorem 1 in [14] applies, and our Theorem IV.1 follows. APPENDIX B PROOF OF LEMMA IV.1 Proof. Theorem IV.1 provides the minimum achievable rate to satisfy d avg ≤ , which is provided by an optimized distribution Q * S,H that minimizes the mutual information I(S; H), and satisfies d avg ≤ . Now, we impose strong coordination between Alice and Bob, by using as target joint distribution Q * S n ,H n = n i=1 Q * S,H , and by Theorem 10 of [13] the same rate I(S; H) of empirical coordination can be achieved, as long as enough common randomness is available. Now, given that Bob's distributionQ S n ,Ĥ n converges in total variation to Q * S n ,H n , so it happens for the marginal, i.e., Q S n ,Ĥ n (s i ,ĥ i ) To prove Theorem IV.2, we observe that Alice now needs to convey her probability distribution Q n H n |S n to Bob using P n H = (P H n ) n to code it. If we indicate with E C,S [K] the average number of bits spent to convey model belief Q using distribution P , Corollary 3.4 in [17] provides the single-shot bounds R(Q, P ) ≤ E C,S [K] ≤ R(Q, P ) + log(R(Q, P ) + 1) + 4, where R(Q, P ) = E C,S [D KL (Q||P )]. We now translate the results to the n-length sequence, and compute its limit as n grows indefinitely. R(Q n , P n ) = E C n ,S n [D KL (Q n ||P n )] = E C n E S n |C n E H n |S n log = E C n E S n |C n E H n |S n log (Q H|S ) n (P H ) n = nR(Q, P ), where (a) is because samples are independent given the nature of the problem, and (b) is because, given the dataset realization, they are also identically distributed by our assumption. Consequently, we can upper bound the total number of bits with E C,S [K] ≤ nR(Q, P ) + log(nR(Q, P ) + 1) + 4, obtaining an average rate of lim n→∞ E C,S [K] n → R(Q, P ). APPENDIX D PROOF OF LEMMA V.1 We now prove the result in Lemma V.1. Let Q be the distribution at the sender, andQ the one at the receiver, i.e., the distributionQ = Q * S,H in the proof of Lemma IV.1. We first bound, for each C ∈ C, d max by noticing that, for each i ∈ {1, . . . , n}, the following holds where (a) is given by strong coordination, (b) is by the combination of the inequalities due to Pinsker and Breatgnolle-Huber, and (c) is Jensen's inequality. We now proceed to bound E C,S D KL (Q Q ) by observing that the average distortion between Q andQ is a linear function of the probabilitieŝ Q, and the setQ = {Q ∈ Φ(H) : d sem (Q,Q) ≤ } satisfying the constraint is convex, and that by the definition of Distortion-Rate function,Q is the distribution inQ minimizing E C,S D KL (Q P ) , where P is the pre-data coding distribution (see Theorem IV.2). Then, Theorem 11.6.1 in [18], also known as the Pythagorean Theorem for the Kullback-Leibler divergence, applies, and we can bound E C,S D KL (Q Q ) ≤ E C,S [D KL (Q P )] − E C,S D KL (Q P ) (a) = R * − R = ∆ R , where (a) is given by Theorem IV.2. Combined together, the two inequalities provide Equation (14). Regarding Scheme 2, it is sufficient to notice that R = I(S;Ĥ 2 )+I(S;Ŝ 2 |Ĥ 2 ) (see Section V), from which Equation (15) follows. We notice that this does not directly apply to d avg , as the distribution Q * solving Equation (9) and providing the rate is not, in general, the one used by Bob to sample actions. This is true for the scheme for d max , in which common randomness is introduced. Fig. 1 . 1The problem of communicating concepts. Theorem IV.1 (Rate-Distortion Theorem for d avg ). For the problem of communicating concepts with distortion d avg , the rate-distortion function satisfies E C,S D KL Q H|S P H − I(S; H) = E C,S [D KL (Q H P H )]. distortion constraint symbol-wise, and so satisfies d max .APPENDIX C PROOF OF THEOREM IV.2 = E C n E S n |C n E H n |S n log n i=1 Q H|Si (P H ) n (b) = E C,S L C (Q) − L C (Q) = E C,S h∈H E z∼C [ (h, z)] Qĥ |S − Q h|S ≤ L max · E C,S h∈HQĥ |S − Q h|S ≤ L max · E C,S TV(Q, Q) (b) ≤ L max · E C,S min 1 2 D KL (Q Q), 1 − e −DKL(Q Q) (c) ≤ L max · min 1 2 E C,S D KL (Q Q) , 1 − e −E C,S [DKL(Q Q)] Information security in big data: Privacy and data mining. 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[ "Weighted Distributed Estimation under Heterogeneity", "Weighted Distributed Estimation under Heterogeneity" ]
[ "Jia Gu \nCenter for Statistical Science\nSchool of Mathematical Science and Guanghua School of Management Peking University\nPeking University\n\n", "Song Xi Chen \nCenter for Statistical Science\nSchool of Mathematical Science and Guanghua School of Management Peking University\nPeking University\n\n" ]
[ "Center for Statistical Science\nSchool of Mathematical Science and Guanghua School of Management Peking University\nPeking University\n", "Center for Statistical Science\nSchool of Mathematical Science and Guanghua School of Management Peking University\nPeking University\n" ]
[]
This paper considers distributed M-estimation under heterogeneous distributions among distributed data blocks. A weighted distributed estimator is proposed to improve the efficiency of the standard "Split-And-Conquer" (SaC) estimator for the common parameter shared by all the data blocks. The weighted distributed estimator is shown to be at least as efficient as the would-be full sample and the generalized method of moment estimators with the latter two estimators requiring full data access.A bias reduction is formulated to the WD estimator to accommodate much larger numbers of data blocks than the existing methods without sacrificing the estimation efficiency, and a similar debiased operation is made to the SaC estimator. The mean squared error (MSE) bounds and the asymptotic distributions of the WD and the two debiased estimators are derived, which shows advantageous performance of the debiased estimators when the number of data blocks is large.
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[ "https://export.arxiv.org/pdf/2209.06482v1.pdf" ]
252,222,462
2209.06482
35f1492f70d88b74422945b7f62820d266ada9fe
Weighted Distributed Estimation under Heterogeneity September 15, 2022 14 Sep 2022 Jia Gu Center for Statistical Science School of Mathematical Science and Guanghua School of Management Peking University Peking University Song Xi Chen Center for Statistical Science School of Mathematical Science and Guanghua School of Management Peking University Peking University Weighted Distributed Estimation under Heterogeneity September 15, 2022 14 Sep 2022* corresponding author 1Keyword: Bias CorrectionDistributed InferenceFederated LearningHeterogeneitySplit and Conquer MethodWeighted Estimation This paper considers distributed M-estimation under heterogeneous distributions among distributed data blocks. A weighted distributed estimator is proposed to improve the efficiency of the standard "Split-And-Conquer" (SaC) estimator for the common parameter shared by all the data blocks. The weighted distributed estimator is shown to be at least as efficient as the would-be full sample and the generalized method of moment estimators with the latter two estimators requiring full data access.A bias reduction is formulated to the WD estimator to accommodate much larger numbers of data blocks than the existing methods without sacrificing the estimation efficiency, and a similar debiased operation is made to the SaC estimator. The mean squared error (MSE) bounds and the asymptotic distributions of the WD and the two debiased estimators are derived, which shows advantageous performance of the debiased estimators when the number of data blocks is large. Introduction Modern big data have brought new challenges to statistical inference. One such challenge is that despite the shear volume of the data, a full communication among the data points may not be possible due to the expensive cost of data communication or the privacy concern. The distributed or the "split-and-conquer" (SaC) method has been proposed to divide the full data sample to smaller size data blocks as data communication is too costly to attain an estimation task in a timely fashion. The SaC estimator is also suited to the situations where the data are naturally divided to data blocks and data communication among the data blocks are prohibited due to privacy concern. The SaC estimation had been considered in [17] for the U-statistics, [30] for the M-estimation, [5] for the generalized linear models, [27] and [4] for the quantile regression, while [2] studied the high dimensional testing and estimation with sparse penalties. [3] studied the estimation efficiency and asymptotic distributions for the general asymptotic symmetric statistics [14] and found differences in the efficiency and the asymptotic distributions between the non-degenerate and degenerate cases. Bootstrap resampling-based methods had been introduced to facilitate statistical inference. [13] proposed the bag-of-little bootstrap (BLB) method for the plug-in estimators by making up economically the full sample for the distributed inference. [23] suggested a sub-sampled double bootstrap method designed to improve the computational efficiency of the BLB. [3] proposed the distributed and the pseudo-distributed bootstrap methods with the former conducted the resampling within each data block while the latter directly resampled the distributed statistics. Privacy has been a major concern in big data applications where people are naturally reluctant to share the raw data to form a pool of big data as practised in the traditional full sample estimation. However, the data holders may like to contribute summary statistics without having to give away the full data information. Federated Learning or the distributed inference with a central host has been proposed to accommodate such reality [12,16,19,28], where summary statistics of the data blocks or the gradients of the objective functions associated with the private data blocks are submitted to a central host for forming aggregated estimation or computation. Homogeneous distribution among the data blocks are assumed in majority of the distributed inference studies with only a few exceptions [6,32]. Federated Learning, on the other hand, was introduced to mitigate many challenges arising from classical distributed optimization. In particular, heterogeneous or Non-IID distributed data across different data blocks is one of the defining characteristics and challenges in the Federated Learning [12,16]. Indeed, it is natural to expect the existence of heterogeneity, especially for data stored in different locations or generated by different stochastic mechanism, for instance mobile phones of different users. However, there has been little published works on the statistical properties of estimators considered in the Federated Learning. This paper considers distributed estimation under heterogeneous distributions among the data blocks, which is closely related to the Federated Learning and especially the multitask learning (MTL) [31]. We consider distributed M-estimation where there is a common parameter shared by the distributions of the data blocks and data-block specific heterogeneous parameters. Our treatment of the heterogeneity is made by explicit parameterization, which is different from the MTL where the heterogeneity is regularized by penalty terms. It is noted that [6] considered a heterogeneous setting, but under a fully parametric likelihood framework. Our study reveals that in the presence of the heterogeneity the full sample M-estimator of the common parameter obtained by requiring full data communication, can be less efficient than the SaC estimator. However, this phenomenon disappears if the objective function of the M-estimation satisfies a generalized second-order Bartlett's identity, which are satisfied by the parametric and quasi likelihoods, and the least square estimation in the parametric regression. We propose a weighted distributed (WD) estimator, which is asymptotically at least as efficient as the full sample and the SaC estimator when the number of data blocks K = o(N 1/2 ) where N is the full sample size. The mean-squared error (MSE) bound and the asymptotic distribution of the proposed WD estimator are derived, as well as the asymptotic equivalence between the WD and the generalized method of moment (GMM) estimator. We propose a debiased weighted distributed (dWD) estimator with a data splitting mechanism on each data block to remove the correlation between the empirical bias correction and the weights used to tackle the heterogeneity. The dWD is asymptotically as efficient as the WD estimator, but with a more relaxed constraint of K = o(N 2/3 ). The bias-correction is also applied to the SaC formulation leading to a more communicationefficient dSaC estimator, which is shown to be more accurate than the subsampled average mixture estimator (SAVGM) [30] in the homogeneous case. The paper is organized as follows. The estimation framework and necessary notations for the study are outlined in Section 2. The relative efficiency between the full sample and the SaC estimators under the heterogeneity is discussed in Section 3 to motivate the construction of the weighted distributed (WD) estimator. The WD estimator is introduced in Section 4 along with its efficiency, asymptotic distribution and MSE bound. Statistical properties of two debiased estimators dSaC and dWD are revealed in Section 5. Section 6 provides numerical verification to the theoretical results. Section 7 concludes with a discussion. Technical details are reported in the supplementary materials (SM). Preliminaries Suppose that there is a large data sample of size N , which is divided into K data blocks of sizes {n k } K k=1 such that N = K k=1 n k and let n = N K −1 be the average sample size of the data blocks. For the relative sample size among data blocks, we assume the following assumption. Assumption 1. There exist constants 0 < c < 1 < C such that c ≤ n k 1 n k 2 ≤ C for all pairs of (k 1 , k 2 ), and if K is a fixed constant we further assume that n k N → γ k ∈ (0, 1) for a set of constants {γ k } K k=1 . The k-th data block consists of a sub-sample {X k,i } n k i=1 which are independent and identically distributed (IID) random vectors from a probability space (Ω, F, P ) to (R d , R d ) with F k as the distribution. The K distributions {F k } share a common parameter φ ∈ R p 1 , while each F k has another parameter λ k ∈ R p 2 specific to F k of the k-th data block. There are maybe other hidden parameters which define F k , which are however not directly involved in the semi-parametric M-estimation, and thus are not of interest in the study. The parameters of interests in the k-th block are θ k = (φ T , λ T k ) T , and the overall pa- rameters of interests are θ = (φ T , λ T 1 , λ T 2 , ..., λ T K ) T ∈ R p 1 +Kp 2 . Suppose there is a common objective function M (X; φ, λ k ) that is convex with respect to the parameter (φ, λ k ) and facilitates the M-estimation of the parameters in each data block. In general, the criteria function can be made block specific, say M k function. Indeed, the presence of the heteroge- neous local parameters {λ k } K k=1 leads to different M k (x, φ) = M (x, φ, λ k ) for the inference on φ, which connects to the multi-task learning (MTL). In the k-th data block the true parameter θ * k = (φ * T , λ * T k ) T is defined as the unique minimum of the expected objective function, namely θ * k = (φ * T , λ * T k ) T = argmin θ k ∈Θ k E F k M (X k,1 ; φ, λ k ).(1) The true common parameter φ * appears in all θ * k , and the block-specific {λ * T k } K k=1 may differ from each other. The entire set of true parameters θ * = (φ * T , λ * T 1 , · · · , λ * T K ) T , can be also identified as θ * = argmin θ∈Θ K k=1 γ k E θ * k M (X k,1 ; φ, λ k ).(2) If the data could be shared across the data blocks, we would attain the conventional full sample M-estimatorθ f ull = argmin θ∈Θ K k=1 n k i=1 M (X k,i ; φ, λ k ),(3) which serves as a benchmark for the distributed estimators. Let ψ φ (X k,i ; φ, λ k ) = ∂M (X k,i ;φ,λ k ) ∂φ and ψ λ (X k,i ; φ, λ k ) = ∂M (X k,i ;φ,λ k ) ∂λ k be the score functions. The estimating equations for the full sample M-estimators are        K k=1 n k i=1 ψ φ (X k,i ; φ, λ k ) = 0, n k i=1 ψ λ (X k,i ; φ, λ k ) = 0 k = 1, ..., K.(4) The above full sample estimation is not attainable for the distributed situations due to privacy or the costs associated with the data communications. The distributed estimation first conducts local estimation on each data block, namely the local M-estimator θ k = (φ k ,λ k ) = argmin θ k ∈Θ k n k i=1 M (X k,i ; θ k ) with the corresponding estimating equations        n k i=1 ψ φ (X k,i ; φ k , λ k ) = 0, n k i=1 ψ λ (X k,i ; φ k , λ k ) = 0.(5) Then, the "split-and-conquer" (SaC) estimator for the common parameter φ iŝ φ SaC = 1 N K k=1 n kφk .(6) The heterogeneity among the distributions and the inference models among the data blocks bring new dimensions to the discussion of the relative efficiency and the estimation errors, which are the focus of this paper. We are to show that the conventionally weighted SaC estimator (6) may not be the best formulation for the estimation of φ. Throughout this paper, unless otherwise stated, · 2 and |||·||| 2 represent the L 2 norm of a vector and a matrix, respectively. Besides, we will use C and C i to denote absolute positive constants independent of (n k , K, N ). An important question is the efficiency and the estimation errors of the SaC estimator φ SaC relative to the full sample estimatorφ f ull . For the homogeneous case, Chen and Peng (2021) [3] found that for the asymptotic symmetric statistics, the SaC estimator (6) Ψ N (X; θ) =          K k=1 n k i=1 ψ φ (X k,i ; φ, λ k ) n 1 i=1 ψ λ (X 1,i ; φ, λ 1 ) . . . n K i=1 ψ λ (X K,i ; φ, λ K )          .(7) Define Ψ θ (θ k ) = (Ψ φ (θ k ) T , Ψ λ (θ k ) T ) T = E∇ θ k M (X k,1 ; θ k ), Ψ θ θ (θ k ) =    Ψ φ φ (θ k ) Ψ λ φ (θ k ) Ψ φ λ (θ k ) Ψ λ λ (θ k )    = E∇ 2 θ k M (X k,1 ; θ k ), J φ|λ (θ k ) = Ψ φ φ (θ k ) − Ψ λ φ (θ k )Ψ λ λ (θ k ) −1 Ψ φ λ (θ k ) and S φ (X k,i ; θ k ) = ψ φ (X k,i ; θ k ) − Ψ λ φ (θ k )Ψ λ λ (θ k ) −1 ψ λ (X k,i ; θ k ). Then we can apply Taylor's expansion and obtain (see Section 1.1 in SM for details) φ f ull − φ * = −{ K k=1 n k N J φ|λ (θ * k )} −1 1 N K k=1 n k i=1 S φ (X k,i ; θ * k ) + o p (N −1/2 ),(8) For the local estimator (φ k ,λ k ) based on the k-th data block that solves (5), by replicating the same derivation leading to (8), we have       φ k − φ * = −n −1 k J φ|λ (θ * k ) −1 n k i=1 S φ (X k,i ; θ * k ) + o p (n −1/2 k ), λ k − λ * k = −n −1 k J λ|φ (θ * k ) −1 n k i=1 S λ (X k,i ; θ * k ) + o p (n −1/2 k ),(9) where J λ|φ (θ k ) = Ψ λ λ (θ k ) − Ψ φ λ (θ k )Ψ φ φ (θ k ) −1 Ψ λ φ (θ k ) and S λ (X k,i ; θ k ) = ψ λ (X k,i ; θ k ) − Ψ φ λ (θ k )Ψ φ φ (θ k ) −1 ψ φ (X k,i ; θ k ).(10) The distributed inference setting is closely related to the Multi-Task Learning (MTL) which fits separate local parameters φ k ∈ R p to the data of different data blocks (tasks) through convex loss functions { k }. In particular, the MTL is formulated as [24]: min Φ,Ω K k=1 n k i=1 k (φ T k X k,i , Y k,i ) + R(Φ, Ω) ,(11) where {(X k,i , Y k,i ), i = 1, 2, · · · , n k } are data in the k-th block, Φ is the matrix with {φ k } K k=1 as column vectors, Ω ∈ R K×K and R(·, ·) measures the extent of the heterogeneity among different data blocks. Choices of R(·, ·) include R(Φ, Ω) = δ 1 tr(ΦΩΦ T ) + δ 2 Φ 2 F for δ 1 , δ 2 > 0 and Ω = I K×K − 1 K 1 K 1 T K such that tr(ΦΩΦ T ) = K k=1 φ k −φ K 2 2 whereφ K = 1 K K k=1 φ k , which leads to the mean-regularized MTL [7]. The second term of R performs regularization on each local model, trying to control the magnitude of the estimates of φ k . The distributed framework is well connected to the MTL in two key aspects. One is that despite we use the same objective (loss) function M over the data blocks, the heterogeneity induced by local parameters {λ k } K k=1 and the distributions effectively define M k (φ, x) = M (x, φ, λ k ) , which is equivalent to the block specific loss functions k used in MTL. Another aspect is that although the MTL assumes different parameters {φ k } over the data blocks, it regularizes them toward a common one. In contrast, we assume there is a common parameter φ shared by the heterogeneous distributions. Full Sample versus SaC Estimation It is naturally expected that the full sample estimatorφ f ull should be at least as efficient as the distributed SaC estimatorφ SaC since the former utilizes the full sample information including the communications among different data blocks. However, we are to show that this is not necessarily true in the presence of heterogeneity. To appreciate this point, we first list more regularity conditions needed in the analysis. Assumption 2. (Identifiability) The parameters θ * k = (φ * , λ * k ) is the unique minimizer of M k (θ k ) = EM (X k,1 ; θ k ) for θ k ∈ Θ k . Assumption 3. (Compactness) The parameter space Θ k is a compact and convex set in R p and the true parameter θ * k is an interior point of Θ k and sup θ k ∈Θ k θ k − θ * k 2 ≤ r for all k ≥ 1 and some r > 0. The true common parameter φ * is an interior point of a compact and convex set Φ ⊂ Θ k . Assumption 4. (Local strong convexity) The population objective function on the k-th fully parametric setting, while in our assumption, we need the overall parameter space to be bounded. This stronger assumption is needed since we do not fully specify the distributions {F k } K k=1 of the random variables and will be useful when we derive the MSE bound for the weighted distributed estimator which will be proposed in Section 4. data block M k (θ k ) = EM (X k,1 ; θ k ) is twice differentiable, and there exists a constant ρ − > 0 such that ∇ 2 θ k M k (θ * k ) ρ − I p×p . Here A B means A−B isfor all k ≥ 1, E ∇ θ k M (X k,1 ; θ * k ) 2v 1 2 ≤ R 2v 1 and E ∇ 2 θ k M (X k,1 ; θ * k ) − ∇ 2 θ k M k (θ * k ) 2v 2 ≤ L 2v . In addition, for any x ∈ R d , ∇ 2 θ k M (x; θ k ) and ∇ θ k M (x; θ k )∇ θ k M (x; θ k ) T are G(x)− and B(x)−Lipschitz continuous, respectively, in the sense that ∇ 2 θ k M (x; θ k ) − ∇ 2 θ k M (x; θ k ) 2 ≤ G(x) θ k − θ k 2 , ∇ θ k M (x; θ k )∇ θ k M (x; θ k ) T − ∇ θ k M (x; θ k )∇ θ k M (x; θ k ) T 2 ≤ B(x) θ k − θ k 2 , for all θ k , θ k ∈ U k := {θ k | θ k −θ * k 2 ≤ ρ} for some ρ > 0, and EG(X k,1 ) 2v ≤ G 2v , EB(X k,1 ) 2v ≤ B 2v for some positive constants G and B. The Lipschitz continuity of the outer product of the first-order derivative is required to control the estimation error when we estimate the asymptotic covariance matrix of the local estimatorθ k , and it can be directly verified under the logistic regression case; see Section 1.2 in the SM for details. √ N (φ SaC − φ * ) d → N (0, K k=1 γ k J φ|λ (θ * k ) −1 Σ k (θ * k )J φ|λ (θ * k ) −1 ), (12a) √ N (φ f ull − φ * ) d → N (0, ( K k=1 γ k J φ|λ (θ * k )) −1 ( K k=1 γ k Σ k (θ * k ))( K k=1 γ k J φ|λ (θ * k )) −1 ), (12b) where J λ|φ (θ * k ) = Ψ λ λ (θ * k ) − Ψ φ λ (θ * k )Ψ φ φ (θ * k ) −1 Ψ λ φ (θ * k ) and Σ k = V ar{S φ (X k,1 ; θ * k )}. Define V (Σ, A) = (A T ) −1 ΣA −1 as a mapping from S p 1 ×p 1 ++ × GL(R p 1 ) to S p 1 ×p 1 ++ , where S p 1 ×p 1 ++ and GL(R p 1 ) denote the symmetric positive definite matrices and invertible real matrices of order p 1 , respectively. Since Σ K k=1 γ k = 1 and γ k > 0, the asymptotic variance of φ SaC can be interpreted as a convex combination of function values {V (Σ k (θ * k ), J φ|λ (θ * k ))} K k=1 and that ofφ f ull can be seen as V ( K k=1 γ k Σ k (θ * k ), K k=1 γ k J φ|λ (θ * k ) ). However, V (·, ·) is not convex with respect to its arguments (Σ, A), which means that the inequality { K k=1 γ k J φ|λ (θ * k )} −1 { K k=1 γ k Σ k (θ * k )}{ K k=1 γ k J φ|λ (θ * k )} −1 K k=1 γ k J φ|λ (θ * k ) −1 Σ k (θ * k )J φ|λ (θ * k ) −1 does not always hold. In other words,φ f ull is not necessarily more efficient thanφ SaC . To gain understanding of Theorem 1 and to motivate the weighted distributed estimator, we consider the errors-in-variables model. Suppose that one observes K blocks of independent data samples {(X k,i , Y k,i )} n i=1 for k = 1, 2..., K and N = nK, where (X k,i , Y k,i ) are IID and generated from the following model:        X k = Z k + e k , Y k = φ * + λ * k Z k + f k ,(13) where {Z k } K k=1 are random variables whose measurements {(X k , Y k )} K k=1 are subject to errors {(e k , f k )} K k=1 , and (e, f ) is bivariate normally distributed with zero mean and covariance matrix σ 2 I 2 and is independent of Z k . Here, φ * is the common parameter across all data blocks while λ * k (λ * k > 0) represents the block specific parameter. We assume that V ar(e) = V ar(f ) to avoid any identification issue arisen when Z is also normally distributed [20]. There is a considerable literature on the regression problem with measurement errors, as summarised in [8,22]. We consider the approach displayed in Example 5.26 of [26] which constructs a kind of marginal likelihood followed by centering to make a bona fide score equation, as detailed in Section 1.3 of the SM. The M-function is M (X k , θ k ) = 1 2σ 2 (1 + λ 2 k ) (λ k X k − (Y k − φ)) 2 ,(14) with the score equation satisfying E∇M (X k,1 , Y k,1 |Z k,1 , θ * k ) = 0 2×1 . For simplicity we assume K = 2, then from Theorem 1 we have        V ar(φ f ull ) ≈ { σ 2 EZ 2 var(Z) 2 1 1+λ * 2 1 + 1 1+λ * 2 2 + σ 4 (EZ) 2 var 2 (Z) 2 (1+λ * 2 1 ) 2 + 2 (1+λ * 2 2 ) 2 ( 1 1+λ * 2 1 + 1 1+λ * 2 2 ) 2 } 1 N , V ar(φ SaC ) ≈ { σ 2 EZ 2 var(Z) (1+λ * 2 1 )+(1+λ * 2 2 ) 2 + σ 4 (EZ) 2 var 2 (Z) } 1 N .(15) Note that the coefficients to σ 2 EZ 2 var(Z) in the first terms of the variances are harmonic and arithmetic means of {1 + λ * 2 1 , 1 + λ * 2 2 }, respectively. By the mean inequality the coefficient in the first term of V ar(φ SaC ) is larger than that in V ar(φ f ull ). The second term of the variances involves (EZ) 2 as a multiplicative factor. Thus, if the unobserved Z has zero mean, the full-sample estimator would be at least as good as the SaC estimator in terms of variation when the sample size goes to infinity. However, the story may change when EZ = 0, because the second term of V ar(φ f ull ) has a factor which is the square of a ratio between the quadratic mean and the arithmetic mean of ( 1 1+λ * 2 1 , 1 1+λ * 2 2 ). The factor is larger than or equal to 1 if and only if λ * 1 = λ * 2 namely the homogeneous case. In the heterogeneous case, by adjusting σ 4 (EZ) 2 var 2 (Z) / σ 2 EZ 2 var(Z) , we can find cases such that λ * 1 = λ * 2 such that the full sample estimator has a larger variance than the SaC estimator. Simulation experiments presented in Section 6 display such cases. Weighted Distributed Estimator The previous section shows that the full sample estimatorφ f ull under heterogeneity may be less efficient than the simple averagedφ SaC . This phenomenon suggests that the conventional wisdom in the homogeneous context case may not be applicable to the heterogeneous case. One may also wonder if the simple SaC estimator can be improved under the heterogeneity. Specifically, how to better aggregate the local estimatorφ k for more efficiency estimation to the common parameter φ is the focus of this section. Formulation and Results Consider a class of estimators formed by linear combinations of the local estimators {φ k }: {φ SaC w |φ SaC w = K k=1 W kφk , W k ∈ R p 1 ×p 1 , K k=1 W k = I p 1 }. We want to minimize the asymptotic variance ofφ SaC w with respect to {W k } K k=1 . According to a generalization of Theorem 1 AsyV ar(φ SaC w ) = K k=1 n −1 k W k A −1 k Σ k (A T k ) −1 W T k ,(16)where A k = J φ|λ (θ * k ) and Σ k = V ar{S φ (X k,i ; θ * k )}. It is noted that the asymptotic variance is defined via the asymptotic normality of the M-estimation. For the time being, A k and Σ k are assumed known and denote H k = A −1 k Σ k (A T k ) −1 . We choose the trace operator as a measure on the size of the asymptotic covariance matrix and this leads to the minimization problem M inimize W k tr K k=1 n −1 k W k H k W T k s.t. K k=1 W k = I p 1 ,(17) which is a convex optimization problem. It can be solved via the Lagrangian multiplier method which gives W * k = ( K s=1 n s H −1 s ) −1 n k H −1 k . If we replace the trace with the Frobenius norm in the objective function (17), the same solution is attained as shown in Section 1.4 of the SM. The SaC estimator under the optimal weights W * k is called the weighted distributed (WD) estimator and denoted asφ W D . By construction, the WD estimator is at least as efficient as the SaC estimator (6). To compare the relative efficiency between φ f ull andφ W D , we note that AsyV ar(φ f ull ) = ( K k=1 n k A k ) T ( K k=1 n k Σ k ) −1 ( K k=1 n k A k ) −1 and AsyV ar(φ W D ) = K k=1 n k A T k Σ −1 k A k −1 .(18) Define F (Σ, A) = A T Σ −1 A. If we can show the convexity of F , an application of Jensen's inequality will establish the relative efficiency of the two estimators. In fact, we have the following lemma. Lemma 1. Suppose H and K are positive definite matrices of order p, and X and Y are arbitrary p × m matrices. Then, Q = X T H −1 X + Y T K −1 Y − (X + Y ) T (H + K) −1 (X + Y ) 0. The lemma implies that ( K k=1 n k A k ) T ( K k=1 n k Σ k ) −1 ( K k=1 n k A k ) K k=1 n k A T k Σ −1 k A k , which means that the WD estimator is at least as efficient as the full sample estimator, and can be more efficient thanφ f ull . That is to say, the simultaneous estimating equations (7), which are obtained from the first-order derivative of the the simple summation of local objectives n k i=1 M (X k,i ; θ k ), are not the best formulation of the M-estimation problem, since the formulation itself does not utilize the heterogeneity existed in the data blocks. In contrast, the WD estimator exploits the potential efficiency gain from the heterogeneity by re-weighting of the local estimators, and this is why the full sample estimator may not be as efficient as the WD estimator. Likelihood and Quasi-likelihood The above results lead us to wonder whether we can attain more efficient distributed estimators than the full sample estimator under the heterogeneity if we restrict to a fully parametric setting. When the distribution of X k,i is fully parametric with density function f (·; φ, λ k ), the Fisher information matrix in the k-th data block is I(θ k ) = I(φ, λ k ) =    I φφ I φλ k I λ k φ I λ k λ k    = −E    ∂ 2 ∂φ 2 logf (X k,1 ; θ k ) ∂ 2 ∂φ∂λ T logf (X k,1 ; θ k ) ∂ 2 ∂λ∂φ T logf (X k,1 ; θ k ) ∂ 2 ∂λ 2 logf (X k,1 ; θ k )    , and the partial information matrix as I φ|λ k = I φφ − I φλ k I −1 λ k λ k I λ k φ . Now, the objective function for the M-estimation (also the maximum likelihood estimation (MLE)) is M (X k,i ; φ, λ k ) = − log f (X k,i ; φ, λ k ). Routine derivations show that Σ k = V ar{S φ (X k,1 ; θ * k )} = I φ|λ k and A k = J φ|λ (θ * k ) = I φ|λ k . Thus, AsyV ar(φ f ull ) = AsyV ar(φ W D ) = K k=1 n k I φ|λ k −1 and AsyV ar(φ SaC ) = 1 N 2 K k=1 n k I −1 φ|λ k . A direct application of Lemma 1 shows that AsyV ar(φ f ull ) = AsyV ar(φ W D ) AsyV ar(φ SaC ). Thus, the full sample MLE can automatically adjust for the heterogeneity and is at least as efficient as SaC estimatorφ SaC . Besides, the weighted distributed estimatorsφ W D can fully recover the efficiency gap of the SaC estimator. The same relationship amongφ f ull ,φ SaC andφ W D also holds for the maximum quasilikelihood estimator (MQLE) with independent observations (see Section 1.5 in the SM for details). If one looks into the asymptotic variances of the MLE and MQLE, it can be found that the underlying reason for (19) is that the two special M-estimation functions satisfy the second order Bartlett's identity [1,18]: E∇M (X k , θ * k )∇M (X k , θ * k ) T = E∇ 2 M (X k , θ * k ). By the variance formula of the asymptotic distribution of the M-estimator and Lemma 1, we readily have that the Bartlett's identity can be relaxed by inserting a factor γ = 0 such that E∇M (X k , θ * k )∇ T M (X k , θ * k ) = γE∇ 2 M (X k , θ * k ).(20) An important example for such a case is the least square estimation for the parametric regression with homoscedastic and non-autocorrelated disturbances ( see Section 1.6 in the SM for details). Otherwise the full sample least square estimator may not be efficient and there is an opportunity for the weighted distributed least square estimation. In summary, as long as the objective function M (x k , θ k ) satisfies (20), thenφ f ull attains the same asymptotic efficiency asφ W D , andφ SaC is at most as efficient as the former two estimators. Relative to Generalized Method of Moment Estimation To provide a benchmark on the efficiency of the WD estimation, we consider the generalized method of moment (GMM) estimator [9]. The GMM estimator possess certain optimal property for semiparametric inference that the weighted distributed estimation can compare with, despite the GMM requires more data sharing than the distributed inference would require. The score functions of the M-estimation on each data block can be aggregated and combined to form the moment equations        n k i=1 ψ φ (X k,i ; φ, λ k ) = 0, n k i=1 ψ λ (X k,i ; φ, λ k ) = 0, k = 1, ..., K.(21) There are pK estimating equations, where the dimension of θ * is pK − (K − 1)p 1 . Thus, the parameter is over-identified which offers potential in efficiency gain for the GMM [9]. The GMM estimation based on the moment restrictions (21) is asymptotically equivalent to solving the following problem: θ GM M = argmin θ k =(φ,λ k )∈Θ k ,1≤k≤Kψ T N (θ)W 0ψN (θ),(22) where W 0 = V ar(ψ N (θ * )) −1 is the optimal weighting matrix [9,29] and ψ N (θ) = ( n 1 i=1 ψ φ (X 1,i ; θ 1 ) T , n 1 i=1 ψ λ (X 1,i ; θ 1 ) T , · · · , n K i=1 ψ φ (X K,i ; θ K ) T , n K i=1 ψ λ (X K,i ; θ K ) T ) T The asymptotic variance of the GMM estimator [9] is AsyV ar (θ GM M ) = (G T 0 W 0 G 0 ) −1 , where G T 0 = E{ ∂ψ T N (θ * ) ∂θ }. A derivation given in Section 1.7 of the SM shows that AsyV ar(φ GM M ) = { K k=1 n k J φ|λ Σ −1 k J φ|λ } −1 .(23) Thus, the weighted distributed estimator's efficiency is the same as that of the GMM estimator. This is very encouraging to the proposed WD estimator as it attains the same efficiency as the GMM without requiring much data sharing among the blocks, which avoids the expenses of the data transmission and preserves the privacy of the data. Estimation of Weights in one round communication To formulate the WD estimator, the optimal weights W * k = ( K s=1 n s H −1 s ) −1 n k H −1 k have to be estimated. By the structure of W * k , we only need to separately estimate H k , the leading principal submatrix of order p 1 of the asymptotic covariance matrixH k ofθ k . It is noted thatH k = (∇Ψ θ (θ * k )) −1 E{ψ θ k (X k,1 ; θ * k )ψ θ k (X k,1 ; θ * k ) T }(∇Ψ θ (θ * k )) −1 =    H k * * *    , where Ψ θ (θ k ) = Eψ θ k (X k,1 ; θ k ). We can construct the sandwich estimator [25] to estimatẽ H k and then H k . The distributive procedure to attain the WD estimator is summarized in the Algorithm 1. The Step 4 in the algorithm is necessary since there is no guarantee that after weighting the estimatorφ W D still belongs to the set Φ as required in Assumption 3. However the event {φ W D ∈ Φ} should happen with probability approaching one. Hence, theφ SAC I(φ W D ∈ Φ) Algorithm 1: Weighted Distributed M-estimator Input: {X k,i , k = 1, ..., K; i = 1, ..., n k } Output:φ W D ,λ k 1 Obtain the initial estimatesθ k = (φ k ,λ k ) based on data block k ; 2 CalculateĤ k (θ k ) in each block, which is the leading principal sub-matrix of order p 1 of (∇ θ kΨ θ k ) −1 (n −1 k n k i=1 ψ θ k (X k,i ;θ k )ψ θ k (X k,i ;θ k ) T )(∇ θ kΨ θ k ) −T wherê Ψ θ k = n −1 k n k i=1 ψ θ k (X k,i ;θ k ); 3 Send (φ k ,Ĥ k (θ k ) −1 ) to a central server and construct φ W D := { K k=1 n kĤk (θ k ) −1 } −1 K k=1 n k (Ĥ k (θ k )) −1φ k ; 4φ W D :=φ W D I(φ W D ∈ Φ) +φ SaC I(φ W D ∈ Φ), whereφ SaC := N −1 K k=1 n kφk . term is negligible compared with that ofφ W D I(φ W D ∈ Φ). We need the following assumption in order to establish the MSE bound and asymptotic properties of the proposed WD estimator. Assumption 6. (Boundedness) There exists constants ρ σ , c > 0 such that for k ≥ 1, |||Σ S,k (θ * k )||| 2 ≤ ρ σ , H k cI p 1 ×p 1 , where Σ S,k (θ k ) = Eψ θ k (X k,1 ; θ k )ψ θ k (X k,1 ; θ k ) T . By the definition of H k (θ k ), we have that |||H k ||| 2 ≤ Ψ θ θ (θ * k ) −1 Σ S,k (θ * k )Ψ θ θ (θ * k ) −1 2 ≤ Ψ θ θ (θ * k ) −1 2 2 |||Σ S,k (θ * k )||| 2 ≤ ρ σ ρ 2 − , which implies H k (θ * k ) −1 ρ 2 − ρσ I p 1 ×p 1 . On the other hand, the above inequality leads to Ψ θ θ (θ E φ W D − φ * 2 2 ≤ C 1 nK + C 2 n 2 + C 3 n 2 K + C 4 n 3 + C 5 K nv , for n = N K −1 andv = min{v, v 1 2 }.v, v 1 ≥ 2, if K = o(n), φ W D − φ * ) T { K k=1 n k H k (θ * k ) −1 } φ W D − φ * ) d → χ 2 p 1 . Although {H k (θ * k )} K k=1 have bounded spectral norms, K k=1 n k N H k (θ * k ) −1 may not con- verge to a fixed matrix in presence of heterogeneity. Thus, we can only obtain the asymptotic normality of the standardized √ N { K k=1 n k N H k (θ * k ) −1 } 1/2 (φ DW − φ * ) . This is why Theorem 3 is formulated in a limiting chi-squared distribution form. The asymptotic normality implies that we can construct confidence regions for φ with confidence level 1 − α as {φ | φ W D − φ) T { K k=1 n kĤk (θ k ) −1 } φ W D − φ) ≤ χ 2 p 1 ,α } (24) after replacing K k=1 n k H k (θ * k ) −1 with its sample counterpart K k=1 n kĤk (θ k ) −1 , where χ 2 p 1 ,α is the upper α quantile of the χ 2 p 1 distribution. The block-specific parameter λ k can also be of interest. Then given the WD estimator of the common parameter φ * , a question is that whether a more efficient estimator of λ * k can be obtained. Specifically, we plug in the WD estimator to each data block and re-estimate λ k . The corresponding updated estimator is denoted asλ (2) k . Actually, the answer is thatλ (2) k is not necessarily more efficient thanλ k . Due to space limit, more discussions on this aspect are available in Section 1.8 in SM. [15] showed that the MLE is asymptotically biased when K/n → C ∈ (0, +∞). This calls for a debias step for the local estimators before aggregation to allow for larger K, which is needed especially in the Federated Learning scenario where the number of users (data blocks) can be much larger than the size of local data. To facilitate the bias correction operation, we have to simplify the notations. Suppose F (θ) is a p × 1 vector function, ∇F (θ) is the usual Jacobian whose l-th row contains the partial derivatives of the l-th element of F (θ). Then the matrices of higher derivatives are defined recursively so that the j-th element of the l-th row of ∇ s L(θ) (a p × p s matrix) is the 1 × p vector f v lj (θ) = ∂f v−1 lj (θ)/∂θ T , where f v−1 lj is the l−th row and j-th element of ∇ v−1 F (θ). We use ⊗ to denote a usual Kronecker product. Using Kronecker product we can express ∇ v F (θ) = ∂ v F (θ) ∂θ T ⊗∂θ T ⊗···⊗∂θ T . Besides, let M n,k (θ k ) = n −1 k n k i=1 M (X k,i ; θ k ),H 3,k (θ k ) = E∇ 2 θ k ψ θ k (X k,1 ; θ k ), Q k (θ k ) = {−E∇ θ k ψ θ k (X k,1 ; θ k )} −1 , d i,k (θ k ) = Q k (θ k )ψ θ k (X k,i ; θ k ) and v i,k (θ k ) = ∇ θ k ψ θ k (X k,i , θ k ) − ∇ θ k Ψ θ (θ k ). According to [21], the leading order bias ofθ k is Bias(θ k ) = n −1 k Q k (θ * k ) Ev i,k (θ * k )d i,k (θ * k ) + 1 2 H 3,k (θ * k )E{d i,k (θ * k ) ⊗ d i,k (θ * k )} .(25)Let B k (θ k ) = Q k (θ k ) Ev i,k (θ k )d i,k (θ k ) + 1 2 H 3,k (θ k )E{d i,k (θ k ) ⊗ d i,k (θ k )} , whose the first p 1 dimension associated with φ are denoted as B 1 k (θ k ). The empirical estimator of B k (θ k ) isB k (θ k ) =Q k (θ k ) n −1 k n k i=1v i,k (θ k )d i,k (θ k ) + 1 2Ĥ 3,k (θ k )n −1 k n k i=1 (d i,k (θ k ) ⊗d i,k (θ k )) (26) whereĤ 3,k (θ k ) = n −1 k n k i=1 ∇ 2 θ k ψ θ k (X k,i ; θ k ),Q k (θ k ) = {−n −1 k n k i=1 ∇ θ k ψ θ k (X k,i ; θ k )} −1 , d i,k (θ k ) =Q k (θ k )ψ θ k (X k,i ; θ k ) andv i,k (θ k ) = ∇ θ k ψ θ k (X k,i ; θ k ). Applying bias correction to each data block, we have the bias-corrected local estimator θ k,bc :=θ k − n −1 kB k (θ k )1 E k,bc ,(27) where E k,bc = {θ k − n −1 kB k (θ k ) ∈ Θ k }. The indicator function here is to ensure thatθ k,bc is within the parameter space. After the local debiased estimators are obtained, we need to aggregate them with estimated weights. However, a direct aggregation will invalidate the bias correction procedure due to the correlation between the estimated weights and the local debiased estimator as they are constructed with the same dataset. The accumulation of the dependence over a large number of data blocks can make the bias correction fail. To remove such correlation between the local estimators and the corresponding esti- mated local weightsŴ k = { K s=1Ĥ s (θ s ) −1 } −1Ĥ k (θ k ) −1 , we first divide each local dataset {X k,i } n k i=1 into two equal-sized subsets D s k = {X (s) k,i } n k /2 i=1 , s = 1, 2. Then, for s = 1, 2 we calculate the local M-estimatorsθ k,s and obtainĤ k,s (θ k,s ), which is the leading principal sub-matrix of order p 1 of i=1 , s = 1, 2 ; 2 Obtain the initial estimatesθ k,s = (φ k,s ,λ k,s ) based on data from D s k , s = 1, 2 ; 3 CalculateĤ k,s (θ k,s ) in each block (s = 1, 2), which is the leading principal sub-matrix of order p 1 of (∇ θ kΨ (s) To provide theoretical guarantee on the bias correction, we need an assumption on the third derivative of the M-function M (see [30]), which strengthens part of Assumption 5. (∇ θ kΨ θ k ) −1 ( 1 n k /2 n k /2 i=1 ψ θ k (X (s) k,i ;θ k,s )ψ θ k (X (s) k,i ;θ k,s ) T )(∇ θ kΨ θ k ) −T , whereΨ θ k = 1 n k /2 n k /2 i=1 ψ θ k (Xθ k ) −1 (2n −1 k n k /2 i=1 ψ θ k (X (s) k,i ;θ k,s )ψ θ k (X (s) k,i ;θ k,s ) T )(∇ θ kΨ (s) θ k ) −T wherê Ψ θ k = 2n −1 k n k /2 i=1 ψ θ k (X(∇ 2 θ k ψ θ k (x; θ k ) − ∇ 2 θ k ψ θ k (x; θ k ))(u ⊗ u) 2 ≤ A(x) θ k − θ k 2 u 2 2 , for all θ k , θ k ∈ U k defined in Assumption 5 and u ∈ R p , where EA(X k,i ) 2v ≤ A 2v for some v > 0 and A < ∞. Theorem 4. Under Assumptions 1 -4 and 6 -7, and Assumption 5 with v, v 1 ≥ 4 , E φ dW D − φ * 2 2 ≤ C 1 nK + C 2 n 2 K + C 3 n 3 + C 4 K nv , wherev = min{v, v 1 2 }.(φ dW D − φ * ) T { K k=1 n k H k (θ * k ) −1 }(φ dW D − φ * ) d → χ 2 p 1 . Note that the reason why Theorem 5 is formulated in the chi-squared distribution form is the same as that when we formulate Theorem 3, and similar confidence region with confidence level 1 − α can be constructed as {φ| φ dW D − φ) T { K k=1 n k H k (θ k ) −1 } φ dW D − φ) ≤ χ 2 p 1 ,α }.(28) The fact that the confidence regions of dWD and WD estimators use the same standardizing matrix K k=1 n k H k (θ k ) −1 reflects that the dWD and WD estimators have the same estimation efficiency. However, the debiased version has more relaxed constraint on K = O(n 2 ) (which is equivalent to K = o(N 2/3 )) than that of the WD estimator at K = o(n) (K = o( √ N )). A more communication-efficient estimator of the common parameter can be defined as the following debiased SaC (dSaC) estimator: φ dSaC = N −1 K k=1 n k (φ k − n −1 kB 1 k (θ k )1 E k,bc ),(29) which only performs bias correction and may be preferable when the heterogeneity is not large. The asymptotic property of the dSaC estimator is summarized in the following proposition. Theorem 6. Under the conditions required by Theorem 4, if K = o(n 2 ), E φ dSaC − φ * 2 2 ≤ C 1 nK + C 2 n 2 K + C 3 n 3 and N 2 (φ dSaC − φ * ) T { K k=1 n k H k (θ * k )} −1 (φ dSaC − φ * ) d → χ 2 p 1 . The corresponding confidence region with confidence level 1 − α can be constructed as {φ|N 2 φ dSaC − φ) T { K k=1 n k H k (θ k )} −1 φ dSaC − φ) ≤ χ 2 p 1 ,α }.(30) It is noted that the dSaC and SaC estimators have the same asymptotic distribution. Hence, the confidence regions based on the SaC estimator can be constructed as (30) witĥ φ dSaC replaced byφ SaC . To compare with the subsampled average mixture method (SAVGM) estimator proposed in [30] which also performs local bias correction but under the homogeneous setting, we have the following corollary to Theorem 6. E θ dSaC − θ * 1 2 2 ≤ 2E ∇ θ 1 Ψ θ (θ * 1 ) −1 ψ θ 1 (X 1,1 ; θ * 1 ) 2 2 nK + C 1 n 2 K + C 2 n 3 ,(31) where θ * 1 is the true parameter for all the K data blocks. The SAVGM estimator resamples rn k data points from each data block k for a r ∈ 1 − r ,(32) whose MSE bound as given in Theorem 4 of [30] is E θ SAV GM − θ * 1 2 2 ≤ 2 + 3r (1 − r) 2 E ∇ θ 1 Ψ θ (θ * 1 ) −1 ψ θ 1 (X 1,1 ; θ * 1 ) 2 2 nK + C 1 n 2 K + C 2 n 3 .(33) Thus, the MSE bound (33) of the SAVGM estimator has an inflated factor 2+3r 2(1−r) 2 > 1 for r ∈ (0, 1), when compared with that of the dSaC estimator, although it is computationally more efficient than the dSaC and dWD estimators as it only draws one subsample in its resampling. For more comparisons between the dSaC estimator and one-step estimators proposed by Huang and Huo (2019) [10], see Section 1.10 in SM. Simulation Results We report results from simulation experiments designed to verify two sets theoretical findings made in the previous sections. One was to confirm the finding in Section 2 that the full sample estimatorφ f ull is not necessarily more efficient than the SaC estimatorφ SaC . The other was to evaluate the numerical performance of the newly proposed weighted distributive (WD) , debaised SaC (dSaC) and debiased WD (dWD) estimators of the common parameter and compare them with the existing SaC and subsampled average mixture method (SAVGM) (with subsampling rate r = 0.05) estimators. Although the SAVGM estimator [30] was proposed under the homogeneous setting, but since its main bias correction is performed locally on each data block k as shown in (32), similar theoretical bounds as formula (33) can be derived without much modifications on the original proof. Throughout the simulation experiments, the results of each simulation setting were based on B = 500 number of replications and were conducted in R paralleled with a single 10-core Intel(R) Core(TM) i9-10900K @3.7 GHz processor. In the first simulation experiment, we simulated the errors-in-variables Model (13) with the objective function (14) to compare the performance of the full sample, the SaC and the WD estimators:φ f ull ,φ SaC andφ W D . The simulation was carried out by first generating IID {Z i,k } from N (µ Z , σ 2 Z ) , and then upon given a Z i,k , (X k,i , Y i,k ) T were independently drawn from N (Z i,k , φ * + λ * k Z i,k ) T , σ 2 I 2×2 . We chose φ * = 1, K = 2, σ 2 = 1 and n 1 = n 2 = 5 × 10 4 = N/2, and λ * 1 , λ * 2 , µ Z and σ 2 Z were those reported in Table 1 under four scenarios. As discussed in Section 2, the relative efficiency ofφ f ull toφ SaC depends on the ratio σ 2 (EZ) 2 /(var(Z)EZ 2 ) as shown in (15). We designed four scenarios according to the above ratio under λ * 1 = λ * 2 and EZ = 0, respectively, which represented the settings where the full sample estimatorφ f ull would be less (Scenario 1) or more (Scenario 2) efficient than the SaC estimator as predicted by the ratio, but not as efficient as the weighted distributed estimatorφ W D . Scenario 3 (λ * 1 = λ * 2 , EZ = 0) was the case whenφ f ull andφ W D would be asymptotically equivalent, and both estimators would be more efficient thanφ SaC . Scenario 4 was the homogeneous case with λ * 1 = λ * 2 in which all the three estimators would have the same asymptotic efficiency. For all the four scenarios, the ARE column of the Table 1 confirmed the relative efficiency as predicted by the asymptotic variances in (15), and was well reflected in the comparison of the RMSEs, as the bias is of smaller order as compared with that of the SD and thus negligible. In the second simulation experiment, we evaluated the numerical performance of the five estimators for the common parameter φ under a logistic regression model. For each of K data block with K ∈ {10, 50, 100, 250, 500, 1000, 2000}, {(X k,i ; Y k,i )} n i=1 ⊂ R p × {0, 1} were independently sampled from the following model: X k,i i.i.d ∼ N (0 p×1 , 0.75 2 I p×p ) and P (Y k,i = 1|X k,i ) = exp(X T k,i θ * k ) 1 + exp(X T k,i θ * k ) , where θ * k = (φ * , λ * T k ) T , φ * = 1, λ * k = (λ * k,1 , λ * k,2 , · · · , λ * k,p 2 ) T and λ * k,j = (−1) j 10(1 − 2(k−1) K−1 ). The sample sizes of the data blocks were equal at n = N K −1 with N = 2 × 10 6 . Two levels of the dimension p 2 = 4 and 10 of the nuisance parameter λ k were considered. A derivation Table 1: Average root mean squared error (RMSE) and the standard deviation (SD), multiplied by 10 2 , of the full sample estimatorφ f ull , the SaC estimatorφ SaC and the WD estimatorφ W D under four scenarios for the errors-in-variables model (12) for N = 10 5 , K = 2 and n 1 = n 2 . AREs (asymptotic relative efficiency) ofφ f ull toφ SaC are calculated from (15 of the bias correction formula for the logistic model is given in Section 1.9 of the SM. We also evaluated the coverage probabilities and widths of the 1−α (α = 0.01, 0.05, 0.1) confidence intervals (CIs) of the common parameter based on the asymptotic normality as given after Theorems 3 and 5. The SAVGM estimator was not included as its asymptotic distribution was not made available in [30]. CIs had comparable coverages with the dWD CIs, their widths were much wider than those of the dWD. This was largely due to the fact that the weighted averaging conducted in the weighted distributed estimation reduced the variation and hence the width of the CIs. The widths of the WD CIs were largely the same with those of the dWD, and yet the coverage levels of the dWD CIs were much more accurate indicating the importance of the bias correction as it shifted the CIs without inflating the width. In addition to the simulation experiments on the statistical properties of the estimators, the computation efficiency of the estimators was also evaluated. Table 3 reports the average CPU time per simulation run based on 500 replications of the five estimators for a range of K and dimension p 2 of the nuisance parameter for the logistic regression model with the total sample size N = 2 × 10 6 . The computation speed of the dSaC and dWD estimators were relatively slower than those of the SaC, WD and SAVGM estimators. The WD estimator was quite fast, which means that the re-weighting used less computing time than the bias-reduction. In comparison, the dWD estimator was the slowest as a cost for attaining the best RMSE among the five estimators in all settings. It is observed in Table 3 that the overall computation time for each estimator first decreased and then increased as K became larger. The decrease in time was because the benefit of the distributed computation, while the increase was due to the increase in the number of optimization associated with the M-estimation performed as K got larger. However, it is worth mentioning that these results did not account for the potential time expenditure in data communication among different data blocks. Discussion This paper investigates several distributed M-estimators in the presence of heterogeneous distributions among the data blocks. The weighted distributed (WD) estimator is able to improve the estimation efficiency of the "Split-And-Conquer" (SaC) estimator for the common parameter. Two debiased estimators ( dWD and dSaC) are proposed to allow for larger numbers of data blocks K. The statistical properties of these three estimators are shown to be advantageous over the SaC and SAVGM estimators. In particular, the WD estimator has good performance for smaller K relative to n, and the debiased WD estimator that conducted both bias correction and weighting offers good estimation accuracy for large K. An important issue for the distributed estimation is the size of K relative to the local average sample size n. This is especially true in Federated Learning setting where the number of clients (data blocks) are usually very large. Both SaC and WD estimators require K = o( √ N ) to preserve the O(N −1 ) convergence rate for its MSE and the √ N rate for the asymptotic variance. The debiased dWD and dSaC relax the restriction to K = o(N 2/3 ) without compromising the convergence rate. The dSaC may be used as a computationally cheaper version of the dWD at the cost of larger variations and wider confidence regions when compared with dWD. Figure 1: Average simulated bias (a, c) and the root mean square errors (RMSE) (b,d) of the weighted distributed (WD) (red circle), the SaC (blue triangle), the debiased SaC (dSaC) (green square), the debiased WD (dWD) (purple cross), the subsampled average mixture SAVGM (pink square cross) estimators, with respect to the number of data block K for the logistic regression model with the dimension p 2 of the nuisance parameter λ k being 4 and 10, respectively with the full sample size N = 2 × 10 6 . Table 2: Coverage probabilities and widths (in parentheses, multiplied by 100) of the 1 − α confidence intervals for the common parameter φ in the logistic regression model based on the asymptotic normality of the SaC, the WD, the debiased SaC and the debiased WD estimators with respect to the number of data blocks K for two dimensions of the heterogeneous parameter p 2 with the full sample size N = 2 × 10 6 . a positive semi-definite matrix. These three assumptions are standard ones on the parameter space and population objective functions as those in Zhang et al. (2013) [30] and Jordan et al. (2019) [11] for the homogeneous case. In the heterogeneous setting, Duan et al. (2021) [6] only requires the parameter space for the common parameter to be bounded, i.e. φ − φ * ≤ r under a Proposition 1 . 1Under Assumptions 1 -4 and Assumption 5 with v, v 1 ≥ 1, and if K kφk andφ f ull are consistent to φ * . Theorem 1. Under Assumptions 1 -4 and Assumption 5 with v, v 1 ≥ 2, if K is a fixed constant, the SaC estimatorφ SaC and the full sample estimatorφ f ull satisfy and this indicate a finite upper bound for the norm of the Hessian matrix, just as that assumed in Jordan et al. (2019) [11] and Duan et al. (2021) [6]. Theorem 2 . 2Under Assumptions 1 -4 and 6, and Assumption 5 with v, v 1 ≥ 2 , the mean-squared error of the WD estimatorφ W D satisfies The v and v 1 appeared in Assumption 5 quantify the moments of the first two orders of the derivatives of the M function and their corresponding Lipschitz functions. When the number of data blocksK = O(n min{1,v −1 2 } ), the convergence rate of MSE ofφ W D is O((nK) −1 ), which is the same as the standard full sample estimator. However, when there are too many data blocks such that K >> n, the convergence rate is reduced to O(n −2 ). Furthermore, if the derivatives of the M function and their corresponding Lipschitz functions are heavy-tailed, sayv < 3, the convergence rate is further reduced to O(Kn −v ). Theorem 3 . 3Under Assumptions 1 -4 and 6, and Assumption 5 with i ;θ k,s ). We then perform the local bias correction to {θ k,s } based on data in subset D s k to attain the debiased estimators {θ bc k,s }. At last, averaged to obtain the final debiased WD (dWD) estimator, whose procedure is summarized in Algorithm 2. That the weight estimation and the debiasing are conducted on different data splits remove the correlation, and realize the gain of biascorrection procedure.Algorithm 2: debiased Weighted Distributed (dWD) Estimator Input: {X k,i , k = 1, ..., K; i = 1, ..., n k } Output:φ dW D 1 For each data block, split the data set into two non-overlapping equal-sized subsets and denote those subsets as D s k = {X i ;θ k,s );4 Calculate the bias corrected estimators in each block (k = 1, 2, · · · , K; s = 1, 2): θ bc k,s :=θ k,s − 2n −1 kB k,s (θ k,s )1 E k,bc,s where E k,bc,s := {θ k,s − 2n −1 kB k,s (θ k,s ) ∈ Θ k }. Denote the first p 1 dimensions ofθ bc k,s asφ bc k,s ; 5 Send {φ bc k,s ,Ĥ k,1 (θ k,s ) −1 , s = 1, 2} to a central server and construct φ dW D s := { K k=1 n kĤk,s (θ k,s ) −1 } −1 K k=1 n k (Ĥ k,s (θ k,s )) Assumption 7 . 7(Strong smoothness) For each x ∈ R p , the third order derivatives of M (x; θ k ) with respect to θ k exist and are A(x)− Lipschitz continuous, i.e. The main difference between the upper bounds in Theorem 4 and that of Theorem 2 for the WD estimator is the disappearance of the O(n −2 ) term for the WD estimator, which has been dissolved and absorbed into the O((n 2 K) −1 ) and O(n −3 ) terms for the dWD estimator. As shown next, this translates to more relaxed K = o(n 2 ) as compared with K = o(n) for the WD estimator in Theorem 3.Theorem 5. Under the conditions required by Theorem 4, if K = o(n 2 ), Corollary 1 . 1Under the homogeneous case such that {X k,i , k = 1, ..., K, i = 1, ..., n; } are IID distributed, and the assumptions required by Theorem 4, Figure 1 1reports the root mean square errors (RMSEs) and absolute bias of the estimators. It is observed that the weighted distributed estimator WD, and the two debiased estimators dSaC and dWD had smaller RMSE than those of the SaC and SAVGM for almost all the simulation settings. Between the SaC and SAVGM, the SAVGM fared better in the lower dimensional case of p 2 = 4, but was another way around for p 2 = 10. It was evidence that the WD estimator had much smaller RMSEs than the SaC and SAVGM estimators for all the block number K, realizing its theoretical promises. In most cases the WD estimator had smaller bias than the SaC estimator although it was not debiased.It also had smaller RMSEs than the debiased SaC estimator dSaC for almost all cases of the block numbers for p 2 = 4, while in the higher dimensional p 2 = 10 the WD estimator was advantageous for K ≤ 250. The latter indicated the need for conducting the bias correction to the WD estimator. Both bias corrected dWD and dSaC were very effective in reducing the bias of the WD and SaC estimators, respectively, especially for larger K when the bias was more severe. The debiased WD attained the smallest RMSEs and the bias in all settings, suggesting the need for conducting both weighting and the bias correction in the distributed inference especially for large K. These empirical results were consistent with Theorems 2 and 4, namely the leading RMSE term of the WD estimator changes from O((Kn) −1 ) to O(n −2 ) when K surpasses the local sample size n, while the leading RMSEs of the dWD is still O((nK) −1 ) until K >> n 2 . ).φf ullφ SaCφW D Scenario (λ * 1 , λ * 2 ) ARE RMSE SD RMSE SD RMSE SD Scenario 1 (0.25,3.25) 0.89 4.55 4.51 4.12 4.09 3.91 3.89 (µ Z = 1, σ 2 Z = 0.1) (0.5,3.5) 0.93 4.65 4.65 4.35 4.35 4.08 4.08 (0.75,3.75) 0.97 4.52 4.52 4.40 4.38 4.13 4.13 Scenario 2 (0.25,2.25) 1.18 2.95 2.95 3.24 3.24 2.89 2.89 (µ Z = 3, σ 2 Z = 0.5) (0.75,2.75) 1.28 3.28 3.26 3.65 3.64 3.17 3.16 (1.25,3.25) 1.31 3.71 3.71 4.16 4.07 3.64 3.61 Scenario 3 (0.25,2.25) 1.97 0.41 0.41 0.61 0.61 0.41 0.41 (µ Z = 0, σ 2 Z = 0.5) (0.75,2.75) 1.92 0.51 0.51 0.70 0.70 0.51 0.51 (1.25,3.25) 1.68 0.64 0.64 0.82 0.82 0.64 0.64 Scenario 4 (0.5,0.5) 1 3.25 3.24 3.31 3.28 3.30 3.26 (µ Z = 4, σ 2 Z = 0.5) (1.0,1.0) 1 3.53 3.53 3.59 3.59 3.59 3.59 (1.5,1.5) 1 4.06 4.03 4.08 4.07 4.06 4.06 Table 2 2reports the empirical coverage and the average width of the CIs. It is observed that for the lower dimensional nuisance parameter case of p 2 = 4 the four types of the CIs all had quite adequate coverage levels when K ≤ 100. However, for K ≥ 250, the SaC CIs first started to lose coverage, followed by those of the WD, while the CIs of the debiased SaC (dSaC) and debiased weighted distributed (dWD) estimators can hold up to the promised coverage for all cases of K. The outstanding performance of the dSaC and dWD CIs was largely replicated for the higher dimensional nuisance parameter case of p 2 = 10, while the other two non-debiased estimator based CIs had their coverage quickly slipped below the nominal coverage levels. Although the dSaC Table 3 : 3Average CPU time for each replication based on B = 500 replications for the SaC, the SAVGM, the WD, the debiased SaC and the debiased WD estimators for the logistic regression model with respect to K and the dimension p 2 of the nuisance parameter. Total sample size N = 2 × 10 6 .K SaC SAV GM W D dSaC dW D p 2 = 4 10 15.65 15.97 18.50 20.00 21.95 50 9.63 9.95 10.66 12.37 14.59 100 8.09 8.63 8.76 10.50 12.05 250 8.49 9.69 9.07 10.84 12.82 500 9.68 11.58 10.25 11.97 14.84 1000 11.67 13.81 12.32 13.93 19.08 2000 15.78 19.68 16.57 18.11 28.55 p 2 = 10 10 34.60 35.19 43.84 50.47 55.35 50 20.13 20.18 24.16 29.99 33.69 100 15.60 16.20 17.74 23.63 24.47 250 10.77 12.61 11.88 18.22 20.39 500 11.55 14.50 12.56 18.80 23.73 1000 15.23 18.27 16.28 22.38 32.24 2000 23.42 27.99 24.62 30.43 48.05 Approximate confidence intervals. M Bartlett, Biometrika. 40Bartlett, M. (1953). Approximate confidence intervals. Biometrika, 40:12-19. Distributed testing and estimation under sparse high dimensional models. H Battey, J Fan, H Liu, J Lu, Z Zhu, The Annals of Statistics. 46Battey, H., Fan, J., Liu, H., Lu, J., and Zhu, Z. (2018). Distributed testing and estimation under sparse high dimensional models. The Annals of Statistics, 46:1352- 1382. 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Rmse, 10RMSE (p 2 = 10) 36) (1.79) (1.51) (1.96) (1.49) (1.25) 250 0. 99 0.94 0.88 0.99 0.94 0.88 0.99 0.93 0.88 (2.38) (1.81) (1.52) (1.96) (1.49) (1.25) (2.38) (1.81) (1.52) (1.97) (1.50) (1.25) 2000 0.01 0.01 0.00 0.99 0.91 0.81 0.98 0.94 0.88 0.99 0.94 0.90 (2.4299 0.93 0.85 0.99 0.95 0.90 0.99 0.96 0.91 0.99 0.95 0.90 (2.36. 1.25) (2.36) (1.80) (1.51) (1.96) (1.49) (1.25) 1000 0.65 0.41 0.28 0.. 1.84) (1.55) (1.96) (1.50) (1.25) (2.42) (1.84) (1.55) (1.98) (1.50) (1.26(2.36) (1.80) (1.51) (1.97) (1.50) (1.26) (2.36) (1.80) (1.51) (1.97) (1.50) (1.26) 100 0.98 0.94 0.91 0.99 0.95 0.91 0.99 0.95 0.91 0.99 0.95 0.91 (2.36) (1.79) (1.51) (1.96) (1.49) (1.25) (2.36) (1.79) (1.51) (1.96) (1.49) (1.25) 250 0.99 0.93 0.85 0.99 0.95 0.90 0.99 0.96 0.91 0.99 0.95 0.90 (2.36) (1.79) (1.50) (1.96) (1.49) (1.25) (2.36) (1.79) (1.50) (1.96) (1.49) (1.25) 500 0.91 0.77 0.66 0.99 0.95 0.88 0.99 0.96 0.90 0.99 0.95 0.89 (2.36) (1.80) (1.51) (1.96) (1.49) (1.25) (2.36) (1.80) (1.51) (1.96) (1.49) (1.25) 1000 0.65 0.41 0.28 0.99 0.94 0.88 0.99 0.94 0.88 0.99 0.93 0.88 (2.38) (1.81) (1.52) (1.96) (1.49) (1.25) (2.38) (1.81) (1.52) (1.97) (1.50) (1.25) 2000 0.01 0.01 0.00 0.99 0.91 0.81 0.98 0.94 0.88 0.99 0.94 0.90 (2.42) (1.84) (1.55) (1.96) (1.50) (1.25) (2.42) (1.84) (1.55) (1.98) (1.50) (1.26)
[]
[ "Expansion of a coherent array of Bose-Einstein condensates", "Expansion of a coherent array of Bose-Einstein condensates" ]
[ "P Pedri ", "L Pitaevskii \nDipartimento di Fisica\nUniversità di Firenze and Istituto Nazionale per la Fisica della Materia, L.go E\nKapitza Institute for Physical Problems\nFermi 2, I50125Moscow, FirenzeRussia, Italy\n", "S Stringari ", "C Fort ", "S Burger ", "F S Cataliotti ", "P Maddaloni ", "F Minardi ", "M Inguscio ", "\nDipartimento di Fisica\nUniversità di Trento and Istituto Nazionale per la Fisica della Materia\nI-38050PovoItaly\n" ]
[ "Dipartimento di Fisica\nUniversità di Firenze and Istituto Nazionale per la Fisica della Materia, L.go E\nKapitza Institute for Physical Problems\nFermi 2, I50125Moscow, FirenzeRussia, Italy", "Dipartimento di Fisica\nUniversità di Trento and Istituto Nazionale per la Fisica della Materia\nI-38050PovoItaly" ]
[]
We investigate the properties of a coherent array containing about 200 Bose-Einstein condensates produced in a far detuned 1D optical lattice. The density profile of the gas, imaged after releasing the trap, provides information about the coherence of the ground-state wavefunction. The measured atomic distribution is characterized by interference peaks. The time evolution of the peaks, their relative population as well as the radial size of the expanding cloud are in good agreement with the predictions of theory. The 2D nature of the trapped condensates and the conditions required to observe the effects of coherence are also discussed.
10.1103/physrevlett.87.220401
[ "https://arxiv.org/pdf/cond-mat/0108004v1.pdf" ]
8,337,901
cond-mat/0108004
c58e03ab9de3bae55e68cdc2ffc4b618facc6178
Expansion of a coherent array of Bose-Einstein condensates 1 Aug 2001 P Pedri L Pitaevskii Dipartimento di Fisica Università di Firenze and Istituto Nazionale per la Fisica della Materia, L.go E Kapitza Institute for Physical Problems Fermi 2, I50125Moscow, FirenzeRussia, Italy S Stringari C Fort S Burger F S Cataliotti P Maddaloni F Minardi M Inguscio Dipartimento di Fisica Università di Trento and Istituto Nazionale per la Fisica della Materia I-38050PovoItaly Expansion of a coherent array of Bose-Einstein condensates 1 Aug 2001(July 7, 2021)PACS numbers: 0375Fi, 3280Pj We investigate the properties of a coherent array containing about 200 Bose-Einstein condensates produced in a far detuned 1D optical lattice. The density profile of the gas, imaged after releasing the trap, provides information about the coherence of the ground-state wavefunction. The measured atomic distribution is characterized by interference peaks. The time evolution of the peaks, their relative population as well as the radial size of the expanding cloud are in good agreement with the predictions of theory. The 2D nature of the trapped condensates and the conditions required to observe the effects of coherence are also discussed. Coherence is one of the most challenging features exhibited by Bose-Einstein condensates. On the one hand it underlies the superfluid phenomena exhibited by these cold atomic gases. On the other hand it characterizes in a unique way their matter wave nature at a macroscopic level. Coherence requires that the system be characterized by a well defined phase, giving rise to interference phenomena. After the first interference measurements carried out on two expanding condensates at MIT [1] the experimental study of interference in Bose-Einstein condensed gases has become an important activity of research opening the new field of coherent atom-optics. The possibility of confining Bose-Einstein condensates in optical lattices has opened further perspectives in the field [2]. Bose-Einstein condensates confined in an optical standing-wave provide in fact a unique tool to test at a fundamental level the quantum properties of systems in periodic potentials. The observation of interference patterns produced by an array of condensates trapped in an optical lattice was already used as a probe of the phase properties of this system [3,4] also allowing to proof the phase relation in an oscillating Josephson current [5]. In [3] the interference effect has been used to explore the emergence of number squeezed configurations in optically trapped condensates. The main purpose of this paper is to investigate the ground state properties of the system of a fully coherent array of condensates. To this aim we have explored the interference pattern in the expanded cloud, reflecting the initial geometry of the sample. The basic phenomenon we want to explore is the atom optical analog of light diffraction from a grating. The analogy is best understood considering a periodic and coherent array of identical condensates aligned along the x-axis. In momentum space the order parameter takes the form Ψ(p x ) = Ψ 0 (p x ) k=0,±1..±kM e ikpxd h = Ψ 0 (p x ) sin[(2k M + 1)p x d/2h] sin p x d/2h(1) where k labels the different sites of the lattice, 2k M + 1 is the total number of sites (in the following we will assume k M ≫ 1) and d is the distance between two consecutive condensates. The quantity n 0 (p x ) = |Ψ 0 (p x )| 2 is the momentum distribution of each condensate (see Eq. (5) below). The momentum distribution of the whole system, given by n(p x ) =| Ψ(p x ) | 2 , is affected in a profound way by the lattice structure and exhibits distinctive interference phenomena. Actually the effects of coherence are even more dramatic than in the case of two separated condensates [6]. Indeed, in the presence of the lattice the momentum distribution is characterized by sharp peaks at the values p x = n2πh/d with n integer (positive or negative) whose weight is modulated by the function n 0 (p x ). Furthermore, differently from the case of two separated condensates, interference fringes appear only if the initial configuration is coherent. In principle the momentum distribution can be directly measured in situ using 2-photon Bragg spectroscopy. This possibility has been already implemented experimentally for a single condensate [7]. However, the very peculiar structure of (1) is expected to influence in a deep way also the expansion of the atomic cloud after the release of the trap. The width of the central peak (n = 0) of the momentum distribution is of the order ∆p x ∼h/R x where R x ∼ k M d is half of the length of the whole sample in the x-direction and the corresponding atomic motion, after the release of the trap, will be consequently slow. On the other hand the peaks with n = 0 carry high momentum and the center of mass of these peaks will expand fast according to the asymptotic law x(t) = ±n 2πh dm t .(2) The occurrence of these peaks is the analog of multiple order interference fringes in light diffraction. We create an array of BECs of 87 Rb in the |F = 1, m F = −1 state by superimposing the periodic optical potential V opt of a far detuned standing-wave on the harmonic potential V B of the magnetic trap. For a more detailed description see [8,9]. The resulting potential is given by V = V B + V opt = 1 2 m ω 2 x x 2 + ω 2 ⊥ (y 2 + z 2 ) + sE R cos 2 (qx + π 2 ) (3) with m the atomic mass, ω x = 2π × 9 Hz and ω ⊥ = 2π × 92 Hz the axial and radial frequency of the magnetic harmonic potential and x lying in the horizontal plane. In (3) s is a dimensionless factor, q = 2π/λ is the wavevector of the laser light creating the standing wave and producing local minima in V opt separated by d = λ/2 and E R =h 2 q 2 /2m ∼ 2πh × 3.6 kHz is the recoil energy of an atom absorbing one lattice photon. By varying the intensity of the laser beam (detuned 150 GHz to the blue of the D 1 transition at λ = 795 nm) up to 14 mW/mm 2 we can vary the intensity factor s from 0 to 5. We calibrated the optical potential measuring the Rabi frequency (Ω R ) of the Bragg transition between the momentum states −hq and +hq induced by the standing wave. The intensity factor is then given by s = 2hΩ R /E R [10]. The procedure to load the condensate in the combined (magnetic+optical) trap is the following: we load 87 Rb atoms in the magnetic trap and cool the sample via rf-forced evaporation until a significant fraction of condensed atoms is produced. We then switch on the laser standing-wave and continue the evaporative cooling to a lower temperature (T≪T c ). Typically, the BEC splits over ∼ 200 wells, each containing 100 ∼ 500 atoms. After switching off the combined potential we let the system expand and take an absorption image of the cloud at different expansion times t exp . In Fig. 1A we show a typical image of the cloud taken at t exp = 29.5 ms, corresponding to a total number of atoms N = 20000 and to a laser intensity s = 5. From the images taken after the expansion we can determine the relative population of the lateral peak with respect to the central one. The experimental results for the relative population of the first lateral peak as a function of the laser intensity s are shown in Fig. 2. The structure of the observed density profiles is well reproduced by the free expansion of the ideal gas where the time evolution of the order parameter, in coordinate space, takes the form: Ψ(x, t) = 1 (2π) 3 dp x Ψ(p x )e ipxx/h e −ip 2 x t/2mh . (4) For a realistic description of Ψ(p x ) we have improved the simple ansatz (1) in order to account for the kdependence of the number of atoms N k contained in each well. Due to magnetic trapping, the central condensates with k ≪ k M will be in fact more populated than the ones occupying the sites at the periphery. We have accounted for the modulation by the simple law N k = N 0 (1 − k 2 /k 2 M ) 2 which will be derived below. For Ψ 0 we have made the Gaussian choice Ψ 0 (p x ) ∝ exp[−p 2 x σ 2 /2h 2 ](5) corresponding, in coordinate space, to Ψ 0 (x) ∼ exp[−x 2 /2σ 2 ]. Using (5) it is immediate to find that the relative population of the n = 0 peaks with respect to the central one (n = 0) obeys the simple law P n = exp[−4π 2 n 2 σ 2 /d 2 ](6) holding also in the presence of a smooth modulation of the atomic occupation number N k in each well. Result (6) shows that, if σ is much smaller than d the intensity of the lateral peaks will be high, with a consequent important layered structure in the density distribution of the expanding cloud. The value of σ, which characterizes the width of the condensates in each well, is determined, in first approximation, by the optical confinement. The simplest estimate is obtained by the harmonic expansion of the optical potential (3) around its minima: V = k (1/2)mω 2 x (x − kd) 2 withω x = 2 √ sE R /h, yield- ing σ = d/(πs 1/4 ). However this estimate is not accurate except for very intense laser fields. A better value is obtained by numerical minimization of the energy using the potential (3) and the wavefunction (5). This gives σ/d = 0.30, 0.27, and 0.25 for s = 3, 4 and 5 respectively. The predicted results for the density distribution n(x) = |Ψ(x)| 2 evaluated for s = 5 and t = 29.5 ms are shown in Fig. 1B (continuous line). From the above calculation we can also determine the relative population P n of the n = 1 peak as a function of the intensity factor s. This is shown in Fig. 2 together with the experimental results. The good comparison between experiment and theory reveals that the main features of the observed interference patterns are well described by this simple model. The 1D model discussed above can be generalized to 3D through the ansatz Ψ 0 (r) = k=0,±1..±kM e −(x−kd) 2 /2σ 2 Ψ k (r ⊥ )(7) which can be used, through a variational calculation, to describe the ground state of the system in the presence of the optical potential, magnetic trapping and two-body interactions. For sufficiently intense optical fields the value of σ is not significantly affected by two-body interactions, nor by magnetic trapping. On the other hand interactions are important to fix the shape of the condensate wave function in the radial direction. Neglecting the small overlap between condensates occupying different sites and using the Thomas-Fermi approximation to determine the wave function in the radial direction we obtain the result | Ψ k (r ⊥ ) | 2 = √ 2 g µ k 1 − r 2 ⊥ (R ⊥ ) 2 k (8) where (R ⊥ ) k = 2µ k /mω 2 ⊥ is the radial size of the k-th condensate, g depends on the scattering length a through the relation g = 4πh 2 a/m, while µ k = 1 2 mω 2 x d 2 k 2 M − k 2(9) plays the role of an effective k-dependent chemical potential. The value of k M is fixed by the normalization condition N = N k and is given by k 2 M = 2hω mω 2 x d 2 15 8 √ π N a a ho d σ 2/5 .(10) In (10) ω = (ω x ω 2 ⊥ ) 1/3 is the geometrical average of the magnetic frequencies, a ho = h/mω is the corresponding oscillator length and a is the s-wave scattering length. From the above equations one also obtains the result N k = N 0 (1 − k 2 /k 2 M ) 2 with N 0 = (15/16)N/k M . Equations (8-10) generalize the well known Thomas-Fermi results holding for magnetically trapped condensates [11] to include the effects of the optical lattice. Neglecting two-body interaction terms in the determination of the Gaussian width in the x-direction is a good approximation only if µ k is significantly smaller than the energyhω x . This condition is rather well satisfied in the configurations of higher lattice potential employed in the experiment. For example, using the typical parameter N = 5 × 10 4 for the total number of atoms and the values ω = 2π × 42 Hz and a/a ho = 3.2 × 10 −3 , we find, for s = 4,ω x ∼ 2π × 14 kHz, µ k=0 ∼ 2πh × 0.5 kHz and k M ∼ 100, corresponding to N 0 ∼ 500. Notice that with these values the condition µ k ≫hω ⊥ required to apply the Thomas-Fermi approximation is rather well satisfied for the central wells. The fact that µ k turns out to be significantly smaller thanhω x not only explains why the interference patterns emerging during the expansion are well described by the ideal 1D model for the array used above, but also points out the 2D nature of the condensates confined in each well. In this context it is worth pointing out that the bidimensionality of these condensates is ensured up to temperatures of the order of k B T ∼hω x , which is significantly higher than the expected value of the critical temperature for Bose-Einstein condensation. Our sample can then be used also to explore the consequence of the array geometry on the critical phenomena exhibited by these optically trapped Bose gases [12]. The above discussion permits also to explain the behaviour of the radial expansion of the gas. In the presence of the density oscillations produced by the optical lattice the problem is not trivial and should be solved numerically by integrating the GP equation. However, after the lateral peaks are formed, the density of the central peak expand smoothly according to the asymptotic law R ⊥ (t) = R ⊥ (0)ω ⊥ t exp , holding for a cigar configuration in the absence of the optical lattice [13]. In Fig. 3 the linear law is plotted using the expression R ⊥ (0) ∼ (R ⊥ ) k=0 = k M dω x /ω ⊥ derivable from Eq. (9) for the condensate occupying the central well. This choice for R ⊥ (0) is justified if the population of the lateral interference peaks is small so that their creation does not affect the radial expansion of the system. Let us finally discuss the conditions required for our system to exhibit coherence. At zero temperature the coherence between two consecutive condensates in the array is ensured if E c ≪ E J , where E c and E J are the parameters of the Josephson Hamiltonian for two adjacent condensates [14]. In particular E c = 2∂µ k /∂N k is the interaction parameter while E J = (h 2 /m) dr ⊥ [Ψ k ∂Ψ k+1 /∂x − Ψ k+1 ∂Ψ k /∂x] x=0 is the Josephson parameter describing the tunneling rate through the barrier separating two consecutive wells. In our case (s = 4 and N = 5×10 4 ) we find E c ∼ 2πh×1 Hz for the most relevant central condensates (k ≪ k M ) while, by solving numerically the Schrödinger equation in the presence of the optical potential V opt of Eq. (3), we find E J ∼ 2πh × 600 kHz. The value of E J is so large that one can safely conclude that the ground state of the system is fully coherent and that the effects of the quantum fluctuations of the phase will be consequently negligible. This reflects the fact that, even for the largest values employed for the laser power the overlap between consecutive condensates is not small enough. The value of E J is also much higher than the values of k B T used in the experiment, so that also the effects of the thermal fluctuations of the phase of the condensate can be ignored. This suggests that the fringes associated with the expansion of the condensate will remain visible up to the highest values of T , corresponding to the critical temperature for Bose-Einstein condensation. We have carried out experiments at different values of T where the signal obtained by imaging the expanding cloud can be naturally decomposed in two parts: an incoherent component due to the thermal cloud which is parametrized by a classical Gaussian Boltzmann distribution, and a Bose-Einstein component exhibiting the interference effects discussed above. In our experiment the interference peaks are visible up to k B T ∼ 2πh × 4.2 kHz. In order to point out the effects of the fluctuations of the phase one should lower the value of E J by orders of magnitude. This can be achieved by increasing significantly the laser power generating the optical lattice. Such effects have been recently observed in the experiment of [3]. In conclusion, we have investigated the consequences of coherence on the properties of an array of Bose-Einstein condensates. We have observed peculiar interference patterns in the density of the expanded cloud, reflecting the new geometry of the sample and discussed on a theoretical basis some key features exhibited by these optically crosses) obtained from the absorption image (A) integrated along the vertical direction. The wings of the central peak result from a small thermal component. The continuous line corresponds to the calculated density profile for the expanded array of condensates for the experimental parameters (s = 5 and texp = 29.5 ms).FIG. 2. Experimental (circles) and theoretical (triangles)values of the relative population of the n = 1 peak with respect to the n = 0 central one as a function of the intensity factor s of the optical potential Vopt. FIG. 3 . 3Radial size of the central peak as a function of the expansion time. Experimental data point are compared with the expected asymptotic law R ⊥ = R ⊥ (0)ω ⊥ texp. work has been supported by the EU under Contracts No. HPRI-CT 1999-00111 and No. HPRN-CT-2000-00125, by the MURST through the PRIN 1999 and PRIN 2000 Initiatives and by the INFM Progetto di Ricerca Avanzata "Photon matter. work has been supported by the EU under Con- tracts No. HPRI-CT 1999-00111 and No. HPRN-CT- 2000-00125, by the MURST through the PRIN 1999 and PRIN 2000 Initiatives and by the INFM Progetto di Ricerca Avanzata "Photon matter". . M R Andrews, Science. 275637M. R. Andrews et al., Science 275, 637 (1997). . B P Anderson, M A Kasevich, Science. 2821686B. P. Anderson and M. A. Kasevich, Science 282, 1686 (1998). . C Orzel, Science. 2912386C. Orzel et al., Science 291, 2386 (2001). . M Greiner, cond-mat/0105105M. Greiner et al., cond-mat/0105105. . F S Cataliotti, Science accepted for publication. F. S. Cataliotti et al., Science accepted for publication. . L Pitaevskii, S Stringari, Phys. Rev. Lett. 834237L. Pitaevskii and S. Stringari, Phys. Rev. Lett. 83, 4237 (1999). . D M Stamper-Kurn, Phys. Rev. Lett. 832876D. M. Stamper-Kurn et al., Phys. Rev. Lett. 83, 2876 (1999). . C Fort, Europhys. Lett. 49C. Fort et al., Europhys. Lett. 49, 8 (2000). . S Burger, Phys. Rev. Lett. 864447S. Burger et al., Phys. Rev. Lett. 86, 4447 (2001). . E Peik, Phys. Rev. A. 552989E. Peik et al., Phys. Rev. A 55, 2989 (1997). . F Dalfovo, S Giorgini, L P Pitaevskii, S Stringari, Rev. Mod. Phys. 71463F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999). . S Burger, in preparationS. Burger et al. in preparation. . Y Castin, R Dum, Phys. Rev. Lett. 775315Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996). . A J Leggett, Rev. Mod. Phys. 73307A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). . L Pitaevskii, S Stringari, cond-mat/0104458L. Pitaevskii and S. Stringari, cond-mat/0104458. . A Cuccoli, A Fubini, V Tognetti, R Vaia, cond- mat/0107387A. Cuccoli, A. Fubini, V. Tognetti, and R. Vaia, cond- mat/0107387
[]
[ "Can the Problem-Solving Benefits of ality Diversity Be Obtained Without Explicit Diversity Maintenance?", "Can the Problem-Solving Benefits of ality Diversity Be Obtained Without Explicit Diversity Maintenance?" ]
[ "Ryan Boldi \nUniversity of Massachusetts Amherst Amherst\nAmherst College Amherst\nMA, MA\n", "Lee Spector [email protected] \nUniversity of Massachusetts Amherst Amherst\nAmherst College Amherst\nMA, MA\n" ]
[ "University of Massachusetts Amherst Amherst\nAmherst College Amherst\nMA, MA", "University of Massachusetts Amherst Amherst\nAmherst College Amherst\nMA, MA" ]
[ "GECCO '23 Companion" ]
When using Quality Diversity (QD) optimization to solve hard exploration or deceptive search problems, we assume that diversity is extrinsically valuable. This means that diversity is important to help us reach an objective, but is not an objective in itself. Often, in these domains, practitioners benchmark their QD algorithms against single objective optimization frameworks. In this paper, we argue that the correct comparison should be made to multiobjective optimization frameworks. This is because single objective optimization frameworks rely on the aggregation of sub-objectives, which could result in decreased information that is crucial for maintaining diverse populations automatically. In order to facilitate a fair comparison between quality diversity and multi-objective optimization, we present a method that utilizes dimensionality reduction to automatically determine a set of behavioral descriptors for an individual, as well as a set of objectives for an individual to solve. Using the former, one can generate solutions using standard quality diversity optimization techniques, and using the latter, one can generate solutions using standard multi-objective optimization techniques. This allows for a level comparison between these two classes of algorithms, without requiring domain and algorithm specific modifications to facilitate a comparison.CCS CONCEPTS• Mathematics of computing → Dimensionality reduction; • Theory of computation → Evolutionary algorithms; • Computing methodologies → Artificial intelligence.
10.48550/arxiv.2305.07767
[ "https://export.arxiv.org/pdf/2305.07767v1.pdf" ]
258,686,537
2305.07767
7da42b4392996f91e26fc3a3fb1e79c48b2d9c8e
Can the Problem-Solving Benefits of ality Diversity Be Obtained Without Explicit Diversity Maintenance? July 15-19, 2023 Ryan Boldi University of Massachusetts Amherst Amherst Amherst College Amherst MA, MA Lee Spector [email protected] University of Massachusetts Amherst Amherst Amherst College Amherst MA, MA Can the Problem-Solving Benefits of ality Diversity Be Obtained Without Explicit Diversity Maintenance? GECCO '23 Companion Lisbon, PortugalJuly 15-19, 202310.1145/3583133.3596336ACM Reference Format: Ryan Boldi and Lee Spector. 2023. Can the Problem-Solving Benefits of Quality Diversity Be Obtained Without Explicit Diversity Maintenance?. In Genetic and Evolutionary Computation Conference Companion (GECCO '23 Companion), July 15-19, 2023, Lisbon, Portugal. ACM, New York, NY, USA, 6 pages. https:// ACM ISBN 979-8-4007-0120-7/23/07. . . $15.00Quality DiversityMulti-Objective OptimizationDimensionality ReductionBenchmarking When using Quality Diversity (QD) optimization to solve hard exploration or deceptive search problems, we assume that diversity is extrinsically valuable. This means that diversity is important to help us reach an objective, but is not an objective in itself. Often, in these domains, practitioners benchmark their QD algorithms against single objective optimization frameworks. In this paper, we argue that the correct comparison should be made to multiobjective optimization frameworks. This is because single objective optimization frameworks rely on the aggregation of sub-objectives, which could result in decreased information that is crucial for maintaining diverse populations automatically. In order to facilitate a fair comparison between quality diversity and multi-objective optimization, we present a method that utilizes dimensionality reduction to automatically determine a set of behavioral descriptors for an individual, as well as a set of objectives for an individual to solve. Using the former, one can generate solutions using standard quality diversity optimization techniques, and using the latter, one can generate solutions using standard multi-objective optimization techniques. This allows for a level comparison between these two classes of algorithms, without requiring domain and algorithm specific modifications to facilitate a comparison.CCS CONCEPTS• Mathematics of computing → Dimensionality reduction; • Theory of computation → Evolutionary algorithms; • Computing methodologies → Artificial intelligence. INTRODUCTION AND BACKGROUND Quality Diversity (QD) optimization is a relatively recent advancement in the evolutionary computation literature where a diverse set of high performing individuals are maintained over the course of the run. This results in divergent, rather than the traditional convergent evolutionary search [21]. There are often two reasons to apply quality diversity optimization. The first of them is to generate a large archive of qualitatively diverse individuals that solve certain problems. For example, finding diverse sets of robot behaviors [3,12] or creating diverse video game or training scenarios [6,7]. The second reason is generally to solve hard exploration problems that often have deceptive reward signals. For example, Lehman and Stanley [15] explore using Novelty Search (NS) to solve a deceptive maze, where there is a single goal, although many ways to solve this goal. They also used NS to evolve a controller for bipedal locomotion that outperformed fitness-based search with the particular fitness function and selection scheme studied. Some more recent examples are the QD-Maze brought forward by Pugh et al. [22], or the modular robotics domain used by Nordmoen et al. [19], which also have a single (real) objective. Despite the only goal for experimenters being solving the single objective, there is the assumption that diversity here is instrumental to solving the task. The intuition behind this makes sense: diverse low-performing solutions might be stepping stones that lead to high performing solutions in the future. This paper focuses on bench-marking Quality Diversity on the second use case, as an exploration algorithm that ultimately solves a single problem, although possibly in unexpected ways. Multi-objective optimization (MOO) is a common paradigm in optimization literature that attempts to optimize for multiple objectives at the same time. For example, non dominated sorting genetic algorithm (NSGA2) [4] and strength pareto evolutionary algorithm (SPEA2) [24] both deal with situations where there are multiple different objectives that often have numerous trade-offs for each other. Lexicase selection, a parent selection technique, has also been found to be useful at optimizing for multiple objectives [13]. There are often many different solutions that can be made by having different trade-offs between objectives, which can be visualized as existing on a Pareto front. These solutions could be highly diverse, as individuals that trade off between objectives differently would likely have qualitatively different behavior [11,14] These methods can therefore be used as a form of implicit quality diversity optimization: the quality comes from solving objectives, and the diversity comes from solving different combinations of objectives. Figure 1: Two deceptive mazes, adapted from [15]. In both maps, the large circle is the starting point of the robot and the small circle represents the goal. Both maps are deceptive in the sense that the path that minimizes euclidean distance to the goal will get stuck at local optima. How do you determine whether a quality diversity algorithm is performing well in hard exploration or deceptive domains? Previous studies compare quality diversity to using single objective optimization in reaching the objective. For example, Lehman and Stanley [15] compare novelty search to single-objective fitness based optimization in maze domains. Pugh et al. [21] compare quality diversity techniques to using only a single distance metric as an objective. Gaier et al. [8] compare using MAP-Elites to a (single) image similarity objective in evolving a target image. There are notably few studies comparing it to multi-objective or multi-modal optimization. Vassiliades et al. [23] compared MAP-Elites and NS to other multi-modal selection schemes for a maze navigation task. Nordmoen et al. [19] compared MAP-Elites [16] to a single-and multi-objective optimization algorithm. However, the multi-objective algorithms simply use diversity as a secondary objective, which does not necessarily provide a fair ground between QD algorithms and MOOs as QD has access to all dimensions of diversity, where MOOs simply get access to an aggregated version of this information. Other work looked at combining QD with MOO, but not on comparing them to each other (partly due to them having different goals in the majority of the cases) [17,20]. When comparing QD's problem solving ability to single objective or limited multi-objective optimization paradigms, the information accessible to both systems is not the same, preventing a fair comparison. In this paper, we consider the problem of providing quality diversity algorithms with the same information as that available to multi-objective optimization. This allows for both MOOs and QD to be compared faithfully in their ability to maximize objective(s) where diversity is instrumental (extrinsic), as opposed to being a goal that we are trying to optimize in and of itself (intrinsic). To do this, we propose the use of a dimensionality reduction technique to generate the behavioral descriptors for the quality diversity methods, and a slightly augmented version of this model to generate the objective values for MOO. This allows for a faithful comparison between both techniques as they will have access to the same performance signal. MOTIVATION To motivate the use of comparison scheme like this, we discuss the potential effects of learning the fitness function from hand-written measures (as it usually is in the literature), and provide a motivating example that shows how optimizing for combinations of these objectives through multi-objective optimization could result in the maintenance of diversity automatically. A common motivating example for quality diversity algorithms aimed at solving deceptive single objective problems is one of the deceptive hard maze. Two mazes that are of medium and hard difficulty can be found in Figure 1. These mazes are both deceptive as an individual greedily moving towards the goal will get stuck at sub-optimal traps. For this reason, diversity must be emphasized to ensure that the goal is actually reached. This fits into scope of the problems discussed in this paper as the sole objective is to reach the goal, yet diversity is intstrumental in reaching this objective. Consider what an arbitrary QD algorithm could do in this scenario. For example, MAP-elites [16] could be implemented with hand written behavioral descriptors such as 1) distance to goal on axis, 2) distance to goal on axis, 3) total distance travelled, and perhaps 4) number of turns taken. Using measure functions like these, solving deceptive mazes like above would be relatively straightforward, as demonstrated by Pugh et al. [21]. Why not single objectives? Single objectives often rely on an aggregation of sub-objectives that each represent qualitatively different goals. This aggregation procedure results in a loss of information regarding the performance of an individual. Instead, we use multi-objective optimization directly on the subobjectives to help facilitate the discovery of high performing individuals. With MOOs, then, finding solutions that optimize combinations of the sub-objectives creates diversity in the population similar to that with QD. However, the diversity that results from this is implicitly optimized for, as opposed to it being explicitly optimized for like in QD. What this means is QD maintains diversity as an objective in-and-of itself, whereas MOOs only optimize for their given objectives. In order to use multi-objective optimization on this domain using the scheme we present in this paper, we make an assumption that the fitness function (proximity to goal) can be approximated as a linear combination of these 4 measures. It is important to note that the raw value of the fitness function need not matter, as long as it matches the original fitness function in ranking the individuals. It is clear that both and distance are negatively correlated with proximity to the goal. For the sake of the example, let us suppose that the total distance travelled could also have a negative (yet smaller in magnitude) correlation to proximity to the goal. Finally, let us say that the number of turns is neutral to fitness (individuals that turn more on average perform no better than those that turn less). We then set each feature as its own objective for multi-objective optimization. Each individual is evaluated on all 4 of these metrics, and individuals are selected based on the multi-objective optimization algorithm being used. Consider an individual that has perfectly matched the location of the goal. This individual will likely be selected regardless of its location, distance, or turn values. Consider an individual that is very far from the goal, yet has covered a large amount of distance. This individual would also be selected, regardless of its distance to the goal. These different trade offs result in a diversity of individuals that solve different combinations of the sub-problems. When using a multi-objective selection scheme like Lexicase selection, this diversity is maintained automatically until convergence to a final goal. [10] Importantly, these systems have access to the same information as the quality diversity optimization algorithms. However, the key difference here is that MOOs could solve these deceptive problems without explicitly maintaining diversity. If QD algorithms can outperform MOOs in this domain, this would mean that the diversity that that specific QD algorithm maintains is instrumental in overcoming the deception in this domain. In the next section, we discuss how to extend this comparison scheme to any domain, regardless of the existence of human-written behavioral descriptors. COMPARISON SCHEME In this section, we will bring together all the ideas presented in this paper into a simple scheme to fairly compare quality diversity algorithms with objective-based algorithms. We operate under the assumption that humans have little intuition about the domain, and that reasonable choices behavioral descriptors or objectives will therefore not be obvious in advance. From Phenotype to Measures Consider learning of a set of measures ( ) as an unsupervised learning task. This can be learned with a variational autoencoder (VAE, [5]) as done in some previous work in QD [2,9]. Figure 2 outlines an example autoencoder architecture that could be used to learn a latent embedding of a given phenotype. Using reconstruction loss, this autoencoder can be used to learn a compressed representation of a phenotype in lower-dimensional space. This makes it possible for QD algorithms such as MAP-Elites to cover a large set of behavioral niches without needing to store a high dimensional archive. Figure 3 shows how the architecture outline in Figure 2 can be used to generate the qualitative measures ( ) that can be used to emphasize diversity using one of many QD techniques. From Measures to Fitness In order to facilitate a comparison between quality diversity techniques and multi-objective optimization techniques, one should try to present them both with as close to the same information as possible. In order to apply multi-objective optimization, we need to extract the sub-objectives from both the phenotype ( ) and the ground truth fitness function ( ). This can be done by simply augmenting the dimensionality reduction architecture used to generate the measures. We assume that the ground truth fitness can be approximated by a linear combination of the measure functions. Although this is a large assumption, it is one that is commonly made in the reward learning literature 1 . In inverse reinforcement learning (IRL), where the task is to learn a reward function that explains an expert trajectory, it is often the case that this reward function is learned as a linear combination of state features [18]. More recent work in IRL uses a set of pretraining algorithms to learn the state features (either with an autoencoder, or predicting a forward dynamics model, etc) before learning the final linear combination of these features that results in a reward function [1]. Figure 4 gives an overview of the training procedure of the fitness prediction model. First, after having trained the autoencoder to a satisfactory level, we freeze the encoder weights, and add a final layer that leads to a single fitness value. We can update the weights of this last layer (through back-propagation, or any other linear regression technique), to accurately predict the fitness value. Then, once the weights are determined, we can de-aggregate them into a vector of sub-fitness values. These values correspond well to sub-objectives for multi-objective optimization algorithms. Bringing It All Together Given a genotype ( ), we can create a phenotype ( ) and a gound truth fitness value ( ). This is domain specific but does not depend on which optimization algorithm being used. Examples of the phenotype could be the trajectory of a reinforcement learning agent, raw sensory information from a robot, or any other characterization of how the genotype behaves after translation. Then, we train a dimensionality reduction model (such as an autoencoder) to learn a lower dimensional representation of the phenotype ( ). Quality Diversity. Using the lower dimensional representation, we can assign the measures to simply be the values of the latent variables (i.e. ( ) = ( )). Then, we can assign the fitness value to be the ground truth fitness ( ). This is very similar to the procedure done by Cully [2] to automatically discover the measure functions. Multi-objective Optimization. We can then augment our measure function to include a final layer that predicts the ground truth fitness. Once this model is trained to convergence, we simply deaggregate the final node, resulting in an element-wise multiplication of the measures and their corresponding weights. These values ì ( ) should sum approximately to the ground truth fitness ( ). These fitness functions can then be used as the multiple objectives for any MOO scheme. Fair Comparison? What metric do we use to evaluate how these methods are performing? Since the domain is one that ultimately does have a single objective, we should measure the performance of the algorithms by how well they solve this objective. Including a comparison about diversity would not be yield a fair performance comparison as only QD methods explicitly optimize for diversity. So, in the context of solving hard exploration problems, we should be measuring how well each algorithm solves these problems. The authors suggest simply using the ground truth fitness function of the individuals produced through both optimization paradigms. Practical Details In practice, using the scheme we presented requires some choices to be made by the user. In this section, we discuss such details and considerations that experimenters attempting to use this comparison scheme should be aware of. Training the Models. In order to train the models, one needs to amass a significant amount of training data. Cully [2] solves this issue by using the archive of individuals produced through a run of MAP-elites to train the dimensionality reduction model. As we are attempting to form a comparison scheme between two classes of algorithms, it is important that the model used does not vary much between these two systems. In order to this, the authors see 2 options: (1) Pre-trained: Generate the training data from a series of prior evolutionary runs (or perhaps exhaustively enumerating all of genotype space, if tractable), and train the models using that. Use the entire dataset to train the VAE, and then do a second pass to learn the weights of the linear layer given the ground truth fitness as a label. (2) Incremental: Train the models with some random phenotypes from randomly sampling from genotype space, and fine-tune the models differently for each system, using the phenotypes cached throughout their respective evolutionary runs and the ground truth fitness. In order to facilitate a fair comparison, the authors recommend method (1). This is because method (2) results in two different models being used to extract the information from phenotype used for selection. This could result in the lack of clarity regarding each system's performance: was it due to actually solving the domain better, or due to generating better training data for the models to later perform better using? Dimensionality of the latent layer. Another practical consideration to be made is the extent to which we shrink dimensions using the dimensionality reduction technique. If the bottleneck layer is too small, it could result in poor reconstruction capability. If the bottleneck layer is too large, it could prohibit effective learning by the optimization algorithms. This is a domain specific consideration, but could be addressed through a series of hyperparameter optimization runs. For example, one could run AURORA [2] using various latent layer sizes, and take the latent layer size that results in the highest QD-Score. As this is simply a benchmark, the cost of training the model can be amortized over many benchmarking tasks. CONCLUSION AND FUTURE WORK We have presented a method to compare quality diversity techniques to objective based techniques for a variety of hard exploration or deceptive problems. The key insight is that the comparison to single objective optimization is not a fair comparison due to the aggregation that occurs when we aggregate an individual's performance to be a single number. Each individual solves different sub-objectives to different extents. Optimizing for different combinations of all these sub-objectives could correspondingly also extrinsically optimize for diversity without needing to explicitly be tasked to. This means we can compare the quality of solutions that are created by explicitly optimizing for diversity to directly optimizing for the sub-objectives in their ability to use extrinsic diversity in their favor. We present a dimensionality reduction technique based on an autoencoder to automatically learn the measure functions from a given phenotype. Then, fitness is approximated as a weighted sum of these measure values. Finally, we remove the summation operation, and set the sub-objectives to simply be an element-wise multiplication of the measure and its weight (i.e. positive weighted measures are increased, negatives decreased). We can perform multiobjective optimization on the sub-objectives, and quality diversity straight on the measures (with access to the ground truth fitness). Future work in improving this benchmark should address the limitation that the sub-objectives might not be a function of the measures. This could be addressed by training a new fitness prediction architecture that does not re-use weights from the measure predicting autoencoder. Simply de-aggregating the penultimate layer on this model would similarly allow for multiple subobjectives that are learned straight from the phenotype. Figure 2 : 2Variational autoencoder architecture used to learn a latent embedding ( ) of a phenotype ( ). As this is a variational autoencoder, ( ) is sampled from a distribution that is parameterized by the output of the encoder (not pictured here for simplicity). Figure 3 : 3Using the encoder of a VAE to predict measures ( ) of a phenotype ( ) Figure 4 : 4Top: Learning the weights for a linear combination of features that sums to an approximation for the true fitness function ( ). These weights can then be used to predict fitness from measures ( ) = ( ), or even straight from the phenotype ( ). Bottom: De-aggregation of last layer of learned fitness model to result in a set of sub-objectives ì ( ) that sum to an approximation of the ground truth fitness ( ). It is possible to relax this assumption to be that fitness is a non-linear function of the measures by incorporating a second multi-layer perception that learns the fitness from the output of the encoder. However, a similar de-aggregation procedure would be necessary on the penultimate layer of the newly added MLP. ACKNOWLEDGMENTSThe authors acknowledge Antoine Cully, Bill Tozier, Edward Pantridge, Li Ding, Maxime Allard, Nic McPhee, Thomas Helmuth, and the members of the PUSH Lab at Amherst College for discussions and inspiration that helped shape this work. Furthermore, the authors would like to thank the anonymous reviewers for their careful reading and insightful comments and discussion.This material is based upon work supported by the National Science Foundation under Grant No. 2117377. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation. Safe Imitation Learning via Fast Bayesian Reward Inference from Preferences. Daniel S Brown, Russell Coleman, Ravi Srinivasan, Scott Niekum, arXiv:2002.09089cs, statDaniel S. Brown, Russell Coleman, Ravi Srinivasan, and Scott Niekum. 2020. 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[]
[ "Nucleosynthesis in multi-dimensional SNIa explosions", "Nucleosynthesis in multi-dimensional SNIa explosions" ]
[ "C Travaglio [email protected] \nMax-Planck Institut für Astrophysik\nKarl-Schwarzschild Strasse 1D-85741Garching bei MünchenGermany\n\nIstituto Nazionale di Astrofisica (INAF)\nOsservatorio Astronomico di Torino\nVia Osservatorio 2010025Pino Torinese (Torino)Italy\n", "W Hillebrandt \nMax-Planck Institut für Astrophysik\nKarl-Schwarzschild Strasse 1D-85741Garching bei MünchenGermany\n", "M Reinecke \nMax-Planck Institut für Astrophysik\nKarl-Schwarzschild Strasse 1D-85741Garching bei MünchenGermany\n", "F.-K Thielemann \nDepartment of Physics and Astronomy\nUniversity of Basel\nB2 CH-4056Klingelbergstrasse, BaselSwitzerland\n" ]
[ "Max-Planck Institut für Astrophysik\nKarl-Schwarzschild Strasse 1D-85741Garching bei MünchenGermany", "Istituto Nazionale di Astrofisica (INAF)\nOsservatorio Astronomico di Torino\nVia Osservatorio 2010025Pino Torinese (Torino)Italy", "Max-Planck Institut für Astrophysik\nKarl-Schwarzschild Strasse 1D-85741Garching bei MünchenGermany", "Max-Planck Institut für Astrophysik\nKarl-Schwarzschild Strasse 1D-85741Garching bei MünchenGermany", "Department of Physics and Astronomy\nUniversity of Basel\nB2 CH-4056Klingelbergstrasse, BaselSwitzerland" ]
[]
We present the results of nucleosynthesis calculations based on multi-dimensional (2D and 3D) hydrodynamical simulations of the thermonuclear burning phase in type Ia supernovae (hereafter SNIa). The detailed nucleosynthetic yields of our explosion models are calculated by post-processing the ejecta, using passively advected tracer particles. The nuclear reaction network employed in computing the explosive nucleosynthesis contains 383 nuclear species, ranging from neutrons, protons, and α-particles to 98 Mo. Our models follow the common assumption that SNIa are the explosions of white dwarfs that have approached the Chandrasekhar mass (M ch ∼ 1.39), and are disrupted by thermonuclear fusion of carbon and oxygen. But in contrast to 1D models which adjust the burning speed to reproduce lightcurves and spectra, the thermonuclear burning model applied in this paper does not contain adjustable parameters. Therefore variations of the explosion energies and nucleosynthesis yields are dependent on changes of the initial conditions only. Here we discuss the nucleosynthetic yields obtained in 2D and 3D models with two different choices of ignition conditions (centrally ignited, in which the spherical initial flame geometry is perturbated with toroidal rings, and bubbles, in which multi-point ignition conditions are simulated), but keeping the initial composition of the white dwarf unchanged. Constraints imposed on the hydrodynamical models from nucleosynthesis as well as from the radial velocity distribution of the elements are discussed in detail. We show that in our simulations unburned C and O varies typically from ∼40% to ∼50% of the total ejected material. Some of the unburned material remains between the flame plumes and is concentrated in low velocity regions at the end of the simulations. This effect is more pronounced in 2D than in 3D and in models with a small number of (large) ignition spots. The main differences between all our models and standard 1D computations are, besides the higher mass fraction of unburned C and O, the C/O ratio (in our case is typically a factor of 2.5 higher than in 1D computations), and somewhat lower abundances of certain intermediate mass nuclei such as S, Cl, Ar, K, and Ca, and of 56 Ni. We also demonstrate that the amount of 56 Ni produced in the explosion is a very sensitive function of density and temperature. Because explosive C and O burning may produce the iron-group elements and their isotopes in rather different proportions one can get different 56 Ni-fractions (and thus supernova luminosities) without changing the kinetic energy of the explosion. Finally, we show that we need the high resolution multi-point ignition (bubbles) model to burn most of the material in the center (demonstrating that high resolution coupled with a large number of ignition spots is crucial to get rid of unburned material in a pure deflagration SNIa model).
10.1051/0004-6361:20041108
[ "https://arxiv.org/pdf/astro-ph/0406281v1.pdf" ]
14,597,168
astro-ph/0406281
8648e4986a65bff75598878c0637d4087891d8e9
Nucleosynthesis in multi-dimensional SNIa explosions 11 Jun 2004 June 19, 2018 C Travaglio [email protected] Max-Planck Institut für Astrophysik Karl-Schwarzschild Strasse 1D-85741Garching bei MünchenGermany Istituto Nazionale di Astrofisica (INAF) Osservatorio Astronomico di Torino Via Osservatorio 2010025Pino Torinese (Torino)Italy W Hillebrandt Max-Planck Institut für Astrophysik Karl-Schwarzschild Strasse 1D-85741Garching bei MünchenGermany M Reinecke Max-Planck Institut für Astrophysik Karl-Schwarzschild Strasse 1D-85741Garching bei MünchenGermany F.-K Thielemann Department of Physics and Astronomy University of Basel B2 CH-4056Klingelbergstrasse, BaselSwitzerland Nucleosynthesis in multi-dimensional SNIa explosions 11 Jun 2004 June 19, 2018Received March **, 2004; accepted **, 2004Astronomy & Astrophysics manuscript no. paperSNIa˙AA˙revised (DOI: will be inserted by hand later)hydrodynamics -nucleosynthesis, nuclear reactions -supernovae: general We present the results of nucleosynthesis calculations based on multi-dimensional (2D and 3D) hydrodynamical simulations of the thermonuclear burning phase in type Ia supernovae (hereafter SNIa). The detailed nucleosynthetic yields of our explosion models are calculated by post-processing the ejecta, using passively advected tracer particles. The nuclear reaction network employed in computing the explosive nucleosynthesis contains 383 nuclear species, ranging from neutrons, protons, and α-particles to 98 Mo. Our models follow the common assumption that SNIa are the explosions of white dwarfs that have approached the Chandrasekhar mass (M ch ∼ 1.39), and are disrupted by thermonuclear fusion of carbon and oxygen. But in contrast to 1D models which adjust the burning speed to reproduce lightcurves and spectra, the thermonuclear burning model applied in this paper does not contain adjustable parameters. Therefore variations of the explosion energies and nucleosynthesis yields are dependent on changes of the initial conditions only. Here we discuss the nucleosynthetic yields obtained in 2D and 3D models with two different choices of ignition conditions (centrally ignited, in which the spherical initial flame geometry is perturbated with toroidal rings, and bubbles, in which multi-point ignition conditions are simulated), but keeping the initial composition of the white dwarf unchanged. Constraints imposed on the hydrodynamical models from nucleosynthesis as well as from the radial velocity distribution of the elements are discussed in detail. We show that in our simulations unburned C and O varies typically from ∼40% to ∼50% of the total ejected material. Some of the unburned material remains between the flame plumes and is concentrated in low velocity regions at the end of the simulations. This effect is more pronounced in 2D than in 3D and in models with a small number of (large) ignition spots. The main differences between all our models and standard 1D computations are, besides the higher mass fraction of unburned C and O, the C/O ratio (in our case is typically a factor of 2.5 higher than in 1D computations), and somewhat lower abundances of certain intermediate mass nuclei such as S, Cl, Ar, K, and Ca, and of 56 Ni. We also demonstrate that the amount of 56 Ni produced in the explosion is a very sensitive function of density and temperature. Because explosive C and O burning may produce the iron-group elements and their isotopes in rather different proportions one can get different 56 Ni-fractions (and thus supernova luminosities) without changing the kinetic energy of the explosion. Finally, we show that we need the high resolution multi-point ignition (bubbles) model to burn most of the material in the center (demonstrating that high resolution coupled with a large number of ignition spots is crucial to get rid of unburned material in a pure deflagration SNIa model). Introduction Type Ia supernovae (SNIa) are known to be stellar explosions with no signs of hydrogen and helium in their spectra, but intermediate mass elements such as Si, S, Ca and Mg near the maximum of their light curves, and many Fe lines at later times. In contrast to massive stars which are the progenitors of Type Send offprint requests to: C. Travaglio II supernovae (SNII), SNIa progenitors are thought to be white dwarfs (WDs) in binary systems (see Whelan &Iben 1973, and for a more recent review). In the canonical model the WD, expected to consist mainly of carbon and oxygen, approaches the Chandrasekhar mass (M ch ) through a not yet known mechanism, presumably accretion from a companion star, and is then disrupted by a thermonuclear explosion. The declining light is powered by the radioactive decay of 56 Ni. A strong argument in favor of this scenario is given by the fact that these explosion models fit quite well the observed light curves and spectra (Leibundgut 2001). Despite the consistency of this general framework with observations the detailed theory of how SNIa evolve and explode is still subject of considerable efforts. Over the last three decades, one-dimensional spherically symmetric models have been used to study the various channels that may give rise to a successful SN Ia in terms of the predicted spectra, light curves, and nucleosynthesis. Much of this work was centered on the M ch scenario wherein a C+O white dwarf accretes H or He from a binary companion (Nomoto, Thielemann, & Yokoi 1984) and ignites explosive carbon burning just before it reaches a critical mass of M ch ∼ 1.39 M ⊙ . The subsequent explosion produces enough 56 Ni (∼ 0.6 M ⊙ ) and intermediate mass elements to reproduce "normal" SN Ia lightcurves and spectra, provided that the amount of C+O burned at any given density is suitably chosen. In 1D models this can be achieved by parameterizing the thermonuclear flame speed and, if desired, the density at which a transition to supersonic burning (detonation) occurs (Khokhlov et al. 1999;Niemeyer 1999). Moreover, some mixing of processed matter had to be assumed in order to fit the observed spectra. Alternative scenarios, including sub-M ch explosions and merging white dwarfs (double degenerates), have met with mixed success (see e.g. Arnett & Livne 1994). More recently it has become possible to perform multidimensional 2D (Livne 1993;Reinecke, Hillebrandt, & Niemeyer 1999;Lisewski et al. 2000) and 3D (Reinecke, Hillebrandt, & Niemeyer 2002a,b;Gamezo et al. 2003) simulations of an exploding M ch -white dwarf. The principal difficulty in these models is the fact that the hydrodynamically unstable and turbulent nuclear flame front develops structures on much smaller length scales than can numerically be resolved. However, this problem can be overcome by "large eddy simulations", i.e. by employing subgrid-scale models for the unresolved scales that provide a guess of the effective turbulent flame speed on the scale of the computational grid (Niemeyer & Hillebrandt 1995a;Röpke, Niemeyer, & Hillebrandt 2003). In this flame model, which is well justified in the thin-flame regime and is tested in experiments with premixed turbulent chemical flames, we do not need a detailed prescription of the nuclear reactions. Instead, the fuel consumption rate is roughly propartional to the surface area of the flame front and its normal (turbulent) velocity. Despite of the need of more detailed studies of such subgrid-scale models, it is important to stress that the multidimensional simulations reach a qualitatively different level of predictive power than 1D models. In particular, the amount of material burned at a given density can not longer be finetuned but is determined by the fluid motions on the resolved scales and a particular choice of the subgrid model (Reinecke et al. 2002a). Therefore, once the flame model has been fixed numerical simulations of the thermonuclear explosion of a given white dwarf can be done by just choosing the ignition conditions, including the chemical composition of the WD, the only remaining (physical) parameter. The undeniable influence of SNIa explosions on, among others, the chemical evolution of galaxies makes the quest for solid theoretical models and nucleosynthetic yields an urgent task. Guided by decades of modeling and nucleosynthesis calculations in spherically symmetric models (the protoype being the W7 model of Nomoto et at. 1984, Iwamoto et al. 1999, Brachwitz et al. 2000, we have begun analyzing the detailed nucleosynthetic yields of our explosion models by post-processing the ejecta. This has been performed adding a "lagrangian component" to our Eulerian scheme in the form of tracer particles passively advected with the flow in the course of the Eulerian calculation. Therefore we record their T and ρ history by interpolating the corresponding quantities from the underlying Eulerian grid. A similar method of tracer particles in an Eulerian code to calculate the nucleosynthesis has been adopted in a previous study of multi-dimensional nucleosynthesis in core collapse SNe by Nagataki et al. (1997), and more recently in calculations for very massive stars (Maeda et al. 2002), for core collapse SNe (Travaglio et al. 2004), and for Type Ia SNe first preliminary results have been discussed by Niemeyer et al. (2003). In this paper we present the nuclear yields resulting from several of our multi-dimensional supernova simulations, and we compare them to the standard W7 (Iwamoto et al. 1999, Brachwitz et al. 2000 results. In Section 2 we summarize the 2D and 3D SNIa calculations, discussing different mode of ignition as well as grid resolution of the hydrodynamic code. In Section 3 we describe our method to perform nucleosynthesis calculations and the nucleosynthesis network adopted for this work. Finally, in Section 4 we present and discuss our nucleosynthesis results. In a first step we have performed resolution studies in 2D consisting of different methods how to distribute the tracer particles, the number of particles used, and grid resolution of the hydrodynamic code. Although 2D simulations cannot be considered to be realistic, as was discussed by Reinecke et al. (2002a), they can serve to guide the more elaborate 3D models. We then discuss the nucleosynthesis resulting from three 3D models, a centrally ignited model and two models with a few and many off-center ignition spots, respectively. It will be shown that the more realistic ignition conditions (central ignition or many ignition spots) also predict nucleosynthesis yields closer to the ones observed in typical SN Ia's. Concerning nucleosynthesis we will in particular analyze the range and distribution of 56 Ni masses we are able to produce with our present models, and the sensitivity of the amount and velocity distribution of unburned material ( 12 C, 16 O, 22 Ne) to the ignition conditions of the explosions which are still a major uncertainty of SN Ia models. Recent multi-dimensional SNIa calculations We have carried out numerical simulations in 2D and 3D, for several different ignition conditions, and for different numerical resolution. Details of these models are given in a series of papers (Reinecke et al. 1999(Reinecke et al. , 2002a(Reinecke et al. , 2002b. A detailed discussion of these models is not the aim of this paper, therefore only a summary of the results essential for our nucleosynthesis calculations will be repeated in this section. As long as the evolution of the white dwarf before the thermonuclear runaway remains largely unexplored (see recent work by Woosley, Wunsch, & Kuhlen 2004 and references therein), only very crude constraints can be put on the flame geometry at the onset of the burning. It appears likely that the deflagration sets in at the surface of quietly burning "hot bubbles". Nevertheless very little is known about the number, size and radial distribution of these hot spots. This is a consequence of the complicated physical processes taking place in the white dwarf's core during the convective smoldering phase prior to ignition lasting for ∼1000 years. The long time scales combined with the relatively slow convective motions make numerical simulations of this phase a daunting task which has not been undertaken in its full complexity so far. Theoretical considerations and simplified simulations carried out by Garcia-Senz & Woosley (1995) suggest that fast burning starts on the surface of many small bubbles (r ≤ 5 km), within 100 km of the star's center. Central ignition is another possible scenario that has been investigated during the last years using multidimensional calculations (Niemeyer & Hillebrandt 1995b;Khokhlov 1995). In this work we present four models: c3 2d 512, a 2D model with central ignition, and grid size of 512 2 ; c3 3d 256, a 3D model with central ignition, and grid size of 256 3 ; b5 3d 256, a 3D model with ignition in 5 bubbles, and grid size of 256 3 ; finally, b30 3d 768, a 3D model with ignition in 30 bubbles, and grid size of 768 3 . This last one is the model with the highest resolution possible to evolve with the computer resources available to us, therefore we will consider it as the 'standard' model for this paper. It achieves a central resolution of 3.33 km, using a grid consisting of 768 3 zones. In the simulated octant of our model b30 3d 768, 30 bubbles with a radius of 10 km were distributed randomly. The bubble locations were drawn from a Gaussian probability distribution with a dispersion of σ = 75 km. Bubbles located more than 2.5 σ away from the center were rejected. In all models we started the simulations with a central ignition density of 2.9×10 9 gr/cm 3 . The simulations have been followed up to 1.5 sec. for all the models, except for the b30 3d 768. Due to a very high consumption in computer time the model b30 3d 768 was stopped when no further energy was released. Fig. 1 shows the energy release for the four models mentioned above. The curves are nearly identical during the first ∼0.5 sec of the simulation. Owing to the small volume of the bubbles, the initial hydrostatic equilibrium is only slightly disturbed. During the first stages the energy release is therefore lower than in previous simulations. Only after the total flame surface has grown considerably (mostly by deformation of the bubbles), vigorous burning sets in. In the late explosion phase (after about 0.5 sec) the total energy differs for the four simulations, and increases moderately with increasing resolution (see Reinecke et al. 2002a,b for more detailed discussion). We also note that the 3D centrally ignited and the five-bubble models are remarkably similar, even if the centrally ignited has a relatively faster burning between ∼0.5 and ∼1 sec. It must be noted that the five-bubble model is not identical to the model b5 3d 256 presented by Reinecke et al. (2002b): due to an oversight during the simulation setup the initial po- sitions of the burning bubbles are not the same. As a consequence, the total energy releases of these two simulations are slightly different. The initial configuration of the front, as well as snapshots of the front evolution at later times are shown in Fig. 2 and Fig. 3 for the models c3 3d 256 and b30 3d 768, respectively. In addition to the total energy release, the mass fraction of unburned material in the central region of the remnant appears to be a good criterion for judging the validity of our simulations, because a high amount of C and O in this region would most likely produce a characteristic signature in the late-time spectra which has not yet been observed. In this respect the results of our earlier calculations were not very encouraging since the ashes rose towards the surface in large structures and left nearly pure fuel in the center. Using many initial bubbles, however, seems to alleviate this problem insofar as the statistical isotropy of the initial flame at least delays the development of large-scale turnover motions. As a consequence, C and O is lower than 20% in the central 0.2 M ⊙ after 0.9 s for model b30 3d 768. For this last model we get ∼40% of the total mass stays unburned (we define unburned the material with T < 1.5×10 9 K). We will discuss this point in more detail when we will present our nucleosynthesis results. Nucleosynthesis in multi-dimensional SNIa The multidimensional SNIa simulations described in Section 2 employed a minimal nuclear reaction network, directly included in the hydrodynamic code, sufficient for a good approximation of the thermonuclear energy release and the predicted gross chemical composition agrees well with the expectations (Reinecke et al. 2002b). It consists of five species (αparticles, 12 C, 16 is instantaneously converted to a NSE of 56 Ni and α-particles at high densities and to 24 Mg at intermediate densities. Below 10 7 g/cm 3 , no burning takes place. Therefore we follow a minimal reaction network directly in the hydrodynamic simulations, and a much more extended network in a post-process step. Here we present the results of the more detailed study of the nuclear abundances in the ejecta obtained by post-processing the output of the four hydrodynamic models discussed above. Since the multidimensional hydrodynamics scheme used in modeling the explosions is an Eulerian one (i.e. the grid does not move with the fluid), in order to record temperature and density evolution as a function of time (the necessary input for nucleosynthesis calculations) we homogeneously distributed ∼10000 marker particles (in 2D models) and ∼20000 marker particles (in 3D models) and followed their T and ρ evolution. The number of particles in the simulation was chosen in order to reproduce in the best way the resolution of the grid (see discussion below). We then calculated the nucleosynthesis experienced by each marker and computed the total yield as a sum over all the markers, after the decay of unstable isotopes. Tracer particles method In one spatial dimension it is nowadays possible to solve reaction networks consisting of hundreds of species online with the hydrodynamics (see e.g. Rauscher et al. 2002 for explosive nucleosynthesis calculations in core collapse SNe). However, it is more common to use reduced networks in order to obtain the (approximate) energy generation rate for the hydrodynamics and to calculate the detailed chemical composition only afterwards in a post-processing step. This is facilitated by the lagrangian nature of nearly all 1D codes employed for explosive nucleosynthesis calculations. In lagrangian schemes, the grid moves with the fluid and therefore it is possible to record the evolution of the temperature and density for different fluid elements (i.e. lagrangian zones), which is required for the postprocessing. In contrast, most grid-based multidimensional hydro schemes are of Eulerian type (i.e. the grid is fixed in space). To obtain the necessary data for the post-processing we added a "lagrangian component" to our Eulerian scheme in the form of marker particles that we passively advect with the flow in the course of the Eulerian calculation, recording their T and ρ history by interpolating the corresponding quantities on the underlying Eulerian grid. In the 3D simulations the star is subdivided into 27 3 grid cells equidistant in the integrated mass M(r), azimuthal angle ϕ and cos θ, so that each grid cell contains the same mass. A tracer particle was placed randomly in each of those grid cells, therefore the total number of tracer particles we used is 27 3 = 19683. After numerical inversion of the function M(r), the (M,ϕ,cos θ) coordinates are mapped onto the Cartesian grid. For the 2D simulation, 100 2 particles are distributed in r and cos θ directions, using the same procedure as above. In all cases the simulation covers one octant, therefore to get the total white dwarf mass we mutiply the mass of each marker by 8 and we sum over all the markers. They are distributed in the way to have identical mass, that is therefore calculated as the ratio between M ch and the total number of tracer particles. The mass of each marker is therefore between ∼10 −3 M ⊙ and ∼10 −4 M ⊙ . The initial distribution of the markers for the b30 3d 768 case is shown in Fig. 4. Finally, we compare the T (t) values for each marker extracted from the hydrodynamical model with the T (t) derived from using the internal energy from the hydro-code (which has contributions from the Boltzmann gas of ionized nuclei, the Planck-spectrum of photons, and relativistic degenerate or non-degenerate electrons positrons). This is done by calculating separately the equation of state for a given maker particle's internal energy, density and composition, and deriving from that the temperature T. As chemical composition for this calculation we use a 16 isotopes network, composed by n, p, 4 He, 12 C, 16 O, 20 Ne, 24 Mg, 28 Si, 32 S, 36 Ar, 40 Ca, 44 Ti, 48 Cr, 52 Fe, 56 Ni, 60 Zn. We found that the T (t) obtained directly from the hydrodynamic model is in average lower by 10% up to 20% for markers with high temperatures (T ≥ 6×10 9 K) as compared with the temperature derived from the energy. This can be understood from the fact that in the hydro-code the internal en- ergy density is the directly computed variable and, therefore, is more accuratly determined than the temperature. Consequently the more precise T (t) distribution is derived from the internal energy and the actual composition. This is what we used for our nucleosynthesis calculations. Nucleosynthesis network The nuclear reaction network employed in computing our postprocess explosive nucleosynthesis calculations contains 383 nuclear species ranging from neutrons, protons, and α-particles to molibdenum. A detailed description of the code we used to solve the nuclear network and the reaction rate library utilized is given by Thielemann et al. (1996) and Iwamoto et al. (1999). Weak interaction rates applied in those calculations were taken from Fuller, Fowler, & Newman (1985). More recently full large-scale shell model calculations for electron capture and β-decays became available also for pf-shell nuclei, i.e. the Fe-group . They have already been included in preliminar calculations by Brachwitz et al. (2000) and Thielemann et al. (2003). We also included the new rates in the calculations presented in this paper. As discussed below more in details, the nuclear reaction rates entering the thermonuclear modeling can play an important role. While large portions of the ejecta which experience maximum temperatures in excess of 6×10 9 K follow nuclear statistical equilibrium (a chemical equilibrium of all strong and electromagnetic reactions), weak interactions occur on a longer timescale and different choice of Fueller et al. (1985) (as used by Iwamoto et al. 1999) or (as used for this work, and by Brachwitz et al. 2000, can strongly affect the results. The initial WD composition we used consists of (mass fraction) 0.475 M ⊙ of 12 C, 0.5 M ⊙ of 16 O, and 0.025 M ⊙ of 22 Ne (in agreement with the W7 initial composition, Iwamoto et al. 1999). With this initial composition we typically simulate a solar metallicity SNIa. When the flame passes through the fuel, 12 C, 16 O and 22 Ne are converted to ashes with different compositions depending on the intial T and ρ. We stop our nucleosynthesis calculations after ∼1.5 sec. When the temperature in the markers dropped at ∼1.5 10 9 K the explosive nucleosynthesis is almost frozen. The distribution of T and ρ as a function of time is shown in Fig. 5 for the model b30 3d 768. The thick lines represent the upper envelope for T and ρ, and with the dashed lines we plot T and ρ histories in some markers taken as examples. As one can notice from Fig. 5, the T distribution is not yet below ∼1.5 10 9 K for all the tracer particles. At T ∼ 3 -4 10 9 K (i.e. the upper values shown in Fig. 5) we still expect some explosive C-burning products. Therefore we linearly extrapolate T and ρ until all the particles have T < 1.5 10 9 K (that corresponds to ∼ 1 sec.). The combination of T and ρ vs. time in each marker is very important for the nucleosynthesis results (as discussed below). Comparing the model shown in Fig. 5 with the T and ρ distribution shown by Iwamoto et al. (1999) for their W7 model, we note differences that can be interesting for the nucleosynthesis calculations. First, the timescale in our models are much smaller (∼1.5 sec) with respect to the W7 timescale (∼6 sec). Then different combinations of T and ρ, in our case rather low T at still high ρ, also give us interesting differences in the nucleosynthesis calculations. Discussion and results In this Section we present the results for nucleosynthesis calculations in the model c3 2d 512, c3 3d 256, b5 3d 256, and our 'standard' b30 3d 768. We also compare them with the W7 calculations by Brachwitz et al. (2000) and Thielemann et al. (2003) Iwamoto et al. 1999, in order to be consitent with the use of elecetron capture rates). We analyze the consequences of different hydrodynamic resolutions on the nucleosynthesis, we compare their different distribution of burned and unburned material, we discuss the trend of the Y e in the markers as a consequence of our nucleosynthesis calculations. Finally, we discuss the velocity distribution of different nuclear species. Nucleosynthesis and yields: comparison between 2D and 3D For the nucleosynthesis calculations the peak temperatures combined with the density distribution achieved during the propagation of the front, are the most important quantities. As shown in Fig. 5, for the b30 3d 768 model the maximum of the T in the markers covers a large range (1.5×10 9 K < T < 8.4×10 9 K). Note that in this Figure each dotted line represents a tracer particle trend (we selected randomly some markers for this plot), and the thick line is the upper envelope of the T and ρ distribution in the total tracer particle sample. At high temperatures all strong interactions and photodisintegrations are so fast that a chemical equilibrium (nuclear statistical equilibrium, NSE) is immediately achieved (in our calculations we assume NSE condition for T ≥ 6.0×10 9 K). The resulting chemical composition is therefore only dependent on ρ, T , and the neutron eccess (that is determined by the total amount of electron captures taking place on longer timescales). An example of the behaviour of the chemical abundances in one marker as a function of time is shown in Fig. 6. For this case the marker is originally located at a radius of ∼150 km, i.e. in the innermost dense zone; the initial density at the position of the marker is ∼ 2.5×10 9 gr/cm 3 (the central density for this model is 2.9×10 9 gr/cm 3 ) and it is heated by the flame front almost immediately (at ∼0.1 s). The NSE conditions are achieved very fast and the temperature reaches a quite high peak T ∼8.5×10 9 K (Fig. 6, upper left panel). The resulting electron fraction Y e drops rapidly (in ∼0.3 s) from the initial value of 0.4989 to 0.468 (Fig. 6, upper right panel). The main O-burning products, 28 Si and 32 S are shown in Fig. 6 (left middle panel), together with the abundance of 24 Mg resulting from C-burning. Furthermore, the rise of the temperature in excess of 6×10 9 K leads to a complete NSE and 28 Si exhaustion within 0.1 s. Due to the high density (typical of explosive Si-burning) normal freeze-out occurs. One of the main products is 56 Fe (Fig. 6, right middle panel). Since T and ρ conditions are very high, also neutron-rich nuclei are built up due to electron captures, and 56 Fe is partly replaced by 54 Fe (Fig. 6, right middle panel) and 58 Ni (Fig. 6, left lower panel). In such details we can follow the nucleosynthesis changes along the time in all the tracer particles. In Fig. 7 we show the distribution of Y e vs. time obtained as a result of the nucleosynthesis calculation in the model b30 3d 768. The two thick lines stand for the upper and lower values of the Y e in the markers. The dashed lines represent the Y e time evolution in some markers randomly taken as examples. The range covers values from ∼0.5 (that represent the typical initial composition as well as the composition of markers with rather low T ), down to ∼0.462, reached by the markers with the highest T (see e.g. the example in Fig. 6). Table 1 lists the synthesized masses for all the stable isotopes up to 68 Zn for the models c3 2d 512, c3 3d 256, b5 3d 256, and b30 3d 768. For comparison we also include in column 2 the calculations for the W7 model (from Thielemann et al. 2003). For the calculations presented in column 5,6,7 we include the nucleosynthesis results starting only when the temperature has reached 90% of the peak. In fact, due to fi- nite numerical resolution in the hydrodynamic simulation, the rise of the temperature is not as steep as it would be in reality. Consequently, when markers reach NSE conditions and weak-interactions start to play the most important role, the nucleosynthesis timescales are fast, and even 0.1 s (i.e. typical timescale we have for the rise of T) are crucial for some reactions to give an important contribution. Nevertheless, as one can see from the Table comparing column 4 (the nucleosynthesis for the c3 3d 256 model has been calculated considering the all rise of the T curve) and column 5 (the nucleosynthesis for the same c3 3d 256 model has been calculated only when T has reached 90% of the peak), the differences in the total yields are very small. This is due to the fact that the amount of markers affected by this inaccuracy is a small fraction of the total. In the Table we do not include isotopes heavier than 68 Zn, even if the network we used was extended up to 98 Mo, since their resulting mass fraction are smaller than ∼10 −15 M ⊙ . An important thing to notice is the difference in the amount of unburned material (as defined at the beginning of this paper, we consider unburned the material with T < 1.5×10 9 K) in the four models. For the 2D model c3 2d 512 ∼60% of the total material remains unburned, instead of 40% we obtain for the 3D model c3 3d 256. Both of these models are centrally ignited, therefore the difference in the amount of unprocessed material is mainly a consequence of the difference in the total energy distribution due to multi-dimension effects (see Fig. 1). In Fig. 8 we show the distribution of burned and unburned material for the model b30 3d 768 (upper panel), c3 3d 256 (middle panel), and b5 3d 256 (lower panel), in terms of number of tracer particles. The distribution plotted in Fig. 8 has been taken at 1.1 sec for all the three models, that is the final time we reached in the b30 3d 768. For all these three simulations the burned component dominates at a radius of ∼5.0×10 7 cm, instead the unburned material is more or less uniformly distributed, with a dominant component in the outermost zones and a tail in the central regions. As also discussed by Reinecke et al. (2002b) the distribution of the unprocessed material depends on the initial conditions for the burning. When the model is centrally ignited, as c3 3d 256, the dominant component of unburned material is in the outermost regions. Instead with a floating-bubble model with comparable resolution, like b5 3d 256, unprocessed material can also be concentrated in the center. Nevertheless the amount of unprocessed material in the center for a floating-bubble model depends on the amount of ignition-spots together with the resolution used. In fact (see Fig. 8) for the b30 3d 768 case most of the unburned component in the innermost regions disappeared. We also tested consequences of burning for longer times. As mentioned in Section 3.2, at ∼1.2 sec the upper limit of the temperature is ∼4×10 9 K (see also Fig. 5), and explosive C-and a bit of Ne-burning can still occur. Therefore, to calculate the nucleosynthesis we extrapolated temperature and density for ∼1 sec. further (when the upper values for T are not higher that 1.5×10 9 K). If we compare the nucleosynthetic yields calculated at the end of the hydro simulation with the nucleosynthetic yields calculated using the extrapolated values of T and ρ, we obtain that only few % of 12 C burns. The consequences can be relevant only for few isotopes, the main products of explosive C-burning, i.e. 20,21 Ne, 23 Na, 25,26 Mg, 27 Al. Therefore only for those isotopes we expect important changes if we could follow the hydrodynamic simulation for longer timescales. It is interesting to notice the difference in the 12 C and 16 O abundance of the W7 model and our multi-dimensional models. The 12 C of the W7 is about a factor of ∼10 lower than in our cases, instead 16 O of W7 is only a factor of ∼3 lower. While 16 O and 12 C we obtain are built up by unprocessed particles, in the W7 C-burning is more efficient and burns a significant amount of 12 C at low T and ρ, with a resulting different C/O ratio. Also, the 56 Fe mass (mainly deriving from the decay of the long-lived 56 Ni) obtained in the W7 model (0.696 M ⊙ , Thielemann et al. 2003) is by far higher than the 56 Fe mass resulting from the multi-dimensional SNIa models. The highest value we can reach in our models is obtained with the highest resolution floating-bubble model b30 3d 768 (0.44 M ⊙ ). We note that the initial conditions (in this case 30 ignition spots) are crucial for a more precise study of the nucleosynthesis, in particular of the innermost regions. Possibly, a model with even higher number of ignition bubbles would give us a still higher 56 Fe mass. We note that neutron-rich isotopes like 48 Ca, 50 Ti, 54 Cr, 54 Fe, 58 Ni are strongly underproduced with respect to the W7 Fig. 9. Nucleosynthetic yields (in mass fraction normalized to the solar value and to the corresponding solar ratio) obtained using 19683 tracer particles in the 3D model b30 3d 768 compared to the W7 yields given by Thielemann et al. (2003). model presented by Iwamoto et al. (1999), instead are in good agreement with the Branchwitz et al. (2000) and Thielemann et al. (2003) predictions. This is due to the differences in the electron capture rates adopted (as just discussed above). As a consequence, electron capture rates of nuclei affects directly the electron fraction Y e (Thielemann, Nomoto, & Yokoi 1986). As shown in Fig. 7, the lowest Y e reached in the markers is 0.462, instead the lowest value in the W7 model by Iwamoto et al. (1999) is ∼ 0.446. For the model by Iwamoto et al. (1999) low Y e values (< 0.46) are reached in the innermost zones, i.e. M < 0.03M ⊙ , where the highest temperatures are reached (∼9.0×10 9 K). The highest T reached in our models, as shown in Fig. 5, are ∼8.4×10 9 K with consequently higher Y e . Finally, in Table 2 we report the synthesized masses of the main radioactive species from 22 Na up to 63 Ni. Bigger differences between our models and W7 are the abundances of 48 Ca and 66 Zn (in this case we use for comparison the W7 data from iwamoto et al. 1999, since the Thielemann et al. (2003) data are not yet available for the long-lived isotopes). As discussed in Thielemann et al. (2003), these isotopes are very sensitive to small variations in the central density of the model. We used 2.9×10 9 gr/cm 3 instead of the 2.0×10 9 gr/cm 3 of the model B2C20 presented by Thielemann et al. (2003) and used in the current paper as a comparison to our model. In Fig. 9 we show the yields obtained for our 'standard' model b30 3d 768 compared with the W7 yields (Brachwitz et al. 2000, scaled to their relative solar abundances and to the 56 Fe abundance. As one can see from the figure, a part from the difference in the relative 56 Fe mass (0.44 M ⊙ for the b30 3d 768 and 0.669 M ⊙ for W7), as well as in the unburned material (i.e. 12 C, 16 O and 22 Ne), the trend for the production of different isotopes is quite similar. In Fig. 10 we plotted the yields of the c3 2d 512 and c3 3d 256 (upper panel), and b5 3d 256 (lower panel), normalized to the 'standard' model b30 3d 768. From Fig. 11 one can notice that with a similar amount of burned material (the difference between the two models in the total burned material it is not more than 10%), the b5 3d 256 produces much more α-elements. Infact the higher efficiency of 12 C burning is clearly shown by a higher production of 23 Na and 40 Ca. Also the b5 3d 256 model has a lower 56 Fe production (0.34 M ⊙ , instead of 0.44 M ⊙ of the b30 3d 768), and Fe-group elements. Radial velocity distribution In Fig. 11 we show the mass fractions of selected isotopes as a function of the radial velocity, taken at the end of our simulation (i.e. ∼1.2 sec.) for the model b30 3d 768 and c3 3d 256. With thick lines we plot the unprocessed material in the form of 12 C, 16 O, and 22 Ne. We note that at the lowest velocities (∼1000 km/s) the dominant component is represented by the unburned material (i.e. 12 C and 16 O) for the floating-bubble high-resolution model, and by 56 Fe for the centrally ignited model. On the opposite site, at the highest velocities (>10000 km/s) the unburned material dominates in the centrally ignited model, instead is comparable to the 56 Fe component in the floating-bubble high resolution model. Maximum velocities reached are about 12000 km/s in both cases. As mentioned in the previous Section, our models at 1.2 sec are not yet in homologous expansion, i.e. pressure and gravity still play a role, changing the velocity distribution and possibly also the density, therefore the velocity distribution of the elements has to be taken with care. However the distribution in velocitites might leave observable features in the spectra that could be used for diagnostic purposes. Finally, by projecting our 3D results on previously obtained 1D ones, the observed evolution of the Si, S, and Ca lines might give us the possibility to understand whether normal SNIa are well mixed deflagrations. Summary and conclusions In this paper we presented the results of nucleosynthesis calculations obtained coupling a tracer particle method to twodimensional and three-dimensional Eulerian hydrodynamic calculations of SNIa explosion. The multidimensional SN Ia simulations described in this work employed a minimal nuclear reaction network, sufficient for a good approximation of the thermonuclear energy release. Although the predicted chemical composition agrees well with the expectations, we presented here the results of a very detailed study of the nuclear abundances in the ejecta obtained by post-processing the output of the hydrodynamic models. Since the multidimensional hydrodynamical scheme applied is of Eulerian type (i.e. the grid does not move with the fluid), we added a lagrangian component to the calculations in the form of tracer particles. In order to record temperature and density evolution as a function of time (necessary input for the nucleosynthesis calculations) we homogeneously distributed ∼20000 marker particles and followed their T and ρ evolution. We then calculated the nucleosynthesis experienced by each marker and computed the total yield as a sum over all the markers including the decays of unstable isotopes. The nuclear reaction network employed in computing the explosive nucleosynthesis contains 383 nuclear species ranging from neutrons, protons, and α-particles to molibdenum. For this work, the initial mixture we used consists of (mass fraction) 0.475 M ⊙ of 12 C, 0.5 M ⊙ of 16 O, and 0.025 M ⊙ of 22 Ne. When the flame passes through the fuel, C, O and Ne are converted to heavier elements, with different compositions depending on the T and ρ history. Nuclear statistical equilibrium conditions are assumed in the marker particles with T > 6 10 9 K. At such temperatures (T > 6 10 9 K) a mixture of 56 Ni and α-particles in NSE is synthesized. Below that temperature burning only produces intermediate mass elements. Once the temperature drops T < 1.5 10 9 K, no burning takes place during the short timescale (≃ 1.5 s) of the explosion ("unburned" material). The current research focused on the sensitivity of the explosion on the ignition conditions and on the detailed nucleosynthetic yields that they predict. We could demonstrate that multi-dimensional explosion models allow us to predict their nucleosynthesis yields with some confidence. It was shown that only 3D models are potentially able to produce enough 56 Ni to explain the light curves of "normal" type Ia supernovae, and that also the ignition conditions (central ignition vs. several offcenter ignition spot) affect the nucleosynthesis yields. Since the number of ignition spots we can put into the numerical models depend on the spatial resolution and since the explosion energy as well as the Ni-mass increase with increasing number of spots, we expect that our best resolved b30 3d 768 model is closest to what a "typical" pure-deflagration supernova might eject. The general nucleosynthesis outcome of SNeIa is dominated by Fe-group elements, involving also sizable fractions of Si-Ca and minor amounts of unburned (C and O) or pure C-burning products (e.g. Na, Ne, Mg). Despite of the fact that differences with respect to the standard W7 nucleosynthesis (Iwamoto et al. 1999, Brachwitz et al. 2000 are found, in particular in the 56 Ni mass produced, as well as in the final amount of unburned material, in general the nuclear yields are consistent with expectations. We can therefore say with some confidence and without parameterization, that the Chandrasekhar mass scenario with a pure turbulent deflagration is a viable candidate for SN Ia explosions. We also note that the significant amount of unburned material ejected by our SNIa models may have an interesting impact on the role of SNIa in the context of Galactic chemical evolution of C (giving a contribution of the order of ∼20% to the total C at the solar composition). In the case of O still the main sources are massive stars. Comparing the nucleosynthesis presented in this paper to observed SNIa spectra, the reader should keep in mind that our models do not reach the homologous expansion phase. We are currently working to modify the combustion hydrocode, using a moving grid that will allow us to follow the evolution much longer. The results will be published elsewhere. We are also performing a detailed parameter study of the variation of the central density and of the initial carbon/oxygen ratio of the SNIa progenitor (Röpke et al., in preparation). Finally, recent calculations by Timmes et al. (2003) indicate large variations of the 56 Ni mass as a function of metrallicity (measured by the original 22 Ne content). An investigation of the metallicity effect on the nucleosynthesis and yields is also in progress (Travaglio et al., in preparation). Thielemann et al. (2003) b This work c This work. In this run we allow the nucleosynthesis calculations for those tracer particles that reach NSE conditions only starting at 90% of the temperature peak. Fig. 1 . 1Total energy evolution for the two-dimensional centrally ignited explosion model (solid line), for the threedimensional low-resolution centrally ignited (dashed line) explosion model, for the three-dimensional low-resolution 5 bubbles (dotted-dotted-dashed line), and for the three dimensional high-resolution 30 bubbles (dotted-dashed line) explosion model. O, 24 Mg, and 56 Ni) and is intended to model the energy release of the thermonuclear reactions only. No reaction rates are employed: all material behind the flame Fig. 2 . 2Snapshots of the front evolution for the centrally ignited model c3 3d 256 at 0 s, 0.2 s, 0.4 s, and 0.6 s. Fig. 3 . 3Snapshots of the front evolution for the floating-bubble model b30 3d 768 at 0 s, 0.1 s, 0.14 s, and 0.2 s. Fig. 4 . 4Radial distribution of the tracer particles in the 3D model at the beginning of the simulation. Fig. 5 . 5Temperature (upper panel) and density (lower panel) history in the tracer particles for the b30 3d 768 model. The thick lines represent the upper envelope of the distribution and the thin dashed lines represent some of the markers randomly taken as examples. Fig. 6 . 6Example of the nucleosynthesis calculation in one tracer particle. T and ρ are plotted in the upper left panel; with T > 6×10 9 K reaches NSE conditions. In the upper right panel the resulting Y e is shown. The other panels give the mass fraction vs. time for16 O, 28 Si, 32 S, 54,56 Fe, 56,57,58 Ni, 63 Cu, and 64 Zn. For 16 O, 28 Si, 32 S the time is plotted only up to 0.2 s since their abundances are zero at later times. Fig. 7 . 7The same asFig. 5for the history of the Y e in the markers. Fig. 8 . 8Distribution of the tracer particles vs. radius at ∼1.2 sec. for b30 3d 768 (upper panel), c3 3d 256 (middle panel), and b5 3d 256 (lower panel) models. The unburned particles (T < 1.5×10 9 K) are plotted with a solid line, and the processed particles with a dashed line. Fig. 10 . 10Nucleosynthetic yield ratio, comparing the model c3 2d 512 (dotted line), and c3 3d 256 (solid line), with our standard model b30 3d 768 (upper panel). Nucleosynthetic yield ratio, comparing the model b5 3d 256 with our standard model b30 3d 768 (lower panel). Fig. 11 . 11Mass fractions of selected isotopes as a function of the radial velocity of the markers (taken at ∼1.2 sec) for the b30 3d 768 model (upper panel) and for the c3 3d 256 (lower panel). The width of each velocity bin is 300 km/s. For each isotope we sum its abundance over all markers in a certain velocity bin. (note that in Figures and Tables we better compare with Brachwitz et al. calculations instead of TABLE 1 S 1M (M ⊙ )  SNI Species W7 (a) c3 2d 512 (b) c3 3d 256 (b) c3 3d 256 (c) b5 3d 256 (c) b30 3d 768 (c) 12 C 5.04E-02 4.09E-01 3.37E-01 3.37E-01 2.49E-01 2.78E-01 13 C 1.07E-06 1.07E-10 9.71E-11 9.68E-11 8.21E-06 3.98E-06 14 N 4.94E-07 2.71E-09 3.96E-09 3.48E-09 1.04E-03 2.76E-04 15 N 1.25E-09 4.40E-11 7.18E-11 6.99E-11 2.48E-05 1.23E-06 16 O 1.40E-01 4.74E-01 4.16E-01 4.17E-01 3.90E-01 3.39E-01 17 O 3.05E-08 1.16E-09 1.29E-09 1.13E-09 7.81E-06 1.31E-06 18 O 7.25E-10 9.49E-11 1.62E-10 1.52E-10 1.15E-04 1.01E-05 19 F 5.72E-10 2.64E-11 3.72E-11 3.34E-11 1.08E-06 2.84E-08 20 Ne 1.97E-03 4.70E-03 7.39E-03 7.10E-03 3.18E-02 6.28E-03 21 Ne 8.51E-06 7.11E-07 1.14E-06 1.03E-06 5.96E-05 2.16E-05 22 Ne 2.27E-03 2.15E-02 1.77E-02 1.77E-02 1.14E-02 1.42E-02 23 Na 6.20E-05 2.99E-05 5.09E-05 5.10E-05 3.49E-03 8.65E-04 24 Mg 1.31E-02 1.04E-02 1.48E-02 1.26E-02 2.35E-02 7.53E-03 25 Mg 4.71E-05 5.49E-05 8.57E-05 7.64E-05 2.41E-03 5.13E-04 26 Mg 3.31E-05 6.60E-05 1.06E-04 1.01E-04 8.56E-04 1.81E-04 27 Al 8.17E-04 7.39E-04 1.08E-03 9.73E-04 2.11E-03 5.85E-04 28 Si 1.52E-01 4.42E-02 5.89E-02 5.39E-02 1.19E-01 5.39E-02 29 Si 7.97E-04 6.47E-04 9.49E-04 9.22E-04 1.81E-03 5.61E-04 30 Si 1.43E-03 1.06E-03 1.48E-03 1.31E-03 2.20E-03 8.03E-04 31 P 3.15E-04 2.02E-04 2.85E-04 2.69E-04 5.24E-04 1.72E-04 32 S 8.45E-02 1.60E-02 2.22E-02 2.57E-02 5.70E-02 2.62E-02 33 S 4.11E-04 1.05E-04 1.42E-04 1.58E-04 3.21E-04 1.21E-04 34 S 1.72E-03 8.68E-04 1.15E-03 1.15E-03 2.30E-03 1.04E-03 36 S 2.86E-07 1.64E-07 2.24E-07 2.47E-07 3.95E-07 1.53E-07 35 Cl 1.26E-04 3.60E-05 4.88E-05 5.90E-05 1.31E-04 4.58E-05 37 Cl 3.61E-05 6.89E-06 8.97E-06 1.27E-05 3.21E-05 1.21E-05 36 Ar 1.49E-02 2.12E-03 3.14E-03 4.09E-03 9.04E-03 4.24E-03 38 Ar 8.37E-04 3.30E-04 4.13E-04 5.12E-04 1.20E-03 5.59E-04 40 Ar 1.38E-08 1.49E-09 2.06E-09 3.04E-09 4.92E-09 1.91E-09 39 K 6.81E-05 1.51E-05 1.84E-05 2.95E-05 7.69E-05 3.24E-05 41 K 6.03E-06 9.03E-07 1.17E-06 2.20E-06 6.03E-06 2.41E-06 40 Ca 1.21E-02 1.68E-03 2.66E-03 3.40E-03 7.08E-03 3.59E-03 42 Ca 2.48E-05 6.66E-06 8.43E-06 1.41E-05 3.61E-05 1.58E-05 43 Ca 1.07E-07 2.26E-08 3.06E-08 3.96E-08 6.37E-08 5.10E-08 44 Ca 9.62E-06 1.80E-06 2.81E-06 3.10E-06 4.52E-06 3.61E-06 46 Ca 2.44E-09 2.58E-12 3.46E-12 1.14E-11 1.91E-11 8.53E-12 48 Ca 1.21E-12 1.99E-17 3.20E-17 1.05E-16 1.54E-16 4.01E-15 45 Sc 2.17E-07 2.16E-08 3.06E-08 6.08E-08 1.65E-07 6.47E-08 46 Ti 1.16E-05 2.80E-06 3.53E-06 5.62E-06 1.47E-05 6.62E-06 47 Ti 5.45E-07 1.38E-07 1.88E-07 2.20E-07 3.61E-07 2.64E-07 48 Ti 2.07E-04 4.11E-05 6.96E-05 7.28E-05 1.32E-04 7.69E-05 49 Ti 1.59E-05 3.28E-06 5.22E-06 5.62E-06 1.10E-05 5.78E-06 50 Ti 1.62E-06 8.22E-10 2.08E-08 2.08E-08 8.67E-10 2.67E-07 50 V 4.58E-09 2.04E-09 3.66E-09 3.50E-09 2.69E-08 2.66E-09 51 V 3.95E-05 1.70E-05 1.90E-05 1.90E-05 2.89E-05 1.95E-05 50 Cr 2.23E-04 1.20E-04 1.20E-04 1.10E-04 1.67E-04 1.19E-04 52 Cr 4.52E-03 1.91E-03 2.80E-03 2.76E-03 3.48E-03 2.58E-03 53 Cr 6.49E-04 4.78E-04 5.18E-04 4.81E-04 5.09E-04 4.83E-04 54 Cr 3.04E-05 3.42E-06 6.38E-06 5.92E-06 4.11E-06 1.22E-05 55 Mn 6.54E-03 5.63E-03 5.93E-03 5.53E-03 4.53E-03 6.38E-03 54 Fe 7.49E-02 6.79E-02 6.61E-02 6.21E-02 4.48E-02 7.33E-02 56 Fe 6.69E-01 2.44E-01 3.28E-01 3.36E-01 3.40E-01 4.39E-01 57 Fe 2.52E-02 1.05E-02 1.35E-02 1.36E-02 1.28E-02 1.86E-02 58 Fe 1.74E-04 8.25E-06 3.16E-05 3.02E-05 8.58E-06 1.05E-04 59 Co 7.66E-04 6.70E-04 7.62E-04 6.81E-04 4.53E-04 7.33E-04 58 Ni 1.02E-01 6.13E-02 7.52E-02 7.31E-02 5.56E-02 9.66E-02 60 Ni 9.22E-03 7.23E-03 9.24E-03 8.16E-03 5.39E-03 7.73E-03 61 Ni 2.69E-04 6.11E-05 8.86E-05 9.26E-05 9.99E-05 1.13E-04 62 Ni 2.31E-03 5.71E-04 7.78E-04 8.16E-04 9.21E-04 1.12E-03 64 Ni 1.84E-07 2.73E-11 1.61E-09 1.61E-09 1.93E-10 5.29E-08 63 Cu 1.59E-06 9.24E-07 9.26E-07 9.27E-07 8.20E-07 9.56E-07 65 Cu 7.72E-07 1.88E-07 2.51E-07 2.61E-07 2.81E-07 3.77E-07 64 Zn 1.50E-05 3.72E-06 4.47E-06 4.65E-06 4.83E-06 6.78E-06 66 Zn 1.31E-08 6.11E-06 7.55E-06 7.86E-06 8.90E-06 1.16E-05 67 Zn 1.18E-11 4.15E-09 5.49E-09 5.68E-09 6.53E-09 7.96E-09 68 Zn 2.66E-10 2.85E-09 3.68E-09 3.86E-09 4.86E-09 5.26E-09 a C.Travaglio, W. 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[ "NUMERICAL SIMULATIONS OF MULTIPHASE WINDS AND FOUNTAINS FROM STAR-FORMING GALACTIC DISKS: I. SOLAR NEIGHBORHOOD TIGRESS MODEL", "NUMERICAL SIMULATIONS OF MULTIPHASE WINDS AND FOUNTAINS FROM STAR-FORMING GALACTIC DISKS: I. SOLAR NEIGHBORHOOD TIGRESS MODEL" ]
[ "Chang-Goo Kim [email protected] \nDepartment of Astrophysical Sciences\nPrinceton University\n08544PrincetonNJUSA\n\nCenter for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA\n\nKim & Ostriker\n\n", "Eve C Ostriker \nDepartment of Astrophysical Sciences\nPrinceton University\n08544PrincetonNJUSA\n" ]
[ "Department of Astrophysical Sciences\nPrinceton University\n08544PrincetonNJUSA", "Center for Computational Astrophysics\nFlatiron Institute\n10010New YorkNYUSA", "Kim & Ostriker\n", "Department of Astrophysical Sciences\nPrinceton University\n08544PrincetonNJUSA" ]
[]
Gas blown away from galactic disks by supernova (SN) feedback plays a key role in galaxy evolution. We investigate outflows utilizing the solar neighborhood model of our high-resolution, local galactic disk simulation suite, TIGRESS. In our numerical implementation, star formation and SN feedback are self-consistently treated and well resolved in the multiphase, turbulent, magnetized interstellar medium. Bursts of star formation produce spatially and temporally correlated SNe that drive strong outflows, consisting of hot (T > 5×10 5 K) winds and warm (5050 K < T < 2×10 4 K) fountains. The hot gas at distance d > 1 kpc from the midplane has mass and energy fluxes nearly constant with d. The hot flow escapes our local Cartesian box barely affected by gravity, and is expected to accelerate up to terminal velocity of v wind ∼ 350 − 500 km s −1 . The mean mass and energy loading factors of the hot wind are 0.1 and 0.02, respectively. For warm gas, the mean outward mass flux through d = 1 kpc is comparable to the mean star formation rate, but only a small fraction of this gas is at velocity > 50 km s −1 . Thus, the warm outflows eventually fall back as inflows. The warm fountain flows are created by expanding hot superbubbles at d < ∼ 1 kpc; at larger d neither ram pressure acceleration nor cooling transfers significant momentum or energy flux from the hot wind to the warm outflow. The velocity distribution at launching near d ∼ 1 kpc better represents warm outflows than a single mass loading factor, potentially enabling development of subgrid models for warm galactic winds in arbitrary large-scale galactic potentials.
10.3847/1538-4357/aaa5ff
[ "https://arxiv.org/pdf/1801.03952v1.pdf" ]
54,752,685
1801.03952
7351cf98ece9e7050561c7bf458e16118ee1c4bd
NUMERICAL SIMULATIONS OF MULTIPHASE WINDS AND FOUNTAINS FROM STAR-FORMING GALACTIC DISKS: I. SOLAR NEIGHBORHOOD TIGRESS MODEL January 15, 2018 11 Jan 2018 Chang-Goo Kim [email protected] Department of Astrophysical Sciences Princeton University 08544PrincetonNJUSA Center for Computational Astrophysics Flatiron Institute 10010New YorkNYUSA Kim & Ostriker Eve C Ostriker Department of Astrophysical Sciences Princeton University 08544PrincetonNJUSA NUMERICAL SIMULATIONS OF MULTIPHASE WINDS AND FOUNTAINS FROM STAR-FORMING GALACTIC DISKS: I. SOLAR NEIGHBORHOOD TIGRESS MODEL January 15, 2018 11 Jan 2018Draft version Typeset using L A T E X twocolumn style in AASTeX61galaxies: ISM -galaxies: star formation -magnetohydrodynamics (MHD) -methods: numerical Gas blown away from galactic disks by supernova (SN) feedback plays a key role in galaxy evolution. We investigate outflows utilizing the solar neighborhood model of our high-resolution, local galactic disk simulation suite, TIGRESS. In our numerical implementation, star formation and SN feedback are self-consistently treated and well resolved in the multiphase, turbulent, magnetized interstellar medium. Bursts of star formation produce spatially and temporally correlated SNe that drive strong outflows, consisting of hot (T > 5×10 5 K) winds and warm (5050 K < T < 2×10 4 K) fountains. The hot gas at distance d > 1 kpc from the midplane has mass and energy fluxes nearly constant with d. The hot flow escapes our local Cartesian box barely affected by gravity, and is expected to accelerate up to terminal velocity of v wind ∼ 350 − 500 km s −1 . The mean mass and energy loading factors of the hot wind are 0.1 and 0.02, respectively. For warm gas, the mean outward mass flux through d = 1 kpc is comparable to the mean star formation rate, but only a small fraction of this gas is at velocity > 50 km s −1 . Thus, the warm outflows eventually fall back as inflows. The warm fountain flows are created by expanding hot superbubbles at d < ∼ 1 kpc; at larger d neither ram pressure acceleration nor cooling transfers significant momentum or energy flux from the hot wind to the warm outflow. The velocity distribution at launching near d ∼ 1 kpc better represents warm outflows than a single mass loading factor, potentially enabling development of subgrid models for warm galactic winds in arbitrary large-scale galactic potentials. INTRODUCTION Galactic scale gas outflows (or winds) are ubiquitous in star forming galaxies (see Veilleux et al. 2005;Heckman & Thompson 2017, for reviews) and believed to be essential to distribution of the gas and metals in galaxies and the circumgalactic/intergalactic medium (CGM/IGM) and hence to regulating cosmic star formation history (see Somerville & Davé 2015;Naab & Ostriker 2017, for reviews). Theoretical models of the stellar mass-halo mass relation constructed by abundance matching of observational stellar mass functions to simulated halo mass functions (e.g., Moster et al. 2013;Behroozi et al. 2013;Rodríguez-Puebla et al. 2017) indicate that galaxies (or dark matter halos) are very inefficient at converting gas into stars. At low redshift, at most ∼ 10−20% of the available baryonic mass has been converted into stars at halo mass of ∼ 10 12 M , while the ratio of stellar mass to halo mass declines steeply toward both lower and higher masses. Recent cosmological hydrodynamic simulations of large volumes of the Universe require strong outflows driven by both star formation and active galactic nuclei feedback to explain low baryonic abundance in galaxies compared to the cosmic fraction (e.g., Springel & Hernquist 2003;Vogelsberger et al. 2014;Schaye et al. 2015). Direct inclusion of feedback processes in large-volume cosmological galaxy formation simulations is still not feasible in practice. For star formation feedback by supernovae (SNe), implementation via simple thermal energy dumps suffers "overcooling," with energy radiated away without preventing in situ star formation or driving winds (e.g., Katz 1992); this is because resolving the cooling radii of SN remnants requires much higher resolution (Kim & Ostriker 2015a) than is practicable in large-volume simulations. In cosmological zoom-in simulations, more careful treatments of SNe allowing for a "momentum prescription" at low resolution can solve at least some aspects of the overcooling problem (e.g., Kimm & Cen 2014;Kimm et al. 2015;Hopkins et al. 2014Hopkins et al. , 2017, especially for dwarfs. However, given the constraints of computational expense, treating unresolved physics with parameterized models is unavoidable in many situations, including in simulations of galaxy groups/clusters, and in the large boxes needed for fullysampled statistics of cosmic galaxy populations (e.g., Schaye et al. 2015;Pillepich et al. 2017). When star formation feedback physics is not directly simulated, galactic winds are not an outcome but an input that is part of the "subgrid" treatment (Somerville & Davé 2015). Currently, however, subgrid models of wind driving by stellar feedback often either adopt highly simplified scaling prescriptions for wind mass loss rates (relative to the star formation rate) and velocities (relative to the halo potential depth), or else are calibrated using empirical results from a limited set of galaxies (and hence are not fully predictive). Better theoretical models are clearly needed. Towards this end, the first step is to provide a physical understanding and detailed characterization of outflowing gas (including both winds and fountains) in galaxies, informed and calibrated based on high-resolution three-dimensional numerical magnetohydrodynamic (MHD) simulations. To fully capture the interaction between stellar feedback and a realistic multiphase interstellar medium (ISM), it is crucial to self-consistently include the gravitational collapse that produces star clusters and to resolve the local injection of energy from individual massive stars. In classical theoretical models of galactic winds motivated by observed starburst galaxies (e.g. Chevalier & Clegg 1985), a steady, adiabatic flow is produced by a central energy source. In this approach, hot, overpressured gas flows are characterized by "mass loading" and "energy loading" factors, respectively defined by the ratios of mass and energy outflow rates to star formation rates and energy injection rates at the wind base. Simple spherical wind models can also be constructed that allow for radiative cooling, such that the temperature precipitously drops at some radius in certain parameter regimes (e.g. Wang 1995;Bustard et al. 2016; Thompson et al. 2016). Observed galactic outflows are multiphase in nature. Systematic observations reveal prevalent multiphase structure of galactic winds with cold molecular (e.g., Weiß et al. 1999;Leroy et al. 2015), neutral (e.g., Heckman et al. 2000;Martin 2005;Rupke et al. 2005;Chen et al. 2010;Contursi et al. 2013), ionized (e.g., Pettini et al. 2001;Shapley et al. 2003;Steidel et al. 2010;Erb et al. 2012;Heckman et al. 2015;Chisholm et al. 2017), and hot gas phases (e.g., Strickland & Stevens 2000;Strickland & Heckman 2007;López-Cobá et al. 2017). For the best studied example, local starburst M82, Leroy et al. (2015, see also Chisholm & Matsushita 2016 have shown a clear signature of decreasing outward mass fluxes in molecular and neutral gas as a function of the distance from the disk midplane, implying a fountain (Shapiro & Field 1976;Bregman 1980) rather than a wind for the cooler gas. In one conceptual framework, warm and cold gas in outflows results when a hot medium accelerated by its own pressure gradient cools radiatively; an alternative concept is that overdense warm and cold ISM clouds are "entrained" by a high-velocity, low-density hot wind. More realistically, both effects can in principle occur, and in general there is a complex interaction between the multiple phases that are present. The mass and energy loading factors are key quantities that characterize winds and quantify their significance in controlling baryonic mass cycles of galaxies. Measuring these loading factors has been of intense observational interest, but uncertainties are still large. In particular, the reported mass loading factor ranges widely from 0.01 to 10 . Depending on the assumed geometry, metallicity, and ionization state, the mass outflow rate can easily be reduced by a factor of 10 (e.g., Chisholm et al. 2016bChisholm et al. , 2017. In addition, uncertain deprojection may result in an overestimate the velocity, incorrectly leading to interpretation of a lowtemperature outflow as a wind rather than a fountain. If gas is not really escaping, the outward mass flux will be a decreasing function of distance from the wind launching region, and mass fluxes estimated at small radii would exceed the true losses from a galaxy. Predicting wind loading factors theoretically requires modeling the interaction between SN remnants (including from clustered SNe) and the ISM. Expansion of superbubbles driven by multiple SNe has been studied by idealized analytic models and simulations (e.g., McCray & Kafatos 1987;Mac Low & McCray 1988;Mac Low et al. 1989;Mac Low & Ferrara 1999;. While these idealized models provide essential physical insight and quantitative estimates, firm theoretical measurements of mass loading in multiphase winds from galactic disks require ISM models with realistic spatiotemporal distribution of SNe and vertical stratification. A number of local stratified-disk simulations, with increasingly high resolution, have modeled the multiphase ISM with SN feedback (e.g., Korpi et al. 1999;de Avillez 2000;de Avillez & Breitschwerdt 2004;Joung & Mac Low 2006;Hill et al. 2012;Gent et al. 2013;Walch et al. 2015;Girichidis et al. 2016b;Li et al. 2017), albeit with SN rates and locations imposed rather responding selfconsistently to star formation. Some recent numerical work has focused specifically on the outflow properties driven by SN feedback (e.g., Creasey et al. 2013;Martizzi et al. 2016;Fielding et al. 2017), although with a cooling cutoff at 10 4 K that does not allow for the full range of ISM phases. Very recently, it has become possible to evolve the turbulent, magnetized, multiphase ISM in local galactic disks with cooling and heating, galactic differential rotation, and self-gravity, including fully self-consistent resolved star formation and SN feedback over durations of several 100 Myr (Kim & Ostriker 2017, Paper I hereafter). A few other recent simulations have also included self-gravity to model SN rates and positions self-consistently with star formation (Hennebelle & Iffrig 2014;Gatto et al. 2017;Peters et al. 2017;Iffrig & Hennebelle 2017), but given their relatively short simulation duration (≤ 100 Myr), they have not reached a statistically quasi-steady state and wind properties may be subject to transient effects from the simulation start-up. In this paper, we analyze our fiducial model from the TIGRESS (Three-phase ISM in Galaxies Resolving Evolution with Star formation and SN feedback) simulation suite introduced in Paper I, in order to provide more comprehensive understanding of multiphase gas outflows in the realistic ISM. Our analysis here mainly focuses on characterizing differences among outflows of different thermal phases. In a subsequent paper, we will analyze models with different galactic conditions to investigate systematic scaling relations of wind properties (e.g., Muratov et al. 2015;Heckman et al. 2015;Heckman & Borthakur 2016;Chisholm et al. 2017). In Section 2, we review equations for steady, adiabatic flows and summarize key physical quantities to be measured from the simulation. We then present an overview of gas flows in the simulation, including overall mass fluxes and vertical profiles of each gas phase; this demonstrates the necessity of a phase-by-phase analysis. In Section 3, we analyze the hot gas component, showing that it is consistent with a wind having well-defined mass flux and specific energy (or Bernoulli parameter) that are approximately constant as a function of distance from the midplane. Section 4 presents an analysis of the warm gas component, showing characteristics of a fountain flow that has decreasing mass and energy fluxes as a function of distance from the midplane. Section 5 provides mass and energy loading factors of each phase, comparing these to previous work and to observations. Section 6 summarizes our main conclusions. OVERVIEW OF GAS FLOWS IN TIGRESS 2.1. Outflow terminology, vertical profiles, and classical adiabatic winds In galactic disks, star formation takes place in dense gas near the midplane, within the scale height of the ISM. Prodigious energy is injected by SNe within this thin layer, and the high-entropy, overpressured gas expands outward. Strong shocks heat and accelerate both dense, cold cloudlets and the warm, diffuse intercloud medium surrounding individual SNRs and superbubbles, with most of this gas cooling back to its original temperature relatively quickly (e.g. Kim & Ostriker 2015a;. Depending on the level of remaining specific energy with respect to the gravitational potential, outflows of cooled, SN-accelerated warm (or cold) gas may either keep moving out of the disk, or may turn around at some height. While most of the energy deposited by SNe is radiated away, some of the hot gas created in strong SN shocks is at low enough density that it has very limited cooling. This accelerates away from the midplane towards higher-latitude, lower-pressure regions, achieving high enough velocity that it can escape from the galactic potential well. In this paper, the term "galactic winds" refers to outflows launched with high enough energy to escape the galactic gravitational potential, while the term "galactic fountains" refers to outflows launched with insufficient energy that eventually fall back. In our simulations (and in real galaxies), the motions of gas are three-dimensional. Within any given patch of the ISM in the disk, followed on its orbit about the galactic center, horizontal averaging can be used to define a mean density, velocity, and other flow properties as a function of height z. In general, the instantaneous mean velocity will have both horizontal (radial-azimuthal) and vertical components. The horizontally-averaged flow quantities may be further averaged over time (with window comparable to an orbit time, so that epicyclic motions are averaged away). If the accretion rate is low, the resulting temporally-averaged velocity at any height will be dominated by vertical motion. Thus, horizontaland temporal-averaging of the gas yields an effectively one-dimensional profile as a function of height, consisting of average values ρ(z) , v z (z) , P (z) , etc. If there is a net outflow sign(v z ) = sign(z), and if there is a net inflow the sign is reversed. Similar to horizontal-and time-averaged flows, classical SN-driven wind solutions are one-dimensional, and the simplest solutions are also adiabatic. For steadystate one-dimensional adiabatic gas flows, the equations of mass and total energy conservation including source terms can be written as ∇ · (ρv) =ρ inj (z),(1)∇ · (ρvB) =ė inj (z),(2) whereρ inj andė inj are the volumetric mass and energy injection rates, respectively, arising from SN feedback. In Equations (1) and (2), z represents the vertical direction for a flow perpendicular to the plane of a galactic disk; an approximately spherical galactic center flow (e.g. Chevalier & Clegg 1985) would instead haveρ inj (r) andė inj (r). Note thatρ inj does not represent SN ejecta itself. Rather, shock heating of ambient ISM gas near SNe increases the mass injection rate above that of the initial SN ejecta, while cooling tends to reduce the rate. Allowing for this shock heating and cooling,ρ inj is simply the mean local rate at which hot material is added the steady flow. Similarly,ė inj represents the mean local energy input rate to the flow, which is bounded above by the initial energy carried by SN ejecta. In Equation (2), the total specific energy (the Bernoulli parameter) is defined by B ≡ v 2 2 + γ γ − 1 P ρ + Φ,(3) consisting of specific kinetic energy, specific enthalpy h = [γ/(γ − 1)]P/ρ, and gravitational potential terms. Note that here, for simplicity, we consider adiabatic, unmagnetized gas, but Appendix A presents the full equations for general MHD flows in a local shearing box, and shows that horizontal-and time-averaging yields a set of simple steady-state 1D flow equations, which can be applied to our simulations. In this paper, the Φ = 0 reference point is at the midplane. For flows in a local Cartesian box, like ours, with energy sources near the midplane, any time-averaged steady winds that may exist are launched vertically along theẑ axis. Through Gauss's Law, volume integration of Equations (1) and (2) gives mass and total energy fluxes through surface area A = L x L y at z = ±d asΣ wind (d) ≡ F M (z = d) − F M (z = −d) =Ṁ inj (d)/A (4) and F E (d) ≡ F E (z = d) − F E (z = −d) =Ė inj (d)/A. (5) Here, d is the distance from the disk midplane, and total mass and energy injection rates within |z| < d arė M inj (d) = d −dρ inj dV andĖ inj (d) = d −dė inj dV . The quantities F M (z) = ρv z and F E (z) = ρv z B stand for mass and total energy fluxes averaged over horizontal area at height z. 1 The above relations assume periodic boundary conditions in both of the horizontal directions; energy terms associated with the background shear from integrals over faces perpendicular tox are discussed in Appendix A. Since SN explosions are usually concentrated within a thin layer near the midplane,Ṁ inj (d) andĖ inj (d) are expected to approach constant values for d H, where H is the disk scale height. Hence, if A is constant as in the local Cartesian coordinates, for a steady wind the (areal) mass and energy fluxesΣ wind and F E would also 1 For the energy flux, we hereafter use subscripts KE, TE, GE, and ME to denote different energy components. These are respectively F KE = ρvzv 2 /2 (kinetic), F TE = ρvzh (thermal, with h enthalpy), F GE = ρvzΦ (gravitational), and F ME = Sz (magnetic; see Section A and Equation (A10)). The subscript E will denote the total energy term, the sum of all four components. approach constants at d H above the source region. For general geometries (e.g., Chevalier & Clegg 1985 for spherical coordinates), however, where the area that encloses a given set of streamlines varies as a function of distance (e.g., A ∝ r 2 for the spherical case), the fluxes would also vary (e.g., ∝ r −2 ). To the extent that horizontal correlations (or anticorrelations) among variables at a given z may be neglected, F E = F M B, and if we may assume symmetry across the midplane this yields B = F E /Σ wind = E inj /Ṁ inj . This implies that beyond the source region whereĖ inj andṀ inj have reached their final values, the Bernoulli parameter becomes constant for a steady (or time-averaged) flow. In the more general case with A an increasing function of distance, because the area perpendicular to streamlines is the same for both mass and energy flux, F E and F M would vary ∝ A −1 but B would still be conserved along streamlines (beyond the source region) irrespective of geometry. Therefore, for steady adiabatic winds (or equivalently for any time-averaged, adiabatic portion in a more general outflow), the Bernoulli parameter is a key quantity that enables robust extrapolation of flow evolution to large distances. 2 As applied to the present problem, this suggests that evaluation of B with our local disk simulations should provide predictions for properties of the hot portion of the wind at large distance (outside our simulation domain) where wind streamlines open up (becoming more radial than vertical). This motivates the need to quantify mass and energy fluxes in the launching region -just above the source region -in our simulations. The simple analysis above provides helpful intuition for gas flows driven by localized energy sources, but the real ISM -and our simulations -is not a single phase, adiabatic gas. In fact, material in the ISM spans a wide range of density and temperature. SN shocks are responsible for generating the hot gas phase (T ∼ 10 6 −10 7 K), which interacts with surrounding warm (T ∼ 10 4 K) and cold (T ∼ 10 2 K) phases. Considering each thermal component individually, radiative heating and cooling as well as mass and energy transfers between components would act as source and sink terms in the conservation equations of each phase. Because SN ejecta strongly interact with the surrounding ISM in extremely complex ways to heat and accelerate gas, some of which may be able to escape from a galaxy, it is not at all obvious how one would estimateṀ inj andĖ inj for individual thermal components of a multiphase outflow. Moreover, star formation and hence SN events are very bursty, and this burstiness may affect yields. Clearly, the total and individual-phaseṀ inj andĖ inj are only quantifiable with self-consistent numerical simulations that capture the full physics of the ISM. Nevertheless, while simulations are essential for obtaining the detailed properties of realistic multiphase outflows, we can still expect certain aspects of classical wind solutions to hold when suitably applied. In particular, as we shall show, the space-time-averaged hot portion of the wind, when considered separately from other phases, shares many similarities with adiabatic one-dimensional winds. TIGRESS simulation model and analysis In Paper I, we presented a novel framework for multiphysics numerical simulations of the star-forming ISM implemented in the Athena MHD code (Stone et al. 2008;Stone & Gardiner 2009). We solve the ideal MHD equations in a local, shearing box, representing a small patch of a differentially rotating galactic disk. This treatment allows us to achieve uniformly high spatial resolution compared to what is possible in a global simulation of an entire galaxy (or galaxies) (e.g., Hopkins et al. 2012;Muratov et al. 2015), which is crucial to resolve both star formation and SN feedback as well as all thermal phases of the ISM both near and far from the midplane. We include gaseous and (young) stellar selfgravity, a fixed external gravitational potential to represent the old stellar disk and dark matter halo, galactic differential rotation, and optically thin cooling and grain photoelectric heating. We utilize sink particles to follow formation of and accretion onto star clusters in dense, cold gas. Massive young stars in these star clusters feed energy back to the ISM, by emitting far-ultraviolet radiation (FUV) and exploding as SNe. The former heats the diffuse warm and cold ISM, while the latter creates hot ISM gas, drives turbulence, and induces outflows. Our simulations yield realistic, fully self-consistent three-phase ISM models with self-regulated star formation. 3 Paper I presented results from a fiducial model with parameters similar to those of the Solar neighborhood. There, we showed that after t ∼ 100 Myr a quasi-steady state is reached. When stars form, massive stars enhance heating in the warm and cold ISM, and the SN rate increases, driving turbulence through-out the ISM. Both feedback processes disperse dense clouds and puff the gas disk up, temporarily shutting star formation off. With the corresponding reduction in star formation feedback, the gas can settle back to the midplane and collect into dense clouds which then form a new generation of stars. In Paper I, we evaluated several basic ISM and wind properties, and demonstrated their convergence as a function of the numerical spatial resolution. In this paper, we analyze detailed properties of galactic winds and fountains for a vertically extended version of the fiducial model of Paper I. The simulation domain covers 1 kpc × 1 kpc horizontally and −4.5 kpc < z < 4.5 kpc vertically, at a uniform resolution ∆x = 4 pc. Representative snapshots displaying a volume rendering of density and velocity vectors during outflow-and inflow-dominated periods are shown in Figure 1 To characterize vertical gas flows, we first construct horizontally-averaged quantities. We then calculate the net mass and energy fluxes that pass through horizontal planes (both upper and lower sides) at d = 1, 2 and 3 kpc (cf. Equations 4 and 5). Figure 3 shows time evolution of (a) mass flux, along with the areal star formation rate (Σ SFR ), and (b) energy flux. Every star formation burst is followed by a burst of energy injection, and this burstiness is reflected in large temporal variations in the mass and energy fluxes. The fluxes can become negative, meaning that the mass and/or energy of gas flowing inward exceeds that of the gas flowing outward, at a given height. As distance d from the midplane increases, the net mass flux significantly decreases, while the net energy flux even through d = 2 and 3 kpc remains large. While net negative mass fluxes (implying fallback) occur at d = 1 kpc after each burst of mass outflow, net energy fluxes almost always remain positive. The differing behavior of mass and energy fluxes is a signature of multiphase flows. Although star formation and hence SN feedback are impulsive rather than continuous, the system approaches a quasi-equilibrium state. This state is a limit cycle mediated by the feedback loop, in which epochs of cooling and collapse alternate with epochs of heating and expansion. 4 Given that a quasi-steady state exists in the present simulation, horizontal-and temporal averages can be constructed to characterize this mean state. Since different thermal phases coexist at all heights, to understand the outflows and inflows of mass and energy it is further necessary to separately construct horizontal averages of each thermal phase. In Figure 4 we plot horizontally-and temporallyaveraged profiles of mass, momentum, and energy distributions for thermally separated gas phases; these profiles average over t = 250 − 500 Myr and also average over upper (z > 0) and lower (z < 0) sides. Profiles show hydrogen number density, n H ≡ ρ/(µ H m H ), outward vertical momentum density, ρv out ≡ ρv z sign(z), and total energy density (excluding gravity) E ≡ 1 2 ρv 2 + P γ − 1 + B 2 8π .(6) Note that this energy density differs from the gas density multiplied by the Bernoulli parameter of Equation (3), as it includes just thermal energy (rather than enthalpy) and also includes magnetic energy density. The four thermal phases plotted in Figure 4 are cold (T < 5050 K), warm (5050 K < T < 2 × 10 4 K), ionized (2 × 10 4 K < T < 5 × 10 5 K), and hot (T > 5 × 10 5 K). 5 Above the warm/cold layer (d > H, where H ≈ 400 pc), the mass density is dominated by the warm component and the energy density is dominated by the hot component. As the individual terms in the energy density are proportional to corresponding terms in the momentum flux (ρv 2 , P ), the hot medium also dominates the momentum flux away from the midplane. The hot medium is the largest contributor to the time-averaged vertical momentum density, which is the same as the timeaveraged net mass flux. Although the mean value of vertical momentum density (or net mass flux) of the warm medium is effectively zero, there are large temporal fluctuations (indicated by the green shaded region) at small and intermediate d because the warm gas contributes significantly to both outgoing and incoming mass fluxes at different times (as in Figure 3). The phase-separated momentum and energy density profiles in Figure 4 (and corresponding profiles of mass and momentum flux) reflect essential differences of gas hold in parts of the parameter space in which the vertical oscillation time (which controls collapse and star formation) is sufficiently long compared to the stellar evolution timescale (which controls feedback and expansion). Figure 1. Sample snapshots illustrating (left) outflow-dominated, and (right) inflow-dominated periods, at t = 300 and 360 Myr, respectively. Gas density is shown in color scale volume rendering and the velocity field is shown with vectors; vector colors (rather than length) indicate velocity magnitudes. A fast-moving, low-density (dark blue) outflow is evident in the left snapshot, while moderate-density (green) gas that was previously blown out to large distances is falling back toward the midplane in the right. Velocity fields are turbulent near the midplane (d < H), but ordered in either outflowing or inflowing directions at large distances. For visual clarity, only the upper half of the simulation box (z = 0 to 4 kpc) is shown, with a full horizontal crossection −512 pc ≤ x, y ≤ 512 pc. flow dynamics between the warm and hot phases, which we separately analyze in the following sections. HOT WINDS In this section, we focus on the hot component, defined by T > 5 × 10 5 K, representing gas that has been shock heated by SN blastwaves. The hot medium fills most of the volume above the disk scale height, and cooling in tenuous hot gas is inefficient. With pressure gradients that accelerate it outward (cf. Figure 4(c)) and a source from SN shocks propagating into the surrounding warm and cold medium near the midplane, the horizontally and temporally averaged hot medium naturally fits the criteria for a (quasi-)steady, adiabatic wind. We therefore might expect that the mass flux and energy flux (and thus the Bernoulli parameter) of the time-averaged hot component would be (approximately) conserved as the gas flows outward. Figure 5 explicitly shows vertical profiles of (a) the mass flux,Σ wind,h and (b) the specific energy, F E,h /Σ wind,h , based on time averages of each horizontally averaged flux. To distinguish between outflows and inflows, for the mass flux in addition to net flux we separately show the flux of outflowing (sign(v z ) = sign(z)) and inflowing (sign(v z ) = −sign(z)) gas. The mass flux of the hot gas shows net outflow (blue line in Figure 5(a)), with negligible inflow flux at all heights (yellow line in Figure 5(a)). As the SNe that create the hot medium are concentrated near the midplane, the hot gas mass flux within the warm/cold layer (d < H; below the vertical dotted line in Figure 5) first increases with d and then decreases as hot SNRs have a maximum size before the onset of cooling. Above the point where the hot gas in the interior of SN remnants (or superbubbles) breaks out of the warm/cold layer the hot gas mass flux is nearly constant with d. The slight decrease of the mass flux at large d is caused by nonzero cooling (radiative and mixing of hot gas with cooler phases). Figure 5(b) shows near constancy of the time-averaged total specific energy B of the hot gas above d = 1 kpc, as expected for a steady, adiabatic wind (given the negligible Poynting flux). We also calculate individual components of the (averaged) energy flux, and divide them by the (averaged) mass flux. This provides the components of the mass-flux-weighted specific energy F E,h /Σ wind,h , consisting of kinetic, thermal (enthalpy), gravitational, and magnetic terms (see Equations 3 and A10). Once the hot gas breaks out of the warm/cold layer, the gas flow approximately preserves mass and energy fluxes because except for limited cooling there are no sources (or sinks) for the hot gas mass and energy. We have checked that the individual cooling and heating terms, including Reynolds and Maxwell stresses arise from the shearing box (see Equation (A7)), are indeed small compared to the SN energy injection. From d = 2 − 4 kpc, the enthalpy of the hot gas implies a temperature in the range 1.2 − 1.5 × 10 6 K (or In SN-driven hot winds, the enthalpy (specific heat) term dominates over other components of the Bernoulli parameter, including the gravitational potential, at heights less than the disk radius. At large distance where streamlines open up in angle, one could expect hot galactic winds with constant B to accelerate past a sonic point to an asymptotic velocity of v wind ≡ √ 2(B − Φ) 1/2 as they adiabatically cool, similar to classical Parker stellar wind solutions. 6 Global simulations indeed show the expected behavior at large distance (e.g. Fielding et al. 2017). In general, hot winds are accelerated by pressure gradients at the same time as enthalpy is converted to kinetic energy, and a sonic transition in a steady wind is . Vertical distributions of mass, vertical momentum, and energy densities, averaged horizontally and over upper (z > 0) and lower (z < 0) sides of the disk and over time t = 250 − 500 Myr. Colored lines for separate thermal components show profiles of (a) hydrogen number density nH , (b) vertical momentum density ρvout (which is the same as the mass flux), and (c) total energy density (excluding gravity) E, each as a function of distance d from the midplane. Color-coded shaded regions represent one-sigma temporal fluctuations. In order to properly visualize both the magnitude and sign of the momentum density, we use a linear scale for |ρvout| < 10 −4 and a log scale for |ρvout| > 10 −4 . The warm and hot phases respectively dominate mass and energy densities above the disk scale height (indicated by vertical dotted line), and the hot component also has the largest vertical momentum density (net outward mass flux). only possible if the crossectional area increases as the flow moves outward (e.g. Shu 1992). In the Cartesian geometry of the present simulations, streamlines cannot open up and there is no associated adiabatic cooling, limiting the pressure and density gradients and therefore the acceleration of the flow. However, the constancy of both the Bernoulli parameter and mass flux with d for the hot medium in our simulation suggests that it properly represents the near-disk regions for a generalized galactic disk wind, in which streamlines emerge from the disk vertically (with A = const when z R disk ) and would open up (A increasing with distance) when z > ∼ R disk . The mass and energy fluxes carried by the hot wind are controlled by the interaction between SN shocks and the warm-cold medium that creates the hot ISM well within the disk, in processes that are unlikely to be affected by large-scale global galactic and CGM properties and geometry. Therefore, the Bernoulli parameter we calculate should be a robust estimator of asymptotic hot wind speed irrespective of the constraints of our Cartesian box. As shown in Figure 3 (and demonstrated by the shaded region in Figure 5(b)), star formation and hence outflows are bursty, resulting in large temporal fluctuations. Based on analysis of B for the hot component within short time ranges, the maximum asymptotic velocity 7 of the hot wind could reach up to v wind ∼ 500 km s −1 , while the mean asymptotic wind speed would be v wind ∼ 350 km s −1 . Defining the speed required to escape to r as v esc (r) ≡ [2(Φ(r) − Φ(0))] 1/2 , in our simulation domain v esc (z = 4 kpc) ∼ 130 km s −1 , so the hot wind easily escapes, i.e. B Φ even at large d for the hot component. A hot wind launched with the local conditions of our simulation would also be able to propagate far into the halo for the Milky Way, where the escape velocities are v esc (50 kpc) ∼ 350 km s −1 and v esc (150 kpc) ∼ 450 km s −1 using MWPotential2014 in galpy (Bovy 2015). More generally, the far-field velocity for a hot wind with given local launching conditions can be estimated based on B and the large-scale galactic potential. Finally, we note that while we have mostly based the discussion above on temporal averages of horizontallyaveraged profiles, the properties of instantaneous profiles are quite similar. To illustrate this, Figure 6 shows the mass flux and specific energy of the hot component at t = 300 Myr, the "outflow" snapshot shown in Fig. (2). Except for local fluctuations, the instantaneous mass flux and specific energy profiles are overall very similar to the corresponding time-averaged profiles. In particular, the specific energy profiles in Fig. 5 and 6 are quantitatively almost the same, while the mass flux is higher in Fig. 6 (3) and (A11)). Mean values are shown as colored lines, and onesigma temporal fluctuations are shown as shaded regions. The specific enthalpy and kinetic energy are much larger than the gravitational potential. The magnetic term plays a minor role. In both (a) and (b) Over the duration of the simulation, the mean net mass flux in warm gas out of the simulation domain is ∼ 1.1 × 10 −4 M kpc −2 yr −1 , about 28 percent of the net mass flux in hot gas. As we shall show, the outward mass flux of the warm medium secularly decreases with d, such that if our box were taller we would expect the mean net mass flux out of the simulation domain to be even smaller. With negligible time-averaged net outward mass flux at large distance, the warm medium at d > ∼ 1 kpc in our simulation is not a true galactic wind. Even though the inflows and outflows of warm gas over long timescales are essentially balanced, time variations in the warm gas flux ρv out are quite large and include both positive and negative values (green shaded region of Figure 4(b)). The fluctuating behavior of the warm medium can be contrasted with the much smaller temporal fluctuations of the mass flux in the hot gas (red shaded region of Figure 4(b)). Evidently, the warm gas does not produce an escaping wind like that in the hot gas but a fluctuating fountain that at any time consists of both outflow and inflow. Figure 3 shows that alternating inflow and outflow dominance in the warm gas is reflected in the alternating signs of the mass flux for the whole medium at d = 1 kpc. In addition, the decrease in the magnitude of the total (phase-integrated) mass flux with increasing d reflects the secular decrease in the net mass flux of the warm fountain (difference between inward and outward fluxes) with distance. To quantify the characteristics of warm fountain flows, we take time averages selectively for outflow (Σ wind (1 kpc) > 0.001 M kpc −2 yr −1 ) and inflow (Σ wind (1 kpc) < −0.001 M kpc −2 yr −1 ) periods. Outflows of warm gas occur when many correlated SNe from a star formation burst lead to a superbubble expanding into the warm and cold layer, while inflows occur when the disk is in a quiescent state with reduced star formation after the cold medium has been dispersed by a previous burst. Figure 7 plots time averaged mass flux (top) and specific energy (bottom) for outflow (left) and inflow (right) periods. Although one flow dominates the other during each period, the opposite flows always exist at all heights and are more significant compared to the case for hot gas ( Figure 5). For the warm medium, the kinetic term in the specific energy exceeds the enthalpy, but remains below the gravitational potential term at d > ∼ 1 kpc. This explains why most of the warm gas outflow turns around at d ∼ 1 − 2 kpc and falls back toward the midplane. During the outflow period, the mean velocity of the warm gas is in the range ∼ 60 − 80 km s −1 for d = 1 − 4 kpc. Overall, the warm medium occupies more volume above d = 1 kpc during the inflow period than the outflow period. This is because the hot gas is mainly generated during the outflow period by shocking the warm gas, and the high-pressure hot gas confines the warm gas into small cloudlets. When the disk becomes quiescent, the warm gas expands into previous hot wind channels (see Figure 2). About a factor of 5 to 10 more volume is occupied by the warm gas in the inflow period than the outflow period for d = 2 and 3 kpc slabs. Although in the current simulations the warm gas is almost entirely confined by the galactic gravitational potential, this would not necessarily be true if the potential were shallower, as in dwarf galaxies. Direct simulations for different galactic conditions, including shallower potentials, are underway using the same TIGRESS framework. However, we can also use our current simulation to provide information on what might be expected by quantifying the fraction of fast-moving warm gas. Figure 8 shows outward velocity probability distribution functions (PDFs) weighted by volume (top), mass (middle), and mass flux (bottom) within slabs of thickness ∆ slab = 20 pc during outflow (left) and inflow (right) periods. We show results at several distances d averaged over both sides of the disk (z = ±d). Note that the volume-weighted PDF is normalized by slab volume (V slab = L x L y 2∆ slab ), and the mass-weighted PDF is normalized by the mean mass within the same volume (M slab ≡ M tot 2∆ slab /L z ) (M tot is the mass in the whole domain at a given time), while the mass flux PDF is in physical units of M kpc −2 yr −1 / km s −1 . During both outflow and inflow periods, Figure 8 shows that the warm gas velocity has a broad distribution with both outward and inward velocities. During the outflow period, Figures 8(a) and (c) show that the volume and mass of high velocity (> 50 km s −1 ) warm gas increases from the midplane to 1 kpc, where the specific kinetic energy is larger than the gravitational potential (see Figure 7(c)). The increase in mass of high-velocity, high-altitude warm gas between the midplane and d = 1 kpc is due to acceleration of the warm medium pushed by expanding superbubbles; this includes warm gas that was shock-heated to the hot phase (and accelerated to high velocity) and subsequently cooled back down. Figures 8(c) and (e) show that as d increases the peaks of the distributions of mass and mass flux move to higher velocity and the overall outflowing gas fraction decreases. This general trend represents dropout/turnaround of warm gas fluid elements with low (and decelerating) velocities that are unable to climb to large d in the gravitational potential. In principle, acceleration of warm clouds driven by hot-gas ram pressure, cooling of fast hot gas, and dropout of low-velocity warm fluid elements could all contribute to the gradual increase of warm-medium specific kinetic energy at d > H shown in Figure 7(c). Figure 8(c) shows, however, that overall the mass of high-velocity warm gas is decreasing with increasing d. This has the important implications that in our simulation (1) warm clouds at large d are not significantly accelerated by ram pressure of the hot, high-velocity gas that is flowing out around them, and (2) relatively little hot gas is converted to the warm phase through cooling at large d. Rather, the warm medium is primarily accelerated via direct energy input from SNe at d < ∼ 1 kpc, and at higher altitudes warm fluid elements slow and turn around according to the competition between the gravitational potential and the kinetic energy they initially acquired at small d. Figure 4(c) and Figure 5(b) are also telling in this regard: the total energy density (and individual components) of the hot medium declines very slowly for d > ∼ 1 kpc, while the energy density of the warm medium declines steeply; since momentum flux terms are proportional to energy density terms, this indicates that there is no significant transfer of momentum from the hot to the warm gas. During the inflow period, the majority of the warm gas is falling. Since SN feedback is never completely turned off, however, some warm gas is still accelerated outward. The fraction of (outgoing) fast-moving warm For the warm medium, the kinetic term exceeds both the thermal and magnetic terms in the specific energy, but is always lower than the gravitational term above ∼ 1 kpc. This explains why the warm medium creates a fountain rather than a wind. The vertical dotted lines in all panels indicates the gas scale height (H = 400 pc). gas (v out > 50 km s −1 ) is reduced by a factor of 5 to 10 in inflow compared to outflow periods. Combined with the total outflowing mass fluxes ofΣ wind,w ∼ 10 −2 and 10 −3 M kpc −2 yr −1 at d = 1 kpc during outflow and inflow periods (see Figure 7(a,b)), respectively, the mass fluxes of the fast-moving warm gas are about 6 × 10 −4 and 10 −5 M kpc −2 yr −1 (see Figure 8(e) and (f)). This can be compared to a mean mass flux of hot gas at the same height of ∼ 1.3×10 −3 and 2×10 −4 M kpc −2 yr −1 during outflow and inflow periods, respectively. Although the "fast" warm outflow has comparable mass flux at d = 1 kpc to that of the hot medium during outflow periods, even at v out > 50 km s −1 the warm medium yields very little mass escaping from the largescale potential in the present simulation, while the hot medium mostly escapes (0.12 and 0.034 M pc −2 of hot and warm gas have respectively escaped during the time interval of 250 Myr). This emphasizes the importance of measuring not just mass fluxes, but mass fluxes and specific energies in comparison to galactic escape speeds. At large distances the large-scale gravitational potential strongly affects the warm-gas velocity distribution by enforcing dropout of lower-velocity material, but closer to the midplane this is less of an issue. At around the disk scale height, the gravitational potential is small compared to the specific kinetic energy of the gas, and global geometric effects are not important yet. We therefore consider the velocity distribution at d = 1 kpc as representative of the launching conditions for a warm outflow, which would apply relatively independently of the global galaxy (e.g. in a dwarf as well as a large galaxy for given local conditions). Here, we find the PDFs during outflow periods of the fast-moving warm gas (v out > 50 km s −1 ) at d = 1 kpc are well fitted by a single exponential function, (z)) probability distributions of the warm medium. Distributions of volume (top), mass (middle), and mass flux (bottom) as a function of velocity are calculated for 20 pc thickness slabs above and below the midplane (z = ±d) centered at d = 0, 0.5, 1, 2, 3 kpc. In the bottom row, the mass flux within each velocity bin is in units of M kpc −2 yr −1 / km s −1 , and negative mass fluxes are shown with dotted lines. Warm gas with both signs of vout is present during both "outflow" and "inflow" periods, but the corresponding mean velocity (for d = 0 kpc) changes sign. Acceleration of the warm gas to high velocities, especially during outflow periods, is evident in the difference of profile shapes between the midplane (d = 0 kpc) and higher latitude, with an exponential tail at high velocity developing by d = 1 kpc. However, an overall deceleration with height in the increasing gravitational potential is also evident in comparison of profiles at increasing d ≥ 1 kpc. The magenta dashed lines in the left column are fits to the fast-moving gas vout > 50 km s −1 at d = 1 kpc given by Equations (7) and (8). df dv out = A f exp − v out v f ,(7) the characteristic velocities are v V = 18 km s −1 and v M = 15 km s −1 . Using the mass PDFs, the mass-flux PDF is given by d(ρv out ) dv out = M slab V slab A M v out exp − v out v M ,(8) where the mean density, M slab /V slab , is given by ρ = 1.2×10 −3 M pc −3 , corresponding to hydrogen number density n H = 0.033 cm −3 . By quantifying the mass flux PDF in warm gas for models with different local conditions, it should be possible to develop a comprehensive quantitative characterization of warm wind launching by star formation feedback. These local results could then be used to make global predictions. For example, integration of Equation (8) for velocities v out > v H for would yield a mass flux (v H /v M + 1) exp(−v H /v M )Σ wind,w (d = 1 kpc). If this holds in general, it means that measurement of the launching-region warm mass fluxΣ wind,w (d = 1 kpc) for given local disk conditions could be used to predict the flux that actually escapes into the halo for a galaxy with arbitrary halo velocity v H . This statistical characterization can be combined with measurements of mass flux and Bernoulli parameter for the hot medium to develop subgrid models of multiphase wind driving for implementation in galaxy formation simulations. MASS AND ENERGY LOADING Simulation Results In this section, we discuss key quantities of gas outflows driven by SNe, mass and energy loading factors. The mass loading factor, β, through the surfaces at d (including both sides of the disk plane) is conventionally defined by the ratio of "outgoing" mass flux to the star formation rate as β ≡Σ + wind Σ SFR ,(9) while the energy loading factor, α, is defined by the ratio of "outgoing" kinetic + thermal (enthalpy) energy flux to the energy production rate of SNe, α ≡ F + KE + F + T E E SN Σ SFR /m * ,(10) where E SN = 10 51 erg is the total energy per SN, and m * = 95.5 M is the total mass of new stars per SN (see Paper I). To computeΣ + wind , F + KE , and F + T E we select only zones with outflowing gas, i.e. with sign(v z ) = sign(z). The areal star formation rate averaged over t = 250 − 500 Myr is Σ SFR = 0.006 M kpc −2 yr −1 ; we use this average Σ SFR in computing all loading factors. Note that a rolling mean of Σ SFR with 100Myr time window gives 50% variation with respect to the mean. Because there is generally a temporal offset between star formation bursts and winds (see Figure 3), the instantaneous ratio of mass or energy fluxes to the star formation rate can significantly over-or under-estimate the true physical loading. The ratio of time-averaged wind fluxes to the time-averaged star formation rate is more meaningful. We decompose the gas into four thermal components and present the loading factors of each thermal phase as a function of d (Figure 9). The energy flux is always dominated by the hot gas, with the energy loading factor of α h ∼ 0.01 − 0.05 above d > 1 kpc. SN feedback causes large outgoing mass fluxes of warm and cold gas within the disk scale height d < H, but the majority of the warm and cold medium has low velocity and cannot travel far from the midplane; this mass flux at small d is best thought of as the "upwelling" of turbulent motions within the disk driven by expanding SNRs and superbubbles. 8 The mass loading factor of the ionized gas also decreases significantly with d. By the time the flow reaches the edge of the simulation domain, β h is at least a factor ∼ 3 larger than all the other components. Therefore, as we have concluded in Sections 3 and 4, only the hot gas forms a genuine"galactic wind," with a mass loading factor of β h ∼ 0.1 that is nearly constant above d > 1 kpc. We note that with the above definitions of α and β, the Bernoulli parameter for the hot wind is given by B = Φ + (α h /β h )E SN /m * . The vertical dependence of the mass loading factors of non-hot phases shown in Figure 9(a) implies that it is important to provide careful distinctions when reporting on the outflows measured in numerical simulations, as "wind" mass loading can be greatly overestimated if this is not done. In particular, the steep decrease of β w with d implies that if this were measured at d ∼ 1 kpc it would overestimate the value at the edge of our box (d ∼ 4 kpc) by a factor ∼ 30, and the true value at larger distance would be even smaller (see also Martizzi et al. 2016, who reached similar conclusions about total β). Thus, a measurement of β w from a simulation with small vertical domain cannot by itself provide a prediction for warm gas mass loss in a galactic wind. However, even with a limited vertical domain, it is possible to discriminate between fountain flow and wind by combining measurements of the warm gas mass flux and its vertical velocity (as in Section 4). Similar considerations apply to observations of gas at T ∼ 10 4 K at high latitudes in edge-on galaxies, but in that case uncertain projection effects make this even more problematic: without an unambiguous measurement of velocity (which is subject to assumed wind geometry), it is impossible to distinguish between fountain flow and wind from observations of the emission measure. More generally, it is essential to decompose outflows in simulations into separate thermal phases to distinguish winds from fountain flows. The integrated β cannot be taken as a true wind mass loading unless the measure- The mass loading factor of the hot gas obtained here is consistent with simple estimates based on idealized experiments of superbubbles driven by multiple SNe in the warm-cold ISM . The shock from an individual SN sweeps up ∼ 10 3 M before it cools and forms a shell (Kim & Ostriker 2015a). For superbubbles created by multiple SNe, the maximum mass in hot gas per SN isM h ∼ 400 − 2000 M prior to shell formation, but subsequently this drops toM h ∼ 10 − 100 M (lower at higher ambient density). When star formation rates are self-regulated (e.g., Kim et al. 2013;Kim & Ostriker 2015b), the mean interval between SNe within projected area πH 2 in the disk of a galaxy is always ∼ 0.3 Myr, and breakout of superbubbles is expected to occur after shell formation (see Section 5 of . For a SN interval ≥ 0.1 Myr, Figure 11 of shows that by the time superbubbles reach a radius of 2H,M h ∼ 10 − 30 M , depending only weakly on ambient density. This corresponds to β h =M h /m * ∼ 0.1 − 0.3. The same idealized superbubble simulations show a hot-gas energy loading factor of a few percent when superbubbles expand beyond ∼ H, since most of energy has already been transferred to acceleration and heating of ambient gas and lost via radiative cooling at the time of shell formation. argued that β h < 1 is expected quite generally unless the temporal and spatial correlations of SNe are extremely enhanced compared to their mean values, requiring more than a factor of 40 elevation compared to the average conditions in self-regulated galactic disks where Σ SFR ≥ 10 −3 M kpc −2 yr −1 . Although our simulation does exhibit large temporal fluctuations (see Figure 3), the peak upward fluctuation compared to the mean star formation rate is only a factor of 5. Further systematic investigations for different galactic conditions will be needed to confirm whether the predictions for low β h apply universally in star-forming galaxies. If local or global conditions make star formation inherently extremely bursty, then β h may be higher. Comparison with Observations The low mass loading factor of the hot gas in our simulations is comparable to (or slightly smaller than) the observed mass loading factor estimated in the best studied local starburst M82. Using Chandra X-ray observations, Strickland & Heckman (2009) constrained the "central" mass loading factor of the hot gas to about β h ∼ 0.3 − 1, with the "central" energy loading factor about α h ∼ 0.3 − 1. Here, they constrained quantities using a large number of hydrodynamical models to explain the observed diffuse and hard X-rays, which come from the central 500pc region. 9 It is difficult to make a direct comparison, but if we consider the state of the hot 9 However, we note that in their modeling the hot wind freely expands into a very tenuous medium rather than expanding into a dense ISM. By comparison, it is evident e.g. in Fig. 8 of that before superbubble breakout from the warm/cold ISM, the hot gas has very high velocity but mostly remains subsonic. This suggests that lack of a dense surrounding medium in a hydrodynamic wind comparison model might lead to a Mach number higher than would be realistic, and hence for density and temperature constrained by emission properties the mass loss rate could be overestimated by a factor of a few. medium within the energy-injecting layer (d < H) in our simulations, we have β h ∼ 0.5 and α h ∼ 0.1 − 0.2. Although there are no systematic observational studies of mass loading factors of the hot gas, β h < 1 is suggested for a wide range of star formation rates from dwarf starbursts to ultraluminous infrared galaxies, utilizing the Chevalier & Clegg (1985) wind model with observational constraints of the X-ray luminosity and star formation rates (Zhang et al. 2014). For the solar neighborhood conditions investigated in the present simulation, the warm gas accelerated by the SNe cannot reach velocities fast enough to escape the gravitational potential. Typically, star formation bursts launch warm outflows with velocity up to ∼ 100 km s −1 , but with most gas at lower velocity (see Figure 8) which can climb to ∼ 1 kpc but no further (compare the kinetic and gravitational curves in Figure 7(c)). Without additional energy and momentum input at high-|z|, SN feedback alone cannot drive warm winds in the Milky Way-type galaxies. Above d = 1 kpc, Figure 9(a) shows that β w < 1. Even though relatively fast warm gas could escape from dwarf galaxies with a shallower potential well than the Milky Way, Equation (7) implies that β w ∼ 1/3 for v out > 50 km s −1 at most. This suggests that achieving β > 1 in dwarfs, as appears necessary to explain current-day galaxy-halo relationships and cosmic history, would require an additional acceleration mechanism such as interaction with an outflowing cosmic ray fluid (e.g., Hanasz et al. 2013;Girichidis et al. 2016a;Ruszkowski et al. 2017). In cosmic-ray driven winds, SN feedback may still be crucial for pushing warm gas out to large d where cosmic ray pressure gradients are sufficient to produce efficient acceleration (Mao & Ostriker 2017, submitted). Observational studies of warm phase outflows (possibly including the ionized phase according to our definition) indicate a wide range of the mass loading factor, β w+i ∼ 0.1 − 10 (e.g., Heckman et al. 2015;Chisholm et al. 2017), with systematically decreasing trends for increasing galaxy mass, circular velocity, and total SFR. As the level of wind mass loading may vary with both local conditions (including Σ gas and Σ * ) as well as the global gravitational potential, it will be quite interesting to measure outflow properties in simulations under widely varying galactic conditions, for comparison with current empirical scaling trends and future observations. Wind Driving Simulation Context Recently, a number of other research groups have performed simulations with similar local Cartesian box setups to study galactic outflows driven by SNe (e.g., Creasey et al. 2013Creasey et al. , 2015Girichidis et al. 2016a,b;Mar-tizzi et al. 2016;Li et al. 2017;Gatto et al. 2017). Most of these simulations have adopted fixed SN rates, while varying the SN placement (e.g. random vs. in high density regions). In contrast, in our simulation, SN rates and locations are self-consistent with star formation, which we believe is crucial in creating realistic multiphase outflows. Girichidis et al. (2016b) ran a set of simulations with solar neighborhood conditions, focusing on the effect of the SN placement and of SN clustering. The authors emphasize that due to the short duration of their simulations (100 Myr), definitive conclusions cannot be drawn regarding wind driving. However, their measurements of outflow in the d = 1 kpc plane can be compared to our fountain flow measurements. For a range of different SN feedback treatments, they report mass loading factors ∼ 5 − 10. Although this is larger than what we find (β w ∼ 1 at d ∼ 1 kpc), their simulations do show a decline in outflow over time, and our measurements are at t > 250 Myr. As pointed out by the authors, the majority of the outflowing mass is relatively dense (n H ∼ 0.1) and moves slowly (v out ∼ 20 − 40 km s −1 ), similar to the properties of our warm fountain at low d. Their fraction of high-velocity gas (v out > 500 km s −1 ) is only 2 − 8 × 10 −5 , which correspondingly reduces the mass loading factor of gas that is certain to escape to large distance. In simulations including cosmic rays, Girichidis et al. (2016a) found slowly moving (v out ∼ 10 − 50 km s −1 ) warm outflows with a mass loading factor near unity. The SN-only comparison model of Girichidis et al. (2016a) had an order of magnitude lower mass loading, with just a fast, low density component passing through the d = 1 kpc surface where the mass flux is measured. While this comparison suggests that cosmic rays may be crucial to accelerating warm outflows, a concern is that the role of SNe may not have been properly captured in Girichidis et al. (2016a). Due to the high computational cost of including cosmic rays, relatively low spatial resolution ∆x ∼ 16 pc was adopted for this pair of comparison simulations. The reported mass outflow rate for SN-only models appears to differ significantly between the high resolution (∆x ∼ 4 pc) simulation of Girichidis et al. (2016b), with mass flux at d = 1 kpc of 4 × 10 −2 M kpc −2 yr −1 , and the lowresolution (∆x ∼ 16 pc) simulation of Girichidis et al. 2016a, with mass flux of 10 −3 M kpc −2 yr −1 . In our own resolution study for a similar parameter regime (see Paper I), we found that numerical convergence of SN driven ISM properties is not guaranteed at ∆x ∼ 16 pc, and the low-resolution outcomes are quite sensitive to the exact prescription for SN feedback. This suggests that higher resolution simulations will be needed to assess the role of interactions between the cosmic ray fluid and gas in driving galactic winds. The Gatto et al. (2017) models are most similar to our simulations in that they self-consistently model star formation and SN feedback, rather than adopting a fixed SN rate. The main conclusion they draw is that the outflow rate strongly depends on the volume filling factor of hot gas. Above a 50% volume filling factor, their measured mass loading factors at d = 1 kpc are ∼ 1 − 100 (but note that this may be exaggerated since loading is evaluated instantaneously rather than based on temporal averages; see discussion after Equation 10 in Section 5.1). While the simulation durations of Gatto et al. (2017) are < 100 Myr, such that star formation and wind mass-loss rates are likely both subject to "startup" transient effects, we agree with the conclusion that significant driving of fountain flows (which dominate at d = 1 kpc based on our work) is associated with prominent superbubbles near the disk midplane. In Paper I, we found the hot gas fills ∼ 20 − 60% of the volume at |z| < H. Martizzi et al. (2016) performed simulations for solar neighborhood conditions, as well as environments with higher gas surface density and SN rates. For highvelocity gas (v out > 300 km s −1 ) that could potentially escape, they reported mass loading factors of 0.02-0.005 (lower for higher gas surface density models). They also noted the absence of wind acceleration from subsonic to supersonic velocities as a limitation of local Cartesian box simulations. We have argued in Section 3 that provided the hot-gas Bernoulli parameter and mass flux have both approached constant values at large d in a Cartesian simulation, they can be combined to make predictions for wind properties at large distance. In this case, the asymptotic mass flux at high velocity would be larger than the near-disk value as high-enthalpy gas is further accelerated when streamlines open, but the total mass flux of hot gas would change little. However, it is not possible to compare hot-gas mass fluxes as Martizzi et al. (2016) did not separate by phase. Li et al. (2017) conducted simulations using fixed SN rates, but decomposed by thermal phase in reporting mass and energy (as well as metal) loading factors. Their measurements are at d = 1 − 2.5 kpc, with their "warm" component including most of what we consider "ionized," and their "hot" extending down to T = 3 × 10 5 K, slightly lower than our "hot" definition. For their solar neighborhood model, the hot and total mass loading factors are about 0.8 and 2−3, which are larger than ours by a factor of 3 to 8 (see also their discussion in comparison to work of Creasey et al. 2013Creasey et al. , 2015Girichidis et al. 2016b). The energy loading factor is also about an order of magnitude larger in their model. The reason for this large difference in mass and energy loading with respect to our findings is mainly due to the difference in the vertical scale height of SNe (relative to the gas scale height), as also pointed out in Li et al. (2017). By placing SNe randomly with a fixed SN scale height of 250pc, we find that we are able to reproduce their results (see Appendix B.2). When we adopt a fixed SN scale height of 250pc, the majority of SN explosions occur outside of the main gas layer. Each SN remnant can then expand into the hot, rarefied disk atmosphere with little interaction with warm gas. Most of the injected energy is carried outward before cooling. The resulting ISM properties are also quite different: the gas scale height is smaller (H ∼ 150 pc), star formation rates are higher, and the hot and warm/cold phases are almost completely segregated (single phase outflow). In Appendix B.2, we also test models with no runaway O stars, and with random SN placement with a smaller scale height of 50pc. Both alternatives result in loading factors and warm gas velocity distribution very similar to the fiducial model. Within the standard TIGRESS framework, the spatial distribution of SNe is subject to the adopted prescription for runaways, but is otherwise self-consistently determined with respect to the gas by the distribution of star formation sites. Better theoretical and observational constraints for runaway OB stars would lead to more accurate modeling of the SN distribution, which in principle could change the wind mass loading. However, our tests suggest that a very large proportion of high-velocity runaways would be needed to significantly increase the wind mass-loading, while the corresponding star formation rate and ISM phase segregation in that case might not be consistent with observations. Finally, we remark that a fine enough grid to spatially resolve both low filling-factor warm gas and high filling-factor hot gas in the wind launching region is crucial for proper physical characterization of galactic winds. If warm and hot gas are artificially mixed, e.g. if the flow in AMR and semi-Lagrangian simulations moves from a higher resolution region near the midplane to a lower resolution region at high latitude, the result can be unphysical in ways that would compromise the implications for real galactic systems. For example, consider mixing of warm and hot flows that have total horizontally-averaged mass, momentum, and energy vertical fluxes of F ρ , F ρv , and F E . If we further assume a steady state and neglect gravity and magnetic fields, the outgoing fluxes of the mixed gas must be the same as the sum of the horizontally-averaged incoming fluxes of the warm and hot gas. That is, F ρ = ρ mix v mix = ρ w v w + ρ h v h , and similarly F ρv = ρ mix v 2 mix + P mix and F E = ρ mix v mix [v 2 mix /2 + γP mix /((γ − 1)ρ mix )] are the sum of "warm" and "hot" terms based on horizontal averages over multiple zones. The post-mixing mean velocity will depend on the total fluxes as v mix = γ γ + 1 F ρv F ρ 1 − 1 − 2(γ 2 − 1) γ 2 F E F ρ F 2 ρv 1/2 . (11) Typically, F ρ (near the disk) will be dominated by the warm medium contribution (see Fig. 9a), F ρv will be dominated by the hot medium contribution (see Fig. 4c), and F E will be dominated by the hot medium contribution (see Fig. 9b). Figure 10 shows this hypothetical post-mixing velocity as calculated from our simulation. This is based on horizontal averages of the outgoing (i.e. zones with sign(v z ) = sign(z)) fluxes, temporally averaged over the outflow period and over both sides of the disk. Also shown, for comparison, are the mean outflow velocities of the warm and the hot gas. The implication of the comparison shown in Figure 10 is that if numerical mixing were to occur at d > 1 kpc, the result would be v mix ∼ 100 km s −1 , which is intermediate between the mean hot and mean warm outflow velocities. Depending on the galaxy/halo global properties, this could have two (opposite) unphysical consequences. On the one hand, if v mix is greater than the galaxy escape speed but the original v w is less than the galaxy escape speed, then the artificially-mixed flow would be able to escape with a much larger mass-loading factor than would be realistic (i.e. ∼ β w rather than β h ). On the other hand, if v mix is smaller than the galaxy escape speed, then the artificially-mixed flow would not be able to escape at all, whereas in reality the hot wind should escape with massloading factor β h , and potentially loaded with more than its share of metals. To avoid these unphysical consequences, it is necessary to separately resolve the multiphase gas even above the dense midplane region. SUMMARY Gas flows blown out of galactic midplane regions by energetic stellar feedback, most notably type II SNe, will either escape as a wind or turn around as a fountain, depending on the specific kinetic and thermal energy of the gas compared to the gravitational potential. Due to the multiphase structure of the ISM, the simplest steady, adiabatic solutions (e.g., Chevalier & Clegg 1985) are not directly applicable, although as- pects of these solutions are informative when considering a phase-decomposed analysis. More generally, the question of "how much mass and energy are carried out by each phase of outflowing gas?" depends strongly on when and where star formation and SNe occur and how feedback transfers energy to the ISM. Given the complexity involved, the properties of galactic winds and fountain flows created by the star-forming ISM must be investigated via fully self-consistent numerical simulations. The TIGRESS implementation described in Paper I provides comprehensive, self-consistent simulations of the multiphase star-forming ISM that can be used for high-resolution investigations of outflows in a wide variety of environments. In this paper, we analyze the solar neighborhood model of Paper I to characterize vertical gas flows phase-by-phase. Our principal finding is that the outflowing gas consists of a superposition of a hot wind and a warm fountain flow. We summarize key conclusions from our analysis of each component below. 6.1. Hot wind 1. The hot gas component, which is created by SN shocks, behaves very similarly to expectations for a steady, adiabatic flow as it expands away from the midplane. For the horizontally-averaged, time-averaged hot component, the mass flux and the Bernoulli parameter are close to constant as a function of the distance from the midplane d above d > 1 kpc, indicating that there is relatively little mass added or subtracted by heating or cooling from/to another phase, and little energy added or subtracted either through shocks, radiation, pressure work on other phases, or mixing. 2. The hot gas mass loading factor, defined as the ratio of outflowing mass flux to the star formation rate per unit area, is β h ∼ 0.1, decreasing by 40% from d = 1 to 4 kpc. The measured value of β h is consistent with estimates from idealized numerical experiments of superbubble expansion in a warm-cold ISM. 3. Above d = 1 kpc, the hot outflow contains a tiny fraction of the energy originally injected in the ISM by SNe. Another tiny fraction is converted to kinetic energy of the warm and cold medium accelerated by superbubble expansion, but most of the original SN energy is radiated away within the disk scale height. The ratio of outflowing hot-gas energy flux to the SN energy injection rate per unit area is α h ∼ 0.02, decreasing by 65% from d = 1 to 4 kpc. For d = 2 − 4 kpc, the mean temperature of the hot gas is ∼ 1.5 × 10 6 K, and its mean velocity is ∼ 200 km s −1 . Magnetic energy in the outflow is negligible, with mean Alfvén speed of ∼ 30 km s −1 . 4. In the present simulation, the energy of the hot medium is carried mainly as heat because the outflow is constrained to remain vertical in our local Cartesian box. Allowing for the opening of streamlines at larger distance (beyond our simulation domain), much of the enthalpy would be converted to kinetic energy. The hot medium is barely affected by gravity within the simulation domain, and with an asymptotic maximum wind speed of ∼ 2(B − Φ) ∼ 350 km s −1 would easily escape to very large distances. More generally, local simulations with varying galactic conditions can provide measures of β h and the Bernoulli parameter B that can be used to provide predictions for hot wind properties at large distances in a wide range of galaxies. 6.2. Warm Fountain 1. The warm (including both neutral and ionized) medium, which dominates the total gas mass in the simulation, acquires energy from SN feedback, mostly in kinetic form. However, the typical outgoing velocity of the warm medium above d = 1 kpc is only v out ∼ 60 km s −1 , which is not enough to overcome the large-scale gravitational potential in the present simulation. As a result, most of the warm gas that is blown out of the midplane eventually turns around and falls back, forming a fountain. 2. Star formation bursts lead to a succession of outflow-dominated and inflow-dominated periods of the fountain flow. During both outflow-dominated and inflow-dominated periods, gas with both signs of velocity (i.e. outflows and inflows) is present on both sides of the disk. Considering just the outflowing gas, owing to deceleration and turnaround the outgoing warm-medium mass loading factor β w is a decreasing function of height d. The mean warm-gas mass flux in outflowing "fountain" gas at d ∼ 1 kpc is β w ∼ 1, but this drops steeply to β w ∼ 0.03 at d = 4 kpc. 3. The value of the warm-medium energy loading factor α w drops from ∼ 0.002 to 10 −4 over d = 1 to 4 kpc. Because α w drops slightly less than β w , the mean specific energy of the warm medium increases with d. The corresponding (volume-weighted) mean velocities of the warm gas at d = 1 and 4 kpc are ∼ 60 and 80 km s −1 . However, it is important to note that the increase of mean velocity in the warm medium with d is primarily due to dropout of low-velocity fluid elements, and does not reflect acceleration with height. Detailed distributions show a secular decrease in the mass and mass flux of high-velocity warm gas with increasing d. 4. A promising way to characterize warm outflows is via the velocity distribution where they are launched at d ∼ H. Here, we find that the high-velocity warm gas has an exponential distribution. For a given PDF in velocity, the portion of the warm gas mass flux that is able to escape as a wind will depend on the halo potential depth. For the large scale galactic potential in the present simulation, very little warm gas escapes, but in a dwarf galaxy the same distribution could lead to a wind with β w ∼ 1/3. Caveats and prospects Our simulations have two main caveats for direct comparison with observations. Firstly, we use a local Cartesian box to achieve high resolution. The uniformly high resolution is crucial for distinguishing different phases and limiting numerical mixing. However, the local Cartesian box prevents us from following the hot outflow's evolution under global geometry with a realistic galactic potential (see Martizzi et al. 2016). Without the opening of streamlines, the hot medium cannot accelerate through a sonic point to reach its asymptotic velocity; we therefore cannot follow this process directly in our simulations. However, proper decomposition of the gas phases allows us to provide well-defined mass and energy loading factors for the hot gas, which can be robustly extrapolated to obtain predictions for asymptotic wind properties at large scales. The hot-gas loading factors at large d are slightly lower than the observational constraints deduced from M82 (Strickland & Heckman 2009). This is likely because in M82 the strong starburst has successfully cleared much of the cooler gas away from the midplane; we find that when SNe explode above the denser phases, the mass and energy loading of hot winds increases. As discussed above, to the extent that the warm phase is ballistic above a scale height, its measured velocity distributions could be used to extrapolate its properties to large distance. However, it is clear that the gravitational potential even within our local box affects the warm-medium properties at large d, so it is important to treat conclusions regarding warmmedium outflows cautiously. Secondly, in the present simulations we do not include photoionization. This affects the properties of the warm and ionized phases, for which the photoionization heating can be important. Inclusion of photoionization is necessary for direct comparison with observations, where the line diagnostics are sometimes better explained by photoionization rather than a shock model (e.g., Chisholm et al. 2016a). As a first step towards this, it may be sufficient to compute ionization in postprocessing, as photoionization is unlikely to be important to the dynamics of high-velocity warm gas even though it may dominate heating. Most observations of highly ionized absorption lines are however based on large apertures, so direct theoretical comparisons would also require simulations with global geometry. We are able to run our simulations over an extended period, long after initial transients that may affect quantitative results reported by others for outflows in simulations with similar physics and resolution to the TIGRESS implementation, but shorter durations. Other recent high-resolution simulations do not have self-consistent star formation and feedback, which may affect outflow loading because this is sensitive to the spatio-temporal correlation of supernovae with ISM gas of different phases. In the future, as more groups run high-resolution simulations with self-consistent star formation and SN feedback for an extended duration, it will be informative to compare phase-separated results for mass and energy loading, Bernoulli parameters, and velocity distributions, for both fountain flows and winds. Application of the TIGRESS implementation to other galactic environments is currently underway, and promises to be quite interesting. Varying the basic model input parameters (especially gas and stellar surface density and metallicity) will enable predictions for multiphase outflow properties in a wide range of galaxies, and will provide detailed information needed to build subgrid models for winds in cosmological simulations of galaxy formation. By extending TIGRESS and other self-consistent multiphase ISM/star formation numerical implementations to include additional feedback (especially radiation and cosmic rays), understanding the complex physics behind galactic winds and fountains is within reach. We are grateful to the referee for helpful report, and to Miao Li and Drummond Fielding for fruitful discussions. This work was supported by grants AST-1312006 from the National Science Foundation, and NNX14AB49G and NNX17AG26G from NASA. Resources supporting this work were provided in part by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center and in part by the Princeton Institute for Computational Science and Engineering (PICSciE) and the Office of Information Technology's High Performance Computing Center. Software: This work made use of the Athena MHD code (Stone et al. 2008;Stone & Gardiner 2009). This work also made use of analysis and visualization softwares including yt (Turk et al. 2011 ∂(ρv) ∂t +∇ · ρvv + P + B 2 8π − BB 4π = −2Ω × (ρv) − ρ∇Φ tot ,(A1) and ∂E ∂t +∇ · 1 2 v 2 + γ γ − 1 P ρ + Φ tot ρv + (B × v) × B 4π = −ρL,(A3) where the total energy density is E ≡ 1 2 ρv 2 + P γ − 1 + B 2 8π .(A4) The total gravitational potential Φ tot may include self-gravity and external-gravity, as well as the tidal potential Φ tidal = −qΩ 2 x 2 (see Paper I). Note that the zero point of Φ tot is at the center of the simulation domain. With the help of shearing-periodic boundary conditions in the horizontal directions, the horizontally-averaged equations are then given by ∂ ρ ∂t + ∂ ∂z ρv z = 0, (A5) ∂ ρv z ∂t + ∂ ∂z ρv 2 z + P + B 2 8π − B 2 z 4π = − ρ ∂Φ ∂z ,(A6) and ∂ E ∂t + ∂ ∂z ρv z B + S z = qΩ L y ρv x δv y − B x B y 4π dy − ρL ,(A7) where the angle brackets denote a horizontal average, q ≡ qdxdy/L x L y . Here, specific energy of the gas, or the Bernoulli parameter, is defined by B ≡ v 2 2 + γ γ − 1 P ρ + Φ,(A8) where Φ = Φ tot − Φ tidal , and the Poynting vector is defined by S ≡ (B × v) × B 4π ,(A9) with its vertical component of S z = v z B 2 − B z v · B 4π .(A10) The right hand side of Equation A7 is the sum of energy source terms due to the Reynolds and Maxwell stresses (on the x-surfaces of the box, from shearing boundary conditions -e.g., Hawley et al. 1996;Stone & Gardiner 2010) and the net cooling. Although the evolution of our simulation is highly dynamic and time-dependent, the fluctuations in the horizontallyaveraged gas properties are with respect to a well-defined equilibrium state (see Figure 4), after an early transient period. We thus take time averages to consider a quasi-steady equilibrium state, dropping the time derivative terms in Equations (A5)-(A7) to analyze characteristics of mean gas flows. We explicitly measured each energy source term for separate thermal phases and confirmed that for the hot gas the energy source terms are negligible, and for the warm gas the cooling dominates the energy source terms. The hot gas can thus be treated nearly adiabatically. Above the region where SN energy is injected, the hot gas mass and energy fluxes remain (roughly) constant, and hence the ratio of the two, B + M z , is also (roughly) constant. Here c) and (e)). The magenta dashed lines in each panel are fits given by Equations (7) and (8). is the specific magnetic energy carried by the magnetic field, which is shown to be small compared to B ( Figure 5). Thus, outside of the energy injection region where the hot component is created, its mean vertical mass flux ρv z and its Bernoulli parameter B are expected to be roughly constant, independent of distance relative to the midplane. We note that more generally, if we consider the time-averaged state of a flow (i.e. ∂(...)/∂t = 0) that is weakly magnetized and has negligible cooling, Equations (A1) and (A3) reduce to ∇ · (ρv) = 0 and v · ∇B = 0. That is, the Bernoulli parameter is conserved along streamlines, regardless of the geometry of the flow. B. NUMERICAL CONVERGENCE TESTS Due to complex and nonlinear interactions between SN feedback and the surrounding medium, satisfaction of simple physical conditions (e.g., resolving the Sedov stage of a single SN blast wave; Kim & Ostriker 2015a) will not guarantee numerical convergence of the simulated system as a whole. It is therefore important to check robustness of our main results for varying numerical resolution. In addition, it has been suggested that the SN placement can significantly impact the resulting ISM structure and feedback efficiency (e.g., Gatto et al. 2015;Li et al. 2015). In TIGRESS, unlike in most other simulations, the SN locations are not set based on a pre-defined vertical distribution, but are self-determined based on the locations where star clusters form and migrate over time. A fraction of the SNe in our simulation are associated with runaways, and we adopt a distribution for the ejection velocity from clusters based on a binary population synthesis model (Eldridge et al. 2011). This distribution, together with the fraction of binary runaways, is in fact not very certain. It is therefore valuable to test how our results might depend on the placement of SNe by comparing different test runs. B.1. Numerical Resolution We analyze the same set of simulations with different numerical resolutions of ∆x =4, 8, and 16 pc. Note that we showed in Paper I that ∆x = 16 pc is a marginal condition for convergence in star formation rates and the ISM properties. Here, we further show detailed velocity distributions of the warm medium ( Figure 11) and mass and energy loading factors of the warm and hot medium ( Figure 12) at varying resolution. These properties, which are key characteristics of hot winds and warm fountains, are evidently converged. B.2. SN Placement In order to compare the effects of SN placement, we consider, in addition to the fiducial model, three models with different prescriptions: (1) no-runaway -all SNe are in star cluster particles without runaways; (2) random-250 -all SNe are randomly located horizontally and consistent with an exponential vertical profile with scale height of z SN = 250 pc vertically, but with the rate determined by star cluster particles; (3) random-50 -the same as random-250, but z SN = 50 pc. By comparing the fiducial model with the no-runaway model for velocity distributions of the warm medium (Figure 13) and mass and energy loading factors of the warm and hot medium (Figure 14), we conclude that presence or absence of runaways with our adopted prescription does not significantly alter the results. With randomly located SNe, the results can change substantially depending on the adopted scale height. For the random-250 model, the hot gas mass and energy loading factors are about 0.8 and 0.2, respectively. In this case, most SNe explode above the gaseous scale height without interacting with the warm and cold medium. The hot SN remnants simply expand outward in the low-density, hot atmosphere. A negligible warm fountain is created. In contrast, for the random-50 model, the majority of SN events happen within the warm/cold gas layer. The hot gas created in SN remnants strongly interacts with the surrounding warm and cold medium, and hot gas is lost in the process. The resulting hot gas mass and energy loading factors in the wind for the random-50 model are similar to the fiducial model. This trend with varying SN scale height was also shown in Li et al. (2017). In short, the mass and energy loading factors of the hot wind are sensitive to the vertical distribution of SNe relative to the gas (mainly the warm medium). In our fiducial model, we assume 2/3 of SNe explode in binaries and eject runaway companions. This gives a runaway SNe fraction of 1/3. Since the ejection velocity distribution is an exponential function with characteristic velocity 50 km s −1 , the fraction of SNe that explode above 200 pc is 15%. This fraction increases to 44% in the random-250 model, while the no-runaway and random-50 models give fractions of 0.4% and 7%, respectively. It is noteworthy that the random-250 model fails to regulate star formation rates at the same level as the fiducial model (the SFR is about 1.5 times higher for the random-250 model). In the random-50 model, an asymmetric vertical distribution of gas develops at later times (more gas is in the upper half of the simulation domain). Since the SNe distribution is set relative to the initial gas distribution rather than the instantaneous gas distribution, the asymmetric gas structure persists and allows more efficient hot and warm outflows in the lower half of the domain. This results in slightly higher mass and energy loading factors of the warm fountain for the random-50 model relative to the fiducial model. . Mass and energy loading factors of the warm (green) and hot (red) phases for different SN prescriptions (same as Figure 9). . Figure 2 displays slices of temperature and vertical velocity through the y = 0 plane for the same snapshots shown in Figure 1. The outflows and inflows seen in Figures 1 and 2 are part of an overall cycle that repeats, representing the response to large amplitude temporal fluctuations in star formation rates (see grey line in Figure 3(a)). Figure 2 . 2Sample slices through y = 0 showing temperature and vertical velocity at t = 300 (left) and 360 Myr (right), representing outflow-and inflow-dominated periods. During outflow-dominated periods (left), the hot component fills most of the volume at high |z| and flows outward at high velocity, while the warm component is confined in small cloudlets. During inflow-dominated periods (right), the warm component occupies most of the volume and falls toward the midplane. As in the volume renderings ofFig. 1, only the upper half of the simulation box (z = 0 to 4 kpc) is shown. Figure 3 . 3Time evolution of (a) mass and (b) energy fluxes driven by bursts in the star formation rate. Mass and energy fluxes per unit area at distances d =1, 2, and 3 kpc from the midplane are measured (Equations 4 and 5). Each burst in ΣSFR (shown in grey in upper panel) leads to outflows in both mass and energy near the midplane. The net mass flux decreases at larger d because most of the outflow near the midplane is warm "fountain" gas with low velocity that turns around (resulting in periods of negative mass flux at d = 1 kpc). Energy fluxes are substantial at all heights, due to the dominance of hot gas that escapes as a wind.sound speed in the range 130 − 150 km s −1 ), and the kinetic energy of the hot gas translates to a velocity of 170 − 200 km s −1 . The Poynting flux contribution to B is negligible, corresponding to an Alfvén speed of 30 − 35 km s −1 . Figure 4 4Figure 4. Vertical distributions of mass, vertical momentum, and energy densities, averaged horizontally and over upper (z > 0) and lower (z < 0) sides of the disk and over time t = 250 − 500 Myr. Colored lines for separate thermal components show profiles of (a) hydrogen number density nH , (b) vertical momentum density ρvout (which is the same as the mass flux), and (c) total energy density (excluding gravity) E, each as a function of distance d from the midplane. Color-coded shaded regions represent one-sigma temporal fluctuations. In order to properly visualize both the magnitude and sign of the momentum density, we use a linear scale for |ρvout| < 10 −4 and a log scale for |ρvout| > 10 −4 . The warm and hot phases respectively dominate mass and energy densities above the disk scale height (indicated by vertical dotted line), and the hot component also has the largest vertical momentum density (net outward mass flux). Figure 5 . 5Time averaged (t = 250 − 500 Myr) vertical profiles of hot gas (a) mass fluxes and (b) specific energies. (a) The net mass flux is shown as a blue line with one-sigma temporal fluctuations as the blue shaded region. The hot gas does not show any significant inflows (yellow line), implying that any decrement of the mass flux is due to a phase transition to cooler phases and not direct inflows of the hot gas. (b) Total Bernoulli parameter B as well as individual components (i.e. kinetic, thermal, gravitational, and magnetic terms; see Equations Figure 6 . 6Mass flux and specific energy profiles for the hot wind as inFigure 5, except at a single instant t = 300 Myr. Figure 7 . 7Vertical profiles of warm gas mass fluxes (top) and specific energies (bottom) averaged over outflow (left; Σ wind (1 kpc) > 0.001 M kpc −2 yr −1 ) and inflow (right;Σ wind (1 kpc) < −0.001 M kpc −2 yr −1 ) periods. Inflows and outflows are comparable with each other, implying little net mass and energy outflows (or inflows) from the simulation domain associated with the warm medium. Note that, for visualization purpose, we take absolute values of the inflow mass flux and net flux during the inflow period; these have negative signs by definition. Figure 8 . 8for f = V /V slab or M/M slab , where the normalization factors for volume and mass weighted PDFs are A V = 0.63/v V and A M = 0.79/v M , respectively, and Outward vertical velocity (vout ≡ vzsign Figure 9 . 9Mass and energy loading factors in each thermal phase. For both cold and warm gas, both mass and energy loading factors stiffly decrease as a function of d. Mass loading also drops off with d for the ionized phase. Only the hot gas is a true wind, with well-defined mass and energy loading factors of β h ∼ 0.1 and α h ∼ 0.02, respectively. ment is made at very large distance in a global simulation. Nevertheless, individual measurements of mass fluxes in different phases, when combined with analysis of the components of their individual Bernoulli parameters, can be used to distinguish fountains from winds even within a limited vertical domain. Figure 10 . 10The hypothetical post-mixing velocity vmix (black) if artificial numerical mixing were to happen at a distance d from the midplane. We use horizontally-averaged outgoing fluxes to calculate vmix using Equation(11), and take a time average over the outflow period. Also shown are mean values of the warm-gas and hot-gas outflowing velocities. The nominal value of vmix is similar to the warm medium velocity at small d, and gets closer to the hot medium velocity at large d. ), astropy (Astropy Collaboration et al. 2013), matplotlib (Hunter 2007), numpy (van der Walt et al. 2011), IPython (Perez & Granger 2007), pandas (McKinney 2010), and VisIt (Childs et al. 2012).APPENDIXA. MHD EQUATIONS IN A SHEARING-BOXThe ideal MHD equations of mass, momentum, and total energy conservation in frame rotating at Ω = Ωẑ are ∂ρ ∂t +∇ · (ρv) = 0, Figure 11 . 11Mass-and mass flux-weighted velocity distributions of the warm medium measured at d = 1 kpc for different numerical resolutions during the outflow period (same asFigure 8( Figure 12 . 12Mass and energy loading factors of the warm (green) and hot (red) phases (same asFigure 9), for different numerical resolutions. Figure 13 .Figure 14 1314Mass-and mass flux-weighted velocity distributions of the warm medium measured at d = 1 kpc for different SN prescriptions during the outflow period (same asFigure 8(c) and (e)). than inFig. 5because the latter includes non-outburst epochs as well as outburst epochs in the temporal averaging.0 1 2 3 4 distance from the midplane, d [kpc] 10 4 10 3 10 2 mass flux, | wind, h | [M yr 1 kpc 2 ] net flux outflow inflow 0 1 2 3 4 distance from the midplane, d [kpc] 10 2 10 3 10 4 10 5 10 6 specific energy, E, h / wind, h [km 2 s 2 ] kinetic enthalpy magnetic grav. , the vertical dotted line indicates the disk scale height (H = 400 pc).0 1 2 3 4 distance from the midplane, d [kpc] 10 4 10 3 10 2 mass flux, | wind, h | [M yr 1 kpc 2 ] net flux outflow inflow 0 1 2 3 4 distance from the midplane, d [kpc] 10 2 10 3 10 4 10 5 10 6 specific energy, E, h / wind, h [km 2 s 2 ] kinetic enthalpy magnetic grav. This holds true for MHD flows if the Poynting flux is negligible, as in our simulations; seeFigure 5. Of course, the absence of global geometry means that we are unable to follow effects of strong noncircular flows in the disk, transport of gas from one radius to another in a fountain (which would also require significant angular momentum exchange), or the transition of hot winds through a sonic point. Nevertheless, the high resolution afforded by our local scope is extremely valuable for limiting artificial mixing, which is essential for understanding key characteristics of multiphase flows. We note, however, that a quasi-steady state is not guaranteed for all galactic conditions (e.g.,Torrey et al. 2017), and may only Note that we omit the "unstable" phase (184 K < T < 5050 K) defined in Paper I and merge it into the "cold" phase since (1) these phases are not of primary interest in this paper since we are focusing on gas above the disk scale height, and (2) the sum of these two phases numerically converges better than the individual phases. Note that for galactic potentials, Φ is generally computed relative to the midplane, whereas for point mass potentials the Φ = 0 reference point is at infinite distance; the zero point in Φ is irrelevant as long as it is consistent in v wind and B. . WARM FOUNTAINS 7 Here, this maximum wind velocity is calculated using Φ at a height d = 4 kpc, although in reality the wind velocity would continue to decrease slowly with distance due to the logarithmic increase of Φ(r) with r at large distances in dark matter halo potentials. 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[ "Asymmetry and tighter uncertainty relations for Rényi entropies via quantum-classical decompositions of resource measures", "Asymmetry and tighter uncertainty relations for Rényi entropies via quantum-classical decompositions of resource measures" ]
[ "Michael J W Hall \nDepartment of Theoretical Physics\nResearch School of Physics\nAustralian National University\nACT 0200CanberraAustralia\n" ]
[ "Department of Theoretical Physics\nResearch School of Physics\nAustralian National University\nACT 0200CanberraAustralia" ]
[]
It is known that the variance and entropy of quantum observables decompose into intrinsically quantum and classical contributions. Here a general method of constructing quantum-classical decompositions of resources such as uncertainty is discussed, with the quantum contribution specified by a measure of the noncommutativity of a given set of operators relative to the quantum state, and the classical contribution generated by the mixedness of the state. Suitable measures of noncommutativity or 'quantumness' include quantum Fisher information, and the asymmetry of a given set, group or algebra of operators, and are generalised to nonprojective observables and quantum channels. Strong entropic uncertainty relations and lower bounds for Rényi entropies are obtained, valid for arbitrary discrete observables, that take the mixedness of the state into account via a classical contribution to the lower bound. These relations can also be interpreted without reference to quantum-classical decompositions, as tradeoff relations that bound the asymmetry of one observable in terms of the entropy of another.
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[ "https://export.arxiv.org/pdf/2304.05704v2.pdf" ]
258,078,926
2304.05704
8b492a585cf0b6f31b1f0674e477692931f416ca
Asymmetry and tighter uncertainty relations for Rényi entropies via quantum-classical decompositions of resource measures Michael J W Hall Department of Theoretical Physics Research School of Physics Australian National University ACT 0200CanberraAustralia Asymmetry and tighter uncertainty relations for Rényi entropies via quantum-classical decompositions of resource measures It is known that the variance and entropy of quantum observables decompose into intrinsically quantum and classical contributions. Here a general method of constructing quantum-classical decompositions of resources such as uncertainty is discussed, with the quantum contribution specified by a measure of the noncommutativity of a given set of operators relative to the quantum state, and the classical contribution generated by the mixedness of the state. Suitable measures of noncommutativity or 'quantumness' include quantum Fisher information, and the asymmetry of a given set, group or algebra of operators, and are generalised to nonprojective observables and quantum channels. Strong entropic uncertainty relations and lower bounds for Rényi entropies are obtained, valid for arbitrary discrete observables, that take the mixedness of the state into account via a classical contribution to the lower bound. These relations can also be interpreted without reference to quantum-classical decompositions, as tradeoff relations that bound the asymmetry of one observable in terms of the entropy of another. I. INTRODUCTION A quantum observable such as position or spin has two generic sources of uncertainty. The first arises when the observable does not commute with the state of the quantum system: this implies that the state is not an eigenstate of the observable, leading to a spread of measurement outcomes. The second source of uncertainty arises when the state is mixed: a loss of classical information due to mixing reduces the predictability of the measurement outcome, leading to an increased spread. It follows that the two sources are, respectively, intrinsically quantum and classical in nature. The first can be useful as a resource, e.g., in allowing a complementary observable to have a small spread, whereas the second is usually not, as mixing contributes noise to all observables. Luo suggested quantifying such quantum and classical contributions to a given measure of uncertainty, M (X|ρ), for a Hermitian operator X and state described by density operator ρ, via a quantum-classical decomposition of the form [1,2] M (X|ρ) = Q(X|ρ) + C(X|ρ), and gave the particular example Var ρ (X) = − 1 2 Tr [X, √ ρ] 2 + Tr [X √ ρX √ ρ] − ⟨X⟩ 2 ρ (2) for the variance of a Hermitian operator. In this example the 'quantum' contribution is the skew information of X with respect to ρ [3], and is clearly nonzero only if the observable does not commute with the state, whereas the 'classical' contribution is nonzero only if the state is mixed, as expected. A second example is a quantumclassical decomposition of the entropy of a Hermitian operator, given by Korzekwa et al. [4]. A simple approach to unifying and generalising such quantum-classical decompositions is discussed in Sec. II, based on the idea that the intrinsically quantum contribution to a given resource measure is maximised when the observer has access to a pure state of the system, and is degraded by classical mixing for a local observer who only has access to a component of such a system. For example, a maximally-entangled pure state of two qubits is a useful quantum resource for dense coding and for estimation of a local rotation, but this usefulness vanishes for an observer who only has access to one of the two qubits (described by a maximally-mixed state). The general approach is described in Sec. II A. It starts with some given measure Q(X, Y, . . . |ρ) of the 'quantumness' of a resource, for operators X, Y, . . . and state ρ. The corresponding 'maximum potential' M (X, Y, . . . |ρ) of the resource and its 'classicality' C(X, Y, . . . |ρ) are then constructed so as to satisfy a decomposition analogous to Eq. (1). Several examples are given in Sec. II B, including the decomposition of variance and covariance matrices with respect to quantum Fisher information; decompositions of Shannon entropy with respect to both asymmetry and the conditional entropy of self-dual communication channels; and, of particular relevance to this paper, the decomposition of Rényi entropy with respect to Rényi asymmetry, where the latter has applications to quantum coherence, time-energy uncertainty relations, quantum information, open quantum systems and quantum metrology [5][6][7][8][9]. It is also shown that measures of quantumness can equivalently be defined via sets, groups or algebras of operators, and generalised to arbitrary discrete observables and to quantum channels. An alternative approach, based on convex and concave roofs, is discussed in Appendix A, with corresponding examples. The main technical results are given in Sec. III, including a strong uncertainty relation for Rényi entropies of discrete observables X and Y represented by positive operator valued measures (POVMs) {X x } and {Y y }: H α (X|ρ) + H β (Y |ρ) ≥ − log µ XY + max{C α (Y |ρ), C β (X|ρ)}.(3) Here the Rényi parameters α, β ∈ [ 1 2 , ∞) are related by α −1 + β −1 = 2; µ XY denotes the maximum eigenvalue of X 1/2 x Y y X 1/2 x over x and y; and the last term takes the mixedness of ρ into account via the classical components of the Rényi entropies. The logarithm base is left arbitrary throughout, corresponding to a choice of units (e.g., to bits for base 2 and to nats for base e). Equation (3) is clearly stronger than the standard uncertainty relation H α (X|ρ) + H β (Y |ρ) ≥ − log µ XY ,(4) for Rényi entropies [10][11][12], which has no classical mixing term. Further, for rank-1 POVMs, with X x = |x⟩⟨x| and Y y = |y⟩⟨y|, Eq. (3) generalises the known uncertainty relation H(X|ρ) + H(Y |ρ) ≥ − log max x,y |⟨x|y⟩| 2 + H(ρ)(5) for Shannon entropies [4,13,14] (which correspond to α = β = 1), that takes the mixedness of the state into account via its von Neumann entropy, H(ρ) := −Tr [ρ log ρ]. It is also shown in Sec. III that Eq. (3) may be written as a direct tradeoff between the asymmetry of one observable and the entropy of another, without any reference to quantum-classical decompositions, leading to a simple lower bound for Rényi asymmetry, as well as to entropic bounds such as H α (X|ρ) ≥ − log µ ρX , α ≥ 1 2 ,(6) for the Rényi entropy of an arbitrary discrete observable X. Some readers may choose to skip directly to Sec. III, and refer back to Sec. II for definitions. Results and possible future work are discussed in Sec. IV. II. BUILDING QUANTUM-CLASSICAL DECOMPOSITIONS A. A general approach As noted in the Introduction, access to a maximallyentangled pure state of two qubits is a useful resource for the estimation of a local rotation, but this usefulness vanishes for an observer who only has access to one of the two qubits, described by a maximally-mixed state. This motivates the more general idea that the quantum contribution to the degree to which a given resource can be exploited is maximised for a notional observer who has access to a pure state of the system, with any classical component arising from the presence of mixing when there is only partial access. Accordingly, if the 'quantumness' of some resource is quantified by a measure Q(X, Y, . . . |ρ), for some set of operators X, Y, . . . and state ρ, the maximum potential of the resource, M (X, Y, . . . |ρ), is defined by M (X, Y, . . . |ρ) := Q(X ⊗ 1 a , Y ⊗ 1 a , . . . |ρ ψ ), (7) where ρ ψ ≡ |ψ⟩⟨ψ| is a purification of state ρ on the tensor product H⊗H a of the system Hilbert space H with an ancillary Hilbert space H a [15]. This immediately yields a corresponding quantum-classical decomposition of the form M (X, Y, . . . |ρ) = Q(X, Y, . . . |ρ) + C(X, Y, . . . |ρ), (8) generalising Eq. (1), with C(X, Y, . . . |ρ) := Q(X ⊗ 1 a , Y ⊗ 1 a , . . . |ρ ψ ) − Q(X, Y, . . . |ρ) representing the classical component of the resource relative to an observer who only has access to the mixed state ρ of the system (rather than to the purified state ρ ψ ). Several examples and generalisations are given below, but first three natural assumptions for Q(X, Y, . . . |ρ) are identified (related requirements have been previously discussed by Luo [1,2] and Korzekwa et al. [4]). (i) It will be assumed that there is no potential 'quantum' resource to exploit if the relevant operators commute with the state of the system, i.e., Q(X, Y, . . . |ρ) = 0 if [X, ρ] = [Y, ρ] = · · · = 0. (9) Hence, the quantumness acts as a measure of noncommutativity. (ii) The decomposition must be well-defined, with the values of the quantum and classical components being independent of the choice of purification. This is guaranteed by the assumption that Q(X, Y, . . . |ρ) is invariant under unitary transformations, i.e., Q(X, Y, . . . |ρ) = Q(U XU † , U Y U † , . . . |U ρU † ) (10) for all unitary transformations U . In particular, noting that any two purifications ρ ψ and ρ ψ ′ can be related by a local unitary transformation U = 1 ⊗ U a acting on the tensor product H ⊗H a [15], where this leaves operators on H invariant, it follows that Eqs. (7) and Eq. (8) are independent of the choice of purification as required. (iii) The classical component must be nonnegative, and vanish for pure states (i.e., when there is no classical mixing). This is equivalent, via Eqs. (7) and (8), to the assumption that the quantumness increases under purification (e.g., when an observer gains access to the full quantum system), i.e., that Q(X, Y, . . . |ρ) ≤ Q(X ⊗ 1 a , Y ⊗ 1 a , . . . |ρ ψ ) (11) for any purification ρ ψ of ρ. In particular, this assumption ensures that C(X, Y, . . . |ρ) ≥ 0, with equality for pure states, as desired. The main advantage of the above construction method is that the maximum potential value of a quantum resource accessible to an observer, M , and the classical contribution due to mixing, C, are fully determined by the choice of the quantumness (or noncommutativity) measure Q. Other advantages are the applicability to arbitrary sets of operators and the straightforward extension to nonprojective observables and quantum channels, as illustrated in the examples below. An alternative method of constructing quantum-classical decompositions, based on convex and concave roofs of pure-state resource measures, is discussed in Appendix A. B. Examples It is convenient in what follows to call a given discrete observable X a projective observable if the corresponding POVM {X x } consists of orthogonal projections, i.e., X x X x ′ = δ xx ′ X x , and a nonprojective observable otherwise. Note that a projective observable X is equivalently represented by the Hermitian operator X ≡ x xX x . The distinction between a projective observable and its corresponding Hermitian operator will always be clear by context. Skew information, variance, and the Fisher information matrix Luo's decomposition of the variance of a projective observable into quantum and classical contributions [1,2], Var ρ (X) = M skew (X|ρ) = Q skew (X|ρ) + C skew (X|ρ) (12) as per Eq. (2), corresponds to choosing the quantumness measure to be the skew information of the observable with respect to the state [3], i.e., Q skew (X|ρ) := 1 2 Tr (i[X, √ ρ]) 2 .(13) The skew information may be physically interpreted as, for example, a measure of the information content of an ensemble that conserves X [3]; a variant of quantum Fisher information in quantum metrology [1,16,17]; a measure of asymmetry [18]; or (for nondegenerate observables) as a measure of coherence [19,20]. Note that assumptions (9) and (10) are trivially satisfied by the skew information, while assumption (11) follows because Q skew (X ⊗ 1 a |ρ ψ ) = Var ρ (X) for any purification of ρ, implying via identity (2) and Eq. (12) that C skew (X|ρ) = Tr ρ 1/4 (X − ⟨X⟩ ρ )ρ 1/4 2 ≥ 0. (14) Hence, properties (i)-(iii) of Sec. II A all hold as required. The above decomposition can be generalised to the case of more than one projective observable, and to other measures of quantum Fisher information, using recent results by Kudo and Tajima [21]. In particular, let Q f F (X 1 , X 2 , . . . |ρ) denote the quantum Fisher information matrix corresponding to a given monotone metric function f , for some set of projective observables X 1 , X 2 , . . . and density operator ρ, where each choice of f corresponds to a measure of the sensitivity of the state to unitary transformations generated by these observables [16,21]. Assumptions (9)-(11) of Sec. II A hold for Q f F as a direct consequence of Theorem 9 of [21], and the construction method yields the quantum-classical decomposition M (X 1 , X 2 , . . . |ρ) = Cov ρ (X 1 , X 2 . . . |ρ) = Q f F + C f F(15) of the symmetrised covariance matrix, with coefficients defined by [Cov ρ (X 1 , X 2 , . . . |ρ)] jk := 1 2 ⟨X j X k + X k X j ⟩ ρ − ⟨X j ⟩ ρ ⟨X k ⟩ ρ .(16) Thus, the covariance matrix has an infinite family of quantum-classical decompositions, with Eqs. (12)- (14) providing just one particular example. It is shown in Appendix A that, for the case of a single projective observable, the alternative construction method given there picks out the particular decomposition having the minimal Fisher information (corresponding to the symmetric logarithmic derivative [16]). Distance, Shannon entropy and standard asymmetry Shannon entropy is a well known resource measure in various information contexts, and Korzekwa et al. have given a corresponding quantum-classical decomposition for the case of projective observables [4], as briefly described below. An alternative decomposition of Shannon entropy is noted in Appendix A 1, based on information properties of self-dual quantum communication channels, i.e., communication channels which are invariant under a duality mapping between the signal ensemble and the receiver measurement [22,23]. The decomposition in [4] corresponds to choosing the quantumness to be a particular measure of the 'distance' from the state of the system to a set of postmeasurement states. More generally, let d(ρ, σ) ≥ 0 be a positive function that vanishes for ρ = σ, and define the corresponding measure of quantumness for a projective observable X with associated projection valued measure {X x } via Q d (X|ρ) := inf σ d(ρ, σ X ) = inf σ:[σ,X]=0 d(ρ, σ),(17) where σ is implicitly restricted to the set of density operators of the system and σ X := X x σX x(18) is the postmeasurement state for a projective measurement of X on state σ. Such distance-based measures of quantumness or noncommutativity are commonly used to quantify asymmetry and coherence resources [18,24]. Note that Q d satisfies assumption (9) of Sec. II A by construction, while assumptions (10) and (11) are satisfied if d(ρ, σ) is invariant under unitary transformations and is nonincreasing under the partial trace operationand in particular if d(ρ, σ) is nonincreasing under general completely positive trace preserving (CPTP) maps [15]. These properties hold in the case of the relative entropy distance function [15], defined by d(ρ, σ) = D 1 (ρ∥σ) := Tr [ρ(log ρ − log σ)] ,(19) for which Q d evaluates to the standard measure of asymmetry [25,26], Q 1 (X|ρ) = D 1 (ρ∥ρ X ) = H(ρ X ) − H(ρ),(20) using the identity D 1 (ρ∥σ X ) = H(ρ X ) − H(ρ) + D 1 (ρ X ∥σ X ). Here the subscript '1' provides the basis for a generalisation to Rényi asymmetry in the next example. The standard asymmetry Q 1 (X|ρ) is equal to the increase in system entropy due to a nonselective measurement of X, and is a useful quantum resource measure in various contexts [4,18,[24][25][26][27][28][29] (these contexts typically have different sets of 'free' states and operations, but the resource measure itself has the same form). Korzekwa et al. simply postulate that the standard asymmetry, Q 1 , is the intrinsically quantum contribution to the Shannon entropy of the observable, H(X|ρ) := − x p(x|ρ) log p(x|ρ) [4]. In contrast, an advantage of the general approach in Sec. II A is that the link to Shannon entropy is derived rather than postulated. In particular, the maximum potential of the standard asymmetry, available to an observer with access to a purification |ψ⟩⟨ψ| of ρ, follows from definition (7) and Eq. (20) as M 1 (X|ρ) = H x X x ⊗ 1 a |ψ⟩⟨ψ| X x ⊗ 1 a − H(|ψ⟩⟨ψ|) = H x ⟨ψ|X x ⊗ 1 a |ψ⟩|ψ x ⟩⟨ψ x | = H(X|ρ),(21) where {|ψ x ⟩} is the set of orthonormal states with |ψ x ⟩ proportional to (X x ⊗1 a )|ψ⟩, and the last line follows via ⟨ψ|X x ⊗1 a |ψ⟩ = Tr [|ψ⟩⟨ψ|X x ⊗ 1 a ] = Tr [ρX x ] = p(x|ρ). Hence, the approach in Sec. II A constructively generates the quantum-classical decomposition H(X|ρ) = Q 1 (X|ρ) + C 1 (X|ρ)(22) given in [4], with the classical contribution evaluating to C 1 (X|ρ) = H(ρ) − x p(x|ρ)H(X x ρX x /p(x|ρ)).(23) Note for any rank-1 projective observable X, with X x ≡ |x⟩⟨x|, that the classical contribution reduces to the von Neumann entropy H(ρ) [4]. More generally, if H(X|ρ) is identified with the entropy of the classical measurement record, then the classical contribution equals the average decrease of the system entropy due to making such a record. Korzekwa et al. use decomposition (22) to obtain a simple proof of entropic uncertainty relation (5) for rank-1 projective observables [4]. The decomposition, uncertainty relation and method of proof are generalised below to Rényi entropies and to nonprojective observables. A rather different decomposition of Shannon entropy, based on a quantumness measure for self-dual communication channels, is discussed in Appendix A 1. Rényi asymmetry and Rényi entropy A particular choice of interest for the distance function d(ρ, σ) in Eq. (17) is the quantum generalisation of the classical Rényi relative entropy [30] to the sandwiched Rényi divergence [31,32], d(ρ, σ) = D α (ρ∥σ) := 1 α − 1 log Tr σ 1−α 2α ρσ 1−α 2α α ,(24) where the index α ranges over [0, ∞). This generalises the relative entropy function in Eq. (19) (which corresponds to the limit α → 1), and introductory expositions of its basic properties may be found, for example, in [31,33]. The most important of these properties for the purposes of this paper is that it is nonincreasing under CPTP maps when α ≥ 1 2 , i.e, the data-processing inequality D α (ϕ(ρ)∥ϕ(σ)) ≤ D α (ρ∥σ), α ≥ 1 2 ,(25) holds for any CPTP map ϕ [33,34]. For projective observables the above choice of d(x, y) generates the corresponding quantumness measure [6] Q α (X|ρ) := inf σ:[σ,X]=0 D α (ρ∥σ) = inf σ D α (ρ∥σ X ) (26) via Eq. (17). This measure generalises the standard asymmetry in Eq. (20), and is known as the Rényi asymmetry of X with respect to ρ. It has found applications in the areas of quantum coherence, quantum information, time-energy uncertainty relations, quantum metrology and open quantum systems [5][6][7][8][9]. Assumptions (9)-(11) hold for the Rényi asymmetry Q α (X|ρ) when α ≥ 1 2 as a consequence of data-processing inequality (25), as may easily be checked. Further, the maximum potential of the Rényi asymmetry is directly related to the Rényi entropy [30] H β (X|ρ) : = 1 1 − β log x p(x|ρ) β(27) of observable X for state ρ, via M α (X|ρ) = Q α (X ⊗ 1 a |ρ ψ ) = H β (X|ρ), 1 α + 1 β = 2.(28) Here the first equality follows via definition (7) and the second equality by showing, via direct minimisation in Eq. (26), that Q α (X| |ψ⟩⟨ψ|) = H β (|ψ⟩⟨ψ| X ) for any pure state |ψ⟩, from which M α (X|ρ) = H β ((ρ ψ ) X⊗1a ) = H β (X|ρ) follows by direct calculation (see also the proof of Theorem 3 in [9]). Hence, the general approach in Sec. II A) yields the quantum-classical decomposition H β (X|ρ) = Q α (X|ρ) + C α (X|ρ), 1 α + 1 β = 2,(29) of Rényi entropy, generalising the case of Shannon entropy in Eq. (22) (which corresponds to the limit α → 1). It follows that the Rényi asymmetry Q α (X|ρ) is upper bounded by the Rényi entropy H β (X|ρ), with equality for pure states [9]. In contrast, the classical component C α (X|ρ) vanishes for pure states by construction, as befits a measure of mixedness, and reaches the upper bound of H β (X|ρ) when X is 'classical' with respect to ρ, i.e., when [X, ρ] = 0. Decomposition (29) will be used in Sec. III to obtain strong uncertainty relation (3) for Rényi entropies. Generalising to nonprojective observables The previous examples are restricted to projective observables. However, it is not difficult to generalise to arbitrary observables, as required for the general results in Sec. III. The idea is to exploit Naimark's extension theorem, that an observable X with POVM {X x } on Hilbert space H, can always be extended to a projective observableX with projection valued measure {X x } on a larger Hilbert spaceH, with X x = PX x P,(30) where P denotes the projection fromH onto H [35][36][37][38][39]. It follows that any given measure of quantumness Q(X|ρ) for projective observables can be extended to nonprojective observables via (notingρ ≡ ρ onH since PρP = ρ) Q(X|ρ) := Q(X|ρ) ≡ Q(X|ρ).(31) While this definition will typically depend on the choice of extension mapping X →X, the results obtained in this paper are valid for any choice. Hence, the mapping can and will be left unspecified in what follows. As a simple example, the extension of the skew information in Eq. (13) to general observables follows, using P ρP = ρ andX xXx ′ = δ xx ′X x , as Q skew (X|ρ) = 1 2 Tr (i[ X, √ ρ]) 2 + Tr ρ(X 2 − X 2 ) ,(32) in terms of the moment operators X m := x x m X x . This is independent of the mapping X x →X x and reduces to Eq. (13) for projective observables. Note that the second term is typically nonzero for nonprojective observables, corresponding to an additional information resource if the extended observable is physically accessible. Remarkably, the maximum potential of the resource is given by the variance for both projective and nonprojective observables, with definition (7) leading to M skew (X|ρ) = Var ρ (X) = ⟨X 2 ⟩ ρ − ⟨X⟩ 2 ρ .(33) Thus, the general decomposition of variance has the same form as Eq. (12), with X replaced byX in Eq. (14). As a second example, the Rényi asymmetry in Eq. (26) generalises to Q α (X|ρ) = inf σ:[σ,X]=0 D α (ρ∥σ) = inf σ D α (ρ∥σX ) (34) where, as always, σ is implicitly restricted to range over the density operators of the system (hence, P σP = σ). Thus, the asymmetry quantifies the distance from the state of the system to the set of postmeasurement states following a measurement ofX. Such postmeasurement states do not typically lie in the Hilbert space H of the system, and hence the asymmetry will typically be nonvanishing for nonprojective observables (see also Corollary 2), corresponding to the sensitivity of ρ to unitary displacements generated byX on the extended Hilbert space. If such displacements are not physically accessible then the generalised asymmetry might more appropriately be referred to as hidden asymmetry (or perhaps super-asymmetry -bazinga!). Note from Eq. (30) and the property P ρP = ρ that Tr[ρX x ] = Tr [ρX x ], implying that H β (X|ρ) = H β (X|ρ). It follows immediately that Eq. (34) generates a quantum-classical decomposition of Rényi entropy having precisely the same form as Eq. (29) for projective observables. Further, it is straightforward to show, using definition (34) and P |ψ⟩ = |ψ⟩, that for a pure state ρ = |ψ⟩⟨ψ| the generalised asymmetry is given by Q α (X| |ψ⟩⟨ψ|) = H β (X| |ψ⟩⟨ψ|), 1 α + 1 β = 2,(35) similarly to the case of projective observables. Hence, Eq. (29) for the general case implies that the classical contribution to the Rényi entropy vanishes for pure states for both projective and nonprojective observables, i.e. C α (X| |ψ⟩⟨ψ|) = 0, α ≥ 1 2 .(36) Finally, it is of interest to note that the general definition of asymmetry in Eq. (34) reduces to Eq. (26) for the case of projective observables, irrespective of the choice of extension mapping X x →X x . In particular, if P EP is a projection for two projections P and E, then writing P ⊥ = 1 − P one has (P EP ⊥ )(P EP ⊥ ) † = P EP − (P EP ) 2 = 0, and so P EP ⊥ = 0 = P ⊥ EP , yielding E = (P + P ⊥ )E(P + P ⊥ ) = P EP + P ⊥ EP ⊥ , from which [P, E] = 0 follows. Hence, if X is a projective observable then [P,X x ] = 0 via Eq. (30). But ρ and σ in Eq. (34) similarly commute with P (since P ρP = ρ and P σP = σ), and evaluation of the right hand side then leads directly to Eq. (26) as claimed. Asymmetry of sets vs groups vs algebras vs channels It is straighforward to generalise the class of distancebased quantumness measures in Eq. (17) from the case of a single projective observable X to any set S of bounded operators, via Q d (S|ρ) := inf σ:[σ,X]=0 ∀X∈S d(ρ, σ) = inf σ∈S ′ d(ρ, σ),(37) where σ again is implicitly restricted to the set of density operators of the system and S ′ := {Y : [X, Y ] = 0 ∀X ∈ S}(38) denotes the commutant of S, i.e., the set of operators that commute with all members of S. This generalised measure is, therefore, the distance from the state of the system to the set of states that commute with the operators in S. Such measures are commonly used to quantify asymmetry resources, i.e., the degree to which the state is noninvariant with respect to members of S, when S is either (i) a unitary representation of some group [7,9,18,25,26,28,29], or (ii) an operator algebra [6,8,40]. Remarkably, while the cases of arbitrary sets, unitary group representations and operator algebras may appear to represent prima facie significant distinctions, this is in fact not so, provided that either S or S ′ in Eq. (37) is closed under the adjoint operation (as is usually the case in practice [6-9, 18, 25, 26, 28, 29, 40]). For example, the standard asymmetry (20) of a projective observable X is equivalently represented in Eq. (37) via any of the set {X}, the group of unitary operators {e iaX }, and the algebra of operators that commute with X. Links between these cases are therefore briefly clarified below for the interested reader, including their unification via the extension of quantumness to quantum channels, and applications to the standard asymmetry measure (other readers may wish to skip directly to Sec. III). Proposition 1. If either S or S ′ is closed under the adjoint operation, then the quantumness measure Q d (S|ρ) in definition (37) is invariant under the replacement of the set S by the von Neumann algebra S ′′ , or by the group G S ′′ of unitary operators in S ′′ (where both replacements are closed under the adjoint operation). Proof. Note that S ′ is closed under the adjoint operation under either of the conditions of the Proposition, since the closure of S under the adjoint implies that [X † , Y ] = 0 for all X ∈ S and Y ∈ S ′ , which is equiv- alent to [X, Y † ] = 0. Further, S ′ is closed under multi- plication and addition (since [X, Y ] = [X, Z] = 0 implies [X, Y Z] = [X, Y + Z] = 0) , and contains the unit operator (since [X, 1] ≡ 0). Hence, S ′ is an algebra, and the closure of S ′ under the adjoint operation guarantees that it is a von Neumann algebra [41]. Now, any von Neumann algebra a is equal to its double commutant a ′′ , and its commutant a ′ is also a von Neumann algebra [41]. Hence, S ′ = (S ′ ) ′′ = (S ′′ ) ′ , implying that Q d (S|ρ) = Q d (S ′′ |ρ) as claimed. Moreover, any element of a von Neumann algebra can be written as a linear combination of (at most four of) its unitaries [41], and hence an operator commutes with the members of the von Neumann algebra S ′′ if and only if it commutes with the members of the group G S ′′ of unitary operators in S ′′ . Thus, S ′ = (S ′′ ) ′ = (G S ′′ ) ′ , implying Q d (S|ρ) = Q d (S ′′ |ρ) = Q d (G S ′′ |ρ) as claimed. Note each replacement is closed under the adjoint operation, since S ′′ is a von Neumann algebra and U −1 = U † for U ∈ G S ′′ . Thus, for example, a distance-based measure of rotational asymmetry has the same value irrespective of whether S in Eq. (37) is chosen to be (a) the set of rotation operators {J x , J y , J z }, or (b) the algebra generated by these operators, or (c) the unitary representation {e iJ·n } of the rotation group [42]. This general link between sets, groups and algebras of operators is further illuminated by extending the definition of Q d to quantum channels. In particular, for a given distance function d(ρ, σ) and CPTP map ϕ, define Q d (ϕ|ρ) := inf σ d(ρ, ϕ(σ)).(39) Thus, Q d (ϕ|ρ) is the distance from ρ to the set of output states of the channel, and may be interpreted as a measure of how closely ρ can be prepared via the channel. Further, this measure reduces to Q d (X|ρ) in Eq. (17) for the channel ϕ X (σ) := σ X . It is therefore natural to ask, for the general case of an arbitrary set of operators S, whether there is a corresponding channel ϕ S such that Q d (S|ρ) = Q d (ϕ S |ρ)(40) i.e., such that the range of ϕ S is equal to the commutant S ′ of S? It turns out that, under the closure assumption of Proposition 1, the answer is largely affirmative. First, if S = {X} for some projective operator X, then ϕ S (σ) = ϕ X (σ) := x X x σX x = σ X(41) as noted above. Second, if the group G S ′′ = {U g } of unitary operators on S ′′ is compact, with normalised measure dg, then ϕ S is the twirling map [25,26] ϕ S (σ) = ϕ G S ′′ (σ) := dgU g σU † g .(42) Third, if the system Hilbert space is finite dimensional, then ϕ S (σ) := E S ′ (σ)(43) where, for a given von Neumann algebra a, E a denotes the conditional expectation map defined via [6,41] Tr [XY ] ≡ Tr [XE a (Y )](44) for X ∈ a and arbitrary Y . These maps agree on their common domains, and cover most situations of interest. Finally, these maps allow the explicit evaluation of the standard asymmetry Q 1 (S|ρ) in most cases. Proposition 2. If S = {X} for a projective observable X and/or G S ′′ is compact and/or the system Hilbert space is finite dimensional, for some set of operators S, and d(ρ, σ) is the relative entropy distance function in Eq. (19), then the standard asymmetry measure Q 1 (S|ρ) defined in Eq. (37) is given by Q 1 (S|ρ) = D 1 (ρ∥ϕ S (ρ)) = H(ϕ S (ρ)) − H(ρ).(45) Proof. The result relies on the composition properties (19) and (39) give ϕ • ϕ = ϕ, ϕ • f = f • ϕD 1 (ρ∥ϕ(σ)) − D 1 (ρ∥ϕ(ρ)) = Tr [ϕ(ρ)(log ϕ(ρ) − log ϕ(σ))] = D 1 (ϕ(ρ)∥ϕ(σ)) ≥ 0, implying Q 1 (ϕ|ρ) = D 1 (ρ∥ϕ(ρ)) + inf σ D 1 (ϕ(ρ)∥ϕ(σ)) = D 1 (ρ∥ϕ(ρ) ). Thus, since Eq. (40) holds under the stated conditions, Q 1 (S|ρ) = Q 1 (ϕ|ρ) = Tr [ρ log ρ] − Tr [ρ log ϕ(ρ)] = −H(ρ) − Tr [ϕ(ρ) log ϕ(ρ)] = H(ϕ(ρ)) − H(ρ) as required. Proposition 2 implies that ϕ S (ρ) is the output state of ϕ S that is closest to ρ, when 'distance' is measured via relative entropy, and significantly generalises Eq. (20) for the standard asymmetry of single observables. It has been previously given by Gao et al. for the case that S is a von Neumann algebra on a finite Hilbert space [6]. Note also that Corollary 2.3 of Gao et al. for strong subadditivity generalises to (using (S ∪ T ) ′ = S ′ ∩ T ′ and Proposition 1) the uncertainty relation Q 1 (S|ρ) + Q 1 (T |ρ) ≥ Q 1 (S ∪ T |ρ),(46) for the standard asymmetry of any two sets S and T for which ϕ S • ϕ T = ϕ T • ϕ S . The following section is concerned with uncertainty relations for Rényi asymmetry. III. TRADEOFF RELATIONS FOR ASYMMETRY AND ENTROPY A. Rényi asymmetry vs Rényi entropy The Rényi asymmetry Q α (X|ρ) in Eqs. (26) and (34) represents the distance between the state of the system and a set of postmeasurement states for X, as measured via the sandwiched Rényi divergence in Eq. (24). It is also a measure of the sensitivity of the state to transformations generated by X (or its Naimark extension), and is the quantum component in the quantum-classical decomposition of Rényi entropy in Eq. (29). It reduces to the standard asymmetry given by the quantum relative entropy in Eq. (20) for the case α = 1, and the general case has recently been found to have useful applications in the areas of quantum information, quantum metrology, quantum coherence, open quantum systems, and time-energy uncertainty relations [5][6][7][8][9]. As the intrinsically quantum contribution to Rényi entropy, Rényi asymmetry should be expected to play a fundamental role in entropic uncertainty relations. This expectation is supported by uncertainty relation (46) of Gao et al. for the standard asymmetry. In particular, for two conjugate rank-1 projective observables X and Y on a d-dimensional Hilbert space, with X x = |x⟩⟨x|, Y y = |y⟩⟨y| and |⟨x|y⟩| 2 = d −1 , one has (σ X ) Y = (σ Y ) X = d −1 1 from Eq. (18), implying ϕ {X,Y } (σ) = d −1 1 from Eqs. (37) and (40), and Eq. (46) simplifies to Q 1 (X|ρ)+Q 1 (Y |ρ) ≥ Q 1 ({X, Y }|ρ) = log d−H(ρ) (47) with Q 1 ({X, Y }|ρ) evaluated via Proposition 2. Noting that H(ρ X ) = H(X|ρ) and H(ρ Y ) = H(Y |ρ) for such observables, and applying Eq. (20), this is equivalent to the tradeoff relation, Q 1 (X|ρ) + H(Y |ρ) ≥ log d,(48) between the standard asymmetry and Shannon entropy of the observables (and to the strong entropic uncertainty relation H(X|ρ)+H(Y |ρ) ≥ log d+H(ρ) for the Shannon entropies of such observables [4,13]). A second example is the analogous tradeoff relation Q α (J z |ρ) + H α (Φ|ρ) ≥ log 2π, α ≥ 1 2 ,(49) for angular momentum and rotation angle [9]. Noting the upper bound Q α (J z |ρ) ≤ H β (J z |ρ) from Eq. (29), this immediately implies and so is stronger than the standard uncertainty relation H α (J z |ρ) + H β (Φ|ρ) ≥ log 2π for the Rényi entropies of these observables [43]. It is shown here that similar tradeoff relations hold for arbitrary pairs of discrete observables, whether projective or nonprojective. However, the case of one projective and one arbitrary observable will be considered first, as the derivation is particularly simple for this case. Theorem 1. For a discrete projective observable X with projection valued measure {X x }, and an arbitrary discrete observable Y with POVM {Y y }, one has the tradeoff relation Q α (X|ρ)+H α (Y |ρ) ≥ − log max x,y λ max (X x Y y X x ), α ≥ 1 2 ,(50) between Rényi asymmetry and entropy, where λ max (A) is the maximum eigenvalue of Hermitian operator A. Proof. The proof is a generalisation of the derivation given by Korzekwa et al. of entropic uncertainty relation (5) for rank-1 projective observables [4]. First, define a 'measure and discard' CPTP map φ Y , from the Hilbert space of the system to the Hilbert space of a record system suitable for registering the result of a measurement of Y , via φ Y (ρ) := y Tr [ρY y ] |y⟩⟨y|,(51) where {|y⟩} is an orthonormal basis for the record system. Equation (18) and data processing inequality (25) for the sandwiched Rényi divergence then give D α (ρ∥σ X ) ≥ D α (φ Y (ρ)∥φ Y (σ X )) = 1 α − 1 log y Tr [ρY y ] α Tr [σ X Y y ] 1−α = 1 α − 1 log y Tr [ρY y ] α Tr σ x X x Y y X x 1−α . The final trace is over a sum of orthogonal subspaces, corresponding to the set of projections X x . Hence, this trace is maximised, for a given value of y, when σ is the eigenstate corresponding to the maximum possible eigenvalue of X x Y y X x over all x, and so is upperbounded by µ = max x,y λ max (X x Y y X x ). Thus, noting that 1 α−1 log y a(y)b(y) 1−α is monotonic decreasing in b(y) for a(y), b(y), α ≥ 0, the Rényi asymmetry in Eq. (26) has the lower bound Q α (X|ρ) = inf σ D α (ρ∥σ X ) ≥ inf σ 1 α − 1 log y Tr [ρY y ] α µ 1−α = − 1 1 − α log y p(y|ρ) α − log µ.(52) Recalling the definition of Rényi entropy in Eq. (27), this is equivalent to the statement of the theorem. Theorem 1 provides a strong link between asymmetry and entropy that significantly extends tradeoff relations (48) and (49) for conjugate observables. It further leads to strengthened uncertainty relations for Rényi entropies, as discussed further below. Note that the proof of the Theorem is relatively elementary, relying on a simple manipulation of data-processing inequality (25) for the sandwiched Rényi divergence, suggesting it may also be used to obtain analogous tradeoff relations for other distance-based measures of quantumness or asymmetry. Noting that the asymmetry Q α (X|ρ) vanishes for [X, ρ] = 0 (consistent with Eq. (9) for measures of quantumness), Theorem 1 gives a lower bound for the Rényi entropy H α (Y |ρ) in this case. A particular choice of X yields the following interesting albeit weak corollary. Corollary 1. For an arbitrary discrete observable X with POVM {X x }, and state ρ with spectral decomposition ρ = j e j E j (i.e., E j is the projection onto the eigenspace corresponding to eigenvalue e j of ρ), one has the lower bound H α (X|ρ) ≥ − log max x,j:ej >0 λ max (E j X x E j ), α ≥ 1 2 ,(53) for Rényi entropy. Proof. Define the projection onto the support of ρ by E := j:pj >0 E j , implying ρ = EρE, and let Y E be the projection of Y onto the support of ρ, with POVM {EY y E}. Replacing X by ρ and Y by Y E in Theorem 1, gives a vanishing asymmetry Q α (ρ|ρ) = 0 via definition (26) and a lower bound µ ρY E = max y,j:pj >0 λ max (E j EY y EE j ) = max y,j:pj >0 λ max (E j Y y E j ) via EE j = E j (= 0) for p j > 0 (= 0). Noting H α (Y E |ρ) = H α (Y |ρ) and changing notation from Y to X then yields the Corollary as desired. Corollary 1 bounds the Rényi entropy of any observable in terms of its degree of incompatibility with the system state. Note that it represents a slight sharpening of Eq. (6) in the Introduction, where the latter does not limit the maximisation to e j > 0, and implies a stateindependent lower bound, − log λ max (X x ), via E j ≤ 1. However, the Corollary does not take the mixedness of the state into account, making it relatively weak in comparison to bounds that do. For example, Corollary 1 is weaker than the known classical lower bound H α (X|ρ) ≥ H ∞ (X|ρ) = − log max x p(x|ρ)(54) for Rényi entropy, following from the monontonic decreasing property H α (X|ρ) ≥ H β (X|ρ) for α < β [44,45], where H ∞ depends on mixedness via the eigenvalues e j of ρ. In particular, the operator E j X x E j is nonzero only on the Tr [E j ]-dimensional unit eigenspace of E j and so has at most Tr [E j ] nonzero eigenvalues, implying that the sum of its eigenvalues, Tr [E j X x E j ], is upper bounded by Tr [E j ] λ max (E j X x E j ). Hence, defining p j := e j Tr [E j ], then j:pj >0 p j = 1 and one has p(x|ρ) = j:ej >0 e j Tr [E j X x E j ] ≤ j:pj >0 p j λ max (E j X x E j ) ≤ max j:ej >0 λ max (E j X x E j ). Maximising over x immediately implies the bound in Corollary 1 is never greater than the bound H ∞ (X|ρ) in Eq. (54), as claimed (with equality for pure states). A second example will be given in Corollary 5 below, which provides a strong lower bound for Rényi entropy that depends on mixedness rather than incompatibility, and which is stronger than both Corollary 1 and the classical bound (54) for sufficiently mixed states. The main technical result of this paper is a generalisation of Theorem 1 to all discrete observables, whether projective or nonprojective, via the generalised definition of asymmetry in Eq. (34). λ max (X 1/2 x Y y X 1/2 x ) = µ Y X(56) denotes the maximum eigenvalue of X 1/2 x Y y X 1/2 x over x and y. Proof. It is first necessary to generalise Y to an observ-ableỸ on the extended Hilbert spaceH for which the Naimark extension (30) for general statesρ onH, where |y ⊥ ⟩ is the record state corresponding to POVM element P ⊥ . For system state ρ on H and general stateσ onH one then has, using data processing inequality (25) and noting that p(y ⊥ |ρ) = Tr ρP ⊥ = 0 via ρ = P ρP , D α (ρ∥σX ) ≥ D α (φỸ (ρ)∥φỸ (σX )) = 1 α − 1 log y Tr [ρY y ] α Tr [σX Y y ] 1−α = 1 α − 1 log y Tr [ρY y ] α Tr σ xX x Y yXx 1−α similarly to the proof of Theorem 1. Also similarly, noting that theX x are orthogonal projections, the final trace is maximised for a given value of y by choosingσ to be the eigenstate corresponding to the maximum over x of the maximum eigenvalue ofX x Y yXx . Now, λ max (X x Y yXx ) = λ max [(X x Y 1/2 y )(X x Y 1/2 y ) † ] = λ max [(X x Y 1/2 y ) † (X x Y 1/2 y )] = λ max (Y 1/2 yXx Y 1/2 y )] = λ max (Y 1/2 y PX x P Y 1/2 y )] = λ max (Y 1/2 y X x Y 1/2 y )] = λ max (X 1/2 x Y y X 1/2 x ),(58) with the second and last lines following from λ max (AA † ) = λ max (A † A), and the fourth and fifth lines from Y y = P Y y P and Eq. (30), respectively. Hence, Q α (X|ρ) = inf σ D α (ρ∥σX ) ≥ inf σ D α (ρ∥σX ) ≥ − 1 1 − α log y p(y|ρ) α − log µ XY(59) similarly to Eq. (52). This is equivalent to the statement of the Theorem, noting that µ XY = µ Y X via the last two lines of Eq. (58). Theorem 2 reduces to Theorem 1 when X is a projective observable, and underpins the strong uncertainty relation for Rényi entropies given in Theorem 3 below. Note that µ XY can also be written as µ XY = max x,y ∥X 1/2 x Y 1/2 y ∥ 2 ∞ = max x,y ∥Y 1/2 y X 1/2 x ∥ 2 ∞ (60) where ∥A∥ ∞ = [λ max (A † A)] 1/2 denotes the operator norm of A, i.e., the largest singular value of A. Theorem 2 immediately provides a simple lower bound for Rényi asymmetry, as per the following corollary. Corollary 2. For an arbitrary discrete observable X with POVM {X x } one has the lower bound Q α (X|ρ) ≥ − log max x λ max (X x ), α ≥ 1 2 ,(61) for the Rényi asymmetry of X. Proof. Choose Y = 1 in Theorem 2 and observe that H α (1|ρ) ≡ 0. The lower bound in Corollary 2 is trivial for projective observables, but typically nonvanishing otherwise, corresponding to the 'hidden' asymmetry discussed following Eq. (34). For example, the lower bound is log 3 2 for the qubit trine observable T discussed following Corollary 5. Finally, it is worth noting here, as a preview to the next Section, that the standard uncertainty relation for Rényi entropies in Eq. (4) of the Introduction has a simple direct derivation via Theorem 2 for the case of Shannon entropy, i.e., α = β = 1, as per the following Corollary. λ max (X 1/2 x Y y X 1/2 x ) = µ Y X (63) denotes the maximum eigenvalue of X 1/2 x Y y X 1/2 x over x and y. Proof. The uncertainty relation follows immediately for any pure state ρ = |ψ⟩⟨ψ|, from Eq. (35) and Theorem 2. It then immediately follows for general states via the concavity of Shannon entropy. Corollary 3 is given for rank-1 projective observables in [10] and for arbitrary observables in [11], and generalised to uncertainty relation (4) for Rényi entropies in [12]. However, the proofs of Theorem 2 and Corollary 3 ultimately rely on data-processing inequality (25), rather than on Riesz's theorem [10][11][12]. This has the advantage of leading to the stronger uncertainty relation in Eq. (3) for Rényi entropies for very little additional work, as will be shown next. B. Strong uncertainty relations for Rényi entropies It is straightforward to strengthen Corollary 3, so as to obtain uncertainty relation (3) of the Introduction, which takes the mixedness of the state into account. Theorem 3. For arbitrary discrete observables X and Y , with corresponding POVMs {X x } and {Y y }, one has the entropic uncertainty relation H α (X|ρ) + H β (Y |ρ) ≥ − log µ XY + max{C α (Y |ρ), C β (X|ρ)}(64) for Rényi entropies with 1/α + 1/β = 2, where C α (Z|ρ) is the classical contribution to H β (Z|ρ), defined via quantum-classical decomposition (29), and µ XY := max x,y λ max (X 1/2 x Y y X 1/2 x ) = µ Y X(65) denotes the maximum eigenvalue of X 1/2 x Y y X 1/2 x over x and y. Proof. Substitution of the quantum-classical decomposition (29) of H β (X|ρ) into Theorem 2 gives H β (X|ρ) + H α (Y |ρ) ≥ − log µ XY + C α (X|ρ)(66) for 1/α + 1/β = 2. Comparing the inequality resulting from swapping X with Y in this expression with the inequality resulting from swapping α with β then yields the Theorem as desired. Recalling that the classical component C α (X|ρ) vanishes for pure states as per Eq. (36), it is seen that Theorem 3 improves on the standard uncertainty relation in Corollary 3 by taking the mixedness of the state into account. For example, for the special case α = β = 1 one has the following corollary. Corollary 4. For an arbitrary rank-1 discrete observable X and an arbitrary discrete observable Y , with respective POVM elements {|x⟩⟨x|} and {Y y }, the Shannon entropies of X and Y satisfy the uncertainty relation H(X|ρ) + H(Y |ρ) ≥ − log max x,y ⟨x|Y y |x⟩ + H(ρ). (67) Proof. For projective observables this result is an immediate consequence of Eq. (23) and Theorem 3 for α = β = 1. More generally, for any discrete observable X, with Naimark extensionX, Eqs. (23) and (31) yield C 1 (X|ρ) = H(ρ) − x p(x|ρ)H(ρ x ), withρ x := X x ρX x /p(x|ρ) defined on the extended Hilbert space. Hence, C 1 (X|ρ) ≤ H(ρ) for all observables X (including X = Y ), with equality whenX is rank-1. But for a rank-1 observable X one can always chooseX to be rank-1. In particular, expressing the projectionX x as an orthogonal sum of rank-1 projections,X x = k |x, k⟩⟨x, k|, the Naimark extension property X x = PX x P in Eq. (30) requires that P |x, k⟩ = 0 for all but one value of k, k = k x say, implying one can replaceX by a rank-1 projective observableX ′ on the extended Hilbert spaceH ′ generated by the span of {|x, k x ⟩}, withX ′ x := |x, k x ⟩⟨x, k x |. Hence, making such a choice, it follows that C 1 (X|ρ) = H(ρ) ≥ C 1 (Y |ρ). Substitution into Theorem 3 with α = β = 1 then gives the Corollary as desired. Corollary 4 generalises the known uncertainty relation for Shannon entropies of rank-1 projective observables in Eq. (5) of the Introduction [4,13], and is given in Eq. (71) of [14] under the assumption of a finite system Hilbert space. Note that H(ρ) appears as the measure of mixedness in this case because it is an upper bound for classicality when α = β = 1 and is saturated for rank-1 observables. Finally, Theorem 3 also yields a strong lower bound for the Rényi entropy of discrete observables that takes mixedness into account. Corollary 5. For an arbitrary discrete observable X with POVM {X x } one has the lower bound H α (X|ρ) ≥ − log max x λ max (X x ) + C α 2α−1 (X|ρ),(68) for Rényi entropy. Proof. Choose Y = 1 in Theorem 3, and note that 0 ≤ C α (1|ρ) ≤ H β (1|ρ) = 0 via quantum-classical decomposition (29) (which is valid for arbitrary observables as discussed in Sec. II B 4). The Corollary also follows directly from the asymmetry lower bound in Corollary 2, again using Eq. (29) (but with α and β swapped). For the case of a rank-1 observable X and α = 1 one has C 1 (X|ρ) = H(ρ) as per the proof of Corollary 4, and the lower bound in Corollary 5 reduces to H(X|ρ) ≥ − log max x ⟨x|x⟩ + H(ρ)(69) This also corresponds to choosing Y = 1 in Corollary 4, and is stronger than the alternative choice Y = X, which replaces H(ρ) by 1 2 H(ρ). For rank-1 projective observables (i.e, ⟨x|x⟩ ≡ 1), it reduces to the known property that the Shannon entropy of such observables is never less than the von Neumann entropy (since p(x|ρ) = j S xj p j where S xj := |⟨x|ψ j ⟩| 2 is a doubly stochastic matrix for any orthogonal decomposition ρ = j p j |ψ j ⟩⟨ψ j | of ρ). It is of interest to compare Corollary 5, which gives a lower bound for Rényi entropy that depends on the sharpness of X and the mixedness of ρ, with classical lower bound (54). As a simple example, consider the projective qubit observable X = σ·n for spin direction n, and state ρ = 1 2 (1+σ·r) with Bloch vector r. Corollary 5 then yields the lower bound 70) for qubit observable X = σ · n, following from Corollary 5, is plotted as a function of the Bloch vector length r (upper solid blue curve), while the classical lower bound (71) is plotted for the cases θn,r = 0, i.e., n · r = r (lower solid red curve), and θn,r = π 3 , i.e., n · r = 1 2 r (dashed black curve). Corresponding plots of the lower bounds for the measurement entropy of the trine observable T in Eqs. (72) and (73) may be obtained by adding log 3 2 to each lower bound. For both observables the classical lower bounds are seen to be weaker than the bound from Corollary 5 for sufficiently mixed states, as discussed more generally in the main text. H(X|ρ) ≥ H(ρ) = − 1 + r 2 log 1 + r 2 − 1 − r 2 log 1 − r 2 ,(70) for the Shannon measurement entropy, with r := |r|. In contrast, the corresponding classical lower bound in Eq. (54) gives H(X|ρ) ≥ log 2 1 + |n · r| = log 2 1 + r| cos θ n,r | , where θ n,r is the angle between n and r. Both lower bounds range from 0 for r = ±n up to log 2 for r = 0. However, as depicted in Fig. 1, the bound in Eq. (70) is always stronger for sufficiently mixed states (as is easily proved noting that Eq. (70) is concave in r and Eq. (71) is convex in r, with equality at r = 0). As an example for the case where X is a nonprojective observable, consider the qubit trine observable X = T with POVM elements T j = 1 3 (1 + σ · m j ), where m j denotes the unit vector (cos(2πj/3), sin(2πj/3), 0) for j = 0, 1, 2 [37,39,46]. Corollary 5 then gives the lower bound H(T |ρ) ≥ log 3 2 + H(ρ),(72) whereas the classical lower bound (54) gives H α (T |ρ) ≥ log 3 2 + log 2 1 + r max j cos θ m j ,r ,(73) where θ n,r is the angle between m j and r. While both lower bounds range from a minimum value of log 3 2 for r = m j , up to the maximum possible value of H α (T |ρ) = log 3 for r orthogonal to the plane of trine directions, the bound in Eq. (72) is always stronger for sufficiently mixed states, as depicted in Fig. 1 (with the common term log 3 2 subtracted). For example, if r is anti-aligned with one of the trine directions, then cos θ m j ,r is maximised for θ m j ,r = π 3 and Corollary 5 outperforms classical lower bound 54 (and hence also Corollary 1) for r ≲ 0.753, corresponding to the intersection of the upper solid and dashed curves in Fig. 1. More generally, for the case of an arbitrary observable X and α = 1, Corollary 5 reduces to the lower bound H(X|ρ) ≥ − log max x λ max (X x ) + H(ρ) − x p(x|ρ)H(ρ x ),(74) for Shannon entropy, withρ x :=X x ρX x /p(x|ρ), for any Naimark extensionX of X as per Eq. (30) (see also the proof of Corollary 4). This case is intriguingly similar in form to the lower bound H(X|ρ a ) ≥ − log max x λ max (X x ) + H(ρ b ) − x p(x|ρ)H(ρ b|x )(75) for Shannon entropy in Eq. (65) of [14] for any joint state ρ ab , with ρ b|x := tr a [ρ ab X x ]/p(x|ρ a ). It would be of interest to explore the connection between these bounds in future work. This is nontrivial given that ρ b = x p(x|ρ a )ρ b|x whereas ρ ̸ = x p(x|ρ)ρ x . The key may lie in the related asymmetry lower bound given in Corollary 2. IV. DISCUSSION The results weave two main threads together. The first is a general approach to quantum-classical decompositions of resource measures for observables (and sets of operators), based on the idea that pure states are the most resourceful (Sec. II A). The quantum contribution to the resource measure arises from the noncommutativity or incompatibility of observables with the state, and the classical contribution from the mixedness of the state. The approach unifies previous decompositions of variance and entropy, and generalises them to the decomposition of the symmetrised covariance matrix with respect to measures of quantum Fisher information, and the decomposition of Shannon entropy with respect to the conditional entropy of self-dual communication channels (Secs. II B 1, II B 2 and Appendix A 1), and is applied to Rényi entropy in particular (Sec. II B 3). In the latter case the quantum contribution is provided by the Rényi asymmetry, with its connection to Rényi entropy arising from duality relation (35) for pure states. The second thread is the generalisation and unification of measures of quantumness and asymmetry to nonprojective observables (Sec. II B 4), and to sets, groups and algebras of operators and to quantum channels (Sec. II B 5). The Rényi asymmetry is a particular example of interest, as it generalises the standard asymmetry measure based on relative entropy and has applications in contexts as diverse as coherence, information, uncertainty relations, metrology, and open quantum systems. The weaving of the two threads is in two natural stages. In the first stage, tradeoff relations between Rényi asymmetry and Rényi entropy are obtained, as per Theorems 1 and 2 (Sec. III A). These have relatively elementary derivations, based on data processing inequality (25) for sandwiched Rényi divergences, and significantly generalise known relations (48) and (49) for the case of conjugate observables. Corollaries 1-3 are simple applications of these Theorems that yield lower bounds for Rényi asymmetry and Rényi entropy, and recover previously known uncertainty relation (4) for the case of Shannon entropy. All these results are independent of any consideration of quantum-classical decompositions. It is in the second stage that the tradeoff relations for Rényi asymmetry are combined with the quantumclassical decomposition of Rényi entropy in Eq. (29), to obtain the strong uncertainty relation for Rényi entropies in Theorem 3 (Sec. III B). This relation is valid for both projective and nonprojective observables, and improves on known uncertainty relation (4) by taking the mixedness of the system state into account via the classical contribution to Rényi entropy. Corollary 4 recovers a general result from Theorem 3 for the case of Shannon entropies, using the property that the classical contribution is bounded by the von Neumann entropy of the system for this case, while Corollary 5 gives a strong bound for Rényi entropy that improves on Corollary 1 and classical lower bound (54) for sufficiently mixed states. It is worthwhile remarking here on a point of terminology. The term 'quantumness' has been used for the measure of noncommutativity Q(X, Y, . . . |ρ) introduced in Sec. II A, given its fundamental role as the inherently quantum contribution to uncertainty and other resource measures [1,2,4]. Note, however, that the nonvanishing of a quantum commutator has a direct classical analogue: the nonvanishing of a classical Poisson bracket. Hence, one can define classical analogues of asymmetry and the like (e.g., as a measure of the distance between a given phase space distribution and the set of phase space distributions invariant under canonical transformations generated by a given classical observable). Nevertheless, this does not extend to give a classical analogue of the approach to quantum-classical decompositions in Sec. II A. In particular, this approach relies on the purification of a given quantum state on a larger Hilbert space, whereas there is no analogous purification of classical phase space distributions to a delta-distribution on a larger phase space. Hence, the term 'quantumness' may be justified as reflecting an essentially quantum feature in the context of quantum-classical decompositions. There are various possible directions for future work, including the following. First, it is known that the bound µ XY in known uncertainty relation (4) can be improved using majorisation properties of Schur-convex functions such as Rényi entropy [47,48]. This suggests looking for similar improvements to the lower bounds in Theorems 1-3. Perhaps, for example, the argument based on a monotonic decreasing function in the proof of Theorem 1 can be strengthened via a majorisation argument. Second, it is of interest to seek analogous tradeoff relations and uncertainty relations for continuous observables such as position and momentum. Here the main technical issue is that the kets appearing in the spectral decompositions of such observables are nonnormalisable, so that the analogue of σ X in Eq. (18) is not a density operator. However, it may be possible to take suitable limits of discrete observables to obtain results for this case, as has been done previously for Shannon entropic uncertainty relations [49,50]. Third, whereas the measures of quantumness considered in the paper are focused on measures of uncertainty and asymmetry, consideration of other resources measures such as entanglement and Bell nonlocality, and related quantum-classical decompositions, is also a possible avenue of exploration. One might, for example, maximise some measure Q( X 1 ⊗ X 2 , Y 1 ⊗ Y 2 , . . . |ρ) or u(X 1 ⊗ X 2 , Y 1 ⊗ Y 2 , . . . |ψ) over sets of local observables to obtain a measure of bipartite entanglement for state ρ or ψ, and apply the approach in Sec. II A or the alternative approach in Appendix A 2 to obtain corresponding quantum-classical decompositions. It would also be of interest to extend Proposition 2 for the standard asymmetry measure, to sets S of two or more (noncommuting) operators on an infinite Hilbert space, for the case that the group of unitary operators on the double commutant S ′′ is noncompact. Fourth and finally, it should be possible to extend or relate the strong uncertainty relation in Theorem 3 to the presence of quantum memory or quantum side information, in which uncertainties are conditioned on systems with which the system of interest is correlated [13,14]. For example, the general method given by Coles et al. in [51] is applicable to Rényi conditional entropies of the form H α (a|b) := − inf σ b D α (ρ ab ∥1 a ⊗σ b ), noting that the sandwiched Rényi divergence (24) satisfies the required properties for α ≥ 1 2 . Hence, noting further the duality property H α (a|b) + H β (a|c) = 0 for 1/α + 1/β = 2 and any pure state ρ abc [31,33], Theorem 1 of [51] yields an uncertainty relation of the form H α (X|b) + H β (Y |c) ≥ − log µ XY , 1 α + 1 β = 2,(76) where X and Y are observables on the first component of a tripartite state ρ abc , ρ Xb : = ⊕ x Tr ac [ρ abc X x ⊗ 1 b ⊗ 1 c ] ≡ φ X (ρ ab ), and ρ Y c := ⊕ y Tr ab [ρ abc Y y ⊗ 1 b ⊗ 1 c ] ≡ φ Y (ρ ac ) (see also Theorem 11 of [31]). It would be of interest to find a connection between quantum memory and the classical contribution to Rényi entropy via a comparison of this uncertainty relation with Theorem 3 (see also the related discussion at the end of Sec. III B). The method of constructing quantum-classical decompositions in Sec. II A is guided by the level of accessibility, and hence usefulness, of a given quantum resource. In particular, the classical component arises due to lack of access to a purified state of the system. Here an alternative approach is briefly described, based on the direct decomposition of the system state into a mixture of pure states. The approach is conceptually simple, but appears to be less general and technically more challenging than the approach given in the main text. The starting point is a function u(X, Y, . . . , |ψ) that quantifies a quantum resource for the pure states of a given system. It is natural to consider such functions in various contexts, e.g., the entropy of the Schmidt coefficients for an entangled pure state. The function u is assumed to satisfies two minimal properties: It is straightforward to check that the above construction satisfies the basic requirements discussed in Sec. II A. Assumption (9) follows from property (A7), noting that commutativity implies there is a pure-state decomposition into simultaneous eigenstates, while assumptions 10 and (11) are direct consequences of property (A8) and definitions (A10)-(A12), respectively. It is seen that the classical mixing contribution arises in this approach directly via the mixing of pure states, in contrast to the approach in Sec. II in which the mixing is generated via a partial trace. a. Example: variance and minimal Fisher information As a first example, choose u to be the variance of a projective observable, i.e., u V (X|ψ) := Var ψ X = ⟨X 2 ⟩ ψ − ⟨X⟩ 2 ψ . The concave and convex roofs of this function are given in [52] and [53], respectively (see also [54]), and the above 'roof' construction then yields the corresponding quantum-classical decomposition Here {|j⟩} is the set of eigenstates of ρ and the summation runs over the eigenstates that generate nonzero denominators. The summation may be recognised, up to a constant factor, as the minimal quantum Fisher information, corresponding to the symmetric logarithmic derivative [16,52,53], and hence this decomposition of the variance is a special case of the decomposition of the covariance matrix in Eq. (15) of Sec. II B. Thus, the 'roof' construction method picks out a preferred quantum Fisher information for the decomposition, in comparison to the method in the main text. b. Example: Shannon entropy and informational power More generally, the convex and concave roofs of a given function will not be amenable to analytic calculation, making this approach reliant on numerical methods. Consider, for example, the case where the function u is the Shannon entropy of observable X, i.e., u H (X|ψ) := H(X|ψ) = − x ⟨ψ|X x |ψ⟩ log⟨ψ|X x |ψ⟩. The roof construction then gives the decomposition where I(X : E) is the Shannon mutual information for observable X and signal ensemble E. Hence the quantum component is directly related to the maximum mutual information of the observable over the set of pure-state signal ensembles for density operator ρ, also called the 'informational power' of the observable for state ρ [55], while the classical component is given by the gap between the maximum and minimum possible mutual informations. In general there are few analytic results for the maximum and minimum mutual informations of a pure state channel with fixed X and ρ, and so the mutual information must be maximised or minimised numerically. However, upper and lower bounds are provided by choosing particular pure state ensembles in P ρ , such as the Scrooge ensemble [56] or (for rank-1 observables) the self-dual ensemble in Eq. (A1) above, which can be used as estimates of the minimum and maximum values, respectively. The von Neumann entropy H(ρ) also provides an estimate of the maximum value via the Holevo bound for mutual information [15]. Hence, for example, for a rank-1 observable X one has the inequalities H(X|ρ) − H(ρ) ≤ Q H (X|ρ) ≤ Q sd (X|ρ), (A20) Q sd (X|ρ) − q(ρ) ≤ C H (X|ρ) ≤ H(ρ),(A21) where Q sd (X|ρ) is the conditional entropy of the self-dual channel in Eqs. (A2) and (A4), and q(ρ) is the subentropy of ρ, i.e., the entropy of the Scrooge ensemble for ρ [56]. for any numerical function f , and the duality property ϕ * = ϕ, which may be checked to hold when ϕ is any of the maps in Eqs. (41)-(43). In particular, these three properties, used in turn, yield Tr [ρ log ϕ(σ)] = Tr [ρ log ϕ • ϕ(σ)] = Tr [ρϕ(log ϕ(σ))] = Tr [ϕ(ρ) log ϕ(σ)]. Hence, definitions Corollary 3 . 3For arbitrary discrete observables X and Y , with corresponding POVMs {X x } and {Y y }, one has the entropic uncertainty relationH(X|ρ) + H(Y |ρ) ≥ − log µ XY(62)for Shannon entropy, where µ XY := maxx,y FIG. 1 . 1Lower bounds on measurement entropy for qubit states. The lower bound in Eq. ( of observables X, Y, . . . , i.e.,u(X, Y, . . . |ψ) = 0 if Xψ = xψ, Y ψ = yψ, . . . XU † , U Y U † , . . . |U ψ) = u(X, Y, . . . |ψ) (A8)for any unitary transformation U . Now, let P ρ denote the set of pure-state decompositions of density operator ρ, i.e.,P ρ := {p j ; ψ j } : p j ≥ 0, j p j |ψ j ⟩⟨ψ j | = ρ , (A9)where |ψ⟩⟨ψ| denotes the density operator corresponding to ψ. The quantum-classical decomposition corresponding to function u is then defined viaM (X, Y, . . . |ρ) := sup {pj ;ψj }∈Pρ j p j u(X, Y, . . . |ψ j ), (A10) Q(X, Y, . . . |ρ) := inf {pj ;ψj }∈Pρ j p j u(X, Y, . . . |ψ j ), (A11) C(X, Y, . . . |ρ) := M (X, Y, . . . |ρ) − Q(X, Y, . . . |ρ). (A12) Thus, M and Q are the concave and convex roofs of u, with the classical component given by their gap. M V (X|ρ) = Var ρ X = Q V (X|ρ) + C V (X|ρ) M H (X|ρ) = Q H (X|ρ) + C H Theorem 2 . 2For arbitary discrete observables X and Y , with corresponding POVMs {X x } and {Y y }, one has the tradeoff relationQ α (X|ρ) + H α (Y |ρ) ≥ − log µ XY ,α ≥ 1 2 , (55) for asymmetry and entropy, where µ XY := max x,y is defined for X. In particular, Y is defined to correspond to the POVM {Y y } ∪ {P ⊥ } oñ H, with P ⊥ :=1 − P , and φ Y in Eq. (51) is generalised to the 'measure and discard' map φỸ (ρ) := Tr [ρY y ] |y⟩⟨y| + Tr ρP ⊥ |y ⊥ ⟩⟨y ⊥ |, (57)y . An alternative roof-based approach to quantum-classical decompositions ACKNOWLEDGMENTSI am grateful to an anonymous referee for suggesting comparing Corollary 1 with classical lower bound(54). Having been recently diagnosed with pancreatic cancer, this may be one of my last physics papers. I therefore take this opportunity to thank the many colleagues I have interacted with over the years, both locally and nonlocally :-) Appendix A: FURTHER EXAMPLES OF QUANTUM-CLASSICAL DECOMPOSITIONS 1. Decomposition of Shannon entropy via a quantumness measure for self-dual channels A quantum-classical decomposition of Shannon entropy is obtained here that is quite different to the decomposition of Korzekwa et al.[4]discussed in Sec. II B 2, based on a choice of quantumness measure that arises in the context of quantum information channels.In particular, consider the communication channel defined by measuring observable X with POVM {X x } at the receiver, on members of an ensemble of signal states with average density operator ρ, for the case of the 'dual' ensemble E X,ρ := {p x ; ρ x } defined by[22,23](note that ρ = x p x ρ x , and the choice of ρ x is left arbitrary if p x = 0). This 'self-dual' channel is invariant under source duality, i.e., interchange of the the signal ensemble and the receiver measurement[22,23]. It provides a strong lower bound for the maximum information obtainable from a measurement of X on an ensemble with density operator ρ[23], and is closely related via the inverse map p x ρ x → X x to the 'pretty good measurement' used in the proof of the Holevo-Schumacher-Westmoreland theorem for the classical capacity of a quantum channel[15]. Each measurement result x of the self-dual channel gives some information about which signal state ρ x ′ was transmitted, with an average uncertainty quantified by the Shannon conditional entropy[15],(A2) Here I(X : E X,ρ ) is the mutual information of the channel, and H(X ′ ) = − x ′ p x ′ log p x ′ = H(X|ρ) is the entropy of the signal-state distribution p x ′ in Eq. (A1). Note that the joint distribution of X and X ′ for state ρ follows via Eq. (A1) asNow, if X is a projective observable, then the conditional entropy of the self-dual channel provides a measure of quantumness satisfying assumption (9) of Sec. II A (since [X, ρ] = 0 implies the signal states in Eq. (A1) are orthogonal, so that one has a noiseless classical channel with H(X, X ′ ) = H(X)). Further, assumptions(10)and(11)are also satisfied (noting for a pure state ρ = |ψ⟩⟨ψ| that the signal states are identical and hence carry no information). Hence, noting p(x, x ′ ) in Eq. (A3) is invariant under Naimark extension (30) (since P ρP = ρ), it follows that the conditional entropy of the self-dual channel in Eq. (A2) is a measure of quantumness for both projective and nonprojective observables, corresponding to (A6) Here the second equality follows by direct calculation[23], where the last term is the Shannon entropy of the joint probability distribution in Eq. 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Vaccaro, Group theoretic formulation of comple- mentarity, in Proceedings of the 8th International Con- ference on Quantum Communication, Measurement and Computing, Tsukuba, Japan, 28 November-3 December 2006, eds. O. Hirota, J. H. Shapiro & M. Sasaki, (NICT Press: Tokyo, Japan, 2006), pp. 421-424 (Eprint avail- able at arXiv:1012.3532). Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules. J A Vaccaro, F Anselmi, H M Wiseman, K Jacobs, 10.1103/PhysRevA.77.032114Phys. Rev. A. 7732114J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Ja- cobs, Tradeoff between extractable mechanical work, ac- cessible entanglement, and ability to act as a reference system, under arbitrary superselection rules, Phys. Rev. A 77, 032114 (2008). Quantifying superposition. J Åberg, 10.48550/arXiv.quant-ph/0612146arXiv:quant-ph/0612146v1J.Åberg, Quantifying superposition, arXiv:quant- ph/0612146v1 (2006). Measuring the quality of a quantum reference frame: the relative entropy of frameness. G Gour, I Marvian, R W Spekkens, 10.1103/PhysRevA.80.012307Phys. Rev. A. 8012307G. Gour, I. Marvian, and R. W. Spekkens, Measuring the quality of a quantum reference frame: the relative entropy of frameness, Phys. Rev. A 80, 012307 (2009). Does nonlinear metrology offer improved resolution? Answers from quantum information theory. M J W Hall, H M Wiseman, 10.1103/PhysRevX.2.041006Phys. Rev. X. 241006M. J. W. Hall and H. M. Wiseman, Does nonlinear metrology offer improved resolution? Answers from quantum information theory, Phys. Rev. X 2, 041006 (2012). On measures of entropy and information. A Rényi, Proceedings of the 4th Berkeley Symposium on Mathematics. J. Neymanthe 4th Berkeley Symposium on MathematicsBerkeley, MSA; Berkeley, MSAUniversity of California PressA. Rényi, On measures of entropy and information, in Proceedings of the 4th Berkeley Symposium on Mathe- matics, Statistics and Probability, Berkeley, MSA, 20-30 June 1961; ed. J. Neyman (University of California Press, Berkeley, MSA, 1961), pp. 547-561. On quantum Rényi entropies: a new generalization and some properties. M Müller-Lennert, F Dupuis, O Szehr, S Fehr, M Tomamichel, 10.1063/1.4838856J. Math. Phys. 54122203M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum Rényi entropies: a new generalization and some properties, J. Math. Phys. 54, 122203 (2013). Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy. M M Wilde, A Winter, D Yang, https:/link.springer.com/article/10.1007/s00220-014-2122-xCommun. Math. Phys. 331M. M. Wilde, A. Winter, and D. Yang, Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative en- tropy, Commun. Math. Phys. 331, 593-622 (2014). Sandwiched Rényi divergence satisfies data processing inequality. S Beigi, 10.1063/1.4838855J. Math. Phys. 54122202S. Beigi, Sandwiched Rényi divergence satisfies data pro- cessing inequality, J. Math. Phys. 54, 122202 (2013). Monotonicity of a relative Rényi entropy. R L Frank, E H Lieb, 10.1063/1.4838835J. Math. Phys. 54122201R. L. Frank and E. H. Lieb, Monotonicity of a relative Rényi entropy, J. Math. Phys. 54, 122201 (2013). Spectral functions of a symmetric operator. M A Naimark, Izv. Akad. Nauk SSSR, Ser. Mat. 4M. A. Naimark, Spectral functions of a symmetric opera- tor, Izv. Akad. Nauk SSSR, Ser. Mat. 4, 277-318 (1940). Probabilistic and statistical aspects of quantum theory. A Holevo, Scuola Normale Superiore. Pisa, Italy2nd ed.. Sec. 2.5A. Holevo, Probabilistic and statistical aspects of quantum theory, 2nd ed. (Scuola Normale Superiore, Pisa, Italy, 2011), Sec. 2.5. A Peres, Quantum theory: concepts and methods. Dordrecht, The Netherlands; SecKluwerA. Peres, Quantum theory: concepts and methods (Kluwer, Dordrecht, The Netherlands, 1993), Sec. 9-6. The broader class of Naimark extensions with the projection P replaced by an isometry. 35] is not considered hereThe broader class of Naimark extensions with the pro- jection P replaced by an isometry [35] is not considered here. Note that a Naimark extensionT for the trine observable T discussed following Corollary 5 is obtained by adding an additional orthogonal basis element |0⟩ to the qubit Hilbert space and defining |j⟩ := (2/3) 1/2 |m j ⟩ + (1/3) 1/2 |0⟩,Tj := |j⟩⟨j| and P =1 − |0⟩⟨0|, where |m j ⟩ denotes the unit eigenstate of σ · m j. for j = 0, 1, 2 [37Note that a Naimark extensionT for the trine observ- able T discussed following Corollary 5 is obtained by adding an additional orthogonal basis element |0⟩ to the qubit Hilbert space and defining |j⟩ := (2/3) 1/2 |m j ⟩ + (1/3) 1/2 |0⟩,Tj := |j⟩⟨j| and P =1 − |0⟩⟨0|, where |m j ⟩ denotes the unit eigenstate of σ · m j for j = 0, 1, 2 [37]. Entropic timeenergy uncertainty relations: an algebraic approach. C Bertoni, Y Yang, J M Renes, https:/iopscience.iop.org/article/10.1088/1367-2630/ab9ee5/metaNew J. Phys. 2283010C. Bertoni, Y. Yang, and J. M. Renes, Entropic time- energy uncertainty relations: an algebraic approach, New J. Phys. 22 083010 (2022). . V F R Jones, 311Von Neumann algebras, available at Course Lecture NotesV. F. R. Jones, Von Neumann algebras, available at Course Lecture Notes (2015), Chapters 3, 11. . Another Choice Is S = {jx, Jy}, Noting For X ∈ S ′ That, Iℏ, X, Jz] = [X, [Jx, Jy]] = 0 via the Jacobi identityAnother choice is S = {Jx, Jy}, noting for X ∈ S ′ that iℏ[X, Jz] = [X, [Jx, Jy]] = 0 via the Jacobi identity. Formulation of the uncertainty relations in terms of the Rényi entropies. I Bialynicki-Birula, 10.1103/PhysRevA.74.052101Phys. Rev. A. 7452101I. Bialynicki-Birula, Formulation of the uncertainty rela- tions in terms of the Rényi entropies, Phys. Rev. A 74, 052101 (2006). Rényi divergence and Kullback-Leibler divergence. T Van Erven, P Harremoës, IEEE Trans. Inf. Theory. 60T. van Erven and P. Harremoës, Rényi divergence and Kullback-Leibler divergence, IEEE Trans. Inf. Theory 60, 3797-3820 (2014). The monotonic decreasing property of Hα with respect to α follows for a finite number of outcomes via Eqs. (2) and Theorem 3 of [44]. More generally, for α < β note that f (x) := x (α−1)/(β−1) is convex (concave) for α < 1 (α > 1). yielding via definition (27) thatThe monotonic decreasing property of Hα with respect to α follows for a finite number of outcomes via Eqs. (2) and Theorem 3 of [44]. More generally, for α < β note that f (x) := x (α−1)/(β−1) is convex (concave) for α < 1 (α > 1), yielding via definition (27) that X|ρ) = (1 − α) −1 log x p(x|ρ)f (p(x|ρ) β−1 ) ≥ (1 − α) −1 log f ( x p(x|ρ) β ) = H β (X|ρ). Hα, Hα(X|ρ) = (1 − α) −1 log x p(x|ρ)f (p(x|ρ) β−1 ) ≥ (1 − α) −1 log f ( x p(x|ρ) β ) = H β (X|ρ). Information-theoretical aspects of quantum measurement. A S Holevo, Probl. Peredachi Inf. 9A. S. Holevo, Information-theoretical aspects of quantum measurement, Probl. Peredachi Inf. 9, 31-42 (1973). Majorization entropic uncertainty relations. Z Pucha La, L Rudnicki, K Życzkowski, https:/iopscience.iop.org/article/10.1088/1751-8113/46/27/272002J. Phys. A. 46272002Z. Pucha la, L. Rudnicki, and K.Życzkowski, Majoriza- tion entropic uncertainty relations, J. Phys. A 46, 272002 (2013). Universal uncertainty relations. S Friedland, V Gheorghiu, G Gour, 10.1103/PhysRevLett.111.230401Phys. Rev. Lett. 111230401S. Friedland, V. Gheorghiu, and G. Gour, Universal un- certainty relations, Phys. Rev. Lett. 111, 230401 (2013). Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information. M J W Hall, H M Wiseman, https:/iopscience.iop.org/article/10.1088/1367-2630/14/3/033040New J. Phys. 1433040M. J. W. Hall and H. M. Wiseman, Heisenberg-style bounds for arbitrary estimates of shift parameters includ- ing prior information, New J. Phys. 14, 033040 (2012). Entropic Heisenberg limits and uncertainty relations from the Holevo information bound. M J W Hall, https:/iopscience.iop.org/article/10.1088/1751-8121/aad50fJ. Phys. A. 51364001M. J. W. Hall, Entropic Heisenberg limits and uncer- tainty relations from the Holevo information bound, J. Phys. A 51 364001 (2018). Uncertainty relations from simple entropic properties. P J Coles, R Colbeck, L Yu, M Zwolak, 10.1103/PhysRevLett.108.210405Phys. Rev. Lett. 108210405P. J. Coles, R. Colbeck, L. Yu, and M. Zwolak, Uncer- tainty relations from simple entropic properties, Phys. Rev. Lett. 108, 210405 (2012). Extremal properties of the variance and the quantum Fisher information. G Tóth, D Petz, https:/journals.aps.org/pra/abstract/10.1103/PhysRevA.87.032324Phys. Rev. A. 8732324G. Tóth and D. Petz, Extremal properties of the variance and the quantum Fisher information, Phys. Rev. A 87, 032324 (2013). Quantum Fisher information as the convex roof of variance. S Yu, arXiv:1302.5311EprintS. Yu, Quantum Fisher information as the convex roof of variance, Eprint: arXiv:1302.5311 (2013). Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices. G Tóth, F Fröwis, https:/journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.013075Phys. Rev. Research. 413075G. Tóth and F. Fröwis, Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices, Phys. Rev. Research 4, 013075 (2022). Informational power of quantum measurements. M Dall&apos;arno, G M D&apos;ariano, M F Sacchi, 10.1103/PhysRevA.83.062304Phys. Rev. A. 8362304M. Dall'Arno, G. M. D'Ariano, and M. F. Sacchi, Infor- mational power of quantum measurements, Phys. Rev. A 83, 062304 (2011). Lower bound for accessible information in quantum mechanics. R Jozsa, D Robb, W K Wootters, 10.1103/PhysRevA.49.668Phys. Rev. A. 49668R. Jozsa, D. Robb, and W. K. Wootters, Lower bound for accessible information in quantum mechanics, Phys. Rev. A 49, 668 (1994).
[]
[ "Mathematics Subject Classification. Primary 11K16, 11N60; Secondary 11N56", "Mathematics Subject Classification. Primary 11K16, 11N60; Secondary 11N56" ]
[ "Verónica Becher ", "Agustín Marchionna ", "Gérald Tenenbaum " ]
[]
[]
Given an integer b 2 and a set P of prime numbers, the set T P of Toeplitz numbers comprises all elements of [0, b[ whose digits (an) n 1 in the base-b expansion satisfy an = apn for all p ∈ P and n 1. Using a completely additive arithmetical function, we construct a number in T P that is simply Borel normal if, and only if, p ∈P 1/p = ∞. We then provide an effective bound for the discrepancy.
null
[ "https://export.arxiv.org/pdf/2304.06850v2.pdf" ]
258,170,078
2304.06850
1b3396db0ddf22b7a7b812bfcd8741c36390245e
Mathematics Subject Classification. Primary 11K16, 11N60; Secondary 11N56 April 12. 2023. 2020 Verónica Becher Agustín Marchionna Gérald Tenenbaum Mathematics Subject Classification. Primary 11K16, 11N60; Secondary 11N56 April 12. 2023. 2020and phrases normal numbersToeplitz sequencesdiscrepancyadditive and multiplicative functions Given an integer b 2 and a set P of prime numbers, the set T P of Toeplitz numbers comprises all elements of [0, b[ whose digits (an) n 1 in the base-b expansion satisfy an = apn for all p ∈ P and n 1. Using a completely additive arithmetical function, we construct a number in T P that is simply Borel normal if, and only if, p ∈P 1/p = ∞. We then provide an effective bound for the discrepancy. Let P denote the set of prime numbers, and let P ⊂ P. Following Jacobs and Keane's definition of Toeplitz sequences in [4], we define the set T P of Toeplitz numbers as the set of all real numbers ξ ∈ [0, b[ whose base-b expansion ξ = n 1 a n /b n satisfies a n = a np (n 1, p ∈ P). For example, 0.a 1 a 2 a 3 . . . is a Toeplitz number for P = {2, 3} if, for every n 1, we have a n = a 2n = a 3n . Then, a 1 , a 5 , a 7 , a 11 , . . . are independent while a 2 , a 3 , a 4 , a 6 , . . . are completely determined by earlier digits. As defined byÉmile Borel, a real number is called simply normal to the integer base b 2 if every possible digit in Z/bZ occurs in its b-ary expansion with the same asymptotic frequency 1/b. A real number is said to be normal to the base b if it is simply normal to all the bases b j , j 1. Borel proved that, with respect to the Lebesgue measure, almost all numbers are normal to all integer bases at least equal to 2. For a presentation of the theory of normal numbers see for example [3,5]. In [1], Aistleitner, Becher and Carton considered the notion of Borel normality under the assumption of dependencies between the digits of the expansion. Thus [1, th. 1] states that, given any integer base b 2 and any finite subset P of the primes, almost all numbers, with respect to the uniform probability measure on T P , are normal to the base b. In the particular case P = {2}, they show [1, th. 2] that almost all elements in T P (still with respect to the uniform measure on T P ) are normal to all integer bases greater than or equal to 2. For P = {2}, a construction of an explicit number in T P that is normal to the base 2 appears in [2]. This construction can be generalized to any integer base b and any singleton P. Let Ω P denote the completely additive arithmetical function defined by Ω P (p) = 1 (P P) (p). Then, Ω P (n) is the sum of the exponents in the canonical factorization of n of those prime factors that do not belong to P. For n 1 and b 2, let a n = a n,b denote the representative of the congruence class Ω P (n) + bZ lying in [0, b[. Thus, given b 2, the real number (1) ξ P = n 1 a n /b n is a an element of T P . Motivated by the question posed in [1] on how to exhibit a normal number in T P for any set P of primes, we construct in this note simply normal numbers for arbitrary bases and a large family of sets P. Theorem. Let P ⊂ P, Q := P P, and let b be an integer 2. The number ξ P is simply normal to the base b if, and only if, (2) p∈Q 1/p = ∞. Moreover, defining, for 0 k < b, ε N,k := 1 N |{1 n N : a n = k}| − 1 b , E(N ) := p N, p∈Q 1 p (N 1), we have (3) ε N,k ≪ e −2E(N )/9b 2 . Our proof rests on the following auxiliary result where we use the traditional notation e(u) := e 2πiu (u ∈ R). Lemma. Let P ⊂ P and let b be an integer 2. The number ξ P is simply normal to the base b if, and only if, So, if (2) holds, then the above lemma implies that ξ P is simply normal to the base b. Notice that {a ∈ Z : |a| 1 2 b} describes a complete set of residues (mod b). The effective bound (3) is then provided by (5). If, on the contrary, condition (4) fails, we apply [7, cor. 2.2], which is an effective version of a result of Delange-see [6,th. III.4.4]. We have (6) p∈Q, p N log p p ≪ η N log N for some η N → 0. A possible choice is η N := min 1 z N log z log N + p∈Q, p>z 1 p . The validity of (6) is then obtained by bounding log p by log z if p z and by log N otherwise. That η N = o(1) follows by noticing that the last sum tends to 0 as z → ∞. Then we get S(N ; a/b) = N log N   p p ν N e(νaΩ P (p)/b) p ν + O η 1/8 N e E(N ) + e E(N ) (log N ) 1/12   , where we are picking the corresponding values from [7, cor. 2.2] as a = 1/8, b = 1/12, and ̺ = 1. To prove that Since the general factor of the last product equals 1 + {e(a/b) − 1}/p + O(1/p 2 ), we deduce from the convergence of p∈Q 1/p and Mertens' formula that σ N ∼ c log N for some c = 0. This yields (7) as required. aΩ P (n)/b = o(1) (a = 1, 2, . . . b − 1, N → ∞). Proof. The necessity of the criterion is clear. We show the sufficiency. Define b k,N = 1 N {1 n N : a n = k} (0 k < b, (4) all inner sums with a = 0 contribute o(N ).We may now embark on the proof of the Theorem. LetS(N ; a/b) := n N e aΩ P (n)/b (a ∈ Z, b 2, N 1).Weaim at necessary and sufficient conditions that ensure S(N, a/b) = o(N ) as N → +∞, and seek effective upper bounds for S(N ; a/b) when such conditions are met. Whenever a and b are coprime, b 2 and |a| b/2, we may apply [7, cor. 2.4(i)] with r = 1, z = e(a/b), ϑ = 2πa/b and κ = 1. Using [7, (7.4)], from which the bound [7, (2.19)] is actually derived, this yields S(N ; a/b) ≪ N e −2a 2 E(N )/(9b 2 ) . have put ν p := 1 + ⌊(log N )/ log p⌋, so that p νp N . Now the double product above Normal numbers with digit dependencies. Christoph Aistleitner, Verónica Becher, Olivier Carton, Trans. Amer. Math. Soc. 3726Christoph Aistleitner, Verónica Becher, and Olivier Carton. Normal numbers with digit dependencies. Trans. Amer. Math. Soc., 372(6):4425-4446, 2019. Finite-state independence. Verónica Becher, Olivier Carton, Pablo Ariel Heiber, Theory Comput. Syst. 627Verónica Becher, Olivier Carton, and Pablo Ariel Heiber. Finite-state independence. Theory Comput. Syst., 62(7):1555-1572, 2018. Distribution Modulo One and Diophantine Approximation. Yann Bugeaud, Series: Cambridge Tracts in Mathematics 193. Cambridge University PressYann Bugeaud. Distribution Modulo One and Diophantine Approximation. Series: Cambridge Tracts in Mathematics 193. Cambridge University Press, 2012. 0 − 1-sequences of Toeplitz type. Konrad Jacobs, Michael Keane, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. 13Konrad Jacobs and Michael Keane. 0 − 1-sequences of Toeplitz type. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 13:123-131, 1969. Uniform distribution of sequences. Lauwerens Kuipers, Harald Niederreiter, Pure and Applied Mathematics. John Wiley & SonsLauwerens Kuipers and Harald Niederreiter. Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Introduction to analytic and probabilistic number theory. Gérald Tenenbaum, Graduate Studies in Mathematics. 163American Mathematical Societythird editionGérald Tenenbaum. Introduction to analytic and probabilistic number theory, volume 163 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, third edition, 2015. Moyennes effectives de fonctions multiplicatives complexes. Gérald Tenenbaum, The Ramanujan Journal. 443The Ramanujan JournalGérald Tenenbaum. Moyennes effectives de fonctions multiplicatives complexes. The Ramanujan Journal, 44(3):641-701, 2017. Correction in: The Ramanujan Journal 53:1:243-244, 2020.
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[ "Invertible Kernel PCA with Random Fourier Features", "Invertible Kernel PCA with Random Fourier Features" ]
[ "Submitted ", "Letters ", "No ", "Xx ", "February " ]
[]
[]
Kernel principal component analysis (kPCA) is a widely studied method to construct a low-dimensional data representation after a nonlinear transformation. The prevailing method to reconstruct the original input signal from kPCA-an important task for denoising-requires us to solve a supervised learning problem. In this paper, we present an alternative method where the reconstruction follows naturally from the compression step. We first approximate the kernel with random Fourier features. Then, we exploit the fact that the nonlinear transformation is invertible in a certain subdomain. Hence, the name invertible kernel PCA (ikPCA). We experiment with different data modalities and show that ikPCA performs similarly to kPCA with supervised reconstruction on denoising tasks, making it a strong alternative.
10.1109/lsp.2023.3275499
[ "https://export.arxiv.org/pdf/2303.05043v1.pdf" ]
257,427,610
2303.05043
935bc99bb36956a0a0fe009bd3d92f465b79c9c7
Invertible Kernel PCA with Random Fourier Features Submitted Letters No Xx February Invertible Kernel PCA with Random Fourier Features 1Index Terms-DenoisingECGkernel PCApre-imageran- dom Fourier featuresreconstruction Kernel principal component analysis (kPCA) is a widely studied method to construct a low-dimensional data representation after a nonlinear transformation. The prevailing method to reconstruct the original input signal from kPCA-an important task for denoising-requires us to solve a supervised learning problem. In this paper, we present an alternative method where the reconstruction follows naturally from the compression step. We first approximate the kernel with random Fourier features. Then, we exploit the fact that the nonlinear transformation is invertible in a certain subdomain. Hence, the name invertible kernel PCA (ikPCA). We experiment with different data modalities and show that ikPCA performs similarly to kPCA with supervised reconstruction on denoising tasks, making it a strong alternative. I. INTRODUCTION P RINCIPAL Component Analysis (PCA) involves finding a projection matrix P that transforms a given input x ∈ R p into a lower-dimensional representation z = P x ∈ R d , with d < p. Conversely, given a lower-dimensional representation, the original input space can be reconstructed with the inverse transformationx = P z. The data x is often assumed to lie on a low-dimensional manifold. In such cases, PCA is beneficial since it enables the extraction of the most important features or directions of maximum variability in the data. The algorithm is optimal [1] in the sense that there is no reconstruction matrix U and reduction matrix V such that the average distance between the original and reconstructed vector x−U V x 2 is smaller than for U = P and V = P . Importantly, the matrix P serves both as a tool for dimensionality reduction and as a means for reconstructing the original input through its transpose P . Kernel PCA (kPCA) builds upon traditional PCA by enabling the study of the principal components after a nonlinear transformation [2]. This allows for the generalization of the assumption that the data lies on a low-dimensional linear manifold to cases where this manifold is nonlinear. Traditional PCA might not be capable of retrieving useful low-dimensional representations z in this scenario, but kPCA might succeed by using PCA after a nonlinear transformation Φ of the input x into a (possibly infinite-dimensional) feature space F z = P Φ(x). (1) Manuscript submitted 27 February 2023. This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation; by Kjell och Märta Beijer Foundation; and by the Swedish Research Council (VR) via the project Physics-informed machine learning (registration number: 2021-04321) All authors are with the Department of Information Technology, Uppsala University, 751 05 Uppsala, Sweden (e-mails: {daniel.gedon, antonio.horta.ribeiro, niklas.wahlstrom, thomas.schon} @it.uu.se) W x + b σ P P σ −1 + W † (α − b) x αᾱ β zβαx kPCA is indeed a natural and valuable idea. However, while the dimensionality reduction can be easily computed it is far from obvious how to obtain a reconstructedx from z. This inverse reconstruction problem is known as the preimage problem. Solutions are proposed based on gradient descent [3], nonlinear optimization [4] or distance constraints in feature space [5]. The most widely disseminated solution by Bakır et al. [6], is to apply (1) to construct a data set D = {(x i , z i )} n i=1 consisting of original inputs and their lowdimensional representations. The goal is to find a nonlinear function f that maps z i back to x i . This approach is available, for instance, in scikit-learn [7] or the multivariate statistics package for Julia [8]. However, there are drawbacks to this approach: Unlike PCA, reconstruction is not an immediate byproduct of kPCA and instead requires solving a supervised learning (SL) problem. Here, we denote this combination as kPCA+SL. Moreover, since the function f needs to be nonlinear, the supervised problem of finding the map between z and x usually results in a non-convex optimization problem. Indeed, direct nonlinear approaches-such as autoencoders [9], [10] and variational autoencoders [11], [12]-that concurrently implement dimensionality reduction and reconstruction, can yield significantly improved performance over kPCA+SL. While deep autoencoders are popular components of generative models, they require solving a non-convex optimization problem. Contrarily, kernel methods and PCA are wellunderstood and widely adopted preprocessing steps. We propose a new formulation of kPCA that provides the reconstruction method as a direct by-product. The method works for any translational-invariant kernel. As we will discuss in Section II, any such kernel can be approximated by a feature map of the type Φ(x) = σ(W x + b) with a nonlinearity σ and W ∈ R r×p . Here, the dimensionality is reduced by the following sequence of computations α = W x + b,(2a)β = σ(α),(2b)z = P β.(2c) The method we propose involves inverting the operations step-by-step, as depicted in Fig. 1. If a particular operation σ(α) cannot be inverted, we decompose the vector α into two arXiv:2303.05043v1 [cs. LG] 9 Mar 2023 components α−ᾱ andᾱ, such that α−ᾱ belongs to a domain where σ is invertible. We can use PCA to compress and decompress the first component, while the second component is bypassed. In this way, we avoid any nonlinear supervised problem and the reconstruction follows directly. II. BACKGROUND Let, k(x, y) be a positive semidefinite kernel k(x, y) = Φ(x), Φ(y) = i φ i (x)φ i (y),(3) where Φ(x) = (φ 1 (x), φ 2 (x), . . . ) denotes a sequence of values that maps the input into the feature space F. A. Kernel PCA For a set of observations {x i } n i=1 the empirical covariance matrix in F × F is given bŷ Σ = 1 n n i=1 Φ(x i )Φ(x i ) . The spectral decomposition of this matrix yieldŝ Σ = i λ i v i v i , such that λ 1 ≥ λ 2 ≥ . . . . We define the projection into the first d components as P = v 1 · · · v d . For which we can obtain the lower dimensional representation z = P Φ(x). B. Infinite dimensional feature maps In practice, the kernel trick enables working with feature spaces of infinite dimension. The method we propose here is however intended for finite-dimensional feature spaces. Hence, when dealing with infinitely dimensional feature maps, we will resort to approximations. Specifically, we will use the feature map truncated to the first r components denoted as Φ(x) = (φ 1 (x), φ 2 (x), . . . , φ r (x)) to approximate the kernel, meaning that we can write k(x, y) ≈ Φ(x), Φ(y) . We follow the development of [13] using random Fourier features to approximate a translation-invariant kernel, i.e. kernels of the form k(x, y) = g(x − y). Bochner theorem guarantees that this kernel is continuous and positive semidefinite iff g(δ) is the Fourier transform of a probability distribution p(ω), possibly re-scaled. Take as an example the Gaussian kernel k(x, y) = exp − x−y 2 2 2 which allows for the decomposition (3) only when considering an infinite dimensional feature space. To approximate this features space with random Fourier features, let W ∈ R r×p be a matrix with random i.i.d. entries drawn from the distribution p(ω) and let b ∈ R r be a vector drawn i.i.d. from U(−π, π). Then, we obtaiñ Φ(x) = √ 2 sin(W x + b), where sin is applied element-wise. It is proved in [13] that, Φ (x),Φ(y) converges uniformly to k(x, y). Moreover, the convergence is exponentially fast in r. III. INVERTIBLE KERNEL PCA Let us consider feature maps of the type Φ(x) = σ(W x + b), where W ∈ R r×p , b ∈ R r and σ is a nonlinearity applied element-wise. The discussion in the previous section motivates how these feature maps can be used to approximate the space associated with any translational-invariant kernel. Next, we detail how to invert the operations, given that the dimensionality reduction was computed according to (2). One of the key challenges is the fact that the activation function σ is in general non-invertible. We describe our solution to deal with these problems next. A. Non-invertible activation functions In most cases of interest, the nonlinear function σ : R → R, σ : α → β is not invertible in the entire domain R, but it might be invertible in a subdomain X ⊂ R. Denote σ X as the function σ restricted to X , then the inverse σ −1 X is well- defined. Letᾱ = α−σ −1 X •σ(α). Consider two examples: First, for the ReLU activation function β = σ(α) = max(α, 0), the invertible domain is X = [0, ∞). Thus, σ −1 X (β) = β and α = min(α, 0). Second, for β = σ(α) = sin α, as used in random Fourier features, we have that σ is invertible in X = (−π/2, π/2]. Thus, σ −1 X (β) = arcsin β andᾱ = (−1) k α+πk for some k ∈ Z. B. ikPCA The reconstruction method inverts the operations in (2) stepby-step. We can write β = P z, (4a) α = σ −1 X ( β) +ᾱ, (4b) x = arg x min W x + b − α 2 2 + λ x 2 2 . (4c) The first step (4a) inverts the dimensionality reduction, and is motivated by the same reasoning as PCA: the projection matrix P is such that the reconstruction error β −P z 2 is minimal. The second step (4b), inverts the nonlinear function σ on the subdomain X and adds the bypassed non-invertible partᾱ. Finally, the last step (4c) inverts the linear map x → W x + b by solving a Ridge regression problem. Notice that for λ → 0 + the last step reduces to Fig. 1. x = W † ( α − b), where W † is the pseudo-inverse of W , as in IV. NUMERICAL EXAMPLES In this section, we outline the experiments to evaluate the performance of the proposed ikPCA method. We focus on the task of denoising inputs of various modalities. Quantitatively, we evaluate the mean square error (MSE) between the denoised test signal and the true non-noisy test signal. We compare ikPCA with PCA and kPCA+SL due to their structural similarity. To ensure a fair comparison, we did not consider denoising autoencoders, which present hierarchical, deep models. In the discussion, we detail how our methodology could be extended to neural networks and a setup that could be better compared with autoencoders. In all experiments, we consider the Gaussian kernel (or its random Fourier feature approximation) and present the results in terms of mean and standard deviation over 20 random runs. Parameters of the methods which are fixed in an experiment were optimized through hyperparameter grid search. For reproducibility, we release our code publicly 1 . A. Synthetic toy data: s-curve We generated synthetic 3-dimensional data points in the shape of the letter 'S' using the s-curve toy problem. For training and testing, we generate n = 2, 000 data points and add Gaussian noise with σ = 0.25. For kPCA+SL we set the kernel width γ = 1 and the reconstruction Ridge strength λ = 1; for ikPCA we set γ = 0.5 and λ = 1. The results are presented in Fig. 2. Our proposed ikPCA method is capable of denoising the data in a comparable manner to kPCA+SL. For this problem, we observe that as few as r = 50 random Fourier features were sufficient to match the performance of kPCA+SL. This provides ikPCA with a computational advantage over kPCA+SL, which needs to invert a n × n matrix. B. USPS Images We utilize the USPS data set which contains handwritten digits in a greyscale format of size 16 × 16 and add Gaussian noise with σ = 0.5. We use n = 1, 000 images for training and 400 images for testing. For kPCA+SL we set γ = 5 · 10 −3 and λ = 10 −2 ; for ikPCA we use 30, 000 random Fourier features and set γ = 10 −4 . In Fig. 3, we vary the regularization parameter λ of ikPCA. Again, our findings show that ikPCA performs similarly to kPCA+SL for an optimal number of principal components d. The figure further suggests that Ridgeless reconstruction behaves comparably in performance to PCA. However, excessive regularization negatively affects the overall reconstruction performance. each method such that the MSE is minimised. All methods demonstrate visually comparable image denoising capabilities, which is supported by the difference in MSE from Fig. 3. C. Electrocardiogram The electrocardiogram (ECG) is a routine, medical test that records the heart's electrical activity, typically used to diagnose various heart conditions. However, noise measured during the recording can complicate the diagnosis. Several methods have been proposed to de-noise the ECG signal. For comparisons, two approaches have been suggested: (1) artificially adding noise to the signal and comparing with the original one itself [14]- [16], or (2) de-noise the existing signal and comparing it to the mean beat as the noise-free reference [17], [18]. We choose the latter approach to account for real-world noise scenarios. We utilize ECGs from the China Physiological Signal Challenge 2018 (CPSC) 2 which contains data between 6 and 60 seconds long [19]. From the 918 ECGs with no abnormalities, we selected the longest recordings and focused on a single lead in this example. To extract the beats, we first remove baseline wander with a high-pass filter. Then, we identify the R-peaks [20], resample the interval between each peak to 512 samples and finally locate the R-peak at the 150th sample following the preprocessing approach of Johnstone et al. [18]. We extracted 70 beats from the selected ECG; 49 for training and 21 for testing. Applying kPCA+SL with γ = 10 and λ = 15, and ikPCA with γ = 5 · 10 −5 and λ = 10, along with the minimum of 512 random Fourier features r, we achieved perfect signal denoising using only the first component, as shown in Fig. 5. Quantitatively over 500 simulations, the MSE for ikPCA was 2.6 ± 0.8 · 10 −5 , similar to that of kPCA+SL, while PCA had a slightly higher MSE. V. COMPUTATIONAL CONSIDERATIONS The computational complexity is not increased by adding the reconstruction stage for ikPCA. The reason is that the cost of obtaining the reconstruction is smaller than the cost of the kPCA decomposition (whenever the input dimension p is lower than the number of samples n). However, our method requires the kernel map to be approximated by r random Fourier features. When r < n this might reduce the computational cost, but when r > n the computational cost is increased by a factor of r/n + 1 compared to that of kPCA. a) kPCA computational cost: Some kernels have closed forms that can be computed in O(1) operations. The cost for kPCA is then dominated by the inversion of the Gram matrix K which requires O(n 3 ) operations. The Gram matrix being the matrix with entry (i, j) equal to k(x i , x j ). b) Computation cost of PCA in the feature space: In ikPCA we approximate the kernels with finite, r-dimensional features Φ(x) = (φ 1 (x), . . . , φ r (x)), and perform PCA on the covariance matrix of the featuresΣ = 1 n n i Φ(x i )Φ(x i ) . Computing the entries of the matrix and its spectral decomposition requires O(r 2 n + r 3 ) operations. Hence, for r < n, approximating the kernel and computing the spectral decomposition ofΣ might be computationally more efficient than working directly with the Gram Matrix K as in kPCA. However, if r > n this advantage is diminished and it can be efficient to work with the Gram matrix K = Φ(X) Φ(X) instead ofΣ. K has the same (nonzero) eigenvalues as nΣ, and its eigenvectors multiplied by Φ(X) yield the eigenvectors of nΣ. The cost in this formulation is O(rn 2 +n 3 ). Therefore, the cost is a factor of r/n + 1 times higher than the cost obtained for kernels with a closed-form solution. According to Claim 1 in [13], r = Ω p 2 log D random Fourier features are required to ensure an approximation error smaller than on a space of diameter D. Thus, r grows linearly with the input dimension p. In the case of the s-curve example, r n, whereas in the USPS example, r > n, due to large p and our method's computational advantage is lost. For highdimensional data like the latter, Nyström approximations [21] could be used and be more efficient in terms of r [22]. c) Cost of reconstruction: The reconstruction cost in ikPCA is dominated by the cost of solving the optimization problem (4c). For this, we require computing the SVD of W one single time with a cost of O(p 3 +p 2 r). The cost of solving the reconstruction is then O(pr + dr) for each newα. VI. CONCLUSION AND DISCUSSION We propose an invertible version of kPCA+SL. While the traditional approach solves a supervised problem to map back from the latent space to the input space, our method obtains this mapping naturally. We approximate the kernel transformation with random Fourier features Φ(x) = σ(W x + b). Although the nonlinear function σ might not be invertible, we observe that it can be inverted in a subdomain. We can exploit this observation by decomposing its input into invertible and non-invertible parts and bypassing the second. We show the effectiveness of our approach for denoising in three examples: an s-curve toy problem, the USPS image data set and ECGs. We compare our method with symmetric kPCA+SL. Symmetry implies here that the kernel for compression and reconstruction are defined identically, which is motivated by implementations in common frameworks [7], [8]. However, the method in [6] is not limited to this by design. Conversely, ikPCA is required to have a symmetric setup due to the natural inversion of the nonlinear transformation in the reconstruction. While our method aligns well with kPCA+SL in the numerical experiments we presented, it remains uncertain how it would compare against a well-tuned non-symmetric kPCA+SL. Despite the simplicity of our method, there is a wide array of possible extensions. To extend the representational power, we can stack multiple layers of Φ(x) = σ(W x + b) transformations in a hierarchical way. Hence, we obtain a structure which is closer to that of a deep autoencoder. This may allow drawing further connections between the theoretically well-established kernel regime and neural networks. In a similar direction, we can view the random Fourier features in our method as an untrained, single-layer neural network. Extending our method to trained neural networks would allow performing reconstruction tasks naturally without re-training. Finally, we experiment with underparameterized data (USPS example with p/n ≈ 0.25) and overparameterized data (ECG example with p/n ≈ 10). This fact, combined with the use of a high number of random Fourier features, raises questions about overparameterization and benign overfitting of denoising models [23], [24]. ACKNOWLEDGMENT The computations were enabled by the supercomputing resource Berzelius provided by National Supercomputer Centre at Linköping University and the Knut and Alice Wallenberg foundation. APPENDIX A. Additional results on s-curve data set The s-curve is generated by the following set of equations where the variable t is often used as a label 3 . For our purposes, we do not require labels but are only concerned with inputs x. Additive Gaussian noise v ∼ N (0, σI 3 ) is added to x. t ∼ U − 3 2 π, 3 2 π , x 1 = sin t, x 2 ∼ U (0, 2) , x 3 = sign(t) (cos t − 1) . Fig. A-1 shows a visualisation of the s-curve data set. Fig A-2 is an extension of Fig 2 for a larger set of random Fourier features r and for a second level of additive noise. For this data set, fewer random Fourier features are necessary than training data points. Hence, our method is numerically faster. Already r = 500 features (less than 1/4 of the number of samples n = 2, 000) are sufficient for the optimal performance curve. We observe that for larger noise values, ikPCA even outperforms kPCA+SL and PCA and that the effect of r is less pronounced. B. Additional results on USPS data set For the following plots, the hyperparameters for kPCA+SL (i.e. kernel width γ and regularization strength λ) were selected such that the lowest reconstruction MSE was achieved. A grid search was utilized. For all results mean (and in error plots also standard deviation) over 20 random runs are presented. Fig. A-3 explores the effect of the number of random Fourier features r for this data set. We observe that generally more random Fourier features yield asymptotically better results. Furthermore, we note that our method ikPCA approaches kPCA+SL for r → ∞ as suggested by the approximation of the kernel. Fig. A-4 explores the effect of the regularization parameter λ for the reconstruction in our ikPCA method. We observe that an optimal trade-off has to be found. For λ → 0 + , ikPCA approaches the performance of PCA. Conversely, for large values of λ, the problem becomes over-regularized and does not generalize anymore. Fig. A-6. We observe that a larger noise value leads to a lower number of optimal principal components d, which is justified as the noise level dominates a larger portion of singular values. Fig. A-6 subsequently shows the MSE values of all three methods when choosing the optimal number of principal components d. We observe that the MSE for optimal tuned methods in this data set is similar for all methods and noise levels. C. Additional results on ECG data Fig. A-8 shows two more examples of reconstructing ECG signals, complementing Fig. 5. The same hyperparameters as in the main text are chosen. In the right plot, we can see that for PCA some reconstructions (red dashed lines) are not optimal, i.e. close to the ground truth line. This leads to a significantly higher MSE. Both kPCA+SL and ikPCA perform similarly. Tab. A-1 compares the MSE values over 500 simulations with different train/test splits for the three ECG traces in Fig. A-8. We observe that ikPCA and kPCA+SL perform similarly in terms of MSE, while PCA has a slightly higher MSE. Hence, ECG denoising is not as good with a purely linear model. Fig. 1 : 1Illustration of our invertible kernel PCA method. Fig. 2 : 2S-curve toy example. Reconstruction MSE for a different number of random features chosen for ikPCA. Fig. 4 Fig. 3 :Fig. 4 : 434displays image denoising results for noise scale σ = 0.25. For ikPCA we chose the optimal regularization λ = 1.3. The number of principal components d is chosen for1 Code is available at https://github.com/dgedon/invertible kernel PCA USPS data. Effect of regularization parameter λ. USPS reconstruction. Fig. 5 : 5Denoising of ECG beats from lead I. The blue area marks the min/max values of the 21 test beats. The red dashed lines show all test reconstructions with ikPCA. Fig. A- 1 : 1Visualization of the s-curve data set. The colour indicates the regression label t. σ = 0.5, samples n = 2, 000 kPCA+SL ikPCA, r = 50 ikPCA, r = 100 ikPCA, r = 500 ikPCA, r = 1, 000 ikPCA, r = 5, 000Fig. A-2: Effect of the number of random features components on reconstruction MSE. Fig. 2 is a modified version of the left figure here. Fig . A-5 shows the combined effect of the additive noise level and the number of principal components d chosen for the latent space. The number of components with the lowest MSE for each method is shown in the left plot of Fig . A-7 is a reconstruction of USPS images for two different noise levels when choosing optimal hyperparameters for all methods. As the quantitative comparison in the right plot ofFig. A-6suggests, the reconstructions are also qualitatively similar. ikPCA, r = 5, 000ikPCA, r = 10, 000 ikPCA, r = 20, 000 ikPCA, r = 30, 000Fig. A-3: Effect of different number of random feature components on reconstruction MSE. FigFig. A- 5 : 5. A-4: Effect of regularization parameter λ on reconstruction MSE. An optimal value has to be chosen. Right plot is a repetition ofFig. 3.Combined effect of noise and latent space dimension on reconstruction MSE. of noise. (Left) The best number of components to achieve the lowest MSE for a certain noise level. (Right) MSE of the three methods for reconstruction choosing the optimal number of components. original, nosiy PCA kPCA+SL ikPCA (a) Noise level σ = 0.25 original, nosiy PCA kPCA+SL ikPCA (b) Noise level σ = 0.5 Fig. A-7: Reconstruction with different methods. Optimal hyperparameters were chosen for each method to achieve the lowest MSE. Left plot is a repetition of Fig. 4. II of an ECG consisting of 40 beats (28/12 for training/test). Fig. A- 8 : 8More ECG reconstruction results.Fig. A-8ais the same example as inFig. 5. PCA TABLE A - A1: Reconstruction MSE for different ECGs. Data is available at http://2018.icbeb.org/Challenge.html See also https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make s curve.html Understanding machine learning: From theory to algorithms. S Shalev-Shwartz, S Ben-David, Cambridge university pressS. Shalev-Shwartz and S. Ben-David, Understanding machine learning: From theory to algorithms. Cam- bridge university press, 2014. Kernel principal component analysis. B Schölkopf, A Smola, K.-R Müller, International Conference on Artificial Neural Networks. B. Schölkopf, A. Smola, and K.-R. 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[ "https://github.com/dgedon/invertible" ]
[ "Quantum Cosmology with Dynamical Vacuum in a Minimal-Length Scenario", "Quantum Cosmology with Dynamical Vacuum in a Minimal-Length Scenario" ]
[ "M F Gusson \nUniversidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil\n", "A Oakes \nUniversidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil\n", "O Gonçalves \nUniversidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil\n", "R G Furtado \nUniversidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil\n", "J C Fabris \nUniversidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil\n", "J A Nogueira \nUniversidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil\n" ]
[ "Universidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil", "Universidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil", "Universidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil", "Universidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil", "Universidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil", "Universidade Federal do Espírito Santo -Ufes Vitória\n29.075-910Espírito SantoBrasil" ]
[]
In this work we consider the effects on a quantum cosmology model of dynamical vacuum due to presence of a minimal length introduced by the GUP (generalized uncertainty principle) related to the modified commutation relation [x,p] := i 1−βP 2 . We determine the wave function of the Universe ψ qp (ξ, t), which is solution of the Modified Wheeler-DeWitt equation in the representation of the quase-position space, in the limit where the scale factor of the Universe is small. Thus, physical results can be obtained directly from ψ qp (ξ, t).16 Note that solution is non-normalized. 17 Remember that b := 3π 4 , Eq. (17).
10.1140/epjc/s10052-021-09114-8
[ "https://arxiv.org/pdf/2012.09158v1.pdf" ]
229,220,743
2012.09158
a87bce054f03c22195f2a7192def0887fa81083a
Quantum Cosmology with Dynamical Vacuum in a Minimal-Length Scenario 16 Dec 2020 M F Gusson Universidade Federal do Espírito Santo -Ufes Vitória 29.075-910Espírito SantoBrasil A Oakes Universidade Federal do Espírito Santo -Ufes Vitória 29.075-910Espírito SantoBrasil O Gonçalves Universidade Federal do Espírito Santo -Ufes Vitória 29.075-910Espírito SantoBrasil R G Furtado Universidade Federal do Espírito Santo -Ufes Vitória 29.075-910Espírito SantoBrasil J C Fabris Universidade Federal do Espírito Santo -Ufes Vitória 29.075-910Espírito SantoBrasil J A Nogueira Universidade Federal do Espírito Santo -Ufes Vitória 29.075-910Espírito SantoBrasil Quantum Cosmology with Dynamical Vacuum in a Minimal-Length Scenario 16 Dec 2020numbers: 9880Qc, 0460-m, 0450Kd, Keywords: Minimal lengthgeneralized uncertainty principlequantum gravityquantum cos- mologydynamical vacuum In this work we consider the effects on a quantum cosmology model of dynamical vacuum due to presence of a minimal length introduced by the GUP (generalized uncertainty principle) related to the modified commutation relation [x,p] := i 1−βP 2 . We determine the wave function of the Universe ψ qp (ξ, t), which is solution of the Modified Wheeler-DeWitt equation in the representation of the quase-position space, in the limit where the scale factor of the Universe is small. Thus, physical results can be obtained directly from ψ qp (ξ, t).16 Note that solution is non-normalized. 17 Remember that b := 3π 4 , Eq. (17). Introduction The Standard Cosmological Model (SCM) is based on an expanding universe which has a very hot and dense origin. The extrapolation of the SCM for very early times leads to energy scales deep in the Planck regime and, strictly speaking, the expanding universe would have a singular initial state. On the other hand, in order to explain many features of the observed universe today, a period of short but rapid, almost exponential, expansion, which is called inflationary phase, is necessary in the very early universe. So, in the primordial Universe the quantum effects were relevant and a quantum approach is necessary to describe the gravitational effects. A quantum approach to gravity is also necessary in the study of black holes. The features of compact astrophysical objects like neutron stars, and even white dwarfs, may be affected by quantum gravity effects. However, we have no quantum gravity theory which is fully acceptable, although there are many proposal for it. An important aspect of almost all proposals for quantization of the gravity is the prediction of the existence of a minimal length. Thus, an effective description of the effects of quantum gravity or a phenomenological approach can be obtained by way of the introduction of a minimal-length scenario [1,2,3,4]. In addition, the introduction of a minimal length could also solve the problems of the initial singularity (big bang singularity) [5,6,7,8,9,10,11,12,13], of the black hole complete evaporation (catastrophic evaporation of a black hole) [3,14,15,16,17,18], of the cosmological constant [12,19,20,21], of the Chandrasekhar limit [4,18,20], of the instability of the Einstein static universe (emergent universe) [22], of the cosmological constant vanishes in multiverse theories [19], and so on. We can obtain a minimal-length scenario by modifying the Heisenberg uncertainty principle (HUP) between the position and momentum operators. There are several different proposals of modification of the HUP which introduce a minimal-length scenario. Those modified uncertainty principles are called Generalized Uncertainty Principle (GUP) 1 . GUP's have been derived from different contexts such as string theory [27,28,29,30,31,32], black hole physics [19,33,34,35] and extra dimensions [36]. Although the most common modified or deformed commutation relation (MCR) associated with a GUP (KMM GUP [37]) is quadratic in the momentum operator, effects of a MCR with a linear term in the momentum operator, which leads to a minimal length and a maximum momentum [38], has been studied in contexts of Cosmology models [10,11,39,40,41] but, nevertheless, apparently this GUP leads to non-unitary theories [40] and it is not compatible with current data available [42]. In the most the inflation is described by a scalar field. So, in an inflationary quantum cosmology theory it is necessary to performed the second quantization of that scalar field. Consequently, in a minimallength scenario that scalar field and its conjugate momentum have to obey to modified commutation relations [43,44,45,46]. As it is well known, the ordinary approach of the third quantization 2 of the Wheeler-DeWitt (WDW) equation leads to the vanishing of the cosmological constant. However, it might be possible to obtain a non-zero cosmological constant in minimal-length scenario [19]. Many authors have studied effects of a minimal length in cosmology using a classical approach in which the Poisson brackets are modified according to the correspondence principle [10,22,47,48,49,50,51]. In the vast majority of the GUP's the parameter related to the minimal length (deformation parameter) is positive. Although GUP's with positive deformation parameter prevent black holes evap-orate completely they remove the Chandrasekhar limit 3 , that is, a white dwarf star could turn out arbitrarily large [52,53]. In [18,20] the author shows that a GUP with a negative deformation parameter can restore the Chandrasekhar limit and in spite of allowing a black hole evaporates completely this takes an infinite amount of time. Even though a negative deformation parameter is unusual, it is consistent with a description in which the universe has an underlying crystal lattice-like structure [54]. Mu-In Park proposed that the HUP can be modified to include the cosmology constant term, which is called extend uncertainty principle (EUP) [35,55]. Whereas in a GUP there is a quadratic term in the momentum uncertainty, a EUP has a quadratic term in the position uncertainty. A GUP that has also a quadratic term in the position uncertainty is called GEUP. In [4] the authors use a GEUP to show that a non-zero cosmological constant can restore the Chandrasekhar limit. Effects of a GEUP in cosmology has been studied in [56]. Modified uncertainty principles that induce a maximum length has been employed in order to describe the cosmological particle horizon [57]. This modified uncertainty principle has terms proportional to even powers in the position uncertainty [58]. Cosmological observational data have been used to obtain constraints for GUP parameters [21,42,48,56,59]. Even though the de Sitter Cosmology Model describes the rapidly expansion phase of the Universe, during which vacuum energy dominates, its traditional treatment considers a system without physical content, since there is only single degree of freedom and one constraint [60]. Schutz formalism [61,62], which describes a relativistic fluid interacting with the gravity field, can be used to regard the vacuum as dynamic entity having different degrees of freedom. Besides this process overcomes the difficulties found in de Sitter Model, it allows a natural way of introducing a variable playing the role of time. Then, the Dynamical Vacuum Model is a model of homogeneous and isotropic universe, filled with vacuum fluid whose state equation is P = −ρ, classically treated according to the Schutz canonical formalism. This process rends a linear Hamiltonian in one of the momenta whose associated degree of freedom of the fluid will play the role of time [63]. In general, with the intention of avoiding the problems caused by the Wheeler-DeWitt equation defined in the superspace 4 (the space of all possible three dimensional metrics), the process of quantization occurs into the mini-superspace approach [64], where an infinite number of degrees of freedom of the gravity field is frozen 5 and remaining degrees of freedom are turned operators. Of course, if the GUP is a fundamental aspect of the nature then not only the position and the momentum have to obey the GUP as well as every others variables which will be quantized [65]. Since a GUP corresponds to a modification of the commutation relation between the operator and its conjugate operator, the Wheeler-DeWitt equation in a minimal-length scenario can be obtained imposing that some or all of those operators (which have come from the quantization of remaining degrees of freedom in the mini-superspace approach) and its conjugate momentum operators satisfy a modified com-mutation relations. In this work, our primary purpose is to determine the corrections in the wave function of the Universe due to the use of a specific generalized uncertainty principle (GUP) in a quantum cosmology model. Hence, we determine the modified Wheeler-DeWitt Equation up to O(β 2 ) considering a quantum cosmology model of dynamical vacuum in a minimallength scenario induced by the commutation relation proposal by P. Pedram [66,67], [X,P ] := i 1 − βP 2 , which induces a minimal uncertainty in the position, ∆x min = 3 √ 3 4 √ β, and an upper bound on the conjugate momentum, P max = 1 √ β . In above equation β is a parameter related to the minimal length. An important aspect of the GUP associated with this modified commutation relation is that it is not perturbative (in the minimal length), consistent with Doubly Special Relativity (DSR) theories which predicts an upper bound for particle momentum [59,68,69,70] and it is in agreement with several theories proposals of quantum gravity [66]. Then, we find the wave function of the Universe for small scale factors in the formal representation of the "position" space, in fact representation of the scale-factor space 6 . However, we can not obtain from that wave function any physical information since the eigenstates of the "position" operators are not physical states [37]. We overcome this problem for obtaining the wave function in the representation of the quase-position space as a superposition of wave functions in the formal representation of the "position" space [71]. Since it may be questioned whether this procedure is correct, we find the Wheeler-DeWitt equation in the representation of the quase-position space and show that the wave function in the quase-position space previously obtained is solution this equation, as we expected. Last but not least, it should be said that the quantization process occurs into the mini-superspace scenario in which the scale factor is the only degree of freedom. As far as we know, the first applications of a minimal-length scenario to a mini-superspace dynamics can be found in [5,7,8]. In short, in this paper we intend to study a dynamical vacuum, using a description given by the Schutz formalism, in a quantum cosmological scenario where the minimallength proposal is explicitly considered. Our main aim is to verify how GUP affects the solutions for the WDW equation with a dynamical vacuum state, in the mini-superspace, in comparison with the usual approach. The paper is organized in the following way. In Section 2, we describe the minimallength scenario used, presenting its main results. In Section 3, we describe the ordinary cosmology model, that is, in a scenario without minimal length. We obtain the gravity action and the fluid action according to the Schutz formalism. In Section 4, we determine the modified Wheeler-DeWitt equation in the formal representation of the "position" space and we find its solution for small values of the scale factor. We also discuss about the validity range at which we may consistently work. Then, we find the physically acceptable solutions, that is, the wave function of the Universe in the representation of the quaseposition space. In Section 5, we present our comments and conclusions. Minimal-Length Scenario In a quantum approach, a minimal length can be introduced by modifying the HUP in order to raise a non-zero minimal uncertainty in the position 7 . There are many proposals for modification of HUP [14,37,38,50,57,66,67,73]. In this work we concern with the GUP ∆x∆p ≥ 2 1 − β [(∆p) 2 + p 2 ] ,(1) proposed by P. Pedram [66,67], which induces a non-zero minimal uncertainty in the position given by ∆x min = 3 √ 3 4 β,(2) where β is a parameter related to the minimal length. We choose the GUP (1) because it is consistent with several proposals for quantum gravity, such as string theory, loop quantum gravity, and it also introduces a maximal measured momentum 8 , P max = 1 √ β ,(3) which is in agreement with the DSR. Another important aspect this GUP is that it is not perturbative. Consequently, if β is small then (1) can be expanded until any order in β we wish to do. Since, ∆x∆p ≥ 1 2 [X,P ] ,(4) then related to the GUP (1) we have the modified commutation relation, [X,P ] := i 1 − βP 2 .(5) Although, the representation of the operators: X =x,(6)P =p + β 3p 3 + β 2 3p 5 + 4β 3 9p 7 + . . . ,(7) 7 It is no trivial to show that a non-zero minimal uncertainty in position can be interpreted as a minimal length [23,37,72]. 8 It is worth noting that this GUP does not induce a maximum uncertainty in the conjugate momentum. wherex andp are the ordinary operators of position and momentum satisfying the canonical commutation relation [x,p] := i , is not an exact representation of the algebra (5), it preserves the ordinary form of the position operator. Thus, in this representation of "position" space we have x|X|ψ(t) = xψ(x, t),(8) x|P |ψ(t) = −i ∂ ∂x + i 3 β 3 ∂ 3 ∂x 3 − i 5 β 2 3 ∂ 5 ∂x 5 + . . . ψ(x, t),(9) where |x are the state eigenvectors of the position operator. This representation of "position" space is only formal since the eigenvalues of the position operator (6) are not physical states and consequently they do not belong to the Hilbert space. This is because the position operator uncertainty vanishes when it is calculated in any of its eigenstates. But that is physically impossible since ∆x ≥ 3 √ β for all physically allowable state in a minimal-length scenario. However, all information on position can be accessible through the maximal localization states, defined as ψ ml ξ |X|ψ ml ξ = ξ(10) and (∆x) |ψ ml ξ = (∆x) min .(11) In the DGS (Detournay, Gabriel and Spindel) approach [74] the maximal localization states are found to be 9 ψ ml ξ (p) = 2 √ 3 2 exp − i ξp cos 3π 4 βp .(12) The representation of quase-position space is obtained by projecting the state vectors onto the maximal localization states, ψ ml ξ |ψ(t) = ψ qp (ξ, t).(13) The action of the position and the momentum operators on the quase-position space are given by ψ ml ξ |X|ψ(t) = ξ + ib β tan −ib β ∂ ∂ξ ψ qp (ξ, t)(14) and ψ ml ξ |P |ψ(t) = ψ ml ξ | p + β 3p 3 + β 2 3p 5 + . . . |ψ(t) ,(15) with ψ ml ξ |p|ψ(t) = −i ∂ψ qp (ξ, t) ∂ξ ,(16) and b := 3π 4 . Hence, up to O(β 2 ) we have ψ ml ξ |X|ψ(t) = ξ + βb 2 ∂ ∂ξ − β 2 b 4 3 ∂ 3 ∂ξ 3 ψ qp (ξ, t),(18) and ψ ml ξ |P |ψ(t) = −i ∂ ∂ξ + i 3 β 3 ∂ 3 ∂ξ 3 − i 5 β 2 3 ∂ 5 ∂ξ 5 ψ qp (ξ, t).(19) Finally, one can show that the wave function in the quase-position space is a superposition of the wave functions in the "position" space, given by ψ qp (ξ) = 1 √ 2 ψ(ξ + b β) + ψ(ξ − b β) .(20) The Model We are going to consider the de Sitter Cosmology Model, which describes the phase of the Universe rapidly expanding, during which the vacuum energy dominates the energy density and gives rise the term corresponding to the cosmological constant Λ. Gravity Action The gravity action more general is given by 10 S g = M d 4 x √ −g(R − 2Λ) + 2 ∂M d 3 x √ hK,(21) where R is the Ricci curvature scalar, K is the trace of the extrinsic curvature K ij , g is the determinant of the metric g µν , h is the determinant of the induced metric over the three-dimensional spatial hypersurface and ∂M is the bounded of the four-dimensional manifold M. According to the Cosmological Principle, that is, for the homogeneous and isotropic universe the metric is that of FLRW (Friedmann-Lemaître-Robertson-Walker), ds 2 = −N 2 (t)dt 2 + a 2 (t) dr 2 (1 − kr 2 ) + r 2 dθ 2 + r 2 sin 2 θdφ 2 ,(22) where N(t) is the lapse function, k = −1, 0, +1 for hyperbolic open, flat open and close universe, respectively, and a(t) is the scale factor of universe. Using the FLRW metric, Eq. (22), the gravity action becomes S g = dt − 6ȧ 2 a N + kNa − 2Na 3 Λ .(23) Since S g = dtL g we have that the gravity Lagrangian is given by L g = − 6ȧ 2 a N + kNa − 2Na 3 Λ.(24) Hence the gravity Hamiltonian, NH g =ȧp a − L g , reads H g = − p 2 a 24a − ka + 2Λa 3 ,(25) where p a = ∂L g ∂ȧ = − 12ȧa N .(26) Fluid Action In this Model the Universe is fully completed by a perfect fluid. Now, we are going to employ the Schutz's formalism [61,62] for the dynamics description this fluid interacting with the gravitational field [63]. For the fluid whose the state equation is p = αρ,(27) where ρ is the density and α depends on the type of fluid, the pression is given by p = α µ α + 1 1+ 1 α e − S α ,(28) where µ is the specific enthalpy and S is the entropy. In the Schutz's formalism the four-velocity is defined by means of six potentials: U ν := 1 µ (φ, ν +ǫη, ν +θS, ν ) .(29) Since ǫ and η are associated with rotation movement, they vanish. From the normalization condition, U ν U ν = −1,(30) we get µ = 1 N φ + θṠ .(31) The fluid action is given by S F = M d 4 x √ −gp.(32) Using Eq. (31) into Eq. (28) and because the model is homogeneous and isotropic, we have S F = dtNa 3 α φ + θṠ N(α + 1) 1+ 1 α e − S α .(33) Thus, the fluid Lagrangian is L F = αa 3 N 1 α (1 + α) 1+ 1 α φ + θṠ 1+ 1 α e − S α .(34) The conjugate canonical momenta to φ and S are p φ = ∂L F ∂φ = a 3 φ + θṠ 1 α e − S α N 1 α (1 + α) 1 α(35) and p S = ∂L F ∂Ṡ = θp φ .(36) Therefore, the fluid Hamiltonian, NH F =φp φ +Ṡp S − L F , is given by H F = p 1+α φ e S a 3α .(37) From Eqs. (36) and (25), it follows that the total Hamiltonian, H = H g + H F , is H = − p 2 a 24a − ka + 2Λa 3 + p 1+α φ e S a 3α .(38) Now, performing the canonical transformation (re-parametrization) T := −p S e −S p −(1+α) φ ,(39) and p T := p 1+α φ e S ,(40) the total Hamiltonian becomes H = − p 2 a 24a − ka + 2a 3 Λ + p T a 3α .(41) Note that p T is linear in Eq. (41). When the quantization occurs, this allows us to consider t = −T as playing role of time and to obtain an equation like Schroedinger. The Quantum Cosmology Model in a Minimal-Length Scenario The modified Wheeler-DeWitt Equation The quantization process takes place in the framework of mini-superspace and according with the Wheeler-DeWitt quantization scheme, p a −→p a(42) and p T −→p T ≡ −i ∂ ∂T ,(43) such that H −→Ĥ and thus satisfying the Wheeler-DeWitt equation, H|ψ = 0.(44) In order to obtain a minimal-length scenario we demand that [â,p a ] := i 1 − βp 2 a .(45) with the representation of the operators as follow: a =x (46) andp a ≡p + β 3p 3 + β 2 3p 5 + O β 3 ,(47) where thex andp operators satisfy the canonical commutation relation, [x,p] := i .(48) Using the representation (46) and (47), the modified Wheeler-DeWitt equation turns out −x (3α−1) 24 p 2 + 2 3 βp 4 + 7 9 β 2p6 + O β 3 − kx (1+3α) + 2Λx 3(1+α) |ψ(t) = − i ∂ ∂t |ψ(t)(49) where we have made T = −t. Now, projecting Eq. (50) onto the formal representation of the "position" space, we have x|Ĥ|ψ(t) = 0,(50)− x (3α−1) 24 − ∂ 2 ∂x 2 + 2 3 β ∂ 4 ∂x 4 − 7 9 β 2 ∂ 6 ∂x 6 + O β 3 − kx (1+3α) + 2Λx 3(1+α) ψ(x, t) = = −i ∂ ∂t ψ(x, t),(51) where we have used Eqs. (8) and (9). Assuming that the solution of the above equation can been written as ψ(x, t) = ϕ(x)τ (t), we found the stationary solutions ψ ω (x, t) = e −iωt ϕ ω (x),(52) and the time independent equation d 2 dx 2 − 2 3 β d 4 dx 4 + 7 9 β 2 d 6 dx 6 + O β 3 ϕ ω (x) + 24x (1−3α) ω − 24kx 2 + 48Λx 4 ϕ ω (x) = 0.(53) Lastly, considering a flat universe (k = 0), without any other matter content, fully completed by a dynamical vacuum with state equation P = −ρ,(54) and Λ = 0 11 , Eq. (53) up to O (β 2 ) becomes 12 d 2 ϕ dx 2 − 2 3 β d 4 ϕ dx 4 + 7 9 β 2 d 6 ϕ dx 6 + 24ωx 4 ϕ = 0.(55) Solution of the modified Wheeler-DeWitt equation It is no possible to solve Eq. (55) analytically since we do not know all initial or boundary conditions 13 , that is, ϕ (N ) (x 0 ), for N = 0, 1, ..., 5. A way to get around this problem is considering x small. This is well reasonable since we are interested in solutions describing the Universe in its quantum regime, that is, in the initial phases of the Universe, when the scale factor was small 14 , and as it is known quantum effects are significant only for small values of the scale factor. Initially, we take into account Eq. (55) in the limit β = 0, which we named ordinary Wheeler-DeWitt equation, d 2 ϕ 0 dx 2 + 24ωx 4 ϕ 0 = 0.(56) 11 Note that if Λ > 0 we can always redefine ω′ := ω + 2Λ. 12 From now on, for the sake of simplicity we are going to omit the subscript ω. 13 In Ref. [2] the author has solved a modified Wheeler-DeWitt equation of forth order d 4 dx 4 using the Sommerfeld polynomial method. However, the solution holds dependent on four parameters that must be determined by the initial conditions, which are not known. 14 Later on, we will discuss more carefully the issue if x can rigorously describe the scale factor. Making ϕ 0 = √ xy(x) and the change of variable ρ := 8ω 3 x 3 , Eq. (55) turns the Bessel equation, d 2 y dρ 2 + 1 ρ dy dρ + 1 − 1 6 2 ρ 2 y = 0.(57) Then, the solution of Eq. (57) is ϕ 0 (x) = A √ xJ 1/6 K 0 x 3 + B √ xN 1/6 K 0 x 3 ,(58) where K 0 := 8ω 3 , J 1/6 (X) and N 1/6 (X) are the Bessel and the Neumann functions of order 1/6, respectively, and A and B are constants. It is easy to see that in the limit for small x J 1/6 (X) ≈ √ x(59) and N 1/6 (X) ≈ 1 √ x .(60) Now, imposing the condition 15 that ϕ 0 (0) = 0 we get B = 0, and the solution is given by ϕ 0 (x) = A √ xJ 1/6 K 0 x 3 .(61) Remembering that J ν (X) = N =0 ∞ (−1) N N!Γ(N + ν + 1) X 2 2N +ν ,(62) and retaining the first three significant terms of ϕ 0 (x) we get ϕ 0 (x) = C 1 x + C 2 x 7 + C 3 x 13 .(63) It is appropriate we rewrite the solution (63) absorbing C 1 in a normalization constant, which we are going to omit without affecting the results, ϕ 0 (x) = x − 4 7 ωx 7 + 8 91 ω 2 x 13 .(64) We can obtain an approximate solution for Eq. (55), in the range of x small, using Eq. (64) into O (β) and O (β 2 ) terms. Thus, d 4 ϕ 0 dx 4 = −480ωx 3 + 10560 7 ω 2 x 9(65) and d 6 ϕ 0 dx 6 = −2880ωx + 760320 7 ω 2 x 7 .(66) With the approximation ϕ ≈ x into Eqs. (65) and (66), Eq. (55) becomes d 2 ϕ dx 2 + 24ωx 4 ϕ + β 320ωx 2 ϕ − 7040 7 ω 2 x 8 ϕ + β 2 −2240ωϕ + 591360ω 2 x 6 ϕ = 0. (67) Subsequently, we are going to find the approximate solution of the above equation in two different ways. Solution: first method Since x = 0 is an ordinary point of Eq. (67) we can find a solution in power series of x. Then, replacing ϕ(x) = ∞ n=0 a n x n into Eq. (67) and retaining only significant terms, we have ϕ(x) = x − 4 7 ωx 7 + 8 91 ω 2 x 13 + β −16Aωx 5 + 5504 385 Bω 2 x 11 + β 2 1120 3 Cωx 3 .(68) The parameters A, B and C are determined by requiring that (68) satisfies Eq. (67) up to order chosen in x. Then, substituting (68) into (67) we have A = 1, B = 38 43 and C = 3 7 . Therefore, the solution reads as ϕ(x) = x − 4 7 ωx 7 + 8 91 ω 2 x 13 + β −16ωx 5 + 4864 385 ω 2 x 11 + β 2 160ωx 3 .(69) Solution: second method The same previous solution can be obtained by taking the approximation following d 2 ϕ dx 2 = −24ωx 4 ϕ 0 + 2 3 β d 4 ϕ 0 dx 4 − 7 9 β 2 d 6 ϕ 0 dx 6 .(70) Integrating twice the above equation we get d 2 ϕ dx 2 = −24ω dx dxx 4 ϕ 0 + 2 3 β d 2 ϕ 0 dx 2 − 7 9 β 2 d 4 ϕ 0 dx 4 + α 1 x + α 2 ,(71) where α 1 and α 2 are integration constants. After that, using (64) into (71) we obtain ϕ(x) = α 2 + α 1 x − 4 7 ωx 7 + 8 91 ω 2 x 13 + β −16A ′ ωx 5 + 64 7 B ′ ω 2 x 11 + β 2 120 3 C ′ ωx 3 .(72) The constants α 1 and α 2 can be determined by demanding that ϕ(x) −→ ϕ 0 (x) when β −→ 0. In the similar way, the parameters A ′ , B ′ and C ′ are found requiring that the solution (72) satisfies Eq. (67). Consequently, we again have the solution (69), as we expected. Validity range In order that the solution (69) works consistently the first disregarded term in ϕ 0 (x) must be much smaller than the last kept terms in O(β) and O (β 2 ), that is, x ≪ β ω 1 8 . Since β is much small, that drastically reduces the validity range for our solution. We can improve this if terms of higher powers in x are taken account in order O (β 0 ) part of the solution. Therefore we can consider √ xJ 1/6 (K 0 x 3 ) as the part in O (β 0 ) of ϕ(x), that is, ϕ 0 (x) . It is clear that terms of order such that x 6n−2 < β 2 ω 1−n lie far outside the validity range and they must throw away. The same reasoning applies in the case of terms in O (β 2 ) when compared to terms in O(β). This way, with the purpose of increasing the validity range of our solution we take account the two first significant terms in O(β). In conclusion, in light of the forgoing the solution turns out 16 ϕ(x) = √ xJ 1/6 K 0 x 3 + β −16ωx 5 + 4864 385 ω 2 x 11 + β 2 160ωx 3 .(73) In dealing with expansions of small parameters (in this case β) it is necessary to take care to consistently work when x goes to an extremely small value. It is easy to see that terms in β 0 are like ω N x 1+6N , terms in β are like βω N x 5+6N and terms in β 2 are like β 2 ω N x 3+6N . Therefore, the terms in βωx 5 and βω 2 x 11 are negligible when x < √ β and the term βω 2 x 11 is negligible when √ β < x < β 1/8 . Note that the term in β 0 x always will be greater than the term in β 2 ωx 3 for x small. Keep in mind that in fact solution (73) is ϕ ω (x), that is, eigenfunctions whose associated eigenvalues are values of the cosmology constant. Physically acceptable solutions As we have already said, we can not obtain directly from ϕ(x) physical results because thex operator eigenstates do not belong to the Hibert space. Nevertheless, the projections of the state vectors |ϕ on the maximal localization states, that is, the wave functions in the representation of the quase-position space do. The wave functions in the quaseposition space can easily be obtained from (73) by using Eq. (20). Therefore, replacing (73) into (20) we have 17 ϕ qp (ξ) = ξJ 1/6 K 0 ξ 3 + β − 16 + 12b 2 ωξ 5 + 4864 385 + 48 7 b 2 ω 2 ξ 11 + β 2 160 − 160b 2 − 20b 4 ωξ 3 .(74) Hence, above quase-position wave function represents the probability amplitude for the Universe being maximally localized around the position ξ. The reader may be questioning if the correct is to determine the solution of the modified Wheeler-DeWitt equation in the representation of the quase-position space, ψ ml ξ | −p 2 a + 24ωâ 4 |ψ(t) = 0. Using Eqs. (18) and (19) dϕ qp dξ − 24ωξ 2 d 2 ϕ qp dξ 2 − 40ωξ 3 d 3 ϕ qp dξ 3 = 0.(76) As can easily be checked (after some algebra) the solution (74) satisfies Eq. (76) up to considered validity order. Conclusion In this work, we saw that the study of the primordial Universe requires a quantum approach, in which the gravitation effects must be taken in consideration. It follows the need to implement a minimal-length scenario, which is carried out by using a GUP. Since we chose the GUP (1) we can expand the representation of the momentum until O(β 2 ) and thus to obtain a modified WDW equation up to O(β 2 ), too. We found the modified WDW equation in the formal representation of "position" space, Eq. (55), and its solution ϕ(x), Eq.(73), because it is simpler than in the representation of quase-position space. However, we can not obtain directly from ϕ(x) physical results. Consequently we obtained the wave function the Universe in the representation of quaseposition space, ϕ qp (ξ), as a superposition of two wave functions of the Universe in the formal representation of "position" space, ϕ qp (ξ) = 1 √ 2 [ϕ(ξ + x min ) + ϕ(ξ − x min )]. With the aim of insuring our result, we found the modified WDW equation in the representation of quase-position space, Eq.(76), and we checked that ϕ qp (ξ) is actually its solution. The ignorance of the initial or boundary conditions, that is, of the derivatives of the wave function does not allow us to find an exact solution for modified WDW equation, what forced us to seek a solution for small values of the scale factor of the Universe. In next works, we are going to study the effects on the scale factor evolution, applications of the obtained results and comparisons with others models and GUP's. we obtaind 2 ϕ qp dξ 2 + 24ωξ 4 ϕ qp +βb 2 − 2 3b 2 d 4 ϕ qp dξ 4 + 144ωξ 2 ϕ qp + 96ωξ 3 dϕ qp dξ + β 2 b 4 − 7 9b 4 d 6 ϕ qp dξ 6 + 24ωϕ qp + 96ωξ Interested reader in finding out more about minimal-length literature may consult References[17,23,24,25,26].2 The 3rd quantization consists of initially considering the Wheeler-DeWitt equation as a classical equation of a classical field (the wave function of the Universe in the 2nd quantization) which then is turned an operator. This way universes can be created and annihilated in a multiverses theory. The maximum mass of a stable white dwarf star.4 As far as we know, there is no known general solution for Wheeler-DeWitt Equation in the superspace.5 In fact, the mini-superspace approach is an approximation in which are only considered the largest gravity field wavelength modes of the order of the size of the Universe. We say representation of the "position" space keeping in mind that it should be the more correct to call represetation of the scale-factor space. Note that it is in the representation of the ordinary momentum space. In units so that c = 1 and 16πG c 4 = 1. That condition is obtained by demanding that the Hamiltonian H must be self-adjoint[63]. That condition should not change, even though it is probably that boundary conditions satisfied by derivatives change. Note that condition does not avoid the initial singularity[8,9]. AcknowledgementsWe would like to thank FAPES, CAPES and CNPq (Brazil) for financial support. A note on theories with a minimal length. 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[]
[ "Black hole thermodynamics as seen through a microscopic model of a relativistic Bose gas", "Black hole thermodynamics as seen through a microscopic model of a relativistic Bose gas" ]
[ "Jozef Skákala \nSchool of Physics\nIndian Institute of Science, Education and Research (IISER-TVM)\n695016TrivandrumIndia\n", "S Shankaranarayanan \nSchool of Physics\nIndian Institute of Science, Education and Research (IISER-TVM)\n695016TrivandrumIndia\n" ]
[ "School of Physics\nIndian Institute of Science, Education and Research (IISER-TVM)\n695016TrivandrumIndia", "School of Physics\nIndian Institute of Science, Education and Research (IISER-TVM)\n695016TrivandrumIndia" ]
[]
Equations of gravity when projected on spacetime horizons resemble Navier-Stokes equation of a fluid with a specific equation of state[7][8][9]. We show that this equation of state describes massless ideal relativistic gas. We use these results and for Schwarzschild black hole we build an explicit and simple molecular model of the fluid. We work with spin zero Bose gas and we show our model makes two predictions: (i) The horizon area/entropy is quantized as given by Bekenstein's quantization rule, (ii) The model explains the correct type of proportionality between horizon area and entropy. The proportionality constant, is however, less than 1/4.
10.1142/s0218271816500474
[ "https://arxiv.org/pdf/1406.2477v2.pdf" ]
118,565,447
1406.2477
10a070516a803115ebdc8110a3a5320a97919779
Black hole thermodynamics as seen through a microscopic model of a relativistic Bose gas arXiv:1406.2477v1 [gr-qc] 10 Jun 2014 Jozef Skákala School of Physics Indian Institute of Science, Education and Research (IISER-TVM) 695016TrivandrumIndia S Shankaranarayanan School of Physics Indian Institute of Science, Education and Research (IISER-TVM) 695016TrivandrumIndia Black hole thermodynamics as seen through a microscopic model of a relativistic Bose gas arXiv:1406.2477v1 [gr-qc] 10 Jun 2014 Equations of gravity when projected on spacetime horizons resemble Navier-Stokes equation of a fluid with a specific equation of state[7][8][9]. We show that this equation of state describes massless ideal relativistic gas. We use these results and for Schwarzschild black hole we build an explicit and simple molecular model of the fluid. We work with spin zero Bose gas and we show our model makes two predictions: (i) The horizon area/entropy is quantized as given by Bekenstein's quantization rule, (ii) The model explains the correct type of proportionality between horizon area and entropy. The proportionality constant, is however, less than 1/4. I. INTRODUCTION There exists a popular viewpoint that general relativity, like fluid mechanics, could be seen as a thermodynamical equilibrium, macroscopic description of a system with completely different microscopic degrees of freedom. The idea traces back to Sakharov [1], however, in the last two decades there has been a surge of activities in this direction, see, for instance, Refs. [2][3][4][5]. Similar ideas also form the core of the analogue gravity programme (see, for instance, the review [6]). Three decades ago, Damour [7] showed that the equations of general relativity when projected on to a black-hole horizon give Navier-Stokes equation of a 2dimensional fluid that lives on the black hole horizon. This allowed Damour to identify various black hole properties with the properties of a 2-dimensional fluid. This approach has now been generalized to arbitrary spacetime horizons in recent works of Padmanabhan [8,9]. (For other results on fluid-gravity correspondence see Refs. [10][11][12].) In case the results of [7][8][9] are not only a curious analogy, one expects that also properties of semi-classical black holes could be holographically obtained from statistical mechanics of the (quantum) (2 + 1)-dimensional fluid living on the black hole horizon. In this work, we use results of Refs. [8,9] and build an explicit and simple molecular model of the (2+1)dimensional fluid. (For details on how the various fluid characteristics arise from the spacetime metric see the above references.) We show that the microscopic fluid model indeed provides a holographic analogue for different phenomena in semi-classical quantum gravity, bringing very interesting insights into the nature of various aspects of black hole thermodynamics. For example, the microscopic model presented here sheds some light on the proportionality between horizon area and entropy, and provides very interesting alternative explanation for the Bekenstein quantization of horizon area (or entropy) [13]. For the sake of simplicity, we consider in this work only the Schwarzschild black hole. * email: [email protected] † e-mail: [email protected] There have been attempts to build microscopic models describing physics of black hole horizons. (Microscopic models outside models of quantum gravity, such as String Theory, or Loop Quantum Gravity.) Specifically, in Ref. [14], the authors have used Bose-Einstein condensate as a microscopic model to describe black hole horizon as a surface of quantum phase transition 1 . In this work, we also use a model of Bose gas, however our approach and its origins are very different from the one in Ref. [14]. In this work we use the metric signature (−, +, +, +) and set G = c = = k B = 1. II. THE MOLECULAR MODEL FOR THE THERMODYNAMICS OF SCHWARZSCHILD BLACK HOLE Model set-up We start with the following general assumptions that link the microscopic theory of the fluid to black hole thermodynamics, (these assumptions are consistent with the original paper of Damour [7]): • since the fluid is a (2+1)-dimensional fluid living on a horizon, the volume of the fluid is naturally the area of the horizon A H , • the temperature of the horizon T H (with the usual normalization of the time-like Killing field) is the temperature of the fluid, • the total energy of the fluid is the black hole Komar energy. (Later, it will be required for the entropy of the fluid to be identified with the black hole entropy.) Damour-Navier-Stokes equation is obtained by projecting Einstein's equations on the horizon and it identifies the fluid's pressure with the horizon parameters as [8] p = κ 8π = T H 4 .(1) (κ is the horizon surface gravity and T H is the Hawking temperature of the horizon.) This result holds true for a general Killing horizon 2 [8] and is also shown to hold from purely thermodynamic reasons by identifying the temporal rate of change of Einstein gravity action with the spacetime entropy production [9]. The equation of state of the fluid (1) will be the starting point of this work. It is interesting to note that for the Schwarzschild black hole the above expression can be somewhat naively derived from the definition of pressure using the black hole entropy, p = ∂S H ∂A H · T H = T H 4 .(2) Eq. (2) boils down to the fact that for Schwarzschild black hole: dM = p · dA H = p · dV.(3) Now let us demonstrate that for a general Killing horizon the equation of state (1) translates into equation of state of an ideal relativistic fluid. The Komar energy of the Killing horizon is computed from E = − 1 8π H ξ a;b dΣ ab ,(4) where ξ is the suitably normalized Killing field generating the horizon. After short calculation this leads to E = T H V 2 .(5) Combining Eq.(1) and Eq.(5) one gets p = T H 4 = E 2V . This is an equation of state of an 2D ideal massless relativistic gas [15]. 2 For the Schwarzschild black-hole and in Schwarzschild radial coordinates, Damour-Navier-Stokes equation leads to the following simple equation p ,i = 0, where i labels the angular coordinates. This is a trivial way to express the zeroth law of black hole thermodynamics, which says that the black hole temperature (or equivalently the horizon surface gravity) is constant across the horizon. For fluid, this means that the pressure is constant across the fluid and also that the viscosity terms vanish, indicating that the fluid looks like an ideal fluid. Our aim is to identify a microscopic model for the Schwarzschild black hole using the fluid-gravity correspondence. For a microscopic theory defined by a Hamiltonian, the entropy is typically a function of three independent parameters S(E, N, V ), where E is system's energy, N is the number of particles and V is the volume. In what will follow we utilize some aspects of the microcanonical ensemble approach. For example the concept of temperature used in the following section will be defined as in the microcanonical ensemble, that is, knowing S(E, N, V ) we will fix N, V and define T = dE/dS. However, the parameter space of Schwarzschild black hole is one-dimensional and can be expressed by a single parameter V = A H (or E.) Therefore, if a microscopic model is supposed to reproduce Schwarzschild black hole, one needs to constrain the space of states of the fluid to only one dimension. This means from the point of view of the relevant states of the system we will not treat the variables E, N, V as independent, but they must fulfill two constrains: E = E(V ) and N = N (V ). We can obtain the constrains as follows: If we interpret the fluid energy as the Komar energy and the volume as the horizon area, they must fulfil the relation imposed by the Schwarzschild solution: V = 16πE 2 .(6) This means Eq.(6) provides the constrain E = E(V ). The second constraint N = N (V ) can be derived from the other well known equation, relating the mass of the black-hole and Hawking temperature: E = 1 8πT .(7) However, to derive N (V ) from Eq. (7), requires information about the microscopic model, and therefore it will be done later. At this stage all one needs to keep in mind is that the fluid's parameter space is fully constrained to one dimension by Eq.(6) and Eq. (7). Furthermore, the constraints mean that for a specific value of V only a specific combination of thermodynamical potentials p, µ, T is allowed. We have shown in the first half of this section that the fluid is an ideal massless relativistic gas. The relativistic gas can be either Bose or Fermi. In the rest of this work, we show that the microscopic model of a massless Bose gas living on a sphere predicts the form of N (V ) that leads to the Bekenstein's black-hole area quantization [13] and gives the correct proportionality between area and black hole entropy. The microscopic description (from the microcanonical point of view) Now let us have a better look at the consequences of the ideal gas model. The energy levels of a free nonrelativistic particles living on a sphere were calculated in [17,18] as: ǫ nr ℓ = ℓ(ℓ + 1) + α 2 2R 2 (8) where R is a radius of the sphere and α 2 is some constant. (In a strict sense there are differences between [17] and [18] on whether the constant is allowed to be non-zero. However any energy quantum spectrum can be always shifted by a constant, which in best case is fixed by gravity considerations, such as these, so physically we stick to the spectrum given by Eq. (8) with arbitrary α.) The energy levels correspond to the Laplacian on the sphere and therefore have the degeneracy of the spherical harmonics, given as g ℓ = 2ℓ + 1. As one can see from the Hamiltonian used in [17], we can obtain spectrum of a massless relativistic scalar particle by a simple transformation: ǫ r ℓ = √ 2ǫ nr = ℓ(ℓ + 1) + α 2 √ 2 · R = 4π{ℓ(ℓ + 1) + α 2 } V . (9) This can be (through our constrains) expressed as a simple function of temperature ǫ r ℓ = {ℓ(ℓ + 1) + α 2 } · T =ǫ ℓ · T,(10)withǫ ℓ = ℓ(ℓ + 1) + α 2 . (ǫ ℓ is independent on the black hole parameters.) For simplicity, in this work, we utilize the spectrum (10) and we model the ideal relativistic massless gas by particles with zero spin, which also means we work with Bose gas. We have constrained our system by equations (6) and (7). One of those equations gives the constraint E(V ), the other equation then implies another constraint N (V ). Let us now obtain the constraint N (V ): Note that in a microcanonical ensemble when T is much larger than spacing between the quantum energy levels, the mean particle's energy linearly decreases with temperature, as prescribed by the equipartition law 3 . It can be shown for a simple harmonic oscillator 4 that the equipartition law ceases to hold for very low temperatures, when the temperature is of a comparable value to the discrete spacing between the energy levels. Furthermore, for temperatures of a value comparable to the spacing between the energy levels the average particle's energy is very close to the ground state energy. From Eq.(10) one observes that, in our constrained system, the temperature always is of a comparable value to the spacing between the energy levels, which means the equipartition law is not applicable. This in turn implies that our free gas is supposed to have mean particle's energy very near the ground state. In such case we can use the spectrum (10) to see that the mean particle's energy must be: E = γ · T, where γ ≥ |α| and γ being approximately of the same order as |α|. Since E = NĒ = N γ · T ≡ (8πT ) −1 , this implies N = 1 8πγT 2 = A H 2γ . Therefore A H = 2γ · N.(11) Eq. (11) is the desired second constraint N (V ). This is one of the main results of this work, and we would like to stress the following points: • The derivation of N (V ) is based on few insights and presuppositions in the microcanonical ensemble approach. In the following section, we derive the same result more rigorously. In particular, in the next section, we will use the result given by Eq. (11) as an Ansatz and it will be shown that the Ansatz is correct in the sense that it fulfills all the constrains of the model. This means the relation (11) is a direct consequence and prediction of our microscopic model. • Eq. (11) is remarkable as it gives Bekenstein's [13] quantization of the black hole horizon area (since N is by definition a natural number). It gives a completely new and independent insight into Bekenstein's result. Furthermore, the insight does not rely on the quantum theory, only on the fluid interpretation of gravity. (The constant |α| is here arbitrary, but can be fixed to obtain the most popular form of Bekenstein type of spectrum as |α| = 4π. This fixing gives wave-length of the particle in the ground state equal to the circumference of the black hole horizon.) Note that the linear proportionality between the horizon area and the number of degrees of freedom is a starting assumption in the grandcanonical statistical analysis of Schwarzschild black hole in Ref. [19]. • The formula (11) has also a deeper meaning; it is problematic to speak about density of fluid's degrees of freedom, as to speak about density one needs pre-defined geometry. However, geometry is in this view a macroscopic property constituted by the fluid. In some sense it is natural to fix the way how the geometry is constituted by the fluid by relating it to the density of the fluid's degrees of freedom as: N A H = (2γ) −1 = const. . This also explains why in the fluid-gravity approach one obtains analogue of Navier-Stokes equation [8] (and energy diffusion equation given by the Raychhadhuri equation), but not the (rest-)mass conservation equation. The fact that the particles are massless, and the density of particles is always the same constant, means that mass conservation equation is fulfilled by a simple identity (its information rests only in how geometry links to the fluid particles). • One of the standard problems with black hole physics is the fact that black holes have negative heat capacity. This means for black holes one cannot define canonical partition function [20] and it seems black holes cannot be represented by well known microscopic models of a fluid. The key observation is that our model is very different: By fixing volume and the number of particles, and computing derivative of E(T ) one obtains positive heat capacity. However since our model is onedimensional constrained model in which change in temperature leads simultaneously to change in all fluid parameters (energy, number of particles and volume), the derivative of E(T ) does not correspond to a standard notion of heat capacity and can be negative. Rigorous statistical ensemble calculation One can confirm the insights from the last section through a more rigorous statistical ensemble calculation of our constrained system. The calculation is a hybrid between canonical and grand-canonical ensemble, therefore we refer it as statistical ensemble calculation. The number of particles is not kept fixed (which makes it look like grand-canonical ensemble), but the system is constrained to only one parameter (which we choose to be the energy), which makes it look as a canonical ensemble. We want to stress that our calculation is still semiclassical and therefore we assume our system to be in a sufficiently low temperature regime. (This is the regime of sufficiently large black holes where semi-classical description is reasonably close to the reality.) More precisely, the sufficiently low temperature regime means that T /T p << 1, where T p is Planck temperature. Since we use Planck units, T p = 1, and therefore in Planck units the condition of semi-classicality translates to T << 1. Let us start from the first principles and assume our system to be in a contact with a huge reservoir. Then the probability of a state with energy E for a Bose system is given by: p(E) ∼ Ω{n ℓ } exp − dS(E, V (E), N (E)) dE · E =(12)Ω{n ℓ } exp − ∂S(E, V (E), N (E)) ∂E · E − ∂S(E, V (E), N (E)) ∂V ∂V ∂E · E − ∂S(E, V (E), N (E)) ∂N ∂N ∂E · E = Ω{n ℓ } exp − E T + f (T ) ∂N ∂E · E = Ω{n ℓ } exp − E T + f (T ) 16π γ · E 2 . Here Ω{n ℓ } = ℓ (n ℓ + g ℓ − 1)! n ℓ !(g ℓ − 1)! and f (T ) = µ − 2γp T , with g ℓ being the degeneracy of the energy level and µ chemical potential. The variables in Eq.(12) are subject to two constrains ℓ ǫ ℓ n ℓ − E = 0,(13)and ℓ n ℓ − N = ℓ n ℓ − 8πE 2 γ = 0.(14) In constraint (14) we substituted for N a function of E using Eq.(11) and Eq.(6). For n ℓ >> 1 one can use Stirling's formula and transform Eq.(12) to: p(E) ∼ exp − E T + 16πf (T )E 2 γ + ℓ [(n ℓ + g ℓ ){ln(n ℓ + g ℓ ) − 1} − n ℓ {ln(n ℓ ) − 1} − ln([g ℓ − 1]!)] .(15) If we assume that the value ofĒ(T ) is given by the extremum of p(E), then we obtain using Lagrange multipliers the following equations ∂ ∂n ℓ ′ : ln n ℓ ′ + g ℓ ′ n ℓ ′ +λ · ǫ ℓ ′ + λ = 0, ∂ ∂E : − 1 T + 32πf (T )E γ −λ − 16πEλ γ = 0, ∂ ∂λ : ℓ ǫ ℓ n ℓ − E = 0, ∂ ∂λ : ℓ n ℓ − 8πE 2 γ = 0. Hereλ and λ are Lagrange multipliers corresponding to the two constraints (13) and (14). Take the first equation and substitute forλ from the second equation to obtain: n ℓ ′ = g ℓ ′ exp 16πEλ γ + 1 T − 32πf (T )E γ ǫ ℓ ′ − λ − 1 .(16) Let us further use the fact that E must be a function of temperature given by Eq. (7). Then equation (16) can be rewritten as n ℓ ′ = g ℓ ′ exp 2λ γ + 1 − 4f (T ) γ ǫ ℓ ′ − λ − 1 .(17) (ǫ ℓ ′ was defined before and is independent on external parameters.) Before we proceed with the rest of the calculations, it may be worth pointing out that the equation (17) has certain limitations. The limitations arise due to the use of Stirling's approximation and the fact that in general for higher occupation numbers (where the numbers are small) the above formula could give inaccurate results. Despite some quantitative modifications (due to changes in the asymptotic behavior ofn ℓ ), in most of the cases one can still expect that the exact results go qualitatively along similar lines as the ones derived via Eq. (17). For example constrains (13) and (14) could fix (slightly different) γ(α) and f (T ), however, the final conclusions can be expected to remain unchanged. A necessary condition for every occupation number is that it is positive for each level. This leads to the condition 2λ γ + 1 − 4f (T ) γ ≥ 0 ∀ T.(18) Now let us explore the possibilities of obtaining divergent number of particles for the asymptotic case T = 0. (This is a necessary condition that our model must fulfill and means that in the limiting case of zero temperature corresponds to a black hole with infinite horizon area and infinite mass.) One possibility would be that 2λ γ + 1 − 4f (0) γ = 0, and λ ≤ 0, which would lead to a constant finite non-zero distribution of particles over states. This would however contradict our assumption that the average particle's energy is close to the ground state. The only remaining possibility (how to obtain the infinite number of particles) is if the infinite number of particles are specifically in the ground state. This happens if: λ = (4f (0) − γ)|α| 2|α| − γ . One can then easily derive from Eq.(17) that: 0 < ∞ ℓ=1n ℓ < ∞ as T = 0.(19) where n 0 would be the occupation number of the ground state ℓ = 0. Since also ∞ ℓ=1ǫ ℓnℓ (the sum of energies above the ground state energy) clearly converges, this means γ = |α| and λ = 4f (0) − |α|.(20) It is important to realize that such a solution, represented by Eq. (20), can be always chosen for λ. This is because the only consistency conditions are the two constrains (13) and (14), in which constraint (13) only leads for T << 1 to our condition γ = |α| and the constraint (14) can always be fulfilled, since it just determines the correct choice of f (T ), which means the correct choice of the chemical potential. Eq.(20) together with Eq.(18) gives the following condition on f (0): f (0) > γ 4 = |α| 4 .(21) Eq. (17) can be further rewritten as n ℓ ′ = g ℓ ′ exp 8f (0) |α| − 1 − 4f (T ) |α| ǫ ℓ ′ − 4f (0) + |α| − 1 , which gives in the T = 0 asymptotic casē n ℓ ′ = g ℓ ′ exp −1 + 4f (0) |α| ǫ ℓ ′ − 4f (0) + |α| − 1 .(22) Now with these observations one can fix the function f (T ) through Eq. (14). If T << 1 the following equation is satisfied: N ≈ 1 exp{−4 · (f (T ) − f (0))} − 1 , (the ground state is non-degenerate). This fixes f (T ) = f (0)− 1 4 ln 1 + 1 N = f (0)− 1 4 ln 1 + |α| · T 2 2π . Therefore [f (T ) − f (0)] ∈ O(T 2 ). Calculating the entropy There is one significant consistency check and this is the fact that we require the fluid's entropy to match the black hole's entropy. As previously mentioned, in the low-temperature limit we need to match the fluid entropy with the semi-classical black-hole thermodynamics results. Let us use the first law of thermodynamics: T dS = dE + p · dV − µ · dN,(23) which leads to (T small again) dS = dE T − f (T ) · dN ≈ dE T − 2f (0) · dE |α|T . This automatically gives entropy to be proportional to the horizon area ∼ A H . However, claiming that entropy must be a growing function of A H gives the following upper bound on f (0): f (0) < |α| 2 . Since the positivity of occupation numbers implies f (0) > |α|/4, it means that f (0) ∈ (|α|/4, |α|/2). This means the right proportionality given by ∼ 1/4 cannot be reached as it would require f (0) = 0. Therefore, our microscopic fluid model predicts that only part of the black hole entropy can be explained by the fluid. All this means is that our model indeed leads to the correct linear proportionality between entropy and the horizon area, but with a lower factor than 1/4. (The proportionality factor depends on the free parameter f (0) and lies in the interval (0, 1/8).) In Appendix A we suggest that similar entropy calculation excludes fermionic model. One would possibly like to derive the same result in a more "proper" manner, from the statistical ensemble calculation. However, it is non-trivial to determine from the partition function what is the entropy of our system for T → 0. It is true that the degeneracy of the ground state, in which dominant number of particles reside for T = 0, is zero, however there are finite occupation numbers when T = 0 for the higher energy levels. These occupation numbers contribute to non-zero entropy. To calculate the entropy (e.g if it is convergent, or divergent at T = 0) from the statistical approach based on Eq.(15) seems to be impossible, as the approximations (e.g Stirling formula) used in this section fail for the higher level occupation numbers. Despite of this, let us make a qualitative observation that as T approaches zero the higher level occupation numbers grow (and approximate a finite constant value) and therefore entropy grows as T → 0. This is consistent with what one observes for Schwarzschild black hole. III. DISCUSSIONS We have shown that equation of state for the holographic fluid living on a spacetime horizon derived by [8] is in fact equation of state of an relativistic ideal gas. This provides better physical understanding of the nature of the fluid. Also the result is valid for generic spacetime horizon and depends only on general relativity theory. Many consequences of our microscopic holographic model for the black hole thermodynamics were discussed in the appropriate places in the text. The first main feature of our model is that it predicts for the black hole horizon an equispaced area / entropy spectrum. (There are many different arguments that lead to this type of spectrum originally suggested by Bekenstein [13]. For overview of these arguments see for example [21].) The spacing in the area spectrum is undetermined by our model, but it is equal to two times ground state energy of a single particle state. One can then proceed further and use either reduced quantization methods [22,23], or quasinormal modes [24,25] to fix the spacing between the area/entropy levels. The most widely agreed outcome will give in Planck units the area spacing as 8π and will lead to the one-particle's ground state energy equal to 4πT . The second prediction is that we recovered with our model the correct type of proportionality between horizon area and entropy. However the proportionality factor is less than one half of what is the result for black holes in Einstein gravity. One can try to "resolve" this problem by suggesting more species of particles whose entropy adds up to the correct proportionality, but this might sound slightly artificial without support of some additional arguments. Anyway, let us keep such a suggestion open for future investigations. There are many more open questions and problems, for example: a) We considered only the simplest, scalar (spin 0) Bose particle model. One does not assume any significant changes for bosons if higher spins s = 1, 2 were employed, but one needs to calculate the oneparticle energy spectra for such cases. Suggesting that the spectra should be similar to spectrum 10, one can also exclude half spin particles (fermions). This calculation is done in Appendix A. However all this needs to be further explored. [9,16,21] suggest that distinguishing between black hole horizon and more general spacetime horizons is somewhat artificial. To what extent can our results be generalized beyond black holes? Also, since our results are for fundamental reasons non-local (energy spectra are a non-local quantity), is there still a chance to somehow reformulate the theory for a local Rindler horizon? d) Another question are generalized theories of gravity. Is a similar microscopic model applicable to black holes / horizons in generalized theories of gravity? The fact that in many generalized theories entropy follows Bekenstein quantization rules, but the horizon area does not [26], suggests that generalization within gravity theories might correspond to a generalization within fluid's equation of state. Is this indeed true? Can one classify gravity theories by the equations of states of the fluid? Also, can one interpret the non-equilibrium thermodynamical effects that occur in higher curvature gravity [27,28] through such a microscopic model of a suitable fluid? e) One can also stick to the model we have and further explore, and discuss, possible consequences of the model, such that could lead to potentially observable predictions. case of zero spin particles, hence the spectrum is given by Eq. (10). [There will be small corrections due to the spin 1/2 nature, but in the present calculation they are not supposed to contribute.] Let us first estimate the function N (E) for temperatures so close to zero, that all the energy states up to Fermi level will be occupied by one particle, and the levels above will have zero occupation number. The energy can be calculated as: E = L ℓ=0 g ℓ · ǫ ℓ = L ℓ=0 (2ℓ + 1) · ℓT. (A1) (Here we approximated the spectrum (10) for higher levels by ǫ ℓ ≈ T ℓ and L labels the Fermi level.) The number of particles N is given as: N = L ℓ=0 (2ℓ + 1).(A2) Calculating the sum in Eq.(A2) one obtains: N = L 2 + 2L. Considering that N >> 1 one further obtains: L ≈ √ N . One can also calculate the sum in Eq.(A1) which leads to: E = L(L + 1)(2L + 1) 3 + L · T. The dominant term gives: E ≈ 2 3 N 3/2 T, and then using our model constraints N = (12π) 2/3 E 4/3 . The state's occupation number can be similarly to the bosonic case derived as: n ℓ = g ℓ exp 1 T + p T 64πE − 4 3 (12π) 2/3 4 3 µ T − λ E 1/3 ǫ ℓ − λ + 1 .(A3) This with our constraints leads to the result: n ℓ = g ℓ exp 3 − 4 3 18π T 1/3 4 3 µ − λT ǫ ℓ − λ + 1 .(A4) b) Only the simplest model of Schwarzschild black hole was explored, it would be nice to see to what extent we can generalize the model to the stationary rotating (Kerr) black hole.c) Many results Bose-Einstein condensates also play important role in the analogue gravity models. The equipartition law was used in the context of Schwarzschild black hole in[16].4 The spacing between the energy levels from Eq.(9) approaches for large ℓ harmonic oscillator's energy levels, for small energy levels the difference between the levels is larger than in the case of oscillator. ACKNOWLEDGMENTSWe would like to thank T. Padmanabhan for discussions. The work is supported by Max Planck-India Partner Group on Gravity and Cosmology. SS is partially supported by Ramanujan Fellowship of DST, India.Appendix A: Fermionic model with energy spectrum(10)In this appendix, we consider fermions assuming that the one-particle energy spectrum will be the same as in Now the first law of thermodynamics leads to:To have the correct asymptotic proportionality between area and entropy µ has to fulfill:The infinite number of particles in the asymptotic T = 0 case can be obtained from Eq.(A3) when:But this yields:which gives from Eq.(A5) entropy decreasing with energy, instead of growing with energy. This argument excludes Fermi particles. Vacuum quantum fluctuations in curved space and the theory of gravitation. A D Sakharov, Sov.Phys.Dokl. 12Gen.Rel.Grav.A.D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Sov.Phys.Dokl. 12 (1968) 1040-1041, (translated) Gen.Rel.Grav. 32 (2000) 365-367 Thermodynamics of spacetime: The Einstein equation of state. 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[ "The QCD Equation of State to O(µ 6 B ) from Lattice QCD", "The QCD Equation of State to O(µ 6 B ) from Lattice QCD" ]
[ "A Bazavov \nDepartment of Computational Mathematics, Science and Engineering and Department of Physics and Astronomy\nMichigan State University\n48824East LansingMIUSA\n", "H.-T Ding \nInstitute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina\n", "P Hegde \nCenter for High Energy Physics\nIndian Institute of Science\n560012BangaloreIndia\n", "O Kaczmarek \nInstitute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina\n\nFakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany\n", "F Karsch \nFakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany\n\nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "E Laermann \nFakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany\n", "Y Maezawa \nYukawa Institute for Theoretical Physics\nKyoto University\n606-8317KyotoJapan\n", "Swagato Mukherjee \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "H Ohno \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n\nCenter for Computational Sciences\nUniversity of Tsukuba\n305-8577TsukubaIbarakiJapan\n", "P Petreczky \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "H Sandmeyer \nFakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany\n", "P Steinbrecher \nFakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany\n\nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "C Schmidt \nFakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany\n", "S Sharma \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "W Soeldner \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n", "M Wagner \nNVIDIA GmbH\nD-52146WürselenGermany\n" ]
[ "Department of Computational Mathematics, Science and Engineering and Department of Physics and Astronomy\nMichigan State University\n48824East LansingMIUSA", "Institute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina", "Center for High Energy Physics\nIndian Institute of Science\n560012BangaloreIndia", "Institute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina", "Fakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany", "Fakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Fakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany", "Yukawa Institute for Theoretical Physics\nKyoto University\n606-8317KyotoJapan", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Center for Computational Sciences\nUniversity of Tsukuba\n305-8577TsukubaIbarakiJapan", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Fakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany", "Fakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Fakultät für Physik\nUniversität Bielefeld\nD-33615BielefeldGermany", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany", "NVIDIA GmbH\nD-52146WürselenGermany" ]
[]
We calculated the QCD equation of state using Taylor expansions that include contributions from up to sixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have been performed with the Highly Improved Staggered Quark action in the temperature range T ∈ [135 MeV, 330 MeV] using up to four different sets of lattice cut-offs corresponding to lattices of size N 3 σ × Nτ with aspect ratio Nσ/Nτ = 4 and Nτ = 6 − 16. The strange quark mass is tuned to its physical value and we use two strange to light quark mass ratios ms/m l = 20 and 27, which in the continuum limit correspond to a pion mass of about 160 MeV and 140 MeV respectively. Sixth-order results for Taylor expansion coefficients are used to estimate truncation errors of the fourth-order expansion. We show that truncation errors are small for baryon chemical potentials less then twice the temperature (µB ≤ 2T ). The fourth-order equation of state thus is suitable for the modeling of dense matter created in heavy ion collisions with center-of-mass energies down to √ sNN ∼ 12 GeV. We provide a parametrization of basic thermodynamic quantities that can be readily used in hydrodynamic simulation codes. The results on up to sixth order expansion coefficients of bulk thermodynamics are used for the calculation of lines of constant pressure, energy and entropy densities in the T -µB plane and are compared with the crossover line for the QCD chiral transition as well as with experimental results on freeze-out parameters in heavy ion collisions. These coefficients also provide estimates for the location of a possible critical point. We argue that results on sixth order expansion coefficients disfavor the existence of a critical point in the QCD phase diagram for µB/T ≤ 2 and T /Tc(µB = 0) > 0.9.
10.1103/physrevd.95.054504
[ "https://export.arxiv.org/pdf/1701.04325v3.pdf" ]
118,809,716
1701.04325
bbb486554a0f667193fc72b2419040abf459b0b7
The QCD Equation of State to O(µ 6 B ) from Lattice QCD March 10, 2022 9 Mar 2017 A Bazavov Department of Computational Mathematics, Science and Engineering and Department of Physics and Astronomy Michigan State University 48824East LansingMIUSA H.-T Ding Institute of Particle Physics Key Laboratory of Quark & Lepton Physics (MOE) Central China Normal University 430079WuhanChina P Hegde Center for High Energy Physics Indian Institute of Science 560012BangaloreIndia O Kaczmarek Institute of Particle Physics Key Laboratory of Quark & Lepton Physics (MOE) Central China Normal University 430079WuhanChina Fakultät für Physik Universität Bielefeld D-33615BielefeldGermany F Karsch Fakultät für Physik Universität Bielefeld D-33615BielefeldGermany Physics Department Brookhaven National Laboratory Upton11973NYUSA E Laermann Fakultät für Physik Universität Bielefeld D-33615BielefeldGermany Y Maezawa Yukawa Institute for Theoretical Physics Kyoto University 606-8317KyotoJapan Swagato Mukherjee Physics Department Brookhaven National Laboratory Upton11973NYUSA H Ohno Physics Department Brookhaven National Laboratory Upton11973NYUSA Center for Computational Sciences University of Tsukuba 305-8577TsukubaIbarakiJapan P Petreczky Physics Department Brookhaven National Laboratory Upton11973NYUSA H Sandmeyer Fakultät für Physik Universität Bielefeld D-33615BielefeldGermany P Steinbrecher Fakultät für Physik Universität Bielefeld D-33615BielefeldGermany Physics Department Brookhaven National Laboratory Upton11973NYUSA C Schmidt Fakultät für Physik Universität Bielefeld D-33615BielefeldGermany S Sharma Physics Department Brookhaven National Laboratory Upton11973NYUSA W Soeldner Institut für Theoretische Physik Universität Regensburg D-93040RegensburgGermany M Wagner NVIDIA GmbH D-52146WürselenGermany The QCD Equation of State to O(µ 6 B ) from Lattice QCD March 10, 2022 9 Mar 2017a [email protected] 2 We calculated the QCD equation of state using Taylor expansions that include contributions from up to sixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have been performed with the Highly Improved Staggered Quark action in the temperature range T ∈ [135 MeV, 330 MeV] using up to four different sets of lattice cut-offs corresponding to lattices of size N 3 σ × Nτ with aspect ratio Nσ/Nτ = 4 and Nτ = 6 − 16. The strange quark mass is tuned to its physical value and we use two strange to light quark mass ratios ms/m l = 20 and 27, which in the continuum limit correspond to a pion mass of about 160 MeV and 140 MeV respectively. Sixth-order results for Taylor expansion coefficients are used to estimate truncation errors of the fourth-order expansion. We show that truncation errors are small for baryon chemical potentials less then twice the temperature (µB ≤ 2T ). The fourth-order equation of state thus is suitable for the modeling of dense matter created in heavy ion collisions with center-of-mass energies down to √ sNN ∼ 12 GeV. We provide a parametrization of basic thermodynamic quantities that can be readily used in hydrodynamic simulation codes. The results on up to sixth order expansion coefficients of bulk thermodynamics are used for the calculation of lines of constant pressure, energy and entropy densities in the T -µB plane and are compared with the crossover line for the QCD chiral transition as well as with experimental results on freeze-out parameters in heavy ion collisions. These coefficients also provide estimates for the location of a possible critical point. We argue that results on sixth order expansion coefficients disfavor the existence of a critical point in the QCD phase diagram for µB/T ≤ 2 and T /Tc(µB = 0) > 0.9. We calculated the QCD equation of state using Taylor expansions that include contributions from up to sixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have been performed with the Highly Improved Staggered Quark action in the temperature range T ∈ [135 MeV, 330 MeV] using up to four different sets of lattice cut-offs corresponding to lattices of size N 3 σ × Nτ with aspect ratio Nσ/Nτ = 4 and Nτ = 6 − 16. The strange quark mass is tuned to its physical value and we use two strange to light quark mass ratios ms/m l = 20 and 27, which in the continuum limit correspond to a pion mass of about 160 MeV and 140 MeV respectively. Sixth-order results for Taylor expansion coefficients are used to estimate truncation errors of the fourth-order expansion. We show that truncation errors are small for baryon chemical potentials less then twice the temperature (µB ≤ 2T ). The fourth-order equation of state thus is suitable for the modeling of dense matter created in heavy ion collisions with center-of-mass energies down to √ sNN ∼ 12 GeV. We provide a parametrization of basic thermodynamic quantities that can be readily used in hydrodynamic simulation codes. The results on up to sixth order expansion coefficients of bulk thermodynamics are used for the calculation of lines of constant pressure, energy and entropy densities in the T -µB plane and are compared with the crossover line for the QCD chiral transition as well as with experimental results on freeze-out parameters in heavy ion collisions. These coefficients also provide estimates for the location of a possible critical point. We argue that results on sixth order expansion coefficients disfavor the existence of a critical point in the QCD phase diagram for µB/T ≤ 2 and T /Tc(µB = 0) > 0.9. The temperature and density dependence of bulk thermodynamic quantities, commonly summarized as the equation of state (EoS), provide the most basic characterization of equilibrium properties of strong-interaction matter. Its analysis within the framework of lattice regularized Quantum Chromodynamics (QCD) has been refined ever since the early calculations performed in pure SU (N ) gauge theories [1]. Quite recently, the continuum extrapolated results for the EoS of QCD with physical light and strange quark masses have been calculated [2,3]. Bulk thermodynamic observables such as pressure (P ), energy density ( ) and entropy density (s) as well as second order quantities such as the specific heat (C V ) and velocity of sound (c s ) have now been obtained at vanishing chemical potentials for the three quark flavors (µ u , µ d , µ s ). In accordance with the analysis of the chiral transition temperature, T c (154±9) MeV [4], bulk thermodynamic observables change smoothly in the transition region. At low temperature they are found to be in quite good agreement with hadron resonance gas (HRG) model calculations, although some systematic deviations have been observed, which may be attributed to the existence of additional resonances which are not taken into account in HRG model calculations based on well established resonances listed in the particle data tables [5,6]. The EoS at vanishing chemical potentials does already provide important input into the modelling of the hydrodynamic evolution of hot and dense matter created in heavy ion collisions. While this is appropriate for the thermal conditions met in these collisions at the LHC and the highest RHIC beam energies, knowledge of the EoS at non-vanishing baryon (µ B ), strangeness (µ S ) and electric charge (µ Q ) chemical potentials is indispensable for the hydrodynamic modelling of the conditions met in the beam energy scan (BES) at RHIC. Due to the well-known sign problem for lattice QCD formulations at non-zero chemical potential a direct calculation of the EoS at non-zero (µ B , µ Q , µ S ) is unfortunately not yet possible. At least for small values of the chemical potentials this can be circumvented by using a Taylor expansion of the thermodynamic potential [7,8]. In this way some results for EoS at non-zero baryon chemical potential have been obtained on coarse lattices [8][9][10]. These calculations have even been extended to sixth order in the baryon chemical potential [11,12]. First continuum extrapolated results for the EoS using second order Taylor expansion coefficients have been obtained within the stout discretization scheme for staggered fermions [13] and simulations at imaginary chemical potential have been used to arrive at a sixth order result for the QCD EoS [14] and up to eighth order for some generalized susceptibilities [15] through analytic continuation. Results for higher order expansion coefficients are clearly needed if one wants to cover the range of chemical potentials, 0 ≤ µ B /T < ∼ 3 that is expected to be explored with the BES at RHIC by varying the beam energies in the range 7.7 GeV ≤ √ s N N ≤ 200 GeV. Of course, the Taylor expansions will break down, should the elusive critical point in the QCD phase diagram [16,17] turn out to be present in this range of baryon chemical potentials. The convergence of the series thus needs to be monitored carefully. This paper is organized as follows. In the next section we briefly discuss Taylor series for a HRG model in Boltzmann approximation. This helps to argue for the significance of sixth order Taylor expansions. In Section III, we present the basic framework of Taylor series expansions, introduce expansions in the presence of global constraints and discuss some details of our calculations and the ensembles used. In Section IV we discuss the 6 th order Taylor expansion of QCD thermodynamics in the simplified case of vanishing strangeness and electric charge chemical potentials. Section V is devoted to the corresponding discussion of strangeness neutral systems n S = 0 with fixed net electric charge (n Q ) to net baryon-number (n B ) ratio, which is of relevance for the description of hot and dense matter formed in heavy ion collisions where typically n Q /n B 0.4. We discuss the relevance of a non-vanishing electric charge chemical potential by considering electric charge neutral (n Q /n B = 0) as well as isospin symmetric (n Q /n B = 1/2) systems. At the end of this section we present a parametrization of the equation of state that can easily be used as input for the modeling of the thermal conditions met in heavy ion collisions. In Section VI we present results on lines of constant pressure, energy density and entropy density and compare their dependence on µ B with empirical results for the freeze-out conditions observed in heavy ion collisions. We comment on the radius of convergence of the Taylor series for the pressure and resulting constraints for the location of a possible critical point in Section VII. Finally we present our conclusions in Section VIII. Details on (A) the statistics and simulation parameters, (B) explicit expressions for the expansions of electric charge and baryon number chemical potentials, and (C) explicit expressions for the expansion parameters of the lines of constant physics are given in three Appendices A-C. II. TAYLOR EXPANSIONS AND THE LOW AND HIGH TEMPERATURE LIMITS OF STRONG INTERACTION MATTER The main aim of this work is to supply a EoS of strong-interaction matter using up to sixth order Taylor expansions for bulk thermodynamic observables. As we will see later at present results on sixth order expansion coefficients in the Taylor series will mainly help to constrain truncation errors in the fourth order expansion rather than providing accurate results on the sixth order contribution to thermodynamic quantities. We will argue that our analysis provides reliable results for the EoS for baryon chemical potentials up to µ B /T 2 at temperatures below T 160 MeV and for an even larger range in µ B /T at higher temperatures. Before turning to a discussion of lattice QCD results on the EoS, it may be useful to analyze truncation effects in the hadron resonance gas (HRG) model, which seems to provide a good approximation for thermodynamics in the low temperature, hadronic regime. For simplicity let us consider the case of vanishing electric charge and strangeness chemical potentials, µ Q = µ S = 0. At temperatures close to the transition temperature T c 154 MeV and for baryon chemical potentials less than a few times the transition temperature, the baryon sector of a HRG is well described in the Boltzmann approximation. In a HRG model calculation based on non-interacting hadrons the pressure may then be written as P (T, µ B ) = P M (T ) + P B (T,μ B ) = P M (T ) + P B (T, 0) + P B (T, 0) (cosh(μ B ) − 1) ,(1) where we introduced the notationμ B ≡ µ B /T and P M (T ) (P B (T,μ B )) denote the meson (baryon) contributions to the pressure. A similar relation holds for the energy density, (T, µ B ) = M (T ) + B (T,μ B ) = M (T ) + B (T, 0) + B (T, 0) (cosh(μ B ) − 1) ,(2)with M/B ≡ T 2 ∂(P M/B /T )/∂T μ B . The µ B -dependent contribution thus is simple and can easily be represented by a Taylor series. Truncating this expansion at (2n)-th order we obtain (∆(P/T 4 )) 2n ≡ (P B (T, µ B ) − P B (T, 0)) 2n T 4 = n k=1 χ B,HRG 2k (T ) (2k)!μ 2k B P B (T, 0) T 4 n k=1 1 (2k)!μ 2k B ,(3) where in the last equality we made use of the fact that in HRG models constructed from non-interacting, point-like hadrons, all expansion coefficients are identical when using a Boltzmann approximation for the baryon sector, i.e. all baryon number susceptibilities are identical, χ B,HRG 2k = P B (T, 0). The ratios of these susceptibilities are unity, χ B,HRG 2k /χ B,HRG 2(k−1) = χ B,HRG 2k /χ B,HRG 2 = 1. Similarly one finds for the net baryon-number density, n B T 3 = P B (T, 0) T 4 sinhμ B = ∞ k=1 χ B,HRG 2k (T ) (2k − 1)!μ 2k−1 B P B (T, 0) T 4 ∞ k=1 1 (2k − 1)!μ 2k−1 B .(4) Higher order corrections are thus more important in the net baryon-number density than in the expansions of the pressure or energy density. For instance, the contribution to µ B n B /T 4 at O(μ 2k B ) is a factor 2k larger than the corresponding O(μ 2k B ) expansion coefficient of the pressure. In Fig. 1 we show results from a Taylor series expansion of the µ B -dependent part of the pressure in a HRG model truncated after leading order (LO), next-to-leading order (NLO) and next-to-next-to-leading (NNLO) order. These truncated expansions are compared to the exact result, i.e. (∆P ) ∞ (T ) = P B (T )(cosh(μ B ) − 1). The insertion shows the deviation of the n-th order truncated Taylor series ((∆P ) n (T )) from the exact result ((∆P ) ∞ (T )). As can be seen already the fourth order Taylor series provides a good approximation for the pressure (and energy as well as entropy density) of a HRG for all µ B ≤ 2T . At µ B = 2T the fourth order Taylor series for the µ B -dependent contribution to the pressure deviates by less than 5% from the exact result. These deviations are, of course, even smaller in the total pressure which in the temperature range of interest is dominated by the meson contribution. Even at T = 170 MeV, which certainly is already above the range of applicability of HRG models, the baryonic contribution to the pressure (energy density) amounts only to about 20% (30%). A 5% truncation error in the µ B -dependent contribution to the pressure or energy density thus amounts to less than a 2% effect in the total pressure or energy density. Similar estimates hold for the more general case of non-vanishing µ Q and µ S . Of course, the good convergence properties of the Taylor series for the pressure in HRG models also reflect that the radius of convergence of this series is infinite. If there exists a critical point in the QCD phase diagram one cannot expect to find that the Taylor series is that well behaved. Still the HRG result provides a benchmark also for the QCD case. If the radius of convergence of the Taylor series for the QCD pressure is finite and, in particular, smaller than µ B 3T , one should find large deviations in the generalized susceptibilities from the corresponding HRG results. Ratios of susceptibilities have to grow asymptotically like, χ B,QCD 2k /χ B,QCD 2(k−1) ∼ k 2 in order to yield a finite radius of convergence for a Taylor expansion. We will come back to a discussion of this asymptotic behavior after having discussed our sixth order calculation of Taylor expansion coefficients. Let us briefly mention also the high temperature limit. At large values of the temperature, the pressure approaches that of a massless ideal gas of quarks and gluons. In this limit the pressure is just a second order polynomial inμ 2 f , P ideal T 4 = 8π 2 45 + f =u,d,s 7π 2 60 + 1 2 µ f T 2 + 1 4π 2 µ f T 4 ,(5) In this limit a fourth order Taylor expansion thus provides the exact results for the basic bulk thermodynamic observables. This also is correct in leading order perturbation theory, i.e. at O(g 2 ) [18]. III. OUTLINE OF THE CALCULATION A. Taylor series in baryon number, electric charge and strangeness chemical potentials Our goal is the calculation of Taylor expansion coefficients for basic bulk thermodynamic observables of stronginteraction matter in terms of chemical potentials µ X for conserved charges (X = B, Q, S). We start with the expansion of the pressure, P , in terms of the dimensionless ratiosμ X ≡ µ X /T , which are the logarithms of fugacities, P T 4 = 1 V T 3 ln Z(T, V,μ u ,μ d ,μ s ) = ∞ i,j,k=0 χ BQS ijk i!j! k!μ i Bμ j Qμ k S ,(6) with χ BQS 000 ≡ P (T, 0)/T 4 . The chemical potentials for conserved charges are related to the quark chemical potentials (µ u , µ d , µ s ), µ u = 1 3 µ B + 2 3 µ Q , µ d = 1 3 µ B − 1 3 µ Q , µ s = 1 3 µ B − 1 3 µ Q − µ S .(7) The expansion coefficients χ BQS ijk , i.e. the so-called generalized susceptibilities, can be calculated at vanishing chemical potential 1 , χ BQS ijk ≡ χ BQS ijk (T ) = ∂P (T,μ)/T 4 ∂μ i B ∂μ j Q ∂μ k S μ=0 .(8) From Eq. 6 it is straightforward to obtain the Taylor series for the number densities, n X T 3 = ∂P/T 4 ∂μ X , X = B, Q, S .(9) This only requires knowledge of the expansion coefficients entering the series for P/T 4 . The energy ( ) and entropy (s) densities, on the other hand, also require derivatives of the generalized susceptibilities with respect to temperature, which are the expansion coefficients of the trace anomaly, ∆(T,μ B ,μ Q ,μ S ) ≡ − 3P T 4 = T ∂P/T 4 ∂T = ∞ i,j,k=0 Ξ BQS ijk i!j! k!μ i Bμ j Qμ k S ,(10) with i + j + k even and Ξ BQS ijk (T ) = T dχ BQS ijk (T ) dT .(11) With this one finds for the Taylor expansions of the energy and entropy densities, T 4 = ∞ i,j,k=0 Ξ BQS ijk + 3χ BQS ijk i! j! k!μ i Bμ j Qμ k S ,(12)s T 3 = + p − µ B n B − µ Q n Q − µ S n S T 4 = ∞ i,j,k=0 Ξ BQS ijk + (4 − i − j − k)χ BQS ijk i!j! k!μ i Bμ j Qμ k S .(13) B. Constrained series expansions In our calculations we generated all generalized susceptibilities up to 6 th order, which are needed to set up the general Taylor series in terms of the three conserved charge chemical potentials as discussed in the previous subsection. In the following we will, however, consider only thermodynamic systems, in which the electric charge and strangeness chemical potentials are fixed by additional constraints and become functions of the baryon chemical potential and temperature. We only consider constraints that can be fulfilled order by order in the Taylor series expansion. That is, for the construction of the 6 th order Taylor series of the pressure in terms ofμ B we need to know the expansion ofμ Q (T, µ B ) andμ S (T, µ B ) up to fifth order inμ B , µ Q (T, µ B ) = q 1 (T )μ B + q 3 (T )μ 3 B + q 5 (T )μ 5 B + . . . , µ S (T, µ B ) = s 1 (T )μ B + s 3 (T )μ 3 B + s 5 (T )μ 5 B + . . . .(14) The above parametrization includes the cases of vanishing electric charge and strangeness chemical potentials, µ Q = µ S = 0, which we are going to discuss in the next section as well as the strangeness neutral case with fixed electric charge to baryon-number ratio, which we will analyze in Section V. 1 We often suppress the argument (T ) of the generalized susceptibilities. We also suppress superscripts and subscripts of χ BQS ijk whenever one of the subscripts vanishes, e.g. χ BQS i0k ≡ χ BS ik . Implementing the constraints specified in Eq. 14 in the Taylor series for the pressure and net conserved-charge number densities one obtains series in terms of the baryon chemical potential only, P (T, µ B ) T 4 − P (T, 0) T 4 = ∞ k=1 P 2k (T )μ 2k B ,(15)n X T 3 = ∞ k=1 N X 2k−1μ 2k−1 B , X = B, Q, S .(16)Usingμ B dP/T 4 dμ B =μ B n B T 3 +μ B dμ Q dμ B n Q T 3 +μ B dμ S dμ B n S T 3 ,(17) and the series expansions ofμ Q andμ S given in Eq. 14 one easily finds the relation between the expansion coefficients for the pressure and number densities, P 2n = 1 2n N B 2n−1 + n k=1 (2k − 1) s 2k−1 N S 2n−2k+1 + q 2k−1 N Q 2n−2k+1 .(18) When imposing constraints on the electric charge and strangeness chemical potentials, these generally become temperature dependent functions as indicated in Eq. 14. The temperature derivative of P/T 4 at fixedμ B in the constraint case and the partial derivative of P/T 4 at fixed (μ B ,μ Q ,μ S ), which defines the trace anomaly ∆ (Eq. 10), thus are related through T dP/T 4 dT = ∆ + Tμ Q n Q T 3 + Tμ S n S T 3 ,(19) where the (total) temperature derivative d/dT is taken at fixedμ B andμ X = dμ X /dT . With this we obtain the Taylor series for the trace anomaly, ∆(T,μ B ) = − 3P T 4 = − 3P T 4 μ B =0 + ∞ n=1 (T P 2n (T ) − h 2n (T ))μ 2n B ,(20) with h 2n = n k=1 s 2k−1 N S 2n−2k+1 + q 2k−1 N Q 2n−2k+1 .(21) We also introduce t 2n = n k=1 s 2k−1 N S 2n−2k+1 + q 2k−1 N Q 2n−2k+1 .(22) With this the Taylor series expansion of the energy and entropy densities for constraint cases, in whichμ Q andμ S satisfy Eq. 14, becomes (T, µ B ) T 4 − (T, 0) T 4 = ∞ n=1 2n (T )μ 2n B ,(23)s(T, µ B ) T 3 − s(T, 0) T 3 = ∞ n=1 σ 2n (T )μ 2n B .(24) with 2n (T ) = 3P 2n (T ) + T P 2n (T ) − h 2n (T ) and σ 2k (t) = 4P 2n (T ) The generalized susceptibilities χ BQS ijk have been calculated on gauge field configurations generated for (2+1)-flavor QCD using the Highly Improved Staggered Quark (HISQ) action [19] and the tree-level improved Symanzik gauge action. + T P 2n (T ) − N B 2n−1 (T ) − h 2n (T ) − t 2n (T ). All calculations are performed using a strange quark mass m s tuned to its physical value. We performed calculations with two different light to strange quark mass ratios, m l /m s = 1/27 and 1/20. The former corresponds to a pseudoscalar Goldstone mass, which in the continuum limit yields a pion mass m π 140 MeV, the latter leads to a pion mass m π 160 MeV. These parameters are fixed using the line of constant physics determined by HotQCD from the f K scale. Using f K = 155.7(9)/ √ 2 MeV allows to determine the lattice spacing a(β) at a given value of the gauge coupling β and the corresponding set of quark masses (m l , m s ), which in turn fixes the temperature on a lattice with temporal extent N τ , i.e. T = (N τ a) −1 . More details on the scale determination are given in [4]. All calculations have been performed on lattices of size N 3 σ N τ with an aspect ratio N σ /N τ = 4. We perform calculations in the temperature interval T ∈ [135 MeV, 330 MeV] using lattices with temporal extent N τ = 6, 8, 12 and 16, which corresponds to four different values of the lattice spacings at fixed temperature. At temperatures T ≤ 175 MeV all calculations have been performed with the lighter, physical quark mass ratio m l /m s = 1/27. In the high temperature region quark mass effects are small and we based our calculations on existing data sets for m l /m s = 1/20, which have previously been generated by the HotQCD collaboration and used for the calculation of second order susceptibilities [20]. These data sets have been extended for the calculation of higher order susceptibilities. Gauge field configurations are stored after every 10 th molecular dynamics trajectory of unit length. All calculations of 4 th and 6 th order expansion coefficients have been performed on lattices with temporal extent N τ = 6 and 8. In these cases we gathered a large amount of statistics. At low temperatures we have generated up to 1.2 million trajectories for N τ = 6 and up to 1.8 million trajectories for N τ = 8. At high temperature less than a tenth of this statistics turned out to be sufficient. The 2 nd order expansion coefficients have been calculated on lattices with four different temporal extends, N τ = 6, 8, 12, 16. At fixed temperature this corresponds to four different values of the lattice cut-off, which we used to extract continuum extrapolated results for the second order expansion coefficients. We also extrapolated results for the higher order expansion coefficients to the continuum limit. However, having at hand results from only two lattice spacings for these expansion coefficients we consider these extrapolations as estimates of the results in the continuum limit. On each configuration the traces of all operators needed to construct up to sixth order Taylor expansion coefficients have been calculated stochastically. For the calculation of 2 nd and 4 th order expansion coefficients we follow the standard approach of introducing a non-zero chemical potential in the QCD Lagrangian as an exponential prefactor for time-like gauge field variables [21], i.e. the chemical potential µ f for quark flavor f is introduced through a factor e µ f a (e −µ f a ) on time-like links directed in the forward (backward) direction. This insures that all observables calculated are free of ultra-violet divergences. For the calculation of all 6 th order expansion coefficients we use the socalled linear-µ approach [22,23]. This becomes possible as no ultra-violet divergences appear in 6 th order cumulants and above. In the linear-µ formulation the number of operators that contribute to cumulants is drastically reduced and their structure is simplified. All operators appearing in the exponential formulation, that involve second or higher order derivatives of the fermion matrix [11], vanish. The remaining operators are identical in both formulations. One thus only has to calculate traces of observables that are of the form, TrM −1 f M f M −1 f M f ....M −1 f M f , where M f is the staggered fermion matrix for light (f = l) or strange (f = s) quarks, respectively, and M f denotes its derivative with respect to the flavor chemical potentialμ f . The final error on these traces depends on the noise due to the use of stochastic estimators for the inversion of the fermion matrices M f , as well as on the gauge noise resulting from a finite set of gauge configurations that get analyzed. We analyzed the signal to noise ratio for all traces of operators that we calculate and identified the operator D 1 = M −1 f M f , as being particularly sensitive to the stochastic noise contribution. This operator has been measured using 2000 random noise vectors. For the calculation of traces of all other operators we used 500 random noise vectors. We checked that this suffices to reduce the stochastic noise well below the gauge noise. The simulation parameters and the statistics accumulated in this calculation are summarized in the tables of Appendix A. All fits and continuum extrapolations shown in the following are based on spline interpolations with coefficients that are allowed to depend quadratically on the inverse temporal lattice size. Our fitting ansatz and the strategy followed to arrive at continuum extrapolated results are described in detail in Ref. [3]. For the current analysis we found it sufficient to use spline interpolations with quartic polynomials and 3 knots whose location is allowed to vary in the fit range. IV. EQUATION OF STATE FOR µQ = µS = 0 Let us first discuss the Taylor expansion for bulk thermodynamic observables in the case of vanishing electric charge and strangeness chemical potentials. This greatly simplifies the discussion and yet incorporates all the features of the more general case. Also the discussion of truncation errors presented in this section carries over to the more general situation. A. Pressure and net baryon-number density For µ Q = µ S = 0 the Taylor expansion coefficients P 2n and N B 2n−1 , introduced in Eqs. 15 and 16, are simply related by P 2n = 1 2n N B 2n−1 = 1 (2n)! χ B 2n .(25) The series for the pressure and net baryon-number density simplify to, P (T, µ B ) − P (T, 0) T 4 = ∞ n=1 χ B 2n (T ) (2n)! µ B T 2n = 1 2 χ B 2 (T )μ 2 B 1 + 1 12 χ B 4 (T ) χ B 2 (T )μ 2 B + 1 360 χ B 6 (T ) χ B 2 (T )μ 4 B + ... ,(26)n B T 3 = ∞ n=1 χ B 2n (T ) (2n − 1)!μ 2n−1 B = χ B 2 (T )μ B 1 + 1 6 χ B 4 (T ) χ B 2 (T )μ 2 B + 1 120 χ B 6 (T ) χ B 2 (T )μ 4 B + ... .(27) In Eqs. 26 and 27 we have factored out the leading order (LO) µ B -dependent part in the series for the pressure as well as the net baryon-number density. This helps to develop a feeling for the importance of higher order contributions and, in particular, the approach to the HRG limit at low temperatures. Note that all ratios χ B 2n /χ B 2 are unity in a HRG and, in the infinite temperature, ideal quark gas limit, χ B 4 /χ B 2 = 2/(3π 2 ) 0.068 is the only non-vanishing higher order expansion coefficient. From Eqs. 26 and 27 it is evident that contributions from higher order expansion coefficients become more important in the number density than in the pressure. Relative to the LO result, the contributions of the NLO and NNLO expansion coefficients for n B /T 3 are a factor two and three larger respectively than for the corresponding expansion coefficients in the pressure series. We show the leading order coefficient χ B 2 (T ) in Fig. 2 and the NLO (χ B 4 ) and NNLO (χ B 6 ) coefficients divided by χ B 2 (T ) in Fig. 3. The left hand part of Fig. 2 shows the leading order contribution χ B 2 in the entire temperature interval used in the current analysis. For the LO expansion coefficients we also used data from simulations on 48 3 × 12 lattices. Here we used existing data for m l /m s = 1/20 [3] and generated new ensembles for m l /m s = 1/27 at nine temperature values below T = 175 MeV. Furthermore, we used data on 64 3 × 16 lattices at a corresponding set of low temperature values. These data are taken from an ongoing calculation of higher order susceptibilities performed by the HotQCD Collaboration 2 . This allowed us to update the continuum extrapolation for χ B 2 given in [20]. The new continuum extrapolation shown in Fig. 2 is consistent with our earlier results, but has significantly smaller errors in the low temperature region. In the right hand part of this figure we compare the continuum extrapolated lattice QCD data for χ B 2 with HRG model calculations. It is obvious that the continuum extrapolated QCD results overshoot results obtained from a conventional, non-interacting HRG model calculations with resonances taken from the particle data tables (PDG-HRG) and treated as point-like excitations. We therefore compare the QCD results also with a HRG model that includes additional strange baryons,which are not listed in the PDG but are predicted in quark models and lattice QCD calculations. We successfully used such an extended HRG model (QM-HRG) in previous calculations [5,6]. As can be seen in Fig. 2 (left), continuum extrapolated results for χ B 2 agree well with QM-HRG calculations. As can be seen in the left hand part of Fig. 3, the ratio χ B 4 /χ B 2 approaches unity with decreasing temperature, but is small at high temperatures where the leading order correction is large. The relative contribution of the NLO correction thus is largest in the hadronic phase, where χ B 4 /χ B 2 1. For temperatures T < ∼ 155 MeV we find χ B 4 /χ B 2 ≤ 0.8. The relative contribution of the NLO correction to the µ B -dependent part of the pressure (number density) in the crossover region and below thus is about 8% (16%) at µ B /T = 1 and rises to about 33% (66%) at µ B /T = 2. At temperatures larger than 180 MeV the relative contribution of the NLO correction to pressure and number density at µ B /T = 2 is less than 8% and 16%, respectively. The relative contribution of the O(μ 6 B ) correction, χ B 6 /χ B 2 , is shown in the right hand part of Fig. 3. The ideal gas limit for this ratio vanishes. Obviously the ratio is already small for all temperatures T > 180 MeV, i.e. χ B 6 /χ B 2 ≤ 0.5. Consequently, forμ B = 2 the correction to the leading order result is less than 2.2% for the µ B -dependent part of the pressure and less than 7% for the net baryon-number density. At lower temperatures the statistical errors on current results for χ B 6 /χ B 2 are still large. However, a crude estimate for the magnitude of this ratio at all temperatures larger than 130 MeV suggests, χ B 6 /χ B 2 ≤ 3. In the low temperature, hadronic regime and forμ B = 2 the O(μ 6 B ) corrections to the µ B -dependent part of the pressure can be about 13%. However, in the total pressure, which also receives large contributions from the meson sector, this will result only in an error of less than 3%. In the calculation of the net baryon-number density, on the other hand, the current uncertainty on O(μ 6 B ) expansion coefficients results in errors of about 40% at temperatures below T 155 MeV. In fact, as discussed already in section II, higher order corrections are larger in the Taylor expansion of the number density. From Eq. 25 it follows for the ratio of NLO and LO expansion coefficients, N B 5 /N B 1 = 3P 6 /P 2 . Clearly better statistics is needed in the low temperature range to control higher order corrections to n B /T 3 . In Fig. 4 we show results for the µ B -dependent part of the pressure (left) and the net baryon-number density (right) calculated from Taylor series up to and including LO, NLO and NNLO contributions, respectively. This suggests that up to µ B 2T results for the pressure at low temperature are well described by a Taylor series truncated at NNLO, while at higher temperature NNLO corrections are small even at µ B 3T . This also is the case for n B /T 3 , although the NNLO correction is large at low temperatures and, at present, does not allow for a detailed quantitative analysis of the baryon-number density in this temperature range. It also is obvious that the Taylor series for the pressure and n B /T 3 in the temperature range up to T 180 MeV are sensitive to the negative contributions of the 6 th order expansion coefficient. The occurrence of a dip in the sixth order expansion coefficient of the pressure has been expected to show up on the basis of general scaling arguments for higher order derivatives of the QCD pressure in the vicinity of the chiral phase transition [24]. It may, however, also reflect the influence of a singularity on the imaginary chemical potential axis [25] (Roberge-Weiss critical point [26]) on Taylor series of bulk thermodynamic observables in QCD. Even with improved statistics it thus is expected that the wiggles, that start to show up in the expansion of pressure and net baryon-number density above µ B /T 2 (see Fig. 4) and reflect the change of sign in the sixth order expansion coefficient, will persist. Getting the magnitude of the dip in χ B 6 /χ B 2 at T 160 MeV under control in future calculations thus is of importance for the understanding of this non-perturbative regime of the QCD equation of state in the high temperature phase close to the transition region. This also indicates that higher order corrections need to be calculated in order to control the equation of state in this temperature regime. µ B /T=2 µ B /T=2.5 µ B /T=1 HRG µ Q =µ S =0 [P(T,µ B )-P(T,0)]/T 4 T [MeV] O(µ B 6 ) O(µ B 4 ) O(µ B 2 ) 0 0.1 0.2 0.µ B /T=2 µ B /T=2.5 µ B /T=1 HRG µ Q =µ S =0 n B (T,µ B )/T 3 T [MeV] O(µ B 6 ) O(µ B 4 ) O(µ B 2 ) B. Net strangeness and net electric charge densities For vanishing strangeness and electric charge chemical potentials the corresponding net strangeness (n S ) and net electric charge (n Q ) densities are nonetheless non-zero because the carriers of these quantum numbers also carry baryon number. The ratios of number densities are given by n X n B = χ BX 11 + 1 6 χ BX 31μ 2 B + 1 120 χ BX 51μ 4 B χ B 2 + 1 6 χ B 4μ 2 B + 1 120 χ B 6μ 4 B , X = Q, S .(28) In a hadron resonance gas the ratios n S /n B and n Q /n B are independent of the baryon chemical potential and, irrespective of the value ofμ B , these ratios approach −1 and 0, respectively, in the T → ∞ limit. One thus may expect that these ratios only show a mild dependence onμ B , which indeed is apparent from the results of the NNLO expansions shown in Fig. 5. For µ Q = µ S = 0 non-vanishing electric charge and strangeness densities only arise due to a non-zero baryonchemical potential. In the low temperature HRG phase n Q and n S thus only receive contributions from charged baryons or strange baryons, respectively. The ratios n Q /n B and n S /n B thus are sensitive to the particle content in a hadron resonance gas and a comparison with PDG-HRG and QM-HRG is particularly sensitive to the differences in the baryon content in these two models. It is apparent from Fig. 5 that at low temperatures the QM-HRG model provides a better description of the lattice QCD results than the PDG-HRG model. C. The energy and entropy densities In order to calculate the energy and entropy densities, defined in Eqs. 23 and 24, we need to extract the temperature derivative of the expansion coefficients of the pressure. We use as a starting point the representation of the pressure given in Eq. 26 and calculate the temperature derivatives of χ B n from the splines used to fit this observable. With this we construct the expansion coefficients B n (T ) and σ B n defined in Eqs. 12 and 13, ∆ /T 4 = (T, µ B ) − (T, 0) T 4 = 3 k=1 2kμ 2k B = 3 k=1 (T P 2k + 3P 2k )μ 2k B ,(29)∆ s/T 3 = s(T, µ B ) − s(T, 0) T 3 = 3 k=1 σ 2kμ 2k B = 3 k=1 ( 2k − (2k − 1)P 2k )μ 2k B .(30) We show the LO and NLO expansion coefficients for energy and entropy densities together with the expansion coefficient for the pressure in Fig. 6. Because of Eq. 25 the expansion coefficients of the net baryon-number density are simply proportional to those of the pressure. Clearly the temperature dependence of the expansion coefficients of the energy and entropy densities shows more structure than in the case of the pressure. Qualitatively this can be understood in terms of the pseudo-critical behavior of bulk thermodynamic observables. Once thermodynamic quantities are dominated by contributions from the singular part of the free energy, which is expected to happen in the transition region, they become functions of (T − T c ) + κμ 2 B . The temperature derivative of the expansion coefficient P 2 , which gives 2 , thus will show properties similar to those of P 4 . The LO correction B 2 /T 4 has a mild peak, which results from the strongly peaked T -derivative of χ B 2 which is qualitatively similar to χ B 4 , and the NLO correction is negative in a small temperature interval above T c , which arises from the negative T -derivative of χ B 4 at high temperature, which resembles the negative part of χ B 6 at high temperature. Although the temperature dependence of n and σ n differs from that of the pressure coefficient, P n , the conclusions drawn for the relative strength of the expansion coefficients are identical in all cases. As can be seen from the inset in Fig. 6 (right) the relative contribution of the NLO expansion coefficients never exceeds 10%. In particular, at temperatures larger than 180 MeV the magnitude of the NLO expansion coefficients never exceeds 2% of the LO expansion coefficients. Again this leads to the conclusion that at µ B /T = 2 and temperatures above 180 MeV the NLO correction contributes less than 8% of the leading correction to µ B -dependent part of the energy and entropy densities. For T < ∼ 155 MeV, however, the NLO contribution can rise to about 30%. A similar conclusion holds for the O(μ 6 B ) corrections, although it requires higher statistics to better quantify the magnitude of this contribution. In Fig. 7 we show results for the total pressure and total energy density. For P/T 4 and /T 4 at µ B = 0 we used the results obtained by the HotQCD Collaboration [3] and added to it the results from the O(μ 6 B ) expansions presented above. This figure also makes it clear that despite of the large error of higher order expansion coefficients, which we have discused above, the error on the total pressure and energy density still is dominated by errors on their values at µ B = 0. V. EQUATION OF STATE IN STRANGENESS NEUTRAL SYSTEMS A. Taylor expansion of pressure, baryon-number, energy and entropy densities We now want to discuss the equation of state for strangeness neutral systems with a fixed ratio of electric charge to baryon-number density, i.e. we impose the constraints [27] n S = 0 , n Q n B = r . These constraints can be realized through suitable choices of the electric charge and strangeness chemical potentials. This thus is a particular case of the constraint expansion discussed in Subsection III B. The expansion coefficients q n , s n , n = 1, 3, 5 needed to satisfy these constraints are given in Appendix B. For r = 0.4 the constrained EoS obtained in this way is usually considered to be most appropriate for applications to heavy ion collisions. We will, however, in the following also comment on other choices of r, including the case of isospin symmetric systems (r = 1/2) and electric charge neutral matter (r = 0). Using the constraints specified in Eq. 31 and the definition of the pressure in terms of generalized susceptibilities, χ BQS ijk , the expansion coefficients P 2n can easily be determined. Here it advantageous to use the relation between the Taylor expansion coefficients of the pressure, P 2n , and number densities, N X 2n−1 , given in Eq. 18, which simplifies considerably for strangeness neutral systems. It now involves only the net baryon-number density coefficients, P 2 = 1 2 N B 1 + rq 1 N B 1 ,(32)P 4 = 1 4 N B 3 + r q 1 N B 3 + 3q 3 N B 1 ,(33)P 6 = 1 6 N B 5 + r q 1 N B 5 + 3q 3 N B 3 + 5q 5 N B 1 .(34) Explicit expressions for all N B n−1 and q n−1 , for n = 2, 4, 6, are given in Appendix B. The resulting expansion coefficients for the pressure are shown in Fig. 8. Also shown in the bottom-right panel of this figure is the ratio of the expansion coefficients for the net baryon-number density, N B n−1 and the appropriately rescaled expansion coefficients of the pressure, nP n . In electric charge neutral systems, r = 0 as well as in the isospin symmetric limit r = 1/2, for which the expansion coefficients q i = 0 vanish for all i, this ratio is unity. In both cases the simple relation given in Eq. 25 holds. Also for other values of r the contribution from terms proportional to r are small. In Fig. 8 Figure 9. Ratio of O(μ 2 B ) expansion coefficients of the pressure in systems with electric charge to net baryon-number ratio r = nQ/nB relative to that of strangeness neutral, isospin symmetric systems (r = 1/2). Triangles show the ratio of the pressure in systems with vanishing electric charge and strangeness chemical potential and the strangeness neutral, isospin symmetric system. Horizontal lines at high temperature show the corresponding free quark gas values. All data points shown are from calculations on lattices with temporal extent Nτ = 8. number-ratio is weak. The O(μ 2 B ) expansion coefficient of the pressure in strangeness neutral systems differs by at most 10% in electric charge neutral (r = 0) and isospin symmetric systems (r = 1/2), respectively. The expansion coefficient P 2 evaluated for different values of r is shown in Fig. 9. For chemical potentialsμ ≤ 2 this amounts to differences less than 1.5% of the total pressure. On the other hand, strangeness neutral systems differ substantially from systems with vanishing strangeness chemical potential. In this case the O(μ 2 B ) expansion coefficients differ by almost 50% in the high temperature limit. For T < 150 MeV this difference is only about 10% reflecting that the different treatment of the strangeness sector becomes less important for the thermodynamics at low temperature. This is also shown in Fig. 9. Compared to the leading O(μ 2 B ) contributions to bulk thermodynamic observables the O(μ 4 B ) and O(μ 6 B ) corrections are smaller in the strangeness neutral case than in the case µ Q = µ S = 0, which we have discussed in the previous section. This is evident from Fig. 10, where we show the ratios 12P 4 /P 2 and 360P 6 /P 2 . These combinations are unity in a HRG with µ S = µ Q = 0 but smaller than unity in the strangeness neutral case. Higher order corrections in Taylor series for strangeness neutral systems thus are of less importance than in the case µ S = 0. This also means that the errors, which are large on e.g. sixth order expansion coefficients, are of less importance for the overall error budget of Taylor expansions in strangeness neutral systems. This is indeed reflected in the µ B -dependence of (P (T, µ B ) − P (T, 0))/T 4 and n B (T, µ B )/T 3 shown in the upper panels of Fig. 11 for the case r = 0.4. As can be seen in these two figures, at low temperatures the µ B -dependent part of the pressure as well as the net baryon-number density agree quite well with HRG model calculations that describe the thermodynamics of a gas of non-interacting, point-like hadron resonances. This agreement, however, gets worse at larger values of µ B . Not unexpectedly, at higher temperatures deviations from HRG model calculations become large already at small values of µ B . This is apparent from the lower two panels of Fig. 11, where we show the ratio of the µ B -dependent part of the pressure and the corresponding HRG model result (left) and the net baryon-number density divided by the corresponding HRG model result (right). In the HRG model calculation (P (T, µ B ) − P (T, 0))/T 4 as well as n B (T, µ B )/T 3 only depend on the baryon sector of the hadron spectrum. The results shown in Fig. 11 thus strongly suggest that HRG model calculations using resonance spectra in model calculations for non-interacting, point-like hadron gases may be appropriate (within ∼ 10% accuracy) to describe the physics in the crossover region of strongly interacting matter at vanishing or small values of the baryon chemical potential, but fail 3 to do so at large µ B /T and/or T > ∼ 160 MeV. At T = 165 MeV QCD and HRG model results for the net baryon-number density differ by 40% at µ B /T = 2. This has consequences for the determination of freeze-out conditions in heavy ion collisions. We will come back to this discussion in Section VI. The µ B -dependent contributions to the energy and entropy densities have been defined in Eqs. 23 and 24. In strangeness neutral systems the expansion coefficients simplify considerably, 2n (T ) = 3P 2n (T ) + T P 2n (T ) − r n k=1 T q 2k−1 N B 2n−2k+1 (35) σ 2n (T ) = 4P 2n (T ) + T P 2n (T ) − N B 2n−1 − r n k=1 (q 2k−1 + T q 2k−1 )N B 2n−2k+1(36) Results for the O(µ 2 B ) and O(µ 4 B ) expansion coefficients are shown in Fig. 12 together with the corresponding expansion coefficients for the pressure and net baryon-number density. Results for the total energy density as well as the total pressure for µ B /T = 0 and 2 are shown in Fig. 13. As discussed in the previous section also here it is evident that current errors on the total pressure and energy density are dominated by errors on these observables at µ B = 0. In Fig. 13 we also show results for the total pressure obtained within the stout discretization scheme. The result for µ B = 0 is taken from [2]. Theμ B -dependent contribution is based on calculations with an imaginary chemical potential [14]. These results have been analytically continued to real values ofμ B using a 6 th order polynomial inμ B . As can be seen the total pressure agrees quite well with the results obtained with a sixth order Taylor expansion, although the results obtained the analytic continuation within the stout discretization scheme tend to stay systematically below the central values obtained from the analysis of Taylor series expansions in the HISQ discretization scheme. B. Parametrization of the equation of state At µ B = 0 the HotQCD Collaboration presented a parametrization of the pressure, obtained as interpolating curves for the continuum extrapolated fit, that also provided an adequate description of all the other basic thermodynamic quantities, i.e. the energy and entropy densities as well as the specific heat and the velocity of sound [3]. Here we want to extend this parametrization to the caseμ B > 0. Similar to what has been done at µ B = 0 it turns out that a ratio of fourth order polynomials in the inverse temperature is flexible enough to describe the temperature dependence of all required Taylor expansion coefficients in the temperature range T ∈ [130 MeV, 280 MeV]. We use such an ansatz for the three expansion coefficients of the net baryon-number density (N B 1 , N B 3 , N B 5 ) and the three electric charge chemical potentials (q 1 , q 3 , q 5 ). This suffices to calculate all thermodynamic observables in strangeness neutral systems. We use a ratio of fourth order polynomials in 1/T as an ansatz for the expansion coefficients of the net baryon- number density, N B k (T ) = N B k,0n + N B k,1nt + N B k,2nt 2 + N B k,3nt 3 + N B k,4nt 4 1 + N B k,1dt + N B k,2dt 2 + N B k,3dt 3 + N B k,4dt 4 , k = 1, 3, 5 .(37) Heret = T c /T and the QCD transition temperature T c = 154 MeV is used as a convenient normalization. Similarly we define the parametrization of the expansion coefficients for the electric charge chemical potential, q k (T ) = q k,0n + q k,1nt + q k,2nt 2 + q k,3nt 3 + q k,4nt 4 1 + q k,1dt + q k,2dt 2 + q k,3dt 3 + q k,4dt 4 , k = 1, 3, 5 .(38) The parameters for these interpolating curves are summarized in Table I. The expansion coefficients of the pressure are then obtained by using Eqs. 32-34. The resulting interpolating curves for P k are shown as darker curves in Fig. 8. All other interpolating curves shown as darker curves in other figures have been obtained by using the above interpolations. In particular, interpolating curves for the energy and entropy densities are obtained by using Eqs. 35 and 36 and calculating analytically temperature derivatives of the parametrizations of P n and q n given in Eqs. 37 and 38. The resulting interpolating curves for the second and fourth order Taylor expansion coefficients are shown in Fig. 12. We also used a ratio of fourth order polynomials to interpolate results for the pressure at µ B = 0. We write the pressure as P (T, µ B = 0) T 4 = p 0n + p 1nt + p 2nt 2 + p 3nt 3 + p 4nt 4 1 + p 1dt + p 2dt 2 + p 3dt 3 + p 4dt 4 .(39) The coefficients p in and p id are also given in Table I. VI. LINES OF CONSTANT PHYSICS TO O(µ 4 B ) We want to use here the Taylor series for bulk thermodynamic observables, i.e. the pressure, energy and entropy densities, to discuss contour lines in the T -µ B plane on which these observables stay constant. It has been argued quite successfully that the thermal conditions at the time of chemical freeze-out in heavy ion collisions can be characterized by lines in the T -µ B plane on which certain thermodynamic observables or ratios thereof stay constant [32,33], although the freeze-out mechanism in the rapidly expanding fireball created in a heavy ion collision is of dynamical origin and will in detail be more complicated (see for instance [34]). While lines of constant physics (LCPs) involving total baryon-number densities, as used in [32,33], are not appropriate for calculations within the framework of quantum field theories, other criteria like lines of constant entropy density in units of T 3 [35] or constant pressure [36][37][38] have been suggested to characterize freeze-out parameters (T f , µ f B ) corresponding to heavy ion collisions at Table I data with model calculations based on some version of a HRG model. We will determine here LCPs from the lattice QCD calculations of pressure, energy and entropy densities and confront them with freeze-out parameters that have been obtained by comparing particle yields, measured at different values of √ s N N , to HRG model calculations. We consider an observable f (T, µ B ), i.e. the pressure, energy density or entropy density which are even functions of µ B . We parametrize a 'line of constant f ' by, T f (µ B ) = T 0 1 − κ f 2 µ B T 0 2 − κ f 4 µ B T 0 4 .(40) In order to determine the expansion coefficients κ f 2 and κ f 4 we need to expand the function f (T, µ B ) up to 4 th order in µ B and up to second order in T around some point (T 0 , 0), f (T, µ B ) = f (T 0 , 0) + ∂f (T, µ B ) ∂T (T0,0) (T − T 0 ) + 1 2 ∂ 2 f (T, µ B ) ∂µ 2 B (T0,0) µ 2 B(41)+ 1 2 ∂ 2 f (T, µ B ) ∂T 2 (T0,0) (T − T 0 ) 2 + 1 2 ∂ ∂T ∂ 2 f (T, µ B ) ∂µ 2 B (T0,0) (T − T 0 )µ 2 B + 1 4! ∂ 4 f (T, µ B ) ∂µ 4 B (T0,0) µ 4 B . Note that we expand here in terms of µ B rather than inμ B ≡ µ B /T . Replacing the temperature T in Eq. 41 by the ansatz for a line of constant f , Eq. 40, and keeping terms up to O(µ 4 B ) gives f (T (µ B ), µ B ) = f (T 0 , 0) + − κ f 2 ∂f (T, µ B ) ∂T (T0,0) 1 T 0 + 1 2 ∂ 2 f (T, µ B ) ∂µ 2 B (T0,0) µ 2 B + − κ f 4 ∂f (T, µ B ) ∂T (T0,0) 1 T 3 0 + 1 2 ∂ 2 f (T, µ B ) ∂T 2 (T0,0) (κ f 2 ) 2 1 T 2 0 − 1 2 ∂ ∂T ∂ 2 f (T, µ B ) ∂µ 2 B (T0,0) κ f 2 1 T 0 + 1 4! ∂ 4 f (T, µ B ) ∂µ 4 B (T0,0) µ 4 B . We then can determine κ f 2 and κ f 4 by demanding that the expansion coefficients at O(µ 2 B ) and O(µ 4 B ) vanish, i.e. κ f 2 = T 0 2 ∂ 2 f (T,µ B ) ∂µ 2 B (T0,0) ∂f (T,µ B ) ∂T (T0,0) ,(42)κ f 4 = 1 2 T 2 0 ∂ 2 f (T,µ B ) ∂T 2 (T0,0) (κ f 2 ) 2 − 1 2 T 3 0 ∂ ∂T ∂ 2 f (T,µ B ) ∂µ 2 B (T0,0) κ f 2 + 1 4! T 4 0 ∂ 4 f (T,µ B ) ∂µ 4 B (T0,0) T 0 ∂f (T,µ B ) ∂T (T0,0) .(43) ∞ k=0 f 2k (T )μ 2k B , the derivatives with respect to µ B appearing in Eqs. 42 and 43 can be replaced by suitable Taylor expansion coefficients of f (T, µ B ), κ f 2 = f 2 (T 0 ) T 0 ∂f0(T ) ∂T (T0,0) (44) κ f 4 = 1 2 T 2 0 ∂ 2 f0(T ) ∂T 2 (T0,0) (κ f 2 ) 2 − T 0 ∂f2(T ) ∂T (T0,0) − 2f 2 (T 0 ) κ f 2 + f 4 (T 0 ) T 0 ∂f0(T ) ∂T (T0,0)(45) We will in the following work out detailed expressions for the quadratic correction coefficient, κ f 2 , for lines of constant pressure (f ≡ P ), energy density (f ≡ ) and entropy density (f ≡ s) in strangeness neutral systems with electric charge to net baryon-number ratio r = 0.4. Details for the quartic coefficient, κ f 4 , are given in Appendix C. pressure f ≡ P : The function f (T, µ B ) is given by P = T 4 n P n (µ B /T ) n , with P 0 = P (T, 0)/T 4 denoting the pressure in units of T 4 at vanishing baryon chemical potential and P n (T ), n > 0, denoting the expansion coefficients of P (T, µ B )/T 4 as introduced in Eq. 15. In the denominator of Eq. 44 we use the thermodynamic relation between pressure and entropy density s = (∂P/∂T ) µ B . The numerator is given by f 2 (T ) = T 4 P 2 (T ). This gives κ P 2 = P 2 s/T 3 ,(46) where s/T 3 is evaluated atμ B = 0. energy density f ≡ : The function f (T, µ B ) is given by = T 4 n n (µ B /T ) n , with 0 = (T, 0)/T 4 denoting the energy density in units of T 4 at vanishing baryon chemical potential . In the denominator of Eq. 44 we use the thermodynamic relation between energy density and specific heat C V = (∂ /∂T ) µ B In the numerator we have f 2 (T ) = T 4 2 (T ). This gives κ 2 = 2 C V /T 3 ,(47) where C V /T 3 is evaluated atμ B = 0. entropy density f ≡ s: The function f (T, µ B ) is given by s = ( + P − µ B n B − µ Q n Q )/T = ( + P − µ B n B (1 + rµ Q /µ B ))/T . As n B is of O(µ B ) we need for the ratio of electric charge and strangeness chemical potentials only the leading order relation µ Q /µ B = q 1 defined in Eq. 14. In the denominator we use, ∂s ∂T = ∂( + P )/T ∂T = − s T + 1 T ∂( + P ) ∂T = C V T .(48) In the numerator we have f 2 (T ) = T 3 ( 2 + P 2 − N B 1 (1 + rq 1 )). With this we get, κ s 2 = T 3 2 + P 2 − N B 1 (1 + rq 1 ) C V = 2 − P 2 C V /T 3 .(49) where we have used Eq. 32 to replace N B 1 in favor of P 2 . We note that κ 2 > κ s 2 , i.e. with increasing µ B the entropy density decreases on lines of constant energy density. The second order coefficients for the lines of constant physics thus can directly be calculated using the continuum extrapolated results for the pressure and energy density obtained at vanishing chemical potential in [3] and the leading order expansion coefficient of the pressure shown in Fig. 10. Similarly we obtain the quartic coefficients from the fourth order expansion of the pressure using the relations given in Appendix C. We show results for κ f 2 and κ f 4 in Fig. 14. Table II). Data points show freeze-out temperatures determined by the STAR Collaboration in the BES at RHIC (squares) [39] and the ALICE Collaboration at the LHC (triangle) [40]. The circles denote hadronization temperatures obtained by comparing experimental data on particle yields with a hadronization model calculation [41]. Also shown are two lines representing the current spread in determinations of the µB-dependence of the QCD crossover transition line (see text). Right: Net baryon-number density on the lines of constant physics for three values of the energy density at µB = 0. Other thermodynamic parameters characterizing these lines are summarized in Table II. In the interval around T c , i.e. T ∈ [145 MeV, 165 MeV] we find, 0.0064 ≤ κ P 2 ≤ 0.0101 , 0.0087 ≤ κ 2 ≤ 0.012 , 0.0074 ≤ κ s 2 ≤ 0.011 .(50) Apparently, at O(µ 2 B ), lines of constant pressure and constant energy or entropy densities agree quite well and they also agree, within currently large errors, with the curvature of the transition line in (2+1)-flavor QCD. The coefficient of the quartic correction for the contour lines turns out to be about two orders of magnitude smaller than the leading order coefficients. This, of course, reflects the small contribution of the NLO corrections to the µ B -dependent part of pressure and energy density. For all fourth order coefficients we find |κ f 4 | ≤ 0.00024 in the temperature interval around T c . For µ B /T ≤ 2 the contribution arising from κ f 4 only leads to modifications of T f (µ B ) that stays within the error band arising from the uncertainty in κ f 2 . The resulting lines of constant physics in the T -µ B plane are shown in Fig. 15 (left) for three values of the temperature, T = 145 MeV, 155 MeV and 165 MeV. These correspond to constant energy densities = 0.203(27) GeV/fm 3 , 0.346(41) GeV/fm 3 and 0.556(57) GeV/fm 3 , which roughly correspond to the energy density of cold nuclear matter, a hard sphere gas of nucleons at dense packing and the interior of a nucleus, respectively. Values of other bulk thermodynamic observables characterizing these LCPs are summarized in Table II. The corresponding net baryon-number densities on these LCPs are shown in Fig. 15 (right). It is apparent from Fig. 15 energy or entropy density agree well with each other up to baryon chemical potentials µ B /T = 2, where the difference in temperature on different LCPs is at most 2 MeV. We also note that the temperature on a LCP varies by about 7 MeV or, equivalently, 5% betweenμ B = 0 andμ B = 2. Thus on a line of constant pressure, the entropy in units of T 3 changes by about 15%. I.e. constant P or constant s/T 3 , which both have been suggested as phenomenological descriptions for freeze-out conditions in heavy ion collisions, can not hold simultaneously, although a change of 15% of one of these observables may phenomenologically not be of much relevance. We also stress that at large values ofμ B the comparison of experimental data with HRG model calculations, e.g. the use of single particle Boltzmann distributions used to extract freeze-out temperatures and chemical potentials, becomes questionable. As shown in Fig. 11 net baryon-number densities extracted from HRG and QCD calculations differ substantially at µ B /T 2. Also shown in Fig. 15 (left) are results on freeze-out parameters and hadronization temperatures extracted from particle yields measured in heavy ion experiments [39][40][41] by comparing data with model calculations based on the hadron resonance gas models. The region µ B /T ≤ 2 corresponds to beam energies √ s N N ≥ 11.4 GeV in the RHIC beam energy scan. Obviously, the freeze-out parameters extracted from the beam energy scan data [39] do not follow any of the LCPs. However, the discrepancy between the freeze-out parameters determined at the LHC [40] and the highest beam energy at RHIC [39] suggests that also these determinations are not consistent among each other. Finally we note that the lines of constant physics discussed above compare also well with the crossover line for the QCD transition. At non-zero values of the baryon chemical potential the change of the (pseudo)-critical temperature has been determined, using various approaches at real [42,43] and imaginary [44][45][46] values of the chemical potential. To leading order one obtains, T c (µ B ) = T c (0) 1 − κ c 2 µ B T c (0) 2(51) with κ c 2 ranging from 0.0066(7) [42,43] to 0.0135(20) [44], 0.0149(21) [45] and 0.020(4) [46]. Lines that cover this spread in curvature parameters are also shown in Fig. 15 (left) for T c (0) = 155 MeV. While a small curvature for the crossover line would suggest that the crossover transition happens under more or less identical bulk thermodynamic conditions a large curvature obviously would indicate that the crossover transition happens already at significantly smaller values of pressure and energy density as µ B /T increases. VII. RADIUS OF CONVERGENCE AND THE CRITICAL POINT As discussed in the previous sections we generally find that the Taylor series for all basic thermodynamic quantities converge well for values of baryon chemical potentials µ B ≤ 2T . Even in the low temperature regime the relative contribution of higher order expansion coefficients are generally smaller than in corresponding HRG model calculations. This, of course, also has consequences for our current understanding of the location of a possible critical point in the QCD phase diagram. The results on the expansion coefficients of the Taylor series for e.g. the pressure can be cast into estimates for the location of a possible critical point in the QCD phase diagram. In general the radius of convergence can be obtained from ratios of subsequent expansion coefficients in the Taylor series for the pressure. Equally well one may use one of the derivatives of the pressure series. As one has to rely on estimates of the radius of convergence that generally are based on a rather short series, it may indeed be of advantage to use as a starting point the series for the net baryon-number susceptibility [47], which diverges at the critical point, but still contains information from all expansion coefficients of the pressure series. The radius of convergence of this series is identical to that of the pressure. Model calculations also suggest that the estimators obtained from the susceptibility series converge faster to the true radius of convergence [48]. For µ Q = µ S = 0 the expansion coefficients of the Taylor series for the net baryon-number and results for this estimator obtained from calculations with an imaginary chemical potential (triangles) [15]. Also shown are estimates for the location of the critical point obtained from calculations with unimproved staggered fermions using a reweighting technique [50] and Taylor expansions [51]. In both cases results have been rescaled using Tc = 154 MeV. susceptibility are again simply related to that of the pressure, χ B 2 (T, µ B ) = ∞ n=0 1 (2n)! χ B 2n+2μ 2n B .(52) From this one obtains estimators for the radius of convergence of the pressure and susceptibility series, r P 2n = (2n + 2)(2n + 1)χ B 2n χ B 2n+2 1/2 , r χ 2n = 2n(2n − 1)χ B 2n χ B 2n+2 1/2 .(53) Both estimators converge to the true radius of convergence in the limit n → ∞. In order for this to correspond to a singularity at real values ofμ B , all expansion coefficients should asymptotically stay positive. Obviously, the estimators r P 2n and r χ 2n are proportional to each other, r P 2n = (2n + 2)(2n + 1)/[2n(2n − 1)]r χ 2n . The difference between these to estimators may be taken as a systematic error for any estimate of the radius of convergence obtained from a truncated Taylor series. In the hadron resonance gas limit one finds for estimators involving sixth order cumulants, r P 4 = 1.58r χ 4 . In the following we restrict our discussion to an analysis of r χ 2n , which at finite n leads to the smaller estimator for the radius of convergence. This seems to be appropriate in the present situation where we only can construct two independent estimators from ratios of three distinct susceptibilities. We thus may hope to identify regions in the QCD phase diagram at small values ofμ B which are unlikely locations for a possible critical point. An immediate consequence of the definitions given in Eq. 53 is that the ratios of generalized susceptibilities need to grow asymptotically like |χ B n+2 /χ B n | ∼ n 2 in order to arrive in the limit n → ∞ at a finite value for the radius of convergence. At least for large values of n one thus needs to find large deviations from the hadron resonance gas results |χ B n+2 /χ B n | HRG = 1. As is obvious from the results presented in the previous sections, in particular from Fig. 3, the analysis of up to sixth order Taylor expansion coefficients does not provide any hints for such large deviations. The ratio χ B 4 /χ B 2 turns out to be less than unity in the entire temperature range explored so far, i.e. for T ≥ 135 MeV or T /T c > 0.87 (6). Below the crossover temperature, T ∼ 155 MeV, the sixth order expansion coefficients also are consistent with HRG model results. They still have large errors. However, using the upper value of the error for χ B 6 /χ B 4 provides a lower limit for the value of the estimator r χ 4 . For temperatures in the interval 135 MeV ≤ T ≤ 155 MeV (or equivalently 0.87(5) ≤ T /T c ≤ 1) we currently obtain a lower limit on r χ 4 from the estimate χ B 6 /χ B 4 χ B 6 /χ B 2 < 3. This converts into the bound r χ 4 ≥ 2, which is consistent with our observation that the Taylor series of all thermodynamic observables discussed in the previous sections is well behaved up to µ B = 2T . A more detailed analysis, using the current errors on χ B 6 /χ B 4 at five temperature values below and in the crossover region of the transition at µ B = 0, is shown in Fig. 16. This shows that the bound arising from r χ 4 is actually more stringent at temperatures closer to T c , where χ 6 starts to become small and eventually tends to become negative. These findings are consistent with recent results for susceptibility ratios obtained from calculations with an imaginary chemical potential [15]. Also in that case all susceptibility ratios are consistent with HRG model results. At present one thus cannot rule out that the radius of convergence may actually be infinite. Results for r χ 4 obtained in Ref. [15] lead to even larger estimators for the radius of convergence than our current lower bound. This is also shown in Fig. 16. The observations and conclusions discussed above are in contrast to estimates for the location of a critical point obtained from a calculation based on a reweighting technique [50] as well as from Taylor series expansion in 2-flavor QCD [49,51]. Both these calculations have been performed with unimproved staggered fermion discretization schemes and thus may suffer from large cut-off effects. Moreover, the latter calculation also suffers from large statistical errors on higher order susceptibilities. Results from Ref. [50] and Ref. [51] are also shown in Fig. 16. We thus conclude from our current analysis that a critical point at chemical potentials smaller than µ B = 2T is strongly disfavored in the temperature range 135 MeV ≤ T ≤ 155 MeV and its location at higher values of temperature seems to be ruled out. Our results suggest that the radius of convergence in that temperature interval will turn out to be significantly larger than the current bound once the statistics on 6 th order cumulants gets improved and higher order cumulants become available. VIII. CONCLUSIONS We have presented results on the equation of state of strong-interaction matter obtained from a sixth order Taylorexpansion of the pressure of (2+1)-flavor QCD with physical light and strange quark masses. We discussed expansions at vanishing strangeness chemical potential µ S = 0 as well as for strangeness neutral systems n S = 0. We have discussed in detail the latter case for a fixed electric charge to net baryon-number ratio, n Q /n B = 0.4, which is appropriate for situations met in heavy ion collisions. The results, however, can easily be extended to arbitrary ratios of n Q /n B . We find that the dependence of basic thermodynamic observables on n Q /n B is small for 0 ≤ n Q /n B ≤ 1/2. This may be of interest for applications in heavy ion collisions where strong external magnetic fields and non-trivial topology in QCD can lead to charge asymmetries in different regions of phase space. We have presented a parametrization of basic thermodynamic observables in terms of ratios of fourth order polynomials in the inverse temperature which is appropriate in the temperature range studied here, i.e. T ∈ [130 MeV, 330 MeV]. We presented results for lines of constant pressure, energy and entropy density in the T -µ B plane and showed that corrections of O(μ 4 B ) are negligible forμ B < 2. For all three observables the curvature term at O(μ 2 B ) is smaller than κ max = 0.012. This suggest that, e.g. energy density and pressure, would drop on the crossover line for the chiral transition, if the corresponding curvature coefficient turns out to be larger than κ max . The Taylor series for pressure and net baryon-number density as well as energy density and entropy density determined for µ S = 0 as well as n S = 0 have expansion coefficients that are close to HRG model results at low temperature. In general ratios of subsequent expansion coefficients approach the corresponding HRG model values from below when lowering the temperature. As a consequence, in the entire temperature range explored so far, the expansions are "better behaved" than the HRG model series, which have an infinite radius of convergence. Assuming that the current results obtained with expansion coefficients up to 6 th order are indicative for the behavior of higher order expansion coefficients and taking into account the current errors on 6 th order expansion coefficients we concluded that at temperatures T > 135 MeV the presence of a critical point in the QCD phase diagram for µ B ≤ 2T is unlikely. in Germany. These grants provided access to resources on Titan at ORNL, BlueGene/Q at ALCF and NIC, Cori I and II at NERSC and Marconi at CINECA. Additional numerical calculations have been performed on USQCD GPU and KNL clusters at JLab and Fermilab, as well as GPU clusters at Bielefeld University, Paderborn University, and Indiana University. We furthermore acknowledge the support of NVIDIA through the CUDA Research Center at Bielefeld University. We are interested in expansion coefficients for strangeness neutral systems in which the net electric-charge is proportional to the net baryon-number. I.e. we introduce the constraint given in Eq. 31. These constraints can be fulfilled order by order in the Taylor expansion of the number densities by choosing the expansion coefficients of the series forμ Q andμ S , given in Eq. 14, appropriately, i.e. the coefficients s n and q n can be determined order by order. We start with the Taylor series for the number densities introduced in Eq. 16 and define the expansion coefficients as N B n = s n χ BS 11 + q n χ BQ 11 + m B n (B1) N Q n = s n χ QS 11 + q n χ Q 2 + m Q n (B2) N S n = s n χ S 2 + q n χ QS 11 + m S n(B3) for n = 1, 3, 5. At each order in the expansion we then have to solve a set of two linear equations, which always have the same structure. We find as solutions s n = − q n χ QS 11 + m S n χ S 2 ,(B4) and q n = −m B n rχ S 2 + m Q n χ S 2 + m S n (rχ BS 11 − χ QS 11 ) (χ QS 11 ) 2 − rχ BS 11 χ QS 11 + rχ S 2 χ BQ 11 − χ S 2 χ Q 2 .(B5) At leading order one finds for the terms m X 1 , m B 1 = χ B 2 , m Q 1 = χ BQ 11 , m S 1 = χ BS 11 ,(B6) and the contributions to the next-to-leading order expansion terms, m X 3 , are given by m B 3 = 1 6 3q 2 1 s 1 χ BQS 121 + 3q 1 s 2 1 χ BQS 112 + 6q 1 s 1 χ BQS 211 + q 3 1 χ BQ 13 + 3q 2 1 χ BQ 22 +3q 1 χ BQ 31 + s 3 1 χ BS 13 + 3s 2 1 χ BS 22 + 3s 1 χ BS 31 + χ B 4 m Q 3 = 1 6 3q 2 1 s 1 χ QS 31 + 3q 1 s 2 1 χ QS 22 + 6q 1 s 1 χ BQS 121 + q 3 1 χ Q 4 + 3q 2 1 χ BQ 13 +3q 1 χ BQ 22 + s 3 1 χ QS 13 + 3s 2 1 χ BQS 112 + 3s 1 χ BQS 211 + χ BQ 31 m S 3 = 1 6 3q 2 1 s 1 χ QS 22 + 3q 1 s 2 1 χ QS 13 + 6q 1 s 1 χ BQS 112 + q 3 1 χ QS 31 + 3q 2 1 χ BQS 121 +3q 1 χ BQS 211 + s 3 1 χ S 4 + 3s 2 1 χ BS 13 + 3s 1 χ BS 22 + χ BS 31 (B7) Finally the contributions to the next-to-next-to-leading order expansion terms, m X 5 , are given by m In (2+1)-flavor QCD calculations the light (u, d) quark masses are taken to be degenerate. A consequence of this degeneracy is that not all generalized susceptibilities χ BQS ijk that enter the above expressions are independent. In a given order n ≡ 2l ≡ i + j + k this results in a set of relations among the expansion coefficients. In general, at order n = 2l, there are l(l + 1) constraints, i.e. for l = 1 this gives rise to two relations, [27] 0 = χ B 2 − 2χ BQ 11 + χ BS 11 0 = χ S 2 − 2χ QS 11 + χ BS 11 ,(B9) Using these constraints it is tedious, but straightforward, to show that in the isospin symmetric case, r = 1/2, indeed all expansion coefficients for the electric charge chemical potential vanish, i.e.μ Q = 0 to all orders in µ B . We show results for the LO expansion coefficients s 1 and q 1 and the ratios of the NLO and LO expansion coefficients, s 3 /s 1 and q 3 /q 1 in Fig. 17. As can be seen the NLO coefficients are already negligible for T > ∼ 170 MeV. The absolute value of the NNLO expansion coefficients s 5 and q 5 never is larger than 1% of the corresponding LO coefficients. In Fig. 17, we also show results from hadron resonance gas (HRG) model calculations. The black curves are the predictions of the usual HRG model which consists of all the resonances listed in the Particle Data Group Tables up to 2.5 GeV (PDG-HRG). The PDG-HRG results for s 1 are substantially smaller than the continuum extrapolated lattice QCD results. It has been argued in [6] that this can be caused by contributions from additional, experimentally not yet observed, strange hadron resonances which are predicted in quark model calculations. A HRG model calculation based on such an extended resonance spectrum (QM-HRG) is also shown in Fig. 17. At finite values of the lattice cut-off we observe significant differences between lattice QCD calculations and both versions of the HRG models. This is in particular the case for the expansion coefficients of the electric charge chemical potentials. One thus may wonder whether these deviations can be understood in terms of taste violations in the staggered fermion formulation which result in a modification of the resonance spectrum and affect most strongly the light pseudo-scalar (pion) sector. In particular, we will give explicit expressions for the case of constant pressure (f ≡ P ), constant energy density (f ≡ ) and constant entropy density(f ≡ s). For the pressure we had the earlier expression (Eq. (15)) P (T, µ B ) − P (T, 0) T 4 = ∞ n=1 P 2nμ 2n B .(C3) Comparing Eqs. (C3) and (C2) we have, f 0 = P (T, 0) ≡ T 4 P 0 , f 2 = T 4 P 2 and f 4 = T 4 P 4 . Thus, ∂f 0 ∂T µ B = ∂P 0 T 4 ∂T µ B = T 3 (T P 0 + 4P 0 ) ≡ s,(C4a)∂ 2 f 0 ∂T 2 µ B = ∂ 2 P 0 T 4 ∂T 2 µ B = T 2 T 2 P 0 + 8T P 0 + 12P 0 ≡ C V T .(C4b) Here s and C V are the entropy density and specific heat per unit volume at vanishing chemical potential. Similarly, ∂f 2 ∂T µ B = ∂P 2 T 4 ∂T µ B = T 3 (T P 2 + 4P 2 ) .(C5) Putting everything together we get, for the pressure: κ P 4 = 1 T P 0 + 4P 0 P 4 − κ P 2 (T P 2 + 2P 2 ) + 1 2 κ P 2 2 T 2 P 0 + 8T P 0 + 12P 0 = T 3 s P 4 (T ) − κ P 2 σ 2 (T ) + 1 2 κ P 2 2 C V T 3 ,(C6) where σ 2 denotes the O(μ 2 B ) expansion coefficient of the entropy density as introduced in Eq. 24. Next we consider κ 4 . Since the energy density is also of dimension four, we only need to replace P 2n with 2n in the first line of Eq. (C6). With this we obtain, κ 4 = 1 T 0 + 4 0 4 − κ 2 (T 2 + 2 2 ) + 1 2 (κ 2 ) 2 T 2 0 + 8T 0 + 12 0 . Since C V ≡ (∂ 0 /∂T ) µ B , the above may be written as κ 4 = T 3 C V 4 − κ 2 (T 2 + 2 2 ) + 1 2 (κ 2 ) 2 1 T 2 ∂C V ∂T .(C8) Finally we consider κ s 4 . Since the entropy density is of dimension three, Eqs. (C4) become ∂(sT 3 ) ∂T µ B = T 2 (T s + 3s) , ∂ 2 (sT 3 ) ∂T 2 µ B = T T 2 s + 6T s + 6s ,(C9) and therefore κ σ 4 = 1 T s + 3s σ 4 − κ σ 2 (T σ 2 + σ 2 ) + 1 2 (κ σ 2 ) 2 T 2 s + 6T s + 6s .(C10) To zeroth order, the specific heat is also given by C V = (∂(T s)/∂T ) µ B . Thus, κ σ 4 = T 3 C V σ 4 − κ σ 2 (T σ 2 + σ 2 ) + 1 2 (κ σ 2 ) 2 1 T 2 ∂C V ∂T . (C11) numbers: 11.10.Wx, 12.38.Gc, 12.38Mh a [email protected] arXiv:1701.04325v3 [hep-lat] 9 Mar 2017 I. INTRODUCTION Figure 1 . 1n-th order Taylor series, (∆P )n for (∆P )∞ = PB(T, 0)(cosh(x) − 1) compared to the exact result. The insertion shows the relative error due to truncation of the Taylor series after n-th order. Note that the sixth order result is hardly visible behind the exact result. Figure 2 . 2The leading order (O(µ 2 B )) correction to the pressure calculated at zero baryon chemical potential. The left hand figure shows the leading order correction in a large temperature range. The right hand part of the figure shows an enlarged view into the low temperature region. In addition to the continuum extrapolation of the lattice QCD results we also show results from HRG model calculations based on all hadron resonances listed by the particle data group (PDG-HRG) and obtained in quark model calculations (QM-PDG). Figure 3 . 3Left: The ratio of fourth and second order cumulants of net-baryon number fluctuations (χ B 4 /χ B 2 ) versus temperature. Right: same as the left hand side, but for the ratio of sixth and second order cumulants of net-baryon number fluctuations (χ B 6 /χ B 2 ). The boxes indicate the transition region, Tc = (154 ± 9) MeV. Grey bands show continuum estimate. Figure 4 . 4The µB-dependent contribution to the pressure (left) and the baryon-number density (right) in the case of vanishing electric charge and strangeness chemicals potential for several values of the baryon chemical potential in units of temperature. The different bands show results including Taylor series results upto the order indicated. Figure 5 . 0 50The ratio of net strangeness and net baryon-number densities (left) and the ratio of net electric charge and net baryon-number densities (right). At low temperatures results from hadron resonance gas calculations at µB = Figure 6 . 6Leading order (left) and next-to-leading order (right) expansion coefficients for the µB-dependent part of pressure, the energy and entropy densities in the case µQ = µS = 0. The inset in the right hand figure shows the ratios of NLO and LO expansion coefficients P4/P2, 4/ 2 and σ4/σ2. Note that the expansion coefficients for the net baryon-number density are directly proportional to those of the pressure series, i.e. N B 1 = 2P2 and N B 3 = 4P4. Figure 7 . 7(Left) The total pressure in (2+1)-flavor QCD in O(μ 6 B ) for several values of µB/T . (Right) The total energy density in (2+1)-flavor QCD in O(μ 6 B ) for several values of µB/T . The results forμB = 0 are taken from Ref. [3]. Figure 8 . 8Expansion coefficients of the pressure (top, and bottom left) and the ratio of net baryon-number density and pressure expansion coefficients (bottom, right) in strangeness neutral systems with r = 0.4. Broad bands show continuum extrapolations as discussed in Section III. The darker lines in the center of the error bands of these extrapolations show the interpolating fits discussed in Subsection V B. At low temperature lines for HRG model calculations based on hadron resonances listed by the Particle Data Group is shown. Figure 10 . 10Ratio of expansion coefficients of the pressure in strangeness neutral systems with r = 0.4. The darker lines in the center of the error bands of these extrapolations show results obtained with the parametrization discussed in Subsection V B. Figure 11 . 11The µB dependent contribution to the pressure (top, left) and the baryon-number density (top, right) for several values of the baryon chemical potential in units of temperature. The lower two panels show these quantities normalized to the corresponding HRG model values, obtained from a calculation with all baryon resonances, up to mass mH = 2.5 GeV, listed in the PDG tables, as function of µB/T for three values of the temperature. Figure 12 . 12Leading order (left) and next-to-leading order (right) expansion coefficients for the µB-dependent part of pressure, the energy and entropy densities in the strangeness neutral case with fixed electric charge to net baryon-number density, nQ/nB = 0.4. The darker lines in the center of the error bands of these extrapolations show the interpolating fits discussed in Subsection V B. The insert in the right hand figure shows the ratios of NLO and LO expansion coefficients N B 3 /N B 1 , P4/P2, 4/ 2 and σ4/σ2. The influence of a non-vanishing electric charge chemical potential, which formally gives rise to deviations from the result in the isospin symmetric limit (N B 1 = 2P2, N B 3 = 4P4), are negligible at O(μ 2 B ) and O(μ 4 B ). For that reason we do not show results for N B 1 and N B 3 . However, we show in the insertion in the left hand figure the ratio N B 3 /N B 1 (black line) which clearly shows that NLO corrections are a factor two larger in the Taylor series for the number density then in the pressure series. Figure 13 . 13The total energy density (upper two curves) of (2+1)-flavor QCD for µB/T = 0 and 2, respectively. The lower two curves show corresponding results for three times the pressure. The dark lines show the results obtained with the stout action from analytic continuation with sixth order polynomials inμB[14]. . Parameters used in the ansatz given in Eq. 37 for the interpolation of the expansion coefficients of the net baryonnumber density of (2+1)-flavor QCD with vanishing net strangeness and a fixed ratio of electric charge and net baryon-number density, nQ/nB = 0.4. These interpolations have been determined for the temperature interval T ∈ [130 MeV, 280 MeV]. Also given are parameters needed for the interpolation of the expansion coefficients for the electric charge chemical potential (Eq. 38) and the coefficients for the parametrization of the pressure at µB = 0 given in Eq. 39. different values of the beam energy ( √ s N N ). Generally such criteria have been established by comparing experimental Figure 14 .Figure 15 . 1415Left: Second order curvature coefficients of lines of constant pressure, energy density and entropy density versus temperature in (2+1)-flavor QCD (bands) and in a HRG model (lines). Right: same as on the left, but for fourth order coefficients. The darker lines in the center of the error bands show the interpolating fits discussed in subsection V B. For κ 4 and κ s 4 only these interpolating curves are shown. Left: Lines of constant pressure, energy density and entropy density versus temperature in (2+1)-flavor QCD for three different initial sets of values fixed at µB = 0 and T0 = 145 MeV, 155 MeV and 165 MeV, respectively (see Figure 16 . 16Estimators for the radius of convergence of the Taylor series for net baryon-number fluctuations, χ B 2 (T, µB), in the case of vanishing electric charge and strangeness chemical potentials obtained on lattices with temporal extent Nτ = 8. Shown are lower bounds for the estimator r χ 4 obtained in this work (squares) ACKNOWLEDGMENTS This work was supported in part through Contract No. DE-SC001270 with the U.S. Department of Energy, through the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Nuclear Physics, the DOE funded BEST topical collaboration, the NERSC Exascale Application Program (NESAP), the grant 05P12PBCTA of the German Bundesministerium für Bildung und Forschung, the grant 56268409 of the German Academic Exchange Service (DAAD), grant 283286 of the European Union, the National Natural Science Foundation of China under grant numbers 11535012 and 11521064 and the Early Career Research Award of the Science and Engineering Research Board of the Government of India. Numerical calculations have been made possible through an INCITE grant of USQCD, ALCC grants in 2015 and 2016, and PRACE grants at CINECA, Italy, and the John von Neumann-Institute for Computing (NIC) Figure 17 . 17The LO Taylor expansion coefficients s1 (top, left) and q1 (top, right) of the expansions ofμS andμQ w.r.t.μB. The bottom set of figures show the ratios of NLO and LO expansion coefficients. The broad bands give the continuum extrapolated results. The curves inside these bands show results obtained with the interpolating curves introduced in Eq. 38. Also shown are the PDG-HRG and QM-HRG results (see text). The solid black lines labeled 'free quark gas' denote the T → ∞ non-interacting massless quark gas result. Appendix C: The coefficient κ f 4 of lines of constant physics at O(µ 4 B ) We will present here results for the expansion coefficient κ f 4 of lines of constant physics defined in Eq. (43), f 2k are defined by f (T, µ B ) = 6% at high temperature. In the infinite temperature ideal gas limit the ratios become N B(bottom, right) we show the ratio N B 2n−1 /nP n for the case r = 0.4 and n = 2 and 4, respectively. At O(μ 2 B ) differences between N B 1 and 2P 2 never exceed 2% and at O(μ 4 B ) the difference between N B 3 and 4P 4 varies between 3% at low temperature and -1 /2P 2 = 1.018 and N B 3 /4P 4 = 0.927, respectively. In general one finds that the dependence of bulk thermodynamic observables on the net electric charge to net baryon 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 140 160 180 200 220 240 260 280 µ S =0, µ Q =0 n S =0, r=0.0 0.2 0.4 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 140 160 180 200 220 240 260 280 T[MeV] P 2,r /P 2,r=1/2 m s /m l =20 (open) 27 (filled) (left) that LCPs for constant pressure,at µB = 0 on LCP T0 [MeV] p/T 4 0 /T 4 0 s/T 3 0 p [GeV/fm 3 ] [GeV/fm 3 ] s [fm −3 ] 145 0.586(80) 3.52(47) 4.11(53) 0.0337(46) 0.203(27) 1.63(21) 155 0.726(95) 4.61(55) 5.34(63) 0.0546(71) 0.346(41) 2.59(30) 165 0.898(110) 5.76(59) 6.66(69) 0.0868(106) 0.556(57) 3.90(40) Table II. Pressure, energy density and entropy density, characterizing lines of constant physics which correspond to the condi- tions met for µB = 0 at T0 = 145 MeV, 155 MeV and 165 MeV. Columns 2-4 give results in appropriate units of temperature, while columns 5-7 give the same results expressed in units of GeV and f m. Table III . IIIEnsemble parameters for calculations with light to strange quark mass ratio m l /ms = 1/20 on lattices of size N 3 σ Nτ with Nτ = 6, 8 and Nσ = 4Nτ . Columns 4 and 8 give the number of gauge field configurations, separated by 10 RHMC steps, that contributed to the analysis of up to sixth order generalized susceptibilities χ BQS ijk . MeV] #conf. 5.980 0.00435 135.29 81200 6.245 0.00307 134.64 180320 6.640 0.00196 134.94 5834 6.010 0.00416 139.71 120790 6.285 0.00293 140.45 172110 6.680 0.00187 140.44 5833 6.045 0.00397 145.05 120770 6.315 0.00281 144.95 138150 6.712 0.00181 144.97 13846 6.080 0.00387 150.59 79390 6.354 0.00270 151.00 107510 6.754 0.00173 151.10 14200 6.120 0.00359 157.17 66180 6.390 0.00257 156.78 135730 6.794 0.00167 157.13 15476 6.150 0.00345 162.28 79660 6.423 0.00248 162.25 115850 6.825 0.00161 161.94 16772 6.170 0.00336 165.98 49760 6.445 0.00241 165.98 120270 6.850 0.00157 165.91 19542 6.200 0.00324 171.15 122700 6.474 0.00234 171.02 139980 6.880 0.00153 170.77 21220 6.225 0.00314 175.76 122730 6.500 0.00228 175.64 133070 6.910 0.00148 175.76 12303Table IV. Same as Table IIIbut for the light to strange quark mass ratio m l /ms = 1/27 and including results for Nτ = 12.Appendix B: Constraints on chemical potential for strangeness neutral systems with fixed electric charge to baryon-number ratioNτ = 6 Nτ = 8 Nτ = 12 β m l T[MeV] #conf. β m l T[MeV] #conf. β m l T[ for l = 2 there are six constraints,0 = χ B 4 − 2χ BQ 31 + χ BS 31 0 = χ S 4 − 2χ QS 13 + χ BS 13 0 = χ BS 22 + χ BS 13 − 2χ BQS 112 0 = χ BS 31 + χ BS 22 − 2χ BQS 211 0 = χ B 4 − 6χ BQ 31 + 12χ BQ 22 − 8χ BQ 13 + 3χ BS 31 + 3χ BS 22 + χ BS 13 − 12χ BQS 211 + 12χ BQS 121 − 6χ BQS 112 0 = χ S 4 + χ BS 31 + 3χ BS 22 + 3χ BS 13 − 8χ QS 31 + 12χ QS 22 − 6χ QS 13 − 6χ BQS 211 + 12χ BQS 121 − 12χ BQS 112 (B10) and for l = 3 there are twelve constraints, 0 = χ B 6 − 2χ BQ 51 + χ BS 51 , 0 = χ BS 15 − 2χ QS 15 + χ S 6 , 0 = χ BS 42 − 2χ BQS 312 + χ BS 33 , 0 = χ BS 33 − 2χ BQS 213 + χ BS 24 , 0 = χ BS 51 − 2χ BQS 411 + χ BS 42 , 0 = χ BS 24 − 2χ BQS 114 + χ BS 15 , 0 = χ B 6 − 6χ BQ 51 + 12χ BQ 42 − 8χ BQ 33 + 3χ BS 51 − 12χ BQS 411 + 12χ BQS 321 + 3χ BS 42 − 6χ BQS 312 + χ BS 33 , 0 = χ BS 33 − 6χ BQS 213 + 12χ BQS 123 − 8χ QS 33 + 3χ BS 24 − 12χ BQS 114 + 12χ QS 24 + 3χ BS 15 − 6χ QS 15 + χ S 6 , 0 = χ BS 42 − 6χ BQS 312 + 12χ BQS 222 − 8χ BQS 132 + 3χ BS 33 − 12χ BQS 213 + 12χ BQS 123 + 3χ BS 24 − 6χ BQS 114 + χ BS 15 , 0 = χ BS 51 − 6χ BQS 411 + 12χ BQS 321 − 8χ BQS 231 + 3χ BS 42 − 12χ BQS 312 + 12χ BQS 222 + 3χ BS 33 − 6χ BQS 213 + χ BS 24 , 0 = χ B 6 − 10χ BQ 51 + 40χ BQ 42 − 80χ BQ 33 + 80χ BQ 24 − 32χ BQ 15 + 5χ BS 51 − 40χ BQS 411 + 120χ BQS 321 −160χ BQS 231 + 80χ BQS 141 + 10χ BS 42 − 60χ BQS 312 + 120χ BQS 222 − 80χ BQS 132 + 10χ BS 33 − 40χ BQS 213 +40χ BQS 123 + 5χ BS 24 − 10χ BQS 114 + χ BS 15 , 0 = χ BS 51 − 10χ BQS 411 + 40χ BQS 321 − 80χ BQS 231 + 80χ BQS 141 − 32χ QS 51 + 5χ BS 42 − 40χ BQS 312 + 120χ BQS 222 −160χ BQS 132 + 80χ QS 42 + 10χ BS 33 − 60χ BQS 213 + 120χ BQS 123 − 80χ QS 33 + 10χ BS 24 − 40χ BQS 114 +40χ QS 24 + 5χ BS 15 − 10χ QS 15 + χ S 6 . We thank the HotQCD Collaboration for providing access to the second order quark number susceptibilities. It has been pointed out that the point-like particle approximation is appropriate in the meson sector but not in the baryon sector at high density. Introducing a non-zero size of hadron resonances[28,29] may, for some observables, improve the comparison with QCD thermodynamics[30,31]. However, it seems that the introduction of several additional parameters will be needed to achieve overall good agreement with the many observables calculated now in QCD in the temperature range of interest, i.e. in the crossover region from a hadron gas to strongly interacting quark-gluon matter. Appendix A: Details on simulation parameters and data sets Our main data sets have been generated on lattices of size N 3 σ × N τ , with N σ /N τ = 4 and N τ = 6, 8 and 12. We performed calculations with two different light to strange quark mass ratios, m l /m s = 1/20 and 1/27, respectively. 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[]
[ "First-passage time statistics for non-linear diffusion", "First-passage time statistics for non-linear diffusion" ]
[ "Przemys Law \nInstitute of Spintronics and Quantum Information\nFaculty of Physics\nAdam Mickiewicz University\nUniwersytetu Poznańskiego 261-614PoznańPoland\n", "Che Lminiak \nInstitute of Spintronics and Quantum Information\nFaculty of Physics\nAdam Mickiewicz University\nUniwersytetu Poznańskiego 261-614PoznańPoland\n" ]
[ "Institute of Spintronics and Quantum Information\nFaculty of Physics\nAdam Mickiewicz University\nUniwersytetu Poznańskiego 261-614PoznańPoland", "Institute of Spintronics and Quantum Information\nFaculty of Physics\nAdam Mickiewicz University\nUniwersytetu Poznańskiego 261-614PoznańPoland" ]
[]
Evaluating the completion time of a random algorithm or a running stochastic process is a valuable tip not only from a purely theoretical, but also pragmatic point of view. In the formal sense, this kind of a task is specified in terms of the first-passage time statistics. Although first-passage properties of diffusive processes, usually modeled by different types of the linear differential equations, are permanently explored with unflagging intensity, there still exists noticeable niche in this subject concerning the study of the non-linear diffusive processes. Therefore, the objective of the present paper is to fill this gap, at least to some extent. Here, we consider the non-linear diffusion equation in which a diffusivity is power-law dependent on the concentration/probability density, and analyse its properties from the viewpoint of the first-passage time statistics. Depending on the value of the power-law exponent, we demonstrate the exact and approximate expressions for the survival probability and the first-passage time distribution along with its asymptotic representation. These results refer to the freely and harmonically trapped diffusing particle. While in the former case the mean first-passage time is divergent, even though the first-passage time distribution is normalized to unity, it is finite in the latter. To support this result, we derive the exact formula for the mean first-passage time to the target prescribed in the minimum of the harmonic potential. *
null
[ "https://export.arxiv.org/pdf/2211.13055v1.pdf" ]
253,802,121
2211.13055
0e2b5de25ad6a72a91c86689f104e636eb981a45
First-passage time statistics for non-linear diffusion 22 Nov 2022 Przemys Law Institute of Spintronics and Quantum Information Faculty of Physics Adam Mickiewicz University Uniwersytetu Poznańskiego 261-614PoznańPoland Che Lminiak Institute of Spintronics and Quantum Information Faculty of Physics Adam Mickiewicz University Uniwersytetu Poznańskiego 261-614PoznańPoland First-passage time statistics for non-linear diffusion 22 Nov 2022(Dated: October 2022) Evaluating the completion time of a random algorithm or a running stochastic process is a valuable tip not only from a purely theoretical, but also pragmatic point of view. In the formal sense, this kind of a task is specified in terms of the first-passage time statistics. Although first-passage properties of diffusive processes, usually modeled by different types of the linear differential equations, are permanently explored with unflagging intensity, there still exists noticeable niche in this subject concerning the study of the non-linear diffusive processes. Therefore, the objective of the present paper is to fill this gap, at least to some extent. Here, we consider the non-linear diffusion equation in which a diffusivity is power-law dependent on the concentration/probability density, and analyse its properties from the viewpoint of the first-passage time statistics. Depending on the value of the power-law exponent, we demonstrate the exact and approximate expressions for the survival probability and the first-passage time distribution along with its asymptotic representation. These results refer to the freely and harmonically trapped diffusing particle. While in the former case the mean first-passage time is divergent, even though the first-passage time distribution is normalized to unity, it is finite in the latter. To support this result, we derive the exact formula for the mean first-passage time to the target prescribed in the minimum of the harmonic potential. * I. INTRODUCTION The first-passage time statistics has attracted much attention of the scientific community for more than the past century [1][2][3] and last until today. Noteworthy is the collection of excellent books [4,5] and review papers [6][7][8][9] that have been published in recent years. The research topic included in these publications reveals that the first-passage phenomena appears in such diverse disciplines as applied mathematics, physics, chemistry, biology and even economics and finance, to name but a few examples. The fundamental concept used in exploration of this kind of phenomena is the first-passage time distribution, according to which the very time is thought of as a random variable. Its average value, in formal terms the first moment of the time distribution, is called the mean first-passage time, also known as hitting time, crossing time or exit time, depending on the specific problem. For this reason, the dynamics of systems studied as part of the above-mentioned disciplines are typically represented by stochastic processes [10,11]. The mean first-passage time is then the first moment when the stochastic process reaches a predetermined state, starting form some initial state. The illustrative example is the mean time for a Brownian particle to hit a prescribed spatial position; the mean time for an enzyme to recognize and interact with a substrate molecule; and the mean time when the stock prise of a product exceeds a certain threshold. A well-known prototype of the stochastic process is a diffusive motion, whose first-passage properties are the main objective of the present paper. In recent decades, the first-passage time statistics has been explored for a variety of diffusive processes such as ordinary diffusion [12], diffusion in external potentials [13], in an Euclidean domains [14][15][16], hierarchical or fractal-like porous media [17,18] and heterogeneous media [19]. Of particular interest were also continuous-time random walk [20], fractional Brownian motion [21], Lévy flights and walks [22,23] and self-similarity of diffusions' first passage times [24]. For a dozen years the current research topic, inter alia in the context of the first-passage problems, are diffusive processes intermittent by stochastic resetting [25,26]. The joint feature of the most listed processes is their space-time dynamics which are formalized in terms of more or less elaborate linear partial differential equations. This fact raises the legitimate question about the first-passage properties of the non-linear diffusive processes. In this paper, we focus on the special variant of the non-linear diffusion equation, known as porous medium equation [27], in which a diffusion coefficient is power-law dependent on the probability density or concentration of particles. In addition, this relation is parameterized by the power-law exponent that will be assumed to be positive and constant. The porous medium equation has found many applications in the study of such disparate transport phenomena as compressible gas flow through porous media [28], heat propagation occurring in plasma [29], groundwater flow in fluid mechanics [30], population migration in biological environment [31,32], the diffusion of grains in granular matter [33] and gravity-driven fluid flow in layered porous media [34]. Here, we study the porous medium equation form yet another perspective, namely the first-passage time statistics. Although such a problem has already been analysed for the fractional non-linear diffusion equation in Ref. [35], some of the results presented in that paper seem to be at least controversial. The reason for our criticism is the improper assumption made by the authors of the cited work that the target point, to which the diffusing object moves and then reaches for the first time, comprises the totally absorbing well. To uphold our objection, we will evidently show that trying to solve the non-linear diffusion equation in the presence of the absorbing boundary condition leads to some kind of contradiction. Instead, we will demonstrate the alternative approach to the first-passage time problem concerning the non-linear diffusion. Let us clarify that throughout this paper the diffusive motion will be restricted to the semi-infinite interval with the target point located at the origin. For such a system we determine the survival probabilities, depending on the power-law exponent characterizing the relation between the diffusion coefficient and the probability density. Surprisingly, this thread has been omitted for some reasons in Ref. [35]. In general, the survival probability defines a likelihood that the first-passage event has not occurred until a given time interval. It is equal to unity at the initial moment of time and then immediately begins to diminish in time. However, by contrast with the ordinary diffusion, the time course of the survival probability for the non-linear diffusion takes place in two phases. Through the first period of time its value constantly remains equal to unity and only in the second phase it monotonically decreases to zero. We show that such a progress of the survival probability in time results from the time evolution of the probability density whose domain is limited to the support of finite extend, outside of which this function disappears. Consequently, it takes some time for the front of the probability density to reach the prescribed target point. Armed with the survival probability, we calculate the first-passage time distribution defined as the time derivative of that former quantity with a minus sign. We point out that due to the long-time tails of the first-passage time distributions obtained for the non-linear diffusion in the semiinfinite interval, their first moment, namely, the mean time to the origin, is divergent. We argue that this hitting time becomes finite as far as a diffusive motion occurs in a bounded domain of a space. To justify our statement, we consider as an example the harmonically trapped particle whose dynamics are determined by the non-linear diffusion equation. The structure of the paper is as follows. In the subsequent section we give a brief overview of the non-linear diffusion equation. Sec. III is reserved to revise the basic concepts of the first-passage time statistics. Here, we also give the reason for which the absorbing boundary conditions are not compatible with the non-linear diffusion equation. In Sec. IV we present the main results regarding first-passage properties of the non-linear diffusion. The analysis of this process in the harmonic potential is performed in Sec. V. We summarize our results in Sec. VI. II. NON-LINEAR DIFFUSION EQUATION In what follows, we restrict our studies of the non-linear diffusion along with its firstpassage time statistics to one dimension. Before we formulate a special type of the equation describing this process, let us firstly consider its more general form (see for example [36]), namely ∂ ∂t p(x, t) = ∂ ∂x D [ p(x, t)] ∂ ∂x p(x, t) .(1) By definition, this is the non-linear partial differential equation for the function p(x, t), which in a physical sense may stand for, depending on the context, the concentration of diffusing particles, where x is the distance from some initial position and t is the time, or the probability density function (PDF) of finding a diffusing particle in the location x at time t. In this paper we will consequently use the latter interpretation. The reason for the non-linear nature of Eq. (1) is a direct dependence of the diffusivity D[p(x, t)] on the PDF through which it also depends on the variables x and t. Therefore, to specify the particular form of Eq. (1) we have yet to establish a specific relationship between the diffusion coefficient and the PDF. Due to many interesting and practical applications that have attracted considerable attention within scientific community [37], we define this relation by the power-law function D = D 0 p(x, t) p 0 σ .(2) In this expression p 0 denotes a constant reference value of a probability density, whereas D 0 is the diffusivity at that reference value. The power-law exponent σ is a certain parameter. Only in the particular case for σ = 0, Eq. (1) converts into the linear diffusion equation with a diffusion constant D 0 . To give Eq. (1) a more convenient form, we now rewrite the diffusion coefficient in Eq. (2) so that D = Dp σ (x, t). Here, the parameter D = D 0 /p σ 0 is the generalized diffusion coefficient of the physical dimension [D] = L σ+2 /T, where L and T are units of the length and the time, respectively. In consequence, the non-linear diffusion equation is as follows: ∂ ∂t p(x, t) = D ∂ ∂x p σ (x, t) ∂ ∂x p(x, t) .(3) A commonly known procedure for solving this class of equations is offered by the method of similarity solutions that utilizes an algebraic symmetry of a differential equation. In order to find its solution, we insert into Eq. (3) a similarity transformation of the algebraic form p(x, t | x 0 ) = 1 T (t) F x − x 0 T (t) ≡ F (z) T (t) , with z = x − x 0 T (t) ,(4) for the PDF of appearing a particle in x at time t, if it was initially localized in the position x 0 at time t = 0. In this way, we effectively reduce the original partial differential equation for the non-linear diffusion to the system of two ordinary differential equations for the separate functions T (t) and F (z) which are relatively easy to solve. We omit detailed calculations here and refer the interested reader to [36], where the discussed method is accessibly explained. Thus, the final result takes the form p(x, t | x 0 ) = 1 T (t) a − bσ 2D x − x 0 T (t) 2 1 σ , with(5)T (t) = [b(σ + 2)t] 1 σ+2 , where a and b are arbitrary integration constants. Their specific values can be determined by adopting suitable boundary conditions. For example, if we set a = 1, with no substantiation for now, and impose the normalization condition ∞ −∞ p(x, t | x 0 )dx = 1 on Eq. (5), performing appropriate integration with a help of the Euler beta function B(ν, µ) = Γ(ν)Γ(µ) Γ(ν+µ) = 2 1 0 z 2ν−1 (1 − z 2 ) µ−1 dz [38] , we find the unknown b and eventually a typical solution of Eq. (3) in the Zel'dovitch-Barenblatt-Pattle algebraic form [39][40][41] p(x, t | x 0 ) = A(σ) (Dt) 1 σ+2 1 − B(σ) (x − x 0 ) 2 (Dt) 2 σ+2 1 σ ,(6) where the two σ-dependent coefficients in the above PDF are A(σ) = σ 2π(σ + 2) Γ 1 σ + 3 2 Γ 1 σ + 1 2 σ+2(7) and B(σ) = σ 1 2(σ + 2) 2 σ+2 √ π Γ 1 σ + 1 √ σ Γ 1 σ + 3 2 2σ σ+2 .(8) The plot in Fig. 1 depicts profiles of the PDF in two different moments of time. A supplementary comment is necessary at this point. The formulae given in Eqs. (5) and (6) do not guarantee that the PDF for the non-linear diffusion is always a real and non-negative func- tion of x as it should be by virtue of a very definition of the probability density p(x, t | x 0 ) 0. For this reason, we need to assume the additional requirement that the PDF can only be determined on the finite support |x − x 0 | B − 1 2 (Dt) 1 σ+2 . Everywhere outside this interval the PDF vanishes and such a property was taken into account when performing integration in the normalization condition to figure out the parameter b. We are still aware choosing a value of the parameter a = 1 without any explanation, which may raise serious reservation. The argument behind such a choice is as follows. Taking the limit σ → 0 and using the assertion that lim z→∞ Γ(z+α) Γ(z+β) z β−α = 1 (see [38]) along with α = 3/2, β = 1 and z = 1/σ, we obtain from Eqs. (7) and (8) that A = 1/ √ 4π and B = lim σ→0 σ/4. Simultaneously, expressing the right-hand site of Eq. (6) through the limit definition of the exponential function e −z = lim n→∞ (1 − z n ) n for n = 1/σ, we immediately retrieve the Gaussian distribution p(x, t | x 0 ) = 1 √ 4πDt exp − (x − x 0 ) 2 4Dt .(9) The above function is a fundamental solution of the linear partial differential equation for the free diffusion given by Eq. (3) with the initial condition p(x, 0 | x 0 ) = δ(x − x 0 ), whenever σ = 0. This result justifies our previous decision to set a = 1. A solution of the partial differential equation is uniquely determined by imposing appropriate boundary conditions. Among them the best known are periodic, reflecting and absorbing, as well as a linear (weighted) combination of the last two boundary conditions. The latter are a simplified version of the more general Robin boundary conditions, which assume that a given function defined on the perimeter of a spacial domain, on which the solution of a partial differential equation is to be found, corresponds to the weighted combination of this solution and its first derivative over the spatial coordinate. Furthermore, the spacial boundary conditions play a significant role in the context of the first-passage processes. A crucial quantity related to this problem is the mean first-passage time (MFPT) or the mean hitting time to a target. This average time is known to be finite when the process proceeds within the domain confined by, for instance, reflecting boundary conditions, and diverges to infinity in an unbounded space. In order to calculate its value, we have to determine either the first-passage time (FPT) distribution, the first moment of which is the MFPT, or the survival probability. Both these functions are directly related to the PDF satisfying the absorbing boundary condition at the target point. For this reason, the MFPT is sometimes called the mean time to absorption. However, associating the absorbing boundary condition with the non-linear diffusion equation rises a serious problem as we will show in the next section. Later, we will explain how to overcome this obstacle in order to construct the basic quantities that quantitatively characterise the first-passage properties of the non-linear diffusion. Finally, we present the main results of this paper. III. SURVIVAL PROBABILITY AND FIRST-PASSAGE TIME DISTRIBUTION Let us imagine a particle that starts from the initial position at x = x 0 > 0 and makes a diffusive motion along a semi-infinite interval 0 x < ∞ with a totally absorbing point at the origin x = 0. What is a chance that the particle survives before reaching the origin for the first time? The quantitative answer to this question is given in terms of a survival probability Q(t | x 0 ). In general, it is defined as a spatial integral of the PDF over a certain region of space where a stochastic process takes place in the presence of an absorbing trap localized somewhere at the perimeter or inside of this region [10]. For diffusion in the semi-infinite interval we have Q(t | x 0 ) = ∞ 0 p(x, t | x 0 ) dx,(10) where the PDF is a solution of a partial differential equation which satisfies the absorbing boundary condition, i.e. p(0, t | x 0 ) = 0 at the origin x = 0. Furthermore, the survival probability is supplemented by additional conditions resulting from natural requirements imposed on the PDF. The first property is a direct consequence of an initial condition p(x, 0 | x 0 ) = δ(x − x 0 ) stating that the particle begins its diffusive motion from the position localized at x = x 0 . Using this property in Eq. (10) along with the normalization condition of the Dirac delta function, i.e. ∞ 0 δ(x − x 0 ) dx = 1, we easily obtain that Q(0 | x 0 ) = 1. The second feature of the survival probability relates to the situation when the initial position of a particle coincides with the absorbing point, i.e. x 0 = 0. In this case the particle remains there forever which means that Q(t | 0) = 0 at any time t > 0. This rule also holds for the PDF. The two properties of the survival probability considered so far are mostly used together with the so called backward diffusion (Fokker-Planck) equation describing the time evolution of this quantity [42]. The last property of the survival probability emerges form our conviction that the particle will be eventually absorbed at the origin for times large enough, so Q(t | x 0 ) → 0 when t → ∞. We should, however, emphasize that this asymptotic limit is not always satisfied. A good example is a biased diffusion in the semi-infinite interval where the behavior of the survival probability depends on whether a drift velocity is positive or negative (see [4]). Armed with the survival probability, we can now consider of how long the diffusing particle will persevere in the semi-infinite interval before reaching the absorbing target for the first time. For this purpose, one needs to calculate the first derivative of the cumulative probability function 1−Q(t | x 0 ) with respect to time, which gives d dt Q(t | x 0 ) = −F (t | x 0 ).(11) The function F (t | x 0 ) specifies the FPT distribution and its first moment determines the MFPT from the initial position at x = x 0 to the target localized at the origin x = 0: T (x 0 ) = ∞ 0 t F (t | x 0 ) dt.(12) Alternatively, inserting Eq. (11) into the above formula and performing an integration per partes under the conditions Q(0 | x 0 ) = 1 and Q(∞ | x 0 ) = 0, we obtain T (x 0 ) = − ∞ 0 t dQ(t | x 0 ) = ∞ 0 Q(t | x 0 ) dt.(13) There are two additional properties regarding the FPT distribution. The first property corresponds to the statement that this density function is by definition normalized to unity. To show this, we have to begin with the integration of Eq. (11) over the time variable in the range from 0 to t. The result is as follows: Q(t | x 0 ) = 1 − t 0 F (τ | x 0 ) dτ,(14) where the property Q(0 | x 0 ) = 1 has been exploited. We will utilize this important equation in Sec. IV A. On the other hand, demanding that t → ∞ and knowing that Q(∞ | x 0 ) = 0, we readily obtain from Eq. (11) the required normalization condition ∞ 0 F (t | x 0 ) dt = 1.(15) This condition implicates the particle is sure to hit the absorbing point, although the mean time, by which such an event occurs, does not necessarily be finite. The second property refers to the relationship between the FPT distribution and the probability current (flux) j(x, t | x 0 ). The latter quantity is, in turn, related to the PDF through the conserved current relation, which is expressed by the continuity equation ∂ ∂t p(t, x | x 0 ) + ∂ ∂x j(x, t | x 0 ) = 0.(16) A combination of this equation with the first derivative of Eq. (10) with respect to time gives that d dt Q(t | x 0 ) = ∞ 0 ∂ ∂t p(x, t | x 0 ) dx = − ∞ 0 dj(x, t | x 0 ) = j(0, t | x 0 ),(17) where we have assumed that the current j(x, t | x 0 ) → 0 for x → ∞ and the non-zero current j(0, t | x 0 ) determines the rate of absorption at the point x = 0. On the basis of Eqs. (11) and (17) we obtain the second property for the FPT distribution F (t | x 0 ) = −j(0, t | x 0 ).(18) This formula allows us to derive the FPT distribution for diffusion in the semi-infinite interval directly from the probability current at the origin. But firstly, we have to solve the corresponding partial differential equation for the PDF with the absorbing boundary condition also imposed at the origin. In what follows, we demonstrate that such a procedure is possible for the linear diffusion equation, while not feasible in the case of the non-linear diffusion equation. A. The linear diffusion equation As we have shown in Sec. II, a typical solution of the linear partial differential equation for the free diffusion is the Gaussian PDF given by Eq. (9). A conventional technique for solving this type of differential equation in the presence of the absorbing point is the image method. Let us clarify that this familiar method emerges form a more general theory of Green's functions and found successful application also in electrostatics. The idea consists in a creation of a virtual system making up of the particle initially in the position x = x 0 > 0 and the fictitious "antiparticle" located in the position x = −x 0 < 0. When the particle begins to diffuse in the semi-infinite interval [0, ∞), then the antiparticle does the same like the mirror image in the semi-infinite interval (−∞, 0]. The free diffusion proceeds until both the particles meet for the first time at the origin, where they disappear due to "annihilation". Owing to the linearity of the diffusion equation, the resulting PDF p(x, t | x 0 ) = 1 √ 4πDt exp − (x − x 0 ) 2 4Dt − exp − (x + x 0 ) 2 4Dt(19) is the combination of two Gaussian distributions and satisfies the absorbing boundary con- j(x, t | x 0 ) = −D ∂ ∂x p(x, t | x 0 ).(20) Hence, calculating the first derivative of the PDF in Eq. (19) with respect to x and setting x = 0, we show that the FPT distribution given by Eq. (18) is as follows: F (t | x 0 ) = x 0 √ 4πD t 3 exp − x 2 0 4Dt .(21) In the long-time limit √ Dt ≫ x 0 , for which the diffusion length is much grater than the initial distance to the origin, the above function reduces to F (t | x 0 ) ∼ x 0 t −3/2 . The existence of this long time tail makes the MFPT from x 0 to the origin infinite, because according to Eq. (12) T (x 0 ) ∼ ∞ 0 t × t −3/2 dt = ∞. On the other hand, the FPT distribution in Eq. (21) fulfills the normalization condition embodied by Eq. (15) which means that the diffusing particle is paradoxically sure to return to the origin. As a consequence of this statement, we can conclude that the survival probability of the particle should vanish in the long time limit. Indeed, inserting Eq. (21) into Eq. (14) and utilizing the integral t 0 τ −3/2 exp − α τ dτ = π α erfc α t [38] , we finally obtain that problem for a more general non-linear diffusion equation [35]. Equation (3) is a simplified version of that fractional and heterogeneous differential equation. Q(t | x 0 ) = 1 − erfc x 0 √ 4Dt = erf x 0 √ 4Dt ,(22) To begin with, let us consider the non-linear diffusion in the semi-infinite interval and establish the absorbing boundary condition p(0, t | x 0 ) = 0 at the origin. Then applying this condition directly to the PDF in Eq. (5), we readily find that p(x, t | x 0 ) = σx(2x 0 − x) 2D(σ + 2)t 1 σ .(23) Having this result to our disposal, we can now determine the appropriate expression for the survival probability given by Eq. (10). An astute look at Eq. (23) convinces us that this PDF can however take negative values and even be imaginary for some values of the parameter σ. On the other site, the integral defining the survival probability becomes divergent when substituting the PDF of the algebraic form. To be sure the PDF is real and non-negative quantity and the integral over a space variable x is convergent, one needs to multiply the PDF by the Heviside unit step function Θ (2x 0 − x) equal to 1, if x 2x 0 , and 0, if x > 2x 0 . The result of this operation is as follows: Q(t | x 0 ) = ∞ 0 Θ(2x 0 − x) p(x, t | x 0 ) dx = 2x 0 0 p(x, t | x 0 ) dx.(24) In the last step, it is enough to put the PDF in Eq. (23) into the above formula and use the integral representation of the Euler beta function, B(ν, µ) [38]. In this way, we show that the survival probability = 1 0 z ν−1 (1 − z) µ−1 dzQ(t | x 0 ) = 2B 1 σ , 1 2 σx σ+2 0 2D(σ + 2) σ+1 t 1 σ .(25) According to Eq. (11) the first derivative of the survival probability in Eq. (25) with respect to time results in the FPT distribution F (t | x 0 ) = 2B 1 σ , 1 2 σ 2 x σ+2 0 2D(σ(σ + 2)t) σ+1 1 σ .(26) The survival probability given by Eq. Nevertheless, let us briefly consider whether all the formulas derived so far are really correct. The functions embodied by Eqs. (25) and (26) undoubtedly tend to zero in the long-time limit and are zero for any time when the initial position of the particle coincides with the origin, i.e. Q(t | 0) = 0 and F (t | 0) = 0. But, it is clearly evident that the survival probability does not satisfy the initial condition Q(0 | x 0 ) = 1. Wang et al did not verify this crucial property in Ref. [35]. In turn, the FPT distribution in Eq. (26) cannot be normalized to unity because the integral expression in Eq. (15) is divergent for any value of the parameter σ. Therefore, what is the main reason that these two fundamental properties of the survival probability and the FPT distribution are broken? The key to unravel this riddle is the absorbing boundary condition imposed on the nonlinear diffusion equation. To show this, we need to compare Eq. (3) with the conserved current relation given by Eq. (16). Hence, the probability current is j(x, t | x 0 ) = −D p σ (x, t | x 0 ) ∂ ∂x p(x, t | x 0 ).(27) Taking advantage of the above equation along with Eq. (18) enables immediate determination of the FPT distribution. In the physical sense, the probability current defines an appropriate measure of the absorption rate. We can easily check, utilizing Eqs. (19) and (20), that the rate of absorption at the point x = 0 is non-zero for the ordinary diffusion. However, the case of the non-linear diffusion equation does not reveal such a behavior. Indeed, by inserting Eq. (23) in Eq. (27), we obtain that j(x, t | x 0 ) = |x − x 0 | (σ + 2)t σx(2x 0 − x) 2D(σ + 2)t 1 σ = |x − x 0 | (σ + 2)t p(x, t | x 0 ),(28) and because p(0, t | x 0 ) = 0 at the absorbing point, so does the probability current j(0, t | x 0 ) = 0. But, the disappearance of this current (flux) means the presence of the reflecting and not absorbing boundary condition at that point. Thus, does this point absorb or reflect the diffusing particle? We are not able to dispel this ambiguity unequivocally. Nevertheless, our analysis shows that the method applied by Wang et al in Ref. [35] is inappropriate and should not be used to explore the first-passage properties of the non-linear diffusion equation. IV. FIRST-PASSAGE PROPERTIES OF NON-LINEAR DIFFUSION A. General method Fortunately, the situation outlined in the previous section is not completely hopeless. In essence, there exists an alternative framework thanks to which the solution of the first-passage problem can successfully be achieved even in the case of the non-linear diffusion. To show this, we will continue our study of the non-linear diffusion in the semi-infinite interval [0, ∞) in the proceeding sections. The method that is at our disposal does not suppose the existence of the absorbing barrier at the target point x = 0. Instead, it treats this point as a "safe marina" to which the particle returns many times after the first visit. To be more precise, the method utilizes a duo of coupled equations. The first equation, we have already met in Eq. (14), constitutes a relation between the survival probability Q(t | x 0 ) and the FPT distribution F (t | x 0 ). The second relation combines the FPT distribution with the PDFs and has the form of the integral equation p(0, t | x 0 ) = t 0 F (τ | x 0 ) p(0, t − τ | 0) dτ.(29) This equation defines the PDF or more precisely the propagator from x 0 to the target at L[f (t) * g(t); t] =f(s)g(s) [44]. Therefore, we can convert Eq. (29) into the algebraic form F (s | x 0 ) =p (0, s | x 0 ) p(0, s | 0) .(30) In turn, performing the Laplace transformation of Eq. (14) yields Q(s | x 0 ) = 1 s 1 −F (s | x 0 ) .(31) The combination of the last two expressions makes the direct relationship between the survival probability and the PDFs in the Laplace domain: Q(s | x 0 ) = 1 s 1 −p (0, s | x 0 ) p(0, s | 0) ,(32) Armed with the above equation and Eq. (13), we can calculate the MFPT T (x 0 ) = lim s→0 ∞ 0 Q(t | x 0 )e −st dt = lim s→0Q (s | x 0 )(33) from the initial position x 0 to the origin of the semi-infinite interval. On the other hand, carrying out the inverse Laplace transformation of Eq. (32), which usually is not trivial operation, allows one to find the survival probability and hence the FPT distribution (see Eq. (11)) in the real space. This is shown in the next section where we will obtain the exact results for the non-linear diffusion with the exceptional values of the parameter σ = 1 and 2, while the approximate formula will be derived for any values of σ falling in-between. B. Results for non-linear diffusion The key quantity appearing in Eqs. 30 and 32 is the Laplace transform of the PDF, i.e. the propagator p(0, t | x ′ ) for the free non-linear diffusion from x ′ to the origin at x = 0, where x ′ corresponds to the initial x 0 or the final position (the target) 0. In the latter case the propagator stands for the so-called return probability density. From the very beginning we posit that the parameter σ in Eq. (2) is assumed to be non-negative. By virtue of this condition, the two σ-dependent coefficients in Eq. (6), namely A(σ) and B(σ) (see Eqs. (7) and (8), respectively), are also non-negative and real. In addition, the following inequality, namely Dt ( B(σ) | x − x 0 |) σ+2 , guarantees that the PDF in Eq. (6) is non-negative and real as the function of time. This necessary condition can be formally expressed through the Heviside unit step function Θ(z) that for z 0 equals to 1 and 0 if z < 0. Because the same requirement must concern the propagator, we conclude that p(0, t | x ′ ) = Θ t − D −1 ( B(σ) |x ′ |) σ+2 A(σ) (Dt) 1 σ+2 1 − B(σ) x ′ 2 (Dt) 2 σ+2 1 σ .(34) Consequently, the Laplace transform of the propagator, which is defined asp(0, s | x ′ ) = ∞ 0 p(0, t | x ′ )e −st dt, takes the following form: p(0, s | x ′ ) = ∞ η(x ′ ,σ) A(σ) (Dt) 1 σ+2 1 − η(x ′ , σ) t 2 σ+2 1 σ exp(−st) dt,(35) where the auxiliary function η(x ′ , σ) = D −1 ( B(σ) |x ′ |) σ+2 allows us to use the shorthand notation in the integrand and the lower limit of the integral. The formula displayed in precisely performed. To continue, we will consider them first. Exact results The first exception that corresponds to the parameter σ = 1 N , where N represents any natural number, is rather trivial and we skip its analysis in the present paper. The next case is relatively simple and refers to the parameter σ = 1. Here, the coefficient A(1) = 3 32 1/3 and the lower limit of integration η(x ′ , 1) = 2|x ′ | 3 9D . Therefore, the Laplace transform of the propagator in Eq. (35) reads p(0, s | x ′ ) = 3 32 1 3 ( √ Ds) − 2 3 Γ 2 3 , 2|x ′ | 3 9D s + 2s 9D 2 3 x ′ 2 Ei − 2|x ′ | 3 9D s ,(36) where Γ(α, z) = ∞ z u α−1 e −u du is the upper incomplete gamma function and Ei(−z) = − ∞ z u −1 e −u du stands for the exponential integral function [38]. Let us note that when x ′ = 0 the above propagator operating in the Laplace domain becomes the return proba- bility densityp(0, s | 0) = 3 32 1 3 Γ 2 3 ( √ Ds) − 2 3 . This formula emerges from the fact that Γ(z, 0) = Γ(z) and the assertion stating that lim z→0 z 2 Ei(−az 3 ) = 0. The latter property is easy verified by making use of the L'Hospital theorem. The aforementioned return probability density can also be obtained from a direct integration in Eq. (35) with η(0, 1) = 0. In contrast to the previous two exceptions, a slightly more difficult is the case when the parameter σ = 2. Here, we only exhibit the final expression for the Laplace transform of the non-linear diffusion propagator with such a parameter. Using the procedure sketched in Appendix A, we readily infer from Eq. 35 that where K ν (z) is the modified Bessel function of the second kind [38]. We also prove there that the Laplace transform of the return probability densityp(0, p(0, s | x ′ ) = π|x ′ | 3 8 √ 2D exp − π 2 x ′ 4 32D s K 3 4 π 2 x ′ 4 32D s − K 1 4 π 2 x ′ 4 32D s ,(37)s | 0) = Γ 3 4 (π √ D) −1/2 s −3/4 . Given the Laplace transforms of the propagators in Eqs. (36) and (37) the Laplace transforms of the survival probabilities for the non-linear diffusion indexed by the power-law exponents σ = 1 and 2 (see Eq. (3)). For this purpose, it is enough to insert each of these formulae into Eq. (32) and straightforwardly obtain the Laplace transform of the survival probability for σ = 1: Q 1 (s | x 0 ) = 1 s − 1 Γ 2 3 1 s Γ 2 3 , 2|x 0 | 3 9D s − 2|x 0 | 3 9D 2 3 1 s 2/3 Γ 0, 2|x 0 | 3 9D s ,(38) where the exponential integral function has been replaced with the upper incomplete gamma function in accordance with the relation Ei(−z) = −Γ(0, z) [38], and the Laplace transform of the survival probability for σ = 2: Q 2 (s | x 0 ) = 1 s − π 3/2 |x 0 | 3 8 √ 2 D 3/4 Γ 3 4 s 1/4 exp − π 2 x 4 0 32D s K 3 4 π 2 x 4 0 32D s − K 1 4 π 2 x 4 0 32D s .(39) At first glance, the above two expressions appear to be quite intricate regarding the performance of the inverse Laplace transform. However, taking advantage of Eqs. (B9) and (B10) derived in Appendix B, we convince ourselves that the exact expression for the survival probability with σ = 1 is Q 1 (t | x 0 ) = 1 + 3 √ 3 2π Θ t − 2|x 0 | 3 9D 9Dt 2|x 0 | 3 − 1 1 3 × 2 F 1 1, 1, 4 3 , 1 − 9Dt 2|x 0 | 3 − 2 F 1 1, 1 3 , 4 3 , 1 − 9Dt 2|x 0 | 3 .(40) In turn, Eqs. (C6) and (C7) derived in Appendix C allow us to Laplace inverse the function in Eq. (39) and eventually obtain the survival probability for the parameter σ = 2, which is as follows: Q 2 (t | x 0 ) = 1 + Γ 2 1 4 |x 0 | 4 √ 2 πD 1/4 Θ t − π 2 x 4 0 16D t − π 2 x 4 0 16D − 1 4 × π 2 x 4 0 16Dt 1 4 − 2 F 1 − 1 4 , 5 4 , 3 4 , 1 − 16Dt π 2 x 4 0 .(41) In the last two formulae, the notation 2 F 1 (a, b, c, z) represents the three-parameter Gaussian hypergeometric function [45]. The time course of the survival probability described by Eq. (40) for σ = 1 is plotted in Fig. 2, whereas the corresponding time course of the survival probability given by Eq. (41) for σ = 2 is shown in Fig. 3. In both the cases the two different distances from the initial position x 0 to the target point at the origin of the semi-infinite interval have been chosen and the diffusion coefficient D = 1.0 has been assumed. We see that the dependence of the survival probability on time consists of two distinct phases. For the first period of time its value constantly remains equal to unity, including the initial condition Q(0 | x 0 ) = 1 at t = 0, and only in the second phase monotonically decreases to reach zero at infinity. This phase appears after the front of the PDF, assigned on a finite support outside of which it disappears (see Fig. 1), has managed to attain the target point for the first time. Prior this event, the probability of finding the diffusing particle at that point amounts exactly zero. It is not a difficult task to prove that these properties emanate directly from the expressions embodied by Eqs. (40) and (41). As noted in Sec. II the first time derivative of the survival probability, preceded with the negative sign, results in the FPT distribution (see Eq. (11)). Here, we present the final formulae for this quantity characterising the first-passage statistics of the non-linear diffusion with the parameter σ = 1 and 2. The method leading to these results is detailed in Appendix D. Thus, recalling Eq. (40) and appealing to Eq. (D13), we show that the FPT distribution for σ = 1 is whereas the FPT distribution for σ = 2 has the following form: F 1 (t | x 0 ) = √ 3 2πt Θ t − 2|x 0 | 3 9D 9Dt 2|x 0 | 3 − 1 − 2 3 1 + 9Dt 2|x 0 | 3 × 2 2 F 1 1, 1, 4 3 , 1 − 9Dt 2|x 0 | 3 − 3 2 F 1 1, 2, 4 3 , 1 − 9Dt 2|x 0 | 3 ,(42)F 2 (t | x 0 ) = Γ 2 1 4 |x 0 | 8 √ 2πt(Dt) 1/4 Θ t − π 2 x 4 0 16D 16Dt π 2 x 4 0 − 1 − 1 4 .(43) The plots exposed in Figs (44) and (45)). Also in this case, the two phases in the time course of these distribution functions can be distinguished. Where the value of the survival probability amounts one, the FPT distribution vanishes. The FPT distribution evaluates the likelihood when the diffusing particle hits the prespecified target for the first time. Its first moment defines the mean time upon which this target might be achieved. We have argued in Sec. III A that the MFPT to the origin of the semi-infinite interval, in which the linear diffusion proceeds, is divergent. Does the same regularity manifest itself in the case of the non-linear diffusion with the parameter σ = 1 and 2? The normalization condition included in Eq. (15) assures the particle will arrive at the target irrespective of the type of diffusive motion. The FPT distributions given by Eqs. (42) and (43) for the non-linear diffusion are also normalized to unity. We can formally demonstrate this property for F 1 (t | x 0 ) by means of the two integrals: ∞ 0 z −µ−1 2 F 1 (α, β, γ, −z) dz = Γ(α+µ)Γ(β+µ)Γ(γ)Γ(−µ) Γ(α)Γ(β)Γ(γ+µ) , that holds for γ = 0, −1, −2, . . . , Re µ < 0, Re(α + µ) > 0 and Re(α + µ) > 0, and ∞ 0 z µ−1 (z + 1) −ν dz = B(µ, ν − µ) = Γ(µ) Γ(ν−µ) Γ(ν) , which is satisfied if 0 < Re µ < Re ν. Only the second integral is needed to confirm the normalization of F 2 (t | x 0 ). The fact that both FPT distributions are normalized to unity implicates that the diffusing particle is sure to reach the origin. Nevertheless, the MFPT turns out to be infinite as in the case of the ordinary diffusion. We can verify this property inserting Eqs. (42) and (43) in Eq. (12), or Eqs. (40) and (41) in Eq. (13), and performing appropriate integration, or finally taking the limit s → 0 in Eqs. (38) and (39) 1, 1, 4 3 , 1 − a z ∝ √ 3 6πa Γ 1 3 Γ 2 3 log(az) z and 2 F 1 1, 2, 4 3 , 1 − a z ∝ 1 3az , where a > 0 is required for both the cases, we show that F 1 (t | x 0 ) ∝ x 2 0 9 ( √ 6D) 2/3 t 5/3 1 + √ 3 π log 27 9Dt 2|x 0 | 3 2 ,(44) when t → ∞, while the following long-time representation F 2 (t | x 0 ) ∝ Γ 2 1 4 x 2 0 16 √ 2πD t 3/2(45) straightforwardly emerges form Eq. (43). Therefore, using Eq. (12) along with Eq. (44) for σ = 1, we indeed obtain that T 1 (x 0 ) = ∞ 0 t F 1 (t | x 0 ) dt ∼ ∞ 0 t × t −5/3 dt = ∞. Here, we did not intentionally include the logarithmic correction with the power-law factor t −5/3 . Similarly, we show taking advantage of Eq. (45) that the MFPT T 2 (x 0 ) = ∞ 0 t F 2 (t | x 0 ) dt ∼ ∞ 0 t × t −3/2 dt = ∞ for σ = 2.p(0, s | x ′ ) ≃ A(σ) D 1 σ+2 s σ+1 σ+2 Γ σ + 1 σ + 2 , s η(x ′ , σ) − 1 σ (s η(x ′ , σ)) 2 σ+2 Γ σ − 1 σ + 2 , s η(x ′ , σ) .(46) In order to obtain this result, we have utilized the following integral Q σ (s | x 0 ) ≃ 1 s 1 − 1 Γ σ+1 σ+2 Γ σ + 1 σ + 2 , sη(x 0 , σ) − 1 σ (sη(x 0 , σ)) 2 σ+2 Γ σ − 1 σ + 2 , sη(x 0 , σ) ,(48) where now η(x 0 , σ) = D −1 ( B(σ) |x 0 |) σ+2 depends solely on the initial position x 0 of the diffusing particle. Surprisingly, despite apparent complexity of this function, we can execute the inverse Laplace transformation in order to find the survival probability depending on the time variable. To this end, it is enough to refer to Eq. (B8) derived in Appendix B and perform simple algebraic operations. The final result reads Q σ (t | x 0 ) ≃ 1 − Θ[t − η(x 0 , σ)] B σ+1 σ+2 , σ+3 σ+2 t η(x 0 , σ) − 1 1 σ+2 2 F 1 1, 1 σ + 2 , σ + 3 σ + 2 , 1 − t η(x 0 , σ) ×   1 − 2 F 1 1, 3 σ+2 , σ+3 σ+2 , 1 − t η(x 0 ,σ) 2 F 1 1, 1 σ+2 , σ+3 σ+2 , 1 − t η(x 0 ,σ)   1 σ .(49) Now we examine whether this survival probability satisfies the two fundamental properties which have been discussed in the previous subsection and re-visited in Sec. III. The first property corresponding to the initial condition Q(0 | x 0 ) = 1 is satisfied by Eq. (49) due to the presence of the Heviside unit step function. The second property concerning the monotonic decrease of the survival probability in the long-time limit, i.e. Q(t | x 0 ) → 0 if t → ∞, is easy to test considering the asymptotic expansions of the Gaussian hypergeometric functions 2 F 1 1, 1 σ+2 , σ+3 σ+2 , 1 − z ∝ z −1/(σ+2) Γ σ+1 σ+2 Γ σ+3 σ+2 and 2 F 1 1, 3 σ+2 , σ+3 σ+2 , 1 − z ∝ z −3/(σ+2) Γ σ+3 σ+2 Γ σ−1 σ+2 / Γ σ σ+2 for z → ∞. Alternatively, we can take advantage of the well known limit theorems and apply them to Eq. (48). The first proposition applicable to the initial condition states that if t = 0, then [44]. In addition, it is suffice to note that lim z→0 Γ(α, z) = Γ(α) and the asymptotic representation of the upper incomplete gamma function Γ(α, z) ∝ z α−1 e −z for |z| → ∞. Q(0 | x 0 ) = lim t→0 Q(t | x 0 ) = lim s→∞ sQ(s | x 0 ), whereas the second proposition states that Q(∞ | x 0 ) = lim t→∞ Q(t | x 0 ) = lim s→0 sQ(s | x 0 ) Then, by virtue of the limit theorems applied to Eq. (48), we immediately conclude that Q(0 | x 0 ) = 1 and Q(t | x 0 ) → 0 for t → ∞. These two properties are reflected in Figs. (6) and (7) where The analysis conducted in Sec. IV B 1 has shown that the FPT distributions for the parameter σ = 1 and 2 are normalized to unity. This means that the diffusing particle will definitely reach the pre-described target, although the MFPT needed to complete this process is infinite. The same conclusion emerges from the formula in Eq. (50), but a direct F σ (t | x 0 ) ≃ Θ[t − η(x 0 , σ)] (σ + 2) B σ+1 σ+2 , σ+3 σ+2 [η(x 0 , σ)] 1/(σ+2) [t − η(x 0 , σ)] (σ+1)/(σ+2) ×   1 − 2 F 1 1, 3 σ+2 , σ+3 σ+2 , 1 − t η(x 0 ,σ) 2 F 1 1, 1 σ+2 , σ+3 σ+2 , 1 − t η(x 0 ,σ)   1 σ 2 F 1 1, σ + 3 σ + 2 , σ + 3 σ + 2 , 1 − t η(x 0 , σ) + F (t, x 0 , σ) σ 2 F 1 1, 1 σ+2 , σ+3 σ+2 , 1 − t η(x 0 ,σ) − 2 F 1 1, 3 σ+2 , σ+3 σ+2 , 1 − t η(x 0 ,σ) ,(50)F (t, x 0 , σ) = 2 F 1 1, 3 σ + 2 , σ + 3 σ + 2 , 1 − t η(x 0 , σ) 2 F 1 1, σ + 3 σ + 2 , σ + 3 σ + 2 , 1 − t η(x 0 , σ) + 2 2 F 1 1, 1 σ + 2 , σ + 3 σ + 2 , 1 − t η(x 0 , σ) 2 F 1 1, 3 σ + 2 , σ + 3 σ + 2 , 1 − t η(x 0 , σ) − 3 2 F 1 1, 1 σ + 2 , σ + 3 σ + 2 , 1 − t η(x 0 , σ) 2 F 1 1, σ + 5 σ + 2 , σ + 3 σ + 2 , 1 − t η(x 0 , σ)(51)F σ (t | x 0 ) ∝ 2 sin 2π σ+2 πσ(σ + 2) t B σ − 1 σ + 2 , 2 σ + 2 η(x 0 , σ) t 2 σ+2 . (52) The derivation of the above formula is detailed in Appendix E. The divergence of the MFPT is then due to the long-time tail of the FPT distribution. In essence, taking advantage of Eq. (12) we obtain that T σ (x 0 ) = ∞ 0 t F σ (t | x 0 ) dt ∼ ∞ 0 t × t −(σ+4)/(σ+2) dt = ∞ for σ > 0. Again, this property is consistent with the divergence of the MFPT for the ordinary diffusion in a semi-infinite interval terminated by the totally absorbing wall [4]. V. MEAN FIRST-PASSAGE TIME FOR NON-LINEAR DIFFUSION IN HAR- MONIC POTENTIAL We have shown in the previous section that the non-linear diffusion equation excludes the finiteness of the MFPT to a target point located somewhere in an unbounded space. Such a property is a part of a more general rule stating that any diffusive motion occurring in the unlimited area of the space makes the MFPT infinite. However, this regularity changes if the diffusing particle moves inside a bounded domain or in a confining potential. Here, we briefly explore the latter scenario and consider the simple version of the non-linear diffusion in the harmonic potential. A more substantial analysis of this issue represented by the following equation ∂ ∂t p(x, t | x 0 ) = ∂ ∂x ∂V (x) ∂x p(x, t | x 0 ) + D p σ (x, t | x 0 ) ∂ ∂x p(x, t | x 0 )(53) and extended to the other types of external potentials V (x) will be a subject of an intense research in the future. Henceforth, we will study the diffusion equation of the harmonically trapped particle, whose non-linearity is determined by the peculiar value of the parameter σ = 1. Without loss of generality, we assume the harmonic potential V (x) = 1 2 αx 2 with the certain stiffness α has the minimum at x = 0. Due to these particular assumptions, Eq. (53) takes the form ∂ ∂t p(x, t | x 0 ) = ∂ ∂x αx p(x, t | x 0 ) + D p(x, t | x 0 ) ∂ ∂x p(x, t | x 0 ) .(54) It is now convenient to substitute in the above differential equation p(x, t | x 0 ) = e αt q(z, τ | z 0 ), where z = x e αt and τ = 1 3α (e 3αt − 1) for t > 0, while z 0 = x 0 and τ = 0 for t = 0. In this manner, we can transform Eq. (54) to much simpler form ∂ ∂τ q(z, τ | z 0 ) = D ∂ ∂z q(z, τ | z 0 ) ∂ ∂z q(z, τ | z 0 ) .(55) This equation is exactly the same as Eq. (3) if the parameter σ = 1. Let us recall that its solution undergoing the normalization condition is manifested by Eq. (6) along with Eqs. (7) and (8). Therefore, inserting there σ = 1, we readily have from Eq. (6) that q(z, τ | z 0 ) = 3 32 1 3 1 (Dτ ) 1/3 − 2 9 2 3 (z − z 0 ) 2 Dτ(56) is the solution of Eq. (55). So, the exact solution of the original Eq. (54) reads p(x, t | x 0 ) = 3 32 1 3 e αt 3α D 1 3 1 (e 3αt − 1) 1/3 − 2 9 2 3 3α D (xe αt − x 0 ) 2 e 3αt − 1 .(57) We can easy check this by a direct substitution of the above function in the partial differential equation. From now on, we restrict our considerations to the non-linear diffusion starting from the initial position at x 0 > 0 and progressing towards the target placed in the minimum x = 0 of the harmonic potential. In this particular case the PDF in Eq. (57) takes the simpler form p(0, t | x 0 ) = 3 32 1 3 3α D 1 3 e αt (e 3αt − 1) 1/3 − 2 9 2 3 3α D x 2 0 e αt e 3αt − 1 ,(58) which allows us to figure out the analytical expression for the MFPT downward the harmonic potential. The PDF must by definition be a non-negative quantity and this property in the case of Eq. (58) will be met if t 1 3α log 1 + 2α|x 0 | 3 3D . For this reason, the Laplace transformation of the PDF is given bỹ p(0, s | x 0 ) = ∞ 0 Θ t − 1 3α log 1 + 2α|x 0 | 3 3D p(0, t | x 0 )e −st dt,(59) where Θ(z) denotes the Heviside unit step function. Plugging Eq. (58) into the above formula and taking advantage of the integral ∞ u z µ−1 (1+α z) ν dz = u µ−ν α ν (ν−µ) 2 F 1 ν, ν − µ, ν − µ + 1, − 1 α u , which proceeds if Re ν > Re µ, we obtain after straightforward calculations that p(0, s | x 0 ) = 3 32 1 3 3α D 1 3 1 s 3D 2α|x 0 | 3 s 3α 2 F 1 s 3α , s 3α + 2 3 , s 3α + 1, − 3D 2α|x 0 | 3 − 2 9 2 3 3α D x 2 0 s + 2α 3D 2α|x 0 | 3 s 3α + 2 3 2 F 1 s 3α + 2 3 , s 3α + 2 3 , s 3α + 5 3 , − 3D 2α|x 0 | 3 .(60) On the other hand, setting above x 0 = 0 and utilizing the Laplace transformation L [(1 − e −at ) ν ] = a −1 B(ν + 1, , a −1 s), which is valid provided that Re a > 0 and ν > −1, we havẽ p(0, s | 0) = 3 32 1 3 3α D 1 3 Γ 2 3 Γ s 3α 3α Γ s 3α + 2 3 ,(61) where the Euler beta function B(x, y) = Γ(x)Γ(y) Γ(x+y) has been simultaneously used. By virtue of Eq. (32), the Laplace transform of the survival probabilitỹ Q(s | x 0 ) =p (0, s | 0) −p(0, s | x 0 ) sp(0, s | 0) .(62) We can now combine Eqs. (60) and (61) in order to insert them in the above equation. This step leads to the following result Q(s | x 0 ) = Γ s 3α + 2 3 Γ 2 3 Γ s 3α + 1 Γ 2 3 Γ s 3α + 1 s Γ s 3α + 2 3 − 1 s 3D 2α|x 0 | 3 s/3α × 2 F 1 s 3α , s 3α + 2 3 , s 3α + 1, − 3D 2α|x 0 | 3 + 2α 3D 2 3 x 2 0 s + 2α × 3D 2α|x 0 | 3 s/3α+2/3 2 F 1 s 3α + 2 3 , s 3α + 2 3 , s 3α + 5 3 , − 3D 2α|x 0 | 3 .(63) The recipe embodied in Eq. (33) allows us to find the MFPT directly from Eq. (63). For this purpose, it is enough to determine the limit of the Laplace transform of the survival probability for s → 0. However, this operation yields the indeterminate form ∞ − ∞ and in consequence the need to apply L'Hospital's rule. Converting the former indeterminate form to 0 0 and using the L'Hospital rule twice, we definitively show that performed with respect to the parameter a, which appears on the right-hand side of Eq. (64), can be determined according to the following formula ∂ ∂a 2 F 1 (a, b, a + 1, z) = bz (a + 1) 2 3 F 2 (a + 1, a + 1, b + 1; a + 2, a + 2; z) , lim s→0 Γ 2 3 Γ s 3α + 1 s Γ s 3α + 2 3 − z s/3α s 2 F 1 s 3α , s 3α + 2 3 , s 3α + 1, −z = − 1 3α ∂ ∂a 2 F 1 a, 2 3 , 1, −z a=0 + log(z) + ψ 2 3 + γ ,(64) where 3 F 2 (a 1 , a 2 , a 3 ; b 1 , b 2 ; z) is a generalized hypergeometric function. Thus, by applying the last two properties to Eq. (63), we obtain the final expression for the MFPT downward the harmonic potential in the case of the non-linear diffusion with the fixed parameter σ = 1: T (x 0 , α) = 1 2α 2 F 1 2 3 , 2 3 , 5 3 , − 3D 2α|x 0 | 3 + D 3α 2 |x 0 | 3 3 F 2 1, 1, 5 3 ; 2, 2; − 3D 2α|x 0 | 3 − 1 3α log 3D 2α|x 0 | 3 + ψ 2 3 + γ .(66) This is the central result of the present section. Fig. 13 illustrates how the MFPT to the target at x = 0 of the harmonic potential depends on the stiffness parameter α. We see that the larger value of α the shorter MFPT for a given distance from the initial position x 0 to the target point. In Fig. 14 the dependence of the MFPT on the stiffness parameter is displayed for two fixed initial positions. We see that for α → 0 and independently of x 0 the MFPT tends to infinity, as it should be in the general case of any diffusive process, and in particular the non-linear diffusion occurred in the unbounded space. In turn, if the initial position x 0 → 0 then the MFPT should vanish. To examine this effect we need to demonstrate that lim x 0 →0 T (x 0 , α) = 0 in Eq. (66). For this purpose, it is enough to prove that the following limit lim x 0 →0 D 3α 2 |x 0 | 3 3 F 2 1, 1, 5 3 ; 2, 2; − 3D 2α|x 0 | 3 − 1 3α log 3D 2α|x 0 | 3 = 1 3α ψ 2 3 + γ ,(67) is correct and then exploit the fact that lim x 0 →0 2 F 1 2 3 , 2 3 , 5 3 , − 3D 2α|x 0 | 3 = 0.(68) To prove the relation in Eq. (67), we take advantage of the asymptotic representation of the generalized hypergeometric function for |z| → ∞, i.e. when x 0 → 0 for fixed D and α: 3 F 2 (a 1 , a 1 , a 3 ; b 1 , b 2 ; −z) ∝ Γ(b 1 )Γ(b 2 )Γ(a 3 − a 1 ) Γ(a 1 )Γ(a 3 )Γ(b 1 − a 1 )Γ(b 2 − a 1 )z a 1 [log(z) + ψ(a 3 − a 1 ) (69) − ψ(b 1 − a 1 ) − ψ(b 2 − a 1 ) − ψ(a 1 ) − 2γ] + O 1 z . For the special values of the parameters a 1 = 1, a 3 = 5 3 and b 1 = b 2 = 2, we obtain from above that 3 F 2 1, 1, 5 3 ; 2, 2; − 3D 2α|x 0 | 3 ∝ α|x 0 | 3 D log 3D 2α|x 0 | 3 + ψ 2 3 + γ .(70) This result implies that the relationship in Eq. (67) is really satisfied. Thus, we can apply it along with Eq. (68) to Eq. (66) and show that the MFPR for the non-linear diffusion equation of the harmonically trapped particle disappears when the initial position x 0 coincides with the target point in the minimum x = 0 of the harmonic potential. VI. CONCLUSIONS It is probably no exaggeration to say that in most of the papers devoted to the studies of first-passage properties of diffusive motion one usually assumes that this process is modeled in terms of the linear partial differential equations. Inspired by such a state of matters, we have centred our efforts to analyse in the present paper the fundamental aspects of first-passage statistics for the non-linear diffusion equation. To be more concrete, we have considered here the special variant of this equation, known as the porous medium equation, in which the diffusion coefficient is power-law dependent on the probability density function of a diffusing particle. Additionally, the corresponding power-law exponent was served as the non-negative and constant parameter. As stated in Introduction, the porous medium At the beginning, we briefly described the basic properties of the non-linear diffusion equation along with its typical solution having the form of the Zel'dovitch-Barenblatt-Pattle algebraic function. Further, the key concepts of the first-passage formalism such as the survival probability, the first-passage time distribution and the mean first-passage time have been concisely revised. In the subsequent two sections we determined these quantities for the ordinary as well as non-linear diffusion that occurred in the semi-infinite interval with the target point located at the origin. While the use of the image method raises no objections in the first case, this technique cannot be applied in the latter case due to the non-linear form of the diffusion equation. Therefore, we had resort to the standard procedure according to which the probability density function undergoes the absorbing boundary conditions at the target point, where it disappears. Although we found the exact formula for the survival probability, it has appeared that this quantity does not meet the crucial property, namely, the initial condition. Then, we have argued that the solution of the non-linear diffusion equation in the presence of the absorbing well entails disappearance of the probability current. Consequently, the totally absorbing well acts as a perfectly reflecting wall and this contradiction leads to ambiguous situation. Therefore, we have chosen an alternative method where instead of the absorbing target there appears the target point to which a diffusing particle arrives for the first time and then can return to it many times. In this way, we were able to obtain the exact and approximate results for the survival probability and hence the first-passage time distribution. The former concern the power-law exponent σ = 1 and 2, whereas the latter correspond to its values in the range between 1 and 2. Moreover, the approximate formulae, even though described by completely different expressions, perfectly agree with the exact results. We have also shown that the time course of the survival probability for the free non-linear diffusion takes place in two consecutive stages. For the initial period of time its value remains equal to unity and only in the second phase of diffusive motion permanently decreases to reach zero in the infinite-time limit. In turn, the first-passage time distribution is always normalized to unity although its first moment, that is the mean first-passage time to the target, diverges to infinity. Such a tendency changes when the non-linear diffusion occurs in the confining potential. We have shown this on the example of the harmonically trapped particle that diffuses downward the potential to reach the target located in the minimum. The first-passage properties of the non-linear diffusion in external potentials will be continued in the future paper. The exploration of first-passage phenomena still attracts unabated attention among scientific community. Despite the immense literature on this subject, our understanding of first-passage dynamics remains incomplete and requires further systematic and in-depth studies. We hope that the present paper has become an inherent part of these investigations, specifically, in the scope of diffusive processes. CONFLICTS OF INTEREST The author declares no conflict of interest. ORCID ID Przemys law Che lminiak, https://orcid.org/0000-0002-0085-9232 1 − π(x − x ′ ) 2 4 √ Dt dt (A1) Now, upon introducing the new notation a(x) ≡ ξ(x, 2) and changing the variable of integration from t to τ , so that t = (τ 2 + a(x)) 2 , we can recast the Laplace transform of PDF in Eq. (A1) to the much simpler form: p(x, s | x ′ ) = 4 π √ D ∞ 0 τ 2 exp −s(τ 2 + a(x)) 2 dτ.(A2) From now on, the rest of our calculations boils down to find the solution of the above integral. It can be obtained by using the similarly looking integral where the factors α, β > 0, while K ν (z) is the modified Bessel function of the second kind. Out of many well-known properties of this function we utilize the two ones: K ν (z) = K −ν (z),(A4) and d dz K ν (z) = −K ν−1 (z) − ν z K ν (z).(A5) First, however, let us differentiate the left and the right side of Eq. (A3) with respect to the parameter β. In this way, we have β 2 2α = β √ β α exp β 2 2α K 1 4 β 2 2α − K 3 4 β 2 2α .(A7) To begin with, let us consider the integral representation of the Gaussian hypergeometric function [45] 2 F 1 (a, b, c, z) = Γ(c) Γ(b)Γ(c − b) 1 0 ν b−1 (1 − ν) c−b−1 (1 − zν) −a dν,(B1) where a, b and c are real numbers and Γ(z) is the Euler gamma function. Upon making the following substitution, where a = 1, b = 1−β, c = 2−(α+β) and z = 1− 1 u , we have 2 F 1 1, 1 − β, 2 − (α + β), 1 − u −1 = 1 B(1 − α, 1 − β) 1 0 u dν ν β (1 − ν) α [ν + u(1 − ν)] ,(B2) where B(x, y) = Γ(x)Γ(y) Γ(x+y) is the Euler beta function. By defining the new variable of integration ξ = ν + u(1 − ν), we can easily convert the right hand side of Eq. (B2) into the following form: 2 F 1 1, 1 − β, 2 − (α + β), 1 − u −1 = u(1 − u) α+β−1 B(1 − α, 1 − β) 1 u dξ ξ(1 − ξ) α (ξ − u) β .(B3) The above expression determines the useful integral 1 u dξ ξ(1 − ξ) α (ξ − u) β = B(1 − α, 1 − β) 2 F 1 (1, 1 − β, 2 − (α + β), 1 − u −1 ) u(1 − u) α+β−1 ,(B4) which will turn out to be essential in the context of our further argumentation. To complete the task, we will proceed as follows. . (B10) The survival probability is by definition a real and non-negative quantity. To ensure the fulfilment of this condition, we have to additionally require that t a. Therefore, it is enough to complete all the three last formulae for the Laplace inverse transformation by multiplying their right hand sides by the Heviside unit step function Θ(t−a). where, as before, Θ(z) means the Heviside unit step function equal to 1, if z 0 and 0 otherwise. According to Ref. [46] the inverse Laplace transformation of the power-law function s −ν multiplied by the modified Bessel function of the second kind K ν (bs) with b > 0 is as follows: L −1 s −ν K ν (bs); s = Θ(t − b) π 2b t 2 − b 2 µ 2 − 1 4 P 1 2 −µ ν− 1 2 t b ,(C2) where P α β (z) corresponds to the associated Legendre function [38]. In the case when z > 1, this special function can be represented by the Gaussian hypergeometric function according to the following formula: P α β (z) = (a − c) 2 F 1 (a − 1, b, c, z) + (c − a − b) 2 F 1 (a, b, c, z) = b (z − 1) 2 F 1 (a, b + 1, c, z) = 0. (D7) Again, converting c → c − 1 in Eq. (D6) and b → b − 1 in Eq. (D7), we connect these two equations to construct the following identity FIG. 1 . 1Probability distribution function for the non-linear diffusion described by Eq. (6) in two consecutive moments of time. The value of the parameter σ = 2.0 has been assumed and the diffusion coefficient D = 1.0 has been established. dition p(0, t | x 0 ) = 0. On the other hand, a construction of the ordinary diffusion equation (see Eq. (3) with σ = 0) on the basis of Eq. (16) requires that the probability current must be of the following form: where erfc(z) = 1 − erf(z) is the complementary error function, whereas erf(z) stands for the error function. This result indicates that the survival probability for the ordinary diffusion in the semi-infinite interval with the absorbing barrier at the origin tends to zero in thelong-time limit, i.e. Q(t | x 0 ) → 0 if t → ∞. This property arises from the Taylor expansion of the error function erf(z) ∼ 2z exp(−z 2 ) √ π for z → 0. The second property of the survival probability, namely that Q(0 | x 0 ) = 1, is also satisfied. To show this, it is enough to use the asymptotic representation of the error function erf(z) ∼ 1 − exp(−z 2 ) √ πz for z → ∞. B. The non-linear diffusion equation The image method allows us to represent the solution of the ordinary diffusion equation in the presence of the absorbing trap as the superposition of probability and "anti"-probability density functions. However, the linear combination of the PDFs can not be the solution of the non-linear diffusion equation displayed in Eq. (3), so we cannot use the image method in this case. For this reason, we will proceed as Wang et al, who analysed the first-passage (25) and the FPT distribution given by Eq.(26), the both of the algebraic form, when inserted, respectively, in Eq. (13) and Eq. (12), make the MFPT to the origin to be divergent for the non-linear diffusion. The same result obviously holds for the ordinary diffusion. x = 0 for any stochastic dynamics as an integral over the first time to reach the point 0 at a time τ t followed by a loop from (0, τ ) to (0, t) in the remaining time t−τ [43]. Note the integral expression in Eq. (29) is a time convolution of two distribution functions, thus a price we must pay to determine the FPT distribution F (t | x 0 ) involves the use of the Laplace transformation. The convolution theorem states that the Laplace transformation, defined asf (s) = L[f (t); t] := ∞ 0 f (t)e −st dt, of the convolution f (t) * g(t) := t 0 f (τ )g(t − τ )dτ of two integrable functions f (t) and g(t) is the product of their Laplace transforms, i.e. Eq. (35) constitutes the starting point for studies of the first-passage time properties of the non-linear diffusion. However, a precise calculation of the integral appearing in Eq. (35) poses a great challenge whenever arbitrary values of the parameter σ > 0 are taken into account. Nevertheless, there exist, at least, three exceptions when this operation can be FIG. 2 . 2Time course of the survival probability given by Eq. (40) for the non-linear diffusion with the parameter σ = 1. Here, two values of the initial position x 0 relative to the target point located at the origin x = 0 of the semi-infinite interval have been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. FIG. 3 . 3with x ′ = x 0 , as well as the corresponding return probability densities with x ′ = 0, we are now set to determine Time course of the survival probability given by Eq. (41) for the non-linear diffusion with the parameter σ = 2. Here, two values of the initial position x 0 relative to the target point located at the origin x = 0 of the semi-infinite interval have been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. FIG. 4 . 4Time course of the FPT distribution given by Eq. (42) for the non-linear diffusion with the parameter σ = 1. The dashed lines refer to the asymptotic representation of the FPT distribution shown in Eq. (44). Here, two values of the initial position x 0 relative to the target point located a the origin x = 0 of the semi-infinite interval have been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. FIG. 5 . 5Time course of the FPT distribution given by Eq. (43) for the non-linear diffusion with the parameter σ = 2. The dashed lines refer to the asymptotic representation of the FPT distribution shown in Eq. (45). Here, two values of the initial position x 0 relative to the target point located a the origin x = 0 of the semi-infinite interval have been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. . 4 and 5 illustrate how the FPT distributions depend on the time, correspondingly, for the parameter σ = 1 and 2. Here, we fixed the diffusion coefficient D = 1.0. All the profiles of these functions are shaped by selection of various distances from the initial position x 0 of the diffusing particle to the target point located at the origin of the semi-infinite interval. The dashed lines correspond to the asymptotic representations of FPT distributions (see Eqs. of the MFPT for the non-linear diffusion with σ = 1 and 2 infers from the long-time tails of the FPT distributions. With the asymptotic expansion of the Gaussian hypergeometric functions at hand, i.e. 2 F 1 duo of exact results derived in Sec. IV B 1 for the non-linear diffusion with the diffusivity specified by the power-law exponent σ = 1 and 2 is exceptional. This circumstance raises the natural question about remaining values of the parameter σ in Eq. (2). FIG. 6 .FIG. 7 . 67Time course of the survival probability given by Eq. (49) for the non-linear diffusion with the parameter σ = 3/2. Here, three values of the initial position x 0 relative to the target point located a the origin x = 0 of the semi-infinite interval have been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. As aforementioned in Sec. IV B the precise integration in Eq. (35) poses serious difficulties for arbitrary values of this parameter. To overcome this problem, we will replace the integrand in Eq. (35) with its Taylor expansion. Such an approach is justified due to the following argumentation. Each PDF is by assumption a non-negative and real (non-complex) quantity. Specifically, the power-law component of the PDF in Eq. 35, i.e. (1 − z) Time course of the survival probability given by Eq. (49) for the non-linear diffusion with three exemplary values of the parameter σ. Here, the value of the initial position x 0 = 2 relative to the target point located at the origin x = 0 of the semi-infinite interval has been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. FIG. 8 . 8The comparison of the approximate formula in Eq. (49) (solid lines) to the exact formulae in Eqs. (40) (circles, σ = 1) and (41) (squares, σ = 2) describing the time course of the survival probability for the non-linear diffusion in the semi-infinite interval. Here, the value of the initial position x 0 = 2 relative to the target point located at the origin x = 0 has been fixed and the value of the diffusion coefficient D = 1.0 has been assumed. z := [η(x, σ)/t] 2/(σ+2) and α = 1/σ, satisfies this crucial requirement if and only if |z| < 1.Consequently, the corresponding inequality η(x, σ) := D −1 ( B(σ) |x|) σ+2 < t must be met, which allows us to approximate the power-law function in Eq. 35 by expanding it in the Taylor series, namely (1 − z) α ≈ 1 − αz. To complement the above reasoning, we would like to emphasize that our preliminary studies confirmed the effectiveness of this approximation as long as the exponent α < 1, what implicates the parameter σ > 1. Consequently, the approximate expression for the Laplace transform of the propagator from a position x ′ > 0 to the origin of the semi-infinite interval is FIG. 9 . 9z u α−1 e −u du = Γ(α, z), representing the upper incomplete gamma function. Setting x ′ = 0 causes that the auxiliary function η(0, σ) = 0 and the second term enclosed by the square bracket in Eq. (46) disappears. But, the remaining upper incomplete gamma function turns into the Euler gamma function, thusp (0, s | 0) Time course of the FPT distribution given by Eq. (50) for the non-linear diffusion with the parameter σ = 3/2. The dashed lines refer to the asymptotic representation of the FPT distribution shown in Eq. (52). Here, two values of the initial position x 0 relative to the target point located at the origin x = 0 of the semi-infinite interval have been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. A direct substitution of Eqs. (46) and (47) in Eq. (32) yields the approximate formula for the Laplace transform of the survival probabilitỹ , where the time courses of the survival probability given by Eq. (49) are displayed, respectively, for various distances between the initial position x 0 of the diffusing particle and the target point settled at the origin x = 0 of the semi-infinite interval, and three disparate values of the parameter σ. Moreover, in Fig. (8), we demonstrate a surprising compatibility of the survival probability described by Eq. (49) with the corresponding exact formulae derived for σ = 1 and 2 in Eqs. (40) and (41), respectively. Given the survival probability in Eq. (49), we are now ready to find the approximate formula representing the FPT distribution for the non-linear diffusion in the semi-infinite interval. Taking into account Eq. (11) and determining the first time derivative of the function shown in Eq. (49), we obtain after quite long calculations that FIG. 10 . 10Time course of the FPT distribution given by Eq. (50) for the non-linear diffusion with three exemplary values of the parameter σ. Here, the value of the initial position x 0 = 1.5 relative to the target point located at the origin x = 0 of the semi-infinite interval has been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. and η(x 0 , σ) = D −1 ( B(σ) |x 0 |) σ+2 . Fig. 9 displays the time course of the FPT distribution with the parameter σ = 3/2 for two different initial positions x 0 = 2.0 and 3.0 relative to the target point established at the origin of the semi-infinite interval, whereas Fig. 10 shows the same behavior from the point of view of three different values of the parameter σ and the fixed value of the initial position x 0 = 1.5. InFig. 11we present the excellent conformity of the approximate expression for the FPT distribution in Eq. (50) to the exact formulae given by Eqs.(42) and(43), correspondingly, for the parameter σ = 1 and 2. FIG. 11 . 11The comparison of the approximate formula in Eq. (50) (solid lines) to the exact formulae in Eqs. (42) (circles, σ = 1) and (43) (squares, σ = 2) describing the time course of the FPT distribution for the non-linear diffusion in the semi-infinite interval. Here, the value of the initial position x 0 = 1.5 relative to the target point located at the origin x = 0 has been fixed and the value of the diffusion coefficient D = 1.0 has been assumed. integration of this function is rather too hard. Instead, we display in Fig. (12) how the time integral of F σ (t | x 0 ) converges to unity with increasing range of numerical integration performed with respect to time for the parameter σ = 3/2, three fixed values of the initial position x 0 = 1.0, 1.5 and 2.0, and the diffusion coefficient D = 1. In turn, utilizing the asymptotic expansion of the FPT distribution given by Eq. (50), we can show a divergence of the MFPT. The asymptotic formula is of the following form: FIG. 12 . 12Numerical test of the normalization condition ∞ 0 F 0σ (τ | x 0 )dτ = 1 for the FPT distribution in Eq. (50) with the chosen parameter σ = 3/2. Here, three values of the initial position x 0 relative to the target point located at the origin x = 0 of the semi-infinite interval has been fixed and the value of the diffusion coefficient D = 1.0 has been assumed. FIG. 13 . 13where ψ(z) = 1 Γ(z) dΓ(z) dz stands for a digamma function and γ ≈ 0.5772 is known as the Euler-Mascheroni constant. The first derivative of the Gaussian hypergeometric function Shown is the dependence of the MFPT on the location of the initial position x 0 relative to the target point placed in the minimum x = 0 of the harmonic potential V (x) = 1 2 αx 2 for the non-linear diffusion given by Eq. (54). Here, two values of the stiffness α have been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. FIG. 14 . 14Shown is the dependence of the MFPT on the stiffness α of the harmonic potential V (x) = 1 2 αx 2 for the non-linear diffusion given by Eq. (54). Here, two values of the initial position x 0 relative to the target point placed in the minimum x = 0 of this potential have been chosen and the value of the diffusion coefficient D = 1.0 has been assumed. equation has found useful applications in such different fields as plasma physics, geophysics and biology. account Eqs. (A4) and (A5), we show that a derivative of the expression enclosed in the square bracket on the right hand side of the above equation Let L[f (t); t] denotes the Laplace transformation of a function f (t) and let L −1 [f (s); s] stands for the inverse transformation of the Laplace transformf (s) := ∞ 0 f (t)e −st dt. According to Ref. [46], the inverse transformation of the Laplece transformf (s) = Γ(β, a s) of the upper incomplete gamma function readsL −1 [Γ(β, a s); s] = a β Θ(t − a) Γ(1 − β)t(t − a) β ,(B5)where a > 0 and Re(β) < 1. Here, Θ(z) is the Heviside unit step function equal to 1, if z 0 and 0 otherwise. In turn, if f (t) = t −α for α < 1 thenf (s) = Γ(1 − α) s α−1 . By virtue of the convolution theorem, which states that the Laplace transformation of the convolution f (t) * g(t) := t 0 f (τ )g(t − τ )dτ of two integrable functions f (t) and g(t) is the product of their Laplace transforms, i.e. L[f (t) * g(t); t] =f(s)g(s), we can express the inverse Laplace transformation of the power-law functionf (s) = s α−1 and the upper incomplete gamma functiong(s) = Γ(β, as), both defined in the Laplace domain, as follows: L −1 [s α−1 Γ(β, a s); s] = t 0 a β Θ(τ − a) Γ(1 − α)Γ(1 − β)τ (τ − a) β (t − τ ) α dτ = a β Γ(1 − α)Γ(1 − β) (t − τ ) α (τ − a) β .(B6)Assuming the new variable of integration ζ = τ t and denoting u = a t , we recast the integral and the remaining part of the formula in the second line of Eq. ζ) α (ζ − u) β . (B7)Let us note that the integral appearing in the above equation has exactly the same form as the integral determined in Eq. (B4). Hence, we conclude that the inverse Laplace transformation of the product of the power-law and the upper incomplete gamma functions is given byL −1 [s α−1 Γ(β,a s); s] use the general result in Eq. (B8) to determine the survival probability for the particular value of the parameter σ = 1 (see the main text). For this purpose, it is enough to set α = 0 and β = 2 3 , which gives from Eq. Appendix C: The inverse Laplace transformation of the product of power-law, exponential and modified Bessel functions Suppose again that L[f (t); t] denotes the Laplace transformation of a function f (t) and let L −1 [f (s); s] stands for the inverse transformation of the Laplace transformf (s) := ∞ 0 f (t)e −st dt. The inverse Laplace transformation off (s), when additionally multiplied by the exponential function e −bs , defined in the Laplace domain with b > 0, is L −1 e −bs f (s); s = Θ(t − b)f (t − b), F 1 (a − 1, b − 1, c − 1, z) − 2 F 1 (a, b − 1, c − 1, z) = − b − 1 Appendix A: The Laplace transform of PDF for the parameter σ = 2The purpose of this supplementary section is to present the detailed derivation of Eq.(37)comprised in the main text. Setting the parameter σ = 2, we get from Eqs.(7)and(8)that two numerical factors appearing in Eq. (35) take the following values, A(2) = 1 √ π and B(2) = π 4 . In addition, the auxiliary function in this equation is ξ(x, 2) = π 4 2 (x−x ′ ) 4 D . Thus, inserting all these quantities into Eq.(35), we find that the Laplace transform of the PDF, or the propagator p(x, t | x ′ ) from x ′ to x at time t, is2)e −st (Dt)1 4If we plugin this derivative back into Eq. (A6) and multiply their both sides by the exponential function exp(− β 2 α ), we obtain that the integralHaving this equation at our disposal and assuming that α = s and β = sa(x), we immediately solve the integral in Eq. (A2). The final result is as followsTo complete this supplement, we should also determine the Laplace transform of PDF for x = x ′ . It can be done in two ways. The first method takes advantage of the following limit values of the modified Bessel functions of the second kind:and lim z→0Calculating these limits in Eq. (A9) we easily show thatThe second method that guarantees the above result consists in the direct calculation of the integral in Eq. (A2) for x = x ′ with ξ(x ′ , 2) = 0. In this case we use the integral representation of the Euler gamma function, i.e. Γ(α) = The Laplace transforms of these functions are well known individually and can be found inRef.[46].Now, combining Eqs. (C1) and (C2) and conducting elementary calculations, we obtain thatNext, it is enough to insert Eq. (C3) into the above formula to get the final result:To determine the inverse Laplace transformation of the survival probability in Eq. (39), we have to set µ = 1 4 and, respectively, ν = 1 4 and ν = 3 4 . In the first case we obtain from Eq. (C5) thatwhere the formula in the second line results from the fact that 2 F 1In both the above expressions, the variable a = πx 2 4 √ 2D depends on the position of the diffusing particle.Appendix D: Exact time derivatives of survival probabilitiesThe objective of the present addendum is to figure out the first derivatives of survival probabilities with respect to time provided the parameter σ = 1 and 2. These results correspond in fact to the exact expressions for the first-passage time distributions contained in Eqs.(42)and(43)of the main text.Let us first consider the case of the parameter σ = 1, for which the survival probability is defined in Eq.(40). Conducting precise calculus we find that its first derivative over the time preceded by the negative sign is as follows:where the auxiliary function τ 1 (x 0 ) = 2|x 0 | 3 9Dt . We show farther how to simplify this highly complex formula. For this purpose, we use the functional identities which reflect intrinsic properties of the Gaussian hypergeometric functions. The first two identities constitute the system of the following equations:By making the change c → c − 1 in Eq. (D2) and a → a − 1 in Eq. (D3), we can combine these two relations to have the single identitySetting the particular values for a = 2, b = 2 and c = 7/3, we obtain from the above formula that 2 F 1 1, 1,The second system of equations is established by the next two functional identitiesa, b, c, z).(D8)By choosing the specific values for a = 2, b = 4/3 and c = 7/3, we see that Eq. (D8) takes the special formTo complete the set of relations embodied by Eqs. (D5) and (D9), we take advantage of the following functional identityand the property according to which if a = 0, then 2 F 1 (0, b, c, z) = 1. Therefore, setting in Eq. (D10) a = 1, b = 1/3, c = 4/3 and converting z → 1 − z lead toIf we now multiply both sides of Eqs. (D5) and (D9) by 3 and change z → 1 − z, as well as apply Eq. (D11), we readily construct the following identity relation between the Gaussian hypergeometric functions:A quick look at Eq. (D1) convinces us that the full expression enclosed in the square brackets of this formula is exactly the same as the left-hand side of the above identity with z = t/τ 1 (x 0 ).In this way, we can reduce Eq. (D1) to the much simpler formNow, we turn to the case of the parameter σ = 2. Exact calculations of the time derivative of the survival probability in Eq. (41) preceded by the negative sign yieldswhere τ 2 (x 0 ) = π 2 x 4 0 16D . To determine the time derivative in the last line of the above formula, we use the system of two equationsBy fixing in Eq. (D15) that a = −1/4, b = 5/4 and c = 3/4, and replacing its right-hand side by Eq. (D16), we obtainLastly, upon conducting appropriate calculations in Eq. (D14) with the help of Eq. (D17)in which z = 1 − t/τ 2 (x 0 ), we achieve the final result: To begin with, let us first rewrite the Laplace transform of the survival probability inEq. (48) as followsBy making the Laplace transformation of Eq. (11) complemented with the initial condition Q σ (0 | x 0 ) = 1 for the survival probability, we havẽwhile the substitution of Eq. (E1) to the above formula simply givesIn order to learn about the behavior of the first-passage time distribution in the long-time limit, that is t → ∞, we need to determine the adequate expression for Eq. (E3) in the limit s → 0. In this case, it is enough to Taylor expand both the upper incomplete gamma functions dependent on s in Eq. (E3). Due to the fact that Γ(α, z) ≃ Γ(α) −α −1 z α for z → 0, they areandIn this way, Eq. (E3) takes the following formIn this point we can take advantage on the inverse Laplace transformations L[1; s] = δ(t)and L[(as) α ; s] = 1 Γ(−α) t a t α for α > 0, as well as the product Γ(α)Γ(−α) = − π α sin(πα) of the Euler gamma functions with opposite values of the argument α. Thus, performing the inverse Laplace transformation of Eq. (E6), we readily obtain thatwhere B(α, β) = Γ(α)Γ(β) Γ(α+β) is the Euler beta function. Considering that σ > 1 and t ≫ 0, we can identify the dominant component in Eq. (E7), which is the expression displayed in the second line of this formula. Therefore, the asymptotic representation of the σ-dependent first-passage time distribution in the first approximation finally reads.(E8) . E Schrödinger, Physik. Z. 16289E. Schrödinger, Physik. Z. 16, 289 (1915). . M Smoluchowski, Physik. Z. 16318M. Smoluchowski, Physik. Z. 16, 318 (1915). . H A Kramers, Physica. 7284H. A. Kramers, Physica 7, 284 (1940). S Redner, A Guide to First Passage Processes. Cambridge University PressS. Redner, A Guide to First Passage Processes (Cambridge University Press, 2001). S Redner, R Metzler, G Oshanin, First Passage Phenomena and Their Applications. SingaporeWorld ScientificS. Redner, R. Metzler, and G. Oshanin, eds., First Passage Phenomena and Their Applications (Singapore: World Scientific, 2014). . A Bray, S Majumdar, G Schehr, Adv. Phys. 62225A. Bray, S. Majumdar, and G. Schehr, Adv. 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R Mahnke, J Kaupužs, I Lubashevsky, Physics of Stochastic Processes. Wiley-VCHR. Mahnke, J. Kaupužs, and I. Lubashevsky, Physics of Stochastic Processes (Wiley-VCH, 2009). . D S Grebenkov, J. of Phys. A: Math. Theor. 4813001D. S. Grebenkov, J. of Phys. A: Math. Theor. 48, 013001 (2015). . D Holcman, Z Schuss, Rev, 56213D. Holcman and Z. Schuss, SIAM Rev. 56, 213 (2014). . D S Grebenkov, Phys. Rev. Lett. 17260201D. S. Grebenkov, Phys. Rev. Lett. 17, 260201 (2016). . D S Grebenkov, R Metzler, G Oshanin, New. J. Phys. 21122001D. S. Grebenkov, R. Metzler, and G. Oshanin, New. J. Phys. 21, 122001 (2019). . S Condamin, O Bénichou, V Tejedor, R Voituriez, J Klafter, Nature. 45077S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, Nature 450, 77 (2007). . B T Nguyen, D S Grebenkov, J. Stat. Phys. 141532B. T. Nguyen and D. S. Grebenkov, J. Stat. Phys. 141, 532 (2010). . G Vaccario, D A , J Talbot, Phys. Rev. Lett. 115240601G. Vaccario and D. A. abd J. Talbot, Phys. Rev. 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A: Math. Theor. 53193001M. R. Evans, S. N. Majumdar, and G. Schehr, J. of Phys. A: Math. Theor. 53, 193001 (2020). J L Vázquez, The porous medium equation. Oxford University PressJ. L. Vázquez, The porous medium equation (Oxford University Press, 2007). G I Barenblatt, M V Entov, V M Ryzhik, Theory of fluid flows through natural rocks. Dordrecht, NetherlandsKluverG. I. Barenblatt, M. V. Entov, and V. M. Ryzhik, Theory of fluid flows through natural rocks (Dordrecht, Netherlands: Kluver, 1990). . J G Berryman, C J Holland, Phys. Rev. Lett. 401720J. G. Berryman and C. J. Holland, Phys. Rev. Lett. 40, 1720 (1978). . P Y Polubarinova-Kochina, Dokl. Acad. Nauk SSSR. 63623P. Y. Polubarinova-Kochina, Dokl. Acad. Nauk SSSR 63, 623 (1948). . M E Gurtin, R C Maccamy, Math. Biosci. 3335M. E. Gurtin and R. C. MacCamy, Math. Biosci. 33, 35 (1977). . J D Murray, Mathematical Biology, I. An Introduction. SpringerJ. D. Murray, Mathematical Biology, I. An Introduction (Springer, 2002). 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Slater, Generalized Hypergeometric Functions (Cambridge University Press, 1961). F Oberhettinger, L Badii, Tables of Laplace Transforms. Berlin, HeidelbergSpringer-VerlagF. Oberhettinger and L. Badii, Tables of Laplace Transforms (Springer-Verlag, Berlin, Hei- delberg, 1973).
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[ "Spin-phonon interactions and magnetoelectric coupling in Co", "Spin-phonon interactions and magnetoelectric coupling in Co" ]
[ "= Nb \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA\n\nSungkyunkwan University\n16419SuwonRepublic of Korea\n", "Ta ) K Park \nDepartment of Chemistry\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n", "J Kim \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA\n", "S Choi \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA\n\nCenter for Integrated Nanostructure Physics\nInstitute for Basic Science\n16419SuwonRepublic of Korea\n\nSungkyunkwan University\n16419SuwonRepublic of Korea\n", "S Fan \nDepartment of Chemistry\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n", "C Kim \nDepartment of Energy and Chemical Engineering\nUlsan National Institute of Science and Technology (UNIST)\nUlsanKorea\n", "D G Oh \nDepartment of Physics\nYonsei University\n03722SeoulKorea\n", "N Lee \nDepartment of Physics\nYonsei University\n03722SeoulKorea\n", "S. -W Cheong \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA\n\nRutgers Center for Emergent Materials\nRutgers University\n08854PiscatawayNew JerseyUSA\n", "V Kiryukhin \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA\n", "Y J Choi \nDepartment of Physics\nYonsei University\n03722SeoulKorea\n", "D Vanderbilt \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA\n", "J H Lee \nDepartment of Energy and Chemical Engineering\nUlsan National Institute of Science and Technology (UNIST)\nUlsanKorea\n", "J L Musfeldt \nDepartment of Chemistry\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n\nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n" ]
[ "Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA", "Sungkyunkwan University\n16419SuwonRepublic of Korea", "Department of Chemistry\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA", "Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA", "Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA", "Center for Integrated Nanostructure Physics\nInstitute for Basic Science\n16419SuwonRepublic of Korea", "Sungkyunkwan University\n16419SuwonRepublic of Korea", "Department of Chemistry\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA", "Department of Energy and Chemical Engineering\nUlsan National Institute of Science and Technology (UNIST)\nUlsanKorea", "Department of Physics\nYonsei University\n03722SeoulKorea", "Department of Physics\nYonsei University\n03722SeoulKorea", "Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA", "Rutgers Center for Emergent Materials\nRutgers University\n08854PiscatawayNew JerseyUSA", "Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA", "Department of Physics\nYonsei University\n03722SeoulKorea", "Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNew JerseyUSA", "Department of Energy and Chemical Engineering\nUlsan National Institute of Science and Technology (UNIST)\nUlsanKorea", "Department of Chemistry\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA", "Department of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA" ]
[]
In order to explore the consequences of spin-orbit coupling on spin-phonon interactions in a set of chemicallysimilar mixed metal oxides, we measured the infrared vibrational properties of Co 4 B 2 O 9 (B = Nb, Ta) as a function of temperature and compared our findings with lattice dynamics calculations and several different models of spin-phonon coupling. Frequency vs. temperature trends for the Co 2+ shearing mode near 150 cm −1 reveal significant shifts across the magnetic ordering temperature that are especially large in relative terms. Bringing these results together and accounting for noncollinearity, we obtain spin-phonon coupling constants of -3.4 and -4.3 cm −1 for Co 4 Nb 2 O 9 and the Ta analog, respectively. Analysis reveals that these coupling constants derive from interlayer (rather than intralayer) exchange interactions and that the interlayer interactions contain competing antiferromagnetic and ferromagnetic contributions. At the same time, beyond-Heisenberg terms are minimized due to fortuitous symmetry considerations, different than most other 4d-and 5d-containing oxides. Comparison with other contemporary oxides shows that spin-phonon coupling in this family of materials is among the strongest ever reported, suggesting an origin for magnetoelectric coupling.Magnetic materials hosting both transition metal centers and heavy elements are contemporary platforms for the study of chemical bonding and novel properties. The strategy is that 3d ions deliver localized orbitals, high spin, and strong electron correlation, whereas 4d and 5d centers contribute more diffuse orbitals, greater hybridization, a tendency toward dimerization, and spin-orbit coupling that competes on an equal footing with electron correlations. 1-4 This competition endows these materials with remarkable properties including ultra-hard magnetism, 5,6 two-sublattice magnetism with frustration 7-9 or independent ground states, 10,11 and mixing across broad energy scales. 12 One important consequence of spin-orbit interactions in these systems is spin-phonon coupling, conventionally described in terms of how the exchange interactions are modulated by particular displacement patterns.[6][7][8][9][13][14][15][16]In addition to revealing how materials communicate across different energy scales, these interactions can drive multiferroicity.[17][18][19]The Co 4 B 2 O 9 (B = Nb, Ta) system is quasi-two dimensional mixed metal oxide with a P3c1 space group [Fig. 1]. 20 This corundum-type structure is derived from Cr 2 O 3 such that four Cr sites are occupied by the mag-netic Co 2+ ions and two nonmagnetic Nb 5+ or Ta 5+ ions reside on the B sites. The octahedrally-coordinated Co 2+ centers are trigonally distorted and both edge-and facesharing. The B ions are arranged into vertical columns and occupy trigonally distorted octahedral sites in the buckled layer. Both systems order antiferromagnetically at T N = 27 and 20 K for Co 4 Nb 2 O 9 and the Ta analog, respectively. 21,22 Originally thought to host collinear abplane spin structure with a small moment along c, 20,23 recent work establishes a noncollinear spin arrangement -with a different magnetic space group. 24-26 A number of teams report magnetostriction across T N . 27-29 Under magnetic field, these materials reveal a spin-flop transition (at 0.3 T for H ab), 21,22,27 large magnetoelectric coupling, 27,28,30-32 magnetodielectric behavior, 32,33 a symmetry reduction with asymmetric distortion, 29 and spin excitations with magnetoelectric characteristics.34The magnetoelectric coupling is approximately α [110] = 20 ps/m for both compounds.27,28,30,31The interaction is nonlinear, 35,36 and a polarization memory effect is observed in the paramagnetic phase. 37 Despite evidence for magnetostriction across T N 23,28,29,38 and proposals that cast spin-phonon coupling as the underlying mechanistic driver for these effects,24,27,30,38,39the fundamental arXiv:2304.04865v1 [cond-mat.mtrl-sci] 10 Apr 2023
10.1063/5.0137903
[ "https://export.arxiv.org/pdf/2304.04865v1.pdf" ]
258,060,083
2304.04865
4113a19b187b7e73672e06d290c6f8cdae78ac70
Spin-phonon interactions and magnetoelectric coupling in Co = Nb Department of Physics and Astronomy Rutgers University 08854PiscatawayNew JerseyUSA Sungkyunkwan University 16419SuwonRepublic of Korea Ta ) K Park Department of Chemistry University of Tennessee 37996KnoxvilleTennesseeUSA J Kim Department of Physics and Astronomy Rutgers University 08854PiscatawayNew JerseyUSA S Choi Department of Physics and Astronomy Rutgers University 08854PiscatawayNew JerseyUSA Center for Integrated Nanostructure Physics Institute for Basic Science 16419SuwonRepublic of Korea Sungkyunkwan University 16419SuwonRepublic of Korea S Fan Department of Chemistry University of Tennessee 37996KnoxvilleTennesseeUSA C Kim Department of Energy and Chemical Engineering Ulsan National Institute of Science and Technology (UNIST) UlsanKorea D G Oh Department of Physics Yonsei University 03722SeoulKorea N Lee Department of Physics Yonsei University 03722SeoulKorea S. -W Cheong Department of Physics and Astronomy Rutgers University 08854PiscatawayNew JerseyUSA Rutgers Center for Emergent Materials Rutgers University 08854PiscatawayNew JerseyUSA V Kiryukhin Department of Physics and Astronomy Rutgers University 08854PiscatawayNew JerseyUSA Y J Choi Department of Physics Yonsei University 03722SeoulKorea D Vanderbilt Department of Physics and Astronomy Rutgers University 08854PiscatawayNew JerseyUSA J H Lee Department of Energy and Chemical Engineering Ulsan National Institute of Science and Technology (UNIST) UlsanKorea J L Musfeldt Department of Chemistry University of Tennessee 37996KnoxvilleTennesseeUSA Department of Physics and Astronomy University of Tennessee 37996KnoxvilleTennesseeUSA Spin-phonon interactions and magnetoelectric coupling in Co (Dated: 12 April 2023) In order to explore the consequences of spin-orbit coupling on spin-phonon interactions in a set of chemicallysimilar mixed metal oxides, we measured the infrared vibrational properties of Co 4 B 2 O 9 (B = Nb, Ta) as a function of temperature and compared our findings with lattice dynamics calculations and several different models of spin-phonon coupling. Frequency vs. temperature trends for the Co 2+ shearing mode near 150 cm −1 reveal significant shifts across the magnetic ordering temperature that are especially large in relative terms. Bringing these results together and accounting for noncollinearity, we obtain spin-phonon coupling constants of -3.4 and -4.3 cm −1 for Co 4 Nb 2 O 9 and the Ta analog, respectively. Analysis reveals that these coupling constants derive from interlayer (rather than intralayer) exchange interactions and that the interlayer interactions contain competing antiferromagnetic and ferromagnetic contributions. At the same time, beyond-Heisenberg terms are minimized due to fortuitous symmetry considerations, different than most other 4d-and 5d-containing oxides. Comparison with other contemporary oxides shows that spin-phonon coupling in this family of materials is among the strongest ever reported, suggesting an origin for magnetoelectric coupling.Magnetic materials hosting both transition metal centers and heavy elements are contemporary platforms for the study of chemical bonding and novel properties. The strategy is that 3d ions deliver localized orbitals, high spin, and strong electron correlation, whereas 4d and 5d centers contribute more diffuse orbitals, greater hybridization, a tendency toward dimerization, and spin-orbit coupling that competes on an equal footing with electron correlations. 1-4 This competition endows these materials with remarkable properties including ultra-hard magnetism, 5,6 two-sublattice magnetism with frustration 7-9 or independent ground states, 10,11 and mixing across broad energy scales. 12 One important consequence of spin-orbit interactions in these systems is spin-phonon coupling, conventionally described in terms of how the exchange interactions are modulated by particular displacement patterns.[6][7][8][9][13][14][15][16]In addition to revealing how materials communicate across different energy scales, these interactions can drive multiferroicity.[17][18][19]The Co 4 B 2 O 9 (B = Nb, Ta) system is quasi-two dimensional mixed metal oxide with a P3c1 space group [Fig. 1]. 20 This corundum-type structure is derived from Cr 2 O 3 such that four Cr sites are occupied by the mag-netic Co 2+ ions and two nonmagnetic Nb 5+ or Ta 5+ ions reside on the B sites. The octahedrally-coordinated Co 2+ centers are trigonally distorted and both edge-and facesharing. The B ions are arranged into vertical columns and occupy trigonally distorted octahedral sites in the buckled layer. Both systems order antiferromagnetically at T N = 27 and 20 K for Co 4 Nb 2 O 9 and the Ta analog, respectively. 21,22 Originally thought to host collinear abplane spin structure with a small moment along c, 20,23 recent work establishes a noncollinear spin arrangement -with a different magnetic space group. 24-26 A number of teams report magnetostriction across T N . 27-29 Under magnetic field, these materials reveal a spin-flop transition (at 0.3 T for H ab), 21,22,27 large magnetoelectric coupling, 27,28,30-32 magnetodielectric behavior, 32,33 a symmetry reduction with asymmetric distortion, 29 and spin excitations with magnetoelectric characteristics.34The magnetoelectric coupling is approximately α [110] = 20 ps/m for both compounds.27,28,30,31The interaction is nonlinear, 35,36 and a polarization memory effect is observed in the paramagnetic phase. 37 Despite evidence for magnetostriction across T N 23,28,29,38 and proposals that cast spin-phonon coupling as the underlying mechanistic driver for these effects,24,27,30,38,39the fundamental arXiv:2304.04865v1 [cond-mat.mtrl-sci] 10 Apr 2023 In order to explore the consequences of spin-orbit coupling on spin-phonon interactions in a set of chemicallysimilar mixed metal oxides, we measured the infrared vibrational properties of Co 4 B 2 O 9 (B = Nb, Ta) as a function of temperature and compared our findings with lattice dynamics calculations and several different models of spin-phonon coupling. Frequency vs. temperature trends for the Co 2+ shearing mode near 150 cm −1 reveal significant shifts across the magnetic ordering temperature that are especially large in relative terms. Bringing these results together and accounting for noncollinearity, we obtain spin-phonon coupling constants of -3.4 and -4.3 cm −1 for Co 4 Nb 2 O 9 and the Ta analog, respectively. Analysis reveals that these coupling constants derive from interlayer (rather than intralayer) exchange interactions and that the interlayer interactions contain competing antiferromagnetic and ferromagnetic contributions. At the same time, beyond-Heisenberg terms are minimized due to fortuitous symmetry considerations, different than most other 4d-and 5d-containing oxides. Comparison with other contemporary oxides shows that spin-phonon coupling in this family of materials is among the strongest ever reported, suggesting an origin for magnetoelectric coupling. Magnetic materials hosting both transition metal centers and heavy elements are contemporary platforms for the study of chemical bonding and novel properties. The strategy is that 3d ions deliver localized orbitals, high spin, and strong electron correlation, whereas 4d and 5d centers contribute more diffuse orbitals, greater hybridization, a tendency toward dimerization, and spin-orbit coupling that competes on an equal footing with electron correlations. [1][2][3][4] This competition endows these materials with remarkable properties including ultra-hard magnetism, 5,6 two-sublattice magnetism with frustration [7][8][9] or independent ground states, 10,11 and mixing across broad energy scales. 12 One important consequence of spin-orbit interactions in these systems is spin-phonon coupling, conventionally described in terms of how the exchange interactions are modulated by particular displacement patterns. [6][7][8][9][13][14][15][16] In addition to revealing how materials communicate across different energy scales, these interactions can drive multiferroicity. [17][18][19] The Co 4 B 2 O 9 (B = Nb, Ta) system is quasi-two dimensional mixed metal oxide with a P3c1 space group [ Fig. 1]. 20 This corundum-type structure is derived from Cr 2 O 3 such that four Cr sites are occupied by the mag-netic Co 2+ ions and two nonmagnetic Nb 5+ or Ta 5+ ions reside on the B sites. The octahedrally-coordinated Co 2+ centers are trigonally distorted and both edge-and facesharing. The B ions are arranged into vertical columns and occupy trigonally distorted octahedral sites in the buckled layer. Both systems order antiferromagnetically at T N = 27 and 20 K for Co 4 Nb 2 O 9 and the Ta analog, respectively. 21,22 Originally thought to host collinear abplane spin structure with a small moment along c, 20,23 recent work establishes a noncollinear spin arrangement -with a different magnetic space group. 24-26 A number of teams report magnetostriction across T N . [27][28][29] Under magnetic field, these materials reveal a spin-flop transition (at 0.3 T for H ab), 21,22,27 large magnetoelectric coupling, 27,28,30-32 magnetodielectric behavior, 32,33 a symmetry reduction with asymmetric distortion, 29 and spin excitations with magnetoelectric characteristics. 34 The magnetoelectric coupling is approximately α [110] = 20 ps/m for both compounds. 27,28,30,31 The interaction is nonlinear, 35,36 and a polarization memory effect is observed in the paramagnetic phase. 37 Despite evidence for magnetostriction across T N 23,28,29,38 and proposals that cast spin-phonon coupling as the underlying mechanistic driver for these effects, 24 excitations of the lattice and their changes across the magnetic ordering transitions are under-explored with only one Raman study of polycrystalline sample identifying modest coupling in two low-frequency modes. 33 Co 4 Nb 2 O 9 and the Ta analog are also ideal platforms for unraveling structure-property relations -not just across simple trends but in higher level coupling processes as well. 40 In order to explore these issues in greater depth, we combine polarized infrared reflectance, a symmetry analysis, and lattice dynamics calculations to reveal the vibrational properties of Co 4 Nb 2 O 9 and Co 4 Ta 2 O 9 across their magnetic ordering transitions. We find that the inplane Co-containing shearing mode couples strongly to the spin system, red-shifting across T N in both materials. The frequency shifts are extremely large, leading to coupling constants of -3.4 and -4.3 cm −1 in Co 4 Nb 2 O 9 and the Ta analog, respectively. Remarkably, analysis of the spin-phonon coupling demonstrates that this displacement pattern modulates only the inter-plane magnetic interactions and that the latter contains competing antiferromagnetic and ferromagnetic terms. In addi-tion to comparisons with other contemporary oxides, we discuss how unique inter-layer spin-phonon interactions drive magnetoelectric coupling in this class of materials. High-quality single crystals of Co 4 Nb 2 O 9 and Co 4 Ta 2 O 9 were grown by flux techniques. 26,36 Near normal reflectance was measured over a wide frequency range (25 -55,000 cm −1 ) using a series of spectrometers including a Bruker 113v Fourier transform infrared spectrometer equipped with a liquid helium cooled bolometer detector, a Bruker Equinox 55 equipped with an infrared microscope, and a Perkin-Elmer Lambda-1050 grating spectrometer. Appropriate polarizers revealed the ab-plane and c-axes response. A Kramers-Kronig analysis was used to obtain the optical constants. 41 We employed a constant low frequency extrapolation and a high frequency extrapolation of ω −1.75 . The infrared absorption, α(ω), and the real part of the optical conductivity, σ 1 (ω), are of primary interest in this work. Open-flow cryostats provided temperature control. The theoretical phonon frequencies were calculated using the VASP code. 42,43 The pseudopotentials are of the projector-augmented-wave type as implemented in VASP, 44,45 with valence configurations 3d 7 4s 2 for Co, 2s 2 sp 4 for O, 4d 3 4p 6 5s 2 for Nb, and 5d 3 6s 2 for Ta. The exchange-correlation functional is described by the Perdew-Burke-Ernzerhof type generalized gradient approximation, 46 with Dudarev type Hubbard U correction 47 on Co 3d orbits by 3 eV. The plane-wave cut-off energy is set to 400 eV. The Brillouin zone sampling grid is 12 × 12 × 4 including the Γ point. Spin-orbit coupling is not taken into account. The structural coordinates are relaxed within a force threshold of 1.0 meV/Å. To obtain the Γ point phonon frequencies, the dynamical matrix is calculated with the primitive hexagonal cell by using density-functional-perturbation theory 48 and is processed with the PHONOPY code. 49 The oscillator strength and dielectric function are calculated by combining Born effective charge tensors and high frequency dielectric constant from the electron response. 50,51 Figure 1(e, f) summarizes the infrared response of Co 4 Nb 2 O 9 . A symmetry analysis reveals 7A 2u + 14E u infrared-active modes and 7A 1g + 15E g Raman-active modes. The doubly-degenerate E u vibrational modes appear in the ab-plane whereas the singly-degenerate A 2u modes vibrate along c. The Ta analog is isostructural with a P3c1 space group and D 3d point group symmetry. As a result, the spectrum of Co 4 Ta 2 O 9 is quite similar to that of the Nb-containing compound [ Fig. S1, Supplementary information]. Overall, the number of infraredactive modes and their peak positions are nearly identical in both Co 4 Nb 2 O 9 and Co 4 Ta 2 O 9 . A summary and detailed comparison between the experimental and theoretical phonon frequencies, symmetries, and displacement patterns is available in Table S6, Supplementary information. The modes related to the heavy Nb and Ta centers appear at the lowest frequencies since ω ∼ k/µ. We also expect features involving Nb to vibrate at slightly higher frequencies than those involving Ta due to simple mass effects. Here, ω is the frequency, k is the spring constant, and µ is the effective mass. We can test this supposition by examining the mode displacement patterns and realizing that the E u modes at 60(58) cm −1 and the A 2u modes at 138(128) cm −1 reflect the presence of Nb vs. Ta, respectively. The E u symmetry phonon near 150 cm −1 that involves shearing of the Co planes against each other will be important in our discussion as well. This structure is marked with a green arrow in Fig. 1(f). We also studied the temperature dependence of the phonons in Co 4 Nb 2 O 9 and the Ta analog. The majority of features move systematically with decreasing temperature. The E u symmetry shearing mode near 150 cm −1 is the only exception. Figure 2(a, c) displays a close-up view of the E u symmetry vibrational mode near 150 cm −1 in both materials. This feature hardens with decreasing temperature in the paramagnetic phase and softens across T N in both systems. It is the only mode that displays this behavior. The Nb compound hosts a sharp frequency shift across the magnetic ordering temperature whereas the Ta ana-log reveals a gradual transition with a sluggish frequency shift and noticeable precursor effect. As shown in Fig. 2(b, d), we fit frequency vs. temperature trends in the paramagnetic phase using a Boltzmann sigmoidal model to capture anharmonic effects: ω(T ) = (ω o − ω h ) (1 + e ((T −Ti)/dT ) ) + ω h .(1) Here, ω h and ω o are the high and low temperature limits; T i and dT are the inflection point and the width of the distribution. As demonstrated by the deviation from the anharmonic phonon model fit below the magnetic ordering transition, this phonon engages in strong spinphonon coupling in both Co 4 Nb 2 O 9 and Co 4 Ta 2 O 9 . 6,52, 53 We also employed line width effects to extract phonon lifetimes. Focusing on the Co 2+ layer shearing mode, we find phonon lifetimes of 0.75 and 0.6 ps at room temperature in Co 4 Nb 2 O 9 and the Ta analog, respectively. They drop slightly with decreasing temperature and rise sharply below T N . The latter is consistent with fewer scattering events due to spin ordering. 54,55 Details are available in the Supplementary information. In magnetic materials, it is well-known that phonon frequencies can be sensitive to the development of nearneighbor spin correlations. 15,16,56,57 In such cases, the frequency is expressed as: 58 ω = ω o + λ S i · S j .(2) Here, ω o is the unperturbed phonon frequency that is nicely revealed from the fit in Fig. 2(b, d) at base temperature, ω is the renormalized phonon frequency due to spin-phonon interactions, S i · S j is the spin-spin correlation function, and λ is the spin-phonon coupling constant. Using Eqn. 2, our frequency vs. temperature trends, and the model fits of ω(T ) shown in Fig. 2(b, d), we extract frequency shifts across the magnetic ordering transitions and calculate spin-phonon coupling constants for Co 4 Nb 2 O 9 and Co 4 Ta 2 O 9 . The displacement pattern of the participating phonon is also key to our analysis. As a reminder, our calculations predict that the 150 cm −1 phonon mode is an out-of-phase E u symmetry displacement involving the Co centers in which the planar and buckled layers vibrate against each other. The Nb and Ta centers do not participate very much in the motion. As a result, the relevant motion contains only one spin-containing center: Co 2+ with S = 3 2 . With this insight and the fact that S i · S j goes as S 2 in the low-temperature limit, we see that the spin-spin correlation function can be approximated as S i · S j ∼ S 2 = ( 3 2 ) 2 = 9 4 . Co 4 Nb 2 O 9 has a frequency shift (∆ω) across T N of -7.5 cm −1 . In other words, the mode softens across the magnetic ordering temperature. This means that the bond between layers gets weaker due to magnetism. We extract a spinphonon coupling constant (λ) of -3.3 cm −1 . Co 4 Ta 2 O 9 has a larger frequency shift than Co 4 Nb 2 O 9 . Using the same estimate for S i · S j and ∆ω = -9.1 cm −1 , we find λ = -4.0 cm −1 . This makes sense because Nb is a 4d element whereas Ta is 5d, so the latter hosts more significant spin-orbit coupling. We can extend this analysis to include additional effects. For instance, we can modify the spin-spin correlation function as S i · S j ≈ S 2 cos 2 (Θ) in order to capture the noncollinearity of the spin states reported in our systems. 25,26 This expression is a simple analog of Malus's rule for the polarization of light. 59 We use θ = 10.5 • and 14 • to obtain λ = -3.4 and -4.3 cm −1 for Co 4 Nb 2 O 9 and Co 4 Ta 2 O 9 , respectively. 25,26 Accounting for non-collinearity increases spin-phonon coupling constants in the Co 4 B 2 O 9 materials by about 10%. These values are an order of magnitude larger than what was extracted for the aforementioned Raman-active modes, 33 so we see that coupling with odd-rather than evensymmetry vibrational modes is significantly more important. We can also analyze the individual interactions between Co 2+ sites. Here, it's important to recall that Co 4 Nb 2 O 9 and the Ta analog are composed of two different layers -akin to a superlattice consisting of planar graphene and buckled SiC [ Fig. 1(c, d)]. Writing down the spin Hamiltonian for the planar layer, the buckled layer, and the interaction terms between layers in a pairwise fashion, we obtain: H spin = i,j J P S i ·S j + i,j J B S i ·S j + i,j J I S i ·S j . (3) Here, the J P 's are in-plane exchange interactions, the J B 's are those in the buckled layer, and the J I 's couple the two layers quantifying interlayer exchange interactions. The S i 's and S j 's are the spins. Interestingly, both ∂ 2 J P /∂u m ∂u n and ∂ 2 J B /∂u m ∂u n are zero because the motion of interest does not modulate these exchange interactions [Section 4, Supplementary Information]. In other words, the in-plane terms can be ignored because the distances and angles do not change as a result of the displacement. Here, the u n,m 's are the displacements (or distances) between Co centers. Writing down the force constant (k n,m ), we find that: We therefore see that spin-phonon coupling in Co 4 Nb 2 O 9 and the Ta analog is entirely an inter-plane effect and that it is the Co shearing mode shown in Fig. 2(e) that modulates the interlayer magnetic interactions. It turns out that there are two primary types of inter-plane interactions in these systems. (We neglect the long interaction between Co centers along c because it is very small.) By analyzing the bond angles and their tendencies toward parallel or anti-parallel alignment based upon Goodenough-Kanamori-Anderson rules along with the number of near-neighbors, we can write: k n,m = ∂ 2Ĥ spin ∂u m ∂u n = ∂ 2 J I ∂u m ∂u n S i ·S j = λ I S i ·S j . (4)λ T otal S i · S j = λ I S i · S j = λ 2,F M S i · S j + 3λ 3,AF M S i · S j .(5) We immediately notice the competition between antiferromagnetic and ferromagnetic interlayer interactions in Eqn. 5 [and Fig. 1(d)], suggesting that Co 4 Nb 2 O 9 and Co 4 Ta 2 O 9 are frustrated. This competition reduces the overall size of the frequency shift across the magnetic ordering transition. It is also why the T N 's are so low. We suspect that the magnitude of λ 2,F M is larger than that of λ 3,AF M 's, but there are more λ 3,AF M 's in the sum leading to a slight preference for an antiferromagnetic ground state. The findings are consistent with softer O-Co-O bond angles and temperature-dependent lattice constants. 27,33 As we discuss below, magnetoelectric coupling in these materials is likely to emanate from inter-layer spin-phonon interactions. Table I summarizes the properties of several representative transition metal oxides, 4d-and 5d-containing systems, and 3d-5d hybrids. The entries are grouped by the electronic state. We immediately notice that some materials have multiple coupled modes and a tendency toward three-dimensional structure whereas others have only a single spin-phonon coupled mode along with a tendency toward layered or chain-like character. As already discussed, the E u symmetry Co 2+ shearing mode near 150 cm −1 is the only feature to engage in spin-phonon coupling in the materials of interest here. We also see that the frequency shifts in Table I have both positive and negative signs. Among the materials with coupled modes that soften, Co 4 Nb 2 O 9 and the Ta analog host the largest relative frequency shift -even though competing antiferromagnetic and ferromagnetic interactions reduce the size of the overall shift. The spin-phonon coupling constant, λ, is often challenging to define, and the procedure for doing so is inconsistent in the literature. We therefore find the relative frequency shifts to be the most reliable for comparison purposes. As pointed out in Ref. 70, the relative frequency shift of a phonon across a magnetic ordering transition ∆ω/ω 0 is usually less than 1% for a transition metal oxide. Co 4 Nb 2 O 9 and the Ta analog are different with relative frequency shifts of -5 and -6%, respectively. These values are unusually large. In fact, the ∆ω/ω 0 's that we find in Co 4 Nb 2 O 9 and Co 4 Ta 2 O 9 [ Fig. 2(b, d)] are more comparable with the 5d oxides and 3d/5d hybrid systems shown in Table I where the relative frequency shifts correlate (in general) with the electronic shell and spin-orbit coupling. The presence of a heavy element at the B site in Co 4 Nb 2 O 9 and Co 4 Ta 2 O 9 naturally raises questions about the spin Hamiltonian and whether additional terms such as anisotropy and Dzyaloshinskii-Moriya interactions contribute to spin-phonon coupling. 14,68,71,73 Recent neutron scattering also demonstrates that the excitation spectrum can not be reproduced without terms that give rise to spin noncollinearity. 24,25 Focusing on Co 4 Nb 2 O 9 , we see that the anisotropies ≥ J's ≥ Dzyaloshinskii-Moriya interaction. 24,25 All of these terms are on the order of 1 meV. Of course, we are not interested in the absolute size of the anisotropy, exchange, or Dzyaloshinskii-Moriya interaction. Instead, we want to know how they are modulated by the 150 cm −1 phonon mode. In other words, we are interested in how these quantities change with respect to the displacement. By using the mode symmetry and details of the displacement pattern, we can identify the terms that are important for spin-phonon coupling as well as those that will likely cancel out. If we modify Eqn. 4 to include these terms, the prefactors that contribute to spin-phonon coupling are ∂ 2 A/∂u 2 and ∂ 2 DM/∂u 2 . Here A is the anisotropy, and DM is the Dzyaloshinskii-Moriya interaction. The question is whether these contributions are large or small. Since A is an on-site term, we do not expect it to change very much with a vibration. Even if it does, we anticipate that shearing of the planar vs. buckled layers against each other will significantly diminish ∂ 2 A/∂u 2 due to their opposite motion. In other words, while ∂ 2 A/∂u 2 for the planar and buckled layers are not exactly equal, they have opposite signs due to the shearing motion of the layers which, when added together, diminish any impact of the overall anisotropy term. Therefore, we argue that in these unique circumstances ∂ 2 A/∂u 2 is small. On the other hand, the ∂ 2 DM/∂u 2 term multiplies a cross product between two sites and has the potential to contribute. However, spin-orbit coupling and noncollinearity derive primarily from the B site, suggesting reduced importance because the displacement pattern does not involve the movement of the B site. Another way to consider the issue is that the Dzyaloshinskii-Moriya interaction is in the Co-O-Co linkage of the buckled layers, and these bond lengths and angles do not change with the motion. 24 This again rules out contributions from the Dzyaloshinskii-Moriya interaction. Magnetoelectric coupling in the Co 4 B 2 O 9 (B = Nb, Ta) family of materials has been of sustained interest. 23,30,35,36 Suggested mechanisms include spinphonon coupling, 28,29 domain and domain wall effects, 30 Dzyaloshinskii-Moriya interactions through the spincurrent model, 24,38 and critical spin fluctuations. 27,30 With the direct observation of large frequency shifts across the magnetic ordering transitions, sizeable spinphonon coupling constants, and microscopic analysis of the competing interlayer interactions, we are in a better position to evaluate how magnetoelectric coupling might emerge from these candidate mechanisms. As established above, our analysis reveals that Co 4 Nb 2 O 9 and the Ta analog have strong spin-phonon interactions that involve competing interlayer exchange interactions modulated by the shearing motion of the Co layers. While there are Raman-active modes that are sensitive to the development of magnetic ordering, 33 the infraredactive E u symmetry Co 2+ layer shearing mode has a fre-quency shift that is an order of magnitude larger, indicating that odd-symmetry motion dominates spin-phonon coupling in this system. This type of exchange striction provides a very natural origin for magnetodielectric coupling. 23,28,29,38 It is also the most likely origin of magnetoelectric coupling 23,28,29 given the fact that the Dzyaloshinskii-Moriya interaction operating through the spin-current model is significantly smaller and does not couple to the Co 2+ layer shearing mode near 150 cm −1 . The change in dipole moment associated with magnetoelectric coupling 74 is consistent with the microscopic nature of this infrared-active mode. To summarize, we measured the vibrational properties of Co 4 Nb 2 O 9 and the Ta analog and compared our findings with lattice dynamics calculations and a detailed model of spin-phonon coupling. In addition to revealing one of the largest relative frequency shifts ever reported, these materials host a Co shearing mode that couples only with the interlayer interactions due to unique symmetry conditions. These interlayer interactions are frustrated. Given the sizable contribution of spin-phonon interactions in these systems, it is likely that magnetoelectric coupling is driven by this effect. I. AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. Author Contributions KP and JLM designed the study. CK, DGO, NL, SWC, and YJC grew the crystals. KP performed the variable temperature measurements. KP and JLM analyzed the spectral data. JK and DV performed DFT calculations. KP, JK, SC, SF, VK, and JLM discussed the measurement results and analysis in detail. KP, JHL, and JLM developed a spin Hamiltonian model. KP, JK, and JLM wrote the manuscript. All authors read and commented on the text. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. view FIG. 2. (a, b) Absorption spectrum of Co4Nb2O9 as a function of temperature and peak position vs. temperature for the 150 cm −1 shearing mode. (c, d) Similar results for Co4Ta2O9. The spectra in (a, c) are shifted for clarity, and the blue dashed lines in (b, d) account for temperature effects. (e) Calculated displacement pattern of the Eu symmetry Co 2+ layer shearing mode near 150 cm −1 . information for the complete description of the vibrational modes, displacement patterns, the temperature dependence of phonons, spinphonon analysis, and phonon lifetimes as a function of temperature. Research at the University of Tennessee is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Science Division under award DE-FG02-01ER45885. SWC was supported by the center for Quantum Materials Synthesis (cQMS), funded by the Gordon and Betty Moore Foundation's EPiQS initiative through GBMF10104 and by Rutgers University. D.V. was supported by the NSF (DMR-1954856). V.K. was supported by the National Science Foundation (DMR-2103625). S.C. was supported by the Institute for Basic Science (IBS-R011-Y3). D. G. Oh, N. Lee, and Y. J. Choi were supported by the National Research Foundation of Korea (NRF-2017R1A5A1014862 (SRC program: vdWMRC center), NRF-2019R1A2C2002601, and NRF-2021R1A2C1006375). J.H.L. at UNIST was supported by Midcareer Researcher (2020R1A2C2103126) and Creative Materials Discovery (2017M3D1A1040828) programs through the National Research Foundation of Korea. Blue, red, and green represent Co, O, and Nb/Ta centers. Octahedra are drawn only for magnetic ions. (c) A simplified view of the planar Co 2+ layer shows a honeycomb-like structure. (d) Schematic illustrating the planar and buckled layers. Only Co 2+ ions are shown. There are two relevant interlayer interactions (λ2,F M and λ3,AF M ). (e) Calculated Eu and A2u phonon frequencies. (f) Polarized infrared absorption of Co4Nb2O9 at 300 K. The response of Co4Ta2O9 is similar and shown in Fig. S1 of the Supplementary information.,27,30,38,39 the fundamental 100 200 300 400 500 600 700 800 0 1 2 ab plane c-polarization Absorption (10 4 cm -1 ) Frequency (cm -1 ) shearing mode (f) (a) buckled planar (b) (c) (e) A 2u E u (d) λ 2, FM λ 3, AFM Co O Nb/Ta FIG. 1. (a, b) Crystal structure of Co4Nb2O9. 22,26,36 TABLE I . ISpin-phonon coupling in Co4Nb2O9 and Co4Ta2O9 compared with representative oxides. An asterisk (*) indicates estimated values. Materials Sites Electronic ω 0 ∆ω ∆ω/ω 0 λ Refs. (cm −1 ) state (cm −1 ) (cm −1 ) (cm −1 ) (cm −1 ) ZnCr 2 O 4 Cr 3+ 3d 3 370 11 3% 6.2 15 CdCr 2 O 4 Cr 3+ 3d 3 365 9 2.5% 4 60 SrMnO 3 Mn 4+ 3d 3 165 30* 18%* 4.8* 61 Fe 1−x CuxCr 2 S 4 Fe 3+ 3d 5 120 to 400 - <-1.5 to 3% - 62 Fe 2 TeO 6 Fe 3+ 3d 5 300 to 800 <1 to 5* <1 to 1.3%* 0.1 to 1.2 63 Sr 2 CoO 4 Co 4+ 3d 5 630 and 410 - - 2 to 3.5 64 MnF 2 Mn 2+ 3d 5 ≈56 to 480.5 2, -1.2, 2.7, 1.5* ≈-0.4 to 3.7%* 0.4, 0.3, 0.3, -0.2 65 FeF 2 Fe 2+ 3d 6 56 to 480.5 - - 0.4, 0.3, -0.5, -1.3 65 Co 4 Nb 2 O 9 Co 2+ 3d 7 144 -7.5 -5% -3.4 This work Co 4 Ta 2 O 9 Co 2+ 3d 7 145 -9.1 -6% -4.3 This work NiO Ni 2+ 3d 8 752.5, 1160 -12.5, 25* -1.7%*, -2.7%* -7.9, 14.7 66 Ni 3 TeO 6 Ni 2+ 3d 8 313, 597.3, 672 -0.4, 0.3, -3.7 -0.1 to <1%* -0.4, 0.3, -3.7 67 Y 2 Ru 2 O 7 Ru 4+ 4d 4 420 and 492 -0.1 and -0.3 -0.2 and -0.6%* -6 and -9 68 NaOsO 3 Os 5+ 5d 3 550 to 800 40 ≈5.7%* 17.8* 69 Cd 2 Os 2 O 7 Os 5+ 5d 3 100 to 800 -4.0 to 20* -1 to 7% -1.8 to 8.9 (with S = 3/2)* 69,70 Y 2 Ir 2 O 7 Ir 4+ 5d 4 333, 425, 500 -0.8*, -1.6*, -6.5* -0.2, -0.4, -1.3% -0.4 to -3.2 (with J eff = 1/2)* 71 Sr 3 NiIrO 6 Ir 4+ 5d 4 133, 310, 534 - - 2, 10, 5 6 Ba 2 FeReO 6 Fe 3+ /Re 5+ 3d 5 /5d 2 390 to 630 ≈30 5.1%* - 72 Sr 2 CrReO 6 Cr 3+ /Re 5+ 3d 3 /5d 2 600 ≈25 4.9%* - 72 . 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Sohn, J.-G. Park, S. J. Moon, and T. W. Noh, npj Quantum Mater. 4, 17 (2019). . A F García-Flores, A F L Moreira, U F Kaneko, F M Ardito, H Terashita, M T D Orlando, J Gopalakrishnan, K Ramesha, E Granado, 10.1103/PhysRevLett.108.177202Phys. Rev. Lett. 108177202A. F. García-Flores, A. F. L. Moreira, U. F. Kaneko, F. M. Ardito, H. Terashita, M. T. D. Orlando, J. Gopalakrishnan, K. Ramesha, and E. Granado, Phys. Rev. Lett. 108, 177202 (2012). . H J Silverstein, K Fritsch, F Flicker, A M Hallas, J S Gardner, Y Qiu, G Ehlers, A T Savici, Z Yamani, K A Ross, B D Gaulin, M J P Gingras, J A M Paddison, K Foyevtsova, R Valenti, F Hawthorne, C R Wiebe, H D Zhou, 10.1103/PhysRevB.89.054433Phys. Rev. B. 8954433H. J. Silverstein, K. Fritsch, F. Flicker, A. M. Hallas, J. S. Gard- ner, Y. Qiu, G. Ehlers, A. T. Savici, Z. Yamani, K. A. Ross, B. D. Gaulin, M. J. P. Gingras, J. A. M. Paddison, K. Foyevtsova, R. Valenti, F. Hawthorne, C. R. Wiebe, and H. D. Zhou, Phys. Rev. B 89, 054433 (2014). . N A Spaldin, 10.1134/S1063776121040208J. Exp. Theor. Phys. 132493N. A. Spaldin, J. Exp. Theor. Phys. 132, 493 (2021).
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In this paper we consider mappings of jet spaces that preserve the module of canonical Pfaffian forms, but are not generally invertible. These mappings are called contact. A lemma on the prolongation of contact mappings is proved. Conditions are found for which mappings transform solutions of some partial differential equations into ones of other equations. Examples of contact mappings of differential equations are given. We consider contact mappings depending on a parameter and give example of differential equation invariant under such maps.
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[ "https://export.arxiv.org/pdf/2305.11422v1.pdf" ]
258,823,302
2305.11422
353dabb377c26bc10a52e8157d218fd92f3d9b05
19 May 2023 19 May 2023Contact mappings of differential equations. Kaptsov O.V.canonical Pfaffian formscontact mappingsinvariant equations In this paper we consider mappings of jet spaces that preserve the module of canonical Pfaffian forms, but are not generally invertible. These mappings are called contact. A lemma on the prolongation of contact mappings is proved. Conditions are found for which mappings transform solutions of some partial differential equations into ones of other equations. Examples of contact mappings of differential equations are given. We consider contact mappings depending on a parameter and give example of differential equation invariant under such maps. Introduction As is well known, contact transformations are used to solve problems of classical mechanics and equations of mathematical physics [1,2,3]. Popular examples of such transformations are the Legendre and Ampere transformations. The theory of contact transformations was developed by S. Lie. At present, there is a large literature devoted to these issues [4,6,5,7]. The contact transformations are diffeomorphisms of the jet space that preserve the contact structure. To integrate differential equations, it is useful to find contact transformations that leave these equations invariant. However, not only contact transformations are applied to integrate differential equations. Leonhard Euler started using differential substitutions, which are not diffeomorphisms, to integrate linear partial differential equations [8]. Now these substitutions are called the Euler-Darboux transformation [9] or simply the Darboux transformation [10]. In this paper, we consider analytic mappings of jet spaces that preserve the modulus of canonical differential forms and call the mappings contact. We prove a lifting lemma that shows how to construct a contact mapping. For applications to differential equations, the mappings is required to transform solutions of the equations into solutions of other equations or act on solutions of given equations. Examples of second-order partial differential equations connected by contact mappings are given. We also study contact mappings depending on a parameter. It's easier to look for such mappings in the form of series in powers of the parameter. As an example, consider the Burgers equation. Parametric contact mappings are found that act on solutions of this equation. These mappings have no inversional maps. Contact mappings of the jet space We begin with notations and definitions. Denote by Z ≥0 the non-negative integer numbers and by N n the set of natural numbers 1, ..., n. The p-th-order jet space [5] with coordinates {x i , u j α : i ∈ N n , j ∈ N m , α ∈ Z n ≥0 } is denoted by J p (R n , R m ) or simply J p . We suppose that J 0 = R n (x) × R m (u). Denote by J ∞ the space of infinite jets and by π p the projection π p : J ∞ −→ J p . Consider a point a ∈ J ∞ and a ring A p a of convergent power series centered at the point a p = π p (a). We write A a = ∞ p=0 A p a . Recall that an operator D on a ring is called a derivation operator if it satisfies the conditions: D(a + b) = Da + Db, D(ab) = D(a)b + aD(b) for all elements a, b of the ring. We say that a derivation operator D k (k ∈ N n ) on the ring A a is the total derivative when the following conditions is satisfied D k (x i ) = δ ik , D k (u j α ) = u j α+1 k , where δ ik is the Kronecker delta and 1 k = (δ 1k , . . . , δ nk ) ∈ Z n ≥0 . So A a is a differential ring with set ∆ of derivation operators D 1 , . . . , D n ; its elements are called differential power series. Any ideal of A a stable under ∆ is called a differential ideal of A a . The differential ideal of the ring A a generated by the set E ⊂ A a is denoted by < E >. The set of differentials {dx i , du j α : i ∈ N n , j ∈ N m ; α ∈ Z n ≥0 , |α| < p} generates a left module Ω ap of differential 1-forms over the ring A ap . As usual, we say that differential forms ω j α = du j α − n i=1 D i (u j α )dx i , j ∈ N m , α ∈ Z n ≥0(1) are canonical. Definition. A submodule of the left module Ω ap generated by canonical forms ω j α (where |α| ≤ p) is denoted by C p a and is called the contact submodule. We describe below a dual transformation of forms [11]. Let A(W ) be the ring of analytic functions on an open set W ⊂ R k and let Ω(W ) be the left module of differential 1-forms on W . Suppose W 1 ⊂ R k 1 , W 2 ⊂ R k 2 are open sets. Then any analytic mapping φ : W 1 −→ W 2 induces a homomorphism φ ⋆ : A(W 2 ) → A(W 1 ), φ ⋆ (f ) = f (φ), and a linear mapφ ⋆ : Ω(W 2 ) → Ω(W 1 ), given bŷ φ ⋆ ( k 2 i=1 f i (y)dy i ) = k 2 i=1 φ ⋆ (f i )dφ i , where φ i is component of φ. It is convenient to think ofφ ⋆ as a module homomorphism over different rings connected by the homomorphism of rings. The maps φ ⋆ andφ ⋆ are usually not distinguished. We now generalize the classical contact transformations [3]. Definition. Let U be a neighborhood of a point a ∈ J p and let V be a neighborhood of a point b ∈ J q (q ≤ p). An analytic mapping φ : U → V is called a contact mapping if the module homomorphismφ ⋆ maps the contact submodule C q b into the contact submodule C p a . The following lifting lemma shows how to construct a contact mapping. Lemma. Let U be a neighborhood of the point a ∈ J p and let φ : U → J 0 be a analytical mapping of the form y = f (x, . . . , u α ), u = g(x, . . . , u α ), |α| ≤ p such that the matrix Df = (D i f j ) 1≤i,j≤n is invertible at the point a. Then there exists an open set U 1 ⊂ J p+1 and a unique contact mapping φ 1 : U 1 → J 1 coinciding with φ on U. Proof. In what follows, we use the following notations dx = (dx 1 , . . . , dx n ), du α = (du 1 α , . . . , du m α ), u α+1 = (D i (u j α )) 1≤i≤n, 1≤j≤m . According to the definition of a contact mapping, the differential form dv − v 1 dx must be represented as dg − v 1 df = B 0 (du − u 1 dx) + · · · + B p (du α − u α+1 dx), |α| = p,(2) where B 0 , ..., B p are m × m matrices. The left-hand side of equation (2) is written g x dx + g u du + · · · + g uα du α − v 1 (f x dx + f u du + · · · + f uα du α ), where g x = ( ∂g j ∂x i ), ( ∂g j ∂u i ), . . . , f uα = ( ∂f j ∂u i ) are the corresponding Jacobian matrices. We collect together the coefficients of like differential terms in (2) and set all of them equal to zero. The result is a system of matrix equations g x − v 1 f x + B 0 u 1 + · · · + B p u α+1 = 0, g u − v 1 f u = B 0 , · · · , g uα − v 1 f uα = B p Substituting B 0 , ..., B p into the first equation of this system, we have Dg = v 1 Df, with matrices Dg = (D i g j ), Df = (D i f k ) where j ∈ N m and i, k ∈ N n . By the condition of our lemma, the matrix Df is invertible. So the lifting formula (first prolongation) is v 1 = (Dg) • (Df ) −1 .(3) The lifting to J 2 , . . . , J k+1 is carried out in a similar way. The recurrent formula has the form v k+1 = (Dv k ) • (Df ) −1 .(4) These formulas are generalizations of the well-known formulas for the lifting (prolongation) of point transformations [1,9]. Definition. If E = {f i } i∈N k is a family of differential series of the ring A p a , then the expression f i = 0, 1 ≤ i ≤ k is called a system of differential equations and denoted by sys[E]. Definition. Let E be a family of differential series in the ring A p a and let V be an open set in R n . We say that a smooth mapping s : V → J p annihilates the family E if s ⋆ (E) = 0,ŝ ⋆ (C p a ) = 0(5) and the rank of s is n at every point of V . If π p 0 is the projection of J p onto J 0 , then the composition π • s is called a solution of sys[E]. The mapping s is lifted so that it annihilates the canonical forms (1) as described above. The lifted map is denoted bys. Proposition. Let E 1 , E 2 be two families of differential series of rings A p a and A p b respectively. Let U be a neighborhood of the point a p ∈ J p and let V be an open set in R n . Assume that a mapping s : V → U annihilates E 1 and φ : U → J q (q ≤ p) is a contact mapping such that φ ⋆ (E 2 ) ⊂< E 1 >, then φ • s annihilates E 2 . Proof. Since φ ⋆ (E 2 ) ⊂< E 1 >,(6) then it is clear thats ⋆ (φ ⋆ (E 2 )) = 0. It follows that s ⋆ (E 2 (φ)) = E 2 (φ • s) = (φ • s) ⋆ (E 2 ) = 0. The equality (φ • s) ⋆ (C p b ) = 0 follows in the same way. Remarks. To put it simply, the contact mapping φ maps solutions of sys[E 1 ] to solutions of sys[E 2 ] if φ ⋆ (E 2 ) ⊂< E 1 >. If we extend the homomorphism φ ⋆ to the ideal < E 2 >, then the condition (6) can be done more invariant φ ⋆ (< E 2 >) ⊂< E 1 > . Definition. Let E be a family of differential series of the ring A p a and let U be a neighborhood of the point a p ∈ J p . A contact mapping φ : U → J q (q ≤ p) such that φ ⋆ (E) ⊂< E > is called a symmetry of sys[E] . Let us give examples of contact mappings connecting partial differential equations. We now use the classical notation. Consider two equations u tt = x n u xx , n ∈ N,(7)v tt = v yy + m y v y , m ∈ R.(8) We want to find a contact mapping that transforms solutions of the equation (7) to solutions of the equation (8). Consider a mapping φ : J 1 → J 0 of the form t ′ = t, y = h(x), v = f (x)u x + g(x)u,(9) where h, f, g are some smooth functions. We will lift this mapping according to the formulas (3), (4) v t = D t v = f u tx + gu t , v tt = f u ttx + gu tt , v y = D x (v) D x h = D x (f u x + gu) h ′ , v yy = D x (v y ) h ′ .(10) Substituting the found expression for v tt into (8), we have f u ttx + gu tt = v yy + m y v y . We can express u tt , u ttx by using (10) and obtain a new equation (x n u xx ) x f + x n u xx g + 1 h ′ D x D x (f u x + gu) h ′ + mD x (f u x + gu) hh ′ = 0.(11) The left-hand side of this equation is a polynomial in u xxx , u xx , u x , u. Collecting the coefficients of like terms in the polynomial and setting all of them equal to zero, we obtain four equations for the functions f, h, g. The two shortest equations are x n (h ′ ) 2 = 1, m(h ′ ) 2 g ′ + hh ′ g ′′ − hh ′′ g ′ = 0. Integrating these equations for n = 2, we find h = ± 2 2 − n x 2−n 2 + c 0 , g = c 1 + c 2 h 1−m where c 0 , c 1 , c 2 are arbitrary constants. The remaining two equations for the function f are easy to integrate. The following two cases arise: c 1 = 0, c 2 = 0 and c 1 = 0, c 2 = 0. In the first case, the function f is equal to ax (a ∈ R). Then the transformation y = ± 2 2 − n x 2−n 2 , v = a(xu x + (n − 1)u), a ∈ R maps solutions of the equation (7) into ones of equation v tt = v yy + 3n − 4 2 − n v y . In the second case, the transformation y = ± 2 2 − n x 2−n 2 , v = ax 2n−3 (xu x + (n − 1)u) maps solutions of the equation (7) into ones of equation v tt = v yy + 5n − 8 n − 2 v y . Parametric contact mappings It is well known that finding symmetries of differential equations can be simplified if we restrict ourselves to the search for one-parameter groups of transformations that leave the equations invariant. In this section, it is assumed that contact mappings depend on the parameter a. More precisely, we seek an expansion of the mappings in powers of a. Next we restrict ourselves to to the case n = 2, m = 1 and use the classical notation for coordinates in the jet spaces J 0 (x, y, u), J 1 (x, y, u, p, q), J 2 (x, y, u, p, q, r, s, t). Consider a mapping of the form x = x + ax 1 + a 2 x 2 + a 3 x 3 + . . . , y = y + ay 1 + a 2 y 2 + a 3 y 3 + . . . , (12) u = u + au 1 + a 2 u 2 + a 3 u 3 + . . . , where x 1 , x 2 , x 3 , y 1 , . . . , u 3 are functions of x, y, ..., u α . To find the first prolongation of the mapping (12)p = p + ap 1 + a 2 p 2 + a 3 p 3 + . . . , q = q + aq 1 + a 2 q 2 + a 3 q 3 + . . . , it is necessary that the differential form ω 0 = dū −pdx −qdȳ(13) vanishes when the Pfaff equation ω 0 = du − pdx − qdy = 0(14) is satisfied. Substituting the expressions (12) into the formω 0 , by using the equality (14), and collecting together all terms that contain a, we obtain the well-known the first prolongation formulas [1] p 1 = D x (u 1 ) − pD x (x 1 ) − qD x (y 1 ), q 1 = D y (u 1 ) − pD y (x 1 ) − qD y (y 1 ). Collecting together all terms that contain a 2 , we find that p 2 = D x (u 2 ) − pD x (x 2 ) − p 1 D x (x 1 ) − qD x (y 2 ) − q 1 D x (y 1 ), q 2 = D y (u 2 ) − pD y (x 2 ) − p 1 D y (x 1 ) − qD y (y 2 ) − q 1 D y (y 1 ). It is important to remark that x 2 , y 2 , u 2 are an arbitrary functions. When we collect together all terms that contain a 3 this leads to p 3 = D x (u 3 ) − pD x (x 3 ) − p 1 D x (x 2 ) − p 2 D x (x 1 ) − qD x (y 3 ) − q1D x (y 2 ) − q 2 D x (y 1 ), q 3 = D y (u 3 ) − pD y (x 3 ) − p 1 D y (x 2 ) − p 2 D y (x 1 ) − qD y (y 3 ) − q 1 D y (y 2 ) − q 2 D y (y 1 ). Similar formulas are valid for p n , q n (n > 3). It is easy to find formulas for the second prolongation r = r + ar 1 + a 2 r 2 + a 3 r 3 + . . . ,s = s + as 1 + a 2 s 2 + a 3 s 3 + . . . , t = t + at 1 + a 2 t 2 + a 3 t 3 + . . . . For this to be accomplished, it is necessary that the differential forms ω 10 = dp −rdx −sdȳω 01 = dq −sdx −tdȳ vanish if ω 10 = dp − rdx − sdy = 0, ω 01 = dq − sdx − tdy = 0. Using arguments similar to those given above, it is easy to obtain the following formulas r 1 = D x (p 1 ) − rD x (x 1 ) − sD x (y 1 ), s 1 = D y (p 1 ) − rD y (x 1 ) − sD y (y 1 ), t 1 = D y (q 1 ) − sD y (x 1 ) − tD y (y 1 ), r 2 = D x (p 2 ) − rD x (x 2 ) − r 1 D x (x 1 ) − sD x (y 2 ) − s 1 D x (y 1 ), s 2 = D y (p 2 ) − rD y (x 2 ) − r 1 D y (x 1 ) − sD y (y 2 ) − s 1 D y (y 1 ), t 2 = D y (q 2 ) − sD y (x 2 ) − s 1 D y (x 1 ) − tD y (y 2 ) − t 1 D y (y 1 ). As example, consider the Burgers equation u y − u xx − uu x = 0.(15) We look for contact mappings such that (15) is invariant under the ones. The symmetry condition implies that the expression uȳ −ūx x −ūūx lies in the ideal < u t − u xx − uu x > . The simplest of these mappings has the form x = x,ȳ = y,ū = u + 2au x au + 1 , a ∈ R. This mapping satisfies the second-order differential equation u aa = 2ū a (aū a −ū) aū + 1 with initial conditions:ū(0) = u,ū a (0) = u x . Recall that in Lie theory, symmetry transformations satisfy first order ordinary differential equations [1]. A more general symmetry mapping is given by the formulas x = x,ȳ = y,ū = u + 2D x (log h), where the function h satisfies the condition D y h − D 2 x h − uD x h ∈< u y − u xx − uu x > .(16) More precisely, the following statement is true. Proposition. Let u be a solution to the equation (15), and let the differential series h satisfy the condition (16). Then the function v = u + 2D x (log h)(17) is also a solution to the Burgers equation v t − v xx − vv x = 0. Indeed, substituting the function v given by (17) into the left-hand side of the last equation, we obtain an expression that can be represented as u y − u xx − uu x + 2D x D y h − D xx h − uD x h h . Thus the Proposition follows from (16). It is important to note that if h satisfies the condition (16), then η = D x h is a solution of the determining equations for the symmetry generator. Therefore, knowing the symmetries of the equation it is easy to find h. In particular, the condition (16) is satisfied by h of the form h = s 0 + a[s 1 (2p + u 2 ) + s 2 u + s 3 (yu + x) + s 4 (2yp + yu 2 + xu)+ s 5 (y 2 (4p + 2u 2 ) + 2xyu + x 2 + 2y)], where a, s 0 , ..., s 5 are arbitrary constants. If s 0 = 0, then the functionū is represented by power series in a. This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement No. 075-02-2023-912). L Ovsiannikov, Group Analysis of Differential Equations. New YorkAcademic PressOvsiannikov, L. V. Group Analysis of Differential Equations, Academic Press, New York, 1982. . V I Arnold, Mathematical Methods of Classical Mechanics. Springer-VerlagArnold, V. I. Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978. . N H Ibragimov, Transformation Groups Applied to Mathematical Physics. ReidelIbragimov, N. H. Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985. G Bluman, Kumei, S. Symmetries and Differential Equations. NY. SpringerBluman, G. Kumei. S. Symmetries and Differential Equations. NY. Springer, 1989 Applications of Lie Groups to Differential Equations. P Olver, SpringerNYOlver P. Applications of Lie Groups to Differential Equations. Springer. NY, 2000 CRC Handbook of Lie Group Analysis of Differential Equations. I-IIIBoca RatonCRC PressIbragimov, N. H. (Editor). CRC Handbook of Lie Group Analysis of Differential Equations. Vol. I-III. CRC Press, Boca Raton, 1995 Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Krasilshchik, I. S. Vinogradov, A. M.AMSKrasilshchik, I. S. Vinogradov, A. M.(eds.). Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. AMS. 1999. Foundations of Integral Calculus. L Euler, Moscowin RussianEuler L. Foundations of Integral Calculus. V. 3, GIFML, Moscow, 1958. (in Russian) Methods for integrating partial differential equations. O V Kaptsov, Moscow, Sciencein RussianKaptsov, O.V. Methods for integrating partial differential equations. Moscow, Science, 2009. (in Russian) Darboux Transformations and Solitons. V Matveev, M Salle, Springer-VerlagBerlinMatveev V, and Salle, M. Darboux Transformations and Solitons. Berlin: Springer-Verlag, 1991. Analysis on Manifolds. J R Munkres, CRC PressBoca RatonMunkres J. R. Analysis on Manifolds. CRC Press. Boca Raton. 1991. Lie's structural approach to PDE systems. O Stormark, Cambridge University PressStormark O. Lie's structural approach to PDE systems. Cambridge University Press. 2000
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[ "COLORINGS OF SOME CAYLEY GRAPHS", "COLORINGS OF SOME CAYLEY GRAPHS" ]
[ "Prajnanaswaroopa S [email protected] " ]
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Cayley graphs are graphs on algebraic structures, typically groups or group-like structures. In this paper, we have obtained a few results on Cayley graphs on Cyclic groups, typically powers of cycles, some colorings of powers of cycles, Cayley graphs on some non-abelian groups, and Cayley graphs on gyrogroups.
10.48550/arxiv.2305.11623
[ "https://export.arxiv.org/pdf/2305.11623v1.pdf" ]
258,823,317
2305.11623
28fb0ba5be3a8babacd9dd496acd867dfbfac0e2
COLORINGS OF SOME CAYLEY GRAPHS 19 May 2023 Prajnanaswaroopa S [email protected] COLORINGS OF SOME CAYLEY GRAPHS 19 May 2023Abstract: Cayley graphs are graphs on algebraic structures, typically groups or group-like structures. In this paper, we have obtained a few results on Cayley graphs on Cyclic groups, typically powers of cycles, some colorings of powers of cycles, Cayley graphs on some non-abelian groups, and Cayley graphs on gyrogroups. Cayley graphs are graphs on algebraic structures, typically groups or group-like structures. In this paper, we have obtained a few results on Cayley graphs on Cyclic groups, typically powers of cycles, some colorings of powers of cycles, Cayley graphs on some non-abelian groups, and Cayley graphs on gyrogroups. Introduction For a simple loopless graph G, we denote by V (G) and E(G) the vertex and edge sets of the graph, respectively. A k-vertex coloring of a graph G is a map c : V (G) → {1, 2, . . . , k} such that c(v i ) = c(v j ), where v i , v j ∈ V (G) are adjacent vertices. The minimum k required to color vertices is called the chromatic number of G, denoted by χ(G). Edge coloring of a graph G is the proper Coloring of the edges of G such that no two edges incident on the same vertex receive the same color. It can also be interpreted as the vertex coloring of its line graph, L(G). In terms of mappings, a k-edge coloring of G is a map c : E(G) → {1, 2, . . . , k} such that c(e) = c(e ′ ) for any two incident edges e, e ′ ∈ E(G). The minimum k required in such a coloring is the edge chromatic number, or the chromatic index of G, denoted by χ ′ (G). By Vizing's theorem ( [10]), it is known that χ ′ (G) is either ∆(G) or ∆(G) + 1, where ∆(G) is the maximum degree of the graph G. The graphs G with χ ′ (G) = ∆(G) are said to be of class I, and those with χ ′ (G) = ∆(G) + 1 are said to be of class II. The total Coloring of a graph G is the Coloring of the elements of G such that no two adjacent vertices, two adjacent edges, or an edge and its incident vertices receive the same color. In other words, a k-total coloring is a map c : V (G) ∪ E(G) → {1, 2, . . . , k} such that c(u) = c(v) for any two adjacent vertices u, v ∈ V (G),c(e) = c(e ′ ) for any two incident edges e, e ′ ∈ E(G) and c(v) = c(e) for any vertex v ∈ V (G) and any edge e ∈ E(G) incident to v. The minimum k required in such a coloring is called the total chromatic number of the graph, denoted by χ ′′ (G). A trivial bound on total Coloring is that χ ′′ (G) ≥ ∆(G) + 1, where ∆(G) is the maximum degree of G. Total Coloring Conjecture(TCC) is the assertion that χ ′′ (G) ≤ ∆(G) + 2 ([1], [9]). The graphs with χ ′′ (G) = ∆(G) + 1 are called type I, and those with χ ′′ (G) = ∆(G) + 2 are said to be type II. In this paper, we obtain some bounds on the total chromatic number of some Cayley graphs on symmetric groups and powers of cycles, which are a class of Cayley graphs on Cyclic groups. We also obtain bounds on the chromatic number, chromatic index, and total chromatic number of some classes of Cayley graphs on gyrogroups. The Cayley graphs on a group/ gyrogroup Γ with symmetric generating set S (a set is called symmetric if both s and s −1 both belong to S) will be denoted by C(Γ, S). As all graphs are loopless, S does not have the identity element of the group/gyrogroup. The symmetric group, which is the group of all bijective functions from an n-element set to itself, will be denoted by S n . C k n will denote the k-th power of n-cycle. 2 Some Results on Cayley graphs of non-abelian groups Theorem 2.1. The graph G = C(S n , S) with S = {(1, 2), (1, 2, . . . , n), (1, n, . . . , 2) with 3|n is type I. Proof. We observe that the graph G is a disjoint union of a perfect matching generated by the element (1,2) and cycles generated by the elements (1, 2, . . . , n), (1, n, . . . , 2) In order to perform the total Coloring, we first color the matching formed by the element (1, 2) by a unique color The remaining graph is then a union of disjoint cycles Now, in order to color the graph totally, we take left cosets of the cyclic group Z n with respect to S n We observe that each of the disjoint n-cycles formed by taking the cosets are 3-total colorable. In addition, the vertex coloring induced from this total Coloring can also apply to the graph as a whole. This is because, g(1, 2, . . . , n) i = g(1, 2), where i ∈ {0, 1, 2, . . . , n − 1}. Therefore, we can color the graph G totally using 4 colors; in other words, G is type I. Proof. The theorem is the fact that the vertex coloring of C(A n−1 , S) can be extended to that of C(A n , S) by taking cosets. First, as the graph C(A n − 1, S) has the triangle by the generating element (1,2,3), the chromatic number should be x ≥ 3. The procedure for taking cosets is as follows. Case:1-n even: The set S contains the element (2, 3, . . . , n), which is an n − 1 cycle. We take right cosets of A n−1 with respect to A n and label the cosets as A n−1 , A n−1 (2, 3, . . . , n), A n−1 (2, 3, . . . , n) 2 . . . , A n−1 (2, n, n − 1, . . . , 3), L, where L = A n−1 (1, 3, . . . , n) is the remaining set of elements. We call A n−1 the principal coset, and the other cosets as non-principal cosets. For coloring the graph C(A n , S), we arrange the cosets by shifting the non-principal cosets alternately one down and two down. As the adjacencies of the principal coset with respect to (1, 2, 3), (1,3,2) are covered in the assumption that C(A n−1 , S) is properly colored with x colors. Therefore the only thing we need to take care is that the principal coset has no neighbors with respect to the elements (2, 3, . . . , n), (2, n, n − 1, . . . , 3) with any element of the non-principal cosets in the shifted arrangement. Now, since both the cosets A n−1 (2, 3, . . . , n) and A n−1 (2, 3, . . . , n) n−2 = A n−1 (2, n, n − 1, . . . , 3) are shifted either one down or two down in the above arrangement, therefore no adjacency clashes will occur from the vertices of the principal cosets to the nonprincipal coset. Similarly, among the non-principal cosets, the possible adjacency clashes can occur with respect to the elements (1, 2, 3), (1, 3, 2) because of the alternate shifting of the cosets. Let us assume that for some elements g 1 (2, 3, . . . , n) i and g 2 (2, 3, . . . , n) j , we have g 1 (2, 3, . . . , n) i (1, 2, 3) = g 2 (2, 3, . . . , n) j , where g 2 = g 1 s and g 1 , g 2 , s ∈ A n−1 . This would imply s = (2, 3, ldots, n) i (1, 2, 3)(2, 3, ldots, n) k with k = n − 1 − j. As s ∈ A n−1 , and (2, 3, . . . , n) i sends n to i + 1 and sends n − i − 1 to n; the only way this could happen is either when n − (i + 2) = k =⇒ j = i + 1, or i = 1 and k = n − 3 as s must fix n. The case i = j + 1 is impossible by the alternate shift arrangement of the cosets; thereby the only case remaining is i = 1 and k = n − 3. In this case, we have (2, 3, . . . , n)(1, 2, 3)(2, 3, . . . , n) n−3 = (1, 2)(3, 4, . . . , n)(2, n − 1, n − 3, . . . n, n − 2, n − 4, . . . , 4) = (1, n − 1, . . . , 2). Since we have taken the cosets of C(A n−1 , S), in which S had this element, again, this case is impossible. Similarly, we see that if for some elements g 1 (2, 3, . . . , n) i and g 2 (2, 3, . . . , n) j , if we have g 1 (2, 3, . . . , n) i (1, 3, 2) = g 2 (2, 3, . . . , n) j with g 1 s = g 2 and g 1 , g 2 , s ∈ A n−1 , then this would imply that s = (2, 3, . . . , n) i (1, 3, 2)(2, 3, . . . , n) k with again k = n − 2 − j. With similar reasoning as before, this would imply either i + 1 = j or i = 2 and k = 1. For this case, computing s = (2, 3, . . . , n) 2 (1, 3, 2)(2, n, . . . , 3) gives us (1, n − 1, . . . , 2) which is, again, impossible as stated earlier. The last coset L is arranged in the same position as that of the principal coset, because we have (2, 3, . . . , n)(1, 3, 2) = (1, 3, . . . , n). Therefore x colors suffice to color the vertices of C(A n−1 , S). Case:2-n odd: We observe that the set S contains the element (1, 2, . . . , n), which is an ncycle. We take right cosets of A n−1 with respect to A n and label the cosets as A n−1 , A n−1 (1, 2, . . . , n), A n−1 (1, 2, . . . , n) 2 . . . , A n−1 (1, n, n − 1, . . . , 2). We call A n−1 the principal coset, and the other cosets as non-principal cosets. For coloring the graph C(A n , S), we arrange the non-principal cosets by shifting the cosets alternately one down, two down, and in the same position. Note the difference from the even case. If 3|n, the last coset is placed one down. As the adjacencies of the principal coset with respect to (1, 2, 3), (1,3,2) are covered in the assumption that C(A n−1 , S) is properly colored with x colors; therefore the only thing we need to take care is that the principal coset has no neighbors with respect to the elements (1, 2, . . . , n), (1, n, n − 1, . . . , 2) with any element of the non-principal cosets in the shifted arrangement. Now, since both the cosets A n−1 (1, 2, . . . , n) and A n−1 (1, 2, . . . , n) n−1 = A n−1 (1, n, n − 1, . . . , 2) are shifted either one down or two down in the above arrangement, therefore no adjacency clashes will occur from the vertices of the principal cosets to the non-principal coset. Similarly, among the non-principal cosets, the possible adjacency clashes can occur with respect to the elements (1, 2, 3), (1, 3, 2) because of the alternate shifting of the cosets. Let us assume that for some elements g 1 (1, 2, . . . , n) i and g 2 (1, 2, . . . , n) j , we have g 1 (1, 2, . . . , n) i (1, 2, 3) = g 2 (1, 2, . . . , n) j , where g 2 = g 1 s and g 1 , g 2 , s ∈ A n−1 . This would imply s = (1, 2, ldots, n) i (1, 2, 3)(1, 2, ldots, n) k with k = n − j. As s ∈ A n−1 , and (1, 2, . . . , n) i sends n to i and sends n − i to n; the only way this could happen is either when i = n−k =⇒ i = j, or the three cases:i = 1, k = n−2, i = 2, k = n−3 and i = 3, k = n−1; as s must fix n. The case i = j is verily impossible by our assumption that C(A n−1 , S) is properly colored, thereby the only cases remaining are i = 1, k = n − 2, i = 2, k = n − 3 and i = 3, k = n − 1. These cases are not possible due to the arrangement of the cosets described; that is, we arrange the cosets one down, two down, or in the same position as the principal coset. Similarly, we see that if for some elements g 1 (1, 2, . . . , n) i and g 2 (1, 2, . . . , n) j , if we have g 1 (1, 2, . . . , n) i (1, 3, 2) = g 2 (1, 2, . . . , n) j with g 1 s = g 2 and g 1 , g 2 , s ∈ A n−1 , then this would imply that s = (1, 2, . . . , n) i (1, 3, 2)(1, 2, . . . , n) k with again k = n − j. With similar reasoning as before, this would imply either i = j or the three cases:i = 1, k = n − 3, i = 2, k = n − 1 and i = 3, k = n − 2. For these cases, as before, the alternate arrangement of cosets yields us contradictions. Therefore, again x colors suffice for vertex coloring in this case. Proof. The above theorem, combined with the fact that C(A 3 , S) has chromatic number 3 gives us an equitable 3-coloring of G, such that any induced bipartite graph formed by taking two independent sets of vertices is regular of degree 2. This can be seen by the fact that (1, 2, 3)(1, 2, 3) = (1, 3, 2), hence if the induced graphs formed by any two independent sets are non-regular, then we would have had edge clashes among the non-principal cosets, as well as by the alternating arrangement of the non-principal cosets. Thus, every independent set of vertices can be extended to total independent sets by taking a perfect matching from the regular graph formed by the other two independent sets. The graph's last two perfect matchings are given two extra colors to give G a full total coloring with 5 colors, thereby proving that G is type I. Some results on Powers of Cycles It is proved that all graphs C k n with n = s(2m + 1) ± 1 , k 2 ≤ m ≤ k, s − even satisfies TCC. We show some related results in the below discussions. Some regard conformability, and some others focus on total Coloring of powers of cycles. The following theorems are also though proved in [12]. We provide shorter proof. Theorem 3.1. Every graph G = C k n with n even is conformable. Proof. If we have k + 1 ≥ n 4 , then we could divide the vertices as [0, n 2 ], [1, n 2 + 1], . . . , [ n 2 − 1, n − 1] to get a conformable coloring of the vertices. On the other hand, if k + 1 < n 2 m , m ≥ 2, we can divide the vertices into x + y classes, where x = ⌊ n 2 m ⌋ and y = n−⌊ n 2 m ⌋ 2 with 2 m vertices in x color classes and 2 vertices in y color classes. The Coloring is conformable as all the independent sets so divided have even parity (including the null independent sets). Theorem 3.2. If k + 1 < n 3 , then the power of cycle graphs C k n are conformable. Proof. The given condition implies that the independence number of the graph C k n is ≥ 3. Therefore, we can put 3 vertices in an independent set. This condition is essential because 3 is the minimum odd number after 1, and the complete graphs of odd order are the only regular and conformable graphs having exactly 1 vertex in all the independent sets. As the parity of n is odd, we must have no null independent sets, for 0 has even parity. So, let us first divide the first 2k + 1 vertices into 2k + 1 independent sets. Now, the induced graph formed by the remaining n − (2k + 1) vertices can be thought of as an induced subgraph of k-th power of a cycle of even order (it is an even power of path P k n−(2k+1) ). Since we have proved that the even powers of cycles are vertex conformable, this implies that the induced subgraph is also vertex conformable. Thus, the graph G, in this case, is also vertex conformable, as we can arrange the remaining n − (2k + 1) vertices like the conformable Coloring of the induced graph of k-th power of even cycle, which ensures odd vertices in all the independent sets of G. The above fact of conformability is strong evidence in favor of the Campos-de Mello conjecture [3]. Though there are graphs that are conformable but still not type I, the high symmetry of powers of cycles seems that conformability implies type I for these graphs. Theorem 3.3. If n = m(k + 1) + 1 for some even integer m and odd k, then G = C k n satisfies TCC. Proof. We show how we can structure the total color matrix as follows. First, we describe the method for m = 2. The remaining cases, that is, m > 2, follow by repeating the copies of the case m = 2 and the connecting edges between two copies connected by a modification of the colors used at the top right end of the matrix, as explained below. This is possible as m is assumed even. The induced graph formed by every set of k vertices starting from 0, that is, the induced graph formed by the vertices 0, 1, . . . , k − 1, the induced graph formed by the vertices k, k + 1, . . . , 2k,. . . are given total Coloring according to the commutative idempotent pseudo-latin square of order k + 1. The pseudo-latin square is not a Latin square, as it has more numbers than its order. It is derived from the commutative idempotent Latin square of order k + 2 by deleting the last row and column. The last vertex is given the color k + 2, so the rightmost and bottom-most entry is k + 2. Now, we fill the remaining entries in the total color matrix using a diagonal pattern; that is, the partial diagonals have the same number throughout. We can say that the remaining portion of the total color matrix can be divided into six parts, of which the first part consists of the super-diagonal starting from the edge 0 − (n − k) and ending the super-diagonal or entry corresponding to edge 0 − (n − 1). The second part consists of the partial super-diagonal starting at the entry corresponding to the edge 1 − (k + 1) and ending at the partial super-diagonal, or entry corresponding to the edge k − (k + 1). The third part corresponds to entries in the last column from the entry corresponding to the edge (n − k − 1) − (n − 1). The fourth, fifth, and sixth parts are symmetric counterparts of the first, second, and third parts. The first part of the total color matrix, that is, the partial super-diagonals starting from the entry corresponding to the edge 0 − (n − k) is given the color k + 3 (same color throughout the super-diagonal); similarly, the next partial diagonal, that is, staring from the entry corresponding to the edge 0 − (n − k + 1) is given the color k + 4 (same color throughout the super-diagonal), and so on so that, the last partial diagonal, which is just one entry corresponding to the edge 0 − (n − 1) is given the color 2k + 2. For the second part of the total color matrix, the partial super-diagonals starting from the entry corresponding to the edge 1 − (k + 1) are given the color 2k + 2 (same color throughout the super-diagonal); similarly, the next partial super-diagonal, that is, staring from the entry corresponding to the edge 2 − (k + 2) is given the color 2k + 1 (same color throughout the super-diagonal) and so on, so that the last partial diagonal closest to the main diagonal, or the entry corresponding to the edge k − (k + 1) is given the color k + 2. The third part of the total color matrix, which consists of the last column entries, is just a continuation of the entries of the k + 1 pseudo-latin square with entries identical to that found in the Latin square of order k + 2. The symmetric counterparts, the fourth, fifth, and sixth parts of the total color matrix, get the same colorings as the first, second, and third parts. In case m > 2, the connecting edges between the odd and even idempotent pseudo-latin squares are given the same Coloring as for the second part of the case m = 2. In contrast, the connecting edges between the even and odd copies of idempotent pseudo-latin squares are given the transpose of the colors used in the first part of the case m = 2. The first part of the case m = 2 is retained as it is for the case m > 2. The procedures above give us a total coloring of G because there are no clashes between the entries in the matrix. To see this, observe that for the case m = 2, the colors given in the pseudo-Latin squares and the super-diagonals are entirely different. The last column entries, part of the k + 2 Latin square, will be distinct from the entries given in the pseudo-latin square and the third and sixth parts of the color matrix. In the case, m > 2, the same properties of the colors (distinctness of colors used in the pseudo-latin squares and the connecting super-diagonals) as for m = 2 are retained, and the super-diagonals (connecting edges) are arranged in such a way that clashes are averted. Thus, the graph G satisfies TCC. Example 3.1. Consider the power of cycle C 5 13 . Here, we have m = 2 and k + 1 = 6. The total color matrix, in this case, is given in Table 1: Table 1: Total Color matrix of C 5 13 Example 3.2. Consider the graph C 5 25 . Here, m = 4. The total color matrix, in this case, is given in Table 2 4 Results on Cayley graphs on Gyrogroups First, we will begin with a lemma that helps determine the isomorphism of Cayley graphs defined on the same group. An exponent distribution of a subset Σ of a generating set S of a group is the set of exponents of the elements of S with respect to the product of elements of Σ. That is, if Σ = {σ 1 , σ 2 , . . . , σ n } and S = {s 1 , s 2 , . . . , s n }; then the exponent distribution of Σ is the set {i 1 , i 2 , . . . , i n }, where s 1 = (σ 1 σ 2 . . . σ n ) i 1 , s 2 = (σ 1 σ 2 . . . σ n ) i 2 , . . .. The following lemma gives an algorithm to determine when two Cayley Graphs on the same group with different generating sets are isomorphic. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 5 2 6 3 7 12 11 10 9 8 1 5 2 6 3 7 4 8 12 11 10 9 2 2 6 3 7 4 1 9 8 12 11 10 3 6 3 7 4 1 5 10 9 8 12 11 4 3 7 4 1 5 2 11 10 9 8 12 5 7 4 1 5 2 6 12 11 10 9 8 6 8 9 10 11 12 1 5 2 6 3 7 7 8 9 10 11 5 2 6 3 7 4 12 8 8 9 10 2 6 3 7 4 1 11 12 9 8 9 6 3 7 4 1 5 10 11 12 10 8 3 7 4 1 5 2 9 10 11 12 11 7 4 1 5 2 6 8 9 10 11 Lemma 4.1. Two Cayley graphs G 1 = C(Γ, S 1 ) and G 2 = C(Γ, S 2 ) are isomorphic if there exist two sets of generating subsets Σ 1 ⊂ S 1 and Σ 2 ⊂ S 2 of G of same cardinality such that their exponent distributions with respect to some permutation of the generating set are the same. Proof. The proof is immediate on noting that one could construct identical graphs by starting from the identity element of Γ and the generating subsets Σ 1 and Σ 2 . The above algorithm could be in polynomial time if the base group Γ is cyclic, as the following corollary shows. Corollary 4.1. If Γ is a cyclic group of order n, then the graphs G 1 = C(Γ, S 1 ) and G 2 = C(Γ, S 2 ) are isomorphic if there exist generating elements of Γ s 1 ∈ S 1 and s 2 ∈ S 2 such that their exponent distributions are same. Proof. The proof is immediate from the previous lemma once it is known that the cyclic groups have a single minimal generator, and the corollary assumes that s 1 , s 2 are those generators. A gyrogroup is a non-associative structure on a set having a left inverse and left identity. It is a magma with a bijective function called the gyroautomorphism that gives an associative-like structure called gyro-associativity to the gyrogroup. Technically, a gyrogroup Γ is a set S with a binary operation ⊕ and a bijective function gyr[a, b] which is an automorphism for (Γ, ⊕) satisfying: i)Left identity: We should have an element 0 ∈ Γ such that for all a ∈ Γ, 0 ⊕ a = a. ii)Left inverse: We should have an element ⊖a ∈ Γ for every element a ∈ Γ, we have (⊖a) ⊕ a = 0. In [6], the authors introduced a gyrogroup with properties resembling dihedral groups' properties. The authors called these 2-gyrogroups. In [7], the author studied several graph theoretic properties of Cayley graphs associated with these gyrogroups. Here, we give a theorem on the total Coloring of a subclass of Cayley graphs on these gyrogroups. Theorem 4.1. Let Γ be the 2-gyrogroup of order n = 2 k presented in [7]. Then, the graph C(Γ, S) with S = {1, 2, . . . , k, m − k, m − k + 1, . . . , m − 1, m 2 + m. Then G satisfies TCC. Further, if the power of cycle C k m is type I, then G is also type I. Proof. For ease of reference, we state the binary operation of the 2-gyrogroup presented in [6] and Example 1 of [7]. The binary operation ⊕ is defined as: i ⊕ j =          i + j (mod m) , i ∈ T 1 ; j ∈ T 1 {i + j (mod m)} + m , i ∈ T 1 ; j ∈ T 2 {i + ( m 2 − 1)j} (mod m)) + m , i ∈ T 2 ; j ∈ T 2 {( m 2 − 1)i + ( m 2 + 1)j (mod m)} + m , i ∈ T 2 ; j ∈ T 2 . Here T 1 = {0, 1, . . . , m − 1}, m = 2 n−1 and T 2 = {m, m + 1 . . . , n − 1}. We note that the subgraph induced by the set T 1 , is isomorphic to C k m , because, we have i ⊕ j = i + j for i, j ∈ {0, 1, . . . , m − 1}. Similarly, the graph C k m is isomorphic to the subgraph induced by the set T 2 with respect to the generating set {( m 2 − 1) + m, 2m − 2, . . . , k( m 2 − 1) + m, 2m − k( m 2 − 1), . . . , 2 + m, ( m 2 + 1) + m}. This is because of two reasons: 1) We have i ⊕ j = i + ( m 2 − 1)j (mod m) for i ∈ T 1 j ∈ T 2 , and 2) We note that the exponent distribution of 1 for {1, 2, . . . , k, n − k, . . . , n − 2, n − 1} and z = ( m 2 − 1) for {( m 2 − 1), m − 2, . . . , k( m 2 − 1)m − k( m 2 − 1), . . . , 2, ( m 2 + 1)} are the same, in the group Z m . Now, note that the element m 2 + m acts as a sort of reflection equivalent for the gyrogroup Γ, in the sense that, we have ( m 2 + m) ⊕2 = ( m 2 + m) ⊕ ( m 2 + m) = ( m 2 + 1)( m 2 + m) + ( m 2 − 1)( m 2 + m) (mod m) ≡ 2 m 2 4 = m( m 2 ) = 0 (mod m) . Therefore, this element induces a perfect matching in the graph C(Γ, S) with the end vertices of the edges being in the two sets T 1 and T 2 , respectively. Now, if i and j and in the same independent set of the subgraph induced by T 1 , then i ⊕ ( m 2 + m) = (i + ( m 2 + m) (mod m)) + m and j ⊕ ( m 2 + m) = (i + ( m 2 + m) (mod m)) + m are also in the same independent set, as i − j = i ⊖ j. Hence, we can divide the vertices of the induced subgraph on T 2 into the same number of independent sets as that of the induced subgraph on T 1 by shifting the translates of an independent set with respect to the reflection element m 2 + m. Thus, the chromatic number of G is the same as that of C k m . As the graphs C k m satisfy TCC by Theorem 16 of Campos-de Mello [3], the induced subgraphs on T 1 and T 2 individually satisfy TCC. Since we have arranged the induced graphs together in the same number of independent sets, only the edge coloring of the connecting perfect matching between the induced subgraphs formed by the element m 2 + m needs to be done in order to complete the total Coloring of G. We give one extra color for this, making G satisfy TCC. This argument also shows why G will be of type I if C k m is also of type I. The above theorem at once gives the following generalization as its corollary. Corollary 4.2. Let Γ be the 2-gyrogroup presented in [7]. Then, if the graph C(Z m , S 1 ) staisfies TCC, then G = C(Gamma, S) with S 1 ∪ { m 2 + m} satisfies TCC. Further, if the graph C(Z m , S 1 ) is type I, then G is also type I. Proof. The proof is immediate once we replace the element 1 in the previous theorem with any suitable generator s ∈ S 1 of the group Z m . Then, the subgraphs induced by the two sets T 1 and T 2 are isomorphic circulant graphs. The element m 2 +m then gives us a perfect matching of elements in G with the end vertices in T 1 and T 2 , respectively. Again, by following the similar argument in the proof of the last theorem, the chromatic number of G is the same as that of the graph C(Z m , S 1 ). Then, giving the perfect matching induced by the element m 2 + m makes G satisfy TCC. The following result, an immediate consequence, gives us the chromatic index of such graphs. Theorem 4.2. Let Γ be the 2-gyrogroup presented in [7]. Then, if the graph C(Z m , S 1 ) satisfies TCC, then G = C(Gamma, S) with S 1 ∪ { m 2 + m} has chromatic number equal to that of the graph χ(C(Z m , S 1 )). In addition, the graph is of class I. Proof. As the circulant graphs C(Z m , S 1 ) are of class I by the Corollary 2.3.1 of [8], we need to only give one extra color to the perfect matching induced by the element m 2 + m. Thus, all the edges of G can be colored in precisely ∆(G) + 1 colors or G is of class I. An immediate generalization of the above result in the context of edge coloring is the following: Theorem 4.3. All Cayley graphs G = Cay(Γ, S) for the 2-gyrogroup described in the theorems above and any generating set S is of class I. Proof. The proof is immediate, once we note that all generating elements of the form s = j + m, where j ∈ {0, 1, . . . , m − 1} are sort of reflections (gyro-reflections), that is, satisfy the property s ⊕2 = s ⊕ s = ( m 2 − 1)s + ( m 2 + 1)s (mod m) ≡ ms (mod s)equiv0. Hence, all the reflections give rise to perfect matchings in the graph G. Since the induced graph on the sets {0, 1, . . . , m − 1} and {m, m + 1, . . . , n − 1} are each circulant, therefore, by [8] Corollary 2.3.1 and the fact that the perfect matchings generated by any elements of the form s can be 1-factorized, the conclusion is immediate. . . . , n), (2, n, . . . , 3)} , if n is even {(1, 2, 3), (1, 3, 2), (1, 2, . . . , n), (1, n, . . . , 2)} , otherwise is colorable with x colors, then C(A n , S) is also colorable with x colors. Corollary 2 . 1 . 21The graph G = C(A n , S) , n ≥ 4 with S = {(1, 2, 3), (1, 3, 2), (2, 3, . . . , n), (2, n, . . . , 3)} , if n is even {(1, 2, 3), (1, 3, 2), (1, 2, . . . , n), (1, n, . . . , 2)} , otherwise is type I. iii) Gyroassociativity: We should have, for three elements a, b,c ∈ Γ, a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ gyr[a, b]c.Thus, Gyrogroups generalize groups, as every group is a gyrogroup (with the gyroautomorphism induced by the identity map). iv)Left loop property: We should have for all a, b ∈ Γ, gyr[a, b] = gyr[(a ⊕ b), b]. Table 2 : 2Total Color matrix of C 525 Graphs and their chromatic numbers. M Behzad, Michigan State UniversityBehzad, M. (1965). Graphs and their chromatic numbers. Michigan State University. On the coequal values of total chromatic number and chromatic index. G Chen, Y Hao, Journal of Combinatorial Theory, Series B. 158Chen, G., Hao, Y. (2023). On the coequal values of total chromatic number and chromatic index. Journal of Combinatorial Theory, Series B, 158, 286-304. A result on the total colouring of powers of cycles. C N Campos, C P Mello, Discrete Applied Mathematics. 1555Campos, C. N., and de Mello, C. P. (2007). A result on the total colouring of powers of cycles. Discrete Applied Mathematics, 155(5), 585-597. The total chromatic number of regular graphs of high degree. Dezheng Xie, Wannian Yang, Science in China Series A: Mathematics. 52Xie, DeZheng, and WanNian Yang. "The total chromatic number of regular graphs of high degree." Science in China Series A: Mathematics 52.8 (2009): 1743-1759. On planarity and colorability of circulant graphs. C Heuberger, Discrete Mathematics. 2681-3Heuberger, C. (2003). On planarity and colorability of circulant graphs. Discrete Mathematics, 268(1- 3), 153-169. Construction of 2-gyrogroups in which every proper subgyrogroup is either a cyclic or a dihedral group. S Mahdavi, A R Ashrafi, M A Salahshour, A A Ungar, Symmetry. 132316Mahdavi, S., Ashrafi, A. R., Salahshour, M. A., and Ungar, A. A. (2021). Construction of 2-gyrogroups in which every proper subgyrogroup is either a cyclic or a dihedral group. Symmetry, 13(2), 316. . R Maungchang, T Suksumran, On Dihedralized Gyrogroups and Their Cayley Graphs. Mathematics. 10132276Maungchang, R., and Suksumran, T. (2022). On Dihedralized Gyrogroups and Their Cayley Graphs. Mathematics, 10(13), 2276. On 1-factorizability of Cayley graphs. R A Stong, Journal of Combinatorial Theory, Series B. 393Stong, R. A. (1985). On 1-factorizability of Cayley graphs. Journal of Combinatorial Theory, Series B, 39(3), 298-307. Some unsolved problems in graph theory. V G Vizing, Russian Mathematical Surveys. 236125Vizing, V. G. (1968). Some unsolved problems in graph theory. Russian Mathematical Surveys, 23(6), 125. On an estimate of the chromatic class of a p-graph. V G Vizing, Diskret analiz. 3Vizing, V. G. (1964). On an estimate of the chromatic class of a p-graph. Diskret analiz, 3, 25-30. Behzad-Vizing conjecture and Cartesian-product graphs. B Zmazek, J Zerovnik, Applied mathematics letters. 156Zmazek, B., and Zerovnik, J. (2002). Behzad-Vizing conjecture and Cartesian-product graphs. Applied mathematics letters, 15(6), 781-784. Even-power of cycles with many vertices are Type 1 total colorable. A Zorzi, C De Figueiredo, R Machado, U S Souza, Electronic Notes in Theoretical Computer Science. 346Zorzi, A., de Figueiredo, C., Machado, R., and Souza, U. S. (2019). Even-power of cycles with many vertices are Type 1 total colorable. Electronic Notes in Theoretical Computer Science, 346, 747-758.
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[ "Federated learning for secure development of AI models for Parkinson's disease detection using speech from different languages", "Federated learning for secure development of AI models for Parkinson's disease detection using speech from different languages" ]
[ "Soroosh Tayebi Arasteh [email protected] \nPattern Recognition Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany\n\nSpeech & Language Processing Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany\n\nDepartment of Diagnostic and Interventional Radiology\nUniversity Hospital RWTH Aachen\nAachenGermany\n", "Cristian David Rios-Urrego \nFaculty of Engineering\nGITA Lab\nUniversity of Antioquia\nMedellínColombia\n", "Elmar Noeth \nPattern Recognition Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany\n", "Andreas Maier \nPattern Recognition Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany\n", "Seung Hee Yang \nSpeech & Language Processing Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany\n", "Jan Rusz \nDepartment of Circuit Theory\nCzech Technical University\nPrague, PragueCzech Republic\n", "Juan Rafael Orozco-Arroyave \nPattern Recognition Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany\n\nFaculty of Engineering\nGITA Lab\nUniversity of Antioquia\nMedellínColombia\n" ]
[ "Pattern Recognition Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany", "Speech & Language Processing Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany", "Department of Diagnostic and Interventional Radiology\nUniversity Hospital RWTH Aachen\nAachenGermany", "Faculty of Engineering\nGITA Lab\nUniversity of Antioquia\nMedellínColombia", "Pattern Recognition Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany", "Pattern Recognition Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany", "Speech & Language Processing Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany", "Department of Circuit Theory\nCzech Technical University\nPrague, PragueCzech Republic", "Pattern Recognition Lab\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nErlangenGermany", "Faculty of Engineering\nGITA Lab\nUniversity of Antioquia\nMedellínColombia" ]
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Parkinson's disease (PD) is a neurological disorder impacting a person's speech. Among automatic PD assessment methods, deep learning models have gained particular interest. Recently, the community has explored cross-pathology and crosslanguage models which can improve diagnostic accuracy even further. However, strict patient data privacy regulations largely prevent institutions from sharing patient speech data with each other. In this paper, we employ federated learning (FL) for PD detection using speech signals from 3 real-world language corpora of German, Spanish, and Czech, each from a separate institution. Our results indicate that the FL model outperforms all the local models in terms of diagnostic accuracy, while not performing very differently from the model based on centrally combined training sets, with the advantage of not requiring any data sharing among collaborators. This will simplify interinstitutional collaborations, resulting in enhancement of patient outcomes.
10.48550/arxiv.2305.11284
[ "https://export.arxiv.org/pdf/2305.11284v1.pdf" ]
258,823,421
2305.11284
ff1da82a6a55ce26e8b269a6bcffda29773907d5
Federated learning for secure development of AI models for Parkinson's disease detection using speech from different languages Soroosh Tayebi Arasteh [email protected] Pattern Recognition Lab Friedrich-Alexander-Universität Erlangen-Nürnberg ErlangenGermany Speech & Language Processing Lab Friedrich-Alexander-Universität Erlangen-Nürnberg ErlangenGermany Department of Diagnostic and Interventional Radiology University Hospital RWTH Aachen AachenGermany Cristian David Rios-Urrego Faculty of Engineering GITA Lab University of Antioquia MedellínColombia Elmar Noeth Pattern Recognition Lab Friedrich-Alexander-Universität Erlangen-Nürnberg ErlangenGermany Andreas Maier Pattern Recognition Lab Friedrich-Alexander-Universität Erlangen-Nürnberg ErlangenGermany Seung Hee Yang Speech & Language Processing Lab Friedrich-Alexander-Universität Erlangen-Nürnberg ErlangenGermany Jan Rusz Department of Circuit Theory Czech Technical University Prague, PragueCzech Republic Juan Rafael Orozco-Arroyave Pattern Recognition Lab Friedrich-Alexander-Universität Erlangen-Nürnberg ErlangenGermany Faculty of Engineering GITA Lab University of Antioquia MedellínColombia Federated learning for secure development of AI models for Parkinson's disease detection using speech from different languages Index Terms: federated learningspeech pathologyParkin- son's diseasedeep learningtrustworthy speech processing Parkinson's disease (PD) is a neurological disorder impacting a person's speech. Among automatic PD assessment methods, deep learning models have gained particular interest. Recently, the community has explored cross-pathology and crosslanguage models which can improve diagnostic accuracy even further. However, strict patient data privacy regulations largely prevent institutions from sharing patient speech data with each other. In this paper, we employ federated learning (FL) for PD detection using speech signals from 3 real-world language corpora of German, Spanish, and Czech, each from a separate institution. Our results indicate that the FL model outperforms all the local models in terms of diagnostic accuracy, while not performing very differently from the model based on centrally combined training sets, with the advantage of not requiring any data sharing among collaborators. This will simplify interinstitutional collaborations, resulting in enhancement of patient outcomes. Introduction Parkinson's disease (PD) is a neurodegenerative disorder that affects the nervous system, leading to the progressive deterioration of motor and non-motor functions, which contribute significantly to decreasing the quality of life of the patient's [1]. PD is characterized by resting tremor, rigidity, bradykinesia, postural instability, and other symptoms [2]. Most PD patients develop speech deficits which are grouped and called hypokinetic dysarthria where the speech is characterized by reduced loudness, monotonous pitch, and changes in voice quality [3,4]. Speech signals can be analyzed objectively to quantify the severity of the disease and track its progression over time, which can be useful in clinical research and treatment monitoring. Among the best motivations to consider the speech signals is that they can be easily collected and analyzed remotely, which can provide greater convenience to patients and reduce the need for frequent clinical visits [5]. This can be especially beneficial for patients who live in remote areas or have limited mobility. In addition, speech signals can provide a complementary source of information to clinical assessment and other diagnostic tests, which can improve the accuracy and reliability of PD diagnosis and treatment [6]. Recently, deep learning (DL)-based methods have particularly gained a lot of attention for analyzing PD speech signals [7,8]. However, a major impediment to developing such robust DL models is the need for accessing lots of training data, which is challenging for many institutions. Thus, benefiting from data from different external institutions could solve this issue. However, strict patient data privacy regulations in the medical context make this infeasible in most cases in realworld practice [9][10][11][12]. Therefore, privacy-preserving collaborative training methods, in which participating institutions do not share data with each other are favorable. Federated learning (FL) [13][14][15], as the golden key to this issue, has been increasingly investigated by researchers and practitioners and received a lot of attention in the medical image analysis domain [11,[16][17][18] as it does not require sharing any training data between participating institutions in the joint training process. To the best of our knowledge, collaborative training methods based on FL have not been addressed in the literature on pathological speech signals yet, despite the availability of similar privacy regulations and restrictions as in the imaging domain [9], especially considering recent literature revealing the vulnerability of pathological speech signals in terms of patient data [19][20][21]. In this paper, for the first time, we investigate the applicability of FL in the privacy-preserving development of DL methods for PD detection using speech signals from three real-world language corpora, each from a separate and independent institution. We hypothesize that utilizing FL will substantially increase the diagnostic performances of networks for each local database while preserving patient privacy by avoiding data sharing between the institutions. Moreover, we assume that the FL model will perform relatively similarly, with only slight degradation compared to the hypothetical and non-privacy-preserving scenario where all the institutions could combine their training sets at a central location. Material and Methods Methodology The methodology addressed in this study consists of the following main stages: data were acquired in different languages (Ger- man, Spanish, and Czech), after, embeddings were extracted from speech signals for each participant using a pre-trained Wav2Vec 2.0 model [7,8], then the extracted embeddings were utilized for the secure FL training of a classification architecture, and finally, a copy of the global model is sent back to each participating site for the classification of PD patients from healthy control (HC) subjects. This methodology is summarized in Fig. 1. Details of each stage are presented below. Data We considered speech corpora in three different languages: Spanish, German, and Czech; each database contains PD patients and HC subjects. The first corpus is PC-GITA which includes recordings of 50 PD patients and 50 HC subjects [22]. All participants were Colombian native speakers. The second corpus contained a total of 176 German native speakers (88 PD patients and 88 HC subjects) [23]. The last database contained recordings of 100 Czech native speakers divided into 50 PD patients and 50 HC subjects [24]. Specialized neurologists evaluated each patient according to the Movement Disorder Society -Unified Parkinson's Disease Rating Scale (MDS-UPDRS-III) [25]. In addition, all recordings were captured in noise-controlled conditions, and the speech signals were downsampled to 16 kHz to feed a deep-learning model. The rapid repetition of the syllables /pa-ta-ka/ was considered in this study. This task allows the evaluation of specific movements required to produce stop consonants (/p/, /t/, /k/). Table 1 shows the demographic information of each database. Feature Extraction To create a representation for each recording, we used Wav2vec 2.0 architecture, a state-of-the-art topology based on transformers proposed in [8]. Wav2Vec 2.0 was trained using a selfsupervised pre-training approach that allows the model to learn representations directly from the raw audio signal without additional annotations or labels. The training process involved two main steps. Firstly, the contrastive pre-training, where the model was trained to distinguish between two versions of the same audio signal including a positive sample (a randomly selected segment of the original audio signal) and a negative sample (a randomly selected segment of a different audio signal). The second stage was fine-tuned based on a specific automatic speech recognition (ASR) task. Particularly in this work, we used a Wav2Vec 2.0 model, pre-trained on 960 hours of un-labeled audio from the LibriSpeech dataset [26], which was derived from English audiobooks and fine-tuned for ASR on the same audio with the corresponding transcripts. Due to the dynamic representation of 768 dimensions for each array with respect to time, we calculated a static vector for each participant from 6 different statistics (mean, standard deviation (std.), skewness, kurtosis, minimum, and maximum), building a speech representation of 4608 dimensions per recording. Federated Learning In order to speed up the collaborative training convergence, the FL process was performed merely for the classification network, i.e., after all the embeddings were locally extracted using the Wav2Vec 2.0 model. Of note, all the data pre-processing and feature extraction steps happened locally by every participating institution without sharing any data with other institutions. Each institution performed a local training round of the classification network and transmitted the network parameters, i.e., the weights and biases, to a trusted server, which aggregated all the local parameters leading to a set of global parameters. In our implementation, we chose each round to be equal to one epoch of training with the full local dataset. Afterward, the server transmitted back a copy of the global network to each institution for another round of local training. The process continued until the convergence of the global network. It is worth mentioning that not only each institution did not have access to any training data from others, but also not even to the network parameters of others, rather only an aggregated network, without the knowledge about the contributions of other participating institutions to the global network. Once the training of the global classification network was converged, every institution could take a copy of the global network and locally utilize it for diagnosing its test data. Classification and Evaluation The classification network architecture contained 4 fullyconnected layers with different sizes: 1024, 256, 64, and 2, respectively. Rectified linear unit (ReLU) activation and batch normalization [27] were considered in each layer, and a Softmax activation function was used at the output. The fully connected network was trained and evaluated following a stratified 10-fold cross-validation strategy. The process was repeated 5 times for a better generalization of the results. The He initialization scheme [28] was applied to all classification network weights and all the biases were initialized with zeros. Crossentropy was chosen as the loss function and the models were optimized using the Adam optimizer [29] with a learning rate of 8 × 10 −5 and weight decays of 5 × 10 −6 . The classification networks were trained for 50 epochs in batches of size 16. Accuracy and area under the receiver-operator-characteristic curve (AUC) were chosen as the main evaluation metrics, while sensitivity and specificity were utilized as supporting metrics. Twotailed paired t-test was employed for determining statistical significance. The significance threshold was set at p-value ≤ 0.05. Experiments and Results For each test database, we compared the diagnostic performances of the methods in three multicentric setups where the network was: i) locally trained using solely the training set of the corresponding database (Local), ii) trained utilizing the combination of all the training sets of different databases at a central location without privacy measures (Central), and iii) trained with all the training sets of different databases based on FL, i.e., without sharing any data and preserving patient privacy information. Furthermore, due to the relatively small test sizes of each database, we repeated each experiment corresponding to each cross-validation step 5 times, including the training and evaluation of the classification network for all 3 setups. Considering the 5 repetitions and 10-fold cross-validation steps, a total of 50 values were obtained for statistical analysis of each experiment. The average evaluation results are reported in Table 2 and details about diagnostic accuracy and classification performance are illustrated in Fig. 2-A. The accuracy of the FL method was significantly higher than local models for Spanish (83.2 ± 10.8% vs. 77.0 ± 13.3; P-value = 0.001) and Czech (76.0 ± 12.2% vs. 70.3 ± 14.6; P-value = 0.020) databases while it was only slightly higher for the German database (75.8 ± 8.3% vs. 74.8 ± 9.1; P-value = 0.455) which contained the largest training set. These results suggest that combining the corpus of the same pathology but in different languages allows generalizing the architecture to classify pathological speech from healthy speech. Furthermore, comparing the non-private "Central" and the secure FL strategies, we can observe that the diagnostic accuracy of the FL method was not significantly different from the "Central" model for Spanish (83.2 ± 10.8% vs. 82.0 ± 11.6; P-value = 0.436) and Czech (76.0 ± 12.2% vs. 77.8 ± 9.2; P-value = 0.334) databases while it was for the German database (75.8 ± 8.3% vs. 78.9 ± 8.3; P-value = 0.023). Moreover, Table 2 shows that the strategy proposed in this work obtained similar results to the state-of-the-art centralized training methods [30], with the advantage of patient privacy preservation by avoiding data exchange between local institutions using an FL strategy. In addition, Fig. 2-B shows a visual comparison between the "Central" and the FL strategies from the receiver-operatorcharacteristic (ROC) curves and the corresponding AUC values obtained in each experiment. Again, when we compared each institution (language) separately, we can observe that the Cen- Table 2: Evaluation results for each database. "Local" represents solely using the training set of the target database, while "Central" means utilizing all training sets when combined with each other at a central location. Values are reported as mean ± std in percentages. The "P-value" is with respect to FL for each database for accuracy values. tral and FL curves have the same trend and show no significant differences. It can also be observed that the Spanish language obtains the best result (AUC of 0.85), followed by German (AUC of 0.79) and Czech language (AUC of 0.78). Finally, Fig. 2-C shows the histogram and the probability density distributions obtained for the classification of German, Spanish, and Czech databases using the FL strategy. It can be observed that all three figures have the highest bins at their extremes, which corresponds to a high probability of the decision taken by the classifier. Moreover, it is possible to observe that in the case of Spanish and Czech, the highest bin is for the HC controls, which is related to the reported specificity (89.2% and 90.8%, respectively); while for Spanish, the highest bin corresponds to PD patients due to a higher sensitivity (90.8%). Discussion In this study, we showed the first successful application of crosslanguage federated learning for PD detection using three patho- logical speech corpora, including a total of 188 PD and 188 HC subjects, covering Spanish, German, and Czech languages. We used a state-of-the-art topology namely the Wav2vec 2.0 [7,8] for obtaining speech representations. We compared the performances in three multicentric setups where the architecture was: i) trained locally and separated by language, i.e., monolingual models, ii) trained utilizing the combination of all the training sets at a central location without privacy measures, i.e., crosslingual model, and iii) trained with all the training sets of different databases based on FL strategy, i.e., without sharing any data and preserving patient privacy. The results indicated that the FL model outperformed all the local models (mono-lingual models) for every test database in terms of diagnostic accuracy, while not requiring any data sharing between institutions. This result is very interesting and encourages the scientific community to further explore techniques for the generalization of models from databases of the same pathology, in different languages, without the need for sharing information between other institutions (cross-lingual model), which has been a major challenge. In addition, comparing the "Central" combination and FL strategies, we observed that in the majority of scenarios, the FL method was not significantly different from the Central method in terms of the model's diagnostic accuracy. This shows that the FL paradigm can considerably help the collaboration of institutions around the world in the creation of DL models with large amounts of data, cross-lingual, and preserving patient privacy by avoiding data exchange between local institutions, a major limitation in real-world practice that was not considered in current state-ofthe-art cross-lingual approaches. Our study has limitations. The collaborative FL training process was implemented in a proof-of-concept mode, i.e., using a single institutional network. Due to strict data protection regulations, the implementation of FL among different institutions would be challenging. However, we simulated a realistic setup where every database corresponded to a separate computing entity and we kept the data strictly independent from each other. As already mentioned, the parameter aggregation mechanism of the central server which was utilized in this study was direct averaging the individual network parameters of each participating database, i.e., the FedAvg algorithm [15], which is the simplest yet the most common aggregation mechanism. Furthermore, the databases utilized in this study were non-independent-and-identically-distributed (non-IID). This was shown to be decreasing the performance of the global model in many different FL applications [31]. Consequently, future work could consider more advanced and taskspecific aggregation methods such as [32][33][34] by accounting for the individual contribution of each participating site by analyzing their gradient updates in each FL training round before aggregation that could potentially increase the performance of the global model. In addition, we considered the most common task of PD detection, i.e., utilizing speech data containing the rapid repetition of the syllables /pa-ta-ka/ for the applicability of FL in pathological speech analysis in this study. In the future, we will extend this by considering further tasks and cross-pathology scenarios. As a side note, we could conclude that the characterization performed by the Wav2Vec 2.0 method is suitable to model different impairments for PD detection. This could be further investigated in the future with other controlled experiments such as at the level of phonemes, words, and phrases that could help interpret the features obtained by this model. Conclusions This paper shows that FL model yields similar or even better results compared to local approaches where mono-lingual models are created for every test database. FL offers the advantage of not requiring any data sharing between institutions, which we hope will encourage researchers and practitioners to improve scientific collaborations among different institutions around the world. The approach shows that FL allows for obtaining competitive results while preserving data privacy. We expect these results to promote simpler and more frequent collaborations between medical institutions, and subsequently, to further improve patient outcomes. Acknowledgments Figure 1 : 1General methodology: each institution pre-processes its local data, extracts the features using a Wav2Vec 2.0 model, and performs one epoch of the classifier network training locally, and transmits its local network parameters to a trusted server. The server aggregates all the parameters from all the institutions and transmits back the resulting global model to each institution for the next round of local training. In the end, each institution takes a copy of the final global model and performs its desired classification locally. Figure 2 : 2Evaluation results. (A) Illustrates the final accuracy values for each test database using the 3 setups, where "Local" represents solely using the training set of the target database, while "Central" means utilizing all training sets when combined with each other at a central location. (B) Shows the receiver-operator-characteristic curves. (C) Shows the histogram and the probability density distributions obtained for the classification of German, Spanish, and Czech databases using the FL strategy. STA was supported by the RACOON network under BMBF grant number 01KX2021. JROA and CDRU were funded by UdeA grant number ES92210001. JR was supported by the National Institute for Neurological Research (Programme EXCE-LES, ID Project No. LX22NPO5107) -funded by the European Union -Next Generation EU. The funders played no role in the design or execution of the study. Table 1 : 1Demographic and clinical information of the participants. [F/M]: Female/Male. Values reported as mean ± std.PD patients HC subjects Spanish Gender [F/M] 25/25 25/25 Age [F/M] 60.7±7/61.3±11 61.4±7/60.5±12 Range of age [F/M] 49-75/33-81 49-76/31-86 MDS-UPDRS-III [F/M] 37.6±14/37.8±22 Speech item (MDS-UPDRS-III) [F/M] 1.3±0.8/1.4±0.9 German Gender [F/M] 41/47 44/44 Age [F/M] 66.2 ±9.7/66.7±8.7 62.6±15.2/63.8±12.7 Range of age [F/M] 42-84/44-82 28-85/26-83 UPDRS-III [F/M] 23.3±12/22.1±10 Speech item (MDS-UPDRS-III) [F/M] 1.2±0.5/1.4±0.6 Czech Gender [F/M] 20/30 20/30 Age [F/M] 60.1±9/65.3±10 63.5±11/60.3±12 Range of age [F/M] 41-72/43-82 40-79/41-77 UPDRS-III [F/M] 18.1±10/21.4±12 Speech item (MDS-UPDRS-III) [F/M] 0.7±0.6/0.9±0.5 Frequency and cooccurrence of vocal tract dysfunctions in the speech of a large sample of Parkinson patients. J Logemann, Journal of Speech and hearing Disorders. 431J. 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[ "A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity", "A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity" ]
[ "M J Luo \nDepartment of Physics\nJiangsu University\n212013ZhenjiangPeople's Republic of China\n" ]
[ "Department of Physics\nJiangsu University\n212013ZhenjiangPeople's Republic of China" ]
[]
The paper proposes a statistical fields theory of quantum reference frame underlying the Perelman's analogies between his formalism of the Ricci flow and the thermodynamics. The theory is based on a d = 4 − ǫ quantum non-linear sigma model (NLSM), interpreted as a quantum reference frame system which a to-be-studied quantum system is relative to. The statistic physics and thermodynamics of the quantum frame fields is studied by the density matrix obtained by the Gaussian approximation quantization. The induced Ricci flow of the frame fields and the Ricci-DeTurck flow of the frame fields associated with the density matrix is deduced. In this framework, the diffeomorphism anomaly of the theory has a deep thermodynamic interpretation. The trace anomaly is related to a Shannon entropy in terms of the density matrix, which monotonically flows and achieves its maximal value at the flow limit, called the Gradient Shrinking Ricci Soliton (GSRS), corresponding to a thermal equilibrium state of spacetime. A relative Shannon entropy w.r.t. the maximal entropy gives a statistical interpretation to Perelman's partition function, which is also monotonic and gives an analogous H-theorem to the statistical frame fields system. A temporal static 3-space of a GSRS 4-spacetime is also a GSRS in lower 3-dimensional, we find that it is in a thermal equilibrium state, and Perelman's analogies between his formalism and the thermodynamics of the frame fields in equilibrium can be explicitly given in the framework. Extending the validity of the Equivalence Principle to the quantum level, the quantum reference frame fields theory at low energy gives an effective theory of gravity, a scale dependent Einstein-Hilbert action plus a cosmological constant is recovered. As a possible underlying microscopic theory of the gravitational system, the theory is also applied to understand the thermodynamics of the Schwarzschild black hole. * Electronic address: [email protected]
10.1142/s0218271823500220
[ "https://export.arxiv.org/pdf/2302.08651v1.pdf" ]
257,019,984
2302.08651
5f29c4670df7c326798f784f73b75eab758c1b3d
A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity 17 Feb 2023 M J Luo Department of Physics Jiangsu University 212013ZhenjiangPeople's Republic of China A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity 17 Feb 2023 The paper proposes a statistical fields theory of quantum reference frame underlying the Perelman's analogies between his formalism of the Ricci flow and the thermodynamics. The theory is based on a d = 4 − ǫ quantum non-linear sigma model (NLSM), interpreted as a quantum reference frame system which a to-be-studied quantum system is relative to. The statistic physics and thermodynamics of the quantum frame fields is studied by the density matrix obtained by the Gaussian approximation quantization. The induced Ricci flow of the frame fields and the Ricci-DeTurck flow of the frame fields associated with the density matrix is deduced. In this framework, the diffeomorphism anomaly of the theory has a deep thermodynamic interpretation. The trace anomaly is related to a Shannon entropy in terms of the density matrix, which monotonically flows and achieves its maximal value at the flow limit, called the Gradient Shrinking Ricci Soliton (GSRS), corresponding to a thermal equilibrium state of spacetime. A relative Shannon entropy w.r.t. the maximal entropy gives a statistical interpretation to Perelman's partition function, which is also monotonic and gives an analogous H-theorem to the statistical frame fields system. A temporal static 3-space of a GSRS 4-spacetime is also a GSRS in lower 3-dimensional, we find that it is in a thermal equilibrium state, and Perelman's analogies between his formalism and the thermodynamics of the frame fields in equilibrium can be explicitly given in the framework. Extending the validity of the Equivalence Principle to the quantum level, the quantum reference frame fields theory at low energy gives an effective theory of gravity, a scale dependent Einstein-Hilbert action plus a cosmological constant is recovered. As a possible underlying microscopic theory of the gravitational system, the theory is also applied to understand the thermodynamics of the Schwarzschild black hole. * Electronic address: [email protected] I. INTRODUCTION Recent works [1,2] show possible relations between Perelman's formalism of the Ricci flow and some fundamental problems in quantum spacetime and quantum gravity, for instance, the trace anomaly and the cosmological constant problem. Perelman's seminal works (the section-5 of [3]) and further development by Li [4,5] also suggest deep relations between the Ricci flow and the thermodynamics system, not only the irreversible non-equilibrium but also the thermal equilibrium thermodynamics of certain underlying microscopic system. In [3] Perelman also declared a partition function and his functionals without specifying what the underlying microscopic ensemble really are (in physics). So far it is not clear whether the beautiful thermodynamic analogies are physical or pure coincidences. On the other hand, inspired by the surprising analogies between the black hole and thermodynamics system, it is generally believed the existence of temperature and entropy of a black hole. Works along this line also showed, in many aspects, the gravitational system would be profoundly related to thermodynamics system (see recent review [6] and references therein), it is generally conjectured that there would exist certain underlying statistical theory for the underlying microscopic quantum degrees of freedom of gravity. It gradually becomes one of the touchstones for a quantum gravity. The motivations of the paper are, firstly, to propose an underlying statistical fields theory for Perelman's seminal thermodynamics analogies of his formalism of the Ricci flow, and secondly, for understanding the possible microscopic origin of the spacetime thermodynamics especially for the Schwarzschild black hole. We hope the paper could push forward the understanding to the possible interplay of the mysterious Perelman's formalism of Ricci flow and the quantum spacetime and gravity. To our knowledge, several tentative works have been devoted to the goal, see e.g. [7][8][9][10], but frankly speaking, the physical picture underlying the Ricci flow is not fully clear, if a fundamental physical theory underlying the Ricci flow and a fundamental theory of quantum spacetime is lacking. Based on our previous works [1,2,[11][12][13][14][15][16] on the quantum reference frame and its relation to Perelman's formalism of the Ricci flow, we propose a statistical fields theory of the quantum reference frame as a possible underlying theory of Perelman's seminal analogies between his geometric functionals and the thermodynamic functions. In section II, we review the theory of quantum reference frame based on a d = 4 − ǫ quantum non-linear sigma model, at the Gaussian approximation quantization, we obtain a density matrix of the frame fields system as a physical foundation to the statistical interpretation of the theory. The induced Ricci flow of the frame fields and the Ricci-DeTurck flow of the frame fields associated with the density matrix is deduced. In section III, we discuss the diffeomorphism and related trace anomaly of the quantum frame fields theory and its profound implications to the irreversible non-equilibrium thermodynamics of the frame fields, for instance, the statistical entropy and an analogous H-theorem of the frame fields, and the effective gravity theory at cosmic scale (especially the emergence of the cosmological constant). In section IV, the thermal equilibrium state of the frame fields as a flow limit configuration (the Gradient Shrinking Ricci Soliton) is discussed, in which the density matrix recovers the thermal equilibrium canonical ensemble density. This section gives a physical foundation to Perelman's seminal thermodynamic analogies. In section V, the framework gives a possible microscopic understanding of the thermodynamics of the Schwarzschild black hole. Finally, we summarize the paper and give conclusions in the section VI. II. QUANTUM REFERENCE FRAME Reference frame is one of the most fundamental notions in physics. Any measurement in physics is performed or described, a reference frame has always been explicitly or implicitly used. In classical physics, the reference frame is idealizationally used via classical rulers and clocks to label the spacetime coordinates, which are classical, external, and rigid without any fluctuation. Even in the textbook quantum mechanics or quantum fields theory, the spacetime coordinates are still classical. But quantum principles tell us that all physical measuring instruments including the rulers and clocks are inescapably quantum fluctuating. Such idealizational and classical treatment of reference frame works not bad in quantum mechanics and quantum fields theory. To a large extent, this is due to the fact that the general coordinates transformation and gravitational effects are not seriously taken into account. Just as expected, when the quantum principles are seriously applied to the spacetime itself and gravitational phenomenon, severe difficulties arise, e.g. information losses (non-unitary), diffeomorphism anomaly and the cosmological constant problems, etc. The quantum reference frame is a recurring theme in literature (not completely list, see e.g. [17][18][19][20][21][22][23][24] and references therein) based on various difference physical motivations, from quantum foundation to quantum information or quantum communication, to quantum gravity. For example, in Ref. [17], the author suggests the general relation between superselection rules and the lack of reference frame. In Ref. [20], it also more practically shows that extra assumptions about the superselection rules can not be avoided from the viewpoint of quantum information and quantum communication theory, if local observers do not share common information about their relative phase or Cartesian frames etc. The extra assumptions of the superselection rules may be also viewed as the weakness of textbook quantum mechanics, which can be overcome by introducing appropriate quantum reference frame. And many models (e.g. [18,22,23]) of quantum reference frame and relational descriptions to the quantum system and the quantum reference frame as a whole are suggest into the quantum foundation. In recent works [24] and the references therein, the authors review three approaches (relational Dirac observables, the Page-Wootters formalism and quantum deparameterizations) of relational quantum dynamics and suggest their equivalence. Other author focus on the possible role of quantum reference frame to the decoherence in quantum gravity [21,25]. Certainly, the works list of the direction is far from complete, which is beyond the scope and ability of the author. Fundamentally, our work shares the similar philosophical viewpoint to the role of quantum reference frame in quantum mechanics, such as considering that an appropriate materialized (but idealized) reference frame obeying the same law of quantum mechanics must be taken into account, and in the full quantum theory a relational description based on an entanglement of a quantum system and the quantum reference frame as a whole must play a fundamental role. However, there are some differences from the past literature that we considered more important. First, we do not simply or merely treat the quantum clock as a quantum mechanical system ( [23,24]) (which is more simple and has less degree of free to deal with as discussed in most quantum reference literature, in fact our early work ( [11,12]) also started from the operationally treatment of quantum clock to get some general conclusions on the vaccum energy and the cosmological constant problem), but in the paper we put both quantum space-rod and clock-time on an equal footing in the framework of quantum statistical fields, so that it makes the theory more appropriate to incorporate gravity, under the assumption of a quantum version of equivalence principle. To my understanding, quantum clock can be viewed as a first step model and far from a theory. Second, based on the quantum spacetime reference frame model (i.e. the d = 4 − ǫ non-linear sigma model), our paper does not treat the genuine relational quantities from the very beginning (as most literature tend to announce), but we prepare the quantum frame fields of reference in a laboratory frame (the d = 4 − ǫ base spacetime of the non-linear sigma model) as the starting reference, and then quantum events are relative to the prepared quantum frame fields. In this sense, the framework equivalently assumes the existence of an external, classical and rigid (free from quantum fluctuation and volume fixed) reference frame to be the laboratory frame, since the non-linear sigma model allows us to assign quantum state of spacetime reference (the target spacetime) to the base spacetime to arbitrary precision. But it can be easily verified that the theory is independent to the laboratory frame (metric, sign etc.) in the non-linear sigma model. The notion of the external and classical laboratory frame is just for convenience, since a quantum statistical fields theory is historically (or maybe more appropriate to) defined on an inertial frame (flat spacetime). So the relational quantities describing the relation between the quantum system and the quantum spacetime reference system is in essential in the framework. Third, also for the reason of the base spacetime independence of the non-linear sigma model, whose Hamiltonian is trivial, so the theory of spacetime reference frame is more properly quantized by using the path integral or functional method rather than the operator methods (e.g. the relational Dirac observables quantization or relational Schrodinger picture in Page-Wootters formalism). And fourth, there is a fundamentally non-unitary relation between two spacetime reference frames under a coordinate transformation due to an irreversible Ricci flow of spacetime reference frame, unlike most approaches in which the coordinate transformation between difference reference frames is assumed unitary. This is considered as a key ingredient of quantum spacetime reference frame that is intrinsically ensemble statistical and thermal. Generally speaking, our approach alongs the general philosophy of the quantum reference frame but is considered independent to the details of the past literature. The framework associates to several elegant physics and mathematical structures that are not discussed in the past literature, such as the non-linear sigma model, Shannon entropy, the Ricci flow and density Riemannian geometry, etc. And our previous works [1,2,[11][12][13][14][15][16] have revealed very rich consequencies of the framework, (e.g. the acceleration expansion of the late epoch universe, the cosmological constant, diffemorphism anomaly, the inflationary early universe, local conformal stability and non-collapsibility, modified gravity, etc.), but frankly speaking, the possible consequencies of the quantum reference frame are still far from fully discovered. The main motivation here for a quantum treatment of a reference frame system is that it might form a foundation to constructing a theory of quantum spacetime and quantum gravity that is analogous to the way it is used to construct the classical general relativity and it is crucial in understanding the microscopic origin of the spacetime thermodynamics. A. Definition In this section, we propose a quantum fields theory of reference frame as a starting point to study a quantum theory of spacetime and quantum gravity, based on an Equivalence Principle extended to reference frame described by quantum state (discussed by a paradox in Section-V-B and in the conclusion of the paper). The generalization of the Equivalence Principle to the quantum level might form another foundation to a quantum reference frame and a quantum gravity. How the Equivalence Principle behaves at the quantum level has many discussions having a long history (e.g. [26][27][28][29][30] and references therein, and [31,32] for an extended thermal version). The Equivalence Principle is the physical foundation of measuring the spacetime by physical material reference frame even at the quantum level, and it is the bridge between the geometric curved spacetime and gravity, and hence the gravity is simply a relational phenomenon that the motion of a test particle in gravity is manifested as a relative motion w.r.t. the (quantum) material reference frame. Without the Equivalence Principle, we would lost the physical foundation of all these concepts. Therefore, the basic argument of the paper is that there are several supports (e.g. uniform quantum origin of the accelerating expansion of the universe posited by myself in previous works [1,2,13], and a consistent incorporating the thermodynamics of the spacetime shown in this work) and the self-consistency of the framework are all possible evidences for its validity for the quantum reference frame. In this framework, a to-be-studied quantum system described by a state |ψ and the spacetime reference system by |X are both quantum. The states of the whole system are given by an entangled state |ψ[X] = ij α ij |ψ i ⊗ |X j(1) in their direct product Hilbert space H ψ ⊗ H X . The state (1) of the to-be-studied system and the reference frame system is an entangled state but a trivial direct product state is for the reason of calibration between them. Usually, a quantum measurement is performed as follows. At a preparation step of a quantum measurement, a one-to-one correlation between a quantum system |ψ i and a reference system |X j (a quantum instrument or ruler) is prepared, called calibration. The step in usual sense is a comparison and adjustment of the measuring instrument |X j by a calibration standard |ψ standard i which is physically similar with the to-be-studied system |ψ i . = |ψ standard i . A wellcalibrated entangled state ij α ij |ψ standard i ⊗ |X j can be used to measure the to-be-studied system |ψ i with the reference to the quantum instrument |X j . In essential, the measurement indirectly performs a comparision between |ψ i and the fiducial state |ψ standard i . So the entangled state |ψ[X] is a superposition of all possible one-to-one correlations. According to the standard Copenhagen interpretation of a quantum state, the to-be-studied quantum system collapsing into a state |ψ i together with the collapsing of the quantum reference system into the corresponding |X j happening by the joint probability |α ij | 2 , meaning that when the state of the quantum instrument is read out being in state |X j , then in this sense the to-be-studied system is inferred to be the corresponding |ψ i . A simple and practical example is the Stern-Gerlach experiment (see [1]). The entangled state generalizes the textbook quantum description of the state |ψ(x) w.r.t. an idealized parameter x of a classical reference system free from quantum fluctuations (in quantum mechanics x is the Newtonian time, in quantum fields theory x a are the Minkowskian spacetime). The entangled state |ψ[X] is inseparable, so that the state can only be interpreted in a relational manner, i.e. the entangled state describes the "relation" between |ψ and |X , but each absolute state. The individual state |ψ has physical meaning only being reference to |X entangled to it. When quantum mechanics is reformulated on the new foundation of the relational quantum state (the entangled state) describing the "relation" between the state of the under-studied quantum system and the state of the quantum reference system, a gravitational theory is automatically contained in the quantum framework without extra assumption. Since the state of reference |X is also subject to quantum fluctuation, so mathematically speaking, the state |ψ[X] can be seen as the state |ψ(x) with a smeared spacetime coordinates, instead of the textbook state |ψ(x) with a definite and classical spacetime coordinates. The state |ψ[X] could recover the textbook state |ψ(x) only when the quantum fluctuation of the reference system is small enough and hence can be ignored. More precise, the 2nd order central moment (even higher order central moments) fluctuations of the spacetime coordinate δX 2 (the variance) can be ignored compared with its 1st order moment of quadratic distance ∆X 2 (squared mean), where ... represents the quantum expectation value by the state of the reference system |X . In this 1st order approximation, this quantum framework recovers the standard textbook quantum mechanics without gravity. When the quantum fluctuation δX 2 as the 2nd order correction of the reference frame system is important and taken into account, gravity as a next order effects emerges in the quantum framework, as if one introduces gravitation into the standard textbook quantum mechanics, details are seen below and in previous works. To find the state |X ∈ H X of the quantum reference system, a quantum theory of the reference frame must be introduced. If the quantum spacetime reference frame |X µ (µ = 0, 1, 2, ...D−1) itself is considered as the to-be-studied quantum system, w.r.t. the fiducial lab spacetime |x a as the reference system, (a = 0, 1, 2, ...d − 1), the entangled state |X(x) = ij α ij |X i ⊗ |x j can be constructed by a mapping between the two states, i.e. |x → |X . From the mathematical viewing point, to defined a D-dimensional manifolds we need to construct a non-linear differentiable mapping X(x) from a local coordinate patch x ∈ R d to a D-manifolds X ∈ M D . The mapping in physics is usually realized by a kind of fields theory for X(x), the non-linear sigma model (NLSM) [33][34][35][36][37][38][39][40] S[X] = 1 2 λ D−1 µ,ν=0 d d xg µν d−1 a=0 ∂X µ ∂x a ∂X ν ∂x a ,(2) where λ is a constant with dimension of energy density [L −d ] taking the value of the critical density (68) of the universe. In the action, x a (a = 0, 1, 2, ..., d − 1), with dimension length [L], is called the base space in NLSM's terminology, representing the coordinates of the local patch. They will be interpreted as the lab wall and clock frame as the starting reference, which is considered fiducial and classical with infinite precision. For the reason that a quantum fields theory must be formulated in a classical inertial frame, i.e. flat Minkowskian or Euclidean spacetime, so the base space is considered flat. Without loss of generality, we consider the base space as the Euclidean one, i.e. x ∈ R d which is better defined when one tries to quantize the theory. The differential mapping X µ (x), (µ = 0, 1, 2, ..., D − 1), with dimensional length [L], is the coordinates of a general Riemannian or Lorentzian manifolds M D (depending on the boundary condition) with curved metric g µν , called the target space in NLSM's terminology. We will work with the real-defined coordinates for the target spacetime, and the Wick rotated version has been included into the general coordinates transformation of the time component. In the language of quantum fields theory, X µ (x) or X µ (x) = D−1 ν=0 g µν X ν (x) are the real scalar frame fields. Here, if not specifically mentioned, we will use the Einstein summation convention to sum over index variable appears twice (Latin index for the lab frame from 0 to d − 1 and Greek index for the spacetime from 0 to D − 1) and dropping the summation notation sigma. From the physical point of view, the reference frame fields can be interpreted as a physical coordinates system by using particle/fields signals, for instance, a multi-wire proportional chamber that measuring coordinates of an event in a lab. To build a coordinates system, first we need to orient, align and order the array of the multi-wires with the reference to the wall of the lab x a , (a = 1, 2, 3). The electron fields (ignoring the spin) in these array of multi-wires are considered as the scalar frame fields. With the reference to the wall of the lab, to locate a position of an event, at least three electron signals X 1 , X 2 , X 3 must be received and read in three orthogonal directions. The location information can be measured from the wave function of the electron fields, e.g. from the phase counting or particle number counting. Usually we could consider the electrons in the wires are free, and the field's intensity is not very large, so that the intensity can be seen as a linear function of the coordinates of the lab's wall, X µ (x) = 3 a=1 e a µ x a , (µ = 1, 2, 3), for instance, here e a µ = δ a µ is the intensity of the signals in each orthogonal direction. Meaning that when the direction µ is the lab's wall direction a, the intensity of the electron beam is 1, otherwise the intensity is 0. Similarly, one need to read an extra electron signal X 0 to know when the event happens, with the reference to the lab's clock x 0 . Thus, the fields of these 3+1 electron signals can be given by X µ (x) = 3 a=0 e a µ x a , (µ = 0, 1, 2, 3). (3) The intensity of the fields e a µ is in fact the vierbein, describing a mapping from the lab coordinate x a to the frame fields X µ . When the event happens at a long distance beyond the lab's scale, for instance, at the scale of earth or solar system, we could imagine that to extrapolate the multi-wire chamber to such long distance scale still seems OK, only replacing the electrons beam in wire by the light beam. However, if the scale is much larger than the solar system, for instance, to the galaxy or cosmic scale, when the signal travels along such long distance and be read by an observer, we could imagine that the broadening of the light beam fields or other particle fields gradually becomes non-negligible. More precisely, the 2nd (or higher) order central moment fluctuations of the frame fields signals can not be neglected, the distance of Riemannian/Lorentzian spacetime as a quadratic form must be modified by the 2nd moment fluctuation or variance δX 2 of the coordinates (∆X) 2 = ∆X 2 + δX 2 .(4) A local distance element in spacetime is given by a local metric tensor at the point, so it is convenient to think of the location point X being fixed, and interpreting the variance of the coordinate affect only the metric tensor g µν at the location point. As a consequence, the expectation value of a metric tensor g µν is corrected by the 2nd central moment quantum fluctuation of the frame fields g µν = ∂X µ ∂x a ∂X ν ∂x a = ∂X µ ∂x a ∂X ν ∂x a + 1 2 ∂ 2 ∂x 2 a δX µ δX ν = g (1) µν (X) − δg (2) µν (X),(5) where g (1) µν (X) = ∂X µ ∂x a ∂X ν ∂x a = e a µ e a ν(6) is the 1st order moment (mean value) contribution to the classical spacetime. For the contribution of the 2nd order central moment δg (2) µν (variance), the expectation value of the metric generally tends to be curved up and deformed, the longer the distance scale the more important the broadening of the frame fields, making the spacetime geometry gradually deform and flow at long distance scale. Since the classical solution of the frame fields (3) given by the vierbein satisfying the classical equation of motion of the NLSM, it is a frame fields interpretation of NLSM in a lab: the base space of NLSM is interpreted as a starting reference by the lab's wall and clock, the frame fields X(x) on the lab are the physical instruments measuring the spacetime coordinates. In this interpretation we consider d = 4 − ǫ, (0 < ǫ ≪ 1) in (2) and D = 4 is the least number of the frame fields. There are several reason why d is not precise but very close to 4 in the quantum frame fields interpretation of NLSM. d must be very close to 4, first, certainly at the scale of lab it is our common sense; Second if we consider the entangled system H ψ ⊗ H X between the to-be-studied physical system and the reference frame fields system, without loss of generality, we could take a scalar field ψ as the to-be-studied (matter) system, which shares the common base space with the frame fields, the total action of the two entangled system is a direct sum of each system S[ψ, X] = d d x 1 2 ∂ψ ∂x a ∂ψ ∂x a − V (ψ) + 1 2 λg µν ∂X µ ∂x a ∂X ν ∂x a ,(7) where V (ψ) is some potential of the ψ fields. It can be interpreted as an action of a quantum fields ψ on general spacetime coordinates X. Since both ψ field and the frame fields X share the same base space x, here they are described w.r.t. the lab spacetime x as the textbook quantum fields theory defined on inertial frame x. If we interpret the frame fields as the physical general spacetime coordinates, the coordinates of ψ field must be transformed from inertial frame x to general coordinates X. At the semi-classical level, or 1st moment approximation when the fluctuation of X can be ignored, it is simply a classical coordinates transformation S[ψ, X] (1) ≈ S[ψ(X)] = d 4 X | det g (1) | 1 4 g (1) µν ∂X µ ∂x a ∂X ν ∂x a 1 2 g (1)µν δψ δX µ δψ δX ν + 2λ − V (ψ) = d 4 X | det g (1) | 1 2 g (1)µν δψ δX µ δψ δX ν − V (ψ) + 2λ ,(8) in which (1) ≈ stands for the 1st moment or semi-classical approximation, and 1 4 g (1) µν ∂X µ ∂xa ∂X ν ∂xa = 1 4 g (1) µν g (1)µν = 1 4 D = 1 has been used. It is easy to see, at the semi-classical level, i.e. only consider the 1st moment of X while 2nd moment fluctuations are ignored, the (classical) coordinates transformation reproduces the scalar field action in general coordinates X up to a constant 2λ, and the derivative ∂ ∂xa is formally replaced by the functional derivative δ δX µ . | det g (1) | is the Jacobian determinant of the coordinate transformation, note that the determinant requires the coordinates transformation matrix to be a square matrix, so at semi-classical level d must be very close to D = 4, which is not necessarily true beyond the semi-classical level, when the 2nd moment quantum fluctuations are important. For instance, since d is a parameter but an observable in the theory, it could even not necessary be an integer but effectively fractal at the quantum level. d not precisely 4 is for the quantum and topological reasons. To investigate this, we note that quantization depends on the homotopy group π d (M D ) of the mapping X(x) : R d → M D . If we consider the (Wick rotated) spacetime M D topologically the S D for simplicity, the homotopy group is trivial for all d < D = 4, in other words, when d < 4 the mapping X(x) will be free from any unphysical singularities for topological reason, in this situation, the target spacetime is always mathematically well-defined. However, the situation d = 4 is a little subtle, since π 4 (S 4 ) = Z is non-trivial, the mapping might meet intrinsic topological obstacle and become singular, i.e. a singular spacetime configuration. When the quantum principle is taken into account, this situation can not be avoided, and by its RG flow the spacetime is possibly deformed into intrinsic singularities making the theory ill-defined at the quantum level and non-renormalizable (RG flow not converge). So at the quantum level, d = 4 should be not precisely, we have to assume d = 4 − ǫ when the quantum principle applies, while at the classical or semi-classical level, considering d = 4 has no serious problem. The above argument is different from the conventional simple power counting argument, which claims the NLSM is perturbative non-renormalizable when d > 2, but it is not necessarily the case, it is known that numerical calculations also support d = 3 and d = 4 − ǫ are non-perturbative renormalizable and well-defined at the quantum level. B. Beyond the Semi-Classical Level: Gaussian Approximation Going beyond the semi-classical or 1st order moment approximation, we need to quantize the theory at least at the next leading order. If we consider the 2nd order central moment quantum fluctuation are the most important next leading order contribution (compared with higher order moment), we call it the Gaussian approximation or 2nd order central moment approximation, while the higher order moment are all called non-Gaussian fluctuations which might be important near local singularities of the spacetime when local phase transition happens, although the intrinsic global singularity can be avoided by guaranteeing the global homotopy group trivial. At the Gaussian approximation, δg (2) µν can be given by a perturbative one-loop calculation [37,38] of the NLSM when it is relatively small compared with g (1) µν δg (2) µν (X) = R (1) µν (X) 32π 2 λ δk 2 ,(9) where R (1) µν is the Ricci curvature given by 1st order metric g (1) µν , k 2 is the cutoff energy scale of the Fourier component of the frame fields. The validity of the perturbation calculation R (1) δk 2 ≪ λ is the validity of the Gaussian approximation, which can be seen as follows. It will be shown in later section that λ is nothing but the critical density ρ c of the universe, λ ∼ O(H 2 0 /G), H 0 the Hubble's constant, G the Newton's constant. Thus for our concern of pure gravity in which matter is ignored, the condition R (1) δk 2 ≪ λ is equivalent to δk 2 ≪ 1/G which is reliable except for some local singularities are developed when the Gaussian approximation is failed. The equation (9) is nothing but a RG equation or known as the Ricci flow equation (some reviews see e.g. [41][42][43]) ∂g µν ∂t = −2R µν ,(10) with flow parameter δt = − 1 64π 2 λ δk 2 having dimension of length squared [L 2 ], which continuously deform the spacetime metric driven by its Ricci curvature. For the Ricci curvature is non-linear for the metric, the Ricci flow equation is a non-linear version of a heat equation for the metric, and flow along t introduces an averaging or coarse-graining process to the intrinsic non-linear gravitational system which is highly non-trivial [44][45][46][47][48]. In general, if the flow is free from local singularities there exists long flow-time solution in t ∈ (−∞, 0), which is often called ancient solution in mathematical literature. This range of the t-parameter corresponds to k ∈ (0, ∞), that is from t = −∞, i.e. the short distance (high energy) UV scale k = ∞ forwardly to t = 0 i.e. the long distance (low energy) IR scale k = 0. The metric at certain scale t is given by being averaged out the shorter distance details which produces an effective correction to the metric at that scale. So along t, the manifolds loss its information in shorter distance, thus the flow is irreversible, i.e. generally having no backwards solution, which is the underlying reason for the non-unitary and existence of entropy of a spacetime. As it is shown in (4), (5), the 2nd order moment fluctuation modifies the local (quadratic) distance of the spacetime, so the flow is non-isometry. This is an important feature worth stressing, which is the underlying reason for the anomaly. The non-isometry is not important for its topology, so along t, the flow preserves the topology of the spacetime but its local metric, shape and size (volume) changes. There also exists a very special solution of the Ricci flow called Ricci Soliton, which only changes the local volume while keeps its local shape. The Ricci Soliton, and its generalized version, the Gradient Ricci Soliton, as the flow limits, are the generalization of the notion of fixed point in the sense of RG flow. The Ricci Soliton is an important notion for understanding the gravity at cosmic scale and studying the the thermodynamics of the Ricci flow at equilibrium. The Ricci flow was initially introduced in 1980s by Friedan [34,35] in d = 2 + ǫ NLSM and independently by Hamilton in mathematics [49,50]. The main motivation of introducing it from the mathematical point of view is to classify manifolds, a specific goals is to proof the Poincare conjecture. Hamilton used it as a useful tool to gradually deform a manifolds into a more and more "simple and good" manifolds whose topology can be readily recognized for some simple cases. A general realization of the program is achieved by Perelman at around 2003 [3,51,52], who introduced several monotonic functionals to successfully deal with the local singularities which might be developed in more general cases. The Ricci flow approach is not only powerful to the compact geometry (as Hamilton's and Perelman's seminal works had shown) but also to the non-compact [53][54][55] and the Lorentzian geometry [15,[56][57][58][59][60][61][62]. C. The Wavefunction and Density Matrix at the Gaussian Approximation So far we have not explicitly defined the quantum state of the reference frame |X in (1). In fact, the previous (2nd order) results e.g. (5), (9) and hence the Ricci flow (10) can also equivalently be given by the expectation value O = X|O|X via explicitly writing down the wavefunction Ψ(X) of the frame fields at the Gaussian approximation. Note that at the semi-classical level, the frame fields X is a delta-distribution and peaks at its mean value, and further more, the action of the NLSM seems like a collection of harmonic oscillators, thus at the Gaussian approximation level, finite Gaussian width/2nd moment fluctuation of X must be introduced. When one performs a canonical quantization to the NLSM at the Gaussian approximation level, the fundamental solution of the wave function(al) (as a functional of the frame fields X µ ) of NLSM takes the Gaussian form, i.e. a coherent state Ψ[X µ (x)] = 1 √ λ(2π) D/4 |det σ µν | 1/4 | det g µν | 1/4 exp − 1 4 |X µ (x)σ µν (x)X ν (x)| ,(11) where the covariant matrix σ µν (x), playing the role of the Gaussian width, is the inverse of the 2nd order central moment fluctuations of the frame fields at point x σ µν (x) = 1 σ µν (x) = 1 δX µ (x)δX ν (x) ,(12) which is also given by perturbative one-loop calculation up to a diffeomorphism of X. The absolute symbol of |X µ σ µν X ν | in the exponential is used to guarantee the quadratic form and hence the determinant of σ µν induced from the Gaussian integral over X positive even in the Lorentzian signature. We can also define a dimensionless density matrix corresponding to the fundamental solution of the wavefunction u[X µ (x)] = Ψ * (X)Ψ(X) = 1 λ(2π) D/2 |det σ µν | | det g µν | exp − 1 2 |X µ (x)σ µν X ν (x)| ,(13)and 1 λ(2π) D/2 √ |det σµν | √ | det gµν | is a normalization parameter, so that λ d D XΨ * (X)Ψ(X) = λ d D Xu(X) = 1,(14) in which we often attribute the flow of the volume form d D X t to the flow of the metric g t , for the volume element d D X t ≡ dV t (X µ ) ≡ |g t |dX 0 dX 1 dX 2 dX 3 . Then the expectation values O can be understood as λ d D X t uO. As the quantum frame fields X are q-number in the theory, precisely speaking, the integral of them should be, in principle, a functional integral. Here the formal c-number integral of them d D X t ... is for the conventional in the Ricci flow literature, in which X is a coarse-grained c-number coordinates of manifolds at scale t. The exact functional integral of X is considered in calculating the partition function and related anomaly of the theory in section-III. Under a diffeomorphism of the metric, the transformation of u(X) is given by a diffeomorphism of the covariant matrix (h is certain function) σ µν →σ µν = σ µν + ∇ µ ∇ ν h.(15) So there exists an arbitrariness in the density u(X) for different choices of a diffeomorphism/gauge. According to the statistical interpretation of wavefunction with the normalization condition (14), u(X 0 , X 1 , X 2 , X 3 ) describes the probability density that finding these frame particles in the volume dV t (X µ ). As the spacetime X flows along t, the volume ∆V t , in which density is averaged, also flows, so the density at the corresponding scale is coarsegrained. If we consider the volume of the lab, i.e the base space, is rigid and fixed by λ d 4 x = 1, by noting (14), we have u[X µ (x), t] = d 4 x d D X t = lim ∆Vt→0 1 ∆V t ∆Vt 1 · d 4 x.(16) We can see that the density u(X, t) can be interpreted as a coarse-grained density in the volume element ∆V t → 0 w.r.t. a fine-grained unit density in the lab volume element d 4 x at UV t → −∞. In this sense, the coarse-grained density u(X, t) is in analogy with the Boltzmann's distribution function, so it should satisfy an analogous irreversible Boltzmann's equation, and giving rise to an analogous Boltzmann's monotonic Hfunctional. In the following sections, we will deduce such equation and the functional of u(X, t). The coarse-grained density u(X, t) has profound physical and geometric meaning, it also plays a central role in analyzing the statistic physics of the frame fields and generalizes the manifolds to the density manifolds. D. Ricci-DeTurck Flow In previous subsection, from the viewpoint of frame fields particle, u(X µ , t) has a coarse-grained particle density interpretation, the eq.(16) can also be interpreted as a manifolds density [63] from the geometric point of view. For instance, u(X, t) associates a manifold density or density bundle to each point X of a manifolds, measures the fuzziness of the "point". It is worth stressing that the manifolds density u(X, t) does not simply a conformal scaling of a metric by the factor, since if it is the case, the integral measure of D = 4-volume or 3-volume in the expectation O = λ d D XuO would scale by different powers. There are various useful generalizations of the Ricci curvature to the density manifolds, a widely accepted version is the Bakry-Emery generalization [64] R µν → R µν − ∇ µ ∇ ν log u,(17) which is also used in Perelman's seminal paper. The density normalized Ricci curvature is bounded from below R µν − ∇ µ ∇ ν log u ≥ σ µν ,(18) if the density manifolds has finite volume. As a consequence, replacing the Ricci curvature by the density normalized one, we get the Ricci-DeTurck flow [65] ∂g µν ∂t = −2 (R µν − ∇ µ ∇ ν log u) ,(19) which is equivalent to the standard Ricci flow equation (10) up to a diffeomorphism. Mathematically, the Ricci-DeTurck flow has the advantage that it turns out to be a gradient flow of some monotonic functionals introduced by Perelman, which have profound physical meanings shown later. The eq. (14) and (16) also give a volume constraint to the fiducial spacetime (the lab), the coarse-grained density u(X, t) cancels the flow of the volume element | det g µν |, so ∂ ∂t u | det g µν | = 0.(20) Together with the Ricci-DeTurck flow equation (19), we have the flow equation of the density ∂u ∂t = (R − ∆ X ) u,(21) which is in analogy to the irreversible Boltzmann's equation for his distribution function. ∆ X is the Laplacian operator in terms of the manifolds coordinates X. Note the minus sign in front of the Laplacian, it is a backwards heat-like equation. Naively speaking, the solution of the backwards heat flow will not exist. But we could also note that if one let the Ricci flow flows to certain IR scale t * , and at t * one might then choose an appropriate u(t * ) = u 0 arbitrarily (up to a diffeomorphism gauge) and flows it backwards in τ = t * − t to obtain a solution u(τ ) of the backwards equation. Now since the flow is consider free from global singularities for the trivialness of the homotopy group, we could simply choose t * = 0, so we defined τ = −t = 1 64π 2 λ k 2 ∈ (0, ∞).(22) In this case, the density satisfies the heat-like equation ∂u ∂τ = (∆ X − R) u,(23) which does admit a solution along τ , often called the conjugate heat equation in mathematical literature. So far (23) together with (19) the mathematical problem of the Ricci flow of a Riemannian/Lorentzian manifolds is transformed to a coupled equations      ∂gµν ∂t = −2 (R µν − ∇ µ ∇ ν log u) ∂u ∂τ = (∆ X − R) u dτ dt = −1(24) and the manifolds (M D , g) is generalized to a density manifolds (M D , g, u) [63,66,67] with the constraint (14). III. THE ANOMALY AND ITS IMPLICATIONS At the semi-classical approximation, see in eq.(8), when the quantum fluctuations of the frame fields or spacetime coordinates are ignored, the general coordinates transformation is just a classical coordinates transformation. We will show that when the quantum fluctuations are taken into account in the general coordinates transformation beyond the semi-classical approximation, quantum anomaly emerges. As is seen in the previous section, the quantum fluctuation and hence the coarse-graining process of the Ricci flow does not preserve the quadratic distance of a geometry, see (4) and (5). The non-isometry of the quantum fluctuation induces a breakdown of diffeomorphism or general coordinate transformation at the quantum level, namely the diffeomorphism anomaly. In this section, we derive the diffeomorphism anomaly of the theory, show its relation to the Shannon entropy whose monotonicity gives an analogous H-theorem of the frame fields system and the Ricci flow. Further more, as the quantum frame fields theory describes a quantum spacetime, together with the generalized quantum Equivalence Principle, the anomaly induced effective action in terms of the Shannon entropy can also be interpreted as a gravity theory, which at low energy expansion is a scale dependent Einstein-Hilbert action plus a cosmological constant. This part has certain overlap with the previous work [2], for the self-containedness of the paper, we hope this section provide a general background and lay the foundation for the subsequent thermodynamic and statistic interpretation of the theory. A. Diffeomorphism at the Quantum Level First we consider the functional quantization of the pure frame fields without explicitly incorporating the matter source. The partition function is Z(M D ) = [DX] exp (−S[X]) = [DX] exp − 1 2 λ d 4 xg µν ∂ a X µ ∂ a X ν ,(25) where M D is the target spacetime, and the base space can be either Euclidean and Minkowskian. Since considering the action or the volume element d 4 x ≡ d 4 x det e (det e is a Jacobian) does not pick any imaginary i factor no matter the base space is in Minkowskian or Euclidean one, if one takes dx (E) 0 → idx (M) 0 then det e (E) → −i det e (M) , so without loss of generality we use the Euclidean base spacetime in the following discussions, and remind that the result is the same for Minkowskian. Note that a general coordinate transformation X µ →X µ = ∂X µ ∂X ν X ν = e ν µ X ν(26) does not change the action S[X] = S[X], but the measure of the functional integral changes DX = x D−1 µ=0 dX µ (x) = x ǫ µνρσ e 0 µ e 1 ν e 2 ρ e 3 σ dX 0 (x)dX 1 (x)dX 2 (x)dX 3 (x) = x |det e(x)| x D−1 a=0 dX a (x) = x |det e(x)| DX,(27) where ǫ µνρσ e 0 µ e 1 ν e 2 ρ e 3 σ = det e a µ = |det g µν | is the Jacobian of the diffeomorphism. The Jacobian is nothing but a local relative (covariant basis) volume element dV (X µ ) w.r.t. the fiducial volume dV (X a ). Note that the normalization condition (14) also defines a fiducial volume element ud 4 X ≡ udV (X µ ), so the Jacobian is nothing but related to the frame fields density matrix u(X µ ) = dV (X a ) dV (X µ ) = |det e µ a | = 1 det e a µ .(29) Here the absolute symbol of the determinant is because the density u and the volume element are kept positive defined even in the Lorentz signature. Otherwise, for the Lorentz signature, it should introduce some extra imaginary factor i into (30) to keep the condition (14). The density so defined followed by (14) is an explicit generalization from the standard 3-space density to a 4-spacetime version. It is the definition of the volume form and the manifolds density ensure the formalism of the framework formally the same with the Perelman's standard form even in the Lorentzian signature. The manifolds density encodes the most important information of a Riemannian or Lorentzian geometry, i.e. the local volume comparison. In this case, if we parameterize a dimensionless solution u of the conjugate heat equation as u(X) = 1 λ(4πτ ) D/2 e −f (X) ,(30) is nothing but a pure real Shannon entropy in terms of the density matrix u N (M D ) = MD d D Xu f + D 2 log(4πτ ) = − MD d D Xu log u.(33) The classical action S[X] is invariant under the general coordinates transformation or diffeomorphism, but the quantum partition function is no longer invariant under the general coordinates transformation or diffeomorphism, which is called diffeomorphism anomaly, meaning a breaking down of the diffeomorphism at the quantum level. The diffeomorphism anomaly is purely due to the quantum fluctuation and Ricci flow of the frame fields which do not preserve the functional integral measure and change the spacetime volume at the quantum level. The diffeomorphism anomaly has many profound consequences to the theory of quantum reference frame, e.g. non-unitarity, the trace anomaly, the notion of entropy, reversibility, and the cosmological constant. The non-unitarity is indicated by the pure real anomaly term, which is also induced by the non-isometry or volume change, and consequently the non-invariance of the measure of the functional integral during the Ricci flow. Because of the real-defined volume form (29) for both Euclidean and Lorentzian signature, the pure real contribution of the anomaly and hence the non-unitarity are valid not only for spacetime with Euclidean but also for the Lorentzian signature, it is a rather general consequence of the Ricci flow of spacetime. Essentially speaking, the reason of the non-unitarity is because we have enlarged the Hilbert space of the reference frame, from a rigid classical frame to a fluctuating quantum frame. The non-unitarity implies the breakdown of the fundamental Schrodinger equation which is only valid on a classical time of inertial frame, the solution of which is in H ψ . A fundamental equation playing the role of the Schrodinger equation, which can arbitrarily choose any (quantum) physical system as time or reference frame, must be replaced by a Wheeler-DeWitt-like equation in certain sense [11], the solution of which is instead in H ψ ⊗H X . In the fundamental equation, the quantum fluctuation of physical time and frame, more generally, a general physical coordinates system must break the unitarity. We know that in quantum fields theory on curved spacetime or accelerating frame, the vacuum states of the quantum fields in difference diffeomorphism equivalent coordinate systems are unitarily inequivalent. The Unruh effect is a well known example: accelerating observers in the vacuum will measure a thermal bath of particles. The Unruh effect shows us how a general coordinates transformation (e.g. from an inertial to an accelerating frame) leads to the non-unitary anomaly (particle creation and hence particle number non-conservation), and how the anomaly will relate to a thermodynamics system (thermal bath). In fact, like the Unruh effect, the Hawking effect [68] and all non-unitary particle creation effects in a curved spacetime or accelerating frame are related to the anomaly in a general covariant or gravitational system. All these imply that the diffeomorphism anomaly will have deep thermodynamic interpretation which is the central issue of the paper. Without loss of generality, if we simply consider the under-transformed coordinates X µ identifying with the coordinates of the fiducial lab x a which can be treated as a classical parameter coordinates, in this situation the classical action of NLSM is just a topological invariant, i.e. half the dimension of the target spacetime exp (−S cl ) = exp − 1 2 λ d 4 xg µν ∂ a x µ ∂ a x ν = exp − 1 2 λ d 4 xg µν g µν = e − D 2 .(34) Thus the total partition function of the frame fields takes a simple form Z(M D ) = e λN (M D )− D 2 .(35) B. The Trace Anomaly The partition function now is non-invariance (32) under diffeomorphism at the quantum level, so if one deduces the stress tensor by T µν = − 2 √ |g| δ log Z δg µν , its trace g µν T µν = 0 is difference from T µ µ = g µν T µν ∆T µ µ = g µν T µν − g µν T µν = λN (M D )(36) known as the trace anomaly. Cardy conjectured [69] that in a d = 4 theory, quantities like T µ µ could be a higher dimensional generalization of the monotonic Zamolodchikov's c-function in d = 2 conformal theories, leading to a suggestion of the a-theorem [70] in d = 4 and other suggestions (e.g. [71,72]). In the following subsections, we will show that the Shannon entropy N and generalizedÑ are indeed monotonic, which might have more advantages, e.g. suitable for a Lorentzian target spacetime and for general D. Note that the Shannon entropy N (M D ) can be expanded at small τ λN (M D ) = λ ∞ n=0 B n τ n = λ B 0 + B 1 τ + B 2 τ 2 + ... (τ → 0).(37) For D = 4 the first few coefficients are B 0 = lim τ →0 N = D 2λ 1 + log √ λ4πτ ,(38)B 1 = lim τ →0 dN dτ = M4 d 4 X |g| R + D 2τ ,(39)B 2 = lim τ →0 1 2 d 2 N d 2 τ = − M4 d 4 X |g| R µν + 1 2τ g µν 2 ,(40) in which B 0 can be renormalized out, and a renormalized B 1 will contribute to the effective Einstein-Hilbert action of gravity, see following subsection D. And B 2 , as a portion of the full anomaly, plays the role of the conformal/Weyl anomaly up to some total divergence terms, for instance, ∆R terms and the Gauss-Bonnet invariant. That is, a non-vanishing B 2 term measures the broken down of the conformal invariance of M D=4 , otherwise, a vanishing B 2 means that the manifold is a gradient steady Ricci soliton as the fixed point of the Ricci-DeTurck flow, which preserves it shape (conformal invariant) during the flows. We note that B 2 as the only dimensionless coefficient measures the anomalous conformal modes, in this sense, N (M D ) indeed relates to certain entropy. However, since the conformal transformation is just a special coordinates transformation, thus it is clearly that the single B 2 coefficient does not measure the total (general coordinates transformation) anomalous modes. Obviously this theory at 2 < d = 4 − ǫ is not conformal invariant, thus as the theory flows along t, the degrees of freedom are gradually coarse-grained and hence the modes-counting should also change with the flow and the scale, as a consequence all coefficients B n in the series and hence the total partition function e λN (M D ) should measure the total anomalous modes at certain scale τ , leading to the full entropy and anomaly. Different from some classically conformal invariant theories, e.g. the string theory, in which we only need to cancel a single scale-independent B k coefficient in order to avoid conformal anomaly. As the theory at higher than 2-dimension is not conformal invariant, the full scale-dependent anomaly N (M D ) is required to be canceled at certain scale. Fortunately, it will show in later subsection that a normalized full anomaly λÑ (M D ) can converge at UV for its monotonicity, thus giving rise to a finite counter term of order O(λ) playing the role of a correct cosmological constant. The idea that the trace anomaly might have a relation to the cosmological constant is a recurring subject in literature [73][74][75][76][77], in the framework, the cosmological constant is naturally emerged in this way as the counter term of the trace anomaly (see subsection-D or [2]). C. Relative Shannon Entropy and a H-Theorem for Non-Equilibrium Frame Fields In the Ricci flow limit, i.e. the Gradient Shrinking Ricci Soliton (GSRS) configuration, the Shannon entropy N taking its maximum value N * , it is similar with the thermodynamics system being in a thermal equilibrium state where its entropy is also maximal. In mathematical literature of Ricci flow, it is often defined a series of relative formulae w.r.t. the extreme values taking by the flow limit GSRS or analogous thermal equilibrium state denoted by a subscript *. In GSRS, the covariance matrix σ µν as 2nd central moment of the frame fields with a IR cutoff k is simply proportional to the metric 1 2 σ µν * = 1 2 δX µ δX ν = 1 2λ g µν |p|=k 0 d 4 p (2π) 4 1 p 2 = k 2 64π 2 λ g µν = τ g µν ,(41) and then σ * µν = (σ µν * ) −1 = 1 2τ g µν ,(42) which means a uniform Gaussian broadening is achieved. And in this gauge, only longitudinal part of fluctuation exists. When the density normalized Ricci curvature is completely given by the longitudinal fluctuation σ µν , i.e. the inequality (18) saturates, giving a Gradient Shrinking Ricci Soliton (GSRS) equation R µν + ∇ µ ∇ ν f = 1 2τ g µν .(43) It means, on the one hand, for a general f (X) = 1 2 |σ µν X µ X ν |, so R µν seems vanish, so the standard Ricci flow equation (10) terminates; and on the other hand, the Ricci-DeTurck flow (19) only changes the longitudinal size or volume of the manifolds but its shape keep unchanged, thus the GSRS can also be seen stop changing, up to a size or volume rescaling. Thus the GSRS is a flow limit and can be viewed as a generalized RG fixed point. In the following, we consider relative quantities w.r.t. the GSRS configuration. Considering a general Gaussian density matrix u(X) = 1 λ(2π) D/2 |det σ µν | | det g µν | exp − 1 2 |X µ σ µν X ν | ,(44) in GSRS limit it becomes u * (X) = 1 λ(4πτ ) D/2 exp − 1 4τ |X| 2 .(45) Therefore, in GSRS, a relative density can be defined by the general Gaussian density u(X) relative to the density u * (X) in GSRSũ (X) = u u * .(46) By using the relative density, a relative Shannon entropyÑ can be defined bỹ N (M D ) = − d D Xũ logũ = − d D Xu log u + d D Xu * log u * = N − N * = − log Z P ≤ 0,(47) where Z P is nothing but the Perelman's partition function log Z P = M D d D Xu D 2 − f ≥ 0,(48) and N * is the maximum Shannon entropy N * = − d D Xu * log u * = d D Xu * D 2 1 + log( √ λ4πτ ) = D 2λ 1 + log( √ λ4πτ ) .(49) Since the relative Shannon entropy and the anomaly term is pure real, so the change of the partition function under diffeomorphism is non-unitary. For the coarse-graining nature of the density u, it is proved that the relative Shannon entropy is monotonic non-decreasing along the Ricci flow (along t), dÑ (M D ) dt = −F ≥ 0,(50)whereF = F − F * ≤ 0 is the GSRS-normalized F-functional of Perelman F = dN dτ = M D d D Xu R + |∇f | 2(51) with the maximum value (at GSRS limit) F * ≡ F (u * ) = dN * dτ = D 2λτ .(52) The inequality (50) gives an analogous H-theorem to the non-equilibrium frame fields and the irreversible Ricci flow. The entropy is non-decreasing along the Ricci flow making the flow irreversible in many aspects similar with the processes of irreversible thermodynamics, meaning that as the observation scale of the spacetime flows from short to long distance scale, the process losses information and the Shannon entropy increases. The equal sign in (50) can be taken when the spacetime configuration has flowed to a limit known as a Gradient Shrinking Ricci Soliton (GSRS), when the Shannon entropy takes its maximum value. Similarly, at the flow limit the density matrix u * eq.(45) takes the analogous standard Maxwell-Boltzmann distribution. D. Effective Gravity at Cosmic Scale and the Cosmological Constant In terms of the relative Shannon entropy, the total partition function (35) of the frame fields is normalized by the GSRS extreme value Z(M D ) = e λN − D 2 e λN * = e λÑ − D 2 = Z −λ P e − D 2 = exp λ M D d D Xu (f − D) .(53) The relative Shannon entropyÑ as the anomaly vanishes at GSRS or IR scale, however, it is non-zero at ordinary lab scale up to UV where the fiducial volume of the lab is considered fixed λ d 4 x = 1. The cancellation of the anomaly at the lab scale up to UV is physically required, which leads to the counter term ν(M D τ =∞ ) or cosmological constant. The monotonicity ofÑ eq.(50) and the W-functional implies [3,78] ν(M D τ =∞ ) = lim τ →∞ λÑ (M D , u, τ ) = lim τ →∞ λW(M D , u, τ ) = inf τ λW(M D , u, τ ) < 0,(54) where W, the Perelman's W-functional, is the Legendre transformation ofÑ w.r.t. τ −1 , W ≡ τ ∂Ñ ∂τ +Ñ = τF +Ñ = d dτ τÑ .(55) In other words, the difference between the effective actions (relative Shannon entropies) at UV and IR is finite ν = λ(Ñ UV −Ñ IR ) < 0.(56) Perelman used his analogies: the temperature T ∼ τ , the (relative) internal energy U ∼ −τ 2F , the thermodynamics entropy S ∼ −W, and the free energy F ∼ τÑ , up to proportional balancing the dimensions on both sides of ∼, the equation (55) is in analogy to the thermodynamics equation U − T S = F . So in this sense the W-functional is also called the W-entropy. Whether the thermodynamic analogies are real and physical, or just pure coincidences, is an important issue discussed in the next sections. In fact e ν < 1 (usually called the Gaussian density [79,80]) is a relative volume or the reduced volumeṼ (M D τ =∞ ) of the backwards limit manifolds introduced by Perelman, or the inverse of the initial condition of the manifolds density u −1 τ =0 . A finite value of it makes an initial spacetime with unit volume from UV flow and converge to a finite u τ =0 , and hence the manifolds finally converges to a finite relative volume/reduced volume instead of shrinking to a singular point at τ = 0. As an example, for a homogeneous and isotropic universe for which the sizes of space and time (with a "ball" radius a τ ) are on an equal footing, i.e. a late epoch FRW-like metric ds 2 = a 2 τ (−dx 2 0 + dx 2 1 + dx 2 2 + dx 2 3 ), which is a Lorentzian shrinking soliton configuration. Note that the shrinking soliton equation R µν = 1 2τ g µν it satisfies and its volume form (29) are independent to the signature, so it can be approximately given by a 4-ball value ν(B 4 ∞ ) ≈ −0.8 [1,2]. So the partition function, which is anomaly canceled at UV and having a fixed-volume fiducial lab, is Z(M D ) = e λÑ − D 2 −ν .(57) Since lim τ →0Ñ (M D ) = 0, so at small τ ,Ñ (M D ) can be expanded by powers of τ N (M D ) = ∂Ñ ∂τ τ + O(τ 2 ) = τF + O(τ 2 ) = M D d D Xu τ →0 R τ →0 + |∇f τ →0 | 2 − D 2τ τ + O(τ 2 ) = M D d D Xu 0 R 0 τ + O(τ 2 ),(58) in which λ d D Xu τ →0 τ |∇f τ →0 | 2 = D 2 (at GSRS) has been used. For D = 4 and small τ , the effective action of Z(M 4 ) can be given by − log Z(M 4 ) = S ef f ≈ M 4 d 4 Xu 0 (2λ − λR 0 τ + λν) (small τ ).(59) Considering u 0 d 4 X = |g t |dV = |g t |dX 0 dX 1 dX 2 dX 3 is the invariant volume element, and using (22) to replace t or τ by cutoff scale k, we have S ef f = M 4 dV |g k | 2λ − R 0 64π 2 k 2 + λν (small k).(60) The effective action can be interpreted as a low energy effective action of pure gravity. As the cutoff scale k ranges from the lab scale to the solar system scale (k > 0), the action must recover the well-tested Einstein-Hilbert (EH) action. But at the cosmic scale (k → 0), we know that the EH action deviates from observations and the cosmological constant becomes important. In this picture, as k → 0, the action leaving 2λ + λν should play the role of the standard EH action with a limit constant background scalar curvature R 0 plus the cosmological constant, so 2λ + λν = R 0 − 2Λ 16πG .(61) While at k → ∞, λÑ → ν, the action leaving only the fiducial Lagrangian D 2 λ = 2λ which should be interpreted as a constant EH action without the cosmological constant 2λ = R 0 16πG .(62) Thus we have the cosmological term λν = −2Λ 16πG = −ρ Λ .(63) The action can be rewritten as an effective EH action plus a cosmological term S ef f = M 4 dV |g k | R k 16πG + λν (small k),(64) where R k 16πG = 2λ − R 0 64π 2 k 2 ,(65) which is nothing but the flow equation of the scalar curvature [43] R k = R 0 1 + 1 4π Gk 2 , or R τ = R 0 1 + 2 D R 0 τ .(66) Since at the cosmic scale k → 0, the effective scalar curvature is bounded by R 0 which can be measured by "Hubble's constant" H 0 at the cosmic scale, R 0 = D(D − 1)H 2 0 = 12H 2 0 ,(67) so λ is nothing but the critical density of the 4-spacetime Universe λ = 3H 2 0 8πG = ρ c ,(68) so the cosmological constant is always of order of the critical density with a "dark energy" fraction Ω Λ = ρ Λ ρ c = −ν ≈ 0.8,(69) which is close to the observational value. The detail discussions about the cosmological constant problem and the observational effect in the cosmology, especially the modification of the Distance-Redshift relation leading to the acceleration parameter q 0 ≈ −0.68 can be found in [1,2,12,13]. If matters are incorporated into the gravity theory, consider the entangled system in H ψ ⊗ H X between the tobe-studied quantum system (matters) and the quantum reference frame fields system (gravity). 2λ term in eq.(8) is normalized by the Ricci flow, by using eq.(60) and eq.(65), a matter-coupled-gravity is emerged from the Ricci flow S[ψ, X] (2) ≈ dV |g k | 1 2 g µν δψ δX µ δψ δX ν − V (ψ) + 2λ − R 0 64π 2 k 2 + λν = dV |g k | 1 2 g µν δψ δX µ δψ δX ν − V (ψ) + R k 16πG + λν(70) IV. THERMAL EQUILIBRIUM STATE A Gradient Shrinking Ricci Soliton (GSRS) configuration as a Ricci flow limit extremizes the Shannon entropy N and the W-functional. Similarly, a thermal equilibrium state also extremizes the H-functional of Boltzmann and the thermodynamic entropy. Thus the process of a generic Ricci flow flows into a GSRS limit is in analogy with the nonequilibrium state evolves into a thermal equilibrium state, they are not merely similar but even equivalent, when the thermal system is nothing but the frame fields system. In this section, following the previous discussions on the nonequilibrium state of the frame fields in 4-dimension, in a proper choice of time, we will discuss the thermal equilibrium state of the frame particle system as a GSRS configuration in lower 3-dimension, in which the temperature and several thermodynamic functions of the system can be explicitly calculated and the manifolds density can be interpreted as the thermal ensemble density of the frame fields particles, giving a statistical interpretation to Perelman's thermodynamic analogies of the Ricci flow. A. A Temporal Static Shrinking Ricci Soliton as a Thermal Equilibrium State When the shrinking Ricci soliton M 4 is static in the temporal direction, i.e. being a product manifolds M 4 = M 3 ×R and δX/δX 0 = 0, where X 0 ∈ R is the physical time, X = (X 1 , X 2 , X 3 ) ∈ M 3 is a 3-space gradient shrinking Ricci soliton of lower dimensions, we can prove here that the temporal static spatial part M 3 is in thermal equilibrium with the flow parameter τ proportional to its temperature, and the manifolds density u of M 3 can be interpreted as the thermal equilibrium ensemble density. According to Masubara's formalism of thermal fields theory, the thermal equilibrium of the spatial frame fields can be defined by the periodicity X(x, 0) = X(x, β) in their Euclidean time of the lab (remind that we start from the Euclidean base space for the frame fields theory), where β = 1/T is the inverse of the temperature. Now the frame fields is a mapping R 3 × S 1 → M 3 × R. Then in such configuration, the τ parameter of the 3-space shrinking soliton M 3 becomes τ = 1 2λ d 3 pdω n (2π) 4 1 p 2 + ω 2 n = 1 2λ T n d 3 p (2π) 3 1 p 2 + (2πnT ) 2 ,(71) where ω n = 2πnT , dωn 2π = T n have been used. The calculation is a periodic-Euclidean-time version of the general eq.(41). Since the density matrix eq.(45) of the frame fields X µ is Gaussian or a coherent state, which the oscillators are almost condensed in the central peak, thus ω 0 = 0 dominants the Masubara sum, τ = 1 2λ T d 3 p (2π) 3 1 p 2 .(72) Different from the naive notion of "temporal static" at the classical level, which means w.r.t. the physical clock X 0 of the quantum reference frame, i.e. δX δX0 = 0. However, the notion "temporal static" is a little subtle at the quantum level. Because there is no "absolute static" at the quantum or microscopic level, since at such microscopic scale the modes are always in motion or vibrating w.r.t. the infinitely precise lab time x 0 , i.e. ∂X(x) ∂x0 = 0. Actually ∂X/∂x 0 is in general non-zero even though its oscillation degrees of freedom are almost frozen (Masubara frequency ω n is zero for the Gaussian wavefunction), while the center of the Gaussian wave pocket of X is in translational motion so p = 0, so its expectation value is in general finite, for instance, ∂X(x) ∂x0 ∼ 3 2 T < ∞ claimed by the equipartition energy of the translational motion in 3-space. In general, whether or not the modes of the spatial frame fields is temporal static depends on the scale to evaluate the average of the physical clock X 0 . The notion of "thermal static" in the sense of statistical physics is approximate at a macroscopic scale rather than a microscopic scale, at which scale the molecules are always in motion (so does the physical clock X 0 ). The macroscopic scale of the thermal static system is at such a long physical time scale δ X 0 ≫ δx 0 that the averaged physical clock is almost frozen ∂x0 ∂ X0 → 0 w.r.t. the infinitely precise lab time x 0 , so that the thermal static condition δX δX0 = ∂X ∂x0 · ∂x0 ∂ X0 → 0 can be achieved. More precisely, when we mention that the 3-space is macroscopic "temporal static", a IR cutoff, for example, H 0 as a macroscopic Hubble scale should be taken into account. The fluctuation modes on the 3-space outside the Hubble scale 0 < |p| < H 0 are frozen and temporal static, while those modes |p| > H 0 inside the Hubble horizon are dynamic. So with this cutoff scale we have τ = 1 2λ T |p|=H0 0 d 3 p (2π) 1 p 2 = C 3 2λ T H 0 = 1 λ 3 T = 1 λ 3 β ,(73) where the 3-space energy density is λ 3 = λ 1 2 C3H0 = 12π 2 λ H0 . Note that if we consider the temporal integral is also cutoff at about a long physical time scale, e.g. the age of the universe O(1/H 0 ), let the temporal direction is normalized as 1 12π 2 12π 2 /H0 0 dx 0 H 0 = 1, then the condition d 4 xλ ≡ 1 gives its 3-space version d 3 xλ 3 = 1,(74) which is the definition of λ 3 on 3-space slice generalizing the critical density λ in a 4-spacetime covariant theory. It is worth stressing that since the spatial slice depends on the definition of time, so the value of λ 3 is not universal (not necessarily equal to above 12π 2 λ H0 in other frame or cutoff, unlike the universal 4-spacetime critical density λ) but frame dependent. If a specific gauge of time or frame is chosen, λ 3 could be considered fix and be used as a proportional to correlate the τ parameter with the temperature of the temporal static frame fields configuration in such a specific gauge of time. The 3-space energy density λ 3 is very useful when we consider a temporal static GSRS spacetime or corresponding thermal equilibrium frame fields ensemble in later discussions. In summary, an important observation is that when M 3 is a shrinking Ricci soliton in a temporal static product shrinking soliton M 3 × R, the global τ parameter of M 3 can be interpreted as a thermal equilibrium temperature defined by the Euclidean time periodic of the frame fields, up to a proportional being a 3-space energy density λ 3 (satisfying eq.(74)) balancing the dimensions between τ and T . Since temperature T is frame dependent, so is the proportional λ 3 . The observation gives us a reason why in Perelman's paper τ could be analogous to the temperature T . The same results can also be obtained if one use the Lorentzian signature for the lab or base spacetime of the frame fields theory (2). In this case the thermal equilibrium of the spatial frame fields instead are subject to periodicity in the imaginary Minkowskian time X(x, 0) = X(x, iβ), but even though the base spacetime is Wick rotated, the path integral does not pick any imaginary i factor in front of the action in (25) as the starting point, so the main results of the discussions retain independent to the signature of the base spacetime. B. Thermodynamic Functions For the thermodynamic interpretation of the quantum reference frame and gravity theory, in this subsection, we derive other thermodynamic functions of the system beside the temperature in the previous subsection, which are similar with the ideal gas. So the frame fields system in the Gaussian approximation can be seen as a system of frame fields gas, which manifests a underlying statistic picture of Perelman's thermodynamics analogies of his functionals. As convention, we all take the temperature T = λ 3 τ , eq.(73), D = 3 and λ replaced by λ 3 , it is equivalent to choose a specific gauge of time for the thermal equilibrium frame fields configuration. When the spatial shrinking soliton M 3 is in temporal static dX 0 = 0 and in thermal equilibrium, the partition function of the thermal ensemble of the frame fields X can be given by the trace/integration of the density matrix, Z * (τ ) = λ 3 d 3 Xu(X) = λ 3 d 3 Xe − X 2 4τ = λ 3 (4πτ ) 3/2 ,(75) the normalized u density can be given by the 3-dimensional version of eq.(45) u * (X) = 1 Z * u(X) = 1 λ 3 (4πτ ) 3/2 e − X 2 4τ .(76) The partition function can also be consistently given by (35) with D = 3 in thermal equilibrium and hence the partition function of the frame fields in the shrinking soliton configuration Z * (τ ) = e λ3N * (M 3 )− 3 2 = exp −λ 3 M 3 d 3 Xu * log u * − 3 2 = λ 3 (4πτ ) 3/2 = V 3 4πλ 1/3 3 β 3/2 = Z * (β),(77) where V 3 = d 3 x is the 3-volume with the constraint λ 3 V 3 = 1. The partition function is identified with the partition function of the canonical ensemble of ideal gas (i.e. non-interacting frame fields gas in the lab) of temperature 1/β and gas particle mass λ 1/3 3 . The interactions are effectively absorbed into the broadening of the density matrix and normalized mass of the frame fields gas particles. The physical picture of frame fields gas in thermal equilibrium lays a statistical and physical foundation to Perelman's analogies between his functionals and thermodynamics equations as follows. The internal energy of the frame fields gas can be given similar to the standard internal energy of ideal gas 3 2 T given by the equipartition energy of translational motion in 3-space. Consider β as the Euclidean time of the flat lab, the internal energy seen from an observer in the lab is E * = − ∂ log Z * ∂β = λ 2 3 τ 2 ∂N * ∂τ = λ 2 3 τ 2 F * = 3 2 λ 3 τ = 3 2 T,(78) in which (52) with D = 3 and λ → λ 3 have been used. The fluctuation of the internal energy is given by E 2 * − E * 2 = ∂ 2 log Z * ∂β 2 = 3 2 λ 2 3 τ 2 = 3 2 T 2 .(79) The Fourier transformation of the density u * (X) is given by u * (K) = d 3 Xu * (X)e −iK·X = e −τ K 2 ,(80) since u satisfies the conjugate heat equation (23), so K 2 is the eigenvalue of the Laplacian −4∆ X + R of the 3-space, taking the value of the F-functional, K 2 = λ 3 d 3 X R|Ψ| 2 + 4|∇Ψ| 2 = λ 3 F ,(81)so u * (K 2 ) = e −λ3τ F .(82) For a state taking energy λ 2 3 τ 2 F = E, the probability density of the state can be rewritten as u * (E) = e − E λ 3 τ = e − E T ,(83) which is the standard Boltzmann's probability distribution of the state. So we can see that the (Fourier transformed) manifolds density can be interpreted as the thermal equilibrium canonical ensemble density of the frame fields. The free energy is given by F * = − 1 β log Z * = −λ 3 τ log Z * = − 3 2 λ 3 τ log(4πτ ),(84) similar with the standard free energy of ideal gas − 3 2 T log T up to a constant. The minus H-functional of Boltzmann at an equilibrium limit and the thermal entropy of the frame fields gas can be given by the Shannon entropy λ 3 N * = S * = −λ 3 d 3 Xu * log u * = 3 2 [1 + log(4πτ )] ,(85) similar with the thermal entropy of fixed-volume ideal gas 3 2 log T + 3 2 up to a constant. The thermal entropy can also be consistently given by the standard formula S * = log Z * − β ∂ log Z * ∂β = 3 2 [1 + log(4πτ )] .(86) which is in analogy with the fact that the W functional is the Legendre transformation of the relative Shannon entropy w.r.t. τ −1 . For this reason, the W functional is also an entropy function related to the (minus) thermodynamics entropy. In summary, we have seen that, under general frame fields (coordinates) transformation the Shannon entropy anomaly N appearing in the partition function (32) (or relative Shannon entropyÑ w.r.t. N * ) has profound thermodynamics interpretations. The Ricci flow of frame fields lead to non-equilibrium and equilibrium thermodynamics of the quantum spacetime, we summarize the comparisons between them in the Table I and II. V. APPLICATION TO THE SCHWARZSCHILD BLACK HOLE In this section, we try to apply the general statistic and thermodynamics interpretation of the quantum frame fields to a physical gravitational system, as one of the touchstone of quantum gravity, i.e. to understand the statistical origin of the thermodynamics of the Schwarzschild black hole. A. The Temperature of a Schwarzschild Black Hole The region in the vicinity of the origin of a Schwarzschild black hole is an example of classical static shrinking Ricci soliton. A rest observer distant from it sees an approximate metric M 3 × R, where the region in the vicinity of the origin of the spatial part M 3 is a shrinking Ricci soliton. The reason is as follows, because the black hole satisfies the Einstein's equation R µν − 1 2 g µν R = 8πGT µν ,(87) where the stress tensor is a point distributed matter in rest with a mass m at the origin x = 0 (seen from the distant rest observer) T 00 = mδ (3) (x), T ij = 0 (i, j = 1, 2, 3),(88) where Latin index i, j is for spatial index in the following. So we have R(x) = −8πGT µ µ = 8πGmδ (3) (x).(89) From the Einstein's equation we have the Ricci curvature of M 3 is proportional to the metric of M 3 R ij (x) = 8πGT ij + 1 2 g ij R = 1 2 8πGmδ (3) (x)g ij (i, j = 1, 2, 3).(90) The equation is nothing but a normalized shrinking Ricci soliton equation (43) for M 3 R ij (x) = 1 2τ g ij (x) (x ≈ 0) (91) with δ (3) (x)τ = 1 8πGm ,(92) where δ (3) (x) plays the role of the 3-space energy density λ 3 in the vicinity of the origin, satisfying d 3 xδ (3) (x) = 1 as eq.(74), so by using the relation between τ and temperature T (73), we can directly read from the equation that a temperature seen by the lab's infinite distant rest observer is T = δ (3) (x)τ = 1 8πGm ,(93) which is the standard Hawking's temperature of the Schwarzschild black hole seen by a distant rest observer. Is the vacuum region outside the origin of the black hole also a shrinking Ricci soliton? One may naively think that the answer is no, since at the classical level, it seems R ij = 0 (not a shrinking soliton eq.(91)), since outside the origin is just vacuum. But as is discussed in the next subsection, we argue that it is not true at the quantum level, if the vacuum and the vicinity region of the origin are in thermal equilibrium, they must be a shrinking Ricci solitons as a whole, i.e. R ij = 1 2τ g ij = 0, eq.(101) in the "vacuum". The above result can be extended to the "vacuum" region outside the origin, the price to pay is that the "vacuum" is full of internal energy corresponding to the Hawking temperature. If the whole spacetime have not been in thermal equilibrium yet, the configuration has to irreversibly go on flowing to a common thermal equilibrium fixed point (a global shrinking Ricci soliton), leading to a global maximized entropy, as the H-theorem asserts. B. The Energy of a Schwarzschild Black Hole In classical general relativity, the mass m is often mentioned as the ADM energy of the black hole m = d 3 xT 00 = d 3 xmδ (3) (x),(94) seen by the distant rest observer (w.r.t. the lab time x 0 ). Here at the quantum level, the coordinates or frame fields and spacetime are quantum fluctuating, which gives rise to the internal energy related to the periodicity of the (Euclidean) lab time x 0 (i.e. β = 1 T ). So, mathematically speaking, the anomaly of the trace of the stress tensor will modify the total ADM mass at the quantum level, see (36). Since the anomaly of the action of the frame fields λ 3 N * representing the spacetime part is always real, the internal energy of the frame fields is given by the (78) E * = − ∂ log Z * ∂β = 3 2 T = 3 16πGm ,(95) in which we have considered the 3-space volume V 3 outside the origin is in thermal equilibrium with the Hawking's temperature at the origin eq.(93), sharing the same equilibrium temperature T in the 3-volume V 3 . We can see that the internal energy E * is an extra contribution to the total energy of the (black hole + "vacuum") system seen by the distant rest observer. Essentially this term can be seen as a quantum correction or a part of the trace anomaly contribution to the stress tensor, thus the total energy of the black hole including the classical ADM energy and the quantum fluctuating internal energy of the metric is m BH = d 3 x T 00 = m + E * = m + 3 16πG 1 m ,(96) where the classical stress tensor T 00 is formally replaced by its quantum expectation value T 00 = mδ (3) (x) + 3 2 T V 3 .(97) A quantum Equivalence Principle should assert that the total energy rather than only the classical ADM mass contributes to the gravitation. For a macroscopic classical black hole, m ≫ 1 G , the first term ADM energy dominants the eq.(96), m BH ≈ m.(98) The second internal energy term is gradually non-negligible for a microscopic quantum black hole. An important effect of the existence of the second term in (96) is, for a microscopic quantum black hole, it makes the total energy bound from below, the minimal energy is of order of the Planck mass m BH ≥ 3 4πG ∼ O(m p ),(99) which seems to prevent the black hole evaporating into nothing. Further more, the internal energy 3 2 T term contributing to the total energy m BH and gravitation also demands that, not only the vicinity of the origin of the black hole is a shrinking soliton (as previous subsection claims), at the quantum level the whole 3-space is also the same shrinking soliton (i.e. satisfying eq.(91) with the identical τ globally and hence the same temperature T everywhere for the whole 3-space), just replacing the δ (3) -density in eq.(93) by the λ 3 -density, which extends the δ (3) -density at the origin to the outside region (the "vacuum"), we have T = λ 3 τ = 1 8πGm , with d 3 xλ 3 = d 3 x T 00 m BH = 1(100) for the whole thermal equilibrium 3-space, although at the classical level the vacuum R ij (x = 0) = 0 is seem not a shrinking soliton outside the origin. The physical reason is transparent that the internal energy's contribution 3T 2V3 in T 00 also plays the role of an additional source of gravity outside the origin. For the whole 3-space with T 00 = 0 and T ij = 0, the Einstein's equation for the whole 3-space is nothing but the Shrinking Ricci Soliton equation (91): R ij = 1 2 R g ij = 1 2 8πG T 00 g ij ≈ 1 2 8πGm T 00 m BH g ij = 1 2T λ 3 g ij = 1 2τ g ij = 0(101) in which R = −8πG T µ µ = 8πG T 00 = 0 is used in the "vacuum" outside the origin. The equation is in fact the spatial components of the Gradient Shrinking Ricci Soliton equation (43) where R ij = R ij + ∇ i ∇ j f , the Gaussian/thermal broadening of the density matrix u contributes to the classical curvature. The vicinity region of the origin plus the "vacuum" outside the origin of the black hole as a whole, is nothing but globally a shrinking Ricci soliton. The "vacuum" is not completely nothing at the quantum level but full of thermal particles T 00 (x = 0) = 0. The Hawking temperature is essentially an Unruh effect, in certain sense, the Gradient Shrinking Ricci soliton equation, eq.(101), might play a more fundamental role than the Unruh's formula, which determines how local acceleration or gravitation gives rise to temperature. The internal energy of the spacetime frame fields is an additional and necessary source of gravity, although macroscopically it is too small to contribute, at the quantum level its contribution is crucial for the 3-space in thermal equilibrium just right being a global shrinking Ricci soliton. The thermal internal energy coming from the quantum fluctuation of the 3-space gravitates normally as the quantum Equivalence Principle will assert. Otherwise, we have to face a paradox as follows. If we consider a frame x having T µν (x) = 0 everywhere, so according to the classical gravity R µν (x) = 0 everywhere, if we transform it to anther accelerating frame x ′ , one expects T µν (x) → T ′ µν (x ′ ) = 0, and hence R ′ µν (x ′ ) = 0 everywhere. However, according to the Equivalence Principle, in the accelerating frame x ′ one should feel equivalent gravity R ′ µν (x ′ ) = 0. It is clearly something is missing, a new dimension of the Equivalence Principle must be considered. In order to solve the paradox and retain the Equivalence Principle, a quantum effect (actually the effect from the diffemorphism anomaly such as the trace anomaly or the Unruh effect) must be introduced so that the accelerating frame must be particles creating from the "vacuum" and be thermal, which plays the role of an equivalent gravitational source making R ′ µν (x ′ ) = 0. The Hawking temperature in the internal energy term of eq.(96) is in essential the Unruh temperature playing such role. In this sense, the validity of the Equivalence Principle should be extended to the reference frame described by quantum state. C. The Entropy of a Schwarzschild Black Hole In the general framework, the entropy of the black hole comes from the uncertainty or quantum fluctuation moment of the frame fields given by the manifolds density u, more precisely, the thermalized black hole entropy is measured by the maximized Shannon entropy in terms of the probability distribution u of the frame fields in the background of the black hole. So in this subsection, we calculate the u density distributed around the Schwarzschild black hole and then evaluate the corresponding entropy as a measure of the black hole entropy. After a proper definition of a zero-point of the Shannon entropy, it gives a standard Bekenstein-Hawking entropy. For an observer in the distant lab rest frame, the contributions to the temporal static u density around the black hole is two folds. Beside the thermal distribution u * in the "vacuum" or bulk outside the black hole horizon, which gives rise to the ideal gas entropy (85) as the background entropy, there is an additionalũ density distribute mostly in a exterior thin shell near the horizon, and sparsely in the bulk outside the horizon, which we will focus on. The reason is as follows. Becauseũ density satisfies the conjugate heat equation (23) on the classical background of the black hole, since the classical scalar curvature R = 0 outside the horizon, and the temperature (equivalently the parameter τ and the mass) can be seen unchange for the thermalized black hole i.e. ∂ũ ∂τ = 0, thus the conjugate heat equation forũ is approximately given by the 4-Laplacian equation on the Schwarzschild black hole ∆ Xũ (X) = 0, (|X| ≥ r H ). (102) Now the temporal static densityũ(X) plays a similar role like a solution of the Klein-Gordon equation on the static background of the black hole. The approximation of the conjugate heat equation is equivalent to interpret the Klein-Gordon modes as a "first"-quantization probability density (not second-quantization fields). As is well-known, there are modes falling into the black hole horizon and hence disappearing from the outside observer's view. Just like the negative Klein-Gordon modes falling into the negative energy states below the groundstate. In a flat background, the amplitudes of the modes falling into and going out of the horizon are identical. So in the second-quantization, the negative mode falling into the horizon can be reinterpreted as a single anti-particle with positive energy modes going out of the horizon with the identical amplitude. However, in a curved background, for instance, the spacetime near the black hole horizon, the statement is no longer true. The two amplitudes differ from each other by a non-unitary equivalent factor. Thus the negative mode falling into the black hole horizon are no longer be reinterpreted as a single anti-particle mode going out, rather than multi-particles thermo-ensemble. At the situation, the densityũ describes the ensemble density of modes going exterior the horizon |X| ≥ r H which can be seen by an outside observer. By a routine calculation of the solution near the exterior black hole horizon resembling a Rindler metric as a starting point, we denote the solutionũ k (ρ), in which k represents the Fourier component/momentum in the direction that are orthogonal to the direction of radius with ρ = log(r − r H ), r the radius, r H = 2Gm the radius of the horizon. The equation becomes − ∂ 2ũ k ∂ρ 2 + k 2 e 2ρũ k = ω 2ũ k ,(103) where ω is the eigen-energy of the modes. By using a natural boundary condition thatũ vanishes at infinity, we can see that each transverse Fourier modeũ k can be considered as a free 1+1 dimensional quantum field confined in a box, one wall of the box is at the reflecting boundary ρ 0 = log ǫ 0 where ǫ 0 ≈ 0, and the other wall of the box is provided by the potential V (ρ) = k 2 e 2ρ ,(104) which becomes large V (ρ) ≫ 1 at ρ > − log k. So we can approximate the potential by the second wall at ρ w = − log k. So the length of the box is given by ∆ρ = ρ w − ρ 0 = − log(ǫ 0 k).(105) Thus the thickness of the horizon is about ∆r ∼ e ∆ρ ∼ ǫ 0 k. The densityũ k (ρ) is located in the box ρ ∈ (ρ 0 , ρ w ). In other words, the solution ofũ density is located mainly in a thin shell near the horizon r ∈ (r H , r H + ǫ 0 k). Furthermore, the modes k is assumed normal distributed (with a tiny width described by the parameter τ ). In this picture, without solving the equation, we can approximately write down the natural solution asũ k (r) τ →0 ≈ δ(|k|)δ(r − r H ), while for finite and small τ , we have a nearly Gaussian form u k (r) ≈ δ(|k|) · 1 (4πτ ) 1/2 e − (r−r H ) 2 4τ ≈ 1 (4π|k| 2 τ ) 1/2 e − (r−r H ) 2 4τ , (r > r H )(106) The exterior horizon solution can be considered as the standing wave solution as the superposition of the modes falling into and coming out of the black hole horizon. Then we have (up to a constant) logũ k (r) r∼rH ≈ − 1 2 log |k| 2 τ .(107) A routine calculation of the relative Shannon entropy or W-functional gives the entropy of each k-mode in the limit in which the width τ is very small, λ 3Ñ (ũ k ) = −λ 3 d 3 Xũ k logũ k = δ(|k|) ∞ rH 4πr 2 dr 1 (4πτ ) 1/2 e − (r−r H ) 2 4τ 1 2 log |k| 2 τ τ ∼0 = δ(|k|) 1 4 A log |k| 2 τ ,(108) where A = 4πr 2 H is the area of the horizon. It is naturally to assume the momentum k in the horizon shell is homogeneous, |k| = |k r | = |k ⊥ |,(109) where k r is the momentum in the radius direction and k ⊥ in the transverse directions on the horizon. When we integrate over all k-modes, we have the total relative Shannon entropy weakly depending to τ λ 3Ñ (ũ) = λ 3 d 3 kÑ (ũ k ) = 1 4 A d 2 k ⊥ (2π) 2 log |k ⊥ | 2 τ dk r δ(k r ) ≈ 1 4 A 1/ǫ 0 2πk ⊥ dk ⊥ (2π) 2 log |k ⊥ | 2 τ = 1 4 A × 1 2πτ − τ 2ǫ 2 1 − log τ ǫ 2 ≈ − A 16πǫ 2 ,(110) in which the transverse momentum is effectively cut off at an inverse of a fundamental UV length scale ǫ 2 . The relative Shannon entropy gives an area law of the black hole entropy. To determine the UV length cutoff ǫ 2 , we need to consider the scale at which the relative entropy is defined to be zero (not only the black hole is locally thermal equilibrium, but also the asymptotic background spacetime is globally thermal equilibrium), thus we need to consider the flow of the asymptotic background spacetime. A natural choice of a thermal equilibrium Ricci flow limit of the background spacetime (the black hole is embedded) is an asymptotic homogeneous and isotropic Hubble universe with scalar curvature R 0 = D(D − 1)H 2 0 = 12H 2 0 at scale t UV where we could consider and normalize the relative entropy to be zero (leaving only the background ideal gas entropy), since there is no information of the local shape distortions in such GSRS background because of the vanishing of its Weyl curvature, while the global curvature is non-zero which codes the information of its global volume shrinking. Under such definition, taking the normalized Shrinking Ricci soliton equation (43) and (22), we have τ UV = −t UV = D 2R 0 = 1 64π 2 λ k 2 UV ,(111) by using the critical density (68), which gives a natural cutoff corresponding to the scale t UV , ǫ 2 = k −2 UV = 1 Dπ G = 1 4π G.(112) This is exactly the Planck scale, which is a natural cutoff scale induced from the Hubble scale H 0 and λ of the framework. However, it is worth stressing that the Planck scale is not the absolute fundamental scale of the theory, it only has meaning w.r.t. the asymptotic Hubble scale. The only fundamental scale of the theory is the critical density λ which is given by a combination of both the Planck scale and Hubble scale, but each individual Planck or Hubble scale does not have absolute meaning. The UV (Planck) cutoff scale could tend to infinity while the complementary (Hubble) scale correspondingly tends to zero (asymptotic flat background), keeping λ finite and fixed. At this point, if we define a zero-relative-entropy for an asymptotic Hubble universe of scalar curvature R 0 , then the black hole in this asymptotic background has a non-zero thermodynamic entropy S = −λ 3Ñ (ũ) = A 4G ,(113) up to the bulk background entropy λ 3 N * = S * ≪ S, eq.(86). Combining the relative Shannon entropyÑ and the bulk thermal background entropy N * , and using the total partition function eq.(35), Z(M 3 ) = e λ3N − 3 2 = e λ3(Ñ +N * )− 3 2 we can also reproduce the total energy of the black hole in (96) m BH = − ∂ log Z ∂β = m + 3 2 T,(114) in which eq.(47) and A = 4πr 2 H = 16πG 2 m 2 = β 2 4π have been used. Different from the holographic idea that the information or entropy are coded in the (infinite thin and 2-dimensional) horizon or boundary of a gravitational system, in this framework where the coordinates of the spacetime geometry are smeared by quantum fluctuation, as a consequence that there is no mathematically precise notion of an infinite thin boundary in a "density manifolds" in general, it is just a semi-classical concept. Note that manifolds density u is mainly distributed at the horizon with a finite thickness (although very small), which contributes most of the anomaly and entropy to the black hole, so although the entropy (113) is proportional to the area, the geometric gravitational entropy given by the framework essentially comes from the 3-volume (note the 3d integral in eq.(108) and eq.(110)) but the 2-surface boundary. Or in other equivalent words, here the area of the horizon is fluctuating (due to its finite thickness) rather fixed, while the total energy and hence the temperature is fixed. In this sense, it is a canonical ensemble but an area ensemble as some ideas might suggest. VI. CONCLUSIONS In this paper, we have proposed a statistical fields theory underlying Perelman's seminal analogies between his geometric functionals and the thermodynamic functions. The theory is based on a d = 4 − ǫ quantum non-linear sigma model, interpreted as a quantum reference frame. When we quantize the theory at the Gaussian approximation, the wavefunction Ψ(X) and hence the density matrix u(X) = Ψ * (X)Ψ(X) eq.(13) can be written down explicitly. Based on the density matrix, the Ricci flow of the frame fields (10) and the generalized Ricci-DeTurck flow (19) of the frame fields endowed with the density matrix is discussed. And further more, we find that the density matrix has profound statistical and geometric meanings, by using it, the spacetime (M D , g) as the target space of NLSM is generalized to a density spacetime (M D , g, u). The density matrix u(X, τ ), satisfying a conjugate heat equation (23), not only describes a (coarse-grained) probability density of finding frame fields in a local volume, but also describes a volume comparison between a local volume and the fiducial one. For the non-isometric nature of the Ricci or Ricci-DeTurck flow, the classical diffeomorphism is broken down at the quantum level. By the functional integral quantization method, the change of the measure of the functional integral can be given by using a Shannon entropy N in terms of the density matrix u(X, τ ). The induced trace anomaly and its relation to the anomalies in conventional gravity theories are also discussed. As the Shannon entropy flows monotonically to its maximal value N * in a limit called Gradient Shrinking Ricci Soliton (GSRS), a relative densitỹ u and relative Shannon entropyÑ = N − N * can be defined w.r.t. the flow limit. The relative Shannon entropy gives a statistical interpretation underlying Perelman's partition function (47). And the monotonicity ofÑ along the Ricci flow gives an analogous H-theorem (50) for the frame fields system. As a side effect, the meanings on the gravitational side of the theory is also discussed, in which a cosmological constant −λν(B 4 ∞ ) ≈ 0.8ρ c as a UV counter term of the anomaly must be introduced. We find that a temporal static GSRS, M 3 , as a 3-space slice of the 4-spacetime GSRS, M 4 = M 3 × R, is in a thermal equilibrium state, in which the temperature is proportional to the global τ parameter of M 3 (73) up to a 3-space energy density λ 3 with normalization d 3 xλ 3 = 1. The temperature and λ 3 both depend on the choice of time R. In the sense that M 3 is in thermal, its Ricci soliton equation eq.(91) or quantum (indistinguishable with thermal) fluctuation eq.(41), can be considered as a generalization of the Unruh's formula, relating the temperature to local acceleration or gravitation. Based on the statistical interpretation of the density matrix u(X, τ ), we find that the thermodynamic partition function (75) at the Gaussian approximation is just a partition function of ideal gas of the frame fields. In this physical picture of canonical ensemble of frame fields gas, several thermodynamic functions, including the internal energy (78), the free energy (84), the thermodynamic entropy (85), and the ensemble density (83) etc. can be calculated explicitly agreeing with Perelman's formulae, which gives an underlying statistical foundation to Perelman's analogous functionals. We find that the statistical fields theory of quantum reference frame can be used to give a possible underlying microscopic origin of the spacetime thermodynamics. The standard results of the thermodynamics of the Schwarzschild black hole, including the Hawking temperature, energy and Bekenstein-Hawking entropy can be successfully reproduced in the framework. And we find that when the fluctuation internal energy of the metric is taken into account in the total energy, the energy of the black hole has a lower bound of order of the Planck energy, which avoid the quantum black hole evaporating into nothing. The internal energy or related temperature of the spacetime frame fields is an additional source of gravity, although macroscopically it is very small, at the quantum level its contribution is necessary for a thermal equilibrium 3-space just right being a GSRS, otherwise, the Equivalence Principle would breakdown. In this paper, the extended quantum Equivalence Principle plays a fundamental role as a bridge from the quantum reference frame theory (as a statistical fields or quantum fields theory on the base/lab spacetime) to the quantum gravity. To sum up, the paper can be seen as an attempt to discuss the deep relations between these three fundamental themes: the diffeomorphism anomaly, gravity and the spacetime thermodynamics, based on the statistical fields theory of quantum spacetime reference frame and the quantum Equivalence Principle. In the spirit of classical general relativity, if we trust the Equivalence Principle, one can not in principle figure out whether one is in an absolute accelerating frame or in an absolute gravitational background, which leads to a general covariance principle or diffeomorphism invariance of the gravitational theory. However, at the quantum level, the issue is a little subtle. If an observer in an accelerating frame sees the Unruh effect, i.e. thermal particles are creating in the "vacuum", which seems leading to the unitarily inequivalence between the vacuums of, for instance, an inertial frame and an accelerating frame, and hence the diffeomorphism invariance is seen breakdown discussed as the anomaly in the paper. The treatment of the anomaly in the paper is that, the anomaly is only canceled in an observer's lab up to UV scale, where the frame can be considered classical, rigid and cold, while at general scale the anomaly is not completely canceled. Whether one can figure out that he/she is in an absolute accelerating frame by detecting the anomaly (ShannonÑ term) at general scale (e.g. by thermodynamic experiments detecting the vacuum thermal particle creation and hence find the non-unitarity)? We argue that if the answer is still "NO!" in the spirit of the general relativity, the anomaly term coming from a quantum general coordinates transformation must be also equivalently interpreted as the effects of spacetime thermodynamics and gravity. Because the 2nd order moment fluctuation of the quantum coordinates or a non-trivial manifolds density u, which gives rise to the diffemorphism anomaly, also contributes to other 2nd order quantities (series coefficients at second spacetime derivative) such as (i) the acceleration (second time derivative of coordinates, e.g. leading to uniform acceleration expansion or other acceleration discrepancies in the universe [1]), (ii) the gravity or curvature (second spacetime derivative of metric, e.g. see (9) and (18)) and (iii) the thermal broadening (second spatial derivative of the manifolds density or the ensemble density, e.g. see (41) and (73)) at the same (2nd) order. In this sense, the validity of the classical Equivalence Principle would be generalized to the quantum level to incorporate the effects of the quantum fluctuation of the spacetime coordinates or frame fields, so that, one in principle still can not figure out and distinguish whether he/she is in an accelerating frame, or in a gravitational field or in a thermal spacetime (as a new dimension of the Equivalence Principle), these three things have no absolute physical meanings and are indistinguishable any more in the framework. The classical Equivalence Principle asserts the equivalence of the first two things at the first order (mean level), the quantum Equivalence Principle asserts the equivalence of the three things even at the second order (variance level), even higher order. Data availability statement All data that support the findings of this study are included within the article. then the partition function Z(M D ) is transformed to Z(M D ) = [DX] exp −S[X] = x |det e| [DX] exp (−S[N (M D ) in the exponential of the change of the partition function Z(M D ) = e λN (M D ) Z(M D ) entropy:Ñ = − d 3 Xũ(X, t) logũ(X, t) Boltzmanian H function: H(t) = d 3 vρ(v, t) log ρ(Boltzmann equation of ideal gas: ∂ρ ∂t = −v · ∇ρ Table I: Frame fields in general Ricci flow at non-flow-limit and the Non-equilibrium thermodynamics. Frame fields at the Ricci flow limit (GSRS) Equilibrium thermodynamics of ideal gas partition function: Z * (τ ) = λ3(4πτ ) 3/2 partition function: Z(T ) = V3(2πmT ) 3/2 GSRS flow parameter: λ3τ temperature: T = β density: u * (K) = e −τ K 2 = e −λ 3 τ F canonical ensemble density: ρ = e − E T −λ3τ log Z * = − 3 2 λ3τ log(4πτ ) free energy: F * = −T log Z(T ) = − 3 2 T log T Shannon entropy: λ3N * = 3 2 [1 + log(4πτ )] thermodynamic entropy: S * = 3 2 (1 + log T ) W functional: W = τ dÑ dτ +Ñ first law of thermodynamics: E * − T S * = F * Monotonicity: dÑ dt ≥ 0 second law of thermodynamics: δS ≥ 0 Table II : IIFrame fields in Gradient Shrinking Ricci Soliton (GSRS) configuration and the equilibrium thermodynamics of ideal gas. . M J Luo, 1907.05217Found. Phys. 51M. J. Luo, Found. Phys. 51, 2 (2021), 1907.05217. . M J Luo, 2106.16150Class. Quant. Grav. 38155018M. J. Luo, Class. Quant. Grav. 38, 155018 (2021), 2106.16150. . G Perelman, arXiv preprint math/0211159G. 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Veli Veliev, arXiv e-prints arXiv:1907.03541 (2019), 1907.03541. . I Bubuianu, S I Vacaru, E V Veliev, 1903.04920Annals of Physics. 423168333I. Bubuianu, S. I. Vacaru, and E. V. Veliev, Annals of Physics 423, 168333 (2020), 1903.04920. . S I Vacaru, 1905.12399European Physical Journal C. 80639S. I. Vacaru, European Physical Journal C 80, 639 (2020), 1905.12399. . W Wylie, D Yeroshkin, arXiv:1602.080001602.08000arXiv e-printsW. Wylie and D. Yeroshkin, arXiv e-prints arXiv:1602.08000 (2016), 1602.08000. D Bakry, M Émery, 978-3- 540-39397-9Diffusions hypercontractives. Berlin Heidelberg; Berlin, HeidelbergSpringerD. Bakry and M. Émery, Diffusions hypercontractives (Springer Berlin Heidelberg, Berlin, Heidelberg, 1985), ISBN 978-3- 540-39397-9. . D M Deturck, Journal of Differential Geometry. 18157D. M. DeTurck et al., Journal of Differential Geometry 18, 157 (1983). . F Morgan, The American Mathematical Monthly. 116134F. Morgan, The American Mathematical Monthly 116, 134 (2009). . I Corwin, Rose-Hulman Undergraduate Mathematics Journal. 72I. Corwin, Rose-Hulman Undergraduate Mathematics Journal 7, 2 (2006). . S M Christensen, S A Fulling, Physical Review D. 152088S. M. Christensen and S. A. Fulling, Physical Review D 15, 2088 (1977). . J L Cardy, Physics Letters B. 215749J. L. Cardy, Physics Letters B 215, 749 (1988). . Z Komargodski, A Schwimmer, Journal of High Energy Physics. 20111Z. Komargodski and A. Schwimmer, Journal of High Energy Physics 2011, 1 (2011). J Perez-Mercader, arXiv: High Energy Physics -Theory. J. Perez-Mercader, arXiv: High Energy Physics -Theory (1993). . J C Gaite, D O&apos;connor, hep-th/9511090Phys. Rev. D. 545163J. C. Gaite and D. O'Connor, Phys. Rev. D 54, 5163 (1996), hep-th/9511090. . N Bilic, B Guberina, R Horvat, H Nikolic, H Stefancic, 0707.3830Phys. Lett. B. 657232N. Bilic, B. Guberina, R. Horvat, H. Nikolic, and H. Stefancic, Phys. Lett. B 657, 232 (2007), 0707.3830. . E T Tomboulis, Nucl. Phys. B. 329410E. T. Tomboulis, Nucl. Phys. B 329, 410 (1990). . I Antoniadis, E Mottola, Phys. Rev. D. 452013I. Antoniadis and E. Mottola, Phys. Rev. D 45, 2013 (1992). I Antoniadis, hep-th/9211055Workshop on String Theory (Preceded by Spring School on String Theory and Quantum Gravity. I. Antoniadis, in Workshop on String Theory (Preceded by Spring School on String Theory and Quantum Gravity 30 Mar -7 Apr) (1992), hep-th/9211055. . H Salehi, Y Bisabr, hep-th/0001095Int. J. Theor. Phys. 391241H. Salehi and Y. Bisabr, Int. J. Theor. Phys. 39, 1241 (2000), hep-th/0001095. . L Ni, Journal of Geometric Analysis. 1487L. Ni, Journal of Geometric Analysis 14, 87 (2004). . H.-D Cao, R S Hamilton, T Ilmanen, arXiv preprint math/0404165H.-D. Cao, R. S. Hamilton, and T. Ilmanen, arXiv preprint math/0404165 (2004). . H.-D Cao, arXiv:0908.2006arXiv preprintH.-D. Cao, arXiv preprint arXiv:0908.2006 (2009).
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[ "Multipole invariants and non-Gaussianity", "Multipole invariants and non-Gaussianity" ]
[ "Kate Land \nTheoretical Physics Group\nImperial College\nPrince Consort RoadSW7 2BZLondonUK\n", "João Magueijo \nTheoretical Physics Group\nImperial College\nPrince Consort RoadSW7 2BZLondonUK\n" ]
[ "Theoretical Physics Group\nImperial College\nPrince Consort RoadSW7 2BZLondonUK", "Theoretical Physics Group\nImperial College\nPrince Consort RoadSW7 2BZLondonUK" ]
[ "Mon. Not. R. Astron. Soc" ]
We propose a framework for separating the information contained in the CMB multipoles, a ℓm , into its algebraically independent components. Thus we cleanly separate information pertaining to the power spectrum, non-Gaussianity and preferred axis effects. The formalism builds upon the recently proposed multipole vectors(Copi, Huterer & Starkman 2003;Schwarz & al 2004;Katz & Weeks 2004), and we elucidate a few features regarding these vectors, namely their lack of statistical independence for a Gaussian random process. In a few cases we explicitly relate our proposed invariants to components of the n-point correlation function (power spectrum, bispectrum). We find the invariants' distributions using a mixture of analytical and numerical methods. We also evaluate them for the co-added WMAP first year map.
10.1111/j.1745-3933.2005.00065.x
[ "https://export.arxiv.org/pdf/astro-ph/0407081v2.pdf" ]
16,111,258
astro-ph/0407081
0c190254f996c22c393eb99a0b2ed282fbae7775
Multipole invariants and non-Gaussianity 2003 Kate Land Theoretical Physics Group Imperial College Prince Consort RoadSW7 2BZLondonUK João Magueijo Theoretical Physics Group Imperial College Prince Consort RoadSW7 2BZLondonUK Multipole invariants and non-Gaussianity Mon. Not. R. Astron. Soc 0002003arXiv:astro-ph/0407081v2 19 Jan 2005 Printed 20 March 2022 (MN L A T E X style file v2.2)cosmic microwave background -Gaussianity tests -statistical isotropy We propose a framework for separating the information contained in the CMB multipoles, a ℓm , into its algebraically independent components. Thus we cleanly separate information pertaining to the power spectrum, non-Gaussianity and preferred axis effects. The formalism builds upon the recently proposed multipole vectors(Copi, Huterer & Starkman 2003;Schwarz & al 2004;Katz & Weeks 2004), and we elucidate a few features regarding these vectors, namely their lack of statistical independence for a Gaussian random process. In a few cases we explicitly relate our proposed invariants to components of the n-point correlation function (power spectrum, bispectrum). We find the invariants' distributions using a mixture of analytical and numerical methods. We also evaluate them for the co-added WMAP first year map. INTRODUCTION The remarkable quality of the WMAP data (Bennett & al 2003) has led us to a new era in observational cosmology. Yet, various claims for "unexpected" non-Gaussian signals cast a shadow on the purity of the data (Copi, Huterer & Starkman 2003;Eriksen & al 2004b;Coles & al 2003;Cruz & al 2003;Hansen & al 2004;Hansen, Banday & Górski 2004;Vielva & al 2003;Komatsu, Spergel & Wandelt 2003;Park 2003;Larson & Wandelt 2004;Mukherjee & Wang 2004;Eriksen & al 2004a;Schwarz & al 2004;Oliveira-Costa & al 2004). Given that there is no plausible theoretical explanation for these signals, a natural worry is that the maps are not, after all, fully free from systematics errors or galactic contamination. Most notably several groups have reported evidence for a preferred axis being selected by large-angle fluctuations, either in the form of multipole planarity (Oliveira-Costa & al 2004;Copi, Huterer & Starkman 2003;Schwarz & al 2004;Coles & al 2003), North-South asymmetries in the power spectrum, three-point function, or bispectrum (Eriksen & al 2004a;Hansen, Banday & Górski 2004;Land & Magueijo 2004), as well as using other methods (Eriksen & al 2004b;Hansen & al 2004). In assessing these asymmetries it is important to distinguish issues of non-Gaussianity (which should be rotationally invariant) from those of anisotropy (existence of a preferred axis). Apart from a subtlety in the definition of ⋆ E-mail:[email protected], [email protected] statistical ensemble (Ferreira & Magueijo 1997) these issues should be clearly separated. Unfortunately no systematic approach for extracting all the independent invariants under rotations from a given set of multipoles, a ℓm , has been proposed. The formalism of Copi, Huterer & Starkman (2003) does not produce invariants. The invariant n-point correlation function (Magueijo 1995;Ferreira, Magueijo & Górski 1998;Magueijo 2000;Magueijo & Medeiros 2003), on the other hand, is awkward to apply and often contains redundant information. In this letter we remedy this deficiency in the current formalism. STATEMENT OF THE PROBLEM Multipoles are irreducible representations of SO(3), so their 2ℓ + 1 degrees of freedom should split into 2ℓ − 2 invariants and 3 rotational degrees of freedom. Ideally we would like to process the {a ℓm } for a given mutipole ℓ into the power spectrum C ℓ (the Gaussian degree of freedom), 2ℓ − 3 invariant measures of non-Gaussianity, and a system of axes (assessing isotropy). In addition one should build 3 invariants per multipole pair, for example the Euler angles relating the two sets of multipole axes. The latter encode inter-scale correlations. No one has ever accomplished this project for general ℓ, but see Magueijo (1995) for a Quadrupole solution. Multipoles are equivalent to symmetric traceless tensors of rank ℓ. The problem is then to extract the independent invariant contractions of these tensors plus a system of axesa generalisation of the concept of eigenvalues and eigen-vectors. Typically the invariants produced by this formalism are related to the n−point correlation function (bispectrum, trispectrum, etc.) The formalism becomes very complicated very quickly, and no systematic breakdown of independent n−point correlation function components has ever been achieved. Alternatively one may extract from the {a ℓm } a length scale and ℓ independent unit vectors (the multipole vectors) as proposed by Schwarz & al (2004), and discussed further by Katz & Weeks (2004). This approach is considerably simpler and unsurprisingly it has been taken further. However we see a (correctable) problem with this approach. The multipole vectors are not rotationally invariant, and so they mix up the issues of isotropy and non-Gaussinity. We propose to correct this shortcoming by taking an appropriate number of independent inner products between these vectors. These are the sought-after invariants, and in some simple cases we relate them to the n−point correlation function. We will also extract from the multipole vectors a system of axes, encoding the multipole directional information. It is these axis that are to be used when testing isotropy. THE EXAMPLE OF THE QUADRUPOLE We start by considering the quadrupole, which as shown in Magueijo (1995) may be written as δT2 = Qijx i x j (1) where Qij is a symmetric traceless matrix and x i are cartesian coordinates on the unit sphere. From this matrix one may extract three eigenvectors and two independent combinations of invariant eigenvalues λi. These are essentially the power spectrum C2 (related to the sum of the squares of λi) and the bispectrum B222 (related to the determinant of the matrix, or the product of λi). In contrast, following the formalism of Katz & Weeks (2004) one writes δT2 = A2 L 1 i L 2 j − 1 3 δij L 1 · L 2 x i x j(2) where A is a scale and {L 1 i , L 2 i } are two units vectors, encoding the information on non-Gaussianity and anisotropy. The vectors {L 1 i , L 2 i } are not invariant, but one can construct an invariant by taking X = L 1 · L 2 . As pointed out to us by Starkman & Schwarz (2004), one may easily check that the eigenvectors of Q are V 1 = L 1 + L 2 (3) V 2 = L 1 − L 2 (4) V 3 = L 1 × L 2(5) These have corresponding eigenvalues A(X + 3)/6, A(X − 3)/6, and −AX/3, respectively. Using results in Magueijo (1995) one may therefore prove that the power spectrum and bispectrum are: C2 = A 2 2 6 (X 2 + 3) (6) B222 = A 3 2 X 2 2 3 3 (9 − X 2 )(7) so that the normalised bispectrum is I2 = B222 C 3 2 2 = X(9 − X 2 ) (X 2 + 3) 3/2(8) These formulae bridge the two formalisms. It was proved in Magueijo (1995) that for a Gaussian process C2, I2 and the eigenvectors are independent random variables, and that I2 is uniformly distributed in the range [−1, 1] (and C2 has a χ 2 5 distribution). We thus can prove that X is statistically independent from C2 (but not from A) and from the eigenvectors. By directly evaluating the Jacobian of the transformation we find that its distribution is P (X) = 27 1 − X 2 (X 2 + 3) 5/2(9) that is, it is not uniform. We have confirmed this result with Monte-Carlo simulations (see Fig. 1, top left panel). This elucidates an interesting feature of multipole vectors. Even though they are algebraically independent (in the sense that they contain no redundant information given a concrete quadrupole realization) they are not statistically independent. If vectors L 1 and L 2 were statistically isotropic (which they are) and statistically independent, then X would be uniformly distributed. As (9) shows this is not the case; hence vectors L 1 and L 2 are statistically correlated. Specifically they prefer being orthogonal to being aligned. Figure 2. The distribution of the modulus of the dot products for multipoles ℓ = 2 − 10, with anchor vectors L 1 and L 2 chosen as the most orthogonal pair of multipole vectors. The solid lines are for dot products X 1i and X 2i , and the dotted line shows the X 12 distribution. The WMAP results are also plotted (short lines along the bottom, dashed line the X 12 result). THE GENERAL PROCEDURE For a general multipole we have ℓ vectors and we could take as invariants all possible inner products between them. This would clearly lead to much redundant information, in contradiction with the requirements laid down in Section 2. A possible way out is to select two "anchor" vectors L 1 and L 2 , consider their dot product X12 = L 1 · L 2 , and then for 3 i ℓ the two products X1i = L 1 · L i and X2i = L 2 · L i . We thus obtain 2ℓ − 3 algebraically independent invariants. In Fig. 1 we plot their distribution for 2 ℓ 10 after the order and direction (±) of the ℓ vectors has been randomised. The anchor vectors L 1 and L 2 are therefore selected at random, so that all Xij invariants for a given multipole have the same distribution. We checked that the invariants X are uncorrelated. As we can see this distribution is ℓ dependent, and the tendency for multipole vectors to seek orthogonal directions is less pronounced for higher multipoles. We have also plotted these invariants as computed for the WMAP coadded masked first year map (details in Land & Magueijo (2004)). No evidence for non-Gaussianity is found, but we defer a closer scrutiny to a future publication. The proposed procedure provides an interesting Gaussianity test. In particular it can be easily applied to extract all the relevant information for high ℓ; in contrast isolating such information from the n-point correlation function is most cumbersome. However the suggested algorithm suffers from the drawback that the anchor vectors L 1 and L 2 are selected at random, and so the procedure is not reproducible. We may correct this by imposing a criterium for their selection, for example L 1 and L 2 may be taken as the most orthogonal among the given ℓ vectors. We may then form the "eigenvectors" (three orthogonal vectors): V 1 = L 1 + L 2 (10) V 2 = L 1 − L 2 (11) V 3 = L 1 × L 2(12) as the natural variables for encoding the multipole directionality. For invariants we may take the power spectrum and the 2ℓ − 3 quantities: X12 = L 1 · L 2 (13) X1i = L 1 · L i (14) X2i = L 2 · L i(15) (for 3 i ℓ). These encode all the relevant non-Gaussian degrees of freedom. The transformation from a ℓm into these variables is invertible (up to discrete uncertainties related to branch choice) and provides a solution to our problem, as phrased in Section 2. The proposed variables for a given multipole ℓ are the power spectrum C ℓ (Gaussian degree of freedom), the 2ℓ − 3 inner products X (non-redundant non-Gaussian invariants), and the orthogonal vectors {V 1 , V 2 , V 3 } (measures of anisotropy). The procedure reduces to the one found for the quadrupole when ℓ = 2. Within this framework the inter-ℓ correlations are measured by the Euler angles relating the systems of axes {V 1 , V 2 , V 3 } associated with each pair of multipoles. These should be uniformly distributed for a Gaussian distribution or indeed for any theory in which the various ℓ are uncorrelated. In Fig. 2 we plot distributions for the X invariants with anchor vectors L 1 and L 2 defined as the most orthogonal. We have used 12,500 realizations to make these histograms. P (X12) peaks around zero. The other X distributions are the same. We also plotted the invariants for the WMAP first year data, again finding no evidence for non-Gaussianity. We have checked that the eigenvectors, V i , are uniformly distributed. Notice that the invariants X cannot be independent variables, since their ranges of variation are interconnected. This is to be compared with the lack of independence among the various vectors L i , as demonstrated in the previous Section. Naturally we could have defined the two anchor vectors L 1 and L 2 in different ways, for example the two most aligned vectors. The invariant X12 would then be peaked around one. We plot the counterpart of Fig. 2 with this alternative definition in Fig. 3. BRIDGING THE TWO FORMALISMS For higher multipoles, relating the proposed formalism and the n−point correlation function, as done in Section 3 for the quadrupole, becomes very involved. For instance, for the octopole the counterpart of (1) and (2) is: δT3 = Q ijk x i x j x k = A3 L 1 i L 2 j L 3 k − 1 3 δijR k x i x j x k(16) where one should note that terms in {i, j, k} are symmetrized. The remainder R k is: Figure 3. The distribution of the modulus of dot products for multipoles ℓ = 2 − 10, with anchor vectors L 1 and L 2 chosen as the most aligned pair of multipole vectors. The solid lines are for dot products X 1i and X 2i , and the dotted line shows the X 12 distribution. The WMAP results are also plotted (short lines along the bottom, dashed line the X 12 result). R k = 3 5 L (1 · L 2 L 3) k(17) Again, T r(Q 2 ) = C3, and therefore one can show that: C3 = A 2 3 6 · 9 (9 + 7S2 + 6S2) where S1 = (L 1 · L 2 ) 2 + (L 2 · L 3 ) 2 + (L 3 · L 1 ) 2 (19) S2 = (L 1 · L 2 ) × (L 2 · L 3 ) × (L 3 · L 1 )(20) For higher multipoles Qi 1 ...i ℓ = A ℓ L (1 i 1 ...L ℓ) i ℓ − 3 5 δ (i 1 i 2 (L (1 · L 2 L 3 i 3 ) ...L ℓ) i ℓ(21) The components of the n-point correlation function may be obtained from the various contractions of Q. They are clearly a function of the dot product of the various multipole vectors. However, the formulae become progressively more complex to derive. With the introduction of our alternative procedure we suggest that these complicated formulae are not necessary. The X variables provide a better basis for describing the data. CONCLUSIONS Harmonic (Fourier) space is the natural arena for comparing theory and observation for Gaussian theories. In several past studies it was also found to be a useful ground for testing the hypothesis of Gaussianity. In this paper we showed further that the degrees of freedom in the spherical harmonic components {a ℓm } can be simply separated into a set of algebraically independent invariants -the power spectrum and a set of 2ℓ − 3 non-Gaussian statistics -and a set of axes encoding the multipole directionality. The Euler angles relating sets of axes associated with pairs of multipoles measure inter-ℓ correlations. This provides an elegant answer to a long unsolved problem -how to process the information contained in a given map into its relevant non-redundant degrees of freedom. Even though we have computed some of the proposed invariants for the WMAP first year map, we defer to a future publication a more systematic application of this framework to real data. We stress that the formalism is easily applicable for high ℓ multipoles. This is to be contrasted with the n-point correlation function. Even though it is trivial to evaluate the bispectrum at high ℓ (Land & Magueijo 2004) it becomes very difficult to distill all the non-redundant information contained in a given multipole in terms of sufficiently high order components of the correlation function. The proposed formalism is far more efficient. Figure 1 . 1The average distribution of the modulus of the 2ℓ − 3 dot products X ij (with the two anchor vectors chosen at random) for multipoles ℓ = 2 − 10. The dotted line corresponds to the analytical expression found for the quadrupole. We have also plotted the invariant values for the WMAP first year data, with the anchor vectors chosen at random (short lines along the bottom of each panel). http://www.eso.org/science/healpix/index.html 2 http://www.phys.cwru.edu/projects/mpvectors/ ACKNOWLEDGEMENTSWe would like to thank Dominik Schwarz, Glenn Starkman and Jeff Weeks for discussion and help with this project. The results in this paper have been derived using the HEALPix 1 package(Górski, Hivon & Wandelt 1998), as well as the publicly available codes described inCopi, Huterer & Starkman (2003)2 . This work was performed on COSMOS, the UK national cosmology supercomputer. . C L Bennett, Astrophys. J. Suppl. 1481Bennett C.L. et al., 2003a, Astrophys. J. Suppl, 148, 1 . P Coles, astro-ph/0310252Coles P. et al., 2003, astro-ph/0310252 . C J Copi, D Huterer, G D Starkman, astro-ph/0310511Copi C.J., Huterer D., Starkman G.D., 2003, astro-ph/0310511. . M Cruz, astro-ph/0405341Cruz M. et al., 2004, astro-ph/0405341 . H K Eriksen, Astrophys. J. 60514Eriksen H.K. et al., 2004a, Astrophys. J, 605, 14 . H K Eriksen, astro-ph/0401276Eriksen H.K. et al., 2004b, astro-ph/0401276 . P Ferreira, J Magueijo, Phys.Rev. 564578Ferreira P., Magueijo J., 1997, Phys.Rev, D56, 4578 . P Ferreira, J Magueijo, K Górski, Astrophys.J. 5031Ferreira P., Magueijo J., Górski K., 1998, Astrophys.J, 503, 1 . K M Górski, E Hivon, B Wandelt, astro-ph/9812350Górski K.M., Hivon E., Wandelt B. 1998, astro-ph/9812350 . F K Hansen, astro-ph/0402396Hansen F.K. et al., 2004, astro-ph/0402396 . F K Hansen, A J Banday, K M Górski, astro-ph/0404206Hansen F.K., Banday A.J., Górski K.M.,2004,astro-ph/0404206 . G Katz, J Weeks, astro-ph/0405631Katz G., Weeks J., 2004, astro-ph/0405631 . E Komatsu, D N Spergel, B D Wandelt, astro-ph/0305189Komatsu E., Spergel D.N., Wandelt B.D., 2003, astro-ph/0305189 . K Land, J Magueijo, astro-ph/0405519Land K., Magueijo J., 2004, astro-ph/0405519 . D L Larson, B D Wandelt, astro-ph/0404037Larson D.L., Wandelt B.D., 2004, astro-ph/0404037 . J Magueijo, Phys. Lett. 342499Erratum-ibidMagueijo J., 1995, Phys. Lett, B342, 32. Erratum-ibid, B352, 499. . J Magueijo, Astrophys. J. Lett. 52857Magueijo J., 2000, Astrophys. J. Lett, 528, 57 . J Magueijo, J Medeiros, MNRAS. 351Magueijo J., Medeiros J., 2004, MNRAS 351, L1-4 . P Mukherjee, Y Wang, astro-ph/0402602Mukherjee P., Wang Y., 2004, astro-ph/0402602 . A De Oliveira-Costa, Phys. Rev. 6963516de Oliveira-Costa A. et al., 2004, Phys. Rev, D69, 063516 . C Park, MNRAS. 349313Park C., 2004, MNRAS, 349, 313 . D Schwarz, astro-ph/0403353Schwarz D. et al., 2004, astro-ph/0403353 A private communication. G Starkman, D Schwarz, Starkman G., Schwarz D., A private communication, May 27 2004. . P Vielva, astro-ph/0310273Vielva P. et al., 2003, astro-ph/0310273
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[ "How Macroscopic Limits on Neutron Star Baryon Loss Yield Microscopic Limits on Non-Standard-Model Baryon Decay", "How Macroscopic Limits on Neutron Star Baryon Loss Yield Microscopic Limits on Non-Standard-Model Baryon Decay", "How Macroscopic Limits on Neutron Star Baryon Loss Yield Microscopic Limits on Non-Standard-Model Baryon Decay", "How Macroscopic Limits on Neutron Star Baryon Loss Yield Microscopic Limits on Non-Standard-Model Baryon Decay" ]
[ "Jeffrey M Berryman [email protected]†[email protected]‡[email protected] ", "Susan Gardner ", "Mohammadreza Zakeri ", "\nDepartment of Physics\nInstitute for Nuclear Theory\nUniversity of California\n94720BerkeleyCAUSA\n", "\nCenter for Neutrino Physics\nPhysics Department\nUniversity of Washington\n98195SeattleWAUSA\n", "\nDepartment of Physics and Astronomy\nVirginia Tech\n24061BlacksburgVAUSA\n", "\nDepartment of Physics and Astronomy\nUniversity of Kentucky\n40506-0055LexingtonKYUSA\n", "\nUniversity of Kentucky\n40506-0055LexingtonKYUSA\n", "Jeffrey M Berryman [email protected]†[email protected]‡[email protected] ", "Susan Gardner ", "Mohammadreza Zakeri ", "\nDepartment of Physics\nInstitute for Nuclear Theory\nUniversity of California\n94720BerkeleyCAUSA\n", "\nCenter for Neutrino Physics\nPhysics Department\nUniversity of Washington\n98195SeattleWAUSA\n", "\nDepartment of Physics and Astronomy\nVirginia Tech\n24061BlacksburgVAUSA\n", "\nDepartment of Physics and Astronomy\nUniversity of Kentucky\n40506-0055LexingtonKYUSA\n", "\nUniversity of Kentucky\n40506-0055LexingtonKYUSA\n" ]
[ "Department of Physics\nInstitute for Nuclear Theory\nUniversity of California\n94720BerkeleyCAUSA", "Center for Neutrino Physics\nPhysics Department\nUniversity of Washington\n98195SeattleWAUSA", "Department of Physics and Astronomy\nVirginia Tech\n24061BlacksburgVAUSA", "Department of Physics and Astronomy\nUniversity of Kentucky\n40506-0055LexingtonKYUSA", "University of Kentucky\n40506-0055LexingtonKYUSA", "Department of Physics\nInstitute for Nuclear Theory\nUniversity of California\n94720BerkeleyCAUSA", "Center for Neutrino Physics\nPhysics Department\nUniversity of Washington\n98195SeattleWAUSA", "Department of Physics and Astronomy\nVirginia Tech\n24061BlacksburgVAUSA", "Department of Physics and Astronomy\nUniversity of Kentucky\n40506-0055LexingtonKYUSA", "University of Kentucky\n40506-0055LexingtonKYUSA" ]
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We investigate how our baryon-loss limits from anomalous binary-pulsar period lengthening can be interpreted microscopically to yield specific constraints on the particle physics of baryon number violation within a neutron star. We focus on the possibility of anomalous baryon disappearance via dark baryon processes and on scenarios in which the produced dark-sector particles do not survive to influence the response of the star to baryon-number-violating effects. We flesh out the conditions for which this may occur, as well as other key assumptions. We then turn to the analysis of particle processes in the dense nuclear medium found at the core of a neutron star, employing the techniques of relativistic mean-field theory. Using our study of in-medium effects and limits on macroscopic baryon number violation we extract limits on in-vacuum baryon-number-violating processes, and we determine them for various equations of state. We conclude by noting the implications of our results for models of dark-sector-enabled baryogenesis. *
null
[ "https://export.arxiv.org/pdf/2305.13377v1.pdf" ]
258,841,504
2305.13377
e3adb5ee361315a7f5fc2c6436469adfa8250629
How Macroscopic Limits on Neutron Star Baryon Loss Yield Microscopic Limits on Non-Standard-Model Baryon Decay 22 May 2023 Jeffrey M Berryman [email protected]†[email protected]‡[email protected] Susan Gardner Mohammadreza Zakeri Department of Physics Institute for Nuclear Theory University of California 94720BerkeleyCAUSA Center for Neutrino Physics Physics Department University of Washington 98195SeattleWAUSA Department of Physics and Astronomy Virginia Tech 24061BlacksburgVAUSA Department of Physics and Astronomy University of Kentucky 40506-0055LexingtonKYUSA University of Kentucky 40506-0055LexingtonKYUSA How Macroscopic Limits on Neutron Star Baryon Loss Yield Microscopic Limits on Non-Standard-Model Baryon Decay 22 May 2023(Dated: May 24, 2023)1 CONTENTS We investigate how our baryon-loss limits from anomalous binary-pulsar period lengthening can be interpreted microscopically to yield specific constraints on the particle physics of baryon number violation within a neutron star. We focus on the possibility of anomalous baryon disappearance via dark baryon processes and on scenarios in which the produced dark-sector particles do not survive to influence the response of the star to baryon-number-violating effects. We flesh out the conditions for which this may occur, as well as other key assumptions. We then turn to the analysis of particle processes in the dense nuclear medium found at the core of a neutron star, employing the techniques of relativistic mean-field theory. Using our study of in-medium effects and limits on macroscopic baryon number violation we extract limits on in-vacuum baryon-number-violating processes, and we determine them for various equations of state. We conclude by noting the implications of our results for models of dark-sector-enabled baryogenesis. * INTRODUCTION The cosmic excess of baryons over antibaryons is well established [1], but the theoretical mechanism by which it is produced is not. The essential theoretical ingredients are thought to be known: baryon number violation (BNV), along with C and CP violation, must all be present in a non-equilibrium environment [2]. Thus BNV would seem to play an essential role, though in the Standard Model (SM) BNV is thought to occur appreciably only at extremely high temperature [3,4] -and the existence of BNV at low energies has as yet to be established. In this paper we continue our scrutiny of such effects through observations of neutron stars, which contain enormous reservoirs of baryons. In earlier work we identified sensitive limits on BNV through the interpretation of precise observations of energy loss in isolated neutron stars and in neutron-star binary systems [5]. These studies limit the baryonnumber-violating effects that occur across the entirety of a neutron star. In this sense they are macroscopic limits. In this paper we interpret these limits in a microscopic way, in that we develop a framework in which they can be translated to limits on the parameters of particular particle physics models that generate baryon-number-violating effects. The particular models to which our studies are most sensitive are those in which baryons decay or otherwise transform to dark-sector fermions, of O(1 GeV) in mass, that carry baryon number. In such cases BNV becomes an apparent, rather than explicit, effect, because the dark-sector particles are unobserved, even if baryon number is not broken. Although the existence of dark matter is certainly established through astrometric observations, both its nature and origin continue to be open questions. It is possible that the origins of dark matter and of the cosmic baryon asymmetry are related, so that the loosely similar value of the cosmic baryon and dark matter energy densities today may follow from a single underlying model [6]. The possibility of baryons that connect to hidden-sector baryons of comparable mass figure in many such explanations. In this paper we constrain this possibility through the study of neutron and hyperon transitions to final states with dark baryons in the neutron star. To our knowledge, an in-depth, quantitative study of non-SM processes within dense nuclear matter has not previously been realized 1 , and its execution necessitates much care. The existence of neutron stars of about 2 M ⊙ in mass speaks to central densities in excess of three times nuclear matter saturation density, so that in this paper we employ relativistic mean field theory in baryonic degrees of freedom for our dense matter description, as its accuracy should improve with increasing density -and thus it should work best at the core of the star. We note that a neutron star may become a hybrid star, i.e., one with a quarkbased core predicated by a finite-density quark-hadron phase transition, if it is sufficiently heavy, and this possibility can also be constructed within this framework [9]. Transitions to dark baryons could also occur within the quark-based core, though we will set aside this possibility in this paper -and revisit it only in offering an assessment of our uncertainties in our concluding summary. The broader possibility of dark decays of the neutron has been noted in explanation [10,11] of the long-standing neutron lifetime anomaly [12], in which the lifetime inferred from counting surviving neutrons is significantly different from that inferred from counting the protons subsequent to ordinary neutron decay. Although the discrepancy may arise from experimental effects, the possibility that dark decays contribute to it in some measure is a continuing possibility [5]. In this paper we provide severe limits on the flavor structure of possible new-physics models with dark baryonic sectors, such as Refs. [10,[13][14][15][16], that arise from the interpretation of neutron-star energy loss constraints we developed in Ref. [5]. In this paper we flesh out the general assumptions of that earlier analysis and note how the specific models we consider can satisfy them. Let us conclude our introduction with a brief outline of the body of our paper. In Sec. 2 we detail the models of baryon dark decays that we are able to constrain through our neutron star studies, and we note how they are distinct from models that we cannot. We also compute baryon dark decay rates in vacuum, for later reference, as well as dark baryon 1 Albeit studies of exotic light particle emission in dense matter, which possesses simplifying aspects, are of long standing [7] and continue to be investigated [8]. removal rates, because our analysis assumes that SM dynamics determine the response of the star in the presence of BNV. In Sec. 3 we consider macroscopic baryon number violation in neutron stars, revisiting our earlier work [5] and fleshing out constraints following from its assumptions in greater detail. In Sec. 4 we develop how to evaluate particle processes within dense matter, employing RMF, as developed in Refs. [17][18][19], to describe the neutron star medium in β-equilibrium [20,21]. In this context, uncertainties in our description of the dense medium are captured through variations in the equation of state (EoS). With these developments in hand, we evaluate particle processes within our framework for the dense nuclear medium of a neutron star in Sec. 5 and use our macroscopic limits on BNV from Sec. 3 to report limits on the parameters of the microscopic models we consider in Sec. 6. In Sec. 7 we consider the implications of our results for models of dark-sector baryogenesis and dark matter, and we offer a summary and outlook in Sec. 8. PARTICLE PHYSICS MODELS OF BARYON DARK DECAYS The possibility of hadronic processes with dark-sector particles naturally emerges in models that explain both the origin of dark matter and the cosmic baryon asymmetry, particularly if the dark sector candidate carries a baryonic charge [13,14,22]. Although it has long been thought that dark matter could also be described as a relic asymmetry [23,24], in these models, rather, the two problems are solved simultaneously [6]. More recently, highly testable scenarios [25] have been developed [16,[26][27][28][29][30], and we probe their flavor structure through the studies of this paper -and in Sec. 7 we consider the implications of the constraints that we find. Since the dark-sector particles are presumably SM gauge singlets, they could be light in mass, potentially with masses comparable to that of the known hadrons, and yet have escaped experimental detection thus far. Our current discussion is loosely inspired by models connected to explanations of the neutron lifetime anomaly [10,16,31], with neutrons decaying to a dark baryon with a photon or an e + e − pair. Models with similar content have been considered for broader purposes [15,[32][33][34], and alternative solutions have also been noted [35][36][37]. The dark channels in the various models would impact the determined bottle lifetime, with a mirror neutron model [35] serving as a rare exception. There, neutron-to-mirror-neutron conversion occurs in a strong magnetic field, impacting the ability to detect protons in the beam-lifetime experiment. This last possibility has been excluded as a complete explanation of the anomaly by a direct experimental search [38]. We note that models that would explain the anomaly through neutron disappearance or decay to dark-sector final states can also be constrained by the close empirical agreement of the neutron lifetime with its measured A decay correlation as interpreted in the SM [5,39,40]. This agreement limits the branching ratio on such exotic processes to [41]: Br(n → exotics) < 0.16% (95% one-sided C.L.) , (2.1) where we note that the neutron lifetime anomaly is roughly a 1% effect [10] 2 . Direct experimental limits on n → χγ [45] and n → χe + e − [46] decays also exist, removing ranges of parameter space as an explanation of the anomaly. We will be able to set much more severe limits through our studies, where we note the limit on Λ → χγ from SN1987 for reference [16]. We regard the neutron lifetime anomaly as a motivation for further investigation of baryon dark decays, with new limits constraining the manner in which the co-genesis of dark matter and the cosmic baryon asymmetry could possibly occur. We now turn to the development of models of dark baryon decays. Following Ref. [33], we introduce a Dirac fermion χ with baryon number B = +1 which interacts with SM quarks via the generic form L χ =χ(i/ ∂ − m χ )χ + u i d j d k χ c L Λ 2 ijk + Q i Q j d k χ c L Λ 2 ijk + h.c. ,(2.2) where i, j, k are generational indices, Q and q denote a left-handed quark doublet and a right-handed quark, respectively -and color and Lorentz indices are left implicit. Such interactions can generate both decay and scattering processes involving dark final states, which we consider closely in this paper. First, though, we address their flavor structure. We could neglect this possibility altogether, dropping all subscript dependence, but simple, renormalizable models that produce Eq. (2.2), at energies below the mass scale of their new physics, show that strong flavor sensitivity can nevertheless exist. Turning to models with leptoquarks [10,15], we consider colored scalars S 1 andS 1 transforming as (3, 1, 1/3) 3 and (3, 1, −2/3), respectively, under the SM gauge groups and SM invariant scalar-fermion interactions. Non-trivial flavor structure follows from the choice of leptoquark in that S 1 2 The most precise measurement of the A correlation coefficient yields the ratio of the axial-vector to vector coupling constants |λ| = 1.27641(56) [42], but recent measurements of the a correlation do not completely fit this picture, yielding |λ| = 1.2677 (28) [43] and |λ| = 1.2796 (62) [44]. 3 This variant was first considered in Ref. [10]. can mediate both n → χγ and Λ → χγ decay at tree level, whereasS 1 can mediate Λ → χγ at tree level but to mediate n → χγ requires a one-loop process with W ± exchange as well [15]. Thus in this paper we strive to probe both n → χγ and Λ → χγ decay processes. These models also readily generate proton decay [10,15,33], noting p → χπ + or p → χK + decay as examples, so that the possible range of χ masses is rather restricted as a result. We note that the stability of the 9 Be nucleus [10], particularly stability against 9 Be → χαα decay [47], requires m χ ≥ 0.937993 GeV , (2.3) slightly in excess of the proton stability constraint m χ > m p −m e , and that atomic hydrogen is stable if m χ > m p +m e = 0.93878 GeV [34]. If either constraint were not satisfied, then the empirical limit on the pertinent lifetime would bound the parameters of the model. Within the SM both systems are absolutely stable, yet empirical tests of that, with a determined lifetime as an outcome, should be possible. We note H lifetime estimates, made finite through a model with a suitably light χ, are made in Ref. [34]. Moreover, the radiative decay H → νχγ, which is subdominant relative to H → νχ, can be probed through measurements at Borexino [34,48]. Similar expectations follow from violating Eq. (2.3) -and a concrete estimate of the 9 Be lifetime can be found in Ref. [49]. In what follows we ignore the possible chiral structure of the quark-χ couplings and simply consider [16] L ⊃ u i d j d k χ Λ 2 + h.c. (2.4) Since the quarks carry electric charge, we have, at the energy scales for which baryonic degrees of freedom are pertinent, L n =n i/ ∂ − m n + g n e 8m n σ αβ F αβ n +χ(i/ ∂ − m χ )χ + ε nχ (nχ +χn) , (2.5) noting g n = −3.826 is the g-factor of the neutron [50]. This form also holds for the Λ upon the replacement n → Λ, taking g Λ = −1.22. After redefining the fields to remove the mixing term in Eq. (2.5), then if ε ≪ m n − m χ , with m χ < m n , we have [10,15] L n→χγ = g n e 8m n ε nχ m n − m χχ σ αβ F αβ n , (2.6) though potentially this operator could also stem from a distinct higher-energy source. Generally, the interaction of Eq. (2.4) can also generate transitions to dark baryon states with mesons, such as the decays n → χ + meson or Λ → χ + meson. Ref. [16] uses chiral effective theory [51] to relate the possibilities. We eschew this path because chiral effective theory ceases to be valid if the density of the neutron star medium much exceeds that of nuclear matter saturation density. Since our particular purpose is to set limits on microscopic models given BNV limits determined from observables associated with the entire neutron star, we set aside the study of final states containing both dark and hadronic degrees of freedom. They are distinct from the final states we do study, and cancellations cannot occur. We thus expect that including these additional decays with hadrons can only make our limits more severe, though the inclusion of hadronic channels would make our estimates less sure. At low energies, the magnetic interaction of Eq. (2.6), employed in Refs. [10,15], can be used to compute n → χγ or Λ → χγ. A pertinent Feynman diagram is illustrated in Fig. 1. Denoting B as either n or Λ, the total decay rate for B → χγ is given by Γ (B → χγ) = g 2 B e 2 ε 2 Bχ 128π (m B + m χ ) 2 m 5 B m 2 B − m 2 χ ,(2.7) in agreement with Ref. [15]. The mixing parameter ε Bχ follows once the UV model is given, and it is what we constrain through the analysis of this paper. To determine the impact of these microscopic processes on the neutron star requires further model building. Thus far, at low energies we have a dark baryon χ, which we take to be a massive Dirac fermion. If it is a stable particle, then it can also be a dark matter candidate. If so, then it may already exist within the material that collapsed to form the protoneutron star, though likely only in small amounts, and through dark decays or adsorption on the star it may accumulate within the star. If it is able to give up its kinetic energy, then it may settle in the core of the star, ultimately impacting its properties and evolution. There are many processes in which χ could participate, though the interactions with baryons are severely limited by the cold, degenerate nature of the interior of the neutron star. In principle, given the nχγ and nχπ 0 effective interactions in the models we have noted, and using N to denote either a neutron or a proton, χ could (i) be produced via nN → χN scattering, (ii) interact elastically with another nucleon via a nN intermediate state, (iii) be formed via the annihilation nn → χχ or (iv) it can decay via χ → p + e − +ν e if it is heavy enough. The reverses of the reactions in (i) and (iii) could also occur. Pauli-blocking effects associated with the cold, dense neutron medium strongly suppress all of the reactions in which nucleons appear in the final state. Moreover, χ − N elastic scattering is further suppressed in that it occurs at O(ε 2 nχ ) at amplitude level. We note Fig. 2 for an illustration. Given this, and our interest in limiting BNV within the star in a model-independent way, implying that the response of the star to BNV ought be controlled by SM dynamics, we think that ensuring χ disappearance is important. Thus we consider two different pathways to do just that. In the first, we add χ-lepton interactions [5], which intrinsically break baryon number and are intrinsically very poorly constrained. We would also want the rate for χ decay to be no less of that for χ production. This path, however, is potentially subject to severe constraints from proton decay experiments. For example, we could have χ → e + e − ν or χ → 3ν, and these channels could give rise to proton decay via an off-shell χ * state as in n N N χ π 0 N χ π 0 N χ π 0 N N N N χ π 0 χ (a) (b) (c)p → π + χ * → π + e + e − ν . (2.8) (Exotic proton decays of just this ilk also emerge in models with quark and lepton compositeness [52].) Admittedly, this process, as well as the p → π + 3ν channel, may evade severe constraints due to the particular nature of existing |∆B| = 1 searches, both because of the final-states studied and the cuts on the final-state particle momenta needed to control backgrounds. Although this path could prove to be viable, we favor an alternate choice: we will allow χ to decay to other dark particles. A simple realization of this is given by [14] L dark ⊃ y dχ ϕ B ξ + h.c. , (2.9) where ϕ B is a complex scalar with B = +1 and ξ is a Majorana fermion -and both are dark-matter candidates. Introducing a Z 2 symmetry, so that χ, n, and p are all Z 2 even, but ϕ B and ξ are Z 2 odd, we see Eq. (2.9) is the only surviving interaction that traces to the visible sector, with n − ξ oscillations, say, forbidden by the Z 2 symmetry. One interesting consequence of this new path is that dark decays can be induced in the scattering with either ϕ B or ξ in the initial state, as developed in Ref. [53] and illustrated in Fig. 3. A similar mechanism, considered in the context of the neutron lifetime anomaly, has been studied in Ref. [36]. The same process can destabilize the proton, with |∆B| = 1 experimental studies constraining the model parameters [53]. We note that theχϕ B ξ interaction can also induce χχ annihilation, as noted and illustrated in decay from the star, yet ϕ B could potentially accumulate in its core -and impact the survival of the neutron star [55]. If we suppose, rather, that ϕ B is light enough to escape the star, then that outcome can be avoided. We now turn to the explicit evaluation of proceses that can remove χ from the neutron star. Fig. 3. Alternatively, χχ annihilation via ϕ B exchange in tchannel would yield a ξξ final state, which could ultimately rematerialize as aχχ pair. A. Dark Baryon Removal Rates If the masses of ξ and ϕ B sum to less than the mass of χ, then the decay χ → ξϕ B is operative. Using Eq. (2.9) and Refs. [54,56], we calculate the width of this decay to be Γ χ→ξϕ B = y 2 d 16πm 3 χ (m χ + m ξ ) 2 − m 2 ϕ B 3/2 (m χ − m ξ ) 2 − m 2 ϕ B 1/2 . (2.10) However, if this decay is operative and if m ϕ B + m ξ < m p − m π , then this allows for proton decay via p + → π + ξϕ B . We avoid potentially running afoul of these constraints by insisting that this decay not be operative and thus require m ξ > m χ . Instead, we focus on possible annihilation processes of χ, where we have assumed that only ϕ B is lighter than χ. Adopting the same tools to compute χχ → ϕ B ϕ B we have: σ χχ→ϕ B ϕ B (s) = y 4 d m 2 ξ 64πs 2 (s − 4m 2 χ )(s − 4m 2 ϕ B ) m 4 ξ + m 2 ξ (s − 2m 2 χ − 2m 2 ϕ B ) + (m 2 χ − m 2 ϕ B ) 2 (2.11) + 4 s + 2m 2 ξ − 2m 2 χ − 2m 2 ϕ B × ln   s + 2m 2 ξ − 2m 2 χ − 2m 2 ϕ B + (s − 4m 2 χ )(s − 4m 2 ϕ B ) s + 2m 2 ξ − 2m 2 χ − 2m 2 ϕ B − (s − 4m 2 χ )(s − 4m 2 ϕ B )   . We note that this cross section goes to zero as m ξ → 0. This must occur, so that this outcome serves as a non-trivial check of our procedure. Our cross section result does not depends on whether the scalar is real or complex, but its interpretation does. If the scalar is real, it cannot carry baryon-number, and χχ annihilation to scalars would then break B by two units. This can only occur if m ξ has a nonzero baryon-number-violating mass. Thus its rate vanishes if m ξ does. We would like to understand how these annihilation processes operate within a neutron star. As we will see in Sec. 3 A, these cross sections would need to be averaged over the true distribution of χs produced in baryon decays within the star. Generically, χs need not be distributed thermally, and the process of thermalization would require self-interactions, which do not appear at tree level in our simple model. The problem of χ transport in the neutron star is beyond the scope of this paper, so that we assume that the thermally averaged cross section is a reasonable estimate of what the true averaged cross section would be. We proceed by employing pertinent results from the seminal Ref. [57]. The thermally averaged cross section ⟨σv⟩ is given formally by ⟨σv⟩ = 1 8m 4 χ T χ K 2 2 (m χ /T χ ) ∞ 4m 2 χ ds σ(s) × (s − 4m 2 χ ) √ sK 1 ( √ s/T χ ), (2.12) where T χ is the χ temperature (which is generically nonzero and may be different from the temperature of the rest of the neutron star) and K 1,2 are modified Bessel functions of the second kind. This expression assumes that it is appropriate to describe the χ fluid as abiding by a Maxwell-Boltzmann distribution; it would be inappropriate to apply this expression to a cold, degenerate population of χ, but such a population does not occur in our framework. To perform the thermal averaging, we expand σ(s) × v in powers of ϵ ≡ s/(4m 2 χ ) − 1: 2.13) this requires that v = 2 ϵ(1 + ϵ)/(1+2ϵ). In the limit in which the χ fluid is nonrelativistic, the thermally averaged cross section can we written in terms of the coefficients a (n) as follows: σv = a (0) + a (1) ϵ + 1 2 a (2) ϵ 2 + . . . ; (⟨σv⟩ = a (0) + 3 2 a (1) T χ m χ + 15 8 a (2) T χ m χ 2 + . . . (2.14) This prescription is expected to be valid as long as T χ ≲ 3m χ [57]. For χχ → ϕ B ϕ B , we find the leading-order contribution to the thermally averaged cross section in T χ to be ⟨σv⟩ χχ→ϕ B ϕ B = 3 2   h 4 m 2 ξ m 2 χ − m 2 ϕ B 8πm χ (m 2 ξ + m 2 χ − m 2 ϕ B ) 2   T χ m χ + . . . (2.15) Since the a (0) term vanishes, we conclude that the s-wave annihilation contribution vanishes, resulting in a suppression at low temperatures. We expect our χs to have a nonzero average kinetic energy from decays, so we do not expect to encounter a scenario in which these annihilations are completely quenched by the low energies of their parents, but it is an interesting feature to note. We conclude by noting some relevant qualitative features of this model. Since χ selfinteractions do arise at the one-loop level as a result of interactions with ϕ B and ξ, we can expect the χ population would thermalize, but that timescale is likely slow relative to that of their annihilation to scalars. There are many more interesting phenomenological consequences of this model that one could explore, but for our purposes, it is enough to assume that the masses and coupling conspire such that χ can be removed from neutron stars quickly enough that our formalism is valid. MACROSCOPIC BARYON NUMBER VIOLATION IN NEUTRON STARS We set out this section by elaborating the main assumptions for our analysis, followed by a description of the resulting formalism, which we flesh out in greater detail than in Ref. [5]. We then discuss the observable effects associated with our framework, along with methods of interpreting pulsar observations to yield limits on BNV in such systems. We use the limits derived at the end of this section to constrain specific baryon dark decay rates in Sec. 6, though we develop our description of dense matter, as well as of particle processes within it, in intervening sections before doing so. A. Assumptions The structure of a neutron star can be approximated by a static and spherically symmetric metric (g µν ) with a line element given by [58] dτ 2 = g µν dx µ dx ν = e 2ν(r) dt 2 − e 2λ(r) dr 2 − r 2 dθ 2 − r 2 sin 2 θ dϕ 2 ,(3.1) in which ν(r), λ(r) are solutions to the Einstein field equations [59], G µν = −8πGT µν , in which G µν is Einstein's tensor, G is the gravitational constant, and T µν is the stress-energy tensor. The rotation effects on the neutron star structure, which are O (Ω 2 / (G M/R 3 )) [60], amount to less than 3% for the fastest spinning pulsar (J1614−2230) that we consider in this work. Furthermore, the inclusion of quasi-static BNV processes, which are sourced by the matter in the star, would keep the spherical symmetry intact and changes to the metric (g µν ) very slow in time, such that the use of Eq. (3.1) is warranted. We also assume that the medium in neutron star can be described by a perfect fluid with T 0 0 = E, T i i = −P (i = 1, 2, 3),(3.2) as the only nonzero components of the stress-energy tensor in which P and E are the local pressure and energy density of the fluid respectively which in general depend on the local baryon number density (n) and temperature (T ) via the EoS. In the standard picture, neutron stars cool down to internal temperatures T ≲ 10 11 K ≪ E F ≲ GeV within a minute after formation [61], such that the thermal contribution to the pressure and energy density can be neglected. The neutron star fluid can then be described as a cold degenerate Fermi gas at β-equilibrium. The existing terrestrial constraints on neutron dark decay, Eq. (2.1), along with the BNV limits we find in Table I, show that BNV rates should be slow with respect to other dynamical processes in the neutron star. We have also devised a model in which χ, the dark baryon-like particle, can be removed efficiently from the star. Thus we expect the deviations from a degenerate state at β-equilibrium due to BNV should be negligibly small, and we leave a more detailed study of possible thermal effects on neutron stars from BNV to future work. In order to be able to apply our model-independent formalism [5], we are going to focus on a subset of models in which the dark contributions to the EoS are negligible relative to the energy density and pressure of the visible sector. In other words, we demand that the following (local) conditions E χ (r) E(r) ≪ 1, and P χ (r) P (r) ≪ 1, (3.3) hold throughout the neutron star at all times, which can be equivalently written as a condition on the local number density of χ: n χ (r) ≪ n(r). This means that χ has to decay or annihilate either back to the visible sector or to some other dark particles that can escape the neutron star. We assume that χ participates in self-annihilation to lighter dark particles that can escape the neutron star (see Sec. 2 A for more details). We can express the condition n χ (r) ≪ n(r) in terms of the BNV rate, Γ BNV , and the annihilation cross section that is averaged over χ distribution, which we denote by ⟨σv⟩. We note that the exact distribution of χ in the neutron star can in principle be found by solving the Boltzmann transport equation in the star, but this is not practical for our estimation purposes. We instead consider two scenarios for χ: one in which the annihilation rate is much faster than the self-interactions which help establish a thermal equilibrium, and another in which self-interactions of χ are much faster than its annihilation rate. We first consider the scenario in which dark particles have a non-thermal distribution at the time of their annihilation. If we ignore the effects due to radial redistribution of χ after their production and prior to their annihilation, their number density (n χ ) would approximately satisfyṅ (3.4) in which n i (t) is the decaying baryon number density which we take to be constant on short timescales, and ⟨σv⟩ is the annihilation cross section averaged over the distribution of χ. χ (t) = n i (t) × Γ BNV − n 2 χ (t)⟨σv⟩, The asymptotic value for χ number density (at times t ≫ 1/ n i Γ BNV ⟨σv⟩) is then equal to n ∞ χ = n i Γ BNV /⟨σv⟩, which relative to the local baryon number density n(r) is given by n ∞ χ (r) n(r) = 1.4 × 10 −15 f i (r) n sat n(r) 1/2 Γ BNV 10 −10 yr −1 1/2 10 −26 cm 3 s −1 ⟨σv⟩ 1/2 , (3.5) in which f i (r) ≡ n i (r)/n(r) < 1 is the fraction of baryon i relative to the total baryon number density, n sat = 0.15 fm −3 is the nuclear saturation density, and we used the scale of the canonical weak-scale cross section (10 −26 cm 3 s −1 ) for comparison. We can see that this ratio is negligible for the reference values in this equation if ⟨σv⟩ ≫ 10 −56 cm 3 s −1 . We can generalize Eq. (3.4) to scenarios in which the redistribution of χ's , after their production and prior to their annihilation, is not negligible, by noting that the total χpopulation satisfiesṄ (3.6) in which B i (t) is the number of decaying baryons of type i, and C ann is the annihilation rate per particle, such that the total annihilation rate is identified as Γ ann ≡ C ann N 2 χ /2. We are interested in short timescales during which B i (t) can be taken as a constant (t ≪ Γ −1 BNV ). In this case, the solutions to Eq. (3.6), assuming N χ (0) = 0, are given by χ (t) = B i (t) × Γ BNV − C ann N 2 χ (t),N χ (t) = B i Γ BNV C ann tanh B i Γ BNV C ann t , t ≪ Γ −1 BNV (3.7) in which the timescale for achieving an equilibrium between the production and annihilation of χ (Ṅ χ (τ ∞ ) ≈ 0) can be identified as τ ∞ = 1/ √ B i Γ BNV C ann , which can be achieved for τ ∞ < t ≪ Γ −1 BNV , if Γ BNV B i ≪ C ann . (3.8) The total number of χs can then be approximated by its equilibrium value given by N ∞ χ = B i Γ BNV /C ann . We can see that if the condition in Eq. (3.8) holds, then N ∞ χ ≪ B i . We now calculate C ann in the scenario in which the annihilation rate of χ is slower than its self-interaction rate, and the χ's are distributed spherically with an average radius of R χ , according to Boltzmann distribution. Using the virial theorem and assuming a radially uniform distribution of background neutron star matter (over R χ ) with an average energy densityĒ, we can write 9) in which k B is the Boltzmann constant, and T χ is the dark sector temperature. The total annihilation rate (Γ ann ) and N χ can then be evaluated as (3.11) in which ⟨σv⟩ is the thermally averaged annihilation cross section. Using the definition of C ann we have 12) and an equilibrium between the production and annihilation can be achieved on timescales n χ (r) = n χ (0) e −r 2 /R 2 χ , R χ = 3 k B T χ 2π GĒ m χ ,(3.Γ ann = 1 2 Rχ 0 4πr 2 [n χ (r)] 2 ⟨σv⟩ dr = 0.24 k B T χ GĒ m χ 3/2 ⟨σv⟩ [n χ (0)] 2 , (3.10) N χ = Rχ 0 4πr 2 n χ (r) dr = 0.78 k B T χ GĒ m χ 3/2 n χ (0),C ann ≡ 2 Γ ann N 2 χ = 0.78 GĒ m χ k B T χ 3/2 ⟨σv⟩,(3.t ≪ Γ −1 BNV , if (see Eq. (3.8)) ⟨σv⟩ ≫ 2 × 10 −55 10 57 B i T χ m χ 3/2 10 15 g/cm 3 E 3/2 Γ BNV 10 −10 yr −1 cm 3 s −1 ,(3.13) in which T χ and m χ have the same units. We can also find the equilibrium value for χ number density at the core by combining the definition of equilibrium number, N ∞ χ = B i Γ BNV /C ann , with Eq. The baryon decay rate (per baryon) in a small volume (V ) in the nuclear matter (n.m.) rest frame (Γ nm ) is defined by d(nV )/dτ = −Γ nm n V , in which τ is the fluid's proper time, and n is the proper baryon number density. We can define a baryon number-flux vector by j µ = u µ n [64], in which u µ is the four-velocity of the fluid (u µ u µ = 1) and use the definition of Γ nm to write j µ ;µ = −n Γ nm , in which ';' denotes the covariant derivative. We then use the relationship √ −g j µ ;µ = ( √ −g j µ ) ,µ [20], in which ',' denotes ordinary partial derivative and g ≡ det|g µν |, to arrive at ∂ ∂t √ −g n u 0 d 3 x + √ −g n u i ,i d 3 x = − d 3 x √ −g nΓ nm ,(3.B 4π ≡ ∂ ∂t R 0 1 − 2M (r) r − 1 2 n(r) r 2 dr = − e ν(r) 1 − 2M (r) r − 1 2 Γ nm (r) n(r) r 2 dr, (3.16) where B is the total baryon number of the neutron star. We have used √ −g = exp(ν(r) + λ(r)) r 2 sin θ, with exp(2λ(r)) = (1 − 2M (r)/r) −1 , and M (r) is the total mass included within radius r: M (r ′ ) = 4π r ′ 0 E(r)r 2 dr. (3.17) Given a particle physics model for BNV we can evaluate Γ nm (r) and use Eq. (3.16) to find the resultingḂ. B. Framework It was shown in Ref. [5] that the conditions in Sec Here, we ignore any possible dependence of O on the angular velocity Ω, i.e., we assume O evolves along a one-dimensional trajectory with Ω = 0 on the general two-dimensional space parameterized by E c and Ω. We can solve forĖ in terms of the rate of baryon loss,Ḃ, such thatȮ O = B O × ∂ Ec O ∂ Ec B Ḃ B ≡ −b(O) × Γ BNV ,(3.19) in which we defined the effective BNV rate Γ BNV ≡ −Ḃ/B and the dimensionless parameter b(O) encodes the relative rate of change in O with respect to Γ BNV . We pick hadronic versions of the DS(CMF) EoS [21] that includes a crust [65] from the CompOSE database [66]. The details of these EoS including their Lagrangians and particle contents are given in C. Observables Baryon loss in pulsars may lead to observable effects on their individual spin-down rate (Ṗ s ), and their orbital period lengthening (Ṗ b ) if they belong to a binary system [5]. The BNV modifications toṖ s are caused by the quasi-equilibrium changes in the moment of inertia (I), and angular momentum loss due to light particles (e.g., ϕ B ) escaping the pulsar. While the first contribution can be expressed in a model-independent manner, the latter depends on the specific BNV model and the masses of particles involved. Therefore, we focus our attention on BNV modifications toṖ b , which can still be formulated in a model independent way. The energy loss due to BNV can modify the orbital period decay rate in a binary system, assuming it is active in one or both of the components. This energy loss can be written as [5]Ṁ eff ≡ d dt M + 1 2 IΩ 2 = b(M ) Ḃ B M + b(I) Ḃ B 2π 2 I P 2 s BNV − 4π 2 IṖ s P 3 s , (3.20) in which b(M ) and b(I) are defined in Eq. (3.19), P s andṖ s are the observed pulsar spin period and its observed rate of change respectively. Note that the rates of change in I due to spin-down, (dI/dΩ)Ω, are negligible in the pulsars that we consider. The relative rate of change in a binary period due to energy loss in its components is given by [67][68][69] Ṗ b P b Ė = −2 Ṁ eff 1 +Ṁ eff 2 M 1 + M 2 , (3.21) in which 1 and 2 refer to the components of the binary system. After plugging Eq. (3.20) into (3.21), we get the following BNV and spin-down contributions to the energy-loss term Ṗ b P b BNV = −2 M 1 + M 2 i=1,2 Ḃ i B i b i (M )M i + b i (I) 2π 2 I i P 2 s,i , (3.22) Ṗ b P b Ω = 8π 2 M 1 + M 2 I 1Ṗs,1 P 3 s,1 + I 2Ṗs,2 P 3 s,2 . (3.23) We should note that the second term in Eq. D. Interpretation The dominant contributions to the observed relative rate of orbital period decay can be written as [70]: Ṗ b P b obs = Ṗ b P b GR + Ṗ b P b Ė intrinsic + Ṗ b P b ext , (3.24) in which the first term is due to gravitational radiation [71], and the third term includes extrinsic effects, e.g., due to the relative motion of a binary pulsar with respect to the solar system barycenter. The numerical values for each of these contributions and the limits oṅ PĖ b (found by subtracting the GR contribution,Ṗ GR b , from the intrinsic orbital-period decay rate,Ṗ int b ≡Ṗ obs b −Ṗ ext b ) are given in Table I for three binary systems. Two of these systems J0348+0432 and J1614−2230) have heavy pulsars that may contain hyperons [72], and the third one is a double pulsar system (J0737−3039A/B) with an extremely high precision in its orbital parameters. 1. PSR J0348+0432: A pulsar-white dwarf binary discovered in 2007 with the Robert C. Byrd Green Bank Telescope [73] with an orbital period of about 2.4 hr. We use the results from the analysis in Ref. [74], in which it was shown that the kinematic, spin-down (Eq. (3.23)), and tidal (Ṗ T b ⪅ 10 −16 ) contributions toṖ b are negligible and the observedṖ b should be mainly caused by the GW emission. We use the value from Ref. [74] for the intrinsic period decay rate,Ṗ int b = −0.275(45) × 10 −12 . 2. PSR J1614−2230: A pulsar-white dwarf binary discovered in 2006 with the Parkes radio telescope [75]. We use the Shapiro delay mass estimates from Ref. [76], and the binary parameters from NANOGrav 12.5 yr data set [77] at 56323 MJD. The observed value ofṖ obs b = 1.57(13) × 10 −12 is dominated by the Doppler shift due to the pulsar motion which is itself mainly caused by the Shklovskii effect [78]: P Shk b = µ 2 d c P b = 1.24(9) × 10 −12 , (3.25) in which we input the value for proper motion µ = 32.4(5) mas yr −1 , and used the parallax distance d = 0.65 ± 0.04 kpc [79]. We use Eq. (16) from Ref. [80] to estimate the contribution due to the Galactic potential, namely, for the period derivative,Ṗ int b = 0.32(16)×10 −12 , is positive at 2σ significance, pointing to a possible underestimation of extrinsic effects and their errors. However, we note that if, for example, we instead assume a negligible value forṖ int b ≈ 0 and double our error estimates, then we would still obtain the same limits. We also evaluate the relatively small GW contribution which for circular orbits is given by [71] Ṗ b P b Gal = − K z | sin(b)| c − Ω 2 ⊙ R ⊙ c cos(l) + β β 2 + sin 2 (l) cos(b), (3.26) in which β ≡ (d/R ⊙ ) cos(b) − cos(l), R ⊙ = 8.0(4) is the Sun's Galactocentric dis- tance, Ω ⊙ = 27.2(9) km s −1 kpc −1 is its Galactic angular velocity, K z is the vertical component of Galactic acceleration approximated by K z 10 −9 cm s −2 ≈ 2.27z kpc + 3.68(1 − exp(−4.31 z kpc )),(3.P GW b = − 192π 5 2πT ⊙ P b 5/3 M p M c (M p + M c ) (1/3) = −4.17(4) × 10 −16 ,(3.28) in which we used the pulsar and white dwarf masses from Ref. [76], T ⊙ = 4.92549094× 10 −6 s, and we neglected the small eccentricity of the orbit e = 1.333(8) × 10 −6 [81]. In estimatingṖΩ b using Eq. (3.23) we assumed the canonical value I = 10 45 g cm 2 for the pulsar's moment of inertia. 3. PSR J0737−3039A/B: A double pulsar discovered in 2003 [82], comprised of two radio pulsars (A and B) with pulse periods of 22.7 ms and 2.8 ms, respectively. We use the data from Ref. [83] and the inferred limits on BNV contributions from Ref. [5]. We can now translate the bounds on (Ṗ b /P b ) BNV from Table I to limits on (Ḃ/B) using Eq. (3.22), which are presented in the last row of Table I. In deriving these limits, we assumed that BNV is only active in the pulsars. We also note that we can only infer a model-independent limit on a linear combination of BNV in pulsars A and B of the double pulsar system (J0737−3039A/B). However, we expect that the rates of BNV (per baryon) would be about the same in both pulsars, i.e., ( In Sec. 6, in which we adopt a specific BNV model (B → χγ), our inferred limits on the mixing parameter (ε Bχ ) are found by evaluating the individual BNV rates in each of the two pulsars J0737−3039A and J0737−3039B, which we then sum to compare to the observational limit on BNV in this system. We also observe that changing between the DS(CMF) EoSs (see Table II) induces variation in, at most, the last significant digit in our limits (see the discussion below Eq. (3.23)). P ext b (10 −12 ) 1.6(3) × 10 −3 1.25(10) −1.68(+11, −10) × 10 −4 P int b (10 −12 ) −0.275(45) 0.32(16) −1.247 752(79) P GR b (10 −12 ) −0.258(+8, −11) −4.17(4) × 10 −4 −1.247 827(+6, −7) (Ṗ b P b )Ė 2σ (yr −1 ) 2.7 × 10 −10 2.7 × 10 −11 8.3 × 10 −13 (Ṗ b P b )Ω (yr −1 ) < 1.4 × 10 −13 ≈ 4.2 × 10 −15 1.04(7) × 10 −13 (Ṗ b P b ) BNV 2σ (yr −1 ) 2.7 × 10 −10 2.7 × 10 −11 7.3 × 10 −13 (Ḃ B ) BNV 2σ (yr −1 ) 1.8 × 10 −10 2.0 × 10 −11 4.0 × 10 −13Ḃ 1 /B 1 ) ≈ (Ḃ 2 /B 2 ), DENSE MATTER CONSIDERATIONS FOR PARTICLE PROCESSES Different lines of evidence reveal that dense matter environments can be discriminating probes of non-SM processes. For example, limits on Λ → χγ, as well as other decay channels with dark particles, follow by noting that the duration of the observed neutrino pulse in SN 1987A should not be significantly impacted by dark sector emission [16]. We, too, have found severe limits on BNV from binary pulsar period lengthening, as shown in Table I. Here we sharpen such studies by computing particle processes within a theoretical framework suitable to the description of the dense matter in the interior of a neutron star. To compute particle processes in dense matter we might first turn to chiral effective theory to describe the low-energy interactions of such hadrons [84,85]. At the simplest level, these studies exploit the symmetries of QCD to systematize the interactions of mesons and baryons in a momentum expansion in powers of (Q/Λ χ ), in which Q is the momentum or pion mass and Λ χ is the chiral-symmetry breaking scale (Λ χ ≈ 1 GeV), with experiments fixing the value of the unknown low-energy constants (LECs) that appear. This framework can also be extended to the determination of the EoS of neutron stars [86,87]. The empirical nature of the LEC determinations limit the applicability of chiral effective theory to densities no more than 2n sat [88]. Moreover, in neutron stars, the central densities can easily exceed that of saturation density by a factor of a few, making the nucleons relativistic. As a result, we turn to relativistic mean-field (RMF) theory in hadronic degrees of freedom to describe the dense matter at the core of a neutron star. In what follows we first describe how a RMF treatment emerges from a simple, covariant quantum field theory description of hadronic interactions before describing the specific chiral mean-field (CMF) EoS that we employ for generating our numerical results, showing how this specific choice maps onto the RMF treatment of the simpler model. We then show how particle decays can be computed within that framework. A. Modelling Dense Matter A prototypical choice is the Walecka model [17][18][19], namely, L φ/V =ψ[(i / ∂ − g V / V ) − (m N − g s φ)]ψ + 1 2 (∂ µ φ∂ µ φ − m 2 s φ 2 ) − 1 4 F µν F µν + 1 2 m v V µ V µ + δL ,(4.1) where F µν = ∂ µ V ν − ∂ ν V µ and a counterterm δL, as the model is renormalizable. It is similar to massive QED with a scalar extension and a conserved current (baryon number). Both a neutral scalar meson (φ) and a neutral vector meson (V µ ), describing the attractive and repulsive features, respectively, of the nucleon-nucleon force appear. The equations of motions (EoMs) take the form / ∂ 2 + m 2 s φ(x) = g sψ ψ (4.2) ∂ ν F νµ + m 2 v V µ = g vψ γ µ ψ (4.3) i / ∂ − g v / V (x) − [m N − g s φ(x)] ψ(x) = 0. (4.4) The EoMs are nonlinear and thus complicated. Working in the mean field limit is grossly simplifying, however. That is, at high baryon number densities, the sources for φ(x) and V µ (x) fields become large, and these field operators can be replaced by their vacuum expec- tation values (VEV) in the n.m. frame: φ(x) → ⟨φ(x)⟩ ≡ φ, and V µ (x) → ⟨V µ (x)⟩ ≡ δ µ0 V 0 . In doing this, we assume rotational invariance and note that in static uniform matter, as in a neutron star, φ and V 0 become constants that only depend on density. The solutions to Eq. (4.4) would take the form of that of the free Dirac equation if the replacements k µ → k * µ ≡ k µ − g v V µ and m → m * ≡ m − g s φ 0 are made. In other words, the medium effects in the RMF limit are captured by a shift in the baryon momenta and masses. In generalizing this result for broader use, we note that the Lagrangian of interactions for a more realistic hadronic model would have more ingredients (e.g., mesons). However, we would still be able to add up the scalar meson VEVs that modify the baryon's mass in a similar manner and denote the effective baryon mass by m * , independent of the specific scalar mesons in our model. Similarly, we can combine all the contributions to the baryon's momentum from vector mesons and denote them by Σ µ , such that in going from the vacuum to the in-medium formalism we would replace k µ → k * µ ≡ k µ − Σ µ . Equipped with this result, we can write the wave-function for a baryon with (canonical) four-momentum k µ (in a uniform medium) as ψ(x) = e −ik·x u(k * , λ), (4.5) in which k * µ ≡ k µ − Σ µ = E * (k * ), ⃗ k − ⃗ Σ is defined to be the kinetic four-momentum and the vector self-energy (Σ µ ) is generated by the vector meson VEVs, with ⃗ Σ = 0 in the n.m. frame. The time-component of k * µ is defined by E * (k * ) ≡ m * 2 + | ⃗ k * | 2 , in which m * is generated by the scalar meson VEVs. The baryon spinor u(k * , λ) satisfies the Dirac equation / k * − m * u(k * , λ) = 0, (4.6) which has the following solution in Dirac-Pauli representation u(k * , λ) = E * (k * ) + m *   1 ⃗ σ· ⃗ k * E * (k * )+m *   χ λ ,(4.7) in which ⃗ σ contains the Pauli matrices, and χ λ is the Pauli spinor with χ ↑ = (1, 0) T and χ ↓ = (0, 1) T . Note that u has a Lorentz-invariant normalization given by u(k * , λ)u(k * , λ) = 2m * . The wave-function for antibaryons can be similarly constructed. The energy spectrum of baryons (k 0 ) is given by E(k) = m * 2 + | ⃗ k − ⃗ Σ| 2 + Σ 0 ; (4.8) in the mean-field approximation, Σ µ and m * do not depend on k µ but they do vary with density. The values for m * and Σ 0 (in the n.m. frame) decrease and increase respectively (see Fig. 6) in such a way that the total energy of baryons in Eq. (4.8) increases at higher densities. As we will see shortly, this brings about in-medium baryon decays to particles that are heavier than the baryon's vacuum mass since E(0) > m B at high densities. In general, the increase in the repulsion between baryons in a RMF framework can be understood by comparing the time-like component of vector (repulsive) interactions, which are proportional to u † u, with scalar (attractive) interactions, which are parameterized by uu = (m * /E * )u † u. As the density increases, m * decreases and the strength of the attractive forces relative to the repulsive ones diminishes [89]. However, we should note that having a highly repulsive nuclear interaction at extremely high densities (compared to n sat ) is a reasonable expectation, regardless of the specific dense matter formalism. Having explained the formalism utilized in this work, we now describe the specific EoS that we use for generating our numerical results. We choose an EoS based on a non-linear hadronic SU(3) CMF model [90], in which the baryonic degrees of freedom include nucleons (n, p), hyperons (Λ, Σ, Ξ) and the spin-3/2 resonances (∆, Σ * , Ξ * , Ω). These baryons interact via exchange of scalar (σ, δ, ζ, χ) and vector mesons (ρ µ , ω µ , ϕ µ ), in which ρ µ and δ are both isovectors. In the RMF limit, the mesons become classical fields, and in the n.m. frame only the zeroth components of vector mesons develop VEV. The Lagrangian density of the CMF model is given by [21] L = L Kin + L Int + L Self + L SB , (4.9) in which L Kin contains the usual kinetic terms for baryons and leptons, L Int is due to the baryon-meson interactions which are given by L Int = − i ψ i g iω γ 0 ⟨ω 0 ⟩ + g iϕ γ 0 ⟨ϕ 0 ⟩ + 2g iρ γ 0 I 3i ⟨ρ 0 3 ⟩ + m * i ψ i . (4.10) We note ψ i denotes a baryon of species i with an effective mass m * i and an isospin 3component I 3i , and the expectation value is evaluated in the ground state. The last two terms in Eq. (4.9), i.e., L Self and L SB , contain the self-interactions of scalar and vector mesons and explicit chiral symmetry breaking terms respectively. The explicit expressions are given in Eqs. (3), (4), and (5) of Ref. [21]. The baryon effective masses are generated by the scalar meson VEVs, except for a small explicit mass term δm i ∼ 150 MeV, and are given by m * i = g iσ ⟨σ⟩ + 2g iδ ⟨δ 3 ⟩I 3i + g iζ ⟨ζ⟩ + δm i , (4.11) in which δ 3 is the isospin 3-component of δ. The time-component of baryon self-energy is given by Σ 0 i = g iω ⟨ω 0 ⟩ + g iρ ⟨ρ 0 3 ⟩I 3i + g iϕ ⟨ϕ 0 ⟩. (4.12) The numerical values for m * and Σ 0 are plotted in Fig. 6. We note that the reduction of the effective baryon masses at high densities as shown in Fig. 6 is due to chiral symmetry restoration at high densities. The coupling constants are chosen [91][92][93] This conventional approach in determining the coupling constants in RMF models relies on an extrapolation from symmetric finite nuclei to infinite neutron matter. We would like to contrast this with an alternative that we may wish to employ in the future, which is based on fitting uniform pure neutron matter properties determined through the use of chiral effective field theory [94]. The latter procedure involves fitting the RMF couplings with the synthetic neutron matter data generated using Quantum Monte Carlo (QMC) many-body methods [95], in addition to reproducing n sat , B/A, and K. , and delta resonances (∆); and the additional vector interactions ("Add. Int.") beyond the standard terms (L Self ) that are included for each EoS respectively. The fourth column represents the assumed value for symmetry energy (E sym ) slope (L). The fifth to eighth columns are the single-particle hyperon potentials, and the last column is the maximum neutron star mass (M max ) that can be generated. EoS DoF Add. Int. L (MeV) U Λ (MeV) U Σ (MeV) U Ξ (MeV) U ∆ (MeV) M max (M ⊙ ) 1 N+Y - 88 −28 5 −18 - 2.07 2 N - 88 −28 5 −18 - 2.13 3 N+Y ωρ 75 −28 5 −18 - 2.00 4 N ωρ 75 −28 5 −18 - 2.05 5 N+Y ωρ + ω 4 75 −27 6 −17 - 2.07 6 N ωρ + ω 4 75 −27 6 −17 - 2.11 7 N+Y+∆ ωρ + ω 4 75 −27 6 −17 −64 2.07 8 N+∆ ωρ + ω 4 75 −27 6 −17 −64 2.09 Our chosen class of EoS has a set of variations that depend on the degrees of freedom that are included, and they are given in Table II In this section, we discuss some of the notable features that emerge in studying processes in the medium, and make comparisons with the vacuum formalism. We start with the quantization of baryon fields in the medium followed by the rate and cross section calculation formalism. We then discuss the electromagnetic form factors of the baryons that are needed The presence of the baryon Fermi sea modifies the quantization procedure of the baryon fields, ψ(x), in medium [18] compared with the usual procedure in vacuum [99]. Once the coefficients behind Fourier modes of ψ(x) are promoted to baryon creation (a † (k)) and annihilation (a(k)) operators (likewise b † (k) and b(k) for antibaryons), we conclude that the action of these operators on the medium ground state |Ω⟩, which contains baryon levels filled to a Fermi momentum (k F ), should be given by b(k)|Ω⟩ =0 ∀ ⃗ k, a † (k)|Ω⟩ =0 | ⃗ k| < k F , a(k)|Ω⟩ =0 | ⃗ k| > k F . (4.13) This leads to a different form (compared to vacuum) for the baryon propagator which is given by [18] G(p) ≡ / p * + m * 1 p * 2 − m * 2 + iε + 2πiδ(p * 2 − m * 2 )θ(p * 0 ) θ k 2 F + p * 2 − p * µ B µ 2 B µ B µ , (4.14) in which θ is the Heaviside step function, B µ is the baryon current density, which in the n.m. frame is given by B µ nm = δ µ0 n B , and the second term in Eq. (4.14) allows for the propagation of holes in the Fermi sea. Using this modified propagator and the spinors in Eq. (4.7), one can derive Feynman rules [18] for calculating the amplitudes for various processes (see Sec. 5 B). However, in calculating rates via phase space integrals, we should first observe that an on-shell (p * 2 = m * 2 ) and positive energy (p 0 > 0) Lorentz-invariant integral over the four-momentum is given by d 4 p δ (p µ − Σ µ ) 2 − m * 2 θ(p 0 )f (p µ ) = d 3 p f |⃗ p − ⃗ Σ| 2 + m * 2 + Σ 0 , ⃗ p 2 |⃗ p − ⃗ Σ| 2 + m * 2 . (4.15) Therefore, we identify the Lorentz-invariant (on-shell) volume element in the medium as d 3 p/2E * (p) . This means that the normalization factors in the in-medium phase space integrals should contain (2E * ) −1 in place of the usual vacuum expression. We also note that the velocity of a baryon is defined in terms of the kinetic momentum as opposed to the canonical one, i.e., v µ ≡ k * µ /E * . This velocity should be used for calculating the cross section of two-body scattering involving a baryon (see App. C). We can explicitly show this by performing an integration over the longitudinal (ẑ) components of the incident beams' momenta (k z A andk z B ). Let us assume for the moment that only one baryon (B) is involved, in which case we have (see Eq. (4.77) of Ref. [100]) 16) in which in the last line we are assumingk z B = p z f −k z A and have identified the baryon velocity using the kinetic momentum, such that |v A − v B | is the relative velocity of the beams as viewed from the laboratory frame. The generalization to the case with two baryons is straightforward. The fact that the velocity of a baryon is zero when ⃗ k * = 0 could have also been deduced by inspecting the kinetic energy component in Eq. (4.8). For this reason, the frame in which ⃗ k * = 0 holds is called the center of velocity (c.v.) frame which is distinct from the center of mass (c.m.) frame defined by ⃗ k = 0. Therefore, the decay rate of a baryon in an arbitrary frame (Γ) is found by boosting (γ) the rate evaluated in the c.v. frame using dk z A dk z B δ k z A +k z B − f p z f δ Ē A +Ē B − f E f = dk z A δ k2 A + m 2 A + k B − ⃗ Σ B 2 + m 2 B + Σ 0 B − f E f kz B = p z f −k z A = k z Ā E A −k z B − Σ z B E * B −1 = k z Ā E A −k * z B E * B −1 ≡ |v A − v B | −1 ,(4.Γ = γ −1 Γ c.v. . Since we study processes that involve electromagnetic (EM) interactions with baryons, the generalization of EM form factors from the vacuum to within the medium should be checked. The in-medium spinors in Eq. (4.7) are different from their vacuum counterparts. Therefore, certain commonly used properties (e.g., Gordon decomposition) in vacuum need to be reestablished. However, we note that the general form of these interactions is determined by the structure of Dirac algebra. While important for formulating our analyses, this is slightly tangential to the broader narrative of this work; we thus relegate the details to the Appendices, but we encourage the reader to study them nonetheless. In App. A, we explicitly show that the vacuum EM vertex form can be generalized to its in-medium form if one replaces m → m * , p → p * , and identifies the electric charge and magnetic moment of a baryon from the scattering amplitudes in the c.v. frame. Our numerical results in Sec. 5 assume the vacuum values for the in-medium form factors F * 1,2 of neutron and Λ. We also derive the non-relativistic limit of baryon's EM interactions and their elastic scattering formalism in App. B. We present the calculations for in-medium Compton scattering in App. C, as a demonstration of the RMF formalism utilized in this work. BARYON DARK DECAY RATES IN DENSE MATTER In this section, we develop the procedures for evaluating particle physics processes, such as neutron decays and neutron-neutron scattering, in the neutron-star medium. Our particular interest is in radiative decays such as B → χγ in the core of the star. In the absence of a matter environment, a common procedure, adopted in many contexts, is to assume the mixing is weak and to redefine the fields, here B i and χ [101], so that they no longer mix, and then to analyze B i → χ transitions in that new basis. In Sec. 5 A, we show why and how this procedure can fail in strongly interacting dense matter, and we argue for a Feynman diagram analysis in its place. Subsequently, starting in Sec. 5 B, we show how the transition rates can be evaluated explicitly and consider their implications. A. General Considerations To illuminate the essential points, we consider the possibility of n-χ mixing in a background field Σ µ , the vector self-energy of a neutron in the neutron-star medium, which interacts with the neutron field ψ n , but not the χ field ψ χ . Thus we adopt the following simple model: L = ψ n i / ∂ − / Σ − m * n ψ n + ψ χ i / ∂ − m χ ψ χ − ε ψ n ψ χ + ψ χ ψ n . (5.1) Under a field redefinition, ψ → ψ ′ , prescribed by   ψ ′ n ψ ′ χ   =   cos θ sin θ − sin θ cos θ     ψ n ψ χ   ,(5.2) Eq. (5.1) becomes L ′ =ψ ′ n i / ∂ − / Σ cos 2 θ − m * n cos 2 θ − m χ sin 2 θ ψ ′ n + ψ ′ χ i / ∂ − / Σ sin 2 θ − m χ cos 2 θ − m n sin 2 θ ψ ′ χ + ψ ′ n sin(2θ) 2 m * n − m χ + / Σ − ε cos(2θ) ψ ′ χ . (5.3) If Σ µ were absent, and with ε real, then for tan (2θ) = 2ε/(m * n − m χ ), L ′ describes two decoupled fields with a modified energy spectrum. These fields can then map to the asymptotic ("in" and "out") states needed to define the S-matrix [102]. To do this, any interactions with these fields should vanish as t → ±∞. For the neutron (and other SM baryons), we note that the effect of the vector self-energy can be absorbed into the definition of a modified single-particle spinor, as discussed in Sec. 4 A, and thus suitable "in" and "out" states can still be constructed. In the current case, Σ µ mediates an interaction between the rotated n and χ fields, putting the utility of our field redefinition procedure into question. After all, even in the mean-field limit, Σ 0 can greatly exceed the n and χ masses at the high densities reached within a neutron star, and it cannot vanish as t → ±∞, since we work within a medium of infinite extent. Since Σ µ is not a Lorentz scalar, we cannot extend our field redefinition approach to include it. Therefore, there would seem to be no advantage to following a field redefinition approach in neutron matter. Moreover, in the small mixing limit (ε ≪ |m * n − m χ |), the mass (n ′ , χ ′ ) and interaction (n, χ) eigenstates are nearly the same. Working with Eq. (5.1), we can treat εψ n ψ χ as a tiny interaction that mediates n ↔ χ transitions within perturbation theory. This Feynman diagram analysis, through the in-medium baryon propagator, Eq. (4.14), naturally includes the impact of momentum dependence and of the neutron self-energy on n-χ mixing. We emphasize that both effects are absent in the field redefinition procedure. As a result, too, we do not have large enhancements in our predictions should the in-medium neutron and χ states become degenerate in energy -the imaginary part of the neutron self-energy effectively eliminates that possibility. Nevertheless, n-χ mixing within the neutron-star medium could potentially lead to effects not possible in terrestrial experiments, and we consider those possibilities more carefully in Sec. 5 C. B. Dark Decay Rate Estimates We now turn to the explicit evaluation of rates of particle processes within the neutronstar medium, with a particular focus on dark decay rates. As long known, the background field associated with matter leads to a spontaneously breaking of Lorentz symmetry, but as a consequence of our Lorentz covariant description, discussed in Sec. 4 A, our expressions always have definite Lorentz transformation properties. In what follows, we exploit our freedom to choose a frame to simplify our analysis. Generally, processes of the form B+{X} → χ+{Y } lead to the following rate of change of the local baryon density n B (with respect to the proper time, τ , referenced to that spacetime point): dn B dτ = − dΠ B {X} dΠ X dΠ χ {Y } dΠ Y (5.4) × f B (⃗ p B ) {X} f X (⃗ p X ) 1 − f χ ( ⃗ k χ ) {Y } 1 ± f Y ( ⃗ k Y ) × |M| 2 × (2π) 4 δ (4) p B + {X} p X − k χ − {y} k Y , where {X} ({Y }) is some set of other states in the initial (final) state -which may be empty. Moreover, dΠ i = d 3 ⃗ p i /[(2π) 3 (2E * i )] is the Lorentz-invariant phase space measure, f (⃗ p) are the species-dependent occupation numbers 4 , and |M| 2 is the spin-summed (as opposed to spin-averaged ) squared matrix element. We denote final-state momenta with k i instead of p i . Consistent with our assumption that there is no appreciable background of χ, we set its occupation factor f χ ( ⃗ k χ ) to zero. All baryonic species abide by zero-temperature Fermi distributions characterized by distinct Fermi momenta p F,B . We briefly discuss important qualitative features of the evaluation of Eq. (5.4) for the decay process B → χγ and present the corresponding results. We relegate details of the calculation to App. E. We work in the interaction basis, so that the decay proceeds via the Feynman diagram containing the n − χ interaction and the baryon magnetic dipole moment operator, which we write as O Bγ = g B e 8m * B Bσ µν BF µν ,(5.5) noting g n = 3.826 and g Λ = −1.226 [50]. This computation is made in a background meanfield of neutron matter, and the associated decay amplitude, as developed in Sec. 4 A, is determined by replacing the canonical momenta of the in-vacuum computation with kinetic momenta as per Eq. (4.5). Labeling canonical momenta as B(p B ) → χ(k χ ) + γ(k γ ), the corresponding spin-summed squared matrix element is |M| 2 = ε 2 Bχ g 2 B e 2 2(m * B ) 2 [(p * B · k χ ) + m * B m χ ] ,(5.dn B dτ = − d 3 ⃗ p B (2π) 3 (2E * B ) d 3 ⃗ k χ (2π) 3 (2E χ ) d 3 ⃗ k γ (2π) 3 (2E γ ) f B (⃗ p B ) × |M| 2 × (2π) 4 δ (4) (p B − k χ − k γ ) ,(5.|) = 1 2m * B d 3 ⃗ k χ (2π) 3 (2E χ ) d 3 ⃗ k γ (2π) 3 (2E γ ) × 1 2 |M| 2 × (2π) 4 δ (4) (p B − k χ − k γ ) , (5.8) where the argument of Γ c.v. follows from our earlier frame choice. 5 Henceforth we abbreviate p B ≡ |⃗ p (n.m.) B |. We have p * ,(c.v.) B = (m * B , 0) and p (c.v.) B = (m * n + Σ (c.v.),0 B , ⃗ Σ (c.v.) B ), with Σ (c.v.dn B dτ = −2 × p F,B 0 p 2 B dp B 2π 2 γ −1 Γ c.v. (p B ) . (5.10) The prefactor of 2 comes from the baryon's two spin degrees of freedom, and the factor of we arrive at the following result: dn B dτ = − ε 2 Bχ g 2 B e 2 128π 3 (m * B ) 2 x F 1 dx √ x 2 − 1 × 1 + 2xσ + σ 2 − µ 2 (1 + 2xσ + σ 2 ) 2 × (1 + 2xσ + σ 2 )(1 + xσ + 2µ) + µ 2 (1 + xσ) ,(5.11) in which This is a simple consequence of larger neutron number fractions at these densities, and the two rates often differ by several orders of magnitude. However, Λs have a further reach in m χ when they are present than neutrons do, owing to the larger total energy of Λs in neutron matter. 6 Of course, the EoS that do not contain hyperons will not lead to Λ → χγ decays within neutron stars. MeV. Note that the color scales are different between the two panels. See text for additional details. x ≡ E * ,(n.m.) B m * B , x F ≡ E * ,(n.m.) F,B m * B , σ ≡ Σ (n.m.),0 B m * B , µ ≡ m χ m * B . C. Medium-Enabled Dark Decay Processes It was shown in Sec. 4 A that baryons in neutron stars have a lower effective mass (m * B ) and a higher self-energy (Σ 0 B ) at higher densities (see Fig. 6), but their overall energy can be much higher than their vacuum rest mass (m B ). In order to illustrate this for a heavy neutron star, we plot the baryon rest-energies (E 0 B ≡ E B (p = 0) in the n.m. frame) for PSR J0348+0432 as a function of radius in Fig. 9. We can see that baryon decays containing a final state χ with m χ > m B , which would be forbidden in vacuum, can occur at the core of heavy neutron stars. This enables a novel way of analyzing models with m χ values for which nuclear and vacuum decays are kinematically forbidden. Furthermore, constraints derived from heavy neutron stars can still be applicable in the vicinity of m χ ≈ m B and beyond that. This should be contrasted with limits derived from processes in vacuum and within nuclei, which diminish at m χ ≈ m B or even at much lower values of m χ due to the binding energy and possible energy cuts on the final states. For example, when inferring limits from n → χγ via detection of γ there is an energy cut E min γ [103], which means m χ values larger than m n − E min γ cannot be constrained. Spontaneous B → χ Conversion The existence of χ raises the possibility that the baryons to which they couple might undergo spontaneous conversion to χ in the neutron-star medium as they propagate. Such an effect could prove loosely analogous to empirically observed matter-enhanced neutrino oscillations [104] or to the possibility of neutron-antineutron oscillations [105][106][107], breaking baryon number by two units. In the latter case the presence of external interactions from matter or magnetic fields modify the energy of the n andn differently, severely reducing the spontaneous oscillation probability for a fixed source of new physics [108], and the crosssection for scattering-mediated n-n conversion is also very small [109]. In this section, we note the distinct features of B-χ conversion. The essential physics is thus: B and χ constitute a two-level quantum system. As we have noted in Sec. 5 A, if the coupling ε Bχ is nonzero, then B and χ constitute the interaction basis, whereas the eigenstates of this Hamiltonian, which we term f 1 and f 2 for this discussion, constitute the mass basis. Formally, the strong interactions that operate in neutron matter only ever produce n -this is what it means for B to be an interaction eigenstate. This B is, however, a coherent superposition of f 1 and f 2 at the moment of its creation. The subsequent evolution of this coherent wavepacket depends on the details of the B −χ system. These details are discussed in depth in App. D; we pick out the most relevant results as they pertain to this discussion. The Hamiltonian that describes our two-state system depends on the local environment: the total energy of the baryon depends on the density through m * B and Σ B , and baryons with different n.m.-frame momenta will mix differently with χ because Lorentz invariance is spontaneously broken by the background. There exists a resonance in this system wherever the condition, which follows from energy-momentum conservation of the canonical momenta, is satisfied. We expect that this condition will occur for at most one value of the (magnitude of the) baryon momentum for a given density. Moreover, Eq. up to O(ε 2 Bχ ) corrections. If the system is far from resonance, then these eigenvalues are well separated. As a result, the B states produced in scattering processes will essentially immediately decohere into their component f 1 and f 2 with, respectively, probabilities of cos 2 θ and sin 2 θ. As such, the state that emerges from the scattering process manifests as either f 1 with probability cos 2 θ ∼ 1 or f 2 with probability sin 2 θ ∼ (ε Bχ /δω) 2 , and the latter may be vanishingly small -and thus so would be any yield in χ. This means that when B is produced in some strong interaction, the wavepacket containing f 1 and f 2 may remain coherent over relatively long timescales. This is analogous to how neutrino mass eigenstates remain coherent as they propagate in terrestrial oscillation experiments, despite being formed in an interaction eigenstate. 7 As in the case of neutrino oscillations, the f 1 and f 2 components of the B state generically evolve with different phases; over time, this leads to nonzero overlap between the evolved state and either B or χ. The state is then measured, in a sense, at its next interaction some time t later, either by its environment or by some experimental apparatus. It is appropriate, in this case, to invoke the concept of an oscillation probability; this is estimated by P B→χ (t) = sin 2 2θ × sin 2 (∆ω * )t 2 . (5.15) When the state is observed, however, it collapses to the combination of f 1 and f 2 appropriate to either B or χ with probabilities given by Eq. (5.15), and the process repeats for further interactions. While the oscillations have a large amplitude (sin 2 2θ ∼ O(1)) in this regime, the probability to convert will remain small if the time between successive measurements δt meas is small, in the sense (∆ω * )(δt meas ) ≪ 1. This is precisely the quantum Zeno effect [112,113]. It remains to determine the timescale of the interactions in the nuclear medium in order to estimate the rate of B → χ conversions. We estimate this to be the light time of the mean interparticle separation around nuclear saturation density: δt strong ∼ n One might expect that this would multiply the large density of baryons to yield a macroscopically relevant rate. However, the near-resonance region occupies a thin shell (parametrically of width ∼ ε Bχ ) within the baryon Fermi sphere; the fraction of baryons relevant for this phenomenon is fantastically small, even in the best case scenario. Thus we summarize by emphasizing that we do not expect B − χ conversion to be a phenomenologically relevant mechanism for the production of χ. D. Total Rates In this section, we report the total baryon decay rates that emerge after integrating our earlier results over the structure of a neutron star with a given central density, n c . For example, in Fig. 10, we show the rates that result from integrating the local BNV rates in INFERRED LIMITS ON BARYON DARK DECAYS We now turn to the task of assessing the limits on the B − χ mixing parameters that emerge from our numerical assessment of the stellar-volume-integrated baryon dark decay rates, as shown in Fig. 11, and the macroscopic baryon number loss limits we have determined from astrophysical observations and their analysis. The latter, namely, are limits on anomalous binary-pulsar period lengthening, to which we refer as "binary spin-down," and they are given in Table I. We show the limits we find for each astrophysical system as well as that associated with a final combined limit. To make our presentation more compact, we first discuss how the individual limits on ε Bχ can be combined before showing all of these results. Note, too, that since our constraint depends on the square of ε Bχ that its sign is left unconstrained -we choose ε Bγ > 0 in reporting our limits. Combining Individual Limits Here we briefly describe our statistical procedure for combining limits on ε Bχ derived from different pulsar binary systems. The limits we show have implicitly been determined as contours of constant χ 2 (m χ , ε Bχ ). Our assumed-true hypothesis is that rate of BNVinduced binary spin-down vanishes in these systems, so we have χ 2 = 0 for ε Bχ = 0. As such, each χ 2 function is generically of the form χ 2 (m χ , ε Bχ ) = Ṗ b P b (m χ , ε 0 ) 2 (ε Bχ /ε 0 ) 4 σ 2 ≡ F (m χ ) × ε 4 Bχ . (6.1) The first equality follows from the fact thatṖ b /P b ∝ ε 2 Bχ , noting Eq. (3.22), and we emphasize that F is a function of m χ only. The limits we have shown correspond to χ 2 = c; 8 we call the resulting curve ε(m χ ). From this, we determine F (m χ ) = c ε(m χ ) 4 ; (6.2) this allows to determine the χ 2 function over the entire parameter space. The combined limit, then, corresponds to the contour along which the sums of the individual χ 2 functions also equals c. Using the definitions above, we determine the combined limit ε comb (m χ ) as follows: χ 2 comb (m χ , ε Bχ ) = i F i (m χ )ε 4 Bχ = i c ε Bχ ε i (m χ ) 4 (6.3) χ 2 comb m χ , ε comb (m χ ) = c =⇒ ε comb (m χ ) = i ε i (m χ ) −4 −1/4 (6.4) This discussion has assumed that all ε i are defined at the same level c, and that the desired combined limit is also at c. This result can be generalized for distinct individual significances c i and combined significance C: ε comb = i c i /C × ε i (m χ ) −4 −1/4 . (6.5) We show our individual pulsar limits as well as our combined limits, realized via our described procedure, for the DS(CMF)-1 EoS in Fig. 12. show the constraint derived for this maximal neutron star. This constraint has been shown in dot-dashing to indicate that it is qualitatively different from the others. We underscore that we have fixed the masses of these neutron stars to their best-fit values to construct these limits. A more statistically complete analysis would propagate the We also show constraints from KamLAND [114], SuperKamiokande [115], and BESIII [116]. light blue. These are as much as twenty orders of magnitude stronger than the constraints we have derived, but we note that these are only operative up to m χ = 920 and 827 MeV, respectively. This is a result of experimental cuts -heavier χs result in less energetic photons in the decay, and eventually these become too soft to be meaningfully detected. We emphasize, in particular, that these experiments cannot probe the region m χ > m n ; while they are more powerful when they are operative, they are fundamentally constrained in ways that astrophysical probes of new physics are not. For Λs, we show the constraint on invisible decays from BESIII [116] in dark cyan. In this case, we find the opposite result: pulsar binaries are able to probe this branching ratio as much as twenty orders of magnitude more severely than laboratory constraints! The caveat is that this requires hyperons to appear in neutron stars, which is still a matter of debate, simply because EoSs without hyperons exist that confront current observational data successfully. However, if hyperons appear in an appreciable amount in these objects, then one can expect vast improvements on laboratory searches. The upper panel of Fig. 14 is incomplete in that there are additional constraints around m χ ≈ m n , a region that has become of interest in recent years as a result of tests of newphysics explanations [10] of the neutron lifetime anomaly [12]. We examine this region more closely in Fig. 15; panel (a) casts these searches in terms of constraints on ε nχ , while panel (b) casts them in terms of constraints on Br(n → χγ). We show in blue the estimated constraint from a direct search for n → χγ using ultra-cold neutrons (UCN) [103], and in green we show a constraint from Borexino from searches for hydrogen decay, both from Ref. [34]. We also show the curve along which the free hydrogen lifetime is supposed to be τ H = 10 32 s in dashed gold, also from Ref. [34]. (The constraints from Ref. [34] are reported at 90% CL, though the differences between those and limits at 2σ should be very small given the ranges shown in the figure.) Clearly, neutron stars are more sensitive to these decays than these (would-be) laboratory constraints by many orders of magnitude. It was noted in Ref. [10] that the existence of χ can destabilize nuclear matter, including 9 Be. This constraint was calculated more precisely in Ref. [49], assuming that the lifetime of 9 Be is longer than 3 × 10 9 years to account for the presence of 9 Be in old, metal-poor stars [117]. This constraint is shown in red in Fig. 15 and is competitive with (if not dominant to) our neutron star constraints in the region of its operation, m χ < 937.993 MeV. We note that other probes of dark decays of nuclei with low neutron separation energies have been discussed in, e.g., Ref. [47]. Particular attention has been paid to decays of 11 Be, with experimental efforts underway at CERN-ISOLDE [118] and ISAC-TRIUMF [119], though we are unaware of any efforts to interpret these experimental results as constraints on new physics. As a side note, it is curious that there are no laboratory constraints, as far as we can tell, on the lifetime of 9 Be. We find the arguments about the presence of 9 Be in old stars compelling and agree that this is a valid constraint, but we are surprised, frankly, that the lifetime is only constrained at the billion-year scale. While experimentalists of yore would have had little reason to interrogate the stability of 9 Be -or indeed, any species thought to be stable in the SM -we regard the observation that the stability of these systems has not been tested in a detailed way in the laboratory as a potentially promising avenue for constraining new physics. We conclude by noting that Ref. [49] has also presented constraints on n → χγ from cosmology and from neutron star cooling. The former is a combination of constraints coming from modifications to Big Bang Nucleosynthesis (BBN) and the Cosmic Microwave Background (CMB); this treatment includes the reverse decay χ → nγ when m χ > m n , and so constrains the region shown. However, in their calculations, χ is assumed to constitute (at least some of) the dark matter. This is unlike our framework, in which we introduced more new states (ξ and ϕ B ) to prevent overaccumulation of χ. Therefore, the limits they derive from BBN and CMB do not apply here, though we agree that this would be an interesting and important avenue to explore. The neutron star cooling constraint derived there, however, makes very rough assumptions about how heat from decays is deposited into the neutron star, with the implicit assumption that increases in the temperature of the core of the neutron star lead to commensurate increases in the observed effective temperature. Yet thermal transport and cooling in neutron stars demands careful investigation; for instance, BNV decays lead to β-disequilibrium, which leads to neutrino cooling via (direct and modified) Urca processes, which impact how the energy released in the decays is deposited back into the SM fluid. While we agree that while old, cold neutron stars should constrain this model, the details are intricate and expected to be sufficiently impactful that we decline to include such constraints here. IMPLICATIONS FOR MODELS OF BARYOGENESIS AND DARK MATTER The prospect of explaining the origins of both the dark matter abundance and the cosmic baryon asymmetry within a single dynamical framework is a beguiling one. Different possibilities have existed for some time, and many share a common feature: there is a dark-sector baryon that carries baryon number and into which SM baryons can decay. A particularly intriguing variant is that of B mesogenesis [14,29,120]. It proceeds in the early universe thesis. Finally, it is an example of a testable mechanism of baryogenesis [25], in that its essential features are subject to direct experimental investigation. Particularly, its reliance on the SM mechanism of CP violation (albeit new CPV sources could enter) implies that the branching ratios of B mesons in SM baryons and the dark fermion (antibaryon) cannot be too small, with the expectation that the branching fractions can roughly be no less than Br(B 0 s,d → χB) ≳ 10 −5 or Br(B + → χB (+) ) ≳ 10 −6 [120]. The expected theoretical window in χ mass is 0.94 GeV < m χ < 4.34 GeV [29]. Studies from Belle [121] and BaBar [122] limit the available parameter space in the mass region of 1 − 4.4 GeV, and it is anticipated that the remaining parameter space can be probed at Belle-II [122]. This model is particularly close to the model we study, in both its visible and hidden-sector components. In this paper we have established severe limits on the ε nχ and ε Λχ mixing parameters for χ masses satisfying m χ ≲ 1400 MeV, as shown in Fig. 13. In this mass region and for the regions of hidden-sector parameter space we have chosen, our limits constrain the flavor structure of models of B-mesogenesis, and we now turn to those and their implications. Different UV completions of B-mesogenesis models fare differently in light of our constraints. Here we consider versions in which only one extra particle is needed. For example, in Ref. [14], a color-triplet, SU(2) L singlet scalar with the SM quantum numbers (3, 1, −1/3) is used, though a scalar of form (3, 1, +2/3) [29] or a vector of form (3, 2, −1/6) [16] are noted alternatives. We do not consider this list exhaustive. The two scalars are just the leptoquarks we have noted in Sec. 2: S * 1 andS * 1 [10,15]. The phenomenology of these specific models has been studied, and in order to explain the baryon asymmetry, the dark matter abundance, and all empirical constraints, including those on |∆F | = 2 meson mixing, a rich flavor pattern of couplings to quarks is needed [29]. To determine the implications of our constraints, we first note the structure of the Lagrangian for each UV completion, following Ref. [16], though we write our 2-spinors as in Ref. [123] and employ the conventions given there. Denoting the new scalars as Y Y and the new vector as X µ , we have L Y 2 3 ⊃ −y dad b ϵ αβγ Y α 2 3 d β a d γ b − y χuc Y α * 2 3 χu α c + h.c. , (7.1) L Y − 1 3 ⊃ −y uad b ϵ αβγ Y α − 1 3 u β a d γ b − y χdc Y α * − 1 3 χd α c − y QaQ b ϵ αβγ Y α − 1 3 Q β a εQ γ b + h.c. , (7.2) L X ⊃ −y Qad b ϵ αβγ X α µ εQ β a σ µ d γ b − y χQc X † α µ Q α c σ µ χ + h.c. ,(7.3) where ε is an antisymmetric tensor in the two-spinor indices and χ is a right-handed field. With the B assignments of −2/3 for the scalars Y 2 3 and Y − 1 3 and B = 1 for χ, the noted interactions conserve baryon number. In Refs. [15,29] y QaQ b (for each a, b) is taken to be zero. The color structure of the first term of Eq. (7.1) requires that the product of dlike quarks be antisymmetric in the generation indices a, b, which follows because we have assumed the scalar is a color triplet. As for the last case, the vector X µ can be written in two-spinor form as [16] X µ = Y µ 2 3 Y µ − 1 3 (7.4) and thus through Eq. (7.3) we see that both scalars couple to left-handed quarks. We have defined our scalar-fermion couplings in the flavor basis, rather than the mass basis, but in the case of couplings to right-handed quarks no distinction needs be made. However, in the case of couplings to left-handed quarks we need to rotate the fields to the mass basis, to parallel the treatment of the charged weak current in the SM. As a result, a flavor diagonal coupling to a left-handed quark of a single flavor can engender a contribution to a flavor-changing neutral current (FCNC). In the example of Z ′ models, satisfying FCNC constraints with a large Z ′ coupling requires nearly flavor-universal couplings [124], where we note that in the flavor universal limit the unitary structure of the CKM matrix makes the FCNC couplings vanish. We will see that this effect does not appear here because our scalars do not ever couple to two left-handed quarks of the same flavor. Replacing a left-handed flavor state d i with a combination of mass states via V ij d j , with V the CKM matrix, we see that the X µ completion does lead to a FCNC of form [16] L X ; FCNC ⊃ −y Qad b Y 2 3 µ V aa ′d c a ′ γ µ P R d b ,(7.5) where we have employed 4-component notation. This interaction engenders not only |∆F | = 2 meson-mixing but also structures such as B (s) →K or B (s) → π 0 at tree level, which can be probed through B decay studies. We also see explicitly that the structure of the vertex does not require a flavor universal coupling to control the size of the effect. Thus there are no particular flavor conspiracies in satisfying the |∆F | = 2 constraints, and to determine the impact of the constraints we have found on the mixing parameters ε nχ and ε Λγ on these models, it suffices to consider the contributions to these quantities from the scalar-fermion couplings with a particular UV complete model. Considering, then, the flavor structure of Eq. (7.1) we see that n → χγ cannot occur at tree level, and a loop graph with W and Y Y exchange is needed to generate the process [15]. The opposite situation is true for Λ → χγ, with Eq. (7.1) and Eq. (7.2) yielding that process at tree level and one loop level, respectively. The pertinent Feynman diagrams are illustrated in Fig. 16, replacing the illustration of Fig.(1). Noting Eqs. (2.5) and (2.7), it is apparent that the mixing parameters ε nχ and ε Λχ depend very differently on the underlying scalar-fermion couplings in the two cases -we refer to Ref. [15] for explicit expressions. In particular, the one-loop diagrams bring in a coupling to the b quark as well, with the following combinations of couplings: y db y χu ; y sb y χu (7.6) y db y χc ; y sb y χc (7.7) y db y χt ; y sb y χt (7.8) each of which could saturate the bound we have found for ε nχ . In regards to the mechanism of B-mesogenesis, operators with the flavor combinations χbud, χbud, χbcd, and χbcs are as discussed in text, after Ref. [15]. χ d d u n γ Y − 1 3 (a) χ d d u n Y2 3 γ W u, c, t s, b (b) χ d u s Λ Y − 1 3 γ W u, c, t d, s, b (c) χ s u d Λ γ Y2 3 (d) pertinent, and they take one of three forms [29] θ (1) ij = (χb)(u i d j ) , θ (2) ij = (χd j )(u i b) , θ (3) ij = (χu i )(d j b) ,(7.9) where i ∈ d, s, j ∈ u, c, and the colors have been contracted to form a color singlet in each case. Taking the couplings in Eq. (7.8) one at a time, we find that saturating our ε nχ constraint we have found limits the coefficient of each of the θ ij operators to be powers of ten smaller than that needed for B-mesogenesis to be successful [29]. We emphasize, however, that this is particular to the mass window in χ and region of hidden-sector parameter space we have noted. For the Y2 3 scalar, those are the operators that would act -and thus we have ruled out this specific model for B-mesogenesis under the conditions we have noted. The other UV completions we have considered are not similarly constrained, because the ε χΛ constraints limit just the flavor combinations y χb y du and y χb y dc pertinent to B-mesogenesis -the other flavor combinations associated with θ (1) ij and θ (2) ij remain unconstrained despite the severity of our limits. SUMMARY BNV has not yet been observed in terrestrial experiments, and its deep ties to explanations of the observationally well-established cosmic baryon asymmetry [2] argue persuasively for its investigation on broader fronts. Previously, we have considered how it might eventually be discovered through precision measurements of neutron star observables, particularly those of changes in the binary-pulsar period, familiar from tests of general relativity [5]. Thus far we have found limits, and they are macroscopic ones, in that they emerge from the consideration of a neutron star as a whole. Such constraints miss a concrete connection to particle physics, and it is badly needed: regardless of whether we continue to constrain or, finally, discern the existence of BNV (in contradistinction to a failure of general relativity) from these studies, further theoretical progress on the problem of BNV requires constraints on the particle physics models of BNV themselves. In this paper, we have developed just such a connection, using a concrete description of the neutron star interior based on a relativistic mean-field theory in hadronic degrees of freedom [17][18][19] that successfully confronts existing macroscopic properties of neutron stars [21]. Within this context, we have developed how to assess the rates for BNV particle processes in dense matter, and we present explicit rates for benchmark processes, particularly B → χγ, considering its rate both at local points within a neutron star as well as its volume rate after integration over the structure of the entire star, up to its crust. Although our in-medium formalism is germane to the evaluation of any particle process in the dense medium of a neutron star, the focus of this paper -noting current sensitivities -is that of apparent BNV through baryon decays to hidden-sector particles. Finally, with this in place, we match the computed rate to our inferred limits on anomalous binary-period lengthening, i.e., how the binary itself spins down, to set one-sided limits at 2σ on the mixing parameters ε Bχ , for individual binary-pulsar systems, as well as a combined limit for all of the studied systems. As a result of these studies, we discover that neutron stars open new windows on the study of BNV, probing m χ parameter space not accessible to terrestrial nucleon decay experiments, due to experimental limitations in the detection of a final-state photon. More than this, the dense nuclear medium admits the study of regions for which m χ exceeds the vacuum mass of the nucleon, as well as the possibility of probing strange baryon decays. Our final limits are reported in Figs. 14 and 15. We observe that in the regions of parameter space to which proton decay (nuclear stability) experiments are sensitive [114,115], they exceed the limits we set by nearly twenty orders of magnitude. In contrast, however, our neutron star limits exceed the sensitivity of those from terrestrial Λ and neutron β-decay experiments by a comparably large amount. Let us emphasize that our limits are likely upper bounds, and hence are conservative, in that they are determined by the electromagnetic decay B → χγ alone, although the particle physics models we study do admit the possibility of B → χ + meson(s) decays as well. This latter set of decays has no reason to be negligible compared to the electromagnetic decays in rate -and we note Ref. [16] for specific examples computed within (in-vacuum) chiral EFT [51]. As a result, we would expect larger B decay rates for fixed ε Bχ , but the challenges in realizing a suitable theoretical assessment of the hadronic channels prompt the conservative approach we have espoused in this paper. We now turn to an assessment of the limitations in our approach. One key question concerns the largest value of ε Bχ , ε max Bχ , we can possibly limit with our formalism, in which the SM drives the dynamical response of the neutron star to BNV. (In our work, dark-sector interactions drive the removal of χ, so that the neutron star survival constraints on the mass of m χ noted in Refs. [101,125,126] do not operate.) We believe a realistic assessment of ε max Bχ requires a study of neutron star heating from relatively fast rates of BNV, the complexities of which lie beyond the scope of this paper. We note, however, the outcomes of terrestrial neutron β-decay searches [45], shown in Fig. 15, as well as limits arising from constraints due to the charged-current structure of the SM [41], noted in Eq. (2.1). Since n → χγ does not derive from a SM weak process in any way, a Br(n → χγ) limit of O(10 −3 ) implies a limit on ε nχ of O(10 −9 )! Thus we think these limits are severe enough that determining ε max Bχ precisely is not an immediate concern, but, rather, an important topic for future investigation. Another potential limitation may be our use of a relativistic mean-field theory framework [17][18][19] in which to describe the nuclear medium within a neutron star. This approach is computationally tractable and readily allows for the treatment of more sophisticated models of the nucleon-nucleon interaction than those in which it was first devised. We have employed the chiral SU(3) hadronic model of Refs. [9,21,91] in this paper. This is admittedly a model that is not QCD, and our ability to assess the errors predicated by this choice is rather limited. We have, however, studied how our results change within a family of EoSs, namely DS(CMF) 1-8 EoSs [92,93], to which it can be connected. Moreover, frankly, there is no other alternative for the treatment of dense nuclear matter, though this may ultimately change [127]. We note that the use of chiral effective theory has been championed in this regard [87], but its applicability does not stretch much beyond that of nuclear saturation density. In the future, it may be advantageous to consider EoSs that blend the chiral effective field theory and relativistic mean-field theory approaches [94]. Nevertheless, given our interest in order-of-magnitude estimates, we believe that our choice is also reasonably realistic. Different paths beckon as opportunities for future work. We believe that studies of neutron star heating from BNV is important not only to discerning the limits of our existing formalism, but also, crucially, to interpreting what a significant observation of anomalous binary spin down might mean. It strikes us that theoretical heating studies and concomitant observational studies of neutron star cooling may be the only tangible way to tell a failure of general relativity, in some undetermined way, from BNV. As for other possibilities, we could consider how our results could change if the neutron star were a hybrid star, containing a quark core [9], or how viable models with a significant χ admixture in the neutron star (albeit constrained by Eq. (2.1) [41]), such as that of Ref. [37], could be addressed through modifications of our formalism. As for future terrestrial experiments that could complement the studies of this paper, it strikes us that empirical studies of the lifetime of SM-stable composites, such as atomic hydrogen, or of the 9 Be nucleus, could yield fruitful results. u(p ′ * ) iσ µν (p ′ ν − p ν ) 2m * u(p * ) = u(p ′ * ) γ µ − p ′ µ + p µ 2m * + Σ µ m * u(p * ), (A.1) in which σ µν ≡ (i/2)[γ µ , γ ν ] and γ µ are the usual Dirac matrices. The in-medium Gordon decomposition is then given by u(p ′ * )γ µ u(p * ) = u(p ′ * ) p ′ µ + p µ 2m * + iσ µν q ν 2m * − Σ µ m * u(p * ) (A.2) in which we defined q ν ≡ p ′ ν − p ν . The general form of a vector interaction vertex, Γ µ , can be written as Γ µ = γ µ A + p ′ µ + p µ B + q µ C + DΣ µ , (A.3) in which A, B, C, D are functions of scalar quantities (e.g., q 2 ). Applying the Ward identity, q µ Γ µ = 0, plus p ′ * 2 = p * 2 = m * 2 and p ′ 2 − p 2 = 2q · Σ, yields C = 0 and 2B = D. The electromagnetic vertex factor can then be written as Γ µ = γ µ F * 1 (q 2 ) + iσ µν q ν 2m * F * 2 (q 2 ), (A.4) in which F * 1,2 are in principle distinct from their vacuum counterparts F 1,2 . We now show how the electric charge can be identified in the scattering amplitude of a baryon from a Coulomb potential A µ = (Φ(x), ⃗ 0). Employing equations u(k * , λ)u(k * , λ) = 2m * and u(k * , λ)γ 0 u(k * , λ) = 2E * (k * ), this amplitude can be written as iM = −ieF * 1 (0)Φ(q) E * m * 2m * χ † χ, (A.5) in which E * = m * 2 + (⃗ p * ) 2 , and χ is the Pauli spinor. The electric charge (q) can then be identified, by considering this scattering in the c.v. frame (⃗ p * = 0), as q = Similarly, we can identify the magnetic moment from the scattering amplitude of a baryon from a static magnetic field potential A µ = (0, ⃗ A) at small momentum transfers (q 2 ≈ 0), which is given by iM = +ieu(p ′ * ) γ i F * 1 (0) + iσ iν q ν 2m * F 2 (0) u(p * )Ã i (0). (A.6) The first term inside the bracket can be written as u(p ′ * )γ i u(p * ) = (E * + m * ) χ † , χ † ⃗ σ · ⃗ p ′ * E * + m *   0 σ i σ i 0     η ⃗ σ·⃗ p * E * +m * η   =χ † σ i ⃗ σ · ⃗ p * + ⃗ σ · ⃗ p ′ * σ i η, (A.7) in which σ i are the Pauli matrices, and χ, η represent the spin states. This expression can be further simplified using σ i σ j = δ ij + iϵ ijk σ k , such that u(p ′ * )γ i u(p * ) = χ † (p * + p ′ * ) i − iϵ ijk (p ′ * − p * ) j σ k η. (A.8) The F 2 term in the scattering amplitude (A.6) already contains a factor of q, and so we can evaluate it using the leading order expansion of the spinors in the non-relativistic limit (⃗ p * ≪ m * ), which is given by u(⃗ p * = 0) = √ 2m * (χ, 0) T . We also note that i 2m * σ ij q j = iϵ ijk 2m * σ k q j , (A.9) i 2m * σ i0 q 0 = q 0 2m *   0 σ i σ i 0   , (A.10) such that the spin-dependent contribution from Eq. (A.10), i.e., u(p ′ * )(σ i0 q 0 )u(p * ) is proportional to q 0 q j , which is subdominant to other terms. The term from Eq. (A.9), i.e., u(p ′ * )(σ ij q j )u(p * ) is given by u(p ′ * ) i 2m * σ ij q j u(p * ) = iϵ ijk q j χ † , 0   σ k η 0   = iϵ ijk q j χ † σ k η. (A.11) The amplitude in Eq. (A.6) can then be written as (note q j = −q j ) iM = − eχ † ϵ ijk q iÃj (0)σ k [F * 1 (0) + F * 2 (0)] η = − 2ie [F * 1 (0) + F * 2 (0)] S kBk , (A.12) in which we defined the magnetic field byB k ≡ −iϵ ijk q iÃj , spin by ⃗ S ≡ (1/2)χ † ⃗ ση, and the baryon g-factor can be identified as g * = 2 [F * 1 (0) + F * 2 (0)]. Appendix B: Nonrelativistic Limit of In-Medium Scattering In this appendix we study the non-relativistic (NR) limit of the RMF model, and derive the elastic scattering formalism in the Born approximation. Since the medium effects in RMF formalism resemble an electromagnetic interaction with a constant EM background field given by eA µ → Σ µ , it is instructive to consider the NR limit of baryon EM interactions in medium. We explicitly show that the NR limit of the modified Dirac (Eq. (4.6)) solutions under the influence of EM interactions, reduces to the two-component Pauli spin theory, with replacements m → m * , eΦ → eΦ + Σ 0 , e ⃗ A → e ⃗ A + ⃗ Σ, in which Σ 0 and ⃗ Σ are the self-energies due to the medium effects, e is the baryon electric charge, with Φ and ⃗ A as the scalar and vector EM potentials respectively. We start from the Schrodinger equation, which can be written by denoting the Dirac wave-function (ψ) in two-component notation [128], ψ = (φ,χ) T , such that we have i ∂ ∂t  φ χ   = ⃗ σ · ⃗ π  χ φ   + eΦ + Σ 0  φ χ   + m *  φ −χ   , (B.1) in Pauli-Dirac representation, in which ⃗ π ≡ ⃗ p − ⃗ Σ − e ⃗ A. Using the definition (φ,χ) = exp(−im * t) (φ, χ), we can rewrite Eq. (B.1) as i ∂ ∂t   φ χ   = ⃗ σ · ⃗ π   χ φ   + eΦ + Σ 0   φ χ   − 2m *   0 χ   . (B.2) We note that in the NR limit, in which kinetic and interaction energies are much smaller than m * , the second component χ is subdominant to the first component φ and is approximately given by χ ≈ ⃗ σ · ⃗ π 2m * φ. (B.3) We also arrive at the Pauli equation governing the first component (φ): i ∂ ∂t φ =    ⃗ p − ⃗ Σ − e ⃗ A 2 2m * − e 2m * ⃗ σ · ⃗ B + eΦ + Σ 0    φ, (B.4) in which ⃗ B = ∇ × A. This expression can be further simplified for a weak uniform magnetic field ( ⃗ A = ⃗ B × ⃗ r/2) as i ∂ ∂t φ = |⃗ p * | 2 2m * − e 2m * ⃗ L * + 2 ⃗ S · ⃗ B + eΦ + Σ 0 φ, (B.5) in which ⃗ p * = ⃗ p − ⃗ Σ is the kinetic three momentum, and ⃗ L * = ⃗ r × ⃗ p * and ⃗ S = ⃗ σ/2 are the baryon's kinetic orbital angular momentum and spin respectively. Note that in the n.m. frame ( ⃗ Σ = 0) the canonical and kinetic three momenta are equal ⃗ p = ⃗ p * . We now construct the elastic scattering formalism off of an arbitrary potential (V ) in −∇ 2 φ + 2i ⃗ Σ · ⃗ ∇φ + | ⃗ Σ| 2 − 2m * E − Σ 0 φ = 0, (B.6) with solutions of the form φ = e −iEt A 1 e i⃗ p.⃗ x + A 2 e −i(⃗ p−2 ⃗ Σ)·⃗ x , (B.7) which can also be written in a more symmetric way in terms of ⃗ p * . If we orient our coordinates such that ⃗ Σ.⃗ x > 0, then for a positive ⃗ p (⃗ p.⃗ x > 0) the first term is a plane wave moving to the right and the second term is a wave moving to the left. Therefore, we pick the first term for incident waves in the elastic scattering problem. Let H 0 be the Hamiltonian used in Eq. (B.4) (with Φ = ⃗ A = 0), and |k (+) ⟩ be the state that satisfies the following Schrodinger equation in the presence of a potential V (E − H 0 ) |k (+) ⟩ = V |k (+) ⟩, (B.8) then, |k (+) ⟩ can be found from the Lippmann-Schwinger equation: |k (+) ⟩ = |k⟩ + 1 E − H 0 + iε V |k (+) ⟩. (B.9) The momentum representation of operator G ≡ (E − H 0 + iε) −1 is given by ⟨⃗ q |G| ⃗ q ′ ⟩ = δ(⃗ q − ⃗ q ′ ) 2m * ⃗ k − ⃗ Σ 2 − ⃗ q − ⃗ Σ 2 + iε , (B.10) and the position space representation is given by ⟨⃗ r |G| ⃗ r ′ ⟩ = d 3 q (2π) 3 2m * ⃗ k − ⃗ Σ 2 − ⃗ q − ⃗ Σ 2 + iε e i⃗ q·(⃗ r−⃗ r ′ ) . (B.11) We define ⃗ R ≡ ⃗ r − ⃗ r ′ , ⃗ ξ ≡ ⃗ k − ⃗ Σ and change the variable from ⃗ q to ⃗ Q ≡ ⃗ q − ⃗ Σ such that ⟨⃗ r |G| ⃗ r ′ ⟩ = d 3 Q (2π) 3 2m * ⃗ ξ 2 − ⃗ Q 2 + iε e i( ⃗ Q+ ⃗ Σ)· ⃗ R = m * e i ⃗ Σ· ⃗ R 4π 2 (iR) ∞ −∞ QdQ ⃗ ξ 2 − ⃗ Q 2 + iε e iQR − e −iQR = −m * 2πR e i ⃗ Σ· ⃗ R e iξR , (B.12) in which we performed the angular integration in the second line, and the complex contour integration in the third line. To characterize the scattering problem at r → ∞ we approximate the above expression for (r ′ /r) → 0 using R = |⃗ r − ⃗ r ′ | ≈ r −r · ⃗ r ′ , such that ⟨⃗ r |G|⃗ r ′ ⟩ = −m * 2πr e i[| ⃗ k− ⃗ Σ |r+ ⃗ Σ·⃗ r ] e −i[| ⃗ k− ⃗ Σ |r+ ⃗ Σ]·⃗ r ′ . (B.13) We now write the asymptotic form of the Lippmann-Schwinger equation in position space as ψ k (⃗ r ) ∼ φ k (⃗ r ) − m * 2πr e i[| ⃗ k− ⃗ Σ |r+ ⃗ Σ·⃗ r ] d 3 r ′ e −i[| ⃗ k− ⃗ Σ |r+ ⃗ Σ]·⃗ r ′ V (r ′ )ψ k (⃗ r ′ ), (B.14) in which ψ k (⃗ r ) ≡ ⟨⃗ r | ⃗ k (+) ⟩ and φ k (⃗ r ) ≡ (2π) −3/2 exp i ⃗ k · ⃗ r . The exponential outside of the integral in the second term is an ellipsoidal wave (stretched along ⃗ Σ) which becomes spherical in the n.m. frame ( ⃗ Σ = 0). The exponent inside the integral is a vector pointing in the direction of | ⃗ k − ⃗ Σ |r + ⃗ Σ, which reduces to the familiar kr term in the n.m. frame. We can see that the gradient of ellipsoidal surface is equal to the vector in the exponent inside the integral since ∇ | ⃗ k − ⃗ Σ |r + ⃗ Σ · ⃗ r = | ⃗ k − ⃗ Σ |r + ⃗ Σ, (B.15) which suggests that the exponent ⃗ k ′ ≡ | ⃗ k − ⃗ Σ |r + ⃗ Σ is the momentum of scattered particle in the direction of an observer at r. Note that the kinetic energy of the scattered particle is given by 16) and the scattering is indeed elastic. We can therefore deduce the scattering amplitude by T (k ′ ) = ⃗ k ′ − ⃗ Σ 2 2m * = ⃗ k − ⃗ Σ 2 2m * = T (k),(B.writing ψ k (⃗ r ) ∼ (2π) −3/2 e i ⃗ k·⃗ r + e i[| ⃗ k− ⃗ Σ |r+ ⃗ Σ·⃗ r ] r f (k ′ |k) , (B.17) in which f (k ′ |k) = −4π 2 m * d 3 r φ * k ′ (⃗ r ) V (r)ψ k (⃗ r), (B.18) which is the Fourier transform of the potential in the Born approximation. Appendix C: In-Medium Compton Scattering In this section we evaluate the Compton scattering cross section of baryons, B(p 1 ) + γ(k 1 ) → B(p 2 ) + γ(k 2 ) (see Fig. 17) in neutron star medium, denoting the photon and baryon energies by ω 1,2 and E 1,2 respectively. We first note that the second term in the k 1 k 2 baryon propagator defined in Eq. (4.14) vanishes since p 1 + k 1 p 1 p 2 k 1 k 2 p 1 − k 2 p 1 p 2(p * 1 + k 1 ) 2 − (m * B ) 2 = 2 E * 1 | ⃗ k 1 | − ⃗ k 1 · ⃗ p * 1 = 2| ⃗ k 1 | (⃗ p * 1 ) 2 + (m * B ) 2 −k 1 · ⃗ p * 1 > 0, (C.1) and similarly, it can be shown that (p * 1 − k 2 ) 2 − (m * B ) 2 is strictly negative. The amplitude for the diagrams shown in Fig. 17 can then be written as iM = iM L + iM R , (C.2) in which iM L = −iu(p 2 ) γ µ F * 1 + iσ µν k 2,ν 2m * B F * 2 ϵ * µ (k 2 ) / p * 1 + / k 1 + m (p * 1 + k 1 ) 2 − (m * B ) 2 × γ ν F * 1 + iσ να k 2,α 2m * B F * 2 ϵ ν (k 1 )u(p 1 ), (C.3) and iM R = −iu(p 2 ) γ ν F * 1 + iσ να k 1,α 2m * B F * 2 ϵ ν (k 1 ) / p * 1 − / k 2 + m (p * 1 − k 2 ) 2 − (m * B ) 2 × γ µ F * 1 + iσ µν k 2,ν 2m * B F * 2 ϵ * µ (k 2 )u(p 1 ), (C.4) in which F * 1,2 are the in-medium form factors. The interaction term in the amplitude can be simplified using γ µ F * 1 + iσ µν k 2,ν 2m * B F * 2 ϵ * µ (k 2 ) = / ϵ * (k 2 )F * 1 − F * 2 2m * B / ϵ * (k 2 )/ k 2 , (C.5) which follows from ϵ µ (k 1 )k µ 1 = ϵ µ (k 2 )k µ 2 = 0. The spin-averaged squared amplitudes simplify to |M L | 2 = 1 4 [(p * 1 + k 1 ) 2 − (m * B ) 2 ] 2 Tr / p * 2 + m * B F * 1 + F * 2 2m * B / k 2 γ µ / p * 1 + / k 1 + m * B γ ν × F * 1 − F * 2 2m * B / k 1 / p * 1 + m * B F * 1 − F * 2 2m * B / k 1 × γ ν / p * 1 + / k 1 + m * B γ µ F * 1 + F * 2 2m * B / k 2 , (C.6) and |M R | 2 = 1 4 [(p * 1 − k 2 ) 2 − (m * B ) 2 ] 2 Tr / p * 2 + m * B F * 1 + F * 2 2m * B / k 1 γ ν / p * 1 − / k 2 + m * B γ µ × F * 1 − F * 2 2m * B / k 2 / p * 1 + m * B F * 1 − F * 2 2m * B / k 2 × γ µ / p * 1 − / k 2 + m * B γ ν F * 1 + F * 2 2m * B / k 1 , (C.7) with the cross-term given by M L M † R = Tr [T LR ] 4 [(p * 1 − k 2 ) 2 − (m * B ) 2 ] [(p * 1 + k 1 ) 2 − (m * B ) 2 ] , (C. 8) in which T LR = / p * 2 + m * B F * 1 + F * 2 2m * B / k 2 γ µ / p * 1 + / k 1 + m * B γ ν F * 1 − F * 2 2m * B / k 1 / p * 1 + m * B × F * 1 − F * 2 2m * B / k 2 γ µ / p * 1 − / k 2 + m * B γ ν F * 1 + F * 2 2m * B / k 1 , (C.9) with M L M † R = M R M † L . We now define the following Mandelstam variables s * ≡ (p * 1 + k 1 ) 2 = (m * B ) 2 + 2p * 1 · k 1 = (m * B ) 2 + 2p * 2 · k 2 (C.10) t * ≡ (p * 2 − p * 1 ) 2 = 2(m * B ) 2 − 2p * 1 · p * 2 = −2k 1 · k 2 (C.11) u * ≡ (k 2 − p * 1 ) 2 = (m * B ) 2 − 2p * 1 · k 2 = (m * B ) 2 − 2p * 2 · k 1 , (C.12) such that s * + t * + u * = 2(m * B ) 2 . We suppress the superscripts ("*") of m * B , F * 1,2 in some of the following equations for convenience. The averaged amplitude-squared can be written as 16) in which we note that I and IV are related via (s ↔ u) replacement. Equation (C.13) can then be written as |M| 2 = 1 16 I (p * 1 · k 1 ) 2 + II (p * 1 · k 1 )(p * 1 · k 2 ) + III (p * 1 · k 1 )(p * 1 · k 2 ) + IV (p * 1 · k 2 ) 2 , (C.13) in which I =8F 4 1 m 4 B + m 2 B (3s + u) − su + 4F 2 2 F 2 1 1 − s m 2 B 2m 4 B − m 2 B (3s + u) + s(3s − u) − F 4 2 m 2 B 2 1 − s m 2 B 3 m 2 B − u , (C.14) II =III = F 4 2 2 − s + u m 2 B su − m 4 B − 8F 4 1 m 2 B 2m 2 B + s + u − 2F 2 2 F 2 1 3m 2 B (s + u) − 2 s 2 + su + u 2 + su(s + u) m 2 B − 2m 4 B , (C.15) IV =8F 4 1 m 4 B + m 2 B (s + 3u) − su + 4F 2 2 F 2 1 1 − u m 2 B 2m 4 B − m 2 B (s + 3u) + u(3u − s) − F 4 2 m 2 B 2 m 2 B − s 1 − u m 2 B 3 , (C.|M| 2 = 2F 4 1 [6m 8 B − m 4 B (3s 2 + 14su + 3u 2 ) + m 2 B (s + u) (s 2 + 6su + u 2 ) − su (s 2 + u 2 )] (m 2 B − s) 2 (m 2 B − u) 2 + F 4 2 [3m 8 B − m 4 B (s 2 + 8su + u 2 ) + 4m 2 B su(s + u) − s 2 u 2 ] 4m 4 B (m 2 B − s) (m 2 B − u) + F 2 2 F 2 1 [2m 6 B − 3m 4 B (s + u) + 2m 2 B (s 2 + su + u 2 ) − su(s + u)] m 2 B (m 2 B − s) (m 2 B − u) . (C.17) We now consider the Compton scattering in the rest (c.v.) frame of B(p 1 ) (see Fig. 18), in which ⃗ p * 1 = 0. We first note that the relationship k 1 · k 2 = p * 1 · (k 1 − k 2 ), written in the c.v. frame, yields ω 1 ω 2 (1 − cos θ) = m * B (ω 1 − ω 2 ). We then arrive at the following kinematics in the c.v. frame ω 2 = ω 1 1 + ω 1 m * B (1 − cos θ) , (C.18) ⃗ p * 2 = [−ω 2 sin θ, 0, ω 1 − ω 2 cos θ] , (C.19) E * 2 = (m * B ) 2 + ω 2 1 + ω 2 2 − 2ω 1 ω 2 cos θ. (C.20) which resembles the familiar Compton's formula. We use these kinematical relationships to write Eq. (C.13) in terms of ω 1 and the scattering angle (θ) in the c.v. frame as The phase space integrals over the final states (see Eq. (4.15)) can be written as |M| 2 =16F 4 1 m B m 3 B 4 (cos(2θ) + 3) + ω 1 sin 2 (θ/2) m 2 B (cos(2θ) + 3) + ω 2 1 [m B (cos(2θ) + 5) − 2ω 1 (cos(θ) − 1)] sin 4 (θ/2) − 8F 2 1 F 2 2 m BdΠ 2 = d 3 k 2 (2π) 3 1 2ω 2 d 3 p * 2 (2π) 3 1 2E * 2 (2π) 4 δ 4 (k 2 + p * 2 − k 1 − p * 1 ) [1 − f B (⃗ p 2 )] = ω 2 dω 2 dΩ 2 16π 2 E * 2 δ ω 2 + (m * B ) 2 + ω 2 1 + ω 2 2 − 2ω 1 ω 2 cos θ − ω 1 − m * B [1 − f B (⃗ p 2 )] = dΩ 2 16π 2 ω 2 E * 2 + ω 2 − ω 1 cos θ [1 − f B (⃗ p 2 )] = dΩ 2 16π 2 ω 2 2 m * B ω 1 [1 − f B (⃗ p 2 )], (C.22) in which dΩ 2 = d cos(θ) dϕ is the differential solid angle of ⃗ k 2 in the c.v. frame, and f B (⃗ p 2 ) is the Pauli blocking factor for the outgoing baryon. The shape of the Fermi surface in a general frame (such as c.v.) changes from being spherical to an ellipsoid. The general form of f B (⃗ p 2 ) in an arbitrary frame is given by and noting v * B = 0, v * A = 1 in our chosen frame (c.v.), the in-medium Compton scattering differential cross section can be written as θ(E * F − p * µ 2 B µ /n B ) [89], in which E * F,B ≡ p 2 F,B + (m * B ) 2 , p F,dσ dΩ 2 = ω 2 2 |M| 2 64π 2 ω 2 1 (m * B ) 2 [1 − f B (⃗ p 2 )], (C.24) in which we recover the Klein-Nishina [100,129] formula if we set f B (⃗ p 2 ) = 0, F 1 = e, F 2 = 0 and replace m * B by m e . Appendix D: Fermion Mixing in Dense Matter In this appendix we evaluate the eigenvalues of a system consisting of a neutral baryon (B) and a dark fermion (χ) with a mixing term between them, in the context of RMF framework. We suppress the superscript ("*") in baryon's effective mass (m * B ) for convenience. The Note that the baryon current J µ B ≡ ψ B γ µ ψ B satisfies ∂ µ J µ B = ∂ µ ψ B γ µ ψ B + ψ B / ∂ψ B = iε ψ χ ψ B − ψ B ψ χ , (D.4) and as expected is not conserved anymore, instead, the combined current J µ ≡ J µ B + J µ χ is conserved. The conserved energy-momentum prescribed by the Noether's theorem [100,130] is given by H = d 3 x ψ B i⃗ γ · ⃗ ∇ + / Σ B + m B ψ B + ψ χ i⃗ γ · ⃗ ∇ + m χ ψ χ + ε ψ B ψ χ + ψ χ ψ B , (D.5) ⃗ P = d 3 x ψ † B −i ⃗ ∇ ψ B + ψ † χ −i ⃗ ∇ ψ χ . (D.6) We expand each of the fields in terms of four modes ω ± 1,2 ( ⃗ k) as in which ψ(x, t) stands for ψ B,χ , a 1,2 and b 1,2 are the annihilation operators for particles and anti-particles, ω stands for ω( ⃗ k), we note the inequality ω( ⃗ k) ̸ = ω(− ⃗ k) if ⃗ Σ B ̸ = 0 (see Eq. (4.8)), and the fact that in the presence of medium (Σ 0 B ̸ = 0) the particle and antiparticle energies are not equal anymore (e.g., see Eq. (2.40) of [89]). The coefficients α and P 2 − m 2 χ (P − Σ B ) 2 − m 2 B ψ B =ε 2 2P µ (P µ − Σ µ B ) + 2m B m χ − ε 2 ψ B , (D.11) in which we note the definition P µ = i∂ µ = (H, − ⃗ P ). We now plug in the field expansion from Eq. (D.7) into Eq. (D.11) to arrive at the equation governing the spectrum of ω ± 1,2 modes: O(δ 2 ) : We can see that the first order term breaks the degeneracy by splitting the energies. ω 2 − k 2 − m 2 χ (ω − Σ 0 B ) 2 − ( ⃗ k − ⃗ Σ B ) 2 − m 2 B = 2ε 2 ω 2 − k 2 − ωΣ 0 B + ⃗ k · ⃗ Σ B + m B m χω 2 1 + 2ω 0 ω 2 (ω 0 − Σ 0 ) 2 − k 2 − m 2 B − 2ω 2 0 m B m χ − Σ 0 ω 0 + ω 2 0 − k 2 + 2ω 2 (ω 0 − Σ 0 ) + ω 2 1 ω 2 0 − k 2 − m 2 χ + 4ω 0 ω 2 1 (ω 0 − Σ 0 ) = 0.u χ (k χ ) 1 / k χ − / Σ B − m * B / ϵ * / k γ u(p * B ) , (E.1) where we note (/ k χ − / Σ B − m * B ) −1 = / k χ − / Σ B + m * B (k χ − Σ B ) 2 − (m * B ) 2 ≡ / k * + m * B (k * χ ) 2 − (m * B ) 2 (E.2) and we define the quantity k * χ ≡ k χ − Σ B = p * B − k γ . Note in the neutron star medium that energy-momentum conservation of the total canonical momentum still holds: p µ B = k µ γ + k µ χ . Consideration of the kinematics show that we need only consider the first term in the full baryon propagator given in Eq. (4.14). We then find the spin-summed matrix element to be |M| 2 = ε 2 Bχ g 2 B e 2 (p * B · k γ ) (m * B ) 2 [(m * B ) 2 − (k * χ ) 2 ] 2 [(m * B ) 2 − (k * χ ) 2 ](k χ · k γ ) + 2(k * χ · k γ )(k * χ · k χ + m * B m χ ) (E.3) = ε 2 Bχ g 2 B e 2 (p * B · k γ ) 2(m * B ) 2 (p * B · k γ ) 2 (p * B · k γ )(k χ · k γ ) + (k * χ · k γ )(k * χ · k χ + m * B m χ ) (E.4) = ε 2 Bχ g 2 B e 2 2(m * B ) 2 [(p * B · k χ ) + m * B m χ ] , (E.5) where we note the useful relations (k * χ ) 2 = (m * B ) 2 − 2(p * B · k γ ) and (k * χ · k γ ) = (p * B · k γ ). Integrated Rates We now address full integral over phase space, dn B dτ = − d 3 ⃗ p B (2π) 3 (2E * B ) d 3 ⃗ k χ (2π) 3 (2E χ ) d 3 ⃗ k γ (2π) 3 (2E γ ) f B (⃗ p B ) × |M| 2 × (2π) 4 δ (4) (p B − k χ − k γ ) . (E. 6) In the main text, we presented the rate as an integral over the baryon Fermi sphere of the dilated widths of individual baryons. Here, we will contrast this approach with a more straightforward evaluation of this integral and demonstrate that these yield consistent results, as expected. Our first step in the evaluation of the rate is to separate the integrals over the χ and γ phase spaces and evaluate these first: dn B dτ = − p F,n 0 p 2 dp 4π 2 E * B G(p); (E.7) G(p) = d 3 ⃗ k χ (2π) 3 (2E χ ) d 3 ⃗ k γ (2π) 3 (2E γ ) |M| 2 × (2π) 4 δ (4) (p B − k χ − k γ ) . (E.8) We have simplified the first integral by noting that it only depends on the magnitude of the three-momentum |⃗ p B | ≡ p, and that we only integrate within the neutron Fermi sphere. We tackle this second integral by computing it in the c.m. frame of the decaying neutron. We note, however, that the matrix element depends on p * B , which has a nonvanishing spatial component, though we will find that this is not relevant for the ultimate evaluation of the integral. We begin by articulating the boost between the n.m. frame and the c.m. frame. We denote the 4-momentum of a baryon in the n.m. frame by p δ √ s − k − k 2 + m 2 χ = δ k − s − m 2 χ 2 √ s E χ √ s . (E.14) Putting these pieces together, we arrive at G(p) = k dk 4πE χ δ k − s − m 2 χ 2 √ s E χ √ s × ε 2 nχ g 2 B e 2 2(m * B ) 2 [(m * B ) 2 + E * B Σ 0 B ](s + m 2 χ ) + 2sm * B m χ 2s (E.15) = ε 2 Bχ g 2 B e 2 32π(m * B ) 2 s − m 2 χ s 2 s[(m * B ) 2 + E * B Σ 0 B + 2m * B m χ ] + m 2 χ [(m * B ) 2 + E * B Σ 0 B ] . (E.16) We can therefore write Eq. (E.6) as (1 + σ 2 + 2xσ) 3/2 (1 + xσ) 1 + σ 2 + 2σx (1 + σx + 2µ) +µ 2 (1 + σx) . dn B dτ = − p F,B 0 p 2 dp 4π 2 E * B G(p) = − E F,B m * B (E * B ) 2 − (m * B ) 2 dE * B 4π 2 G(p),(E. (E. 18) We note that if the self-energy were to vanish (σ = 0) we would recover the vacuum decay rate reported in Eq. (2.7). FIG. 1 . 1(Color Online) Illustration of n → χγ decay in the degrees of freedom of Eq. (2.4). The decay Λ → χγ follows from the replacement of one d quark with a s quark. FIG. 2 . 2Illustration of various χ-nucleon (N ) processes at low energies, with the heavy black dot denoting the nχπ 0 effective vertex noted in the text, namely, (a) n−N scattering to produce χ−N , (b) χ − N elastic scattering, and (c) N − N annihilation to produce χ − χ. Processes with γ in place of π 0 are also possible. The reverse of reactions (a) and (c) should be strongly suppressed by Pauli-blocking effects in the interior of a neutron star. Fig. 4 .FIG 4The scope of possibilities can be limited through judicious choices of the parameters of the dark sector. For example, if m ξ > m n then dark-matter induced nucleon decay will not occur, and Fig. 4 depicts the only possible tree-level annihilation channel. This last effect acts to remove χ produced through neutron . 3. (Color Online) Feynman diagram contributing to induced neutron decay via aχϕ B ξ interaction, as per Ref. [54] -a p decay channel follows from the replacement of d → u in the spectator quark. FIG. 4 . 4Feynman diagram contributing to χ-χ annihilation via ξ exchange to yield B-carrying scalars, as per the conventions of we defined B 57 ≡ B i /10 57 ∼ O(1). Assuming the reference values in this equation, we can see that this ratio is about 10 −13 (MeV/T χ ) 3/4 for m χ ∼ O(GeV). Therefore, we have shown that the self-annihilation of χ can be very effective at keeping its concentration negligible. This concludes the analysis of the necessary conditions on χ self-annihilation cross section imposed by Eq. (3.3). The explicit forms of decay and annihilation rates for χ in terms of our model parameters are presented in Sec. 2 A.With the assumptions set forth in this section, we only need to specify the EoS of hadronic matter to find the neutron star structure. Once the EoS is specified, the Tolman-Oppenheimer-Volkoff (TOV)[62,63] equations can be integrated with the initial conditions M (0) = 0 and E(0) = E c up to the surface of the star, corresponding to P (r) = 0. In other words, to study BNV effects on neutron stars generated by a fixed EoS, we focus on the unique family of stars, each parameterized by its central energy density (E c ), known as the single-parameter sequence[20] of stars. FIG. 5 . 5. 3 A are necessary for a modelindependent analysis of BNV effects on neutron star. These conditions can be summarised as: (1) BNV is slower than chemical and dynamical responses in the neutron star, and (2) the contributions to the EoS from any new particles (e.g., χ) are negligible (see Eq. (3.3)). The overall effect of BNV within this framework is to relocate the neutron star along its singleparameter sequence prescribed by the chosen baryon-number-conserving EoS. The rate of change in any neutron star observable O as a result of this quasi-equilibrium evolution can be written asȮ ≡ dE c dt ∂O ∂E c . (3.18) (Color online) (a) The set of observable quantities (O): mass (M ), radius (R), baryon number (B), and moment of inertia (I) for a sequence of neutron stars as a function of the central energy density (E c ) relative to their canonical values (O * ): M ⋆ = 1.4 M ⊙ , R ⋆ = 12 km, B ⋆ = 10 57 , I ⋆ = 70 (M ⊙ km 2 ) assuming DS(CMF)-1 EoS. (b) The b(O) factors for three parameters (O = M, R, I) as a function of neutron star masses for DS(CMF)-1 EoS. See Sec. 4 A for more details on our choice of EoS. Sec. 4 4A. In order to evaluate b(O) factors, we generate a sequence of neutron stars on a grid of E c values, and then find the derivative of O using the central finite difference method. The resulting b(O) is plotted in Fig. 5 for various observable quantities as a function of neutron star masses for DS(CMF)-1 EoS. In Sec. 3 C, we use this formalism to show how changes in neutron star parameters due to BNV would affect pulsar-binary orbital decay rates. (3.22), which is due to changes in the moment of inertia, is O(10 −3 ) × b(I) M ⊙ for J1614−2230 and even smaller for the other two systems considered in this work. Given that b(M ) ≈ 1 and |b(I)| ∼ O(1), and the spin-down contributions from Eq. (3.23) are usually subdominant, we conclude that our limits would be mainly controlled by the first term in Eq. (3.22). For this reason, our inferred limits oṅ B/B are not sensitive to the specific choices of EoS. We can use the observed pulsar binary period decay rate to limit the contributions from Eq. (3.22), but first we need to identify other sources of binary orbital decay. heights z ≡ |d sin(b)| ≤ 1.5 kpc. We use the pulsar's coordinates (l, b) = (352.64 • , 20.19 • ) to find z = 0.223(14) kpc, andṖ Gal b = 1(5) × 10 −14 . These extrinsic effects combine to yieldṖ ext 10) × 10 −12 . Our resulting estimate since their masses are very close and the composition of light neutron stars ought not change much over 0.1 M ⊙ . to reproduce the hadron vacuum masses, the nuclear saturation properties (density n sat = 0.15 fm −3 , binding energy per nucleon B/A = −16.00 MeV, compressibility K = 300 MeV), the asymmetry energy (E sym = 30 MeV), and hyperon potentials. Furthermore, the pion and kaon decay constants constrain the scalar meson VEVs. FIG . 6. (Color Online) The effective masses (a), vector self-energies (b), and the energy of baryons at rest in the n.m. frame (c) as a function of density in the DS (CMF)-1 EoS. The horizontal lines correspond to the vacuum masses of baryons, and the vertical lines indicate the central number density (n c ) of the pulsars we consider in this work. for our calculations in Sec. 5 B. can be computed in different ways. The Lorentz-invariance of each measure dΠ i affords the opportunity of performing different parts of the integration in different frames. We note that the integration over dΠ B is simplest in the n.m. frame: there, we have f B (⃗ p , and the integrand is isotropic. Contrariwise, it is simplest to evaluate the integration over dΠ χ dΠ γ in the c.v. frame. Moreover, as discussed in Sec. 4 B, the width of an individual baryon is most simply interpreted in its respective c.v. frame, since the baryon is not moving. The baryon width in the c.v. frame takes the form Γ c.v. (|⃗ p (n.m.) B self-energy in the c.v. frame. The results in different frames are connected by Lorentz boosts, yielding Γ c.m. (p B ) = m * B E * ,(c.m.) B Γ c.v. (p B ) = E * ,(n.m.) B E * ,(c.m.) B Γ n.m. (p B ) . (5.9) The total rate of baryon loss in the n.m. frame is then given by integrating over all baryons in the local fluid, accounting for the contraction of their individual widths by a factor γ −1 = m * B /E * B , with E * B = |⃗ p (n.m.) B | 2 + (m * B ) 2 the n.m.-frame energy associated with the kinetic momentum: 2π 2 in the denominator ensures that if the function γ −1 Γ c.v. (p B ) were a constant, then the result would be (γ −1 Γ c.v. )×n B . Combining Eqs. (5.8) and (5.10), we find that this procedure leads to the same result as the direct evaluation of Eq. (5.7) shown in App. E. In either case, 5 The additional factor of 1 2 in Eq. (5.8) arises because |M| 2 has been spin-summed and not spin-averaged. c.m.-frame single baryon decay rate Γ c.m. (p B ) is given in Eq. (E.18). We illustrate the rates of B → χγ as a function of m χ and n in Fig. 7, for both neutrons (a) and Λs (b). This calculation is for the DS(CMF)-1 EoS, but the results are qualitatively similar for the other EoS in this family. 6 The vertical axes have been normalized to the value of nuclear saturation density in this EoS, n sat = 0.15 fm −3 . The respective color scales are shown at right, assuming ε Bχ = 10 −16 MeV; the units are fm −3 s −1 and we emphasize that these rates scale as ε 2 Bχ . To guide the eye, we have also added black contours every quarter order of magnitude. Solid contours correspond to integer numbers; dot-dashed contours correspond to half-integer numbers; and dotted lines correspond to quarter-integer numbers. In either panel, the dashed vertical line indicates the vacuum mass of the corresponding baryon. The dotted horizontal line corresponds to the central density of the heaviest stable neutron star within this EoS, corresponding to M TOV ≈ 2.07 M ⊙ ; the region above this line has been grayed out because these densities do not occur in a stable neutron star. Similarly, the dot-dashed horizontal lines correspond to the central densities of neutron stars with the masses of J0348+0432, J1614-2230 and J0737-3039A/B. We observe that when n → χγ is operative, it is almost always numerically larger than the rate of Λ → χγ (for ε nχ = ε Λχ ). FIG. 7 . 7(Color Online) The proper rates for B → χγ decays, (−dn B /dτ ) (in fm −3 s −1 ), for neutrons (a) and Λs (b) assuming DS(CMF)-1 EoS. In either panel, we fix the corresponding ε Bχ to be 10 −16 FIG. 8 . 8(Color Online) The baryon proper decay (B → χγ) rate (−dn B /dτ ) (per unit volume) assuming ε Bχ = 10 −16 MeV for (a) neutrons and (b) Λs as a function of radius for four pulsars using DS(CMF)-1 EoS. The numbers next to curves (colors) indicate different values of m χ = {0.5, 1.0, 1.2, 1.3} GeV. FIG. 9 . 9m 2 χ + |⃗ p (n.m.) | 2 ≈ (m * B ) 2 + |⃗ p (n.m.) | 2 + Σ 0,(n.m.) (Color Online) The particle composition (a), and the energy of baryons at rest in the n.m. frame (b) as a function of radius in PSR J0348+0432 assuming the DS (CMF)-1 EoS. The horizontal lines correspond to the vacuum masses of baryons. We choose models for which we expect the steady-state admixture of dark states to be completely negligible. ( 5 . 513) cannot be satisfied if Σ 0,(c.v.) Bis complex. In what follows, we set this latter possibility aside, because, as we will see, other effects act to suppress the likelihood of B-χ conversion. At resonance, the offset between the interaction and mass bases is maximal, corresponding to a mixing angle of θ = 45 • ; however, if Eq. (5.13) is violated by more than a few times ε Bχ , then the mixing angle is parametrically small: left-and right-hand sides of Eq. (5.13), respectively (cf. Eqs. (D.14) and (D.15)).We first consider what happens when the system is not close to resonance. In this case, the eigenvalues of the system are given by Eqs. (D.25) and (D.27), which are very nearly The situation is richer if the state is close to resonance. In this case, if the canonical momentum of the baryon is fixed by Eq. (D.17), then the eigenvalues of the Hamiltonian are better given by Eq. (D.32). They are nearly identical to each other, but they are c −1 ∼ O(10 −23 ) s. For a benchmark value ε Bχ = 10 −16 MeV, the argument of the latter sine function in Eq. (5.15) is ∼ O(10 −39 ) MeV s ∼ O(10 −18 ); this is safely approximated as small, and we see that the quantum Zeno effect is indeed operative under these conditions. Therefore, even if the mixing angle is large, we estimate the probability to be P B→χ (δt strong ) ∼ O(10 −36 ) an approximate per-baryon conversion rate of R B→χ (δt strong ) ∼ O(10 −13 ) s Fig. 7 1 .FIG 71over the neutron star volume using Eq. (3.16) and report these results as a function of m χ and n c . Panel (a) is for neutron decays, while panel (b) is for Λ decays; similar to Fig. 7, the contours correspond to constant (base-ten log of the) integrated rate of B → χγ. We have again fixed ε Bχ = 10 −16 MeV, and note that the results are in s −We . 10. (Color Online) The volume-integrated rates for B → χγ decays, (−dB/dt) (in s −1 ) for neutrons (a) and Λs (b) assuming the DS(CMF)-1 EoS. As in Fig. 7, we have fixed ε Bχ = 10 −16 MeV. The right panels (c & d) show the relationship between the neutron star mass and its central density. black contours in the same way as in Fig. 7, and we have again indicated the central densities of J0348+0432, J1614-2230, and J0737-3039A/B within this EoS. The right panels, (c) and (d), contextualize these results by showing the neutron star mass, M NS , on the horizontal axis as a function of the central density on the vertical axis. Note that Figs. 7 and 10 together imply that J0737-3039A/B are both too light to contain hyperons. FIG . 11. (Color Online) The total baryon loss rate per baryon (−Ḃ/B) due to B → χγ decays assuming ε Bχ = 10 −16 MeV for neutron and Λ as a function of m χ for four pulsars using the DS(CMF)-1 EoS. . 12. (Color Online) Exclusion constraints at 2σ on the n − χ (upper panel) and Λ − χ (lower panel) mixing parameters as functions of m χ for the DS(CMF)-1 EoS. The blue, red and green curves correspond to the 2σ limits derived from J0348+0432, J1614-2230, and J0737-3039A/B, respectively. The dashed black curves correspond to the combined limits, realized as per the discussion in the text. The vertical dashed lines indicate the in-vacuum baryon mass in each case. Figure 13 13depicts our results for the constraints on ε nχ (upper panel) and ε Λχ (lower panel) as functions of m χ , calculated for each of the eight EoS in the DS(CMF) family.Equations of state that do not include hyperons are indicated with dashed curves in the upper panel. We also note that the DS(CMF)-3 EoS formally cannot support a neutron star with a mass of 2.01M ⊙ -its maximum TOV mass is 2.00M ⊙ . However, this is within 1σ of the observed mass of J0348+0432; we therefore elect to include it in this figure, but instead 115 115. 13. (Color Online) Combined exclusion constraints at 2σ on the n − χ (upper panel) and Λ − χ (lower panel) mixing parameters as functions of m χ for the eight equations of state in the DS(CMF) family. uncertainty in the inferred masses of the observed pulsars into the determinations of their central densities (within the context of a given EoS), and thus into the predicted baryon loss and binary spin-down rates. The mass uncertainties on J0348+0432 and J1614-2230 are O(10 −2 )M ⊙ ; we anticipate that there would be O(1) corrections to the limits whenever these are the only operative constraints, though the orders of magnitude are expected to be correct. That said, the uncertainties on the masses of J0737-3039A/B are O(10 −5 )M ⊙ [83], so that we expect the limits on ε nχ for m χ ≲ 1000 MeV, noting Fig. 12, to be quite robust. In Fig. 14, we reinterpret our constraints on ε Bχ as constraints on the branching fractions for B → χγ in vacuum and contrast them against laboratory constraints, with neutrons (Λs) in the upper (lower) panel. For neutrons, we also show the KamLAND constraint on invisible neutron decay [114] in red and the SuperKamiokande constraint on n → νγ [. 14. (Color Online) Exclusion limits at 2σ on the vacuum branching fraction for B → χγ for neutrons (upper panel) and Λs (lower panel). The results for each EoS are color-coded as in Fig. 13. 15. (Color Online) Exclusion limits at 2σ on the vacuum branching fraction for n → χγ as per Fig. 14, in the particular χ mass region pertinent to an explanation of the neutron lifetime anomaly. Additional constraints and expected limits have been included as detailed in the text,after Ref.[34]. from late time, out of equilibrium production of B mesons (with equal fractions of b and b quarks) that evolve under SM CP-violating processes before decaying to a SM baryon and a dark fermion carrying the opposite sign of baryon number. Thus: no new sources of CP violation of the SM are required; the baryon number of the universe is conserved -it is just sequestered into visible and dark sectors with opposite baryon number; and it occurs late in the history of the universe in that occurs after the QCD phase transition, making it possible to realize hadronic states, and before the epoch of big-bang nucleosyn- FIG. 16 . 16(Color Online) Feynman diagrams for n → χγ and Λ → χγ decays as mediated by the baryon-number-carrying scalars Y ACKNOWLEDGMENTS We thank the Network for Neutrinos, Nuclear Astrophysics (N3AS), and Symmetries(N3AS) for an inspiring environment. J.M.B. acknowledges support from the National Science Foundation, Grant PHY-1630782, the Heising-Simons Foundation, Grant 2017-228 and the U.S. Department of Energy Office of Science under award number DE-SC00018327. J.M.B. also thanks the Institute for Nuclear Theory at the University of Washington for its kind hospitality and stimulating research environment. This research was supported in part by the INT's U.S. Department of Energy grant No. DE-FG02-00ER41132. S.G. and M.Z. acknowledge partial support from the U.S. Department of Energy Office of Science under contract DE-FG02-96ER40989. S.G. thanks N3AS and the Theoretical Division at Fermilab for gracious hospitality and lively environments and acknowledges support from N3AS and the Universities Research Association during sabbatical visits while completing this paper. Appendix A: In-Medium Electromagnetic Form Factors In this appendix we derive the general form for electromagnetic interactions of baryons in the context of hadronic RMF models, and explicitly show how the electric charge and magnetic moment are to be identified from the scattering amplitudes of baryons off of electric and magnetic potentials respectively. We start from the Dirac equation (4.6), / pu(p * ) = (m * + / Σ)u(p * ), in which we suppressed the spin index λ, to write F * 1 1(0)(E * ,(c.v.) /m * ) = F * 1 (0). This can also be understood from the time component of spin-independent conserved EM current J 0 = ψγ 0 ψ = 2E * , with the Lorentz invariant electric charge defined in the c.v. frame (E * ,(c.v.) = m * ). this NR limit, by turning off the EM fields, i.e., ⃗ A = Φ = 0, for the rest of this discussion. From Eq. (B.4), we deduce the energy eigenvalues E = |⃗ p * | 2 /2m * + Σ 0 , which agree with the NR expansion of Dirac energy eigenvalues given in Eq. (4.8). The energy eigenfunctions in position space satisfy Figure 1 1: x FIG. 17. Feynman diagrams for the baryon Compton scattering B(p 1 ) + γ(k 1 ) → B(p 2 ) + γ(k 2 ). FIG. 18 .= 0 . 180The Compton scattering in the rest (c.v.) frame of B(p 1 ). Note that even though there is a specific direction to the canonical momentum ⃗ p Therefore, we have the freedom to choose the z-axis in the direction of the incoming photon. Evaluating the integrated cross section in Eq. (C.24) will require specifying the Fermi ellipsoid in the c.v. frame, which depends on ⃗ p (n.m.) 1 (see Eq. (C.23)). (2θ) − 8 cos(θ) + 7] sin 2 (θ/2) − m B ω 2 1 [5 cos(θ) − 7] sin 2 (θ/2) + 8ω 1 m 2 B sin 2 (θ/2) + 2m 3 B + F 4 2 m B ω 2 1 [5 − cos(2θ)] [m B − ω 1 (cos(θ) − 1)] . (C.21) B is the Fermi momentum defined in the n.m. frame, B µ is the baryon current density defined below Eq. (4.14), and n B is the baryon number density. Evaluating the invariant argument of the step-function in the n.m. frame yields θ(E * F − E * ,(n.that even though the amplitude in the c.v. frame depends only on θ, integrating over the azimuthal angle (ϕ) requires the explicit coordinates of the initial baryon B(p 1 ) momentum in the n.m. frame, ⃗ p (n.m.) 1 , in our chosen coordinate in Fig. 18. Using Eq. (4.16), / Σ B − m B ψ B =εψ χ , (D.2) i / ∂ − m χ ψ χ =εψ B . (D.3) 1 (k, s)u 1 (k, s)e −iω (+) 1 t+ik·x + b † 1 (k, s)v 1 (k, s)e −iω (−) 1 t−ik·x +β a 2 (k, s)u 2 (k, s)e −iω (+) 2 t+ik·x + b † 2 (k, s)v 2 (k, s)e −iω ( β can be found by requiring that the Hamiltonian in Eq. (D.5) is diagonal. The spinors u(p, s) and v(p, s) satisfy (see Eq. (2.33) of Ref.[18])u † (p, s)u(p, s ′ ) =v † (p, s)v(p, s ′ ) = δ ss ′ ,m stands for m B , m χ , and we make the replacements p → p * B , E → E * B for the baryon. We can combine Eqs. (D.2) with (D.3) and arrive at the following equation after multiplying the left-hand-side by / P + m χ / P * + m B : the solutions in the absence of mixing (ε = 0) by ω 0 , and solve Eq. (D.12) using a perturbation series in powers of δ ≡ ε/ω 0 : ω = ∞ i=0 ω i δ i , with the zeroth order equation yieldingO(δ 0 ) : ω 0 − Σ 0 B 2 − ( ⃗ k − ⃗ Σ B )rest of this discussion we consider the energy spectrum in the n.m. frame ( (n.m.) B by Σ 0 for convenience, we can rewrite Eq. (D.12) asω 4 − 2Σ 0 ω 3 − 2k 2 + m 2 B + m 2 χ − Σ 2 0 + 2ε 2 ω 2 + 2Σ 0 k 2 + m 2 χ + ε 2 ω + k 2 + m 2 χ k 2 + m 2 B − Σ 2 0 + ε 2 2k 2 − 2m B m χ + ε 4 = 0.(D.16) We note that both of the parenthesis in Eq. (D.13) can be simultaneously equal to zero, if k satisfies the following condition at zeroth order: our solutions into two sets: the normal solutions for which the zeroth order condition in Eq. (D.17) is not satisfied, and those for which Eq. (D.17) is satisfied, which we denote by "*" superscript. First we write down the general equations to third order in perturbation. The first order equation is given byO(δ 1 ) : 2ω 0 ω 1 (ω 0 − Σ 0 ) 2 − k 2 − m 2 B + 2ω 1 (ω 0 − Σ 0 ) ω 2 0 − k 2 − m 2 χ = 0.(D.18) Given the O(δ 0 ) equation in Eq. (D.13), we conclude that either ω 1 = 0 or the condition in Eq. (D.17) is satisfied. The second order equation yields: + m 2 χ m χ (m B + m χ ) − Σ 0 k 2 2ωωω+ 2the third order equation is given byO(δ 3 ) : (2ω 1 ω 2 + 2ω 0 ω 3 ) (ω 0 − Σ 0 ) 2 − k 2 − m 2 B + 2ω 1 ω 2 1 + 2ω 0 ω 2 (ω 0 − Σ 0 ) + (2ω 3 (ω 0 − Σ 0 ) + 2ω 1 ω 2 ) ω 2 0 − k 2 − m 2 χ − 2ω 2 0 ω 1 (2ω 0 − Σ 0 ) + 2ω 1 ω 0 2ω 2 (ω 0 − Σ 0 ) letus assume that the condition in Eq. (D.17) doesn't hold, in which case O(δ 1 )condition yields ω 1 = 0. We then solve forω 2 in O(δ 2 ): + + m 2 χ m χ (m B + m χ ) + Σ 0 k 2 (B) = − k 2 + m 2 B − Σ 0 2 m B (m B + m χ ) − Σ 0 k 2 + m 2 B k 2 + m 2 B Σ 0 Σ 0 − 2 k 2 + m 2 B + m 2 B − m 2 χ , (D.24)which after plugging into the O(δ 3 ) equation yields ω 3 = 0, and so the energies of χ and Bto third order are given byω (+) (χ) = k 2 + m 2 χ + ε 2 m χ (m B + m χ ) − Σ 0 k 2 (−) (χ) = − k 2 + m 2 χ − ε 2 m χ (m B + m χ ) + Σ 0 k 2 (+) (B) =Σ 0 + k 2 + m 2 B + ε 2 m B (m B + m χ ) + Σ 0 k 2 (−) (B) =Σ 0 − k 2 + m 2 B − ε 2 m B (m B + m χ ) − Σ 0 k 2 + m 2 B k 2 + m 2 B m 2 B − m 2 χ − Σ 0 2 k 2 + m 2 B − Σ 0 + O(δ 4 ),(D.28) in which the negative energy solutions would be interpreted as antiparticles. We now consider the second set of solutions assuming that Eq. (D.17) holds. The zeroth order equation yields order equation O(δ 1 ) is trivial, and the second order equation yields: ω * 1 = ± (m B + m χ ) 2Σ 2 0 m B + (m χ − m B ) (m B + m χ ) m χ − m B ) (m B + m χ ) 2 Σ 0 (m χ − m B ) (m B + m χ ) 3 (m B − m χ ) 2 − Σ O(δ 3 ). (D.32) the quantities in the n.m. frame. The boost from the n.m. to the c.m. writep * ,(c.m.) B · k (c.m.) χ = [(m * B ) 2 + E * B Σ 0 B ](s + m 2 χ ) 2s − (|⃗ p B |Σ 0 B )(s − m 2 χ ) 2scos θ * , (E.13) in which cos θ * is the angle between ⃗ k 17) which after using the definitions in Eq. (5.12) turns into the expression given in Eq. (5.11).Using Eq. (5.9), we can also write the individual baryon decay rate in the c.m. frame Γ c.m. TABLE I . IThe relevant binary parameters for J0348+0432 [74], J1614−2230 [76, 77], and J0737−3039A/B [83]. See the discussion in Sec. 3 D for more details. TABLE II . IIThe set of CMF EoS variants taken from Ref.[92,93]; we refer to them as DS(CMF)-1 through DS(CMF)-8, respectively, in later use. The second and third columns describe the degrees of freedom (DoF); nucleons (N), hyperons (Y) for convenience. Thus in order to explore how our results vary with EoS, we employ the choices given there. The set of EoSs that we utilize has also been extended to include crusts based on a zero-temperature unified EoS[65] at β-equilibrium with similar values of the symmetry energy slope (L), in which the effective interactions are Skyrme forces Rs[96] (EoS 1-2) and SkMP[97] (EoS 3-8) with cluster energy functionals taken from Ref.[98].B. Medium Effects: Effective Masses and Beyond The occupation factor for Y depends on whether or not Y is a boson (+) or fermion (−). We point the interested reader to Ref.[110] for a discussion about the role of (de)coherence in understanding neutrino oscillations specifically, as well as Ref.[111] for a comparative analysis of neutrino oscillations with adiabatic conversion. 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[ "DERIVATION AND WELL-POSEDNESS FOR ASYMPTOTIC MODELS OF COLD PLASMAS", "DERIVATION AND WELL-POSEDNESS FOR ASYMPTOTIC MODELS OF COLD PLASMAS" ]
[ "Diego Alonso-Orán ", "Angel Durán ", "Rafael Granero-Belinchón " ]
[]
[]
In this paper we derive three new asymptotic models for an hyperbolic-hyperbolicelliptic system of PDEs describing the motion of a collision-free plasma in a magnetic field.The first of these models takes the form of a non-linear and non-local Boussinesq system (for the ionic density and velocity) while the second is a non-local wave equation (for the ionic density). Moreover, we derive a unidirectional asymptotic model of the later which is closely related to the well-known Fornberg-Whitham equation. We also provide the well-posedness of these asymptotic models in Sobolev spaces. To conclude, we demonstrate the existence of a class of initial data which exhibit wave breaking for the unidirectional model. 2020 Mathematics Subject Classification. 35R35, 35Q35, 35S10, 76B03.
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[ "https://export.arxiv.org/pdf/2305.13922v1.pdf" ]
258,841,647
2305.13922
41ed5384b7e45461b97f20510c3094243e4b15a4
DERIVATION AND WELL-POSEDNESS FOR ASYMPTOTIC MODELS OF COLD PLASMAS Diego Alonso-Orán Angel Durán Rafael Granero-Belinchón DERIVATION AND WELL-POSEDNESS FOR ASYMPTOTIC MODELS OF COLD PLASMAS In this paper we derive three new asymptotic models for an hyperbolic-hyperbolicelliptic system of PDEs describing the motion of a collision-free plasma in a magnetic field.The first of these models takes the form of a non-linear and non-local Boussinesq system (for the ionic density and velocity) while the second is a non-local wave equation (for the ionic density). Moreover, we derive a unidirectional asymptotic model of the later which is closely related to the well-known Fornberg-Whitham equation. We also provide the well-posedness of these asymptotic models in Sobolev spaces. To conclude, we demonstrate the existence of a class of initial data which exhibit wave breaking for the unidirectional model. 2020 Mathematics Subject Classification. 35R35, 35Q35, 35S10, 76B03. Introduction The motion of a cold plasma in a magnetic field consisting of singly-charged particles can be described by the following system of PDEs [2,12] n t + (un) x = 0, (1a) u t + uu x + bb x n = 0, (1b) b − n − b x n x = 0 (1c) where n, u and b are the ionic density, the ionic velocity and the magnetic field, respectively. Moreover, it has also been used as a simplified model to describe the motion of collission-free two fluid model where the electron inertial, charge separation and displacement current are neglected and the Poisson equation (1c) is initially satisfied, [2,20]. In (1) the spatial domain Ω is either Ω = R or Ω = S 1 (i.e. x ∈ R or x ∈ [−π, π] with periodic boundary conditions) and the time variable satisfies t ∈ [0, T ] for certain 0 < T ≤ ∞. The corresponding initial-value problem (ivp) consists of the system (1) along with initial conditions n(x, 0) = n 0 (x), u(x, 0) = u 0 (x), (2) which are assumed to be smooth enough for the purposes of the work. System (1) was introduced by Gardner & Morikawa [12]. Furthermore, Gardner & Morikawa formally showed that the solutions of (1) converge to solutions of the Korteveg-de Vries equation (see also the paper by Su & Gardner [25]). Berezin & Karpman extended this formal limit to the case where the wave propagates at certain angles with respect to the magnetic field [2], i.e. for angles satisfying certain size conditions. Later on, Kakutani, Ono, Taniuti & Wei [20] removed the hypothesis on the angle. This formal KdV limit was recently justified by Pu & Li [24]. 1.1. Contributions and main results. The purpose of this paper is two-fold. First, we derive three asymptotic models for the hyperbolic-hyperbolic-elliptic system of PDEs describing the motion of a collision-free plasma in a magnetic field given in (1). The method to obtain the new asymptotic models relies on a multi-scale expansion (cf. e. g. [1, 3-5, 9, 13, 14]) which reduces the full system (1) to a cascade of linear equations which can be closed up to some order of precision. More specifically, writing n = 1 + N, U = u, b = 1 + B,(3) and, for ε > 0, introducing the formal expansions N = ∞ ℓ=0 ε ℓ+1 N (ℓ) , B = ∞ ℓ=0 ε ℓ+1 B (ℓ) , U = ∞ ℓ=0 ε ℓ+1 U (ℓ) ,(4) then the first model is an O(ϵ 2 ) approximation of (1) and takes the form of the following Boussinesq type system h t + (hv) x + v x = 0, (5a) v t + vv x + [L , N h] h + N h = 0,(5b) for h = εN (0) + ε 2 N (1) , v = εU (0) + ε 2 U (1) . The nonlocal terms in (5) are given by L = −∂ 2 x (1 − ∂ 2 x ) −1 , N = ∂ x (1 − ∂ 2 x ) −1 (so ∂ x N = −L ),(6) which are Fourier multiplier operators with symbols L h(ξ) = ξ 2 1 + ξ 2ĥ (ξ), N h(ξ) = iξ 1 + ξ 2ĥ (ξ). (7) (where h(ξ) denotes the Fourier transform of h at ξ) and [L , ·] · denotes the commutator [L , f ] g = L (f g) − f L g.(8) The extra assumption U (0) = N (0) in (4) leads to the formal derivation of the second asymptotic model, as a bidirectional single non-local wave equation h tt + L h = (hh x + [L , N h] h) x − 2 (hh t ) x .(9) The formal reduction of (9) to the corresponding unidirectional version, cf. [29], yields h t = − 1 2 (3hh x − [L , N h]h − N h − h x ) ,(10) being the third asymptotic model introduced in the present paper. We note that the unidirectional equation (10) has strong similarities with the well-known equation u t + 3 2 uu x = N u,(11) proposed by Fornberg & Whitham as a model for breaking waves [11]. The latter equation has been intensively studied during the last decades and several results regarding the well-posedness of the ivp in different functional spaces as well as various wave-breaking criteria have appeared in the literature [15][16][17][18][19]30]. A significant difference between the structure of system (11) and the unidirectional equation (10) derived in the present paper is the emergence of the nonlocal commutator-type term. The second purpose of this work is the study of several analytical properties of the models (5), (9), and (10). They are mainly concerned with the existence of conserved quantities, wellposedness (in the sense of existence and uniqueness of solutions of the corresponding ivp), and the formation of special solutions. This paper will be focused on the first two points, while the existence and dynamics of solitary-wave solutions will be the object of a separate forthcoming study. Specifically, the results shown in this paper can be summarized as follows: • The system (5) and the unidirectional model (10) admit several quantities preserved by the solutions in suitable spaces, including a Hamiltonian structure. On the other hand, (9) can be written in a conservation form, leading in a natural way to the existence of a conserved quantity. • The system (5) is locally well posed on a modified Sobolev space involving the operator L . • The bidirectional non-local wave equation (9) has a unique local solution close to the equilibrium and for initial data with sufficiently small L ∞ norm. • The ivp for the equation (10) is locally well-posed in Sobolev spaces H s (R) for s > 3 2 . Furthermore, a blow-up criteria for the solution by means of a logarithmic Sobolev inequality is provided. In addition, smooth solutions of (10) are shown to exhibit wave breaking under suitable hypothesis on the initial condition. 1.2. Structure of the paper. In Section 2 we present the asymptotic derivation of the three models studied in this paper from the motion of a cold plasma by means of a multi-scale expansion. Section 3 is devoted to the study of conservation properties of the models derived in Section 2. Focused on well-posedness, the nonlocal Boussinesq system (5) is analyzed in Section 4, while local existence for the bidirectional non-local wave equation (9) close to the equilibrium state is studied in Section 5. Concerning the unidirectional model (10), well-posedness and blow-up criteria for the solutions are derived in Section 6. These results are finished off in Section 7, where wave breaking of some smooth solutions for the unidirectional model is shown. 1.3. Preliminaries and notation. Let us next introduce some notation that will be used throughout the rest of the paper. The functional spaces. For 1 ≤ p ≤ ∞, let L p = L p (R) be the usual normed space of L pfunctions on R with || · || p as the associated norm. For s ∈ R, the inhomogeneous Sobolev space H s = H s (R) is defined as H s (R) ≜ f ∈ L 2 (R) : ∥f ∥ 2 H s (R) = R (1 + ξ 2 ) s | f (ξ)| 2 < +∞ , with norm ∥f ∥ H s = ∥f ∥ 2 L 2 + ∥f ∥Ḣ s , where ∥f ∥ 2Ḣ s = ∥Λ s f ∥ L 2 and Λ s is defined by Λ s f (ξ) = |ξ| s f (ξ), where f is the Fourier transform of f . The space of functions with bounded mean oscillation BMO(R) (cf. [26,27]) is defined by BMO(R) ≜ f ∈ L 1 loc (R) : ∥f ∥ BMO(R) = sup r>0,x 0 ∈R x 0 +r x 0 −r f (x) −f (x) dx < +∞ , wheref (x) = 1 2r x 0 +r x 0 −r f (y) dy. Next, let us introduce two lemmas estimates regarding Sobolev spaces. The first one deals with the so called Kato-Ponce commutator estimate. 21,22]). If f, g ∈ H s W 1,∞ with s > 0, then for p, p i ∈ (1, ∞) with i = 1, . . . , 4 and 1 Lemma 1.1 ( [p = 1 p 1 + 1 p 2 = 1 p 3 + 1 p 4 , we have ∥ [Λ s , f ] g∥ L p ≤ C s,p (∥∂ x f ∥ L p 1 ∥Λ s−1 g∥ L p 2 + ∥Λ s f ∥ L p 3 ∥g∥ L p 4 ), and ∥Λ s (f g)∥ L p ≤ C s,p (∥f ∥ L p 1 ∥Λ s g∥ L p 2 + ∥Λ s f ∥ L p 3 ∥g∥ L p 4 ). The second gives a logarithmic Sobolev inequality. ∥f ∥ L ∞ ≤ C 1 + ∥f ∥ BMO 1 + log(1 + ∥f ∥ H s (R) ) , holds for all f ∈ H s . The Helmholtz operator. We denote by Q the operator (1 − ∂ 2 x ) −1 which acting on functions f ∈ L 2 (R) has the representation Qf (x) = [G ⋆ f ](x) = R G(x − η)f (η) dη, G(x) = 1 2 e −|x| , x ∈ R.(12) Furthermore, the Fourier symbol of Q is Qf (ξ) = 1 1 + |ξ| 2 f (ξ), and by a simple computation we have that Qf ∈ H 2 if f ∈ L 2 and ∂ 2 x Qf (x) = (Q − I) f (x), x ∈ R,(13) where I denotes the identity operator. Constants. Throughout the paper C = C(·) will denote a positive constant that may depend on fixed parameters and x ≲ y (x ≳ y) means that x ≤ Cy (x ≥ Cy) holds for some C. Derivation of the asymptotic models In this section, we derive the three asymptotic models (5), (9), and (10) of system (1) by means of a multi-scale expansion (cf. [3-5, 9, 13, 14]). 2.1. The non-local Boussinesq model. Using (3), the system (1) can be equivalently written as N t + (U N ) x + U x = 0,(14a)U t + U U x + (1 + B)B x 1 + N = 0, (14b) B − N − B x 1 + N x = 0.(14c) In this new variables, the initial data (2) takes the form N (x, 0) = n 0 (x) − 1, U (x, 0) = u 0 (x).(15) Furthermore, we can rewrite (14b) as U t + U U x + B x + BB x + U t N + N U U x = 0. Similarly, (14c) can be expanded B − N − B xx 1 + N + B x (1 + N ) 2 N x = 0, and then it takes the similar form B − N + BN 2 − N 3 + 2N B − 2N 2 − B xx − N B xx + B x N x = 0. Then, (14) becomes N t + U x = −(N U ) x ,(16a)U t + B x = −U U x − BB x − U t N − N U U x ,(16b)B − N − B xx = −BN 2 + N 3 − 2N B + 2N 2 + N B xx − B x N x .(16c) Now, from the ansatz (4) and equating in powers of ϵ, the system (16) leads to a cascade of linear equations for the coefficients N (ℓ) , B (ℓ) , and U (ℓ) . The first terms satisfy N (0) t + U (0) x = 0,(17a)U (0) t + B (0) x = 0,(17b)B (0) − N (0) − B (0) xx = 0,(17c) with initial data from (15). System (17) can be explicitly decoupled and we find N (0) t + U (0) x = 0,(18a)U (0) t + ∂ x (1 − ∂ 2 x ) −1 N (0) = 0,(18b)B (0) = (1 − ∂ 2 x ) −1 N (0) .(18c) Then, (17a), (17b) lead to N (0) tt + L N (0) = 0, U (0) tt + L U (0) = 0.(19) The second term in the expansion solves N (1) t + U (1) x = −(N (0) U (0) ) x ,(20a)U (1) t + B (1) x = −U (0) U (0) x − B (0) B (0) x − U (0) t N (0) ,(20b)B (1) − N (1) − B (1) xx = −2N (0) B (0) + 2(N (0) ) 2 + N (0) B (0) xx − B (0) x N (0) x .(20c) Note that, using (17c), the equation (20c) can be written as B (1) − N (1) − B (1) xx = −2N (0) (B (0) − N (0) − B (0) xx ) − N (0) B (0) xx − B (0) x N (0) x = −(N (0) B (0) x ) x . Then, from (18c), we have B (1) = (1 − ∂ 2 x ) −1 N (1) − N (N (0) B (0) x ) = (1 − ∂ 2 x ) −1 N (1) − N (N (0) N N (0) ).(21) Now, using (18b-c), equation (20b) can be written as U (1) t + B (1) x = −U (0) U (0) x − B (0) (N N (0) ) + (N N (0) )N (0) . Furthermore, from (18b), note that the last two terms can be written as −B (0) (N N (0) ) + (N N (0) )N (0) = −N N (0) B (0) xx , and using (18c) again we obtain U (1) t + B (1) x = −U (0) U (0) x + N N (0) L N (0) .(22) Substitution of B (1) from (21) into (22) leads to U (1) t + N N (1) = −L (N (0) N N (0) ) − U (0) U (0) x + N N (0) L N (0) = −U (0) U (0) x − L , N N (0) N (0) ,(23) where the commutator is given by (8). The approximate model (5) for the truncations h = εN (0) + ε 2 N (1) , v = εU (0) + ε 2 U (1) , is derived from (23) after neglecting O(ε 3 ) terms. 2.2. The non-local single wave equation model. From (19) we observe that N (0) and U (0) satisfy the same linear nonlocal wave equation. Thus, under the extra assumption of having the same initial data, we can conclude that U (0) = N (0) .(24) This will allow to further simplify the system (5). Taking the time derivative of (20a) and using (23) we find that N (1) tt − N N (1) x = (U (0) U (0) x ) x − L , N N (0) N (0) x − (N (0) U (0) ) xt . Furthermore, from (24) and (6), we conclude that N (1) tt + L N (1) = N (0) N (0) x + L , N N (0) N (0) x − 2 N (0) N (0) t x . Considering now the truncation h = εN (0) + ε 2 N (1) , and neglecting contributions of order O(ε 3 ) in the last expression, the single wave equation (9) emerges. 2.3. The unidirectional non local wave model. In this section, we derive the unidirectional asymptotic model (10). We introduce the following far field variables χ = x − t, τ = εt.(25) Using the chain rule we have that ∂ 2 ∂t 2 h(χ(x, t), τ (t)) = h χχ − εh χτ − εh τ χ + ε 2 h τ τ . On the other hand, using the representation (12) of the Helmholtz operator Q, it is not hard to see that Q = (1 − ∂ χχ ) −1 . Therefore, from the change of variables (25) and neglecting terms of order O(ε 3 ) (notice that by construction h ∼ O(ε)), we find that equation (9) becomes (h χ − 2εh τ ) χ + (N h) χ = (3hh χ − [L , N h]h) χ which after integrating in χ, reordering terms and going back by abuse of notation to variables x and t we obtain (10). Conserved quantities In this section we derive some conserved quantities of the models above. We start with the system (5). Note first that we can write [L , N h] h = L (hN h) + 1 2 ∂ x (N h) 2 .(26) Property (26) leads to the formulation of (5) in conservation form ∂ t h v + ∂ x f (h, v) = 0, where f (h, v) = v(1 + h) v 2 2 − N (hN h) + 1 2 (N h) 2 + Qh . Then, if u = Qh and we assume that h, v, u, u x → 0 as |x| → ∞, we obtain the preservation of I 1 (h, v) = R hdx, I 2 (h, v) = R vdx. On the other hand, the following lemma is used below to derive a third conserved quantity. Lemma 3.1. If h → 0 as |x| → ∞, then: R hL (hN h) dx = − 1 2 R h∂ x (N h) 2 dx, R hN hdx = 0. (27a) Proof. We use the Fourier symbols of the operators L , N , the relation ∂ x N = −L , and Plancherel identity to have the following identities: R hL (hN h) dx = R hN hL hdx = − R hN h∂ x N hdx = − 1 2 R h∂ x (N h) 2 dx, R hN hdx = − R (N h) hdx. □ Proposition 3.2. Let h, v be solutions of (5) with h, v → 0 as |x| → ∞ and let I(h, v) = R hvdx = R (uv + u x v x )dx,(28) where u = Qh. Then d dt I(h, v) = 0. Proof. Using (26), we write (5) in the form h t + (hv) x + v x = 0, (29a) v t + vv x + L (hN h) + 1 2 ∂ x (N h) 2 + N h = 0. (29b) Multiplying (29a) by v, (29b) by h, adding these two amounts, using Lemma 3.1, and the hypotheses on h and v, we have R (h t v + v t h + v((1 + h)v) x + vv x h) dx = 0. Finally, using that h, v → 0 as |x| → ∞ again, note that (5) is the Hamiltonian formulation. The proof is direct. R (v((1 + h)v) x + vv x h) dx = R (vv x + v(hv) x + vv x h) dx, = R ∂ ∂ x v 2 2 − v x hv + v x hv dx = 0. □ A final result on Theorem 1. The system (5) admits a Hamiltonian structure ∂ t h v = J δE(h, v), where the solution pair (h, v) is smooth enough and vanishes at infinity, δE = δE δh , δE δv T , denotes the variational derivative, J = −∂ x 0 1 1 0 , and E(h, v) = 1 2 R v 2 (1 + h) + (Bh) 2 + h(N h) 2 dx, Q = B 2 . On the other hand, using (26), the bidirectional model (9) can be written in a conservation form ∂ t (h t + ∂ x h 2 ) − ∂ x N h + ∂ x h 2 2 − ∂ x N (hN h) + ∂ x (N h) 2 2 = 0, which, assuming that h is sufficiently smooth and that h x vanishes at infinity, implies that d dt R (h t + ∂ x h 2 )dx = 0. As far as the unidirectional model (10) is concerned, using again (26), the model is written in conservation form h t + ∂ x 3 4 h 2 + 1 2 N (hN h) − 1 4 (N h) 2 − 1 2 Qh − h 2 = 0,(31) which implies, when h → 0 as x → ±∞, the preservation in time of R hdx. If, in addition, we multiply (31) by h, integrate on R and use Lemma 3.1, then the L 2 norm R h 2 dx, is the second conserved quantity. Finally, the unidirectional model (10) also admits a Hamiltonian structure h t = 1 2ϵ ∂ x δE(h), where now δE = δE δh and E(h) = 1 2 R h 2 − h 3 + (Bh) 2 + h(N h) 2 dx,(32) and where the phase space for (32) involves smooth enough functions h vanishing at infinity. Well-posedness for the non-local Boussinesq system In this section we will show the well-posedness of system (5). Due to the coupled nature of the equations when performing the a priori energy estimates we need to symmetrize the system. To this end, let us introduce the following functional space X = {(h, v) : ∥(h, v)∥ X = ∥(h, v)∥ 2 L 2 (R)×L 2 (R) + √ L ∂ 2 x h 2 L 2 (R) + ∂ 3 x v 2 L 2 (R) < ∞}.(33) If m(ξ) = ξ 2 1+ξ 2 denotes the Fourier multiplier of the operator L (cf. (7)) then in (33) T = √ L denotes the operator with Fourier symbol m(ξ), and therefore it formally satisfies T 2 = L . Then the norm introduced in the definition of X in (33) is related to a classical seminorm in H k (R) as follows. Lemma 4.1. Let k ∈ N and f ∈ L 2 (R) be smooth enough. Then, there exists a constant C > 0 such that ∂ k x f L 2 ≤ C ∥f ∥ L 2 + √ L ∂ k x f L 2 . (34) Proof. Let R > 0 and B(0, R) = {x ∈ R : |x| ≤ R}. Using Parseval identity we have that ∂ k x f 2 L 2 = ∂ k x f 2 L 2 = R ξ 2k |f (ξ)| 2 = B(0,R) ξ 2k |f (ξ)| 2 dξ + R\B(0,R) ξ 2k |f (ξ)| 2 dξ. The first integral can be bounded by B(0,R) ξ 2k |f (ξ)| 2 dξ ≤ R 2k R |f (ξ)| 2 dξ = R 2k ∥f ∥ 2 L 2 .(35) On the other hand, note that for ξ ∈ R \ B(0, R) we have the pointwise bound ξ 2 1 + ξ 2 ≥ R 2 1 + R 2 . Then it holds that R\B(0,R) ξ 2k |f (ξ)| 2 dξ ≤ 1 + R 2 R 2 R\B(0,R) ξ 2+k 1 + ξ 2 ξ k |f (ξ)| 2 dξ ≤ 1 + R 2 R 2 √ L ∂ k x f 2 L 2 . (36) Therefore, choosing for instance R = 1, (35), (36) yield (34). □ Then we see that X is a modified version of H 2 × H 3 . Theorem 2. For (h 0 , v 0 ) ∈ H 2 × H 3 there exist a time 0 < T max and a unique solution (h, v) ∈ C((0, T max ), H 2 × H 3 ) of the ivp of (5) with h(x, 0) = h 0 (x), v(x, 0) = v 0 (x), x ∈ R. Proof. We first focus on obtaining some a priori estimates. We define the energy E(t) = ∥(h, v)∥ 2 L 2 ×L 2 + √ L ∂ 2 x h 2 L 2 + ∂ 3 x v 2 L 2 ,(37) where the norm considered in the space L 2 (R) × L 2 (R) is given by ∥(f, g)∥ L 2 ×L 2 = ||f || 2 L 2 + ||g|| 2 L 2 1/2 , f, g ∈ L 2 (R). Multiplying the first equation of (5) by h, the second by v, adding the resulting equalities and integrating on R we have 1 2 d dt ∥(h, v)∥ 2 L 2 ×L 2 = − R ((hv) x + v x ) h dx − R (vv x + [L , N h]h + N h) v dx. Now integration by parts, the application of Hölder and Young inequalities, and the Sobolev embedding H 1 2 +ϵ (R) → L ∞ (R) for ϵ > 0 lead to 1 2 d dt ∥(h, v)∥ 2 L 2 ×L 2 ≤ C ∥v x ∥ L ∞ ∥h∥ 2 L 2 ≤ ∂ 3 x v 3 L 2 + ∥h∥ 3 L 2 .(38) Next, we deal with the other terms in (37). Multiplying the first equation in (5) by L ∂ 4 x h and integrating we have R L ∂ 4 x hh t dx = − R (hv) x L ∂ 4 x h dx − R v x L ∂ 4 x h dx.(39) On the other hand, multiplying the second equation in (5) by ∂ 6 x v and integrating we obtain R v t ∂ 6 x v dx = − R vv x ∂ 6 x v dx − R [L , N h]h∂ 6 x v dx − R N h∂ 6 x v dx.(40) Since −L = ∂ x N , integrating by parts we find that the last term in (39) is given by − R v x L ∂ 4 x h dx = R v x N ∂ 5 x h dx = − R ∂ 6 x vN h dx.(41) Moreover, we have that R h t L ∂ 4 x h dx = 1 2 d dt √ L ∂ 2 x h 2 L 2 , − R v t ∂ 6 x v dx = 1 2 d dt ∂ 3 x v 2 L 2 .(42) Then, adding (39) and (40), and using (41),(42) lead to 1 2 d dt √ L ∂ 2 x h 2 L 2 + ∂ 3 x v 2 L 2 = − R (hv) x L ∂ 4 x h dx I 1 + R vv x ∂ 6 x v dx I 2 + R [L , N h]h∂ 6 x v dx I 3 . (43) We now estimate each of the integrals I i in (43). First, notice that integration by parts yields I 2 = − R ∂ 2 x (vv x )∂ 3 x v dx = − 3 2 R (∂ 3 x v) 2 v x dx, and thus |I 2 | ≤ C ∥∂ x v∥ L ∞ ∂ 3 x v 2 L 2 ≤ C ∂ 3 x v 3 L 2 ,(44) where in the second inequality we used the Sobolev embedding H 1 2 +ϵ (R) → L ∞ (R) for ϵ > 0. Integrating by parts the first term I 1 we find that I 1 = − R ∂ 3 x (hv)∂ 2 x L h dx = − R ∂ 3 x vh + ∂ 2 x vh x + v x ∂ 2 x h ∂ 2 x L h dx J 1 − R v∂ 3 x h∂ 2 x L h dx J 2 . We use Hölder inequality to estimate J 1 as |J 1 | ≤ C ∥h∥ L ∞ ∂ 3 x v L 2 + ∂ 2 x v L ∞ ∥∂ x h∥ L 2 + ∥v x ∥ L ∞ ∥h xx ∥ L 2 ∂ 2 x L h L 2 ≤ C ∥h xx ∥ L 2 ∂ 3 x v L 2 ∂ 2 x L h L 2 .(45) As for J 2 , using the identity (cf. (13)) L = −∂ 2 x (1 − ∂ 2 x ) −1 = Id − D,(46) we obtain that J 2 = − R v∂ 3 x h ∂ 2 x h − Q∂ 2 x h dx = − 1 2 R v∂ x (∂ 2 x h) 2 dx + R v∂ 3 x hQ∂ 2 x h dx. Using integration by parts in both terms we infer that |J 2 | ≤ C ∥v x ∥ L ∞ ∥h xx ∥ 2 L 2 ≤ ∂ 3 x v L 2 ∥h xx ∥ 2 L 2 .(47) In a similar way, expanding the commutator and using (46) again lead to I 3 = R (L (N hh) − N hL h) ∂ 6 x v dx = − R Q(N hh)∂ 6 x v dx K 1 + R N hQh∂ 6 x v dx K 2 . We rewrite both terms K 1 and K 2 as K 1 = − R (N hh) x L ∂ 3 x v dx, and K 2 = − R (N hQh) xxx ∂ 3 x v dx = − 1 2 R ∂ 4 x (Qh) 2 ∂ 3 x v dx = − R ∂ 2 x (Qh x ) 2 − QhL h ∂ 3 x v dx. Using the fact that, cf. (7), √ L f L 2 ≤ C ∥f ∥ L 2 , f ∈ L 2 ,(48) then the term K 1 can be bounded by |K 1 | ≤ C ∥h xx ∥ 2 L 2 L ∂ 3 x v L 2 ≤ ∥h xx ∥ 2 L 2 ∂ 3 x v L 2 ,(49) for some constant C. As far as K 2 is concerned, by expanding the derivatives, tedious but a straightforward computation and using (48) again shows that |K 2 | ≤ C ∥Qh∥ L ∞ ∂ 3 x v L 2 √ L ∂ 2 x h L 2 ≤ C ∂ 3 x v L 2 ∥h xx ∥ L 2 √ L ∂ 2 x h L 2 ,(50) for some constant C. From (44), (45), (47), (49), and (50) we conclude that 1 2 d dt √ L ∂ 2 x h 2 L 2 + ∂ 3 x v 2 L 2 ≤ C ∂ 3 x v 3 L 2 + ∂ 3 x v L 2 ∥h xx ∥ 2 L 2 + ∂ 3 x v L 2 ∥h xx ∥ L 2 √ L ∂ 2 x h L 2 . (51) Applying Lemma 4.1 and Young's inequality to (51) leads to 1 2 d dt √ L ∂ 2 x h 2 L 2 + ∂ 3 x v 2 L 2 ≤ C √ L ∂ 2 x h 3 L 2 + ∥h∥ 3 L 2 + ∂ 3 x v 3 L 2 ,(52) for some constant C. Combining estimates (38) and (52) yields E ′ (t) ≤ CE 3 2 (t),(53) which ensures a local time of existence t * > 0 such that E(t) ≤ 4E(0), 0 < t < t * . In order to construct the solutions, we first define the approximate problems using mollifiers (cf. proof of Theorem 4). More precisely, the regularized system is given by h ϵ t + J ϵ (J ϵ h ϵ J ϵ v ϵ ) x + J ϵ J ϵ v ϵ x = 0, (54a) v ϵ t + J ϵ (J ϵ v ϵ ∂ x J ϵ v ϵ ) + [L , N h ϵ ] h ϵ + N h ϵ = 0.(54b) By the properties of the mollifiers we can repeat the previous energy estimates and provide the same a priori bounds for the regularized system of (54a)-(54b). Hence, we will find a uniform time of existence T max > 0 for the sequence of regularized problems. To conclude the argument, we pass to the limit. Furthermore, the continuity in time for the solution (instead of merely weak continuity) is obtained as follows: first, the energy estimate (53) yields the strong right continuity at t = 0. Moreover, it is easy to check that changing variablest = −t provides the strong left continuity at t = 0 and hence the continuity in time for the solutions. To conclude let us remark that the uniqueness follows by a classical contradiction argument as in Theorem 4. □ Well-posedness in Sobolev spaces for the bidirectional non-local wave equation In this section, we provide the local-well posedness on the bidirectional non-local wave equation (9). Theorem 3. Let (h 0 , h 1 ) be such that (h 0 − 1, h 1 ) ∈ H 4 × H 3 and ∥h 0 − 1∥ L ∞ < 1/2. Then, there exist 0 < T and a unique solution to (9) (h − 1, h t ) ∈ C([0, T ], H 4 × H 3 ), with initial condition (h 0 , h 1 ). Proof. As before, existence and uniqueness of solutions of (9) are based on deriving useful a priori energy estimates. To this end, we write h = 1 + w and then equation (9) becomes w tt + L w = ((1 + w)w x + [L , N w] w) x − 2 ((1 + w)w t ) x .(55) We define the energy E(t) = ∥w t ∥ 2 H 3 + √ L ∂ 3 x w 2 L 2 + ∥w∥ 2 H 4 .(56) Testing equation (55) against w t and integrating by parts we have 1 2 d dt ∥w t ∥ 2 L 2 + √ L w 2 L 2 = R (w 2 x + (1 + w)w xx )w t dx + R ([L , N w] w) x w t dx − 2 R (1 + w)w tx w t + w x w 2 t dx ≤ CE(t) 3/2 . In particular, d dt ∥w t ∥ 2 L 2 ≤ CE(t) 3/2 . Furthermore, from Cauchy-Schwarz inequality and (56) d dt ∥w∥ 2 L 2 (R) = 2 R w t w dx ≤ 2 ∥w∥ L 2 ∥w t ∥ L 2 ≤ E(t). On the other hand, it holds that 1 2 d dt ∂ 3 x w t 2 L 2 + √ L ∂ 3 x w 2 L 2 = − R (w 2 x + (1 + w)w xx )∂ 6 x w t dx M 1 − R [L , N w] w∂ 7 x w t dx M 2 + 2 R (1 + w)w tx ∂ 6 x w t + w x w t ∂ 6 x w t dx M 3 . In order to estimate each of the integrals M i , we first notice a hiding energy extra term in M 1 . Integrating by parts we obtain M 1 = − 1 2 d dt ∂ 4 x w 2 L 2 − R w 2 x ∂ 6 x w t dx M 12 − R ww xx ∂ 6 x w t dx M 13 . It is easy to check that after integration by parts |M 12 | ≤ R ∂ 3 x (∂ x w) 2 ∂ 3 x w t dx ≤ C ∂ 4 x w L 2 ∥w x ∥ L ∞ + ∂ 3 x w L 2 ∥w xx ∥ L ∞ ∂ 3 x w t L 2 . (57) Similarly, integrating by parts we have that M 13 = − R ∂ 3 x (ww xx )∂ 3 x w t dx = − R w∂ 5 x w∂ 3 x w t dx + l.o.t,(58) where |l.o.t| ≤ C ∂ 3 x w L 2 ∥w xx ∥ L ∞ + ∥w xx ∥ L ∞ ∥w xx ∥ L 2 + ∥w x ∥ L ∞ ∂ 4 x w L 2 ∂ 3 x w t L 2 ,(59) and, after integration by parts again − R w∂ 5 x w∂ 3 x w t dx = 1 2 R w∂ t (∂ 4 x w) 2 dx + R w x ∂ 4 x w∂ 3 x w t dx.(60) The last term in (60) can be bounded as R w x ∂ 4 x w∂ 3 x w t dx ≤ C ∥w x ∥ L ∞ ∂ 4 x w L 2 ∂ 3 x w t L 2 .(61) Thus, from (58), estimates (59),(61) imply that M 13 = 1 2 R w∂ t ∂ 4 x w 2 dx + l.o.t (62) where by means of Young's inequality we find that |l.o.t| ≤ C ∂ 3 x w L 2 ∥w xx ∥ L ∞ + ∥w xx ∥ L ∞ ∥w xx ∥ L 2 + ∥w x ∥ L ∞ ∂ 4 x w L 2 ∂ 3 x w t L 2 ≤ CE(t) 3/2 . In order to estimate the commutator term M 2 , let us recall that from (13) we have M 2 = − R [Q, Qw x ] w∂ 7 x w t dx = − R Q (Qw x w) ∂ 7 x w t dx + 1 2 R ∂ x (Qw) 2 ∂ 7 x w t dx. Using a dualityḢ −4 −Ḣ 4 argument and the fact that Q is continuous between H s (R) and H s+2 (R) for any s ∈ R we readily check that |M 2 | ≤ C ∥Q(Qw x w)∥Ḣ 4 + ∂ x (Qw) 2 Ḣ4 ∂ 7 x w t Ḣ−4 ≤ C ∥Qw x w∥Ḣ 2 + (Qw) 2 Ḣ5 ∂ 3 x w t L 2 ≤ C ∥w x ∥ L 2 ∥w xx ∥ L 2 + ∂ 3 x w 2 L 2 ∂ 3 x w t L 2 ≤ CE(t) 3/2 ,(63) from some constant C, where in the last inequality we used that H s is a Banach algebra for s > 1 2 and Young's inequality. Similarly, splitting M 3 and integrating by parts we infer that |M 3 | = 2 R w tx ∂ 6 x w t + ww tx ∂ 6 x w t + w x w t ∂ 6 x w t dx ≤ C ∂ 3 x w L 2 ∥w tx ∥ L ∞ + ∥w xx ∥ L ∞ ∥w txx ∥ L 2 + ∥w x ∥ L ∞ ∂ 3 x w t L 2 + ∂ 4 x w L 2 ∥w t ∥ L ∞ + ∂ 3 x w L 2 ∥w xt ∥ L ∞ × ∂ 3 x w t L 2 ≤ CE(t) 3/2(64) Hence, combining estimates (57)-(64) and from (56) we conclude that d dt E(t) ≤ C E(t) 3/2 + E(t) + 1 2 R w∂ t ∂ 4 x w dx. Integrating in time leads to E(t) ≤ E(0) + C t 0 E 3/2 (s) + E(s) ds + 1 2 t 0 R w(x, s)∂ s ∂ 4 x w(x, s) 2 dxds. To deal with the latter integral we use Fubini and integrate by parts in time which yields E(t) ≤ E(0) + C t 0 E 3/2 (s) + E(s) ds − 1 2 t 0 R w s (x, s)(∂ 4 x w(x, s)) 2 dxds − 1 2 R w 0 (x)(∂ 4 x w 0 (x)) 2 dx − R w(x, t)(∂ 4 x w(x, t)) 2 dx .(65) Defining, for T > 0 E T = sup 0≤t≤T E(t) = sup 0≤t≤T ∥w t ∥ 2 H 3 + √ L ∂ 3 x w 2 L 2 + ∥w∥ 2 H 4 , taking supremum in time in (65) and Hölder's inequality lead to E T ≤ E 0 + Ct E 3/2 T + E T + 1 2 ∥w 0 ∥ L ∞ ∂ 4 x w 0 2 L 2 + 1 2 sup 0≤t≤T ∥w∥ L ∞ ∂ 4 x w 2 L 2 + 1 2 sup 0≤t≤T t 0 ∥w s ∥ L ∞ ∂ 4 x w 2 L 2 ds.(66) Furthermore, writing w(x, t) = w 0 (x) + w(x, t) − w 0 (x) = w 0 (x) + t 0 ∂ s w(x, s) ds, then Sobolev embedding yields sup 0≤t≤T ∥w∥ L ∞ ≤ ∥w 0 ∥ L ∞ + t sup 0≤t≤T ∂ 3 x w t L 2 .(67) Therefore combining (66), (67), and Young's inequality we find that E T ≤ E 0 + E 3/2 0 + Ct E 3/2 T + E T + ∥w 0 ∥ L ∞ sup 0≤t≤T ∂ 4 x w 2 L 2 ,(68) for some constant C. Now taking w 0 with ∥w 0 ∥ L ∞ = 1 2 , from the definition of E T and (68) it holds that E T ≤ 2 E 0 + E 3/2 0 + 2Ct E 3/2 T + E T , which is valid for t ∈ (0, t 1 ), t 1 = min{1, T }. (Sharper inequalities can be obtained, if necessary, from smaller choices of ∥w 0 ∥ L ∞ ). Noticing that E q T ≤ E 2 T + 1 for q = 1 and q = 3 2 and the fact that t ∈ (0, t 1 ), t 1 = min{1, T } (in particular 0 < t < 1), we find that the polynomial estimate E T ≤ N 0 + 4CtE 2 T ,(69) holds. Here we have used the notation N 0 = 2 E 0 + E 3/2 0 + 2C . Similar polynomial estimates as (69) have been derived in [8]. Let us define the polynomial P t (y) = 4Cty 2 − y + N 0 , so that P t (E T ) ≥ 0. The roots of P t are computed as Taking 0 < t ≪ 1 (for instance 0 < t = 1 32CN 0 ) we have (cf. Figure 1) y ± = 1 ± √ 1 − 16CtN 0 8Ct .0 < y − = 4N 0 1 − 1/2 < y + = 4N 0 1 + 1/2 and 0 < y − < 4N 0 < y + . Furthermore, the application t → E t is continuous for t ∈ [0, T ). This, together with the fact that E 0 < N 0 and P t (N 0 ) > 0, implies that ∀t ∈ (0, t ⋆ ), E T < 4N 0 ,(70) where t ⋆ = min{ 1 32CN 0 , 1, T }. Similarly as before, in order to perform the a priori estimates rigorously and construct the local existence of solutions we follow a regularization procedure. More precisely, the regularized version of (55) takes the form w ϵ tt + L w ϵ = (J ϵ ((1 + J ϵ w ϵ )J ϵ w ϵ x ) + [L , N w ϵ ] w ϵ ) x − 2J ϵ ((1 + J ϵ w ϵ )J ϵ w ϵ t ) x .(71) Repeating the same a priori estimates for the regularized equation (71) we can deduce the analogue of (70) namely ∀t ∈ (0, t ⋆ ), E T < 4N 0 ,(72) where t ⋆ = min{ 1 32CN 0 , 1, T ϵ }. Notice that the time of existence of the solution depends a priori on the regularization parameter ϵ. Nevertheless, the following argument shows that (72) holds independently of the parameter ϵ. Indeed, take T ϵ the first time such that E T ϵ = 8N 0 . We observe that the precise choice of the quantity 8N 0 is not special. One could choose also any other quantity bigger than 4N 0 to make the argument work. If the previous equality does not hold, that is, E T ϵ < 8N 0 , this implies that T ϵ = ∞, and hence we conclude (since t ⋆ is the minimum) that t ⋆ is independent of ϵ. On the other hand if T ϵ is finite then t ⋆ = min{ 1 32CN 0 , 1, T ϵ } = min{ 1 32CN 0 , 1}. Indeed, this assertion follows from invoking the continuity of E T of the regularized problem, and using that by construction E T ϵ = 8N 0 and E t ⋆ < 4N 0 . This concludes the proof. □ 6. Well-posedness in Sobolev spaces for unidirectional non-local wave model The main goal of this section is to prove the local well-posedness of equation (10) in L 2 based Sobolev spaces. Note first that, using (13), we have −L = Q − 1, and (10) can be rewritten as (we set ε = 1 for simplicity) h t = − 1 2 (3hh x + [Q, Qh x ]h − Qh x − h x ) .(73) For this equation, we show the following local well-posedness result. Theorem 4. Let s > 3 2 and h 0 ∈ H s (R) with mean zero. Then there exist T max > 0 and a unique local solution h ∈ C([0, T max ), H s (R)) of (73). Proof. The proof follows from the combination of appropriate a priori energy estimates and the use of a suitable approximation procedure using mollifiers, cf. [23]. Thus, we first focus in deriving a priori energy estimates and later comment briefly on the approximation procedure to construct the solution. We begin by reminding (cf. Section 3) that the mean property is conserved in time by the solutions R h(x, t) dx = R h 0 (x) dx, as well as the L 2 norm R h 2 (x, t) dx = R h 2 0 (x) dx. Furthermore, applying Λ s to (73), multiplying by Λ s h and integrating we obtain 1 2 d dt ∥h∥ 2Ḣ s = − 3 2 R Λ s (hh x )Λ s h dx − 1 2 R Λ s ([Q, Q x h]h) Λ s h dx(74)+ 1 2 R Λ s Qh x Λ s h dx + 1 2 R Λ s h x Λ s h dx. Using the self-adjointness of the operator Q, a straightforward computation shows that the last two terms on the right hand-side of (74) are zero. The first term can be estimated by means of the classical Kato-Ponce commutator as follows. Using integration by parts, we rewrite R Λ s (hh x )Λ s h dx = R [Λ s , h]h x Λ s h dx + R hΛ s h x Λ s h dx = R [Λ s , h]h x Λ s h dx − 1 2 R h x |Λ s h| 2 dx. Now, invoking the first equation in Lemma 1.1 with p = 2, p 1 = ∞, p 2 = 2, we have that R Λ s (hh x )Λ s h dx ≲ ∥[Λ s , h]h x ∥ L 2 ∥Λ s h∥ L 2 + ∥h x ∥ L ∞ ∥Λ s h∥ 2 L 2 ≲ ∥h x ∥ L ∞ ∥Λ s h∥ 2 L 2 .(75) The second term on the right hand side in (74) can be bounded as follows. Expanding the commutator and using the self-adjointness of Q yield R Λ s ([Q, Q x h]h) Λ s h dx = R Λ s (Qh x h)Λ s Qh dx − R Λ s (Qh x Qh)Λ s h dx.(76) Then applying the second estimate in Lemma 1.1 with p = 2, p 1 , p 3 = ∞, p 2 , p 4 = 2, the Sobolev embedding H 1 2 +ϵ (R) → L ∞ (R) for ϵ > 0, and the fact that Q is continuous between H s (R) and H s+2 (R) for any s ∈ R we find that R Λ s (Qh x h)Λ s Qh dx ≲ ∥Λ s (Qh x h)∥ L 2 ∥Λ s Qh∥ L 2 ≲ (∥Λ s Qh x ∥ L ∞ ∥h∥ L 2 + ∥Qh x ∥ L ∞ ∥Λ s h∥ L 2 ) ∥Λ s Qh∥ L 2 ≲ ∥h∥ H s− 1 2 +ϵ ∥h∥ L 2 + ∥h∥ H − 1 2 +ϵ ∥h∥ H s ∥h∥ H s−2 . Similarly, one can show that R Λ s (Qh x Qh)Λ s h dx ≲ ∥h∥ H s− 1 2 +ϵ ∥h∥ H −2 + ∥h∥ H − 1 of regularized problems. Using compactness arguments and passing to the limit conclude the proof of existence. The continuity in time for the solution (instead of merely weak continuity) is obtained by classical arguments (cf. [23]): On the one hand, the differential equation (81) yields the strong right continuity at t = 0. On the other hand, it is easy to check that changing variables t = −t, we can repeat once again the same bounds and provide the strong left continuity at t = 0. Combining both arguments shows the continuity in time of the solution. As for uniqueness, let h 1 , h 2 ∈ C([0, T max ), H s (R)) be two solutions corresponding to the same initial condition and denote h = h 2 − h 1 . Then h satisfies ∂ t h = − 1 2 3h 2 h 2 x − 3h 1 h 1 x + [Q, Qh 2 x ]h 2 − [Q, Qh 1 x ]h 1 − Q h x − h x .(83) Then, multiplying (83) by h and integrating, we have 1 2 d dt h 2 L 2 = − 3 2 R hh 2 x + h 1 h x h dx + R [Q, Qh 2 x ]h 2 − [Q, Qh 1 x ]h 1 h dx. Using (27b), it is not hard to check that each commutator can be rewritten as a Burgers-type nonlinear-term and therefore R [Q, Qh 2 x ]h 2 − [Q, Qh 1 x ]h 1 h dx ≤ C h 1 L ∞ + h 2 L ∞ h 2 L 2 ≤ C, for some constant C. Similarly, integrating by parts, R hh 2 x + h 1 h x h dx ≤ C ∂ x h 1 L ∞ + ∂ x h 2 L ∞ h 2 L 2 . Defining β(t) = h 1 L ∞ + h 2 L ∞ + ∂ x h 1 L ∞ + ∂ x h 2 L ∞ , then 1 2 d dt h 2 L 2 ≤ Cβ(t) h 2 L 2 . Uniqueness holds applying Grönwall inequality. □ Remark 1. One can readily check that a direct consequence of the derived estimates in the proof of Theorem 4 provides the following blow-up criterion when combined with the logarithmic Sobolev inequality (cf. Lemma 1.2) which we state as a theorem for the sake of clarity. Proof. The result follows as a direct consequence of estimate (79). Using the logarithmic inequality in Lemma 1.2 we find that d dt ∥h∥ H s ≤ C 1 + ∥h x ∥ BMO 1 + log(1 + ∥h∥ H s ) ∥h∥ H s ,(85) for some constant C. From (85), it holds that 1 + ∥h∥ H s ≤ (1 + ∥h 0 ∥ H s )exp (1 + CT max ) exp(C Tmax 0 (∥hx(τ )∥ BMO dτ ) . Therefore, if there exists L > 0 such that Tmax 0 ∥h x (τ )∥ BMO dτ < L, then 1 + ∥h∥ H s ≤ (1 + ∥h 0 ∥ H s )exp (1 + CT max ) exp(CL) . On the contrary, if In this section, we investigate the possible wave breaking phenomena for equation (10), that is, the formation of an infinite slope in the solution in the x-direction. As we did for the wellposedness result in Section 6, it is more convenient to work with the alternative formulation (73). The following lemma shows that for the maximal time of existence T max > 0, we have that the solutions remains bounded. More precisely, ∥h(t)∥ L ∞ (R) < ∞.(86) Proof. To establish the L ∞ bound, we follow a pointwise method (cf. [6,7]). Due to Theorem 4, we have that h ∈ C([0, T max ), H s ) ∩ C 1 ([0, T max ), H s−1 ), hence by the Sobolev embedding theorem, if s > 7/2 we have that h ∈ C 1 ([0, T max ) × R). In particular, m(t) = inf x∈R h(x, t) = h(x t , t), M (t) = sup x∈R h(x, t) = h(x t , t), for t > 0, (for some x t , x t ) are Lipschitz functions. Following [7], one can readily check that M (t) satisfies |M (t) − M (s)| = h(x t , t) − h(x s , s) if M (t) > M (s) h(x s , s) − h(x t , t) if M (s) > M (t) ≤ h(x t , t) − h(x t , s) if M (t) > M (s) h(x s , s) − h(x s , t) if M (s) > M (t) ≤ |∂ t h(x t , z)||t − s| if M (t) > M (s) |∂ t h(x s , z)||t − s| if M (s) > M (t) ≤ max y,z |∂ t h(y, z)||t − s|. Similarly |m(t) − m(s)| ≤ max y,z |∂ t h(y, z)||t − s|. From Rademacher's theorem it holds that M (t) and m(t) are differentiable in t almost everywhere. Furthermore M ′ (t) = lim δ→0 h(x t+δ , t + δ) − h(x t , t) δ = lim δ→0 h(x t+δ , t + δ) − h(x t , t) ± h(x t+δ , t) δ ≤ lim δ→0 h(x t+δ , t + δ) − h(x t+δ , t) δ ≤ ∂ t h(x t , t). In a similar fashion, we obtain that M ′ (t) = lim δ→0 h(x t+δ , t + δ) − h(x t , t) δ = lim δ→0 h(x t+δ , t + δ) − h(x t , t) ± h(x t , t + δ) δ ≥ lim δ→0 h(x t , t + δ) − h(x t , t) δ ≥ ∂ t h(x t , t). As a consequence M ′ (t) = ∂ t h(x t , t) a.e.(87) Similarly m ′ (t) = ∂ t h(x t , t) a.e. Therefore, from (87), evaluating (73) at x = x t and noticing that h x (x t , t) = 0, we have M ′ (t) = − 1 2 ([Q, Qh x (x t )]h(x t ) − Qh x (x t )) .(88) Moreover, from (12), the estimates ∥G∥ L ∞ = 1 2 , ∥∂ x G∥ L ∞ = 1 2 , and Young's inequality, it holds that Qh(x t ) ≤ ∥G∥ L ∞ (R) ∥h∥ L 2 (R) = 1 2 ∥h∥ L 2 (R) ,(89)[Q, Qh x (x t )]h(x t ) ≤ ∥G∥ L ∞ (R) ∥Qh x h∥ L 2 (R) + ∥Qh x ∥ L ∞ (R) ∥Qh∥ L ∞ (R) ≤ 1 2 ∥h∥ 2 L 2 .(90) From (88), the estimates (89), (90) and the preservation of the L 2 norm by the solutions of (73) imply that M ′ (t) ≤ 1 4 1 + ∥h 0 ∥ 2 L 2 , and therefore M (t) ≤ M (0) + 1 4 1 + ∥h 0 ∥ 2 L 2 t.(91) for t ∈ [0, T max ). In a similar way, we have m(t) ≥ m(0) − 1 4 1 + ∥h 0 ∥ 2 L 2 t.(92) for t ∈ [0, T max ). Thus combining (91) and (92) ∥h(t)∥ L ∞ (R) ≤ ∥h 0 ∥ L ∞ (R) + 1 4 1 + ∥h 0 ∥ 2 L 2 T max < ∞,(93) and (86) holds. □ Next, let us state the wave breaking result. Theorem 6. Let s > 9 2 , h 0 ∈ H s (R), and let h be the solution of the ivp of (73) with initial condition h 0 . Assume that inf x∈R h 0,x (x) ≤ −H 0 ,(94) for some positive constant H 0 , which depends on ∥h 0 ∥ L 2 (R) , ∥h 0 ∥ L ∞ and specified below. Then there exists T b < ∞ such that lim inf t→T b inf x∈R h x (x, t) = −∞.(95) Proof. Similarly as in the proof of Lemma 7.1, we follow a pointwise method to derive an ODE which breaks down in finite time. Again, due to Theorem 4, we have that h ∈ C([0, T max ), H s ) ∩ C 1 ([0, T max ), H s−1 ), hence by the Sobolev embedding theorem, if s > 9/2 then h x ∈ C 1 ([0, T max )× R). We begin by setting m(t) = inf x∈R h x (x, t) = h x (x t , t), for t ∈ [0, T max ), for some x t . Arguing as in Lemma 7.1, one can check that m(t) is Lipschitz and hence via Radamacher's Theorem it is inferred that m ′ (t) = ∂ t h x (x t , t), a.e. Differentiating (73) with respect to x and using (46) yield h xt = − 1 2 3h 2 x + 3hh xx + ∂ x [Q, Qh x ]h − Qh + h − h xx .(96) Evaluating (96) at x = x t and noticing that h xx (x t , t) = 0 we have m ′ (t) = − 3 2 m(t) 2 + K(t),(97) with K(t) = 1 2 ∂ x [Q, Qh x ]h(x t , t) − Qh(x t , t) + h(x t , t) . We now estimate the quantity K(t). First notice, from (12), that ∥Qh + h∥ L ∞ ≤ ∥G ⋆ h∥ L ∞ + ∥h∥ L ∞ ≤ ∥G∥ L ∞ ∥h∥ L 2 = 1 2 ∥h∥ L 2 + ∥h∥ L ∞ .(98) On the other hand, from the Sobolev embedding H 1 2 +ϵ (R) → L ∞ (R) for ϵ > 0 and the fact that Q is continuous between H s (R) and H s+2 (R) for any s ∈ R, then the commutator term is bounded by ∂ x [Q, Qh x ]h(x t , t) ≤ ∥[Q, Qh x ]h∥ The first term on the right hand-side of (99) is bounded by ∥Qh x h∥ H − 3 2 +ϵ ≤ C ∥h∥ 2 L 2 (R) ,(100) and the second term can be estimated from the Sobolev algebra property as (Qh) 2 H 3 2 +ϵ ≤ C ∥Qh∥ 2 H 3 2 +ϵ ≤ C ∥h∥ 2 L 2 (R) ,(101) for some constant C. The bounds (98)-(101), the L ∞ estimate (93), and the time preservation of the L 2 norm by h lead to K(t) ≤ C ∥h∥ 2 L 2 (R) + ∥h∥ L 2 (R) + ∥h∥ L ∞ (R) ≤ C ∥h 0 ∥ L ∞ (R) + 1 + ∥h 0 ∥ 2 L 2 (t + 1) . Using (102), then (97) implies m ′ (t) ≤ − 3 2 m(t) 2 + A + Bt,(103) where A and B are the positive constants, depending on ∥h 0 ∥ L ∞ , ∥h 0 ∥ L 2 , determined in (102). In order to show the wave breaking result, let us assume that the initial data is such that (cf. [28]) m(0) ≤ −C,(104) for some positive constant C such that C 2 > 4A. Then 2A − m(0) 2 /2 < 0 and m ′ (t) t=0 ≤ −m(0) 2 − 1 2 m(0) 2 + 2A ≤ −m(0) 2 < 0. Therefore there exists a sufficiently small 0 < δ such that m ′ (t) < 0, for 0 ≤ t < δ. This implies m(t) < m(0) ≤ −C and similarly 2A − m(t) 2 /2 < 0 for 0 ≤ t < δ. Thus, if 0 < t ≤ min{δ, A/B} from (103) it holds that m ′ (t) ≤ −m(t) 2 − 1 2 m(t) 2 + 2A ≤ −m(t) 2 . This leads to m(t) ≤ m(0) m(0)t + 1 .(105) Taking the initial datum such that 1 −m(0) ≤ A B ,(106) note that (104), (106) define a constant H 0 for which m(0) satisfies (94), and (105) implies the existence of a time T b where (95) holds. □ Lemma 1. 2 2( [10]). Let s > 1 2 . There exists a constant C = C(s) > 0 such that Figure 1 . 1Positive real roots of the polynomial P t (y) for t ≪ 1. Theorem 5 ( 5Blow-up criteria). Let s > 3 2 , h 0 ∈ H s (R) with zero mean and let T max > 0 be the lifespan associated to the solution h to (73) with h(x, 0) = h 0 (x). Then h blows-up in finite time T max if and only ifTmax 0 ∥h x (τ )∥ BMO dτ = ∞. (t)∥ BMO dt = ∞, by means of the embedding H 1 2 (R) ⊂ BMO(R), we deduce that the solution h(x, t) will blow up in finite time and (84) follows. □ 7. A wave breaking result for the unidirectional non-local wave model Lemma 7.1. Let s > 7/2, h 0 ∈ H s (R), and let T max > 0 be the maximal time of existence of the unique solution h of (73) given by Theorem 4. Thensup t∈[0,Tmax) we conclude thatsup t∈[0,Tmax) +ϵ ∥h∥ H s−2 ∥h∥ H s . Thus, where we have used that ∥h∥for some constant C. Therefore, combining estimates (75), (77) leads towhich in particular gives the following inequalityUsing the Sobolev embedding Hfor some constant C independent of t. Defining E(t) = ∥h∥ 2 H s , estimate (80) leads to the following differential equationwith C = C s > 0 which ensures a uniform time of existence T max > 0 such thatOnce this uniform time of existence has been obtained, the local existence result follows classical regularization procedure (cf.[23]) as follows. First, we consider a symmetric and positive mollifier J ∈ C ∞ c , J (x) = J(|x|) such that R J = 1. 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Departamento de Análisis Matemático y Instituto de Matemáticas y Aplicaciones (IMAULL)Spain. Email address: [email protected] de Análisis Matemático y Instituto de Matemáticas y Aplicaciones (IMAULL), Uni- versidad de La Laguna C/. Astrofísico Francisco Sánchez s/n, 38200 -La Laguna, Spain. Email address: [email protected] Los Castros s/n. Matemáticas Departamento De, Santander, SpainEstadística y Computación, Universidad de Cantabria. AvdaDepartamento de Matemáticas, Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n, Santander, Spain.
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[ "Data-Driven Convex Approach to Off-road Navigation via Linear Transfer Operators", "Data-Driven Convex Approach to Off-road Navigation via Linear Transfer Operators" ]
[ "Joseph Moyalan ", "Yongxin Chen ", "Umesh Vaidya " ]
[]
[]
We consider the problem of optimal control design for navigation on off-road terrain. We use a traversability measure to characterize the difficulty of navigation on off-road terrain. The traversability measure captures terrain properties essential for navigation, such as elevation maps, roughness, slope, and texture. The terrain with the presence or absence of obstacles becomes a particular case of the proposed traversability measure. We provide a convex formulation to the off-road navigation problem by lifting the problem to the density space using the linear Perron-Frobenius (P-F) operator. The convex formulation leads to an infinite-dimensional optimal navigation problem for control synthesis. We construct the finite-dimensional approximation of the optimization problem using data. We use a computational framework based on the data-driven approximation of the Koopman operator. This makes the proposed approach data-driven and applicable to cases where an explicit system model is unavailable. Finally, we apply the proposed navigation framework with single integrator dynamics and Dubin's car model.
10.1109/lra.2023.3262200
[ "https://export.arxiv.org/pdf/2210.00723v2.pdf" ]
252,683,019
2210.00723
277121042021f8225f4da1bba8e5ab35039752ef
Data-Driven Convex Approach to Off-road Navigation via Linear Transfer Operators May 2023 Joseph Moyalan Yongxin Chen Umesh Vaidya Data-Driven Convex Approach to Off-road Navigation via Linear Transfer Operators May 20231Index Terms-Motion and Path PlanningOptimization and Optimal ControlModel Learning for Control We consider the problem of optimal control design for navigation on off-road terrain. We use a traversability measure to characterize the difficulty of navigation on off-road terrain. The traversability measure captures terrain properties essential for navigation, such as elevation maps, roughness, slope, and texture. The terrain with the presence or absence of obstacles becomes a particular case of the proposed traversability measure. We provide a convex formulation to the off-road navigation problem by lifting the problem to the density space using the linear Perron-Frobenius (P-F) operator. The convex formulation leads to an infinite-dimensional optimal navigation problem for control synthesis. We construct the finite-dimensional approximation of the optimization problem using data. We use a computational framework based on the data-driven approximation of the Koopman operator. This makes the proposed approach data-driven and applicable to cases where an explicit system model is unavailable. Finally, we apply the proposed navigation framework with single integrator dynamics and Dubin's car model. I. INTRODUCTION N AVIGATION problem is one of the most critical research fields in the robotics community. More recently, the problem of off-road navigation, driven by robotics applications in an unstructured environment, has received much attention. The objective is to drive a robot/vehicle from some initial set to the desired target set through a terrain where traversability varies continuously over the entire domain of interest. This is in contrast to navigation in the presence of obstacles where the regions with obstacles are prohibited and hence not traversable. There is extensive literature on navigation in the presence of obstacles. Navigation function and potential function are used for navigation in the presence of obstacles [1]- [4]. While the potential function could have local minima preventing the navigation from initial set to the target, the navigation function is hard to find. The control barrier functions (CBFs) are also used for navigation with safety constraints [5]. CBFs combine ideas from the control Lyapunov function and barrier certificates for invariance to ensure safety. However, finding CBFs suffer from the same challenges as finding control Lyapunov function and cannot be easily adapted for navigation in off-road terrain where the definition of safety itself is nebulous. The problem of off-road navigation has attracted more interest recently. In [6], perception is used to determine the terrain traversability and local control strategy for navigation. Most of the current literature on this topic has been using sensor data from LIDAR, cameras, and GPS/IMUs to map the off-road terrain to generate a traversability map [7]- [10]. An existing algorithm such as A ⋆ is used to design traversable paths in the off-road environment. However, due to the nonconvex nature of the traversability map and hence the cost, the problem becomes nonconvex and, therefore, difficult to solve with no guarantee of global optimality. One of this paper's main contributions is providing a convex formulation to the off-road navigation problem. The convex formulation is made possible by transforming the problem in the dual space of densities. The formulation of optimal control problem in the dual space of densities is proposed in [11], [12], and its extension to navigation problem in the presence of deterministic and stochastic obstacles is studied in [13]- [15]. This paper focuses on the off-road navigation problem for a given traversability map. The terrain traversability map includes information about the difficulty level in navigating. The terrain's traversability measure depends on terrain parameters such as elevation map, roughness, slope, and texture. Therefore, we have utilized the normalized elevation map while constructing the traversability measure. The convex formulation leads to an infinite-dimensional convex optimization problem for the off-road navigation problem. First, we use data to construct the finite-dimensional approximation of the infinite-dimensional convex problem. Then, we use a computational framework based on the datadriven approximation of linear Koopman and Perron-Frobenius (P-F) operators for the finite-dimensional approximation of the infinite-dimensional convex optimization problem. The second main contribution is providing a numerically efficient computational algorithm for the data-driven approximation of the P-F operator, preserving some natural properties of this operator. Finally, we demonstrate the application of the developed framework for off-road navigation of vehicle dynamics with the Dubin car model. We also compare the results obtained using our proposed approach with the existing A ⋆ algorithm. The study's main finding is that the traversability cost associated with A ⋆ is more than one computed using our proposed approach. The rest of the paper is structured as follows. Section II consists of problem formulation, and we discuss the main results in Section III. Then, in Section IV, we develop the computational framework based on the linear operator framework. Finally, we present the simulation results in Section V, and a conclusion is in Section VI. II. PROBLEM FORMULATION FOR OFF-ROAD NAVIGATION This section defines the traversability map, which will be later used in the convex formulation of the navigation problem. We will also motivate the choice of the cost function for off-road navigation. Let us consider the following dynamical system in control affine form aṡ x = f (x) + g(x)u(1) where x ∈ X ⊂ R n and u ∈ U ⊂ R m are the states and control input respectively. We assume that f (x), g(x) ∈ C 1 (X, R n ), i.e. the space of continuously differentiable functions on X. The dynamical control system is assumed to model the control dynamics of the vehicle. The control-affine form is not restrictive, and this will typically be the case for the robotics and vehicle dynamics application [16], [17]. Notations: We consider B(x) to be the Borel σ-algebra on X and M(X) as the vector space of real-valued measures on B(X). Let L ∞ (X) and L 1 (X) be the space of essentially bounded and integrable functions on X respectively. The notations in bold and lower case will represent vectors and notations in bold and upper case will represent matrices. Also, s t (x) is the notation for the trajectory of feedback systeṁ x = f (x) + g(x)k(x) starting from initial condition x at time t ∈ R, where u = k(x) ∈ C 1 (X, R m ) is the feedback input. Similarly, s −t (x) represents the closed-loop trajectory as the function of initial condition x backward in time. A. Traversability Map We assume that the traversability description of the terrain is captured by a nonnegative function b(x) ∈ L 1 (X). We assume that the function b(x) captures the information of the elevation map, terrain roughness, slope, and terrain texture. The construction of such a map is an active area of interest where onboard sensors on the vehicles such as vision, LIDAR, and IMU, as well as drone sensory images, can be used to construct such maps [18]- [20]. We propose the following definition of traversability measure, which captures the relative degree of difficulty of traversing unstructured terrain. Definition 1. Let µ b ∈ M(X) be the associated traversability measure, i.e., dµ b (x) = b(x)dx, where b(x) ≥ 0 is assumed to be an integrable function and is zero on the final target set, X T . For any set A ∈ B(X), the traversability of the set A is defined using b(x) as Trav(A) := A b(x)dx =: µ b (A).(2) Trav(A) captures the relative difficulty of traversing through the region A ⊂ X. In particular, if µ b (A 1 ) < µ b (A 2 ) where A i ∈ B(X), then the region A 2 is more difficult to traverse than region A 1 . It is easy to see that the above definition of traversability measure also captures the information of binary obstacles. In particular, if X u is an obstacle set, then we can describe it using b(x) = 1 λ(X u ) 1 Xu (x)(3) where λ(·) is the Lebesgue measure and 1 Xu is the indicator function of the set X u . The main objective of this paper involves determining the control inputs u to navigate the vehicle dynamics from some initial state X 0 to some final target set X T while keeping the traversability cost below some threshold, say γ, i.e., ∞ 0 b(x(t))dt ≤ γ(4) where x(t) is the trajectory of the control system (1). In this paper, we are interested in the asymptotic navigation problem, where the objective is to find the shortest distance path to the target and the control cost. In particular, we consider the following cost function min u V (x) = min u ∞ 0 q(x(t)) + u ⊤ Rudt.(5) where q(x) is the distance function which is zero at the target set X T , and R > 0 is the positive definite matrix. Instead of minimizing the cost function from every initial condition x as in (5), our proposed convex formulation relies on minimizing the following cost function averaged over all states x ∈ X 0 . min u J(µ 0 ) = min u X V (x)dµ 0 (x)(6) where µ 0 is the measure capturing the distribution of the initial state. In particular, for the initial state of the vehicle in set X 0 , we have measure µ 0 supported on set X 0 . The form of the cost function where V (x) is averaged over the state x ∈ X 0 plays a fundamental role in the convex formulation of optimal navigation problem in the space of density. In the rest of the paper, we will assume that µ 0 is absolutely continuous with density function h 0 , i.e., dµ0 dx = h 0 (x). For example if µ 0 is supported on initial set then h 0 (x) = 1 X0 (x) i.e., indicator function of set X 0 . The objective is to find the feedback controller k(x) to minimize the cost function in (6). Appropriate conditions on the initial measure µ 0 are necessary to ensure the cost function is finite. We make sure that the density function h 0 is finite and positive semi-definite on X and h 0 ∈ L 1 (X) ∩ C 1 (X). Along with minimizing the cost function, it is also of interest to avoid certain obstacle sets, X u , and limit the control authority, i.e., |u j | ≤ L j . The obstacle avoidance constraints for almost every trajectory starting from the initial set X 0 are written as X 1 Xu (x(t))dµ 0 (x) = 0, ∀t ≥ 0 where x(t) is the solution of system (1) starting from initial condition x. With the above definition, we can state the problem statement for optimal off-road navigation using the traversability map as given below. Problem 1. (Optimal off-road Navigation Problem) Navigate almost every system trajectory for (1) starting from the initial set X 0 to the target set X T while avoiding the obstacle set X u such that cost for traversing is kept below some threshold (γ) and the following cost is minimized. min u X ∞ 0 q(x(t)) + u ⊤ Ru dtdµ 0 (x) (7a) s.t. X 1 Xu (x(t))dµ 0 (x) = 0, ∀t ≥ 0 (7b) X ∞ 0 b(x(t))dtdµ 0 (x) ≤ γ (7c) |u j | ≤ L j , j = 1, . . . m. (7d) x = f (x) + g(x)u, lim t→∞ x(t) ∈ X T . (7e) where q(x) is the distance from x to the target set X T . So the objective is to find the shortest path to the target set while keeping the traversability cost below a certain threshold γ. The following section proves that the optimal off-road navigation problem, defined in Problem 1, can be written as a convex optimization over the density space. III. CONVEX FORMULATION OF OPTIMAL OFF-ROAD NAVIGATION We make the following assumption on the control dynamical system and the controller for the system (1). Assumption 1. For the optimal off-road navigation problem, we assume that the optimal control input is feedback in nature, i.e., u = k(x) ∈ C 1 (X), such that the cost function corresponding to this input is finite. With the above assumption, we can write the feedback control system in the forṁ x = f (x) + g(x)k(x) =: F c (x).(8) Now, we will consider the following definition of almost everywhere (a.e.) stability as introduced in [21]. Definition 2. [Almost everywhere (a.e.) stability] The equilibrium point or an attractor set of the system (8) represented by A is said to be a.e. stable w. r.t. measure µ 0 ∈ M(X) if µ 0 {x ∈ X : lim t→∞ s t (x) / ∈ A} = 0.(9) This paper's main results ensure that the feedback control obtained for the navigation is a.e. stable with respect to the target set of (8). We next introduce the following definitions of linear operators [22]. Definition 3 (Koopman Operator). U t : L ∞ (X) → L ∞ (X) for (8) is given by [U t φ](x) = φ(s t (x)),(10) where φ is a test function in the lifted function space L ∞ (X) C 1 (X). The Koopman generator for (8) is given by lim t→0 [U t φ](x) − φ(x) t = F c (x) · ∇φ(x) =: U Fc φ.(11) Definition 4 (Perron-Frobenius Operator). P t : L 1 (X) → L 1 (X) for (8) is given by [P t ϕ](x) = ϕ(s −t (x)) ∂s −t (x) ∂x ,(12) where |.| stands for the determinant and ϕ is a test function. The P-F generator for (8) is given by lim t→0 [P t ϕ](x) − ϕ(x) t = −∇ · (F c (x)ϕ(x)) =: P Fc ϕ. (13) These two operators are dual to each other as X [U t φ](x)ϕ(x)dx = X [P t ϕ](x)φ(x)dx.(14) These two operators enjoy positivity and Markov properties used in the finite-dimension approximation. A. Obstacle avoidance constraints The first result of this paper allows us to write the obstacle avoidance constraints (7b) in the integral form. Lemma 1. For the dynamical system (8), if ∞ 0 X 1 Xu (s t (x))dµ 0 (x)dt = 0,(15)then X 1 Xu (s t (x))h 0 (x)dx = 0, ∀t ≥ 0,(16) i.e., the amount of time system trajectories spends in the region X u starting from the positive measure set of initial condition corresponding to the initial set, X 0 , with density h 0 (x) is equal to zero. Proof. Proof by contradiction. Assume (16) is not true, i.e., there exists some time t 0 for which X 1 Xu (s t0 (x))h 0 (x)dx = X [U t0 1 Xu ](x)h 0 (x)dx > 0 Then using the continuity property of the Koopman semigroup, we know there exists a ∆ such that t0+∆ t0 X [U t0 1 Xu ]((x))h 0 (x)dxdt > 0. We have 0 < t 0 +∆ t 0 X [Ut 0 1X u ](x)h0(x)dxdt ≤ ∞ 0 X [Ut 0 1X u ](x)h0(x)dxdt = 0(17) Remark 1. Following the results of Lemma 1, we can replace the obstacle avoidance constraints (7b) as ∞ 0 X 1 Xu (s t (x))dµ 0 (x)dt = 0(18) Remark 2. In the rest of the paper, we will use the notation X 1 = X \ N ǫ , where N ǫ is the ǫ neighborhood of the target set. With no loss of generality, we assume the target set X T is locally stable with the domain of attraction containing N ǫ . B. Convex formulation The following theorem presents the main result of this paper. Theorem 1. Under Assumption 1, the optimal off-road navigation problem can be written as the following convex optimization problem in terms of optimization variables ρ ∈ L 1 (X 1 ) ∩ C 1 (X 1 , R ≥0 ) andρ ∈ C 1 (X 1 , R) J ⋆ = inf ρ,ρ X1 q(x)ρ(x) +ρ (x) ⊤ Rρ(x) ρ(x) dx (19a) s.t. X1 1 Xu (x)ρ(x)dx = 0 (19b) X1 b(x)ρ(x)dx ≤ γ (19c) |ρ j (x)| ≤ L j ρ(x) (19d) ∇ · (f ρ + gρ) = h 0 , a.e. x ∈ X 1 (19e) The solution to the above optimization problem is used to recover the optimal feedback control input as follows: k ⋆ (x) =ρ ⋆ (x) ρ ⋆ (x)(20) where (ρ ⋆ ,ρ ⋆ ) are the solution of (19). The proof is provided in the Appendix. Remark 3. The convex structure of the optimization problem presented in (19) comes from the fact that the decision variables ρ andρ enter the cost quadratically and the constraints linearly and hence led to the infinite-dimensional convex optimization problem. We discuss the data-driven approach for the finite-dimensional approximation of the infinite-dimensional problem in Section IV. C. Control constraints Other constraints on the state and control input can be written convexly in terms of the optimization variables ρ andρ. One such constraint is the curvature constraint. For example, consider a Dubin's car model x 1 = u 1 cos θ,ẋ 2 = u 1 sin θ,θ = u 2 . Aside from the kinematic constraints imposed by nonholonomy, most often, the additional constraint on the radius of curvature of the paths of the vehicle must be considered [23]. The curvature constraints themselves will be a function of the terrain properties. The curvature constraints close the kinematic model of Dubin's car to reality. We can capture the curvature constraints as follows. |u 2 | |u 1 | ≤ 1 C(21) where u 2 is the angular velocity of Dubin's car model. The constraint in (21) ensures that the control design is realistic and that the angular velocity of the kinematic model is bounded. The (21) can be easily added to the optimization problem of theorem 1 convexly as follows. Following the general formulation specialized for two input systems, we can write u 1 =ρ 1 ρ and u 2 =ρ 2 ρ , hence |u 2 | |u 1 | ≤ 1 C =⇒ C|ρ 2 | − |ρ 1 | ≤ 0(22) which is linear and hence convex inρ 1 andρ 2 . IV. COMPUTATIONAL FRAMEWORK In this section, we utilize Naturally structured dynamic mode decomposition (NSDMD) [24] and provide a modification approximation, namely Approximate NSDMD. Then we formulate our problem with the given computational framework. A. Data-Driven Approximation: Approximate NSDMD The NSDMD algorithm introduced in [24] incorporates the natural properties of the linear operators, namely the positivity and the Markov properties. However, using the NSDMD algorithm is computationally expensive. Also, executing the NSDMD algorithm to find the P-F operator involves convex optimization with linear constraints and does not admit an analytical solution. Our proposed transfer operator theoretical framework for the control design relies on these operators' positivity and Markov properties. In our proposed modification of the NSDMD algorithm, we gain numerical efficiency at the expense of preserving these properties approximately. We call this modified algorithm "Approximate NSDMD (A-NSDMD)," and it can be solved analytically as it is a least square problem, as shown below. For the continuous-time dynamical system (1), consider snapshots of the data set obtained as time-series data from single or multiple trajectories. X = [x 1 , x 2 , . . . , x Q ], Y = [y 1 , y 2 , . . . , y Q ],(23) where x i ∈ X and y i ∈ X such that y i = s ∆t (x i ). In our proposed data-driven computation, we obtain the pair of two consecutive snapshots using the system with no control input (s 0 ∆t (x i )) and with the step input (s 1 ∆t (x i )). Let Ψ = [ψ 1 , . . . , ψ P ] ⊤ be the set of basis functions. We obtain the finite-dimensional approximation of the Koopman operator (U) through EDMD as the result of the following least square problem. min U GU − A F ,(24)G = 1 Q Ψ(X)Ψ(X) ⊤ , A = 1 Q Ψ(X)Ψ(Y) ⊤ ,(25) with U, G, A ∈ R P ×P , · F stands for Frobenius norm. The analytical solution of the above least square problem is U = G † A.(26) The EDMD algorithms provide convergence results with respect to the number of data points and basis functions [25], [26]. In this paper, we work with Gaussian Radial Basis Functions (RBF), which are the positive basis. The positivity constraints utilized in NSDMD are avoided in the A-NSDMD based on the approximated positivity of the Koopman operator when RBFs are used. This means that all the negative elements of the Koopman matrix obtained are 1e-6 or lower. Under the assumption that the basis functions are positive, the A-NSDMD obtainsÛ from U using row normalization, i.e., the entries of the matrixÛ are obtained as: [Û] ij = [U] ij j [U] ij(27) where [Û] ij is the (i, j) th entry of the matrixÛ. The above modification helps us to avoid Markov constraints. As a result, the A-NSDMD is just a least square problem with an analytical solution as given by (26). We obtain the P-F matrix asP = U ⊤ . The generator of the P-F operator is obtained as P f ≈P − I ∆t =: M.(28) where I is the identity matrix. Note that we did not strictly enforce the positivity property on the linear operators in the above construction. However, the numerical evidence suggests that the entries of the matrix U obtained using the EDMD algorithm with a positive basis function are predominantly positive. Table I compares the computation time of calculating the P-F operator for different numbers of RBFs utilized in lifting the dynamics. This provides empirical proof of the computational efficiency of A-NSDMD over NSDMD. B. Approximation of Optimization Problem This section discusses finite-dimensional approximation of the infinite-dimensional optimal navigation problem (19a)-(19c). We use the approximation of the generator for the vector field f and g. The generator approximation of the two vector fields is given as follows P f ≈ M 0 , P g ≈ M 1 .(29) The generator approximation of P f starts with obtaining openloop time series data {x 0 k } = {x 0 0 , x 0 1 , . . . , x 0 Q } by substituting u = 0 in (1). Similarly, the approximation of P gi is obtained by getting (1) where u i is a column vector consisting of all zeros except at i th position where the element value is 1. Then we calculate the matricesÛ 0 andÛ i by utilizing (24)- (27). The P-F matrix is then obtained asP 0 =Û 0⊤ andP i =Û i⊤ . Then we calculate M 0 and M i as follows: {x i k } = {x i 0 , x i 1 , . . . , x i Q } from substituting u = u i inP f ≈P 0 − I ∆t =: M 0 , P f+gi ≈P i − I ∆t(30) and P gi = P f+gi − P f =: M i Approximate NSDMD algorithm outlined in Section IV-A is used to approximate with Ψ(x) = [ψ 1 (x), ψ 2 (x), . . . , ψ P (x)] ⊤ as the basis functions. Let h 0 = Ψ ⊤ m, ρ = Ψ ⊤ v,ρ j = Ψ ⊤ w j .(32) Assuming R is the identity matrix, we perform the following finite-dimensional approximation inf ρ,ρ X1 q(x)ρ(x) +ρ (x) ⊤ρ (x) ρ(x) dx ≈ min wj ,v D 1 v + w ⊤ D 2 w v (33a) where D 1 = X1 q(x)Ψ ⊤ (x)dx , D 2 = X1 Ψ(x)Ψ ⊤ (x) dx and division is assumed element-wise. Similarly, we can write X1 1 Xu (x)ρ(x)dx ≈ X1 1 Xu (x)Ψ ⊤ (x)dx v = d 1 v (34) X1 b(x)ρ(x)dx ≈ X1 b(x)Ψ ⊤ (x)dx v = d 2 v (35) where d 1 := X1 1 Xu (x)Ψ ⊤ dx and d 2 := X1 b(x)Ψ ⊤ dx. Assumption 2. We consider all the basis functions to be positive and linearly independent. Remark 4. The Gaussian Radial Basis Functions (RBF) is used in this paper for the simulation results i.e., ψ k (x) = exp(− x−c k 2 2σ 2 ) where c k is the center of the k th Gaussian RBF. Therefore the ONP in Theorem 1 can be written as min wj ,v D 1 v + w ⊤ D 2 w v s.t. −M 0 v − j M j w j = m d 1 v = 0, d 2 v ≤ γ, |w j | ≤ L j v V. SIMULATION RESULTS The centers of the RBF used in the simulation results are chosen to be uniformly distributed in the state space at a distance of d. The RBF's standard deviation (σ) is selected as 0.5 d. We perform all the simulations using MATLAB R2021b on a Dell computer with 64 GB Random Access Memory (RAM) and an Intel(R) i7-10700K processor (3.80 GHz). In this paper, we use the data to do the linear approximation of the single integrator dynamics and Dubin's car model. The CVX toolbox is used for solving an optimization problem. The computational time to obtain the proposed algorithm's feedback control ranges from 5-15 minutes. Example 1. Consider the single integrator dynamicṡ x 1 = u 1 ,ẋ 2 = u 2(36) The initial and terminal sets are labeled as X 01 , X 02 and X T 1 . These sets are described as follows. • X {x ∈ R 2 : −3 ≤ x 1 ≤ 3, −3 ≤ x 2 ≤ 3} • X 01 {x ∈ R 2 : 1 − (x 1 − 1.5) 2 − (x 2 + 2.25) 2 ≥ 0} • X 02 {x ∈ R 2 : 1 − (x 1 + 1.5) 2 − (x 2 + 2.25) 2 ≥ 0} • X T 1 {x ∈ R 2 : 1 − x 2 1 − (x 2 − 2) 2 ≥ 0} • X u1 {x ∈ R 2 : 1 − x 2 1 − x 2 2 ≥ 0} The linear operators are approximated using 625 RBF and the time-step of discretization ∆t = 0.01. The value of γ is 0.6. We have obtained the results for the traversability maps labeled by b 1 (x). The optimal navigation problem given in (19a) -(19e) can be utilized to avoid any particular predetermined region, such as an enemy camp, lake, or pits. In Fig. 1, we show plot for initial conditions in initial sets X 01 and X 02 navigating successfully to terminal set X T 1 with traversability map given by b 1 (x). We notice that the trajectories navigate through a region where the traversability map takes smaller values while trying to avoid hard obstacle set X u1 . Example 2. Consider the Dubin's car dynamicṡ x 1 = u 1 cos θ,ẋ 2 = u 1 sin θθ = u 2(37) The initial and terminal sets are labeled as X 03 and X T 3 . These sets are described as follows. • X {x ∈ R 3 : −3 ≤ x 1 ≤ 9, −3 ≤ x 2 ≤ 9, −3 ≤ x 3 ≤ 3} • X 03 {x ∈ R 2 : 1 − (x 1 − 5.5) 2 − (x 2 − 7.5) 2 ≥ 0} • X T 3 {x ∈ R 2 : 1 − x 2 1 − x 2 2 ≥ 0} • X u5 {x ∈ R 2 : 1 − (x 1 − 3) 2 − (x 2 − 4) 2 ≥ 0} The linear operators are approximated using 2500 RBF and the time-step of discretization ∆t = 0.01. The value of γ is 0.7. We have obtained the results for the traversability map given by b 3 (x). We show the optimal navigation plots with a hard obstacle set X u5 in Fig. 3. The control plots for the trajectories given in Fig. 3 are displayed in Fig. 4. The control limits are set to be ±3 and are seen to be met by the control. The problem of data-driven approximation of linear operators is an active area of research. There are mainly two sources of errors: first, due to the finite amount of data, and second, due to the finitely many basis functions used in the approximation. In [25]- [27], sample complexity results for the finite-dimensional approximation of the linear operator are developed, where for a fixed number of basis functions, the approximation error is shown to decay as 1 √ M , where M is the number of data points. Characterizing approximation error as the function of the number of basis functions is difficult as the results will depend on the underlying system dynamics. The sample complexity-based error bounds for the approximation of linear operator can be used to characterize the convergence rate of the solution obtained from the finitedimensional approximate optimization problem to the true optimal solution [28] due to the proposed convex formulation of the optimization problem. The computational framework for the finite-dimensional approximation of linear operators suffers from scalability issues for a system involving high dimensional state space. However, there are ways to address the computational burden for the system with reasonable (5-10) state-space dimensions. In particular, methods based on exploiting the sparsity structure of the linear operators, spectral properties of linear operators, and using the back-stepping approach for control design are currently being investigated to extend the applicability of the results to the system of practical interest [29]- [31]. Fig. 3: Optimal navigation with initial points centered around (5.5, 7.5) in initial set X 04 to terminal set X T 3 with traversability map given by b 3 (x).The traversability map includes hard obstacles centered around (3,4). We also compare our approach with A ⋆ algorithm to show its efficiency over the existing approach. We use MATLAB Fig. 4: Optimal control plots for u 1 corresponding to different initial conditions with the configuration of initial and final sets as in Fig. 3. 2021b as the simulation platform for the comparison. The A ⋆ algorithm [32] is a search-based algorithm for path planning which uses heuristic functions for selective search in the workspace. The cost function of A ⋆ algorithm consists of two terms: g(x), the actual cost, and h(x), the heuristic cost. Here, g(x) represents the transition cost from the initial state to the current state, whereas h(x) represents the cost to go from the current state to the target state. When A ⋆ is implemented with a grid map, then g(x) would represent the grid values. For our comparison purposes, we selected g(x) to be equal to the traversability map b(x) and h(x) to be equal to the Euclidean distance between the current state and target state. The trajectory comparison between A ⋆ algorithm and the P-F approach is shown in Fig. (2). Here,γ = 10 for the P-F-based approach. Table II contains the traversability cost of trajectories obtained from A ⋆ algorithm and the P-F approach. The traversability cost is calculated by summing the traversability values, b(x(t)), along the trajectories generated by the A ⋆ and P-F based approach. We observe that our propsoed approach performs better than the A ⋆ algorithm as it leads to lower cost of traversability. VI. CONCLUSION The problem of navigation on off-road terrain is considered. We use traversability measures to describe the relative degree of difficulty of navigation. A convex formulation for the optimal navigation problem is constructed using the traversability information of the terrain. The convex formulation leads to an infinite-dimensional convex optimization problem for navigation. Furthermore, we utilize the data-driven approximation of the linear P-F operator for the finite-dimensional approximation of the optimization problem. Finally, simulation results are showcased to show the validity of the proposed method. Future research efforts will incorporate the framework's uncertainty arising from vehicle dynamics and vehicle-terrain interaction. VII. APPENDIX Proof. We use the fact that dµ 0 = h 0 (x)dx, and the definition of the Koopman operator to write the cost function (7a) as J = X1 ∞ 0 [U t (q + k ⊤ Rk)](x) dth 0 (x)dx = X1 ∞ 0 (q + k ⊤ Rk)(s t (x)) dth 0 (x)dx Performing change of variable y = s t (x) or x = s −t (y) and using the definition of P-F operator, we can write the above as J = X1 ∞ 0 (q + k ⊤ Rk)(x)[P c t h 0 ] dtdx = X1 (q + k ⊤ Rk)ρ(x)dx (38) where ρ(x) := ∞ 0 [P t h 0 ](x)dt.(39) Since the cost function is assumed to be finite (Assumption 1), the ρ(x) is well-defined for a.e. for a.e. x. Furthermore, since the P-F operator is a positive operator and h 0 (x) ≥ 0, we have ρ(x) ≥ 0. Hence (38) can be written as J(µ 0 ) = X1 (q(x)ρ(x) +ρ (x) ⊤ Rρ(x) ρ(x) )dx(41) whereρ(x) := ρ(x)k(x). This shows that we can write the cost function in (7) in the form (19). We next discuss the constraints. Following the results of Lemma 1, we know that (15) implies (16). Hence, the (7b) are implied by ∞ 0 X 1 Xu (s t (x))dµ 0 (x)dt = 0 Again performing change of variables y = s t (x) and using the definition of P-F operator, (42) can be written as X1 1 Xu (x)ρ(x)dx = 0 with ρ(x) as defined in (39). Similarly, it follows that (7c) can be written as X1 b(x)ρ(x)dx ≤ γ. The control constraints |u j | ≤ L j for j = 1, . . . , m can be written as |ρ j (x)| ≤ L j ρ(x) follows from the fact that u = k(x) =ρ(x)/ρ(x) and ρ(x) is positive. We next show that the ρ(x) as defined in (39) satisfies the constraints (19e). Substituting (39) in the constraint of (19e), we obtain ∇ · (F c (x)ρ(x)) = ∞ 0 ∇ · (F c (x)[P t h 0 ](x))dt = ∞ 0 − d dt [P t h 0 ](x)dt = −[P t h 0 ](x) ∞ t=0 = h 0 (x),(43) In deriving (43) we have used the infinitesimal generator property of P-F operator Eq. (13) and the fact that lim t→∞ [P t h 0 ](x) = 0 following (40). Next, we show that the target set X T is a.e. stable w.r.t measure µ 0 supported on the initial set X 0 . Consider the set S ℓ S ℓ = {x ∈ X 1 : s t (x) ∈ X 1 , for some t > ℓ} and let S = ∞ ℓ=1 S ℓ . The set S contains points, some of whose limit points lie in X 1 , and for almost every stability of the target set, we need to show that µ 0 (S) = 0. From the construction of the set S, it follows that s t (S) = S, where s t (S) = {s t (x) : x ∈ S}. µ 0 (S) = X1 1 S (x)h 0 (x)dx = X1 1 S (s t (x))h 0 (x)dx = X1 1 S (x)[P t h 0 ](x)dx(44) where we have use the fact that x ∈ S iff s t (x) ∈ S. Since the above is true for all t ≥ 0, we obtain using dominated convergence theorem Financial support from NSF under grants 1942523, 2008513, 2031573 and NSF CPS award 1932458 is greatly acknowledged. J. Moyalan and U. Vaidya are with the Department of Mechanical Engineering, Clemson University, Clemson, SC; {jmoyala,uvaidya}@clemson.edu. Y. Chen is with the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA; {yongchen}@gatech.edu ρ(x) = [ρ 1 (x), . . . ,ρ m (x)] ⊤ and w = [w 1 , . . . , w m ] ⊤ . Now let us express the following terms as combinations of basis functions Fig. 1 : 1Optimal navigation with initial points centered around (1.5, −2.25) in initial set X 01 and centered around (−1.5, −2.25) in initial set X 02 to terminal set X T 1 with traversability map given by b 1 (x).The traversability map includes hard obstacle centered around (0, 0). Fig. 2 : 2Comparison of P-F approach with A ⋆ algorithm with traversability map given by b 2 (x). The trajectories start from two initial points: point A (4.5, 3.2) and point B (1.7, 4.2) and end at target point given by (0.7, 0.5). x and is an integrable function. The function [P t h 0 ](x) is a uniformly continuous function of time which can be inferred from the definition of the P-F operator and the assumption made on function h 0 . 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Vaidya, "Spectral analysis of koopman operator and nonlinear optimal control," in 2022 IEEE 61st Conference on Decision and Control (CDC). . IEEE. IEEE, 2022, pp. 3853-3858. Improved a-star algorithm for long-distance offroad path planning using terrain data map. Z Hong, P Sun, X Tong, H Pan, R Zhou, Y Zhang, Y Han, J Wang, S Yang, L Xu, ISPRS International Journal of Geo-Information. 1011785Z. Hong, P. Sun, X. Tong, H. Pan, R. Zhou, Y. Zhang, Y. Han, J. Wang, S. Yang, and L. Xu, "Improved a-star algorithm for long-distance off- road path planning using terrain data map," ISPRS International Journal of Geo-Information, vol. 10, no. 11, p. 785, 2021. Systemes d'équations différentielles d'oscillations non linéaires. I Barbalat, Rev. Math. Pures Appl. 42I. Barbalat, "Systemes d'équations différentielles d'oscillations non linéaires," Rev. Math. Pures Appl, vol. 4, no. 2, pp. 267-270, 1959.
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[ "Momentum-resolved charge excitations in high-T c cuprates studied by resonant inelastic x-ray scattering", "Momentum-resolved charge excitations in high-T c cuprates studied by resonant inelastic x-ray scattering" ]
[ "K Ishii \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n", "M Hoesch \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n\nEuropean Synchrotron Radiation Facility\n38000GrenobleFrance\n", "T Inami \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n", "K Kuzushita \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n", "K Ohwada \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n", "M Tsubota \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n", "Y Murakami \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n\nDepartment of Physics\nGraduate School of Science\nTohoku University\n980-8578SendaiJapan\n", "J Mizuki \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n", "Y Endoh \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n\nInternational Institute for Advanced Studies\n619-0025KizuKyotoJapan\n", "K Tsutsui \nSynchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan\n\nInstitute for Materials Research\nTohoku University\n980-8577SendaiJapan\n", "T Tohyama \nInstitute for Materials Research\nTohoku University\n980-8577SendaiJapan\n\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan\n", "S Maekawa \nInstitute for Materials Research\nTohoku University\n980-8577SendaiJapan\n\nCREST\nJapan Science and Technology Agency\n4-1-8 Honcho332-0012KawaguchiJapan\n", "K Yamada \nInstitute for Materials Research\nTohoku University\n980-8577SendaiJapan\n", "T Masui \nDepartment of Physics\nGraduate School of Science\nOsaka University\n560-0043ToyonakaJapan\n\nSuperconducting Research Laboratory\nISTEC\n135-0062TokyoJapan\n", "S Tajima \nDepartment of Physics\nGraduate School of Science\nOsaka University\n560-0043ToyonakaJapan\n\nSuperconducting Research Laboratory\nISTEC\n135-0062TokyoJapan\n", "H Kawashima \nDepartment of Physics and Mathematics\nAoyama-Gakuin University\n229-8558SagamiharaKanagawaJapan\n", "J Akimitsu \nDepartment of Physics and Mathematics\nAoyama-Gakuin University\n229-8558SagamiharaKanagawaJapan\n" ]
[ "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "European Synchrotron Radiation Facility\n38000GrenobleFrance", "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "Department of Physics\nGraduate School of Science\nTohoku University\n980-8578SendaiJapan", "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "International Institute for Advanced Studies\n619-0025KizuKyotoJapan", "Synchrotron Radiation Research Unit\nJapan Atomic Energy Agency\n679-5148HyogoJapan", "Institute for Materials Research\nTohoku University\n980-8577SendaiJapan", "Institute for Materials Research\nTohoku University\n980-8577SendaiJapan", "Yukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan", "Institute for Materials Research\nTohoku University\n980-8577SendaiJapan", "CREST\nJapan Science and Technology Agency\n4-1-8 Honcho332-0012KawaguchiJapan", "Institute for Materials Research\nTohoku University\n980-8577SendaiJapan", "Department of Physics\nGraduate School of Science\nOsaka University\n560-0043ToyonakaJapan", "Superconducting Research Laboratory\nISTEC\n135-0062TokyoJapan", "Department of Physics\nGraduate School of Science\nOsaka University\n560-0043ToyonakaJapan", "Superconducting Research Laboratory\nISTEC\n135-0062TokyoJapan", "Department of Physics and Mathematics\nAoyama-Gakuin University\n229-8558SagamiharaKanagawaJapan", "Department of Physics and Mathematics\nAoyama-Gakuin University\n229-8558SagamiharaKanagawaJapan" ]
[]
We report a Cu K-edge resonant inelastic x-ray scattering (RIXS) study of high-Tc cuprates. Momentum-resolved charge excitations in the CuO2 plane are examined from parent Mott insulators to carrier-doped superconductors. The Mott gap excitation in undoped insulators is found to commonly show a larger dispersion along the [π, π] direction than the [π, 0] direction. On the other hand, the resonance condition displays material dependence. Upon hole doping, the dispersion of the Mott gap excitation becomes weaker and an intraband excitation appears as a continuum intensity below the gap at the same time. In the case of electron doping, the Mott gap excitation is prominent at the zone center and a dispersive intraband excitation is observed at finite momentum transfer.
10.1016/j.jpcs.2008.06.041
[ "https://arxiv.org/pdf/0708.3534v1.pdf" ]
53,680,684
0708.3534
b5d8feca5e43b272be4ae04df060d93add7561e1
Momentum-resolved charge excitations in high-T c cuprates studied by resonant inelastic x-ray scattering 27 Aug 2007 K Ishii Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan M Hoesch Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan European Synchrotron Radiation Facility 38000GrenobleFrance T Inami Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan K Kuzushita Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan K Ohwada Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan M Tsubota Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan Y Murakami Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan Department of Physics Graduate School of Science Tohoku University 980-8578SendaiJapan J Mizuki Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan Y Endoh Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan International Institute for Advanced Studies 619-0025KizuKyotoJapan K Tsutsui Synchrotron Radiation Research Unit Japan Atomic Energy Agency 679-5148HyogoJapan Institute for Materials Research Tohoku University 980-8577SendaiJapan T Tohyama Institute for Materials Research Tohoku University 980-8577SendaiJapan Yukawa Institute for Theoretical Physics Kyoto University 606-8502KyotoJapan S Maekawa Institute for Materials Research Tohoku University 980-8577SendaiJapan CREST Japan Science and Technology Agency 4-1-8 Honcho332-0012KawaguchiJapan K Yamada Institute for Materials Research Tohoku University 980-8577SendaiJapan T Masui Department of Physics Graduate School of Science Osaka University 560-0043ToyonakaJapan Superconducting Research Laboratory ISTEC 135-0062TokyoJapan S Tajima Department of Physics Graduate School of Science Osaka University 560-0043ToyonakaJapan Superconducting Research Laboratory ISTEC 135-0062TokyoJapan H Kawashima Department of Physics and Mathematics Aoyama-Gakuin University 229-8558SagamiharaKanagawaJapan J Akimitsu Department of Physics and Mathematics Aoyama-Gakuin University 229-8558SagamiharaKanagawaJapan Momentum-resolved charge excitations in high-T c cuprates studied by resonant inelastic x-ray scattering 27 Aug 2007resonant inelastic x-ray scatteringcharge excitationhigh-Tc cuprates We report a Cu K-edge resonant inelastic x-ray scattering (RIXS) study of high-Tc cuprates. Momentum-resolved charge excitations in the CuO2 plane are examined from parent Mott insulators to carrier-doped superconductors. The Mott gap excitation in undoped insulators is found to commonly show a larger dispersion along the [π, π] direction than the [π, 0] direction. On the other hand, the resonance condition displays material dependence. Upon hole doping, the dispersion of the Mott gap excitation becomes weaker and an intraband excitation appears as a continuum intensity below the gap at the same time. In the case of electron doping, the Mott gap excitation is prominent at the zone center and a dispersive intraband excitation is observed at finite momentum transfer. Introduction Strongly correlated electron systems have attracted much attention because they display a wide variety of fascinating physical properties, such as high-T c superconductivity in cuprates. The unveiling of their electronic structure is mandatory to clarify the mechanisms underlying these physical behaviors. Essentially, one can think of the electronic structure of high-T c cuprate superconductors as that of a doped Mott insulator in two dimensions, in which charge and spin degrees of freedom of the electron are crucial. Neutron scatting is a powerful tool to investigate the spin dynamics. On the other hand, the photon is a good probe for the charge sector. Conventional optical methods, such as optical conductivity measurement, can give infor-mation about the momentum-conserved excitations, but momentum-resolved experiments are required for our complete understanding of the charge dynamics. Recent developments of x-ray sources from synchrotron radiation make this possible by inelastic x-ray scattering. Especially, resonant inelastic x-ray scattering (RIXS) has the advantage of element selectivity and it enables us to elucidate excitations related to the Cu orbitals by tuning the incident photon energy to the Cu K absorption edge. While angle-resolved photoemission spectroscopy (ARPES) yields one-particle excitation from the occupied state [1], RIXS provides the two-particle excitation spectra, from which one can explore both occupied and unoccupied states. In the last decade, the importance of the RIXS technique has been increasingly recognized for the investigation of the electronic structure of strongly correlated electrons systems. In this paper, we report a comprehensive RIXS study on high-T c cuprates from parent Mott insulators to carrier- doped superconductors. We focus on the Mott gap excitation and the intraband excitation below the gap. The latter emerges when the material is carrier doped. Evolution of the electronic structure upon carrier doping is a central issue in the physics of doped Mott insulator and we examine here how finely it can be probed using RIXS. Material dependence, including a peculiar resonance condition in Mott insulators, is discussed as well through a comparison of our results with previous work. Mott gap excitation in RIXS In this section, we briefly describe how the Mott gap excitation occurs in the RIXS process. Figures 1 show a schematic energy diagram of the CuO 2 plane and the RIXS process at the Cu K-edge. The parent materials of high-T c superconductors are charge-transfer insulators [2]. The Cu 3d x 2 −y 2 level splits into a lower Hubbard band (LHB) and an upper Hubbard band (UHB) by strong electron correlation and the O 2p σ band is situated between them. In addition, Cu 1s and Cu 4p levels, which participate in the x-ray absorption and emission processes, are included in the diagram. Fig. 1(a) shows the ground state of a parent Mott insulator. X-ray absorption at the left Cu site brings about a dipole transition from the Cu 1s to the 4p level [ Fig. 1(b)]. The 1s core hole predominantly scatters a valence electron. The energy of the Cu 3d x 2 −y 2 of the left Cu site is lowered by an attractive Coulomb interaction with the Cu 1s core hole. Then an electron in the O 2p σ transfers to the UHB to screen the core-hole potential [ Fig.1(c)]. Finally, the Cu 4p electron goes back to the 1s level again and a photon is emitted [ Fig. 1(d)]. The hole in the O 2p σ and the right Cu site form a Zhang-Rice singlet (ZRS) [3]. We call this scattering event a Mott gap excitation in the CuO 2 plane, i.e., an excitation from the ZRS band to the UHB. Experimental details The RIXS experiments were performed at BL11XU in SPring-8, where a specially designed inelastic x-ray scattering spectrometer was installed [4]. Using a Si (111) doublecrystal monochromator and a Si (400) channel-cut secondary monochromator, an incident energy resolution of about 220 meV was obtained. Horizontally scattered x-rays were analyzed in energy by a spherically bent Ge (733) crystal. The total energy resolution estimated from the full width at half maximum of the elastic line was about 400 meV. All spectra were measured at room temperature. In Continuation of our previous work on superconductors [5,6], their parent Mott insulators, Nd 2 CuO 4 and YBa 2 Cu 3 O 6 were measured. We selected Ca 2−x Na x CuO 2 Br 2 as a hole doped sample. The advantage of this material in RIXS experiments is the absence of rare earth atoms that reduces the absorption of x-rays by the sample itself and a larger scattering intensity can be expected with this advantage. For the study of electron doping, we measured Nd 2−x Ce x CuO 4 . Single crystals were prepared for all the samples. The Na concentration (x) in Ca 2−x Na x CuO 2 Br 2 was estimated as x = 0.2 from the superconducting transition temperature (T c = 14K) [7]. YBa 2 Cu 3 O 6 and Ca 1.8 Na 0.2 CuO 2 Br 2 were sealed in a beryllium cell filled with inert gas in order to avoid reaction with oxygen or water in the air. The surface of the crystals is perpendicular to the c-axis. The crystals were mounted with the c-axis in the scattering plane. The strong two dimensionality of the electronic structure of the CuO 2 plane ensures that the momentum dependence along the c * -axis is small. Therefore we fixed the c * component in the absolute momentum transfer ( Q) at a value where the scattering angle (2θ) is close to 90 • . This enables us to reduce the elastic scattering by the polarization factor of the Thomson scattering [6]. We measured the momentum dependence of the CuO 2 plane in the transverse geometry. A schematic view of the RIXS experimental configuration is shown in the inset of Fig. 2(d). The polarization of the incident x-rays ( ǫ i ) contains nearly equal out-of-plane ( ǫ i c) and in-plane ( ǫ i ⊥ c) components. (d) Prominent peaks in the absorption spectra are labeled by the same letters A-N as in the reports [8,9]. In Nd 2 CuO 4 , the peaks B and C in the ǫ c spectrum are assigned to the 1s-4p π transitions. The peaks D and E in the ǫ a spectrum are the 1s-4p σ transitions. In each pair, the peaks at lower energy (B and D) and those at higher energy (C and E) correspond to well-screened and poorly-screened corehole final states, respectively [8]. On the other hand, the assignment of the features in the YBa 2 Cu 3 O 6 spectra is rather complicated [9] because there are two distinct Cu sites. One type of Cu atoms is in the monovalent Cu(1) site, which forms the one-dimensional chains in YBa 2 Cu 3 O 7 . The other is the divalent Cu(2) site in the CuO 2 plane. The peaks of A and D in the ǫ a spectrum are absent in YBa 2 Cu 3 O 6.95 and they are attributed to the transitions at the Cu(1) sites. The peak A is a proof of the existence of a monovalent Cu atom [10,11]. The peaks M and N are the 1s-4p σ transitions of the Cu(2) site. A pair of peaks of the 1s-4p π transitions of the Cu(2) site should appear in the ǫ c spectrum. It can be assigned to the peaks C and F but the 1s-4p σ transition of the Cu(1) site may overlap. Results and Discussion YBa 2 Cu 3 O 6 A C D F M N ε//a ε//c RIXS k i k f Q ab c ε i s a m p le In YBa 2 Cu 3 O 6 , a large RIXS intensity is observed around 5 eV when the incident photon energy is close to the peak A in the absorption spectra, indicating that this excitation is related to the Cu(1) site. In order to show the resonant feature of the Mott gap excitation more precisely, the RIXS intensity in Fig. 2 (a) and (b) is integrated between 1 and 3 eV and plotted as a function of the incident photon energy in Figs. 2(c) and (d). Clear difference is observed in the incident energy profile of the Mott gap excitation between Nd 2 CuO 4 and YBa 2 Cu 3 O 6 , that is, there are three well-pronounced resonances for the former and only one peak with a bump on the lower-energy side for the latter. Each resonant energy of the RIXS intensity is found to correspond to a peak in the absorption spectra. We note that both out-of-plane ( ǫ i c) and in-plane ( ǫ i ⊥ c) components coexist in the incident photon polarization in our experimental condition. In contrast to the two cases, La 2 CuO 4 shows two distinct resonances with respect to the polarization condition; one is at E i = 8990 eV in the outof-plane condition and the other is at E i = 8994 eV in the in-plane condition [12]. As shown in Fig. 1, the screening process by the charge transfer from the ligand oxygen to the UHB in the intermediate state is important for the Mott gap excitation. This means that the Mott gap excitation is enhanced when the incident energy is tuned to the well-screened state in the x-ray absorption spectra [13,14]. This rule is applicable to La 2 CuO 4 [12] as well as to the in-plane condition of Nd 2 CuO 4 [peak D in Fig. 2(c)] and YBa 2 Cu 3 O 6 [peak M in Fig. 2(d)]. In contrast, two resonances appear in the out-ofplane condition of Nd 2 CuO 4 [peak B and C in Fig. 2(c)]. In the configuration-interaction picture, the well-screened and poorly-screened states are expressed by linear combinations of |1s, 3d 9 , 4p π and |1s, 3d 10 , L, 4p π , where 1s and L denote a hole in 1s level and ligand oxygen, respectively. Generally, |1s, 3d 10 , L, 4p π is dominant in the well-screened state. The two resonances in the out-of-plane condition of Nd 2 CuO 4 indicate that |1s, 3d 10 , L, 4p π has relatively large weight at the absorption labeled C in Fig. 2(c), though it has been assigned to the poorly-screened state. We have no explanation for the weakness or disappearance of the resonance in the out-of-plane condition of YBa 2 Cu 3 O 6 so far. It may be related to a structural difference, namely, the Cu atom in the CuO 2 plane is fivefold coordinated by four planar and one apical oxygens in YBa 2 Cu 3 O 6 , which breaks the mirror symmetry perpendicular to the c-axis, while the symmetry is present in La 2 CuO 4 and Nd 2 CuO 4 . Further systematic studies are necessary to understand the difference in the out-of-plane condition. The RIXS intensity is almost zero at ∼ 1 eV and a clear Mott gap can be seen at all momenta. Taking advantage of the experimental condition of 2θ ≃ 90 • , the gap is much clearer in our spectra than that in the previous RIXS works on parent materials, Ca 2 CuO 2 Cl 2 [17] and La 2 CuO 4 [16] where the scattering intensity below the gap does not reach close to zero due to the tail of the elastic scattering. Momentum dependence of the Mott insulators Here, we discuss the momentum dependence of the overall spectral weight rather than that of the peak positions. A recent detailed study of HgBa 2 CuO 4+δ and La 2 CuO 4 demonstrated that a multiplet of peaks with a weak dispersion can be elucidated by utilizing the subtle dependence of the cross section on the incident photon energy [18,12]. Though the peak position of each peak shows a weak dispersion, the overall spectral weight containing the intensity of the multiplet shifts as a function of momentum transfer. Moreover, such a multiplet is not clearly observed in our data, which may be due to different experimental conditions with Ref. [18,12], such as polarization and absolute momentum transfer. Focusing on the spectral weight, we notice from Figs. 3 (a) and (b) that the changes in the excitation spectra are small along the [π, 0] direction while the spectral weight clearly shifts to higher energy as the momentum transfer increases along the [π, π] direction. The center of gravity of the RIXS spectra between 1 and 4 eV is plotted as a function of momentum transfer in Fig. 3(c), which corresponds to the dispersion relation of the spectral weight of the Mott gap excitation. The momentum dependence of the Mott gap excitation shows a large anisotropy; the dispersion along the [π, π] direction is larger than that along the [π, 0] direction. The larger dispersion along the [π, π] direction was predicted by a theoretical calculation based on a single band Hubbard model with long range hoppings (t-t ′ -t ′′ -U model) [19]. Besides, it has been experimentally observed in Ca 2 CuO 2 Cl 2 [17]. In La 2 CuO 4 [16], the authors claimed that the spectral weight of the lowest excitation (labeled A in the reference) becomes small at q = (π, π) and this results in a shift of the spectral weight to higher energy. In Fig. 3(d), we compare the RIXS spectra of La 2 CuO 4 in Ref. [16] with our data. The momentum dependence of La 2 CuO 4 is qualitatively similar to that of Nd 2 CuO 4 and YBa 2 Cu 3 O 6 . Therefore, we conclude that the larger dispersion along the [π, π] direction than that along the [π, 0] direction is a common character of the Mott gap excitation in the insulating CuO 2 plane. Quantitatively, the RIXS spectra depend on the material. Comparing the spectra at the zone center in Fig. 3(d), the gap energy is found to systematically change with the number of oxygens coordinated around Cu, that is, it is largest in the octahedral structure of La 2 CuO 4 and smallest in the square structure of Nd 2 CuO 4 . YBa 2 Cu 3 O 6 , with its pyramidal coordination geometry, lies between the two. This systematic change of the gap has been already reported from an optical conductivity experiment [20], which is consistent with our RIXS results. The magnitude of the dispersion also displays material dependence. Experimentally, the dispersion along the [π, 0] direction is apparent in La 2 CuO 4 [16,12,15], while that of Nd 2 CuO 4 and YBa 2 Cu 3 O 6 is fairly small. The dispersion of La 2 CuO 4 in Ref. [15], where momentum-dependent RIXS spectra along the [π, 0] direction are presented, is superimposed in Fig. 3(c). We can also ascertain from the spectral weight that the dispersion along [π, 0] direction is larger in La 2 CuO 4 . On the other hand, it is demonstrated theoretically that the long-range hoppings t ′ and t ′′ play an important role for the dispersion of RIXS spectra [19,21]; the dispersion along the [π, 0] direction becomes larger without t ′ and t ′′ . Accordingly, our experimental results suggest that long range hopping parameters of Nd 2 CuO 4 and YBa 2 Cu 3 O 6 are larger than those of La 2 CuO 4 . Comparing relatively between Nd 2 CuO 4 and La 2 CuO 4 , it is consistent with the parameters which are obtained from the shape of the Fermi surface in ARPES experiments [22]. Furthermore the experimental result that YBa 2 Cu 3 O 6 has larger long range hopping than La 2 CuO 4 agrees with a theory which proposed that materials with higher T c at optimal doping have larger hopping term [23]. Carrier-doping effect In this section, we discuss RIXS spectra of carrier-doped superconductors and compare them with those of Mott insulators. Fig. 4(a) shows the momentum dependence of RIXS spectra obtained for hole-doped Ca 1.8 Na 0.2 CuO 2 Br 2 . The incident photon energy is 8984 eV which corresponds to the well-screened state of the out-of-plane condition in the absorption spectrum. A large spectral weight remains located above 2 eV, which indicates that the Mott gap persists even in the hole-doped superconductor. At the same time, the gap is filled by a continuum intensity below 2 eV. While the former excitation is the interband excitation across the Mott gap, the latter is related to the dynamics of doped holes in the Zhang-Rice singlet band and we call it intraband excitation. (π,π) (π/2,π/2) (0,0) (π/2,0) (π,0) When the interband excitation of Ca 1.8 Na 0.2 CuO 2 Br 2 is compared with that of Mott insulators, the spectral weight around 2 eV decreases by hole doping except for the spectrum at q = (π, π) where the intensity around 2 eV is weak even in Mott insulators. As a result, the total spectral weight of the interband excitation shifts to higher energy and the dispersion becomes weaker. In Fig. 3(c), the center of gravity of the spectral weight between 2-4 eV is shown as a function of momentum. It is apparent that the dispersion is weaker in the hole-doped Ca 1.8 Na 0.2 CuO 2 Br 2 than in the Mott insulators. Such a change is also observed in La 2−x Sr x CuO 4 [24]; a well-defined peak feature at 2-3 eV for x = 0 (labeled A in Ref. [16]) decreases in intensity or disappears at x = 0.17 and the overall spectral weight (labeled AB in Ref. [24]) shows weaker dispersion. Furthermore, the excitation from the CuO 2 plane in optimally-doped YBa 2 Cu 3 O 7−δ also shows very weak dispersion [5,25]. These characteristic changes of the Mott gap excitation by hole doping are consistent with a theoretical prediction [26,21]. The shift of the spectral weight to higher energy is attributed to a shift of the Fermi energy upon hole doping. On the other hand, it is suggested in the theory that the weaker dispersion in the hole-doped case compared with that in the undoped case is related to the reduction of antiferromagnetic correlation. With realistic parameters used in the theory, the antiferromagnetic correlation is strongly suppressed by hole doping and the dispersion of the Mott gap excitation becomes weaker. In contrast, the short-range antiferromagnetic correlation is kept in the electron-doped case, and the magnitude of the dispersion of the Mott gap excitation is almost the same as that of the undoped case. This result indicates that the underlying magnetism governs the charge dynamics in the high-energy region up to a few eV. It is noted that the two-leg ladder shows a different behavior upon hole doping. A recent RIXS study of (La,Sr,Ca) 14 Cu 24 O 41 demonstrated that the momentum dependence of the Mott gap excitation in the two-leg ladder is insensitive to the hole doping as well as the spin gap state [27]. Next, we discuss the results of electron doping. In Fig. 4(b), RIXS spectra for the x = 0, 0.075, and 0.15 in samples of Nd 2−x Ce x CuO 4 are compared. The incident photon energy is fixed at 8991 eV for all samples. The spectra are normalized to the intensity at 4-5 eV. The doping dependence is apparently different between the zone center and finite q. At the zone center, the Mott gap excitation at 2 eV is unchanged upon electron doping, which means that the charge gap due to the electron correlation persists even in the metallic phase. Two theoretical calculations of the electron doped CuO 2 plane have been reported. One is based on a single band Hubbard model using a numerically exact diagonalization technique on a small cluster [26]. The other adopts a d-p model and the Hartree-Fock approximation is applied [28]. In the former theory, the Mott gap excitation persists upon electron doping, and especially, it is enhanced in the spec-trum at the zone center. On the other hand, the latter suggests a collapse of the gap. Our experimental observation of the 2 eV feature remaining in the electron-doped metallic state supports the validity of the former theory. At finite momentum transfer, large spectral weight appears below the gap. Its intensity is roughly proportional to the number of doped electrons (x). Hence it is related to the intraband dynamics of the doped electrons in the UHB. Because details of the intraband excitation have already been published [6,29], we do not discuss them here. Finally, we show the difference of resonance conditions between the undoped and electron-doped cases. In Fig. 4(c), RIXS spectra measured at two incident photon energies are compared for Nd 2 CuO 4 . The lower 8984 eV is close to the well-screened state of the out-of-plane condition [peak B in Fig. 2(c)], while the higher 8991 eV corresponds to the poorly-screened state [peak C in Fig. 2(c)]. The scale factors are common for the spectra at the three momenta. The spectra strongly resemble between the two incident photon energies. On the other hand, the spectra of Nd 1.85 Ce 0.15 CuO 4 in Fig. 4(d) show a difference near the zone center, q = (π/10, 0); the 2 eV feature is suppressed at 8983 eV. This is an additional proof that the excitation at 2 eV at the zone center is qualitatively different from the intraband excitation observed at the finite momentum transfer. In the electron-doped case, only the intraband excitation is expected to be enhanced at the incident photon energy where the core hole is created at the doped site [6,21]. This energy should exists below the well-screened state and 8983 eV in Nd 1.85 Ce 0.15 CuO 4 may be close to it. Summary We have performed a Cu K-edge RIXS study of high-T c cuprates from parent Mott insulators to carrier-doped superconductors. While the resonance condition depends on materials, an anisotropic dispersion, that is, larger dispersion along the [π, π] direction than the [π, 0] direction, is commonly observed in undoped Mott insulators. Upon hole doping, the dispersion of the Mott gap excitation becomes weaker, while the Mott gap excitation is prominent at the zone center in the electron doped case. At the same time, an intraband excitation emerges below the gap in both holeand electron-dopings. These characteristics are consistent with a theoretical calculation based on the Hubbard model. Fig. 1 . 1Schematic energy diagram of CuO 2 plane and RIXS process at the Cu K-edge. Two Cu sites and one O 2pσ level are shown. Filled and open circles are electron and hole, respectively. 4. 1 . 1Incident energy dependence and resonance condition of the Mott insulators Figures 2 2(a) and (b) show the incident energy dependence of the RIXS spectra of Nd 2 CuO 4 and YBa 2 Cu 3 O 6 , respectively. The reduced momentum transfer in the ab plane ( q) is (π, 0). The spectral weight around 2 eV arises from the excitation across the Mott gap and its intensity is resonantly enhanced at some incident energies.Because the intermediate state of RIXS corresponds to the final state of x-ray absorption, it is interesting to compare absorption spectra with the incident Fig. 2 . 2Incident photon energy dependence of RIXS spectra of (a) Nd 2 CuO 4 and (b) YBa 2 Cu 3 O 6 . The absolute momentum transfer is Q = (0.5, 0, 12.5) and (0.5, 0, 12.3) for Nd 2 CuO 4 and YBa 2 Cu 3 O 6 , respectively. The incident energy for each scan can be read from the vertical axis. X-ray absorption spectra of (c) Nd 2 CuO 4 and (d) YBa 2 Cu 3 O 6 . RIXS intensities integrated between 1 and 3 eV are also shown as a function of the incident photon energy. energy dependence of RIXS. Solid and dashed lines in Figs. 2(c) and (d) are the absorption spectra of Nd 2 CuO 4 and YBa 2 Cu 3 O 6 obtained by the total fluorescence yield. Figures 3 3(a) and (b) show the momentum dependence of Nd 2 CuO 4 and YBa 2 CuO 6 , respectively. The incident photon is fixed at the resonance energy derived from Figs. 2. Fig. 3 . 3Momentum dependence of RIXS spectra of (a) Nd 2 CuO 4 , (b) YBa 2 Cu 3 O 6 . The incident photon energy is 8991 eV for Nd 2 CuO 4 and 8996 eV for YBa 2 Cu 3 O 6 . Filled circles are the raw data. The data in the anti-Stokes region are folded at the origin and plotted as open circles, from which we can estimate the elastic tail and the background. (c) Momentum dependence of the Mott gap excitation. The center of gravity of the RIXS spectra between 1 and 4 eV is plotted for Nd 2 CuO 4 , YBa 2 Cu 3 O 6 , and La 2 CuO 4 . The data of La 2 CuO 4 are taken from the Ref. [15]. For Ca 1.8 Na 0.2 CuO 2 Br 2 , the center of gravity between 2 and 4 eV is used to exclude contribution from the intraband excitation. The thick lines are drawn assuming a linear dispersion. (d) Comparison of three Mott insulators. The symbols are the same as in (c). The data of La 2 CuO 4 are taken from the Ref. Fig. 4 . 4(a) RIXS spectra of hole-doped Ca 1.8 Na 0.2 CuO 2 Br 2 . The incident photon energy is 8984 eV. Filled circles are the raw data. The data in the anti-Stokes region are folded at the origin and plotted as open circles. (b) RIXS spectra of Nd 2−x CexCuO 4 , which show electron-doping dependence. The incident photon energy is 8991 eV. (c) and (d) Comparison of RIXS spectra of Nd 2 CuO 4 and Nd 1.85 Ce 0.15 CuO 4 measured at two incident photon energies. Acknowledgments The authors thank Dr. I. Jarrige for proofreading of the manuscript. This work was performed under the inter-university cooperative research program of the Institute of Materials Research, Tohoku University and financially supported by the Grant-in-Aid for Scientific Research on Priority Areas "Invention of anomalous quantum materials" from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. K. I. was also supported by Grant-in-Aid for Young Scientists from JSPS. The crystal growth of YBa 2 Cu 3 O 6 was supported by the New Energy and Industrial Technology Development Organization (NEDO) as the Collaborative Research and Development of Fundamental Technologies for Superconductivity Applications. . A Damascelli, Rev. Mod. Phys. 75473A. Damascelli, et al., Rev. Mod. Phys. 75 (2003) 473. . J Zaanen, Phys. Rev. Lett. 55418J. Zaanen, et al., Phys. Rev. Lett. 55 (1985) 418. . F C Zhang, Phys. Rev. B. 373759F. C. Zhang, et al., Phys. Rev. B 37 (1988) 3759. . T Inami, Nucl. Instrum. Methods Phys. Res. A. 4671081T. Inami, et al., Nucl. 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[]
[ "Efficient Large-scale Scene Representation with a Hybrid of High-resolution Grid and Plane Features", "Efficient Large-scale Scene Representation with a Hybrid of High-resolution Grid and Plane Features" ]
[ "Yuqi Zhang \nThe Chinese University of Hong Kong\nShenzhen\n", "Guanying Chen \nThe Chinese University of Hong Kong\nShenzhen\n", "Shuguang Cui Fnii \nThe Chinese University of Hong Kong\nShenzhen\n", "Sse \nThe Chinese University of Hong Kong\nShenzhen\n" ]
[ "The Chinese University of Hong Kong\nShenzhen", "The Chinese University of Hong Kong\nShenzhen", "The Chinese University of Hong Kong\nShenzhen", "The Chinese University of Hong Kong\nShenzhen" ]
[]
c) Hash-grid (a) GT (b) Ours (d) TensoRF (e) Dense-grid (f) MLP Large-scale scene 7UDLQLQJWLPHPLQ 3615G% Mega-NeRF (one day) 2XUV +DVKJULG 'HQVHJULG 7HQVR5) 0/3 Figure 1. We propose a hybrid feature representation for neural radiance fields (NeRF) to enable efficient large-scale unbounded scene modeling. Compared with results of models that replace the hybrid representation with hash-grid [31], dense-grid [11, 45], TensoRF [7] or MLP [30], our hybrid representation achieves faster convergence and higher accuracy. Notably, our method can finish training in 1.5 hours on a single GPU while achieving results better than Mega-NeRF [52] that requires about one day's training with 8 GPUs.AbstractExisting neural radiance fields (NeRF) methods for large-scale scene modeling require days of training using multiple GPUs, hindering their applications in scenarios with limited computing resources. Despite fast optimization NeRF variants have been proposed based on the explicit dense or hash grid features, their effectivenesses are mainly demonstrated in object-scale scene representation. In this paper, we point out that the low feature resolution in explicit representation is the bottleneck for large-scale unbounded scene representation. To address this problem, we introduce a new and efficient hybrid feature representation for NeRF that fuses the 3D hash-grids and highresolution 2D dense plane features. Compared with the dense-grid representation, the resolution of a dense 2D plane can be scaled up more efficiently. Based on this hybrid representation, we propose a fast optimization NeRF variant, called GP-NeRF, that achieves better rendering results while maintaining a compact model size. Extensive experiments on multiple large-scale unbounded scene datasets show that our model can converge in 1.5 hours using a single GPU while achieving results comparable to or even better than the existing method that requires about one day's training with 8 GPUs. Our code can be found at https://zyqz97.github.io/GP NeRF/.
10.48550/arxiv.2303.03003
[ "https://export.arxiv.org/pdf/2303.03003v2.pdf" ]
257,364,886
2303.03003
ef91b32ea9dc7deada8f6c37ced8da92074bf72d
Efficient Large-scale Scene Representation with a Hybrid of High-resolution Grid and Plane Features Yuqi Zhang The Chinese University of Hong Kong Shenzhen Guanying Chen The Chinese University of Hong Kong Shenzhen Shuguang Cui Fnii The Chinese University of Hong Kong Shenzhen Sse The Chinese University of Hong Kong Shenzhen Efficient Large-scale Scene Representation with a Hybrid of High-resolution Grid and Plane Features c) Hash-grid (a) GT (b) Ours (d) TensoRF (e) Dense-grid (f) MLP Large-scale scene 7UDLQLQJWLPHPLQ 3615G% Mega-NeRF (one day) 2XUV +DVKJULG 'HQVHJULG 7HQVR5) 0/3 Figure 1. We propose a hybrid feature representation for neural radiance fields (NeRF) to enable efficient large-scale unbounded scene modeling. Compared with results of models that replace the hybrid representation with hash-grid [31], dense-grid [11, 45], TensoRF [7] or MLP [30], our hybrid representation achieves faster convergence and higher accuracy. Notably, our method can finish training in 1.5 hours on a single GPU while achieving results better than Mega-NeRF [52] that requires about one day's training with 8 GPUs.AbstractExisting neural radiance fields (NeRF) methods for large-scale scene modeling require days of training using multiple GPUs, hindering their applications in scenarios with limited computing resources. Despite fast optimization NeRF variants have been proposed based on the explicit dense or hash grid features, their effectivenesses are mainly demonstrated in object-scale scene representation. In this paper, we point out that the low feature resolution in explicit representation is the bottleneck for large-scale unbounded scene representation. To address this problem, we introduce a new and efficient hybrid feature representation for NeRF that fuses the 3D hash-grids and highresolution 2D dense plane features. Compared with the dense-grid representation, the resolution of a dense 2D plane can be scaled up more efficiently. Based on this hybrid representation, we propose a fast optimization NeRF variant, called GP-NeRF, that achieves better rendering results while maintaining a compact model size. Extensive experiments on multiple large-scale unbounded scene datasets show that our model can converge in 1.5 hours using a single GPU while achieving results comparable to or even better than the existing method that requires about one day's training with 8 GPUs. Our code can be found at https://zyqz97.github.io/GP NeRF/. We propose a hybrid feature representation for neural radiance fields (NeRF) to enable efficient large-scale unbounded scene modeling. Compared with results of models that replace the hybrid representation with hash-grid [31], dense-grid [11,45], TensoRF [7] or MLP [30], our hybrid representation achieves faster convergence and higher accuracy. Notably, our method can finish training in 1.5 hours on a single GPU while achieving results better than Mega-NeRF [52] that requires about one day's training with 8 GPUs. Abstract Existing neural radiance fields (NeRF) methods for large-scale scene modeling require days of training using multiple GPUs, hindering their applications in scenarios with limited computing resources. Despite fast optimization NeRF variants have been proposed based on the explicit dense or hash grid features, their effectivenesses are mainly demonstrated in object-scale scene representation. In this paper, we point out that the low feature resolution in explicit representation is the bottleneck for large-scale unbounded scene representation. To address this problem, we introduce a new and efficient hybrid feature representation for NeRF that fuses the 3D hash-grids and highresolution 2D dense plane features. Compared with the dense-grid representation, the resolution of a dense 2D plane can be scaled up more efficiently. Based on this hybrid representation, we propose a fast optimization NeRF variant, called GP-NeRF, that achieves better rendering results while maintaining a compact model size. Extensive experiments on multiple large-scale unbounded scene datasets show that our model can converge in 1.5 hours using a single GPU while achieving results comparable to or even better than the existing method that requires about one day's training with 8 GPUs. Our code can be found at https://zyqz97.github.io/GP NeRF/. Introduction Recent neural scene representation methods have achieved great success in 3D scene reconstruction and novel-view synthesis [30,33,43,63]. Specifically, neural radiance fields (NeRF) [30] models the scene with a multilayer perception (MLP) that takes a 3D point location as well as the view direction as input, and predicts its color and density. The pixel color of a view ray can then be computed by volume rendering using colors and densities of points sampled in that ray. However, existing methods mainly focus on object-scale scenes. Compared with object-scale scenes, large-scale scenes (e.g., urban-scale environments) can cover more than 1, 000, 000m 2 areas [52], which often require hours or even days to capture thousands of high-resolution images (see Table 1 for examples). Existing methods for large-scale scene modeling focus on improving the scalability of NeRF. They adopt a distributed training strategy that divides the scene into multiple partitions, each partition is represented by a large MLP model to increase the rendering quality [47,52]. However, these methods require days of training with multiple GPUs, significantly hindering their applications in users with lim- ited computational resources. Recently, many NeRF variants have been proposed to speed up the optimization speed [31,45,64]. One of the key ideas is to store local features in 3D dense voxel grid [45,64] or hash-grid [31], such that most of the time-consuming MLP computation can be replaced with the fast feature interpolations. However, the effectivenesses of these methods are mainly demonstrated in object-scale scene representation. A straightforward idea is to adopt the grid representation to speed up the large-scale scene optimization. But as the parameter number of dense-grid representation grows cubically as O(N 3 ) with resolution N , existing methods [45,64] often use a relatively small resolution (e.g., N = 160 in DVGO [45]) during optimization, making them not suitable for representing large-scale scene 1 . Multi-resolution hashgrid [31] is an efficient grid structure that applies a hash function to randomly map 3D points into a hash table. By limiting the hash table size (e.g., T = 2 19 ) in each resolution, the resolution of hash-grid can be set in a much larger number. However, in the presence of hash collision, directly applying hash-grid for large-scale scene modeling leads to sub-optimal results. Therefore, it is important to develop efficient high-resolution feature representation to represent a large-scale scene. Motivated by the recent success of 2D plane features whose parameter grows quadratically with the resolution as O(N 2 ) in 3D-aware image synthesis [6], in this work, we introduce an efficient hybrid representation for large-scale unbounded scene modeling. The key idea of our representation is to enhance the 3D hash-grid feature with multiple orthogonally placed high-resolution dense 2D plane features. The structured dense plane features are complementary with the randomly-mapped hash-grid feature, especially for surface regions with collisions. Compared with directly scaling up the hash table size in the hash-grid representation, the proposed hybrid representation can achieve higher accuracy with comparable runtime while using much fewer parameters. Based on the proposed hybrid representation, we propose an efficient NeRF method, called GP-NeRF, for large-scale unbounded scene representation that can finish training with significantly fewer times (see Fig. 1). To summarize, our key contributions are: 1 e.g., if representing a 500 × 250m 2 scene in a grid with the resolution of 250 3 , each voxel will be responsible for an area of 1 × 2m 2 . • We propose a new fast optimization NeRF variant, called GP-NeRF, that is specifically designed for large-scale unbounded scene modeling. • We introduce a new hybrid feature representation that integrates the complementary features from 3D hashgrid and dense 2D planes to enable efficient and accurate large-scene modeling. • Experiments show that our method can finish training in 1.5 hours on a single GPU while achieving results comparable to or even better than the existing method that requires about one day's training on 8 GPUs. Related Work Large-scale 3D Reconstruction. Large-scale scene reconstruction from multi-view images is a classic problem in computer vision [1,12,22,39,44,70]. Traditional methods often rely on the structure-from-motion (SFM) pipeline to estimate the camera poses [40], and apply dense multi-view stereo [13,14] to generate the 3D model of the scene. In this work, our method adopts the recent neural representation [30] for scene reconstruction and novel-view synthesis. Neural Scene Representation. Neural scene representation has revolutionized the problem of scene reconstruction and novel-view synthesis [42,48,49,60]. In particular, neural radiance fields (NeRF) [30] has attracted considerable attention for its photo-realistic render quality. Many follow-up methods are then developed to improve its robustness [2,29,66], generalization [8,51,54,61,65], dynamic modeling [23,36,50], scene decomposition [4,62,69], etc. Large-scale Scene Rendering. There are methods extending NeRF for handling large-scale scenes. NeRFusion [68] proposes a progressive update scheme for large indoor scene reconstruction by gradually constructing the local volumes to build the final global volumes. Urban Radiance Fields [38] makes use of the LIDAR data and RGB images to achieve better reconstruction results for streetview environments. BungeeNeRF [59] employs the residual learning strategy to train a NeRF-based model for rendering multi-scale data of a city from Google Earth Studio. Some methods adopt a divide-and-conquer strategy to handle large-scale scenes. Block-NeRF [47] decomposes the street views of a city into several blocks, and each block is represented by an individual NeRF. Mega-NeRF [52] divides the space according to the camera distributions to render a large-scale scene captured by a drone. Similarly, Wu et al. [58] stack multiple tiles (each tile consists of two MLPs) based on a global mesh proxy for scalable indoor scene rendering. However, these methods merely consider the ability to reconstruct large scenes, and suffer from a long training time and low efficiency. Note that the divide-and-conquer strategy can also be integrated with our method. Efficient NeRF Rendering and Optimization. To improve the rendering speed of NeRF, some follow-up works have been proposed [18,24,27,28,32,53,56,57,64]. NSVF [27] constructs a sparse voxel field, in which each vertex of the voxel stores local attributes. PlenOctrees [64] utilizes spherical harmonics instead of directly predicting RGB values, and adopts an Octree-based data structure to accelerate the rendering speed. KiloNeRF [37] decomposes the NeRF into thousands of MLPs. Besides, FastNeRF [16] factorizes the original NeRF and leverages caching to speed up rendering. Recently, several approaches for fast optimization of NeRF have been proposed [7,11,15,17,31,45]. DVGO [45] and Plenoxels [11] adopt the explicit voxel-grid representation to replace the time-consuming MLP queries with the fast grid interpolation operation. Instant-NGP [31] further encodes the feature of a 3D point by multi-resolution hash encoding to support high-resolution grids with low memory consumption. In addition to grid-based methods, there are some other works on fast optimization. Efficient-NeRF [19] accelerates the training by removing redundant sample points based on the coarse reconstruction. Ten-soRF [7] represents the radiance fields as 4D tensor fields and utilizes the tensor factorization to speed up the training phase. EG3D [6] proposes a tri-plane representation to enable high-resolution multi-view consistent image generation. Concurrent to our work, some method utilizes plane features for dynamic scene modeling [5,10,41]. In contrast to these methods, we focus on enhancing the hash-grid feature with high-resolution plane features to enable efficient large-scale unbounded scene modeling. Preliminary Neural Radiance Fields. Given a set of images with calibrated camera poses, NeRF [30] represents the scene through the weights of a multilayer perceptron (MLP). Taking a 3D point position x i = (x, y, z) and the view direction d as input, the NeRF MLP can predict the corresponding density σ i and color c i of the input: (σ i , c i ) = MLP(x i , d).(1) To render the pixel color of a view ray, multiple points are sampled along the ray and fed into the MLP to query their densities and colors. Volume rendering [20] is then apply for accumulating the discrete color values to compute the pixel color:Ĉ (r) = N −1 i=0 T i (1 − exp(−σ i δ i )c i ),(2) where T i = exp(− i−1 j=0 σ j δ j ) denotes the accumulated transmittance and δ i denotes the distance between adjacent sample points. Fast Feature Interpolation for Acceleration. To produce high-quality rendering, NeRF adopts a large MLP model to represent the scene (i.e., 9 fully-connected layers with a channel number of 256). As millions of MLP queries are needed in each iteration, a long optimization time is required, especially for large-scale scenes. To speed up the training, methods based on the explicit representation are proposed [7,11,31,45]. These methods replaced most of the time-consuming MLP computation with a fast feature interpolation operation on the explicit representation G (e.g., a dense voxel grid or hash-grid). Taking the interpolated feature f as input, a small MLP model can be applied to regress the density and color, significantly reducing the optimization: f i = Interpolate (G, x i ), (3) (σ i , c i ) = MLP small (f i , d).(4) As the parameter number of the voxel grid grows dramatically with the increase of resolution N , these methods adopt a small resolution (e.g., 160 in DVGO [45] and 300 in TensoRF [7]). Although working well on object-scale scenes, the problem of low-resolution grid features become noticeable in a large-scale scene. Although the hash-grid representation [31] can set a much larger resolution by fixing the hash table size, the collision effect will become more severe. Therefore, it is important to develop efficient high-resolution feature representation for fast optimization of large-scale scenes. Method Our goal is to develop an efficient high-resolution feature representation to improve the capacity of fast optimization NeRF methods in representing a large-scale scene. We start by revisiting the recent efficient hash-grid [31] representation for scene modeling and identify its strengths and limitations. Based on the discussion, we introduce an efficient hybrid representation that enhances the 3D hashgrid feature with high-resolution dense 2D plane features, just with a small increase in the parameters and runtime. Then, we parameterize a large-scale unbounded scene in a contract space [3] to enable a compact space representation. Last, we introduce the network architecture and optimization strategy (see Fig. 2). Multi-resolution Hash-grid Representation Multi-resolution hash-grid [31] is an efficient data structure to represent a scene with high-resolution grids (e.g., 1024 or even higher resolution) without significantly increasing the parameter number by randomly mapping 3D points to a linear hash table with a fixed size. The parameter number of a multi-resolution hash-grid is bounded by L · T · F , where L is the number of the resolution, T and Overview. First, we sample 3D points along the rays emitted from pixels. Second, we parameterize the space into a contracted space for a compact space parameterization. Then, querying the position into the proposed hybrid-representation to extract hash-grid feature and multi-resolution plane features, which are fed into a light-weight MLP to predict the sigma σ and the color values c (color MLP requires the additional view directions, appearance embeddings, and plane feature input). Finally, the image colors can be computed by volume rendering. F are the hash table size and feature dimension in each resolution. The suggested configuration is (L = 16, T = 2 19 , F = 2) to balance the trade-off between the capacity and efficiency [31], leading to a bound of 2 24 parameters. Our experiment shows that the hash-grid representation with a highest resolution of 2048 clearly outperforms the low-resolution dense-grid [45] and TensoRF [7] representation in large-scale scene rendering, demonstrating the resolution of feature resolution plays a very important part in large-scale scene modeling. However, due to the collision problem in random hash mapping, the interpolated feature inevitably contains mixed information for different surface points, limiting the performance of NeRF model. We also observe that increasing the hash table size T can improve the results, but at the cost of a significant increase in parameter number and a longer optimization time [31]. This discussion motivates us to introduce an efficient strategy to boost the hash-grid feature for large-scale scenes just using a small number of parameters and runtime. High-resolution 2D Plane Features Observing that 2D plane features whose parameter grows quadratically with the resolution as O(N 2 ) can provide strong information to enable 3D-aware imagesynthesis [6] and 3D reconstruction [35], we propose to enhance the hash-grid feature with high-resolution dense plane features for large-scale scene representation. Orthogonally Placed 2D Plane Features. We design the plane features as three orthogonally placed planes with a resolution of N and a feature dimension of F , then the parameter number for each plane is N 2 · F . For a queried 3D point, we first orthogonally project it on these three planes, and obtain the 2D plane features with bilinear interpolation. We then concatenate the three interpolated features to form a feature vector of length 3 · F . We are able to scale up the resolution of the dense 2D planes to 1024. By doing this, our plane representation can provide features in a resolution that "equivalent to" a densegrid with 2 30 parameters, which is difficult to achieve in a dense-grid due to the significant memory consumption. Efficient Multi-resolution Design. However, the feature dimension cannot be too large as the parameters will increase sharply (e.g., setting F = 8 already results in a parameter number of 8 × 1024 × 1024 = 2 23 ), or too small (e.g., F = 2) as the plane features will not provide useful information (see our ablation study in Table 5). To fulfill the goal of providing high-resolution dense features, we design multi-resolution planes following [31]. Specifically, we adopt a four resolutions configuration N ∈ {128, 256, 512, 1024}, each has a feature dimension of 2, resulting in an 8 dimension multi-resolution features. This design can effectively boost the hash-grid feature while maintaining a low parameter numbers. Moreover, as in a large-scale scene, the height of a scene is often smaller than the horizontal length. To reduce the waste of features in the vertical planes, we scale the vertical planes using the camera altitude measurements. Relation to Previous Work. Our method is inspired by EG3D [6] which proposes a tri-plane representation, where each plane has a shape of 256 × 256 × 32, and the interpolated features from three different planes are fused by an add operation for GAN-based image generation. In contrast, our goal is to design efficient high-resolution features for largescale scene modeling by scaling up the plane feature to 1024 and using a multi-resolution strategy to maintain a low parameter numbers. Moreover, we use a simple concatenation operation to combine the three interpolated features, as we experimentally find that applying the add operation leads to worse results, which might be explained by the small feature dimension in our representation (i.e., 8 in ours and 32 in EG3D). The recent TensoRF [7] decomposes the radiance fields into several matrices and vectors. However, the matrix has a shape of N 2 · R · F , where R = 192 is the component number. Compared with the shape of N 2 · F in our plane feature, the matrices in TensoRF representation cannot be efficiently scaled to a high resolution (e.g., 1024), making it not suitable for large-scale scene modeling. Hybrid of Grid and Plane Features Given a queried 3D point, we interpolate the hash-grid to get a 32-dimension grid feature and the orthogonally placed planes to get a 24-dimension feature. Then these two features are concatenated to form a hybrid feature to be the input of NeRF MLP. Note that there are more sophisticated feature fusion strategies (e.g., MLP fusion) to fuse the hashgrid and plane features. However, to maintain a simple and efficient architecture and also to better demonstrate the effectiveness of our plane features, we just use a simple concatenation. Compared with the hash-grid that adopts a random mapping hash-function, the orthogonally placed plane representation introduces a structured and high-resolution dense feature, which is complementary to the hash-grid feature, especially in regions with hash collision. As a result, the proposed hybrid feature representation can provide better features for later density and color prediction. Scene Space Parameterization As a large-scale scene covers a wide spatial range and is always unbounded, it is important to parameterize it in a way that it can be represented by a grid-like data structure. We adopt the mip-NeRF 360 parametrization [3,46] to represent a unbounded scene. Considering a scene whose center is the origin, we divide the scene into foreground and background regions, separating by a pre-define bound B. Given a 3D point, we will first normalize it by x = x/B, and then applied the space contraction: x = x, x p ≤ 1 ; 1 + b − b x p x x p , x p > 1 ,(5) where p denotes p-norm, b is to control the size of background space. Following [3], we set p = 2 and b = 1. Figure 3. Illustration of the scene parameterization. According to the pre-defined bound, we adopt linear sampling and invert linear sampling for the foreground and background areas, respectively. Then, the sampled points are contracted into a compact space. Bound +∞ -1-b 1+b 1 -1 (a) Euclidean space (b) Contracted space By doing this, the distribution of points is more compact, where the foreground point (i.e., x p ≤ 1) is unchanged and a point at infinity will be mapped to the sphere with a radius of 1 + b. This compact representation enable us to represent the scene with the proposed hybrid feature representation (see Fig. 3). Model Optimization After defining the hybrid representation and scene parameterization for large-scale scenes, we introduce our network architecture and optimization strategy. Network Architecture. Given a 3D point position x i and its direction d as input, our method first extract the features from the hybrid feature representation. These hybrid features are then fed into a 64-channel two-layer MLP network that regresses the view-independent density value σ i and a 15-channel geometry feature. Then, the view direction encoded with the spherical harmonics function, the per-frame appearance embedding [29], and the plane feature are concatenated with the geometry features to be the input of a 64-channel three-layer network to predict the view-dependent color value. The per-frame appearance embedding is adopted to account for the illumination changes in the captured images. We experimentally found that include the plane feature to the color MLP leads to better results, as discussed in the supplementary material. Point Sampling. We adopt a linear sampling strategy for foreground and an invert sampling strategy for the background [46] (see Fig. 3), which satisfies the need of highquality foreground rendering and covers a wide range of background. In details, we sample N f = 128 and N b = 64 points per-ray for these two areas, respectively. We also adopt the coarse-to-fine sampling strategy [30], where the coarse and fine stages have the same number of sampled points, but use the same model to reduce the model size. Independent Background Modeling. Existing methods show that representing the foreground and background regions independently leads to better results [52,66]. To increase the model capacity, we utilize two separate models for the foreground and background regions. For the foreground region, we use the proposed hybrid feature representation for feature encoding. For the background region, we only use a hash-grid with a hash table size of T bg = 2 19 . Our design can improve the performance without an obvious increase of time costs. Then, points in different regions will be processed by the corresponding models. Loss Function. Given multiple 3D points sampled from a view ray r, volume rendering will be utilized to produce the pixel color. We adopt the mean square error (MSE) between the rendered colorĈ(r) and the ground-truth color C(r) as the loss function: Loss = r∈R Ĉ (r) − C(r) 2 2 ,(6) where R is the sampled ray set. Experiments Datasets. Following Mega-NeRF [52], we evaluate our GP-NeRF on three public large-scale datasets, namely the Mill19 [52], Quad6k [9], and UrbanScene3D [25] datasets, with the camera poses refined by PixSFM [26]. Mill19 dataset is proposed by Mega-NeRF [52], which includes Mill19-Building and Mill19-Rubble, covering about 250 × 400 m 2 large areas. Quad6k dataset is a large-scale Structure-from-Motion (SfM) dataset containing about 5100 images of Cornell University Arts Quad. Following [52], we adopt three urban scenes from Ur-banScene3D dataset, which provide high-resolution images from drones. Implementation Details. We implemented our method with PyTorch [34] and used the Adam optimizer [21] with default parameters. Our model was trained with 100K iteration using a batch size of 5 × 1024 rays. The learning rate was set to 0.001. Following the practice in [31], the hyper-parameter of hash table size is set to L = 16, T = 2 19 , F = 2. The lowest and highest resolutions are set to 16 and 2048, respectively. The runtime was measured on an NVIDIA GeForce RTX 3090 GPU. Evaluation Metrics. We evaluate the existing methods and our method in terms of PSNR, SSIM [55], and the VGG implementation of LPIPS [67]. Evaluation Comparison with Existing Methods. We first compare our method with existing methods to verify its effectiveness. Specifically, we compare with the state-of-the-art Mega-NeRF [52] and fast NeRF methods (Plenoxels [11], TensoRF [7], and Instant-NGP [31]). The training time are measured by training on the same amount of data. We can see from Table 2 that our method achieves the best results on five of the six scenes. Compared with Mega-NeRF which requires about one day's training with 8 GPUs, our method achieves a fast convergence speed of 1.5 hours using a single GPU, while maintaining comparable or even better results. The reason is that our method only use a single 5-layer network with 64 channels and an efficient hybrid feature representation for feature interpolation, while Mega-NeRF uses large 9-layer MLPs with 256 channels for predictions. Despite enjoying fast training convergence, existing fast NeRF methods [7,11,31] achieve poor rendering quality. It is because these methods are not designed for largescale unbounded scenes with illumination changes and suffer from low-resolution representation. Comparison with Other Feature Representations. To further demonstrate the effectiveness of our hybrid feature representation, we design three baseline methods. Specifically, we replace the proposed hybrid feature representation of our method with the dense-grid (resolution is 160 3 ) [11,45] and pure hash-grid [31] representation. In addition, we adopt the VM-192 variant of TensoRF with a resolution of 300 3 voxels [7]. To have a fair comparison, other parts of the methods are kept the same. We also apply the feature space scaling according to the camera altitude measurement for the baseline methods for fair comparisons. Quantitative results in Table 3 show that the results of methods with dense-grid and TensoRF representation are worse, indicating that they are not suitable for large-scale scenes as their feature resolutions are low. Compared with these two methods, our method achieves much more accurate rendering results, with comparable or smaller parameters (i.e., ours is 33.0M, Ours w/ TensoRF is 34.8M, and Ours w/ dense-grid is 98.4M). Our method also consistently outperforms Ours w/ hashgrid on all six datasets. We notice that the hash-grid method finishes training with less time, we therefore design experiments to train the hash-grid method with the same time Table 3. Comparison between our hybrid-representation and existing feature representations. We compare with three baselines that replace the hybrid-representation in our method with the dense-grid [11,45], the pure hash-grid [31], and the TensoRF (VM-192) [7]. Mill as ours for a fair comparison, namely Ours w/ hash-grid (01: 30) in Table 3. The results show that the performance of hash-grid method improves with a longer training time, but its results are still far behind our method, further demonstrating the effectiveness of our method. Figure 4 shows the qualitative comparison. We can see that the baselines based on low-resolution features, e.g. Ours w/ TensoRF and dense-grid, produce blurry results, while Ours w/ hash-grid produces better results with highresolution hash-grid. However, at the presence of collision, the rendering details are still blurry. In contrast, we design the multi-scale high-resolution dense plane and utilize the hybrid-representation to enhance the hash-grid feature, leading to better results (e.g. the railway track in the Mill19-Rubble dataset). Ablation study In this section, we conduct ablation studies to evaluate the proposed GP-NeRF on two representative scenes, Mill19-Rubble [52] and UrbanScene3D-Residence [25]. Effectiveness of the Plane Feature. To verify the effectiveness of the proposed hybrid feature representation, we integrate our plane features with hash-grid with different hash table sizes (i.e., from 2 19 to 2 22 ) and compare them with methods using only hash-grid features, and the results are summarized in Fig. 5. We can see that as the hash table size improves, the results of hash-grid only method in-1XPEHURISDUDPHWHUVPLOOLRQV 3615G% $ % crease but with a significant increase in model size. It can be clearly seen that the performance of models with a hybrid of hash-grid feature (with different hash table sizes) and plane features exceed the corresponding pure hash-grid results. Notably, the hybrid representation can achieve better results using much fewer number of parameters than a pure hash-grid with higher table size (e.g., comparing the point A and B in Fig. 5). To better verify the effectiveness of our plane feature, we remove the hash-grid and only use plane features for feature encoding. Experiment with ID 0 in Table 4 shows that the pure plane features can also achieve respectable results (see Fig. 6 for visual results). Furthermore, we replace the plane feature in our hybrid representation with the densegrid (90 3 ) and TensoRF (VM-192, 210 3 ) that has comparable parameters as our method (i.e., 33.0M). Experiments with IDs 1-4 in Table 4 shows that integrating hash-grid with the low-resolution dense-grid or TensoRF cannot consistently improve the results and might lead to worse results (see the average PSNR), indicating that the high-resolution plane features help to resolve the collision problem and the improvement comes from our design rather than an increase of parameters. Both the quantitative and qualitative results reiterate that the plane features provide strong information for the scene geometry and appearance and is complementary to the hash-grid features. Effects of Plane Feature Dimension. In Table 5, we first conduct experiments with IDs 0-1 to investigate the influence of feature dimension in a single-resolution plane representation. We can see that the result of a single-resolution plane with small feature dimension (e.g., 2) is worse, and using a high feature dimension (e.g., 8) can achieve good PSNR results, but the increase in the number of parameters is also significant. Experiments with ID 2-3 in Table 5 show that multi-resolution planes with small resolution can also achieve good results while maintaining a small parameter number. These results justify the design of our multiresolution plane features, which strikes a good balance between the performance and parameters. Effects of Plane Feature Resolution. To investigate the influence of the resolution of plane features, we compare results of plane features with different highest resolutions (the base resolution is 128). Table 6 shows that there is a tendency that the larger the highest resolution is, the better performance achieves, which is a trade-off between the increase of parameters and performance. This also demonstrates the effectiveness of plane features. GP-NeRF as the Local Radiance Field As a general NeRF variant, our GP-NeRF can also be adopted as the local radiance field in Mega-NeRF's partition training framework [52] to further improve the per- formance. We trained two Mega-NeRF variants with GP-NeRF and Instance-NGP [31] (for purpose of comparison) as the local NeRF. Table 7 shows that adopting our GP-NeRF as the local radiance field achieves the best results in 1.5 hours, clearly outperforms Mega-NeRF and the Instant-NGP variant, verifying the design of our method. Conclusion In this paper, we have presented a hybrid feature representation for the neural radiance fields to enable efficient large-scale scene modeling. Our hybrid representation enhances the hash-grid feature with orthogonally placed high-resolution and dense plane features, especially for surface regions with hash collisions. Compared with directly scaling up the hash table size in the hash-grid, our representation can achieve higher accuracy with comparable runtime while using much fewer parameters. Based on our hybrid representation, we propose a new variant of NeRF, called GP-NeRF, to represent large-scale unbounded scenes. Compared with the existing method [52] that requires one day's training on 8 GPUs, our method achieves comparable or even better results with 1.5 hours on a single GPU, which is especially useful for scenarios with limited computing resources. Limitations. Despite the significant acceleration of optimization achieved for large-scale scene representation, our method still cannot achieve real-time scene reconstruction. Moreover, our method does not explicitly model dynamic objects and thus cannot represent a dynamic scene. We would like to address these two problems in the future. Figure 1 . 1Figure 1. We propose a hybrid feature representation for neural radiance fields (NeRF) to enable efficient large-scale unbounded scene modeling. Compared with results of models that replace the hybrid representation with hash-grid [31], dense-grid [11, 45], TensoRF [7] or MLP [30], our hybrid representation achieves faster convergence and higher accuracy. Notably, our method can finish training in 1.5 hours on a single GPU while achieving results better than Mega-NeRF [52] that requires about one day's training with 8 GPUs. Figure 2 . 2Figure 2. Overview. First, we sample 3D points along the rays emitted from pixels. Second, we parameterize the space into a contracted space for a compact space parameterization. Then, querying the position into the proposed hybrid-representation to extract hash-grid feature and multi-resolution plane features, which are fed into a light-weight MLP to predict the sigma σ and the color values c (color MLP requires the additional view directions, appearance embeddings, and plane feature input). Finally, the image colors can be computed by volume rendering. Figure 4 . 4Qualitative results on the Mill19-Rubble and UrbanScene3D-Campus dataset. Figure 6 . 6Effectiveness of the plane feature. Table 1 . 1Statistics of large-scale scene datasets reported in[52].Datasets Resolution Num of images Coverage area Mill19 -Building 4608 × 3456 1940 262 × 438 m 2 Mill19 -Rubble 4608 × 3456 1678 206 × 248 m 2 Quad 6k 1708 × 1329 5147 285 × 420 m 2 UrbanScene3D -Residence 5472 × 3648 2582 291 × 491 m 2 UrbanScene3D -Sci-Art 4864 × 3648 3019 373 × 317 m 2 UrbanScene3D -Campus 5472 × 3648 5871 1346 × 1542 m 2 Table 2 . 2Quantitative comparison on the testing datasets. * indicates that Mega-NeRF's results are computed with its released models. ↑PSNR ↑SSIM ↓LPIPS ↑PSNR ↑SSIM ↓LPIPS ↑PSNR ↑SSIM ↓LPIPS ↑PSNR ↑SSIM ↓LPIPS ↑PSNR ↑SSIM ↓LPIPS ↑PSNR ↑SSIM ↓LPIPS ↓Time (h)Mill 19 -Building Mill 19 -Rubble UrbanScene3D -Residence UrbanScene3D -Sci-Art UrbanScene3D -Campus Quad 6k Average Method Plenoxels [11] 17.75 0.419 0.670 20.48 0.462 0.658 18.27 0.517 0.579 18.93 0.638 0.528 20.40 0.456 0.780 15.76 0.520 0.665 01:30 TensoRF (VM-192) [7] 18.19 0.416 0.697 20.77 0.452 0.675 18.32 0.498 0.607 18.39 0.594 0.585 17.22 0.418 0.842 14.56 0.512 0.679 01:31 Instant-NGP [31] 18.68 0.439 0.614 21.18 0.497 0.579 18.91 0.554 0.543 19.00 0.644 0.528 16.31 0.527 0.729 14.91 0.509 0.732 01:22 Mega-NeRF* [52] 20.93 0.547 0.504 24.06 0.553 0.516 22.08 0.628 0.489 25.60 0.770 0.390 23.42 0.537 0.618 17.66 0.536 0.616 20:10 Ours 20.99 0.565 0.490 24.08 0.563 0.497 22.41 0.659 0.451 25.56 0.783 0.373 23.46 0.544 0.611 17.67 0.521 0.623 01:35 Quantitative results of integrating the plane feature with hash-grids that have a hash table size T ranged from 2 19 to 2 22 .2 19 2 22 2 19 2 22 KDVKSODQH KDVKRQO\ 1XPEHURISDUDPHWHUVPLOOLRQV 3615G% $ % 2 19 2 22 2 19 2 22 KDVKSODQH KDVKRQO\ (a) Rubble (b) Residence Figure 5. Table 4 . 4Effectiveness of the model design.Rubble Residence Average ID Model # Param PSNR SSIM PSNR SSIM PSNR SSIM 0 Plane only 16.8M 22.75 0.493 20.61 0.554 21.68 0.524 1 Hash-grid (2 19 ) 24.6M 23.77 0.546 22.05 0.634 22.91 0.590 2 Hash-grid + Dense 33.3M 23.82 0.554 21.86 0.639 22.84↓ 0.597 3 Hash-grid + TensoRF 33.1M 23.77 0.547 22.02 0.634 22.90↓ 0.591 4 Hash-grid + Plane (Ours) 33.0M 24.08 0.565 22.41 0.659 23.25↑ 0.612 Table 5 .Table 6 . 56Effects of the feature dimension per resolution and the multi-resolution design in the plane feature representation. 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[]
[ "Rotation Invariant Quantization for Model Compression", "Rotation Invariant Quantization for Model Compression" ]
[ "Joseph Kampeas \nTel-Aviv Research Center\nIsrael\n", "Yury Nahshan \nTel-Aviv Research Center\nIsrael\n", "Hanoch Kremer \nTel-Aviv Research Center\nIsrael\n", "Gil Lederman \nTel-Aviv Research Center\nIsrael\n", "Shira Zaloshinski \nTel-Aviv Research Center\nIsrael\n", "EmirZheng Li \nTel-Aviv Research Center\nIsrael\n", "Haleva Huawei \nTel-Aviv Research Center\nIsrael\n" ]
[ "Tel-Aviv Research Center\nIsrael", "Tel-Aviv Research Center\nIsrael", "Tel-Aviv Research Center\nIsrael", "Tel-Aviv Research Center\nIsrael", "Tel-Aviv Research Center\nIsrael", "Tel-Aviv Research Center\nIsrael", "Tel-Aviv Research Center\nIsrael" ]
[]
Post-training Neural Network (NN) model compression is an attractive approach for deploying large, memory-consuming models on devices with limited memory resources. In this study, we investigate the rate-distortion tradeoff for NN model compression. First, we suggest a Rotation-Invariant Quantization (RIQ) technique that utilizes a single parameter to quantize the entire NN model, yielding a different rate at each layer, i.e., mixed-precision quantization. Then, we prove that our rotation-invariant approach is optimal in terms of compression. We rigorously evaluate RIQ and demonstrate its capabilities on various models and tasks. For example, RIQ facilitates ×19.4 and ×52.9 compression ratios on pretrained VGG dense and pruned models, respectively, with < 0.4% accuracy degradation. Code: https://github.com/ehaleva/RIQ.
10.48550/arxiv.2303.03106
[ "https://export.arxiv.org/pdf/2303.03106v1.pdf" ]
257,364,938
2303.03106
fa11639c93456438209aa0231c98f8a0aab36af1
Rotation Invariant Quantization for Model Compression Joseph Kampeas Tel-Aviv Research Center Israel Yury Nahshan Tel-Aviv Research Center Israel Hanoch Kremer Tel-Aviv Research Center Israel Gil Lederman Tel-Aviv Research Center Israel Shira Zaloshinski Tel-Aviv Research Center Israel EmirZheng Li Tel-Aviv Research Center Israel Haleva Huawei Tel-Aviv Research Center Israel Rotation Invariant Quantization for Model Compression Post-training Neural Network (NN) model compression is an attractive approach for deploying large, memory-consuming models on devices with limited memory resources. In this study, we investigate the rate-distortion tradeoff for NN model compression. First, we suggest a Rotation-Invariant Quantization (RIQ) technique that utilizes a single parameter to quantize the entire NN model, yielding a different rate at each layer, i.e., mixed-precision quantization. Then, we prove that our rotation-invariant approach is optimal in terms of compression. We rigorously evaluate RIQ and demonstrate its capabilities on various models and tasks. For example, RIQ facilitates ×19.4 and ×52.9 compression ratios on pretrained VGG dense and pruned models, respectively, with < 0.4% accuracy degradation. Code: https://github.com/ehaleva/RIQ. Introduction Deep neural networks are widely used for various tasks, such as computer vision, Natural Language Processing (NLP), and recommendation systems. Nevertheless, while performance continuously improves, the models become larger with a massive increase in the number of parameters. In fact, modern Neural Network (NN) models may have billions and even trillions of parameters, which makes the deployment of these models a challenging task (Chang et al., 2020). One way to mitigate this issue is compressing the model's parameters to reduce its overall memory footprint while satisfying an accuracy constraint. Namely, obtaining a smaller model that is (almost) as capable as the original model. The most common model compression techniques are weight pruning, quantization, knowledge distillation, and low-rank decomposition. Such optimizations strive to find a smaller model while keeping the original model's accuracy, overlooking the potential inherent in its entropy limit. In the context of NN models, the entropy value is of particular interest as it provides the (theoretical) number of bits required for representing the model parameters. The optimal compression asymptotically attains this entropy limit. In this context, lossy compression gives considerable merit as it facilitates reducing the NN size significantly with negligible accuracy degradation. The key steps of this approach are the quantization and the compression (Polyanskiy & Wu, 2014, Ch. 25). In the quantization phase, the number of unique weight values (symbols) is reduced, consequently reducing the model's entropy. Modifying the symbols' statistics, however, introduces distortion (i.e., quantization error) with respect to the original model. Hence, proper quantization methods are substantial as they determine both the resulting entropy and the distortion of the model's output. In the compression phase, redundant information is removed, reaching the most compact representation possible without introducing further errors. Thus, the only distortion originates from altering the symbols' statistics, during the quantization. Finding a solution that quantizes the model at the lowest possible bit rate while satisfying a certain distortion requirement is at the heart of quantization optimization problems, and is known as the rate-distortion problem (Cover & Thomas, 2006). Typically, NN model quantization optimizes a certain distortion (or accuracy) for a given target rate, which is suitable for reducing the computational complexity of the NN model to a lower precision. However, it is reasonable to consider the dual problem, which optimizes the compression rate while satisfying a certain distortion (or accuracy) requirement. This paper addresses the latter. In this study, we investigate the rate distortion for NN model compression, where the distortion is measured by a cosine distance between the outputs of the original and the quantized model (i.e., angular deviation). In particular, we for-1 mulate the model compression as an optimization problem, where the goal is maximizing the model compression ratio subject to a given deviation constraint. Our focus is mixedprecision solutions, where each layer gets quantized at a possibly different rate. Specifically, the main contribution is as follows. • We design a post-training Rotation-Invariant Quantization (RIQ) method which compresses NN models subject to a deviation constraint. The main theme of our approach is picking the quantization bin width to be proportional to the layers' norm. Since both norm and cosine distance are invariant to rotations, this yields the optimal solution in terms of rate distortion. To find the optimal quantization parameter efficiently, we first derive the scale in which the rate increases with the deviation and then suggest a searching paradigm that leverages our finding. • To analyze the optimality of the RIQ algorithm, we introduce a surrogate model that depicts quantization in terms of rotation of the model weights. Its analysis reveals that the rate-distortion minimizing distribution for NN models is a spherical (rotation invariant) distribution constructed by the product of layers' spherical distribution. Due to convexity, the rate achieved under this product distribution is bounded by a rate achieved under the layers' average spherical distribution. Since RIQ optimizes the rate only over spherical distributions, it reaches the optimal solution efficiently. • We rigorously evaluate the RIQ and demonstrate its capabilities on various models and tasks. RIQ attains a remarkable compression ratio with a negligible accuracy loss in all assessments, surpassing recent results in the literature. Related Work This section is devoted to prior work on model compression that is most relevant to this study. Roughly speaking, typical model compression methods can be classified into four categories. Weight pruning, quantization, knowledge distillation (Sarfraz et al., 2021;Walawalkar et al., 2020), and low-rank decomposition (Idelbayev & Carreira-Perpinán, 2020;Lin et al., 2018;Lee et al., 2019). Even though such methods strive to find a smaller model while retaining the model's accuracy, they often tend to neglect the potential inherent in the entropy limit. In this study, we seek to minimize the model entropy by quantization and then attain this entropy limit by compression while satisfying a distortion requirement at the model's output. Quantization is a prominent method for compressing NN models. In Wu et al. (2020); Banner et al. (2019), the authors considered fixed-bit quantization methods, where all layers are quantized at the same integer bit rate. In this paper, on the other hand, we consider post-training mixed-precision solutions. Bhalgat et al. (2020); Wang et al. (2019); Idelbayev et al. (2021) focused on quantization-aware training, where the weights quantization is performed during the training. To attain lower quantization rates, Fan et al. (2020); Baskin et al. (2021) suggested training the models with noise. Although quantization-aware training methods may achieve better results than the post-training approaches, they are time-consuming, and thus, may not be applicable for deployment purposes. Cai et al. (2020); Hubara et al. (2021) proposed knowledge distillation techniques to minimize Kullback-Leibler (KL) divergence between the layers of the quantized and original model, using a small calibration set. Nevertheless, these works determine the quantization bin width to optimize the distortion, which is dual to the problem dealt herein. Specifically, RIQ optimizes the rate for a given distortion requirement. The idea of pruning NN connections based on informationtheoretic ideas was explored already in the seminal works (LeCun et al., 1989;Hassibi et al., 1993). Later, Han et al. (2015) used magnitude threshold to remove redundant parameters, and then, utilized Huffman's entropy coding to compress these parameters. Since then, pruning techniques gained popularity, searching for effective methods to prune parameters Frankle & Carbin, 2018;Lee et al., 2019). For assessing pruning and quantization algorithms, Gao et al. (2019); Isik et al. (2022) provided a rate-distortion theory framework, showing that entropy reduction during training is beneficial, as low-entropy models are more amenable to compression (Oktay et al., 2019;Baskin et al., 2019). This study continues these guidelines, using rate-distortion theory to analyze rotation-invariant solutions and further provides enhancements that find the optimal solution efficiently. Preliminaries In this section, we rigorously define the model compression optimization problem and the relevant known results on quantization and the rate-distortion theory. Throughout, w and w (bold) denote scalar variable and vector, respectively, unless stated otherwise. · and ·, · denotes the standard 2 -norm and the inner product, respectively. We use p w (w) to denote the probability distribution of a random variable w. Hereafter, w [1:L] = {w 1 , ..., w L } ∈ R N , denotes the weights of a pretrained model with L layers, where w ∈ R n are n weights of layer and N = Problem Statement Let f : R nx → R ny be a pretrained model that characterizes the prediction of input x to an output y ∈ R ny . The model can be decomposed into L intermediate mappings called layers, such that f (x) = f w L f w L−1 (· · · f w1 (x)) , where w ∈ R n are the weights of layer . We further assume that each layer comprises an affine operation (e.g., convolution) followed by a nonlinear activation operator (e.g., ReLU). Our goal is to obtain the smallest (quantized and compressed) version of this modelf , whose output is as close as possible to the output of f . To assess the fidelity of the quantization, a sample x is sent through f andf , and the deviation between the outputs f (x) andf (x) is measured. In this study, we focus on the cosine distance as distortion measure. That is, d f,x (w [1:L] ,ŵ [1:L] ) 1 − f (x),f (x) f (x) · f (x) (1) This distortion reflects the rotation angle that is required to alignf (x) with f (x), and according to our experiments, it serves as a decent proxy to various scoring methods such as accuracy and perplexity, (Usino et al., 2019). Hereafter, the term deviation is used to depict d f,x (w [1:L] ,ŵ [1:L] ), which is the cosine distance between the outputs, whereas the term distortion refers to the cosine distance d(w ,ŵ ) between the weights w and their quantized representationŵ . As x passes the first layer of f andf , it is rotated (and scaled) by w 1 andŵ 1 , respectively. Due to the quantization, w 1 andŵ 1 acts differently on x, and thus, yields unequal outputs. These unequal outputs are rotated (and scaled) by the next layer's weights, and so on, reaching the output of f andf . The resulting deviation in Equation (1) relates to the distortions gathered through the layers. In particular, each quantized layer produces a rotation distortion in its output, and this distortion keeps propagating and accumulating through the layers until reaching the model's output. Thus, characterizing the connection between the deviation to the distortion in each layer is a key to optimizing the model quantization. Though finding the exact link is intricate in general, intuitively, if the bin widths grow jointly, the deviation increases monotonically with the distortion of the layers. When setting the bin width of each layer proportionally to the layer's norm, the distortions are invariant to the orientation of the weights, as the norm is invariant to rotations. In terms of rate-distortion, the rotation-invariant property minimizes the rate and hence is optimal. This motivates a searching paradigm over rotation-invariant solutions, where the layers' bin widths are jointly adjusted. Formally, given a trained model f , and a sample x, we wish to find a quantized modelf whose weightsŵ [1:L] solves the following optimization problem. min Rate(ŵ [1:L] ) s.t. d f,x (w [1:L] ,ŵ [1:L] ) ≤ D for some deviation requirement D, where the rate is the number of bits per symbol allocated to the parameters of the quantized modelf . In this work, we characterize the properties of the minimum rate and devise a searching method that finds the minimum rate that satisfies D efficiently. In this formulation, we consider mixed-precision quantization solutions, where the weights w of each layer are quantized at a different rate R . That is, the average number of bits per symbol, R , varies over the layers. Moreover, the rate values R are not limited to integer values as typically considered. Allowing non-integer quantization rates may seem peculiar at first sight, however, using fewer symbols is often beneficial in terms of the entropy of the quantized weights. Specifically, to obtain the smallest model, we must pick a solution whose entropy is minimal. This entropy can be achieved by compressing the layers' weights of the modelf with an entropy achieving encoder, e.g., an arithmetic encoder such as the Asymmetric Numeral Systems (ANS) (Duda, 2013). The resulting compression ratio, assuming 32 bits representation of the source symbols, is Compression Ratio ≈ 32 · L =1 n L =1 n · H(ŵ ) + |T |(2) Where |T | denotes the coding table size of layer and H(ŵ ) is the empirical entropy (Cover & Thomas, 2006). Rate-Distortion Theory The rate-distortion theory determines the minimum number of bits per symbol, or simply the minimum bit rate, required for describing a random variable with a certain (average) distortion. In particular, to quantize a sequence of n independent realizations w w w = (w 1 , · · · , w n ), generated by a source w with distribution p w (w w w), w w w ∈ W n into R bits, encoding and decoding functions are utilized. The encoder E : W n → {0, 1} R maps the sequence to one of 2 R possible indices, and the decoder D : {0, 1} R →Ŵ n maps the given index into an estimated (quantized) sequencê w w w = (ŵ 1 , · · · ,ŵ n ). Thus, the rate-distortion pair (R, D) are the resulting rate R and distance D = d(w w w,ŵ w w) between the original sequence and the quantized sequence. In general, we wish to minimize both the rate and the distortion, however, there is an inherent tradeoff, characterized by rate-distortion function as (Cover & Thomas, 2006, Ch where I(w;ŵ) = H(w) − H(w|ŵ) is the mutual information between the source vector w and its reconstructionŵ (Cover & Thomas, 2006, Ch. 2.4), and d(·, ·) is a predefined distortion metric, such as the cosine distance in Equation (1). Thus, the rate-distortion function determines the infimum rate R that achieves a given distortion D. This infimum is attained by minimizing overall conditional distributions p(ŵ w w|w w w) for which distortion D is satisfied under p(w w w). Uniform Scalar Quantization The rate-distortion theory tells that it is optimal to describe the whole sequence jointly, using one of 2 R indices, even when the variables are i.i.d. Yet, in terms of entropy, Koshelev (1963) showed that uniform scalar quantization is (asymptotically) optimal when one intends to further compress the quantized data losslessly. Since this paper considers the latter approach, this section briefly discusses uniform scalar quantization and its analysis. For a random variable w ∈ [−A/2, A/2], where A ∈ R, uniform quantization partitions the range [−A/2, A/2] into N bins uniformly, such that each bin has width ∆ = A/N . Thus, any realization of w is encoded (rounded) into an integer value, w/∆ , that corresponds to its bin index. The decoder then reconstructs its value bŷ w = w/∆ · ∆(4) The fidelity of this quantization is typically measured by a distortion measure, such as the Mean Squared Error (MSE) criterion, defined as D(N ) = E|w −ŵ| 2 . To analyze, it is more convenient to examine the quantization in terms of rate R = log 2 N . In high-rate regime (i.e., R 1), the probability density in each bin is nearly flat, and consequently, the expected distortion is (Polyanskiy & Wu, 2014, Ch. 25.1) E|w −ŵ| 2 = ∆ 2 /12(5) Further, the resulting entropy of the quantized symbol is (Cover & Thomas, 2006, Theorem 8.3.1) H(ŵ) = H(w) − log(∆) [bits/symbol](6) where H(w) = − w∈W p(w) log p(w) is the entropy function (Cover & Thomas, 2006, eq. (2.1)). In other words, quantization reduces the entropy by log ∆, and thus, a larger ∆ yields a lower entropy, and hence, potentially, a higher compression ratio by Equation (2). Rotation-invariant Mixed-Precision Quantization In this section, we present the RIQ method, which yields a different quantization rate in each layer (i.e., mixedprecision solution) while satisfying the deviation requirement in Equation (1). Then, we use the rate-distortion theory to analyze its performance. The analysis reveals that it is optimal to address a single (average) distortion, which is driven by a single quantization parameter. The RIQ Algorithm The key motivation for rotation-invariant quantization is its optimality. In particular, given a model and a deviation requirement, it is sufficient to optimize the rate over rotationinvariant solutions. Typical quantization methods which optimize the distortion for a given rate, on the other hand, determine ∆ according to the range max(w ) − min(w ), whose value depends on the orientation of w , and hence, do not fulfill the rotation invariant requirement, which is substantial for optimality. Accordingly, RIQ designs the bin width, ∆ , in proportion to the length w in each layer. Since length is invariant to rotations, the resulting ∆ is indifferent to the orientation of w . Consequently, the resulting distortion is also indifferent to the orientation of w , as the bin width dictates the distortion. Let θ be the rotation angle from w toŵ such that w w ,ŵ ŵ cos(θ ). Then, to examine the relation between ∆ , the length w , and the resulting distortion d(w ,ŵ ) = 1 − cos(θ ) for each layer , the following lemma is useful. Lemma 4.1. Let 1 − cos(θ ) be the distortion of layer . Then, the quantization bin width asymptotically satisfies ∆ = √ · w · 24/n The proof of Lemma 4.1 is elaborated in Appendix A.1. Lemma 4.1 links between angular distortion, bin width, norm, and dimension. For instance, from the lemma follows that 1 − cos(θ ) = n ·∆ 2 24· w 2 , i.e., the distortion scales linearly with the dimension n . Interestingly, this connection expands to the entire model as follows. cos(θ [1:L] ). Assuming ŵ = w + o( w ), then, cos(θ [1:L] ) = L =1 w 2 w [1:L] 2 cos(θ ) + o w 2 w [1:L] 2 The proof is deferred to Appendix A.2. In words, a rotation of w [1:L] translates to a convex combination of the layers' rotation, and vice versa. Interestingly, due to convexity of the rate-distortion, it is beneficial to average over as many rotations as possible, which means considering partitioning the model parameters into shorter vectors as we show in the sequel. Still, the most natural partition of the model is simply to it layers, which is considered herein. Note that when √ = 1 k n 24 , where k is a parameter to be optimized, the resulting bin width is ∆ (k) = w /k. Since ∆ (k) is proportional to the layer's length it is indifferent to the orientation of w (i.e., rotation invariant). Further, letting the bin width grow linearly with w results in distortion = 1 − cos(θ ) = n 24·k 2 , and hence by Corol- lary 4.2, 1 − cos(θ [1:L] ) = 1 24·k 2 L =1 n · w 2 w [1:L] 2 . Namely, the distortion of the entire parameters scales as O(1/k 2 ). Remarkably, the deviation also scales as O(1/k 2 ). Proposition 4.3. In the high rate regime, the deviation in Equation (1) under RIQ scales as O(1/k 2 ). The proof is deferred to Appendix A.3. Figure 1 validates the analysis of Proposition 4.3. In particular, we measure the deviation for ResNet-50 as a function of k using both random data and a calibration set. Then, we compare the resulting deviations with a fitted curve of the form y = a/k 2 . Clearly, the monotonicity allows to search efficiently the optimal solution. Essentially, since ∆ (k) and are monotonically decreasing with k, then by Equation (6), the entropy increases with k. Thus, the objective of RIQ is to find the smallest k (i.e., minimum entropy) solution that satisfies the deviation requirement in Equation (1). Next, we introduce an efficient iterative searching algorithm for finding the optimal k. In each iteration, the algorithm refines the searching range until reaching the smallest k (up to a small constant) that satisfies the deviation requirement. Practically, however, as k increases, ∆ (k) → 0. To prevent this, we add a small constant 0 to √ , which bounds the value of the smallest ∆ (k). In this case, setting √ = 1 k n 24 + 0 , yields, ∆ (k) = w · 1 k + 0 · 24 n(7) Moreover, since optimizing k over unbounded search space is intricate and time-consuming, we derive searching bounds for the optimal k in the following proposition. Let k * be the optimal (smallest) k that satisfies constraint D, and let * Initialize k min = √ n * /24 1− 0 , k max = √ n * /24 √ 0· 0 , k = k min , step = √ k max − k min . while k ≤ k max do for = 1 to L do ∆ = w · 1 k + 0 · 24 n ŵ = w ∆ · ∆ if d f,x (w [1:L] ,ŵ [1:L] ) ≤ D then if step ≤ Stop condition threshold then compress to H(ŵ [1:L] ) with entropy encoder end if k max = k step = √ step k = k − step · step end if k = k + step end for end while be the index of the layer with the largest n in f . Proposition 4.4. The optimal k * satisfies the following bounds: √ n * /24 (1− 0) ≤ k * ≤ √ n * /24 ( 0 · √ 0) . The proof is deferred to Appendix A.4. To further improve the search time, a nested refinement is utilized. Specifically, at each stage, only |O(k)| values of k in ascending order are considered. Once a certain value of k satisfies the deviation requirement D, this k becomes the new upper limit for the search, and the search region is refined within a smaller region of k to consider, again with only |O(k)| potential values to inspect. This repeatedly continues until the search step is sufficiently small. These refinements enable fast convergence in relatively few iterations. See Algorithm 1 for a detailed description of RIQ. Remark 4.5. The additional degree of freedom that 0 gives is substantial. For example, it facilitates enforcing quantization to R bits (e.g., R = 8 bits) for low precision runtime, by setting the limit k → ∞ at 0 ( ) = max(w )−min(w ) 2 R −1 / 24· w 2 n . For simplicity, in the sequel we apply the same small common constant value 0 to all layers. RIQ Rate-Distortion Analysis In this section, we provide theoretical justification for the optimality of RIQ. We introduce a surrogate model for which the rate-distortion analysis with cosine distance is tractable, showing that the minimizing distribution of the mutual in-formation is indifferent to the orientation of w , and is characterized by a single parameter k, as RIQ suggests. First, extending Equation (4) to NN model quantization, where layer is encoded uniformly, yieldŝ w = w /∆ · ∆(8) For tractability, it is common to analyze the rate-distortion for Equation (8) by a surrogate model in which the distortion is modeled as a random additive noise (Kipnis & Reeves, 2021;Marco & Neuhoff, 2005). Yet, when considering angular deviation such representation hinders the rate-distortion analysis since the relation between additive noise to cosine distance that we wish to examine is intricate. Accordingly, we suggest a reparameterization to the additive noise model, which represents quantization as random rotation (and scale) of the weights in each layer. This enables to analyze the rate-distortion of Equation (8) for the deviation in Equation (1) in the sequel. Surrogate Model. Let w be the weights of layer , and letŵ denote their quantized representation. Let θ be a random rotation angle from w toŵ , such that w w ,ŵ ŵ = cos(θ ), and let U(θ w ) ∈ SO (n ) be a random orthogonal transformation corresponding to a random rotation that is θ away from w . Then, w = ŵ · U(θ w ) w w(9) models the quantized weightsŵ . Intuitively, U(θ w ) randomly rotates any given vector uniformly on a sphere, where one degree of freedom is lost due to the requirement of being θ away from w . To obtain w in Equation (8), the realization of U(θ w ) should rotate the unit vector w / w in the plane generated by w andŵ , and then, stretches it into the length ŵ . In other words, this model describes a random vectorw that is uniformly distributed on a cone that is θ away from w , for which a single realization matches Equation (8). The merit of this model is its tractable analysis, from which spherically symmetric distribution emerges to depict the quantized weights (Fang et al., 2018, Definition 2.1). Accordingly, each layer obtains a randomly rotated version of w , which translates to a joint rotation at angle θ [1:L] of all the parameters by Corollary 4.2. Consequently, in the high rate regime, where the support of θ is sufficiently small, the distortion and the deviation decrease at same scale with k by Proposition 4.3. Proposition 4.6. Let w be the weights of layer , and let w model the quantized representation of those weights, modeled by Equation (9). Then,w w have a spherical (rotation-invariant) distribution. A detailed proof is given in Appendix A.5. Essentially, the strength of Proposition 4.6 is twofold. First, it proves that the distribution ofw w in each layer does not change when arbitrary rotations are applied to it. Second, it holds for any distribution of w and θ . The following theorem extends the results of Proposition 4.6 to multiple layers, showing that spherical distribution is also the minimizing distribution of the rate-distortion problem of NN models. is a product distribution constructed as the product of the layers' spherical distribution. Consequently, the infimum rate is characterized by a single parameter. The detailed proof is given in Appendix A.6. The key steps of the proof are, first, showing that the minimizing distribution is a product distribution. Then, due to the convexity of the rate-distortion in Equation (10), we bound the mutual information with a convex combination of θ distributions, given in Corollary 4.2. Consequently, the problem is simplified to a single (average) layer optimization, which is governed by a single quantization parameter. Remarkably, the joint minimizing distribution of the model's weights p w w w [1:L] w w w [1:L] is also spherical since any partitioning of spherical distribution (naturally occurring by the model's layers) remains spherical (Fang et al., 2018, Theorem 2.6). Further, due to the convexity of the mutual information, it is beneficial to consider as many partitions as possible, which can only reduce the mutual information. Practically, however, running the model layer-by-layer, where each layer is quantized by a single scalar, has merit due to its simplicity. In this sense, RIQ is optimal as it considers only rotation invariant solutions, from which the optimal solution emerges. Empirical results In this section, we evaluate the compression ratio and model accuracy of RIQ with ANS and compare them to relevant baseline results. In all experiments, we use pre-trained models for their relevant tasks. Specifically, for classification tasks we use VGG, Simonyan & Zisserman (2014), ResNet-50, He et al. (2016), and ViT, Dosovitskiy et al. (2020) models, from the torchvision library 1 on the Ima-geNet data-set (I1k). For detection tasks, we use YOLOv5, Jocher et al. (2022), from Neural-Magic library 2 on the COCO dataset (Lin et al., 2014). For NLP task, we use the DistilBERT model, Sanh et al. (2019), on SQuAD dataset, (Rajpurkar et al., 2016). Following RIQ, each quantized layer is encoded with the ANS encoder that achieves asymptotically the entropy limit. An efficient implementation of ANS on GPU 3 was demonstrated by Weißenberger & Schmidt (2019), reaching a decoding rate of over 20 GB/s. For reproduction purposes, we provide a Python code of our algorithm 4 which includes both the quantization phase (RIQ) and compression phase (ANS). Additional results are given in Appendix A.8. To measure the resulting deviation by Equation (1) on a validation set as a function of a deviation requirement D, we use two types of calibration data: (a) three real images, sampled from the training, and (b) randomly generated data that follows the Gaussian distribution. In Figure 2(a) we present the deviation measurements on three models: ResNet-50, VGG, and ViT, where the identity line (black-dotted) is given for reference. As we see, even a small calibration set of three images (solid lines) is sufficient to predict the deviation on the validation set. Further, we see that the randomly generated data may not predict well the resulting deviation on the validation set, leading to either a less compressed model (ResNet-50 and VGG) or a higher deviation (ViT). In Figure 2(b), we evaluate the effect of RIQ on the accuracy and the inverse compression ratio (i.e., the reciprocal of Equation (2) 2019), which requires further training for fine-tuning. Indeed, RIQ provides superior results compared to post-training quantization methods, yet, falls short compared to HAQ. This is since retraining the quantized model creates a different model with a different rate-distortion curve, which is out of the scope of this paper. In Table 1, we compare RIQ to other relevant baseline methods on the VGG-16 and ResNet-50 models. In this table, we optimized the dual problem of the rate-distortion, i.e., minimizing the deviation (accuracy drop) for a given rate requirement (compression). Thus, the RIQ in Algorithm 1 was slightly modified to obtain a stop condition based on the compression ratio rather than a deviation constraint, and then we measured the resulting accuracy of the quantized models for the VGG and ResNet-50. Noticeably, RIQ outperforms the baseline techniques, yielding better accuracy per compression ratio. Typical compression ratio and score degradation achieved by RIQ are presented in Table 2 for a variety of models and tasks. Note that the constraint on the quantization output, in the form of a cosine distance in Equation (1) does not translate identically to the drop in each score. While the first is a general distortion metric, the latter is a task-specific measurement. Yet, in general, the scores improve monotonically as the cosine distance decreases. To further assess the potential of RIQ, we evaluate our method on sparse models taken from the Neural-Magic 2 . Notably, the resulting compression of sparse models is significantly higher with relatively small degradation in accuracy. (Nagel et al., 2020) 75.23 76.07 0.84 S-AdaQuant (Hubara et al., 2021) 75.10 77.20 2.10 BRECQ (Li et al., 2021) 76.29 77.00 0.71 GPFQ (Zhang et al., 2022) 74 Conclusion In this paper, we have investigated a post-training quantization method that strives to minimize the rate of the model's parameters subject to a deviation constraint. A rotation-invariant quantization scheme (RIQ) was introduced, which quantizes each layer in proportion to the layer's norm, searching for the optimal solution over the family of spherical distributions. To find the solution efficiently, we derived the scale in which the rate increases with the deviation and then suggest a searching paradigm that bounds the search space based on our findings. The rate-distortion curve was thoroughly analyzed, showing that the minimizing distribution is a product distribution, constructed as the product of the layer's spherical distribution, which coincides with the RIQ approach. Experiments conducted with RIQ showed minor accuracy degradation for both computer vision and NLP tasks. When applying RIQ on sparse models, the compression ratio reached remarkable values (up to ×52.9) with a score degradation of less than 1%. These superb compression rates together with the negligible accuracy drop, are attractive and practical. In particular, as RIQ minimizes the rate of each layer, it is suitable for per-layer execution of compressed models, thus, facilitating the deployment of large models on lean servers or edge devices with limited memory. A. Appendix In this section, we provide rigorous proofs for the theorems and the statements herein. Further, we present additional results for RIQ. A.1. Proof of Lemma 4.1 Proof. Let w be the realization of the weights vector of layer , andŵ be the quantized representation of those weights, where θ denotes the angle between those vectors. Before diving into the cosine distance analysis, let us revisit the mean squared error analysis of the uniform quantizer in Section 3.3, and extend it to the multivariate case. In this case, the distortion is w −ŵ 2 = n · 1 n n i=1 |w ,i −ŵ ,i | 2 (a) = n · E |w ,j −ŵ ,j | 2 + o(1) (b) = n · ∆ 2 /12 + o(1)(11) where (a) follows from the law of large numbers, and (b) follows by the analysis of the scalar uniform quantizer, given in (Polyanskiy & Wu, 2014, Ch. 25.1). This observation is useful for analyzing the cosine distance between w andŵ . Specifically, since w −ŵ 2 = w 2 + ŵ 2 − 2 w · ŵ cos(θ ), assuming ŵ = w + o( w ), yields w −ŵ 2 = 2 w 2 + o( w 2 ) − 2 w 2 cos(θ ) + o( w 2 ) = 2 w 2 · (1 − cos(θ )) + o( w 2 ). Hence, normalizing both sides by 2 w 2 , we obtain that (1 − cos(θ )) = w −ŵ 2 2 w 2 + o(1). Combining the analysis of Equation (11) with Equation (12), we obtain (1 − cos(θ )) = w −ŵ 2 2 w 2 + o(1) = ∆ 2 · n 24 · w 2 + o(1). By denoting = 1 − cos(θ ), and omitting the little order o(1), the lemma follows. = L =1 w ,ŵ w [1:L] · ŵ [1:L] = L =1 w · ŵ · w w ,ŵ ŵ w [1:L] · ŵ [1:L] = L =1 w 2 ŵ [1:L] 2 cos(θ ) + o( w 2 ŵ [1:L] 2 ) Since L =1 w 2 w [1:L] 2 = 1, the parameters' distortion is simply a convex combination of the layers' distortion. A.3. Proof of Proposition 4.3 Proof. By Equation (12), 1 − cos(θ f ) = f (x) − f (x) 2 2 · f (x) 2 + o(1). where θ f is the deviation angle obtained by Equation (1). To make the dependence of f (x) on its weights explicit, let us denote f (x) = f x (w [1:L] ) andf (x) = f x (ŵ [1:L] ). Since the denominator is independent of the quantization, it is sufficient to focus on the enumerator. Accordingly, we wish to examine f x (ŵ [1:L] ) − f x (w [1:L] ) 2 = f x (w [1:L] + ) − f x (w [1:L] ) 2 g( ) where is (random) quantization errors. Our goal is to prove that E [g( )] is monotonically decreasing in k (i.e., higher rate must reduce the quantized model deviation). Using Taylor expansion at = 0 0 0, we have g( ) = g(0 0 0) + ∇g(0 0 0) · + 1 2 T H(g(0 0 0)) + o( 3 ) = 1 2 T H(g(0 0 0)) where the last step follows since g(0 0 0) = 0 0 0, and noting that ∇g(0 0 0) = 0. Finally, omitting the little order o( 3 ), which is negligible in the high rate regime. Recall that in the high rate regime the error in entry i, i ∼ U [−∆/2, ∆/2] is i.i.d. uniformly distributed. Accordingly, in each layer the corresponding sub-vector satisfies E T = ∆ 2 /12 · I n , where I n is the n × n identity matrix. Since the errors are independent with zero mean and variance ∆ 2 /12, we can utilize the Hutchinson (1989) trick. Accordingly, let N = L =1 n , then 1 2 E T H(g(0 0 0)) = 1 2 E   N i=1,j=1 i H(g(0 0 0)) ij j   = N i=1,j=1 H(g(0 0 0)) ij E [ i j ] = N i=1 H(g(0 0 0)) ii E 2 i where the last step follows since E [ i · j ] = 0∀i = j. Finally, letting ∆ ∆ 2 1 12 , . . . , ∆ 2 L 12 T = 1 k 2 w 1 2 12 , . . . , w L 2 12 T , we can present the last step vectorially as ∆ T diag (H(g(0 0 0))) = 1 12 · k 2 · w 1 2 , . . . , w L 2 · diag (H(g(0 0 0))) . Since g(0 0 0) is a quadratic function with minimum at = ( ( (0), thus H(g(0 0 0)) is positive definite, which means that its diagonal entries are real and non-negative. Further, the entries of ∆ are positive, hence, this dot product is monotonically decreasing in k, as O(1/k 2 ), which completes the proof. Note that this proposition holds for any x. A.4. Proof of Proposition 4.4 Proof. The layer whose quantization error converges last to 0 dictates when to stop the search. Specifically, when k is sufficiently large in Equation (7), the error in layer * reaches √ * = o( 0 ) + 0 , where o(·) denotes little order of magnitude. That is where 0 becomes dominant. At this point, we say that the error has converged for all layers (as it converged even at the largest layer * ). Since 0 ≤ 1 in the cosine distance criterion, we choose the little order of magnitude to be o( 0 ) = 0 · √ 0 , and hence, k can be bounded from above by 1 k n * /24 + 0 ≥ o( 0 ) + 0 , which happens when k ≤ n * /24/( 0 · √ 0 ). In our experiments, we let 0 = 0.01, hence, the upper limit is simply k ≤ 1000 · n * /24 5 For a lower bound, we use again the fact that ≤ 1. Thus, focusing on layer * , we observe that 1 k · n * /24 + 0 ≤ 1, which happens as long as k ≥ n * /24/(1 − 0 ). This completes the proof. A.5. Proof of Proposition 4.6 Proof. By Fang et al. (2018, Theorem 4.3), a necessary and sufficient condition forw w to have a spherical rotationinvariant distribution on a cone that is θ away from w is when p ( w , v 1 | w , v 2 , w ) d = p ( − w , v 1 | w , v 2 , w ) , for any pair of perpendicular vectors v 1 = 0 and v 2 = 0 that are orthogonal to w . Consider the model in Equation (9), any orthogonal transformation U(θ w ) can be represented by an orthonormal basis, obtained by the Gram-Schmidt process. That is, finding two orthonormal vectors u 1 and u 2 that span the plane of rotation generated by w and some w that is θ away from w , and then, extend this basis to R n . This allows us to consider the rotation in the plane generated by those vectors, with respect to the extended basis (, https://stephenmontgomerysmith.github.io/). Accordingly, let u 1 = w w and u 2 = w − u1,w u1 w − u1,w u1 , then U(θ w ) = I n − u 1 u T 1 − u 2 u T 2 + [u 1 , u 2 ] R θ [u 1 , u 2 ] T ,(13) where I n is the n × n identity matrix and R θ is the rotation matrix R θ = cos(θ ) − sin(θ ) sin(θ ) cos(θ ) , that rotates at a scalar angle θ , and [u 1 , u 2 ] is n × 2 matrix whose columns are u 1 and u 2 , respectively. Plugging Equation (13) to Equation (9), and noting that w w = u 1 , we obtaiñ w = ŵ · (cos(θ )u 1 + sin(θ )u 2 )(14) To simplify notation, let u (cos(θ )u 1 + sin(θ )u 2 ), and note that for any perpendicular pair v 1 , v 2 that are orthogonal to w , the vector u can be decomposed to u = u + u ⊥ , where u resides in the plane generated by v 1 and v 2 , and u ⊥ resides in the null-space of this plane. For illustration, see Figure 3. Hence, Accordingly, we have w = ŵ · (u + u ⊥ )(15p ( w , v 1 | w , v 2 , w ) (16) (a) = p ŵ · (u + u ⊥ ), v 1 ŵ · (u + u ⊥ ), v 2 , w (b) = p ŵ · u , v 1 ŵ · u , v 2 , w (c) = p ŵ · u · cos(φ) ŵ · u · sin(φ), w (d) = p ŵ · u · cos(π − φ) ŵ · u · sin(π − φ), w (e) = p − ŵ · u · cos(φ) ŵ · u · sin(φ), w = p (− ŵ , v 1 | ŵ , v 2 , w )(17) where (a) follows by Equation (15). (b) follows by the linearity of the inner product and since u ⊥ is perpendicular to both v 1 and v 2 (c) follows due to the orthogonality of the basis v 1 and v 2 , where φ is the angle between u and v 1 . (d) follows since the angle between u and an arbitrary v 1 is arbitrary, and hence, every angle has the same distribution. (e) follows by trigonometric identities for the cosine and sine function. Thus, Proposition 4.6 follows. A.6. Proof of Theorem 4.7 Proof. Consider the rate-distortion function = L =1 H (w ) − L =1 H w | w [1: −1] ,w [1:L] (19) ≥ L =1 H (w ) − L =1 H ( w |w ) (20) = L =1 I (w ;w ) (21) ≥ L =1 R(D )(18) where Equation (20) follows since conditioning reduces entropy. Note, however, that Equation (20) can be attained with equality by letting p w [1:L] w [1:L] = L =1 p w w . Consequently, the minimizing distribution in Equation (10) is a product distribution (Polyanskiy & Wu, 2014, Theorem 5.1 (2)). Interestingly, Equation (22) implies that the optimal solution has a simple form of a layer-by-layer solution, which significantly simplifies the problem at hand. Finding the solution, of course, requires formulating the relation between the deviations D and D of each layer , and hence, the resulting rate R(D ) for each layer. Moreover, since each layer obtains a different rate, it implies that the optimal solution is indeed a mixed-precision solution, where each layer can be considered independently, and hence, the minimizing distribution is product distribution. First, let us consider the deviation. Let p w 1 w 1 · p w 2 w 2 · · · p w L w L be a distribution that satisfies the deviation requirement D, for which the induced cosine distance (distortion) in each layer is at most , for = 1 − cos(θ ). By Corollary 4.2, the distortion over the entire parameters is hence (θ )). In words, convex combination of the angles also satisfies the deviation constraint D. Generally, cosine distance is a rotation-invariant distance, as the angle between vectors does not change when they are rotated together. This further hints that the minimizing distribution should also be a rotation-invariant distribution as follows. Next, let us address p(w |w ). By Equation (14), given w (and hence, u 1 ), the probability ofw is determined by the probability of the rotation angle θ and the length ŵ . Specifically, for any vector s ∈ R n , the density function of this product, if exists, is (Melvin Dale, 1979, Ch. 4 .1) pw |w (s ) = ∞ 0 p ŵ |w (h) · p (cos(θ )u1+sin(θ )u2)|w (s /h) · 1 h dh,(23) where the rotation θ occurs on R 2 , rotating about (n − 2)-dimensional subspace. Further, note that the dimension n is dictated only by the given w . Apparently, since each layer resides at a different dimension n , it is impossible to consider the convex combination of the layers' distribution directly, as done for the vector case, e.g., as considered in Polyanskiy & Wu (2014, Ch. 5). Nevertheless, since the rotation of θ is done on R 2 in each layer , which is described by the rotation matrix R θ in Equation (13), it is still beneficial to consider a convex combination of θ distributions over the layers, to allow a similar treatment to Polyanskiy & Wu (2014, Ch. 5), as follows. To bound the mutual information, the density of (cos(θ )u 1 + sin(θ )u 2 ) |w should be expressed first in terms of the density of cos(θ )|w . Examining Equation (14), we note that by the transformation of random variables formula, p (cos(θ )u1+sin(θ )u2)|w (s /h) = p cos(θ )|w u T 1 s /h .(24) Considering the high rate regime, where ∆ is sufficiently small, and thus, ŵ = w + o(1), then, the density function in Equation (23) becomes pw w (s ) = p ŵ ·(cos(θ )u1+sin(θ )u2)|w (s ) (25) = ∞ 0 p ŵ |w (h) · p (cos(θ )u1+sin(θ )u2)|w (s /h) · 1 h dh (26) (a) = ∞ 0 p ŵ | w (h) · p (cos(θ )u1+sin(θ )u2)|w (s /h) · 1 h dh (27) (b) ≈ ∞ 0 δ (h − w ) · p (cos(θ )u1+sin(θ )u2)|w (s /h) · 1 h dh (28) (c) = ∞ 0 δ (h − w ) · p cos(θ )|w u T 1 s /h · 1 h dh (29) (d) = p cos(θ )|w u T 1 s / w · w −1(30) where (a) follows since the norm w is a function of the given w . (b) follows since the uncertainty about ŵ given w is negligible, and hence, p ŵ | w (h) ≈ δ(h − w ), i.e., the conditional density is approximately the Dirac delta function. (c) follows by Equation (24). (d) follows by the characteristics of the Dirac delta function. Hence, it is possible to consider a convex combination of p w w over the layers, where each layer is embedded in possibly different n , by considering a convex combination of the rotations' probability p cos(θ )|w , since all rotations are done in R 2 . Accordingly, let where I p (·; ·) denotes explicitly the mutual information under probability p. Thus, the infimum rate has a form of a scalar (single-letter) rate. pw [1:L] w [1:L] (s [1:L] ) L =1 w 2 w [1:L] 2 · p cos(θ )|w w T s w 2 · w −1 .(31) Moreover, since averaging over more rotations should further reduce the mutual information by its convexity, then, the minimizing p w [1:L] w [1:L] can be chosen to be rotation-invariant (Polyanskiy & Wu, 2014, Ch. 5.2). Consequently, the unique minimizing distribution p w [1:L] is also rotation-invariant. Remarkably, Fang et al. (2018, Theorem 2.6) states that when partitioning a spherical rotation-invariant distribution (naturally, according to the layersw ), then its components also have a spherical rotation-invariant distribution. This coincides with Proposition 4.6, which proves that the partitioning satisfies this property. Accordingly, the unique minimizing distribution p(w [1:L] w [1:L] ) of the rate-distortion function is a product distribution over the layers, where each term is a spherical rotation-invariant distribution. This completes the proof. A.7. Relation to Other Error Criteria Remark A.1. The proof of Lemma 4.1 in Appendix A.1 may serve as a proxy to other error criteria such as the Signal to Quantization Noise Ratio (SQNR), (Caffarena et al., 2010). Specifically, similar to the proof of Lemma 4.1, the resulting connection between the scale ∆ and the SQNR in each layer is w w w −ŵ w w w w w = ∆ 2 12 · n w w w 2 Or, equivalently, ∆ = w w w 12/n A.8. Additional Results A.8.1. DECOMPOSING THE RATE-DISTORTION CURVE The key steps of lossy compression are quantization and compression. In the quantization phase, the RIQ approach is minimizing the overall model's entropy by allocating a small number of unique symbols for large-norm layers. To achieve (asymptotically) this entropy limit, we utilize the ANS (lossless) entropy encoder. In this section, we evaluate the contribution of each step to the rate-distortion tradeoff. Namely, the average rate per (quantized) symbol before and after ANS. At run-time, when a certain layer is required, it is decoded and represented at a rate according to RIQ. If this rate is below 8 bits/symbol, it enables significant acceleration by performing 8 bits integer operations, as discussed in Appendix A.8.2. Figure 4(a) depicts the rate-distortion curve for ResNet-50, decomposed to the quantization step (dashed lines) and the resulting compression step, following the quantization step (solid lines). As a baseline, the uniform scalar quantization (red color) is given for comparison with RIQ (green color). Interestingly, RIQ (dashed green line) outperforms the uniform quantization (dashed red line) by about ∼ 4 bits/symbol and even its resulting compressed size by about ∼ 1 bit/symbol. Indeed, the latter indicates that uniform quantization does not minimize the model's entropy. Applying the ANS compression following RIQ reduces additional ∼ 3 bits/symbol (solid green line), which according to our analysis is the minimum entropy possible for a given distortion. Moreover, our method achieves a reduction of about ∼ 8 bits/symbol compared to uniform scalar quantization alone, and an additional ∼ 4 bits/symbol when ANS is applied to the uniformly quantized weights. The rate-distortion curves for various models, in particular, the VGG (green circles), ResNet-50 (red squares), ViT (blue diamonds), and DistilBERT (orange triangles) are given in Figure 4(b). As expected, the curves decrease monotonously, reaching an impressive compression rate of less than 8 bits/symbol on average even for extremely low cosine distance in all presented models. A.8.2. ROTATION INVARIANT QUANTIZATION WITH QUANTIZED ACTIVATIONS Quantizing both the NN model's weights and its activations can further accelerate the inference, (Wu et al., 2020;Nagel et al., 2020;Krishnamoorthi, 2018). Nevertheless, in this case, the quantization error of both the weights and the activation affects the model's output. In the seminal work of (Wu et al., 2020), the authors utilized the KL distance for quantizing the activations to minimize the information loss at the output. In this section, we examine the RIQ approach, combining it with activation quantization. To demonstrate, we use the (NVIDIA, 2021) quantization library for the ResNet-50 model with a "mini ImageNet" validation set, which comprises one image per class and a total of 1000 images. We evaluate this library's performance as a baseline, where the activations are quantized by the KL-distance criterion, and the weights are quantized to 8-bit linearly. The resulting cosine distance at the output of this baseline is 0.69%. For comparison, this reference value is given as the deviation requirement to RIQ. In particular, to integrate RIQ, the activations are quantized as the baseline, and then, we run RIQ according to Algorithm 1. This way, RIQ is aware of the activations' quantization error during its search for the single-letter solution. Note that to facilitate the acceleration of int8 operations, RIQ must yield a quantization rate of up to 8 bits/symbol. In case a certain layer requires a higher rate, we simply perform linear uniform quantization to 8 bits (without clipping), as the baseline does. Figure 5(a) characterizes the compression ratio as a function of cosine distance. The leftmost point reflects a cosine distance of 0.0069 achieved by the baseline of (NVIDIA, 2021). Remarkably, the RIQ attains superior compression with relatively low deviation even when the activations are quantized. In run-time, of course, the reconstructed values are represented again by 8-bit value, and hence, the significant acceleration of (Wu et al., 2020) is still valid. A.8.3. COMPARISON WITH MULTI-OBJECTIVE BAYESIAN OPTIMIZATION In this section, we utilize the Multi-Objective Bayesian Optimization (MOBO) tool, described in (Daulton et al., 2020) to compress NN models, and compare results with RIQ. To compress models with MOBO, we set two objective functions for it. The first objective is minimizing the cosine distance in Equation (1). The second objective is maximizing the compression ratio in Equation (2). Then, we let MOBO optimize the rate-distortion tradeoff (i.e., the Pareto frontier surface). Nonetheless, MOBO is quite complex and requires strong computing capabilities for exploration and exploitation. Particularly, reaching the optimal solution may take days and even weeks, using multiple GPUs. Even on small NN models, to address the high-dimensional search spaces, we apply sparse axis-aligned subspace priors for Bayesian optimization (qNEHVI + SAASBO), with the batch Noisy Expected Improvement (qNEI) acquisition function, as suggested by (Eriksson . 10 ) 10R(D) = min p(ŵ w w|w w w):E[d(w w w,ŵ w w))]≤D I(w;ŵ) Corollary 4 . 2 . 42Let w [1:L] be a vector representation of the weights andŵ [1:L] denotes its quantized representation, and let θ [1:L] be the rotation angle from w [1:L] toŵ [1:L] Figure 1 . 1Validation of Proposition 4.3 on ResNet-50. Theorem 4. 7 . 7Let f (·) be a NN model with L layers whose weights are w [1:L] , and letw [1:L] be the quantized representation of those weights. Then, the unique minimizing distribution p(w w w [1:L] w w w [1:L] ) of the rate-distortion function R(D) = min p w w w [1:L] w w w [1:L] : E[d f,x (w [1:L] ,w [1:L] )]≤D I w [1:L] ;w [1:L] (10) Figure 2 . 2(a) Cosine distance of the validation dataset as a function of the deviation constraint (on the calibration dataset). Models include VGG (green circles), ResNet-50 (red squares), and ViT (blue diamonds). (b) Accuracy vs Compression for ResNet-50 model. RIQ (green) vs. RAP (orange), OCS (red), linear quantization (cyan), and HAQ (purple). HAQ, however, requires training after the quantization. ) for a pre-trained ResNet-50 model. Interestingly, the rate-distortion curve reflects well the accuracy-compression tradeoff. For comparison, we depict the accuracy-compression results of the Relaxed Advanced Pipeline (RAP) method,Hubara et al. (2021), Outlier Channel Splitting (OCS),Zhao et al. (2019), and the Hardwareaware Automated Quantization (HAQ),Wang et al. ( A. 2 . 2Proof of Corollary 4.2 Proof. Let θ [1:L] be the rotation angle from w [1:L] toŵ [1:L] such that w [1:L] w [1:L] ,ŵ [1:L] ŵ [1:L] cos(θ [1:L] ). Assuming ŵ = w + o( w ), then, Figure 3 . 3Illustration of the projection ofw onto the arbitrary perpendicular vectors v1 and v2. E [d f,x (w [1:L] ,w [1:L] )]≤D I w [1:L] ;w [1:L] Assuming the weights of each layer w are statistically independent of the weights of the other layers, then, by the properties of the mutual information, we have I w [1:L] ;w [1:L] = H w [1:L] − H w [1:L] w [1:L] = 1 . 1Assuming [1:L] ≤ D, then due to the convexity of the cosine distance for |θ | ≤ π/2, by Jensen inequality 1 − cos Figure 5 . 5(a) The compression ratio as a function of cosine distance. The left-bottom red triangle depicts the resulting distance of 0.0069 achieved by the baseline with a compression ratio of ×4. The orange upside-down triangles depict the cosine distance and compression ratio attained by RIQ with the ANS compression. The orange line depicts the trend line. (b) MOBO optimization process. Interestingly, MOBO converges at the few last iterations to ×12 compression, with a highest value of ×12.61. On the other hand, RIQ reaches practically the same compression ratio in a few seconds. Algorithm 1 The RIQ algorithmInput: model weights w [1:L] , deviation requirement D, minimum error 0 Table 1 . 1Comparison of Top-1 accuracy on the ImageNet dataset for various quantization techniques.Model Comp. Method Quant. Acc. (%) Ref. Acc. (%) Acc. Drop (%) VGG-16 32/3 GPFQ (Zhang et al., 2022) 70.24 71.59 1.35 RIQ (Ours) 71.58 71.59 0.01 32/4 MSE (Banner et al., 2019) 70.50 71.60 1.10 OMSE (Choukroun et al., 2019) 71.48 73.48 2.00 GPFQ (Zhang et al., 2022) 70.90 71.59 0.69 RIQ (Ours) 71.55 71.59 0.04 32/5 GPFQ (Zhang et al., 2022) 71.05 71.59 0.54 RIQ (Ours) 71.58 71.59 0.01 ResNet-50 32/3 GPFQ (Zhang et al., 2022) 70.63 76.13 5.50 RIQ (Ours) 74.76 76.14 1.38 32/4 MSE (Banner et al., 2019) 73.80 76.10 2.30 OMSE (Choukroun et al., 2019) 73.39 76.01 2.62 AdaRound Then, by the convexity of the rate-distortion functionCover & Thomas (2006, Theorem 2.7.4),p w [1:L] w [1:L] can only reduce the mutual information in Equation(21). Specifically,L =1 I p w w (w ;w ) ≥ L · Ip w [1:L] w [1:L] w [1:L] ;w [1:L] Figure 4. (a) Rate-distortion curve for ResNet-50 model obtained for RIQ (green circles) as well as Uniform linear quantization (red squares). Rates are presented for both the quantized model (dashed) as well as following an ANS compression. (b) Rate distortion curves obtained by RIQ + ANS, for a variety of models: VGG (green circles), ResNet-50 (red squares), ViT (blue diamonds), and DistilBERT (orange triangles).0 0.02 0.04 0.06 0.08 0.1 Deviation (cosine distance) 2 4 6 8 10 12 14 Rate (bits/symbol) RIQ + ANS RIQ only Uniform + ANS Uniform ResNet-50 (a) 0 0.02 0.04 0.06 0.08 0.1 Deviation (cosine distance) 0 1 2 3 4 5 6 7 Rate (bits/symbol) VGG ResNet-50 ViT DistilBERT (b) https://pytorch.org/vision/stable/ models.html 2 https://sparsezoo.neuralmagic.com https://github.com/weissenberger/ multians 4 https://github.com/ehaleva/RIQ When the original weights are represented with R bit symbols, then, choosing 0 = 0, yields a trivial upper bound kmax, which is the largest number that can be represented with R bits, e.g., kmax ≤ 2 31 , when using 32 bits integer. & Jankowiak, 2021;Daulton et al., 2021b;a). Moreover, since the two objectives are not within the same range the cosine similarity objective had to be scaled accordingly to converge to the optimal solution, where a calibration set of 4 images are used during 30 iterations of exploration/exploitation. Accordingly, we pick a (relatively) small model for comparison (with a size of 112 KB), letting MOBO to find for each layer its optimal bin width and quantize accordingly. We emphasize that the MOBO solution does not rely on the rotation invariant insights. InFigure 5(b), the optimization process of MOBO is presented, where each dot depicts experiment results, and its color indicates the iteration in which this result was attained. The compression results of RIQ are presented for comparison. Remarkably, RIQ and MOBO attained almost identical results of ×12.6 and ×12.61, respectively, with a cosine distance of 0.005. This indicates that RIQ reaches the optimal solution. 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Y Idelbayev, M A Carreira-Perpinán, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionIdelbayev, Y. and Carreira-Perpinán, M. A. Low-rank com- pression of neural nets: Learning the rank of each layer. In Proceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition, pp. 8049-8059, 2020. Optimal quantization using scaled codebook. Y Idelbayev, P Molchanov, M Shen, H Yin, M A Carreira-Perpinán, J M Alvarez, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionIdelbayev, Y., Molchanov, P., Shen, M., Yin, H., Carreira- Perpinán, M. A., and Alvarez, J. M. Optimal quantization using scaled codebook. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12095-12104, 2021. An information-theoretic justification for model pruning. B Isik, T Weissman, A No, International Conference on Artificial Intelligence and Statistics. PMLRIsik, B., Weissman, T., and No, A. An information-theoretic justification for model pruning. In International Confer- ence on Artificial Intelligence and Statistics, pp. 3821- 3846. PMLR, 2022. G Jocher, A Chaurasia, A Stoken, J Borovec, Nanocode012, Y Kwon, Taoxie, J Fang, K Michael, V Lorna, A Montes, D Nadar, J Laughing, P Skalski, Z Wang, A Hogan, C Fati, L Mammana, Alexwang1900, D Patel, D Yiwei, F You, J Hajek, L Diaconu, Minh , M T , 10.5281/zenodo.6222936ultralytics/yolov5: v6.1 -TensorRT, Ten-sorFlow Edge TPU and OpenVINO Export and Inference. Jocher, G., Chaurasia, A., Stoken, A., Borovec, J., NanoCode012, Kwon, Y., TaoXie, Fang, J., imyhxy, Michael, K., Lorna, V, A., Montes, D., Nadar, J., Laughing, tkianai, yxNONG, Skalski, P., Wang, Z., Hogan, A., Fati, C., Mammana, L., AlexWang1900, Pa- tel, D., Yiwei, D., You, F., Hajek, J., Diaconu, L., and Minh, M. T. ultralytics/yolov5: v6.1 -TensorRT, Ten- sorFlow Edge TPU and OpenVINO Export and Infer- ence, February 2022. URL https://doi.org/10. 5281/zenodo.6222936. Gaussian approximation of quantization error for estimation from compressed data. A Kipnis, G Reeves, 10.1109/TIT.2021.3083271IEEE Transactions on Information Theory. 678Kipnis, A. and Reeves, G. Gaussian approximation of quan- tization error for estimation from compressed data. IEEE Transactions on Information Theory, 67(8):5562-5579, 2021. doi: 10.1109/TIT.2021.3083271. Quantization with minimal entropy. V Koshelev, Probl. Pered. Inform. 14Koshelev, V. Quantization with minimal entropy. Probl. Pered. Inform, 14:151-156, 1963. Quantizing deep convolutional networks for efficient inference: A whitepaper. R Krishnamoorthi, abs/1806.08342ArXiv. Krishnamoorthi, R. Quantizing deep convolutional net- works for efficient inference: A whitepaper. ArXiv, abs/1806.08342, 2018. Optimal brain damage. Y Lecun, J Denker, S Solla, Advances in neural information processing systems. 2LeCun, Y., Denker, J., and Solla, S. Optimal brain damage. Advances in neural information processing systems, 2, 1989. D Lee, S J Kwon, B Kim, G.-Y Wei, arXiv:1905.10145Learning low-rank approximation for CNNs. arXiv preprintLee, D., Kwon, S. J., Kim, B., and Wei, G.-Y. Learn- ing low-rank approximation for CNNs. arXiv preprint arXiv:1905.10145, 2019. Y Li, R Gong, X Tan, Y Yang, P Hu, Q Zhang, F Yu, W Wang, S Gu, Brecq, arXiv:2102.05426Pushing the limit of post-training quantization by block reconstruction. arXiv preprintLi, Y., Gong, R., Tan, X., Yang, Y., Hu, P., Zhang, Q., Yu, F., Wang, W., and Gu, S. Brecq: Pushing the limit of post-training quantization by block reconstruction. arXiv preprint arXiv:2102.05426, 2021. Holistic cnn compression via low-rank decomposition with knowledge transfer. S Lin, R Ji, C Chen, D Tao, J Luo, IEEE transactions on pattern analysis and machine intelligence. 41Lin, S., Ji, R., Chen, C., Tao, D., and Luo, J. Holistic cnn compression via low-rank decomposition with knowl- edge transfer. IEEE transactions on pattern analysis and machine intelligence, 41(12):2889-2905, 2018. Microsoft coco: Common objects in context. T.-Y Lin, M Maire, S Belongie, J Hays, P Perona, D Ramanan, P Dollár, C L Zitnick, 978-3-319-10602-1Computer Vision -ECCV. Fleet, D., Pajdla, T., Schiele, B., and Tuytelaars, T.ChamSpringer International PublishingLin, T.-Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ramanan, D., Dollár, P., and Zitnick, C. L. Microsoft coco: Common objects in context. In Fleet, D., Pajdla, T., Schiele, B., and Tuytelaars, T. (eds.), Computer Vi- sion -ECCV 2014, pp. 740-755, Cham, 2014. Springer International Publishing. ISBN 978-3-319-10602-1. The validity of the additive noise model for uniform scalar quantizers. D Marco, D Neuhoff, 10.1109/TIT.2005.846397IEEE Transactions on Information Theory. 515Marco, D. and Neuhoff, D. The validity of the additive noise model for uniform scalar quantizers. IEEE Transactions on Information Theory, 51(5):1739-1755, 2005. doi: 10.1109/TIT.2005.846397. The algebra of random variables. Melvin Dale, S , WileyMelvin Dale, S. The algebra of random variables. Wiley, 1979. Up or down? adaptive rounding for post-training quantization. M Nagel, R A Amjad, M Van Baalen, C Louizos, T Blankevoort, PMLR, 2020. NVIDIA. Pytorch quantization -functionalities. docs.nvidia.comInternational Conference on Machine Learning. Nagel, M., Amjad, R. A., Van Baalen, M., Louizos, C., and Blankevoort, T. Up or down? adaptive rounding for post-training quantization. In International Conference on Machine Learning, pp. 7197-7206. PMLR, 2020. NVIDIA. Pytorch quantization -functionalities. docs.nvidia.com, 2021. URL https://docs. Scalable model compression by entropy penalized reparameterization. D Oktay, J Ballé, S Singh, A Shrivastava, arXiv:1906.06624arXiv preprintOktay, D., Ballé, J., Singh, S., and Shrivastava, A. Scalable model compression by entropy penalized reparameteriza- tion. arXiv preprint arXiv:1906.06624, 2019. Lecture notes on information theory. Lecture Notes for ECE563 (UIUC) and. Y Polyanskiy, Y Wu, 67Polyanskiy, Y. and Wu, Y. Lecture notes on information theory. Lecture Notes for ECE563 (UIUC) and, 6(2012- 2016):7, 2014. Squad: 100,000+ questions for machine comprehension of text. P Rajpurkar, J Zhang, K Lopyrev, P Liang, arXiv:1606.05250arXiv preprintRajpurkar, P., Zhang, J., Lopyrev, K., and Liang, P. Squad: 100,000+ questions for machine comprehension of text. arXiv preprint arXiv:1606.05250, 2016. V Sanh, L Debut, J Chaumond, T Wolf, Distilbert, arXiv:1910.01108a distilled version of bert: smaller, faster, cheaper and lighter. arXiv preprintSanh, V., Debut, L., Chaumond, J., and Wolf, T. Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter. arXiv preprint arXiv:1910.01108, 2019. Knowledge distillation beyond model compression. F Sarfraz, E Arani, B Zonooz, 2020 25th International Conference on Pattern Recognition (ICPR). IEEESarfraz, F., Arani, E., and Zonooz, B. Knowledge distillation beyond model compression. In 2020 25th International Conference on Pattern Recognition (ICPR), pp. 6136- 6143. IEEE, 2021. K Simonyan, A Zisserman, arXiv:1409.1556Very deep convolutional networks for large-scale image recognition. arXiv preprintSimonyan, K. and Zisserman, A. Very deep convolu- tional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. Document similarity detection using k-means and cosine distance. W Usino, A S Prabuwono, K H S Allehaibi, A Bramantoro, A Hasniaty, Amaldi , W , International Journal of Advanced Computer Science and Applications. Usino, W., Prabuwono, A. S., Allehaibi, K. H. S., Bra- mantoro, A., Hasniaty, A., and Amaldi, W. Document similarity detection using k-means and cosine distance. International Journal of Advanced Computer Science and Applications, 2019. Online ensemble model compression using knowledge distillation. D Walawalkar, Z Shen, M Savvides, European Conference on Computer Vision. SpringerWalawalkar, D., Shen, Z., and Savvides, M. Online ensem- ble model compression using knowledge distillation. In European Conference on Computer Vision, pp. 18-35. Springer, 2020. Hardware-aware automated quantization with mixed precision. K Wang, Z Liu, Y Lin, J Lin, S Han, Haq, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionWang, K., Liu, Z., Lin, Y., Lin, J., and Han, S. Haq: Hardware-aware automated quantization with mixed pre- cision. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8612- 8620, 2019. Massively parallel ans decoding on gpus. A Weißenberger, B Schmidt, Proceedings of the 48th International Conference on Parallel Processing. the 48th International Conference on Parallel ProcessingWeißenberger, A. and Schmidt, B. Massively parallel ans de- coding on gpus. In Proceedings of the 48th International Conference on Parallel Processing, pp. 1-10, 2019. Integer quantization for deep learning inference: Principles and empirical evaluation. H Wu, P Judd, X Zhang, M Isaev, P Micikevicius, arXiv:2004.09602arXiv preprintWu, H., Judd, P., Zhang, X., Isaev, M., and Micikevi- cius, P. Integer quantization for deep learning infer- ence: Principles and empirical evaluation. arXiv preprint arXiv:2004.09602, 2020. Post-training quantization for neural networks with provable guarantees. J Zhang, Y Zhou, R Saab, arXiv:2201.11113arXiv preprintZhang, J., Zhou, Y., and Saab, R. Post-training quantization for neural networks with provable guarantees. arXiv preprint arXiv:2201.11113, 2022. Exploration and estimation for model compression. Y Zhang, S Gao, H Huang, Proceedings of the IEEE/CVF International Conference on Computer Vision. the IEEE/CVF International Conference on Computer VisionZhang, Y., Gao, S., and Huang, H. Exploration and esti- mation for model compression. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 487-496, 2021. 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[ "https://github.com/ehaleva/RIQ.", "https://github.com/weissenberger/", "https://github.com/ehaleva/RIQ" ]
[ "Improvement of quantum key distribution protocols", "Improvement of quantum key distribution protocols" ]
[ "Guihua Zeng \nNational Key Laboratory on ISDN of XiDian University\n710071Xi'anChina\n", "Xinmei Wang \nNational Key Laboratory on ISDN of XiDian University\n710071Xi'anChina\n" ]
[ "National Key Laboratory on ISDN of XiDian University\n710071Xi'anChina", "National Key Laboratory on ISDN of XiDian University\n710071Xi'anChina" ]
[]
The security of the previous quantum key distribution protocols,which is guaranteed by the nature of physics law, is based on the legitimate users. However, the impersonation of Alice or Bob by eavesdropper, in practice, will be existed in a large probability. In this paper an improvement scheme for the security quantum key is proposed.Quantum cryptography [1] is a recently developed technique that permits two parties, who share no secret information initially, to communicate over an open channel and to
null
[ "https://arxiv.org/pdf/quant-ph/9810021v1.pdf" ]
117,133,806
quant-ph/9810021
5721ff0af1527d41c2595fad1072058d47b3b0fd
Improvement of quantum key distribution protocols 7 Oct 1998 Guihua Zeng National Key Laboratory on ISDN of XiDian University 710071Xi'anChina Xinmei Wang National Key Laboratory on ISDN of XiDian University 710071Xi'anChina Improvement of quantum key distribution protocols 7 Oct 1998Quantum cryptography The security of the previous quantum key distribution protocols,which is guaranteed by the nature of physics law, is based on the legitimate users. However, the impersonation of Alice or Bob by eavesdropper, in practice, will be existed in a large probability. In this paper an improvement scheme for the security quantum key is proposed.Quantum cryptography [1] is a recently developed technique that permits two parties, who share no secret information initially, to communicate over an open channel and to establish between themselves a shared secret sequence of bits. Quantum cryptography is provably secure against eavesdropping attack, in that, as a matter of fundamental principle, the secret data can not be compromised unknowingly to the legitimate users of the channel. Three ingenious protocols [2][3][4] in quantum cryptography have been proposed. The first, by Bennett et al, relies on the uncertainty principle of quantum mechanics to provide key security. The security guarantee is derived from the fact that each bit of data is encoded at random on either one of a conjugate pair of observables of quantummechanical object. Because such a pair of observables is subjected to the Heisenberg uncertainty principle [5], measuring one of the observables necessarily randomizes the 1 other. A further elegant technique has been proposed by Ekert, which relies on the violation of the Bell inequalities [6] to provide the secret security. And the third technique, devised by Bennett, is based on the transmission of nonorthogonal quantum states. Raw quantum cryptography is useless in practice because limited eavesdropping may be undetectable, yet it may leak some information, and errors are to be expected even in the absence of eavesdropping. Also, we must protect against an eavesdropper who would impersonate Alice for Bob and Bob for Alice. For these reasons, quantum cryptography must be supplemented by classical tools such as privacy amplification [7], error correction Obviously, the tools from steps b) to d) are classic supplement. For demonstration we use the quantum cryptographic protocols known as BB84 or four-state protocols. In general, the BB84, Ekert92, and B92 protocols possess the same process, the different is only in the method of quantum transmission, the process is described in figure 1. In the first step of establishing the key, Alice sends a random sequence of signal built up from the four possibly signal state, each appearing with equal probability. Bob possesses two measurements apparatuses adapted to the two sets of signal states. He may distinguish either between vertical and horizontal linear polarize photons or between right and left circular polarize photons. For each of the signals sent to him by Alice he chooses with equal probability an apparatuses to use. After Bob's receiving and measurement, he sends publicly the measurement base to Alice, and Alice compares the base between Alice and Bob's base. After Alice and Bob obtain what is call the raw data by the quantum transmission, the raw data must be sifted because it consists of those bits which Bob either did not receive at all or did not correctly measure in the basis used to transmit them. By comparison publicly the basis between Alice and Bob, the data sifting procedure is completed. 2 The Third step is the data correction. A distinct feature of error correction in quantum cryptography is that the error correction process is public, while the transmission itself is secret. In other words, Alice and Bob must conduct a public discussion to identify, with a high degree of confidence, all errors in their data, while at the same time leaking as little information as possible about the data. The basic idea is that Alice and Bob compute and Obviously, the above procedure is based on the legitimate users, refereed to as Alice and Bob. However, the practice existence of impersonation of Alice or Bob by eavesdropper, make us have to take some action to against the eavesdropper, an efficient way is to verify the communicators' identity. In the follows, we improve the previous quantum key distribution scheme to guarantee the security of quantum key for truly legitimate users . After the privacy amplification, a compressed key is obtained, but it can not be acted as the final key because of the impersonation. So the fifth step for identity verification 3 following the previous schemes, which is described in figure 1, must be added for the security quantum key. The improving scheme is shown in figure 2. Of course, the identity verification step can also be inserted in the front of the privacy amplification according to the sequence: quantum transmission −→ data sifting −→ error correcti on −→ identity verification −→ privacy amplification, the schematic diagram is described in figure 3. It is more practicable according to the latter sequence, because if one of the communicators is impersonation, the procedure may be over before the step of privacy amplification. The key problem of the identity verification is to obtain the authentication key, it can be established by the technique that divides the initial quantum secret key K (it may be called Raw Key) into two parts, i.e.,K = K a ⊕ K m , where the sign ⊕ represents the logic plus, the key K a is used for identity verification, while the key K m is as a final shared secret key between Alice and Bob. The K a may be obtained by two techniques. A single method is to choose the bits from K according to a proper 'rule', which is adapted publicly by users Alice and Bob, for example, one may take the bits in odd position in K. The guaranteed security of the quantum key K keeps the taken bits a high degree of security, although the 'rule' is chosen publicly. Then Alice and Bob constructs independently the authentication key K a . At last Alice and Bob correct the K a like that techniques used in the second step (Data Sifting) of quantum key distribution. More complexity, one can adopt the privacy amplification technique again or the hash function to obtain a shorter key as K a from the "Raw key" K. In this way, the K a is more secure and the quantum key will be not influenced. As shown in figure 2, after obtaining the shared dynamical-key K a , Alice and Bob use it to verify themselves identity, the technique may be like that base on the symmetric cryptosystem. If the processes of identity verification give the 'yes', the K m may act as the final key, otherwise the communication is over or re-set up. Reference 4 [ 8 ] 8. To obtain more high security quantum privacy key, in general, four processes has be included in the quantum key distribution: exchange a series of block check sums of their data and proceed by bi-section to locate the error in each of the problem blocks. After their block check sums agree several times in a row, Alice and Bob conclude that all transmission errors have been removed. Each disclosed check sum is presumed to have been recorded by eavesdropper Eve, and to be worth one bit of Renyi information to Eve. The number of iterations required, and hence the amount of Renyi information leaked, depends on the desired confidence level, the initial error rate, and the manner in which Alice and Bob select their check sum blocks.By the distillation art of secret key, the so called privacy amplification, a final secure quantum key is generated and distributed. The basic principle of privacy amplification is as follows. Let Alice and Bob shared a random variable W , such as a random n-bit string, while an eavesdropper Eve learns a corrected random variable V , providing at most t < n bits of information about W , i.e., H(W |V ) ≤ n − t. Eve is allowed to specify an arbitrary distribution P V W (unknown to Alice and Bob) subject to the only constraint thatR(W |V = v) ≤ n − twith high probability (over values v), where R(W |V = v) denotes the second-order conditional Renyi entropy of W , given V = v. For any s < n − t, Alice and Bob can distill r = n − t − s bits of the secret key K = G(W ) while keeping Eve's information about K exponentially small in s , by publicly choosing the compression function G at random from a suitable class of maps into {0, 1} n−t−s . C H Bennett, G Brassard, S Breidbart, S Wiesner, Advances in cryptology: Proceedings of Crypto'82. D.Chaum, R.L.Rivest and A.T.ShermanNew YorkPlenumC.H.Bennett, G.Brassard, S.Breidbart and S.Wiesner, In Advances in cryptology: Proceedings of Crypto'82, edited by D.Chaum, R.L.Rivest and A.T.Sherman (Plenum, New York, 1983). Experimental quantum cryptography. C H Bennett, F Bessette, G Brassard, L Salvail, J Smolin, J.Cryptology. 53C.H.Bennett,F.Bessette, G.Brassard, L.Salvail and J.Smolin, Experimental quantum cryptography, J.Cryptology 5, 3 (1992). Quantum cyptography bases on Bell's theorem. A K Ekert, Phys. Rev. Lett. 67661A.K.Ekert, Quantum cyptography bases on Bell's theorem, Phys. Rev. Lett. 67, 661(1991). Quantum cryptography using any two non-orthogonal states. C H Bennett, Phys. Rev. Lett. 683121C.H.Bennett,Quantum cryptography using any two non-orthogonal states, Phys. Rev. Lett. 68, 3121(1992). A Bohm, quantum mechanics. Springer-Verlag New York IncA.Bohm, quantum mechanics, Springer-Verlag New York Inc. 1979. . J S Bell, Physics. 1195J.S.Bell, Physics (Long Island City, N.Y.) 1, 195(1965). Generalized privacy amplification. C H Bennett, G Brassard, C Crepeau, U M Maurer, IEEE Trans. Inform. Theory. 411915C.H.Bennett, G.Brassard, C.Crepeau and U.M.Maurer, Generalized privacy ampli- fication, IEEE Trans. Inform. Theory, 41, 1915(1995). Schematic diagram of the quantum key distribution system with identity verification. The identity verification step is in front of privacy amplification step a) and acts as the last step b). • Figure 2, • Figure 2. Schematic diagram of the quantum key distribution system with identity verification. The identity verification step is in front of privacy amplification step a) and acts as the last step b).
[]
[ "Stellar population astrophysics (SPA) with the TNG The Arcturus Lab", "Stellar population astrophysics (SPA) with the TNG The Arcturus Lab" ]
[ "C Fanelli \nDipartimento di Fisica e Astronomia\nUniversità degli Studi di Bologna\nvia Piero Gobetti 93/240129BolognaItaly\n\nINAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly\n", "L Origlia \nINAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly\n", "E Oliva \nINAF-Osservatorio Astrofisico di Arcetri\nLargo Enrico Fermi 550125FirenzeItaly\n", "A Mucciarelli \nDipartimento di Fisica e Astronomia\nUniversità degli Studi di Bologna\nvia Piero Gobetti 93/240129BolognaItaly\n\nINAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly\n", "N Sanna \nINAF-Osservatorio Astrofisico di Arcetri\nLargo Enrico Fermi 550125FirenzeItaly\n", "E Dalessandro \nINAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly\n", "D Romano \nINAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly\n" ]
[ "Dipartimento di Fisica e Astronomia\nUniversità degli Studi di Bologna\nvia Piero Gobetti 93/240129BolognaItaly", "INAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly", "INAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly", "INAF-Osservatorio Astrofisico di Arcetri\nLargo Enrico Fermi 550125FirenzeItaly", "Dipartimento di Fisica e Astronomia\nUniversità degli Studi di Bologna\nvia Piero Gobetti 93/240129BolognaItaly", "INAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly", "INAF-Osservatorio Astrofisico di Arcetri\nLargo Enrico Fermi 550125FirenzeItaly", "INAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly", "INAF-Osservatorio di Astrofisica e Scienza\nvia Piero Gobetti 93/3, 40129BolognaItaly" ]
[]
Context. High-resolution spectroscopy in the near-infrared (NIR) is a powerful tool for characterising the physical and chemical properties of cool-star atmospheres. The current generation of NIR echelle spectrographs enables the sampling of many spectral features over the full 0.9-2.4 µm range for a detailed chemical tagging. Aims. Within the Stellar Population Astrophysics Large Program at the TNG, we used a high-resolution (R=50000) NIR spectrum of Arcturus acquired with the GIANO-B echelle spectrograph as a laboratory to define and calibrate an optimal line list and new diagnostic tools to derive accurate stellar parameters and chemical abundances. Methods. We inspected several hundred NIR atomic and molecular lines to derive abundances of 26 different chemical species, including CNO, iron-group, alpha, Z-odd, and neutron-capture elements. We then performed a similar analysis in the optical using Arcturus VLT-UVES spectra. Results. Through the combined NIR and optical analysis we defined a new thermometer and a new gravitometer for giant stars, based on the comparison of carbon (for the thermometer) and oxygen (for the gravitometer) abundances, as derived from atomic and molecular lines. We then derived self-consistent stellar parameters and chemical abundances of Arcturus over the full 4800 − 24500 Å spectral range and compared them with previous studies in the literature. We finally discuss a number of problematic lines that may be affected by deviations from thermal equilibrium and/or chromospheric activity, as traced by the observed variability of He I at 10830 Å.
10.1051/0004-6361/202039397
[ "https://export.arxiv.org/pdf/2011.12321v1.pdf" ]
201,125,229
2011.12321
71452554c1d10f518411cfdddb1a140e04ec6d07
Stellar population astrophysics (SPA) with the TNG The Arcturus Lab December 13, 2021 C Fanelli Dipartimento di Fisica e Astronomia Università degli Studi di Bologna via Piero Gobetti 93/240129BolognaItaly INAF-Osservatorio di Astrofisica e Scienza via Piero Gobetti 93/3, 40129BolognaItaly L Origlia INAF-Osservatorio di Astrofisica e Scienza via Piero Gobetti 93/3, 40129BolognaItaly E Oliva INAF-Osservatorio Astrofisico di Arcetri Largo Enrico Fermi 550125FirenzeItaly A Mucciarelli Dipartimento di Fisica e Astronomia Università degli Studi di Bologna via Piero Gobetti 93/240129BolognaItaly INAF-Osservatorio di Astrofisica e Scienza via Piero Gobetti 93/3, 40129BolognaItaly N Sanna INAF-Osservatorio Astrofisico di Arcetri Largo Enrico Fermi 550125FirenzeItaly E Dalessandro INAF-Osservatorio di Astrofisica e Scienza via Piero Gobetti 93/3, 40129BolognaItaly D Romano INAF-Osservatorio di Astrofisica e Scienza via Piero Gobetti 93/3, 40129BolognaItaly Stellar population astrophysics (SPA) with the TNG The Arcturus Lab December 13, 2021Astronomy & Astrophysics manuscript no. 39397corrTechniques: spectroscopic -stars: abundances -individual stars: Arcturus -stars: late-type Context. High-resolution spectroscopy in the near-infrared (NIR) is a powerful tool for characterising the physical and chemical properties of cool-star atmospheres. The current generation of NIR echelle spectrographs enables the sampling of many spectral features over the full 0.9-2.4 µm range for a detailed chemical tagging. Aims. Within the Stellar Population Astrophysics Large Program at the TNG, we used a high-resolution (R=50000) NIR spectrum of Arcturus acquired with the GIANO-B echelle spectrograph as a laboratory to define and calibrate an optimal line list and new diagnostic tools to derive accurate stellar parameters and chemical abundances. Methods. We inspected several hundred NIR atomic and molecular lines to derive abundances of 26 different chemical species, including CNO, iron-group, alpha, Z-odd, and neutron-capture elements. We then performed a similar analysis in the optical using Arcturus VLT-UVES spectra. Results. Through the combined NIR and optical analysis we defined a new thermometer and a new gravitometer for giant stars, based on the comparison of carbon (for the thermometer) and oxygen (for the gravitometer) abundances, as derived from atomic and molecular lines. We then derived self-consistent stellar parameters and chemical abundances of Arcturus over the full 4800 − 24500 Å spectral range and compared them with previous studies in the literature. We finally discuss a number of problematic lines that may be affected by deviations from thermal equilibrium and/or chromospheric activity, as traced by the observed variability of He I at 10830 Å. Introduction The enhanced sensitivity of IR observations to intrinsically red (i.e. cool) and/or reddened (by dust extinction) objects make near-IR (NIR) spectrographs the ideal instrument for studying the physics, chemistry, and kinematics of cool giant and supergiant stars in galaxy fields as well as in star clusters. Cool giant and supergiant stars are among the brightest populations in any stellar systems and are easily observable at IR wavelengths out to large distances. They are also easy to detect in heavily reddened environments, such as the inner disk and bulge regions, where observations in the visual range are prohibitive. These stars are important tracers of the star formation and chemical enrichment history of their hosts. High-resolution spectroscopy of these stars is crucial to obtain an exhaustive description of their detailed chemistry and nucleosynthesis. Different chemical elements are synthesised in stars with different initial masses and thus released into the in-Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. This study is part of the Large Program titled SPA -Stellar Population Astrophysics: the detailed, age-resolved chemistry of the Milky Way disk (PI: L. Origlia), granted observing time with HARPS-N and GIANO-B echelle spectrographs at the TNG. terstellar medium with different time delays with respect to the onset of star formation. The detailed chemical tagging of key elements is therefore crucial to constrain formation and chemical enrichment scenarios of the Milky Way and other nearby stellar system, in which these stars can be individually resolved. In the past two decades high-resolution NIR spectroscopy has experienced a burst of activity in terms of newly commissioned spectrographs and stellar surveys. However, the precise identification and characterisation of the optimal atomic and molecular lines for abundance analysis, as well as their modelling over the entire NIR range, is still work in progress. To this purpose, high-resolution spectroscopy in both visual and NIR spectral ranges of suitable chemical calibrators is mandatory. Arcturus is such a calibrator for giant stars, and we present a comprehensive study in the YJHK NIR bands using the echelle spectrum at R=50000 that has recently been obtained with the GIANO-B spectrograph (Oliva et al. 2012b,a;Origlia et al. 2014b;Tozzi et al. 2016) at the Telescopio Nazionale Galileo (TNG). Arcturus (α Boo, HR5340, HIP69673, or HD124897) is a luminous, nearby K1.5 IIIp giant star that is often used as a calibrator in chemical studies of cool stellar populations. Differential chemical analysis of giant stars relative to Arcturus can indeed largely minimise most of systematic errors due to atmospheric Article number, page 1 of 16 arXiv:2011.12321v1 [astro-ph.SR] 24 Nov 2020 A&A proofs: manuscript no. 39397corr (2011) However, it is challenging to take a spectrum of Arcturus because of its apparent ultra-bright luminosity. Most of the chemical studies of Arcturus are based on high-resolution spectroscopy. The reference Arcturus spectrum covering the entire spectral range from the UV to the IR is the one made available by Hinkle & Wallace (2005, and references therein). This spectrum has been built using three different instruments: the Space Telescope Imaging Spectrograph (STIS) mounted on board of the Hubble Space Telescope in the 1000-3000 Å range, the echelle optical spectrograph in the 3100-9000 Å range, and the Fourier transform spectrometer in the 0.9-5 µm range mounted at the Kitt Peak National Observatory (KPNO) 4 m telescope. Ryde et al. (2009) (2011) provided atmospheric parameters and abundances for several metals by mostly using the Hinkle & Wallace (2005) optical spectrum and the line list by Asplund et al. (2009). Smith et al. (2013) and Shetrone et al. (2015) used the H-band portion of the Hinkle & Wallace (2005) spectrum and their detailed line list prepared for the Sloan Digital Sky Survey III Apache Point Galactic Evolution Experiment (APOGEE) (Majewski et al. 2007) to provide stellar parameters, Fe, CNO and other elemental abundances. Arcturus was also studied by Fulbright et al. (2006) using an optical spectrum taken with the Hamilton spectrograph at the 0.6m CAT telescope of the Lick Observatory. Kondo et al. (2019) have analysed a ZYJ spectrum of Arcturus at R 28000 taken with the WINERED spectrograph at the 1.3 m Araki Telescope at Koyama Astronomical Observatory. They derived Fe I abundances and microturbolence using two different line lists: the Vienna Atomic Line Database (VALD3) (Ryabchikova & Pakhomov 2015), and the public line list provided by Meléndez & Barbuy (1999). Finally, we mention the works by Maas et al. (2017);D'Orazi et al. (2011) andOverbeek et al. (2016), who discussed the Arcturus abundances of P, Y, and Dy, respectively. Table 1 This paper is organised as follows. In Section 2 we describe the observation and the data reduction of the Arcturus spectrum. In Section 3 we discuss the method we adopted for spectral anal-ysis, and in Section 4 we describe the procedure we used to determine the stellar parameters for Arcturus using new NIR diagnostics. In Section 5 we report the results of our chemical analysis in the optical and NIR range, and in Section 6 we compare them with those from previous studies and draw our conclusions. Observations and data reduction Arcturus was observed on July 2, 2018, with GIANO-B, the high-resolution (R=50000) NIR (9500-24500 Å) spectrometer at the TNG (Origlia et al. 2014a). The observation was part of the Large Program called Stellar population astrophysics: detailed age-resolved chemistry of the Milky Way disk (PI: L. Origlia). Spectra were collected using a grey filter that attenuates the light by about 5 magnitudes. Nodding was used to optimise the subtraction of the background and other detector patterns: we collected several pairs of exposures with the star alternatively positioned at 1/4 (position A) and 3/4 (position B) of the slit length. The integration time was 60 seconds per A,B position, with a mean seeing of ∼ 0.7 . The raw spectra were reduced using the data reduction pipeline software GOFIO , which processes calibration (darks, flats, and U-Ne lamps taken in daytime) and scientific frames. The main feature of the GOFIO data reduction is the optimal spectral extraction and wavelength calibration based on a physical model of the spectrometer that accurately matches instrumental effects such as variable slit tilt and order curvature over the echellogram (Oliva et al. 2018). The data reduction package also includes bad pixel and cosmic removal, sky and dark subtraction, flat-field and blaze correction. The spectrum was corrected for telluric absorption using the spectra of an O-type standard star taken at different air masses during the same night. The normalised spectra of the telluric standard taken at low and high air-mass values were combined with different weights to match the depth of the telluric lines in the Arcturus spectrum. Figs. A.1,A.2,A.3,and A.4 in the appendix show the rest-frame normalised spectra corrected for telluric absorption. The average signal-to-noise ratio of the reduced and telluric-corrected spectrum is about 150 per pixel. We also analysed two optical spectra of Arcturus retrieved from the ESO archive, in order to cross-check chemical abundances over the widest possible spectral range. These optical spectra were collected with the high-resolution spectrograph UVES at the ESO Very Large Telescope (VLT) at a resolution of R∼ 80000, using the CD3 Red Arm 580 and CD4 Red Arm 860, which cover the 4800 -6800 Å and 6700 -10400 Å wavelength ranges, respectively. Spectral analysis Accurate and precise stellar parameters and chemical abundances of Arcturus were determined by means of spectral synthesis technique applied to the observed spectra. Synthetic spectra were computed by using the radiative transfer code TURBOSPECTRUM (Alvarez & Plez 1998;Plez 2012) with MARCS models atmospheres (Gustafsson et al. 2008), the atomic data from VALD3 and the most updated molecular data from the website of B. Plez, https://www.lupm.in2p3.fr/ users/plez/. The synthetic spectra were convoluted with a Gaussian function in order to reproduce the observed broader profile that corresponds to an equivalent resolution of 32, 000. The additional broadening is mainly due to the macro-turbulence velocity (≈ 6 kms −1 ) because the projected rotational velocity of Arcturus is negligible (ξ rot = 2.4 kms −1 , Gray 1981). For the abundance analysis, we used a selected list of C I, Na I, Mg I, Al I, Si I, P I, S I, K I, Ca I, Sc I, Ti I, V I, Cr I, Mn I, Fe I, Fe II, Co I, Ni I, Cu I, Zn I, Y I; Y II, Ce II, Nd II, and Dy II atomic lines and CO, OH, CN, and HF molecular lines. Each line was carefully checked against possible blending with close contaminants. For this purpose, we developed a code called TurboSLine that identifies as potential contaminants any atomic or molecular j-th line whose centroid λ j is within one full width at half maximum (FWHM) from the centroid λ i of the analyzed i-th line. For each of these potential contaminants, we computed the theoretical line equivalent width (EW) and the amount of contamination using the following approximation: C j = EW j × 1 − |λ i − λ j | FWHM j .(1) If C j > 0.1 × EW i , the i-th line was classified as blended, and it was not normally used for abundance analysis. Some other lines were later on rejected because they are contaminated by the wings of nearby strong photospheric and/or by deep telluric lines by visual inspection. In our chemical analysis we also rejected strong lines because of the uncertainty in the modelling of their wings, non-local thermal equilibrium (NLTE), and chromospheric effects, etc. Tables A.1-A.4 1 provide the complete list of optical and NIR atomic and molecular lines used for the abundance analysis. For the computation of the chemical abundances we used SALVADOR, a tool developed by A. Mucciarelli (priv. comm.), which performs a χ 2 minimisation between observed and synthetic spectra while the normalisation of the observed spectrum around each line is optimised interactively. As a further check, we also computed the line EWs and derived the corresponding abundances. The latter were found to be practically coincident with those obtained from the spectral synthesis, with eventually only a slightly higher dispersion, likely because lines with some impurity provide slightly more uncertain abundances when the EW method is used. Stellar parameters Arcturus belongs to a kinematic group of several dozen old stars (Eggen 1971). Its metallicity ([Fe/H] −0.5 dex) and some enhancement of the alpha elements suggest that it likely formed in the thick disk of our Galaxy (Ramírez & Allende Prieto 2011;Bensby et al. 2014), although an extragalactic origin has also been proposed (Navarro et al. 2004). From a kinematic perspective, the Arcturus total velocity v t = √ (U 2 + V 2 + W 2 ) = 106 km s −1 and its location in the Toomre diagram (see Fig. 1) suggest that it is a thick disk star (Bensby et al. 2014 Previous determinations of the Arcturus stellar parameters (cf. Table 1) suggested temperatures in the 4275-4290 K range, log(g) in the 1.55-1.70 dex range, and microturbulence in the 1.20-1.85 kms −1 range. Using the Dartmouth web-tool at http://stellar.dartmouth.edu (Dotter et al. 2008 (2011)), we found that old (≥10 Gyr) ages are consistent with an effective temperature T e f f in the 4260-4310 K range and gravity log(g) in the 1.60-1.70 dex range. We used these photometric ranges for T e f f and log(g) to also constrain the microturbulence velocity range with the standard approach of minimising the slope between the iron abundance and the reduced EW of the measured lines log(EW/λ) (see the discussion in Mucciarelli 2011). We find microturbulence velocities in the 1.50 and 1.70 kms −1 range. We finally adopted the value of ξ = 1.60 ± 0.05 kms −1 that best minimizes any trend between the abundances and the reduced EW of about 400 iron lines distributed over the 4800 − 23400 Å spectral range, as shown in Fig. 2. This value of microturbolence also minimises the trend when the optical and the NIR lines are taken separately. As described in the following, the photometric values of T e f f and log(g) were fine-tuned using the spectroscopic data. log(EW/ ) 6.5 7.0 7.5 Fig. 2. Iron abundances as a function of the reduced EW for all the measured optical (blue circles) lines in the UVES spectra and NIR (red circles) lines in the GIANO-B spectrum. The dotted line marks the bestfit median abundance. C-thermometer We defined a new powerful diagnostic tool to derive T e f f in oxygen-rich cool stars, based on the balance between the carbon abundance, derived from atomic lines (see Table 2), and molecular CO roto-vibration transitions. Hereafter, we refer to this method as the C-thermometer. The basic principle of this thermometer follows from the very high dissociation potential of the CO molecule (11.1 eV). Thus the CO/C 0 relative abundance of carbon in molecular (CO) and atomic (C 0 ) form has a very strong exponential dependence on temperature. Like for other diatomic molecules, the abundance ratio also depends linearly on the gas pressure (i.e. on gravity); but this effect is far weaker. Figure 3 shows the behaviour of the Saha equilibrium abundances for gas temperatures and pressures relevant for this work. When the spectrum is modelled, CO and C I lines must provide the same abundances. If this does not happen, it is because the model temperature is incorrect and must be tuned until the two abundances match. However, the analysis is complicated (2013) showed that the 10683/10685/10691 Å multiplet in the Y band requires NLTE correction for a star like Arcturus. In particular, the 10691 Å line needs to be corrected by ∆A(C) = −0.23 dex. NLTE corrections are sensitive to the adopted stellar model and depend on temperature and gravity, hence they cannot be safely used to define a reliable thermometer. It is therefore desirable to use only those C I lines that are not affected by NLTE effects. 4. C abundances as a function of temperature from the C I lines in the H band (green circles), and the forbidden [C I] line at 8727.14 Å (empty triangle). The shaded green region is the corresponding best-fit ±1σ trend of the C I lines, while the shaded red region is the best-fit ±1σ trend of the CO lines. The large black dot marks the intersection of the two curves, and its x-coordinate provides the best-fit temperature. (eV) 6.5 7.0 7.5 For this purpose, given that forbidden lines do not suffer from NLTE (e.g. Alexeeva & Mashonkina 2015), we used the [CI] at 8727.14 Å measurable in our UVES red spectrum to derive a proxy of the atomic carbon abundance in LTE. Then, we computed LTE abundances for all the C I lines measurable in our GIANO-B spectrum (Table 2), and we checked their abundances against that from the [C I] line. We found that the C I lines in the H band with excitation potentials above 9 eV provide similar C abundances, and the C I lines in the Y and J bands with lower excitation potentials give LTE abundances that are systematically (∼ 0.3 dex) higher. The reason most likely are NLTE effects. We therefore used only the C I lines in the H band for the Cthermometer. Fig. 4 shows the variation in carbon abundance from C I and CO lines in the Arcturus GIANO-B spectrum as a function of T e f f . Both diagnostics are very sensitive to T e f f , but have opposite trends. The two curves intersect at T e f f = 4283 ± 33 K. The quoted error of ±33 K corresponds to a ±1σ variation in the derived C abundances from CI and CO. The C-thermometer is virtually independent of the other parameters within the uncertainties. Variations of ±0.05 kms −1 in microturbulence velocity or ±0.06 dex in log(g) have a negligible effect on temperature (≤ 15 K) and C abundance (≤0.02 dex). Even for a stronger variation of log(g) (up to 0.2 dex) and ξ (up to 0.2 kms −1 ) does the corresponding variation in temperature lie within the error. Interestingly, the C-thermometer also works if the molecular carbon abundance is derived from spectral synthesis of the ∆v=3 CO band-heads in the H band because it is fully consistent with the one derived from individual CO roto-vibration lines (see Sect. 5.1). Noticeably, the temperature of 4283 ±33 K derived from the C-thermometer also allows us to minimise any trend between iron abundances from neutral lines and their excitation potential within the errors, as shown in Fig. 5, which is the standard spectroscopic method for inferring the effective temperature. O-gravitometer The relative abundance of OH and atomic oxygen (O 0 ) depends linearly on the gas pressure (i.e. on gravity), while it has a weak dependence on temperature (see Figure 3) because of the low dissociation potential of the OH molecule (4.4 eV). Therefore, the OH/O 0 ratio can be used to estimate the gravity when the temperature is constrained. After fixing the temperature at T e f f = 4283 K, as derived from the C-thermometer, we can fine-tune the gravity by balancing the O abundances derived from the forbidden transition [O I] (blue) and from the OH lines (red) with gravity. The black dot marks the intersection of the two curves, and its x-coordinate provides the best-fit log(g)=1.67±0.06. Bottom panel: best-fit trends of Iron abundances from Fe II (shaded) and Fe I lines (red) with gravity. The black dot marks the intersection of the two curves, and its x-coordinate provides the best-fit log(g)=1.68 ± 0.10. at 6300.3 Å, and from the numerous OH lines measurable in the NIR spectrum. Fig. 6 (top panel) shows the variation in O abundance from the [O I] and OH lines in the Arcturus optical and NIR spectra, respectively, as a function of log(g). The intersection of the two curves occurs at log(g)= 1.67 ± 0.06 dex. The quoted error of ±0.06 dex corresponds to a ±1σ variation in the derived O abundances from [OI] and OH. Variations of ± 33 K in temperature affect gravity by ±0.10 dex, while variations of ±0.05 kms −1 in microturbulence velocity affect log(g) by ∓0.02 dex. Using the standard method for inferring spectroscopic log(g), that is, minimising the difference between the iron abundances from neutral and ionised optical lines, we obtained a very similar best-fit log(g)=1.68 ± 0.10 dex (see Fig. 6, bottom panel). Chemical analysis The adopted stellar parameters for the chemical analysis of Arcturus are summarised in Table 3. Abundance errors from the un- Grevesse & Sauval (1998) and Asplund et al. (2009). certainties in the stellar parameters were estimated by computing elemental abundances with varying T e f f by ±33 K, log(g) by ±0.06 dex, and ξ by ±0.05 kms −1 (see Table 3). Notes. On average, these errors amount to a few hundredths of a dex at most. The only notable exception is the higher sensitivity of OH and HF lines to the effective temperature: a variation of ±33 K indeed implies an error in the derivation of oxygen and fluorine abundances of ±0.06 and ±0.07 dex, respectively. In the computation of these errors we did not include the interdependence between the C, N, and O abundances that contribute to the formation of the measured molecular lines nor the effect of the abundances of the main electron donors on those derived from ionised species. We estimate that these effects normally yield errors below 0.1 dex (see also e.g. Ryde et al. 2009). Measurement errors include uncertainty in the continuum positioning and photon noise. For elements with more than two measurable lines, we computed the dispersion around the mean abundance, while for those with one or two measurable lines, we computed the dispersion from a Monte Carlo simulation, taking into account an error in the measured EW of ≈ ±FWHM/signal-to-noise ratio for a line FWHM sampled with 2-3 pixels. The measurement errors quoted in Table 4 are the σ dispersion divided by the square root of the number of lines. In the course of the chemical analysis we checked a few problematic lines against possible NLTE effects using the online web tool http://nlte.mpia.de/gui-siuAC_secE.php, M. Kovalev et al. 2018. Abundances and corresponding measurement errors for all the sampled chemical elements in the log(N X /N H )+12 and in the [X/H] solar scales, adopting as solar reference both Grevesse & Sauval (1998) (Gre98) and Asplund et al. (2009) (Aspl09), are listed in Table 4. In Fig. 5.2 we compare the derived abundances from NIR and optical lines for all the measured elements. CNO and fluorine In our chemical analysis of Arcturus we first computed the abundances of CNO and then those of the other elements. CNO are the most abundant metals, and in red giant and supergiant star spectra the many molecular CO, CN, and OH lines are the most important potential contaminants. Following Ryde et al. (2009) and Smith et al. (2013), we adopted an iterative method to derive CNO abundances in order to consider the interplay among these three elements in setting the molecular equilibrium. The resulting abundances are listed in first three lines of Table 4. Most interestingly, the carbon abundance derived from the ∆v=3 CO band-heads visible in the H band is 8.02 and equal within the errors with the values derived from atomic C I lines and isolated CO roto-vibrational transitions. This result indicates that the ∆v=3 CO band-heads could be effectively used to measure carbon abundances in cold stars where the single CO lines are severely blended or too weak for a reliable abundance analysis. Following a similar approach, we estimated the abundance of 13 C using two ∆v=3 and three ∆v=2 band-heads of 13 CO. This yielded a 12 C/ 13 C isotopic abundance ratio of 7 ± 1, consistent with the effects of the second dredge-up coupled to some additional mixing in low-mass giants (see e.g. Charbonnel & Lagarde 2010). The fluorine abundance was derived from one HF molecular line at 23358.33 Å in the K band using the transition parameters of Jönsson et al. 2014. Iron-peak elements The iron abundance was derived for more than 400 lines of Fe I and a dozen Fe II lines. Fully consistent values were obtained from all these line sets, with a σ dispersion of about 0.1 dex. In Fig. 7 we show the inferred iron abundances from the measured iron lines over the full 4800-24500 spectral range as a function of the line central wavelength. We adopted the average value of 6.93 ± 0.01 (see Table 4) as the reference iron abundance of Arcturus. However, it is interesting to note that many of the lines in the YJ bands provide abundances that are systematically lower (on average by 0.04 dex) than the reference abundance. We inspected some of these lines for possible NLTE effects and found indeed that some positive corrections of ≤0.1 dex should be applied to their LTE abundances. For the other iron-peak elements, we were also able measure both optical and NIR lines and derived suitable abundances. In particular, for chromium and nickel, we used a few dozen lines both in the optical and NIR range. For vanadium we used a few dozen lines in the optical with hyperfine splitting (hereafter HFS) and only a few in the NIR. For manganese we used the few measurable lines both in the optical and NIR. In particular, in the NIR we used the J-band lines at 13218.49 Å and 13415.64 Å that need an NLTE correction of ∆A(Mn) = −0.16 and ∆A(Mn) = −0.04, respectively, and two lines in the H band with HFS. For cobalt we used a few dozen optical lines with HFS and a few NIR lines. The latter show a larger scatter that can be explained with small NLTE effects. For example, we applied an NLTE correction of +0.06 dex to the LTE abundance of the 16757.64 Å line. For copper we were only able use one optical and one NIR line. The NIR line at 16005.75 Å has HFS and needs to be used with caution because it might be blended. For zinc we used one line in the optical and two in the NIR. The NIR line at 11054.28 Å is partially blended with CN, but the line at 13053.63 Å is free of contamination. Altogether, the iron-peak elements show fully consistent optical and NIR abundances. They homogeneously scale as iron, with the possible exception of copper and zinc, which are slightly enhanced. α-capture elements Dozens of unblended lines of Si I, Ca I, Ti I, Mg I, and S I are available in the NIR spectrum of Arcturus for an abundance analysis. The NIR Mg I and S I lines are known to experience NLTE effects (Zhang et al. 2017;Takeda et al. 2016). However, at the metallicity, temperature, and gravity of Arcturus, the corrections are negligible. Sulphur also shows a forbidden line [S I] Article number, page 7 of 16 A&A proofs: manuscript no. 39397corr 6 7 8 9 1 1 1 2 1 3 1 4 1 5 1 6 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 9 5 8 6 0 6 6 Atomic Number at 10821 Å that provides a fully consistent abundance with the one derived from the selected S I lines. This further proves that NLTE effects are negligible. The inferred NIR abundances for these alpha elements are fully consistent with the optical ones, as detailed in Table 4 and shown in Fig. 5.2. The derived abundances of Mg and Si, and to a lesser extent, of Ti, S, and Ca, suggest some [α/Fe] enhancement, as for oxygen. This is typical of thick-disk stars. C N O F N a M g A l S i P S K C a S c T i V C r M n F e C o N i C u Z n Y C e N d D Z-odd elements A few optical and NIR lines of Na, K, and Sc can be safely used to derive reliable abundances. We found consistent optical and NIR solar-scaled abundances of Na and Sc and some enhanced K abundance. However, the K lines can show significant NLTE effects with negative corrections to the LTE abundances (see e.g. Zhang et al. 2006;Osorio et al. 2020), thus implying a lower [K/Fe] relative abundance, about solar or even subsolar scaled. Two NIR lines of phosphorus at 10529.52 Å and 10581.58 Å are also measurable, giving an abundance of 5.14±0.08. A third line at 10596.90 Å gives a unexpected higher abundance (see also Maas et al. 2017), probably blended because its profile is clearly asymmetric. We therefore rejected it. For aluminium, ten optical lines and four NIR lines at 10782.05 Å, 10768.37 Å, 10872.97 Å, and 10891.77 Å in the Y band with small (if any) NLTE corrections provide homogeneous abundances that are higher by almost a factor of two than the solar-scaled value. For the NIR lines we used log(gf) from NIST, which is slightly different from the lines adopted in VALD3. The strong lines at 13123.41 Å and 13150.75 Å and the Kband line at at 21163.76 Å have HFS, show significant NLTE effects (Nordlander & Lind 2017), and require a negative abundance correction of 0.25 − 0.30 dex. Although when these lines are corrected for NLTE, they provide Al abundances that are reasonably consistent with those of the Y band and optical lines, we did not use them. The three strong lines at 16718.96 Å, 16750.56 Å, and 16763.36 Å also show NLTE effects and have HFS. The line at 16750.56 Å also has strong and blended wings. In Arcturus-like stars, the abundances derived from these lines can be quite uncertain, therefore we did not use them (see Sect. 5.6). Neutron-capture elements We measured NIR lines for five neutron-capture elements: yttrium (mostly an s-process element), cerium (an s-process element), neodimium (mostly an s-process element), and dysprosium (an r-process element). The NIR Ce II and Nd II lines were identified for the first time by Cunha et al. (2017) and Hasselquist et al. (2016), respectively. One neutral and five ionised lines of yttrium were measured in the optical, but only one ionised line is measured in the NIR Y band (see also Matsunaga et al. 2020). We find that the Y abundance is slightly depleted with respect to the solar scale value, in agreement with the disk chemistry at the Arcturus metallicity (see e.g. Reddy et al. 2006;Bensby et al. 2014). We finally used a few optical and NIR-ionised lines of cerium, neodymium, and dysprosium, and we derived about solar-scaled Ce and Nd abundances and slightly enhanced Dy with respect to the solar-scaled value. Chromospheric activity We realised that some strong lines in the GIANO-B spectrum are deeper than the corresponding lines in the FTS winter and summer spectra of Arcturus by Hinkle et al. (2000). A few examples are shown in Fig. 9. Chromospheric activity can fill the core of strong lines and mimic shallower absorptions (e.g. Shcherbakov et al. 1996). We therefore wondered whether a variation for the chromospheric activity in Arcturus might cause the different line depth in the GIANO-B and FTS spectra. For this purpose, we used the He I line at 10830 Å, which is a good indicator of chromospheric activity (Danks & Lambert 1985). As shown in Fig. 9, when the winter and especially the summer FTS spectra were acquired, chromospheric activity was higher, as suggested by some He I emission and shallower photospheric lines, while when the GIANO-B spectrum was acquired, the activity was low, without He I emission and with deeper photospheric lines. Because strong lines can be problematic also because they might be weakened by this chromospheric activity, they should be used with great caution for an abundance analysis. We excluded these lines from our abundance analysis. Discussion and conclusions Detailed high-resolution optical and NIR spectroscopy of stellar calibrators is fundamental for defining optimal diagnostics for atmospheric parameters and chemical analysis of stars and stellar populations with different ages, metallicities, and evolutionary properties. While diagnostic tools from high-resolution optical spectroscopy are well established and have been calibrated for a long time, those from NIR spectroscopy have been begun to be explored only recently with the new generation of NIR echelle spectrographs, whose performances are suitable for such quantitative studies. We used Arcturus as a laboratory to explore optical and NIR spectroscopic diagnostics for chemical analyses over the full spectral range from 4800 to 24500. We then provided a comprehensive and self-consistent determination of the stellar parameters and chemical abundances of Arcturus. The value of this combined optical and NIR study is multifold and is summarised below. i) The study maximises the set of diagnostic lines so that almost all the chemical elements of interest can be sampled from a statistically significant number of lines for most of them. ii) The study enables sampling lines of a given species at different wavelengths, which extends the range of excitation potentials and transition probabilities for a better understanding of the physics of line formation and the modelling of the observed spectrum. iii) The study drives the analysis towards a physical selfconsistent solution over the entire spectrum of the degeneracy problem among stellar parameters and chemical abundances. Taking advantage of our optical and NIR analysis, i) we were able to set an optimal value for the microturbulence velocity that works over the full spectral range from 4800 to 24500 Å, and ii) we were able to define a new spectroscopic thermometer and new spectroscopic gravitometer for cool giants, based on atomic and NIR molecular diagnostics of carbon and oxygen abundances, as detailed in Sect. 3 and Figs. 4 and 6. Using these diagnostic tools, we infer a temperature and gravity for Arcturus that are fully consistent with photometric estimates and with the values obtained from the standard tools used in optical spectroscopy (see Figs. 5 and 6). As discussed in Sect. 5 and shown in Fig. 5.2, we find fully consistent optical and NIR abundances for all elements we analysed. This demonstrates that i) the current generation of NIR echelle spectrographs is fully adequate to deliver high-quality data for quantitative spectroscopy as in the optical, and ii) the available atomic and molecular data for the NIR lines are generally accurate enough for a reliable chemical abundance analysis. Carbon, sodium, potassium, and iron-peak elements (with the exception of copper and zinc, which are slightly higher) are consistent with solar-scaled values, with abundances between one-fourth and one-third solar. Nitrogen, oxygen, fluorine, and alpha elements (with the only exception of Ca, which is about solar-scaled), are higher by a few tenths of a dex than the corresponding solar-scaled values. The values inferred for [F/O] and [F/Fe] agree with literature values for thick-disk giant stars of similar metallicity (see e.g. Grisoni et al. 2020, their figure 2). Among the neutron-capture elements, cerium and neodymium are about solar-scaled, dysprosium is slightly higher, and Yttrium is slightly lower than the corresponding solar-scaled values. Our inferred stellar parameters and chemical abundances are normally fully consistent with those obtained in previous optical or NIR studies, as shown in Fig. 10. We stress here that abundance differences of a few hundredths dex in different studies is intrinsic to the analysis process because different studies may use different codes, model atmospheres, and/or line lists and transition probabilities as well as some different assumptions for the stellar parameters. In particular, when we compare our optical abundances with those obtained by Ramírez & Allende Prieto (2011), we found some notable discrepancies only for C and Mn abundances. They determined the C abundance from four C I lines. In our analysis we rejected these lines because the two at 9078 Å and 9111 Å are affected by NLTE and the other two at 8335 Å and 5380 Å are blended. We thus used only the forbidden line at 8727 Å. As discussed in Sect. 4.1, our [C I] abundance is fully consistent with the C abundance derived from CO and C I lines in the H band. Regarding Mn, the authors mostly measured blue lines that are absent from our UVES spectrum, while we measured lines in the red part of the optical spectrum. As discussed in Sect. 5.2, our optical Mn abundance is fully consistent with our NIR estimates and the values obtained in other NIR studies. It is also interesting to compare our results on stellar parameters and iron abundances with those obtained by the NIR studies of Kondo et al. (2019) in the in the 9300−13100 Å spectral range and of Smith et al. (2013) in the 15000 − 17000 Å spectral range, at about half the spectral resolution of our study. Kondo et al. (2019) a microturbulence velocity of 1.2 kms −1 , which is lower than any previous study and also lower than our adopted value of 1.6 kms −1 . They also used two different line lists, VALD3 and the list by Meléndez & Barbuy (1999) These rejected lines give an average abundances that differs by more than 0.1 dex and has a significantly larger (>0.2 dex) dispersion. C N O F N a M g A l S i P S K C a S c T i V C r M n F e C o N i C u Z n Y C e N d D We also used 30 lines in the YJ bands that are not in the VALD3 Kondo et al. (2019) line list. On average, they provide abundances that are a few hundredths dex higher than those from the the lines in common. As a result, our average VALD3 iron abundance from our selection of YJ band lines is 0.08 dex higher than in Kondo et al. (2019) and only a few hundredths dex lower than our reference abundance from the full set of optical and NIR lines. As mentioned in Sect. 5.2, the correction for NLTE effects can significantly mitigate if not solve the problem. When we use the MB99 astrophysical log(g f ), which are given for lines in the 10, 000 − 18, 000 Å range and are on average lower by 0.2-0.3 dex than those in VALD3, we find corresponding larger abundances. When compared to the iron abundances obtained in previous optical studies (see Table 1) and also in the present one, the abundances from the MB99 log(g f ) are in excess by more than 1σ in the YJ band and more than 2σ in the H band. When we also use the lower microturbulence velocity of Kondo et al. (2019), we obtain even larger and unlikely iron abundances. Smith et al. (2013) used a microturbulence velocity of 1.85 kms −1 and nine H-band lines with astrophysical log(g f ) cali-brated on the Sun and Arcturus IR FTS spectra by Livingston & Wallace (1991) and Hinkle & Wallace (2005), respectively. Their log(g f ) values are somewhat in between those of VALD3 and MB99. In our analysis we used 84 H band lines. Five out of the nine lines listed by Smith et al. (2013) are in common with our sample. The other four lines in the Smith et al. (2013) list have been rejected because they are problematic (i.e. partially blended and/or with strong wings). Our and their average abundances from the five lines in common are very similar. The lower log(g f ) values used by Smith et al. (2013) are somewhat compensated for by their slightly higher microturbulence velocity. The slightly revised iron abundance in Shetrone et al. (2015) is practically coincident with our estimate. The problem of the imperfect modelling of a line is highly degenerate. A given variation in the abundance from a given line can be obtained by modifying its log(gf), but also by slightly varying the stellar parameters and/or by using different prescriptions for the damping, HFS, NLTE corrections, etc. This degeneracy (see also e.g. Takeda 1992) cannot be easily removed. The astrophysical calibration of the log(gf) is becoming very popular. However, this calibration is model dependent (i.e. it depends on the adopted model atmospheres, spectral code, lines, etc.), and it also depends on the calibrator itself, that is, on the selected star and observed spectrum as well as on the adopted stellar parameters. None of the proposed astrophysically calibrated log(gf) can therefore be safely adopted in studies that use recipes and tools for chemical analyses that are different from those used for the astrophysical calibration. Our combined optical and NIR analysis shows that it is not necessary to systematically tune the log(gf) of the NIR lines in the VALD3 database to obtain reliable abundances if the appropriate set of lines is chosen and self-consistent stellar parameters are derived. 50 1.66 ± 0.10 1.70 ± 0.05 −0.57 ± 0.04 15100 − 17000 22300 * * Shetrone et al. (2015) 4286 ± 35 1.64 ± 0.06 1.22 ± 0.12 −0.69 ± 0.06 9300 − 13100 28000 * * * Kondo et al. (2019) a 4286 ± 35 1.64 ± 0.06 1.20 ± 0.11 −0.49 ± 0.04 9300 − 13100 28000 * * * Kondo et al. (2019) b Notes. (a) Using VALD3 line list: http://vald.astro.uu.se (b) Using Meléndez & Barbuy (1999) line list (*) Spectrum observed with the 0.6 m CAT telescope at the Lick Observatory and the Hamilton spectrograph (**) Spectrum from Hinkle & Wallace (2005) (***) Spectrum observed with WINERED mounted at the 1.3 m Araki Telescope at Koyama Astronomical Observatory. parameters (e.g. McWilliam & Rich 1994; Worley et al. 2009; Alves-Brito et al. 2010; Ramírez & Allende Prieto 2011). ), we computed isochrones with [Fe/H]=−0.5, [α/Fe]=+0.20 and different ages. At the bolometric luminosity of Arcturus (L bol = 174 ± 7 L , Smith et al. (2013), see also Ramírez & Allende Prieto Fig. 3 . 3Selected pressure values are representative of the line-forming regions for surface gravities log(g)=2.0, 1.5, and 1.0 by the fact that some C I lines may show significant departure from LTE (see e.g. Fabbian et al. 2006; Takeda & Takada-Hidai 2013), while CO lines form under LTE conditions (e.g. Hinkle & Lambert 1975; Ryde et al. 2009). For example, Takeda & Takada-Hidai Fig. 5 . 5Iron abundances from neutral lines in the optical UVES spectra (blue circles) and in the NIR GIANO-B spectrum (red circles) as a function of their excitation potential. The dotted line marks the derived best-fit median iron abundance. has been derived by combining the measurement of 5 C I atomic lines and 14 CO molecular lines. N, O, and F abundances have been derived from the measurement of molecular lines only. Fig. 7 . 7Iron abundances for all the measured optical and NIR lines as a function of the line central wavelength. The dashed line marks the average abundance. Fig. 8 . 8Derived [X/Fe] chemical abundances for Arcturus from the GIANO-B NIR spectrum (red symbols) and UVES optical spectra (blue symbols). The circles indicate the neutral species, and the triangles indicate the ionised species. Error bars are fromTable 4. Fig. 9 . 9Chromospheric He I line in the Y band and a few strong photospheric Al, Si, and Mg lines in the in the GIANO-B spectrum (top panel) and in the winter, January (middle panel) and summer, June (bottom panel) FTS spectra byHinkle & Wallace (2005). (hereafter MB99). Our list has 48 lines in common with the Kondo et al. (2019) VALD3 line list. Our and the Kondo et al. (2019) VALD3 abundances are similar, although we adopted a 0.4 kms −1 higher microturbulence velocity. The other 20 lines in the Kondo et al. (2019) VALD3 line list have not been used in our analysis because they are blended or contaminated by nearby strong photospheric or telluric lines. FigFigFig . A.2. Observed J-band Arcturus spectrum (black line) with the telluric correction (red line) . A.3. Observed H-band Arcturus spectrum (black line) with the telluric correction (red line) . A.4. Observed K-band Arcturus spectrum (black line) with the telluric correction (red line) Table 1 . 1Stellar parameters and metallicity of Arcturus inferred from different optical and NIR studies.T e f f log(g) ξ [Fe/H] Range Res Re f erences K dex kms −1 dex Å λ/∆λ 4283 ± 39 1.55 1.61 ± 0.03 −0.55 ± 0.07 5370 − 7880 45000 * Fulbright et al. (2006) 4290 1.55 1.70 −0.50 5000 − 7000 50000 * * Ryde et al. (2009) 4286 ± 30 1.66 ± 0.05 1.74 −0.52 ± 0.02 5000 − 9300 100000 * * Ramírez & Allende Prieto used a portion of the Hinkle & Wallace (2005) H-band spectrum, from 15326 to 15705 Å, to study several clean molecular lines of CO, CN, and OH and derive C, N, and O abundances. Ramírez & Allende Prieto lists the stellar parameters and metallicity [Fe/H] of Arcturus inferred from different optical and NIR studies. ).200 150 100 50 0 50 V (km/s) 25 50 75 100 125 150 175 200 (U 2 + W 2 ) 1/2 (km/s) Arcturus Thick disk Thin disk Halo Fig. 1. Toomre diagram for thin-disk (blue dots), thick-disk (red dots), and halo (black crosses) stars from Reddy et al. 2003 and Reddy et al. 2006 and for the disk stars of Bensby et al. 2014 (grey dots). The black star marks the position of Arcturus, and the dashed lines delineate con- stant total space velocities with respect to the LSR of V tot = 85 and 180 km s −1 , respectively (e.g. Nissen 2004, and references therein). Table 2 . 2Measurable atomic carbon lines in the NIR spectrum of Arcturus.λ air χ log(gf) (a) Note Å eV 8727.140 1.26 −8.165 Forbidden 9658.435 7.49 −0.280 NLTE 10683.080 7.48 +0.079 NLTE 10685.340 7.48 −0.272 NLTE 10691.245 7.49 +0.344 NLTE 10707.320 7.48 −0.411 NLTE 10729.529 7.49 −0.420 NLTE 11748.220 8.64 +0.375 NLTE 11753.320 8.65 +0.691 NLTE 11754.760 8.64 +0.542 NLTE 17234.463 9.70 +0.293 LTE 17448.535 9.00 +0.012 LTE 17672.039 7.95 −1.974 LTE 17768.910 9.71 +0.420 LTE 17793.158 9.71 −0.045 LTE Notes. (a) For all the tabulated lines we used the NIST log(gf) values as listed in the VALD3 database. Fig. 6. Top panel: best-fit trends of Oxygen abundances from the [O I] line8.5 8.6 8.7 8.8 A(O) [O I] OH 1.5 1.6 1.7 1.8 1.9 log(g) 6.8 7.0 A(Fe) FeI FeII Table 3 . 3Stellar parameters for ArcturusParamater Value Error T e f f 4283 K 33 K log(g) 1.67 dex 0.06 dex ξ 1.60 kms −1 0.05 kms −1 [Fe/H] * −0.57 dex 0.01 dex Notes. (*) We used the solar A(Fe) =7.50, as in Table 4 . 4Arcturus chemical abundances and associated measurement errors from NIR and optical lines.NIR used a standard temperature and gravity, but0.50 0.25 0.00 0.25 0.50 0.75 [X/Fe] OPT Fulbright+2007 Ramirez+2011 D'orazi+2011 Overbeek+2016 Cunha+2017 Fig. 10. Derived [X/Fe] chemical abundances for Arcturus from some optical (top panel) and NIR (bottom panel) studies quoted inTable 1and in Sect. 1. Blue symbols are our abundances from the optical UVES spectra, and red symbols our abundances from the NIR GIANO-B spectrum. Blue and red dots refer to neutral species, and blue and red triangles to ionised species.y 0.50 0.25 0.00 0.25 0.50 0.75 [X/Fe] NIR Ryde+2009 Smith+2013 Shetrone+2015 (DR12) Maas+2017 6 7 8 9 1 1 1 2 1 3 1 4 1 5 1 6 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 9 5 8 6 0 6 6 Atomic Number Tables A.1-A.4 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http: //cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ Acknowledgements. We thank the anonymous referee for his/her detailed report and useful suggestions. C. Fanelli would like to thank A. Minelli for useful discussions. We acknowledge the support by INAF/Frontiera through the "Progetti Premiali" funding scheme of the Italian Ministry of Education, University, and Research. We acknowledge support from the project Lighton-Dark granted by MIUR through PRIN2017-000000 contract and support from the mainstream project SC3K -Star clusters in the inner 3 kpc funded by INAF. . 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[]
[ "Integrability and exact solution of an electronic model with long range interactions", "Integrability and exact solution of an electronic model with long range interactions" ]
[ "K E Hibberd ", "J R Links ", "\nDepartamento de Física Teórica\nCentre for Mathematical Physics\nUniversidad de Zaragoza\n50009ZaragozaSpain\n", "\nThe University of Queensland\nQLD\n4072Australia\n" ]
[ "Departamento de Física Teórica\nCentre for Mathematical Physics\nUniversidad de Zaragoza\n50009ZaragozaSpain", "The University of Queensland\nQLD\n4072Australia" ]
[]
We present an electronic model with long range interactions. Through the quantum inverse scattering method, integrability of the model is established using a one-parameter family of typical irreducible representations of gl(2|1). The eigenvalues of the conserved operators are derived in terms of the Bethe ansatz, from which the energy eigenvalues of the Hamiltonian are obtained.
null
[ "https://export.arxiv.org/pdf/cond-mat/0209265v1.pdf" ]
119,039,494
cond-mat/0209265
e85c76a1a130b96c9f013fcab9d87f5b4c57317f
Integrability and exact solution of an electronic model with long range interactions 11 Sep 2002 March 22, 2022 K E Hibberd J R Links Departamento de Física Teórica Centre for Mathematical Physics Universidad de Zaragoza 50009ZaragozaSpain The University of Queensland QLD 4072Australia Integrability and exact solution of an electronic model with long range interactions 11 Sep 2002 March 22, 2022 We present an electronic model with long range interactions. Through the quantum inverse scattering method, integrability of the model is established using a one-parameter family of typical irreducible representations of gl(2|1). The eigenvalues of the conserved operators are derived in terms of the Bethe ansatz, from which the energy eigenvalues of the Hamiltonian are obtained. Introduction The Quantum Inverse Scattering Method (QISM) [1] is one of the most powerful tools in the exact study of quantum systems. It can be applied in a number of contexts, including both one-dimensional systems with nearest neighbour interactions such as the Heisenberg [1] and Hubbard [2] models, and also for the analysis of models with long range interactions such as the Gaudin Hamiltonians [3] and extensions [4]. These latter constructions in particular have received renewed attention as it has been realised that the reduced BCS model, which was recently proposed to describe superconducting correlations in metallic grains of nanoscale dimensions [5], can be shown to be integrable through the use of Gaudin Hamiltonians in non-uniform external fields [6]. Formulating the reduced BCS model in the framework of the QISM reproduces the exact solution originally obtained by Richardson and Sherman [7], and opens the way for the calculation of form factors and correlation functions [8]. Motivated by this result, one can investigate to what extent the construction can be generalised to yield new classes of models, with some examples already given in [9]. Here, we will consider a case where an underlying superalgebraic structure (i.e., one with both bosonic and fermionic degrees of freedom) is employed to yield an electronic model. The supersymmetric formulation of integrable systems can be traced back to the work of Kulish [10], and recently supersymmetric Gaudin Hamiltonians have been analysed in detail in [11]. In this article we present a Hamiltonian derived through the QISM from a solution of the Yang-Baxter equation (YBE) associated with a typical irreducible representation of the Lie superalgebra gl(2|1), which has the explicit form H = D j ǫ j n j − g D j,k σ=± Q † jσ Q kσ .(1) Above, the energy levels ǫ j are two-fold degenerate, g is an arbitrary coupling parameter and D is the total number of distinct energy levels. Also, n j is the fermion number operator for energy level ǫ j and for the parameters α j , j = 1...D we define Q jσ = c jσ α j + 1.X n j,−σ j , with X j = α j /(α j + 1). The c σ , c † σ , σ = ±, are two-fold degenerate Fermi annihilation and creation operators. The Hamiltonian has a similar form to the reduced BCS model [5]. There, Cooper pairs are scattered into vacant energy levels while the one particle states are blocked from scattering. In the Hamiltonian above, there is correlated scattering depending on the occupation numbers. One of the features of this model is that the scattering couplings can be varied through the choice of the parameters α i . Via the algebraic Bethe ansatz method and using the minimal typical representation of gl(2|1), from which these free parameters arise, we establish the exact solvability of the model. Here we outline the necessary definitions and constructions, while full details will be presented elsewhere. The Lie superalgebra gl(2|1) has generators E i j , i, j = 1, 2, 3 with supercommutator relations [E i j , E k l ] = δ k j E i l − (−1) ([i]+[j])([k]+[l]) δ i l E k j . Above, the BBF grading = S + = c † + c − , E 2 1 = S − = c † − c + , E 1 1 = −α − n + , E 2 2 = −α − n − , E 3 3 = 2α + n, E 1 3 = Q † + , E 2 3 = Q † − , E 3 1 = Q + and E 3 2 = Q − , and we set S z = (n + − n − )/2. The Casimir invariant of the algebra, C = 3 i,j=1 E i j ⊗ E j i (−1) [j] , which commutes with all the elements of gl(2|1), will also be needed, and has the eigenvalue ξ C = −2α(α + 1) in the above representation. Below we let V (α) denote the four-dimensional model on which the representation acts, with the basis |+− , |+ , |− , |0 . The Yang-Baxter equation and integrability To construct the model, we use the supersymmetric formulation of the QISM [10]. We take the following solution of the YBE which acts on W ⊗ W ⊗ V (α), where W denotes the three-dimensional vector module of gl(2|1), R 12 (u − v)L 13 (u)L 23 (v) = L 23 (v)L 13 (u)R 12 (u − v)(2) with R(u) = I ⊗ I + η u 3 m,n=1 (−1) [n] e m n ⊗ e n m ,(3) and the L-operator is given by L(u) = I ⊗ I + η u 3 m,n=1 (−1) [n] e m n ⊗ E n m .(4) The representations taken for the operators E n m are as stated above, the variable u represents the rapidity η is arbitrary and I is the identity operator. By the usual procedure of the QISM, we define a transfer matrix acting on the D-fold tensor product space (for distinct α i ) V (α 1 ) ⊗ V (α 2 ) ⊗ ... ⊗ V (α D ) via t(u) = str 0 (G 0 L 0D (u − ǫ D )...L 01 (u − ǫ 1 )) , which gives a mutually commuting family satisfying [t(u), t(v)] = 0. Above, str 0 denotes the supertrace taken over the auxiliary space labelled by 0 and G can be any matrix which satisfies [R(u), G ⊗ G] = 0. For the BBF grading we choose G = diag(exp(βη), exp(βη), 1) and by employing the algebraic Bethe ansatz method the eigenvalues of the transfer matrix are found to be (cf. [10]) Λ(u) = exp(βη) D i 1 − ηα i (u − ǫ i ) P j a(v j − u) + exp(βη) D i 1 − ηα i (u − ǫ i ) P j a(u − v j ) M k a(γ k − u) − D i 1 − 2ηα i (u − ǫ i ) M j a(γ j − u),(5) where a(u) = 1 + η/u. The parameters v i , w j satisfy the Bethe ansatz equations M k a(γ k − v j ) = − P i a(v i − v j ) a(v j − v i ) , D i γ l − ǫ i − 2ηα i γ l − ǫ i − ηα i = exp(βη) P j a(γ l − v j ). We now introduce the operators T j = lim u→ǫ j (u − ǫ j ) η 2 t(u), which satisfy [T j , T k ] = 0.(6) By taking the quasi-classical expansion T j = τ j + o(η), this leads to τ j = −βψ j + D i =j θ ji ǫ j − ǫ i where θ = 3 m,n E n m ⊗ E m n (−1) [m] and ψ = E 3 3 . It is easily deduced that these operators satisfy [τ j , τ k ] = 0. Writing K = D i,j (S + i S − j + S − i S + j + 2S z i S z j ) , which satisfies [K, τ j ] = 0, ∀ j, we define the Hamiltonian as follows; H = 1 2β 2 D j (1 + 2βǫ j )τ j + 1 4β 3 D j,k τ j τ k + 1 2β D j C j − K 2β + 2 D j ǫ j (α j + 1) = 1 2β D j D k =j θ jk − 1 2β D j (1 + 2βǫ j )ψ j + 1 4β D i,j ψ i ψ j − 1 β D j α j (α j + 1) − K 2β + 2 D j ǫ j (α j + 1). The term involving θ jk may be simplified using the Casimir invariant and the commutation relations of the algebra gl(2|1) D j D k =j θ jk = D k,j θ jk − D j C j , = K − 1 2 D j,k ψ j ψ k + D j ψ j − 2 D j,k σ=± Q + jσ Q kσ + 2 D j α j (α j + 1). For g = 1/β we obtain the Hamiltonian (1), which establishes integrability since [H, τ j ] = 0, ∀ j. From (5,6) we obtain the eigenvalues of τ j for the BBF grading, λ j = −2βα j + α j M i 1 γ i − ǫ j − 2 D i =j α j α i ǫ j − ǫ i ,(7) as the quasi-classical limit of the eigenvalues of the transfer matrix. The corresponding Bethe ansatz equations are β + P j 1 γ l − v j = D i α i ǫ i − γ l , M l 1 γ l − v j = 2 P i =j 1 v i − v j .(8) For a given solution of the Bethe ansatz equations we find that the number of electrons, N = 2D − M, n + − n − = M − 2P and the eigenvalue of K reads ξ K = 1 2 (M − 2P )(M − 2P + 2).(9) The energy eigenvalues can be computed using (7,8,9) and are given by E = 2 D j ǫ j − M l γ l − 2g D j α j − gM. Similar results have been obtained for the FBB and BFB gradings, which will appear elsewhere. chosen and the elements are realised in terms of the Fermi operators through (cf.[12]) E 1 2 AcknowledgementsWe thank Petr Kulish for useful discussions. Jon Links acknowledges the Australian Research Council for financial support and Katrina Hibberd is supported by project number BFM2000-1057 from the Ministerio de Ciencia y Tecnologia, Spain. . L D Faddeev, Int. J. Mod. Phys. 101845L.D. Faddeev, Int. J. Mod. Phys. A10, (1995) 1845. . B S Shastry, Phys. Rev. Lett. 562453B.S. Shastry, Phys. Rev. Lett. 56, (1986) 1529, 2453. . M Gaudin, J. Phys. (Paris). 371087M. Gaudin, J. Phys. (Paris) 37, (1976) 1087. . K Hikami, P P Kulish, M Wadati, J. Phys. Soc. Jpn. 613071K. Hikami, P.P. Kulish and M. Wadati, J. Phys. Soc. Jpn. 61, (1992) 3071. . J Delft, D C Ralph, Phys. Rep. 34561J. von Delft and D. C. Ralph, Phys. Rep. 345, (2001) 61; . J Delft, Ann. Phys. (Leipzig). 10219J. von Delft, Ann. Phys. (Leipzig) 10, (2001) 219. . M C Cambiaggio, A M F Rivas, M Saraceno, Nucl. Phys. 624157M.C. Cambiaggio, A.M.F. Rivas and M. Saraceno, Nucl. Phys. A624, (1997) 157. . R W Richardson, Phys. Lett. 382R.W. Richardson, Phys. Lett. 3, (1963) 277; 5, (1963) 82; . R W Richardson, N Sherman, Nucl. Phys. 52253R.W. Richardson and N. Sherman, Nucl. Phys. 52, (1964) 221, 253. . H.-Q Zhou, J Links, R H Mckenzie, M D Gould, Phys. Rev. B. 6560502R)H.-Q. Zhou, J. Links, R.H. McKenzie and M.D. Gould, Phys. Rev. B 65, (2002) 060502(R). . J Links, H.-Q Zhou, M D Gould, R H Mckenzie, J. Phys. A: Math. Gen. 356459J. Links, H.-Q. Zhou, M.D. Gould and R.H. McKenzie, J. Phys. A: Math. Gen. 35, (2002) 6459; . J Links, K E Hibberd, Int. J. Mod. Phys. B. 16J. Links and K.E. Hibberd, Int. J. Mod. Phys. B 16, (2002) 2009; . J Links, H.-Q Zhou, R H Mckenzie, M D Gould, Int. J. Mod. Phys. B. to appearJ. Links, H.-Q. Zhou, R.H. McKenzie and M.D. Gould, Int. J. Mod. Phys. B, to appear; . X.-W Guan, A Foerster, J Links, H.-Q Zhou, Nucl. Phys. B. to appearX.-W. Guan, A. Foerster, J. Links and H.-Q. Zhou, Nucl. Phys. B, to appear. . P P Kulish, J. Sov. Math. 352648P.P. Kulish, J. Sov. Math. 35, (1986) 2648. . P P Kulish, N Manojlovic, Lett. Math. Phys. 5577P.P. Kulish and N. Manojlovic, Lett. Math. Phys. 55, (2001) 77; . J. Math. Phys. 4264J. Math. Phys. 42, (2002) 64. . A J Bracken, M D J R Gould, Y.-Z Links, Zhang, Phys. Rev. Lett. 742768A.J. Bracken, M.D. Gould. J.R. Links and Y.-Z. Zhang, Phys. Rev. Lett. 74, (1995) 2768.
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[ "Impact of solvent on state-to-state population transport in multistate systems using coherences", "Impact of solvent on state-to-state population transport in multistate systems using coherences" ]
[ "Amartya Bose \nDepartment of Chemical Sciences\nTata Institute of Fundamental Research\n400005MumbaiIndia\n", "Peter L Walters \nDepartment of Chemistry and Biochemistry\nGeorge Mason University\n22030FairfaxVirginiaUSA\n" ]
[ "Department of Chemical Sciences\nTata Institute of Fundamental Research\n400005MumbaiIndia", "Department of Chemistry and Biochemistry\nGeorge Mason University\n22030FairfaxVirginiaUSA" ]
[]
Understanding the pathways taken by a quantum particle during a transport process is an enormous challenge. There are broadly two different aspects of the problem that affect the route taken. First is obviously the couplings between the various sites, which translates into the intrinsic "strength" of a state-to-state channel. Apart from the inter-state couplings, the solvents affecting the energies of the state, and their relative coupling strengths and time-scales form the second factor. This impact of dissipative media is significantly more difficult to analyze. Building on recently derived relations between coherences and population derivatives, we present an analysis of the transport that allows us to account for both the effects in a rigorous manner. We demonstrate the richness hidden behind the transport even for a relatively simple system, a 4-site coarsegrained model of the Fenna-Matthews-Olson complex. The effect of the local dissipative media is highly non-trivial. We show that while the impact on the total site population may be small, there are dramatic changes to the pathway taken by the transport process. The ability to untangle the dynamics at a greater granularity opens up possibilities in terms of design of novel systems with an eye towards quantum control.
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[ "https://export.arxiv.org/pdf/2301.12712v5.pdf" ]
257,038,534
2301.12712
c0ba33f0c4ef8c1724550ec8e7b4639490eae8d7
Impact of solvent on state-to-state population transport in multistate systems using coherences Amartya Bose Department of Chemical Sciences Tata Institute of Fundamental Research 400005MumbaiIndia Peter L Walters Department of Chemistry and Biochemistry George Mason University 22030FairfaxVirginiaUSA Impact of solvent on state-to-state population transport in multistate systems using coherences Understanding the pathways taken by a quantum particle during a transport process is an enormous challenge. There are broadly two different aspects of the problem that affect the route taken. First is obviously the couplings between the various sites, which translates into the intrinsic "strength" of a state-to-state channel. Apart from the inter-state couplings, the solvents affecting the energies of the state, and their relative coupling strengths and time-scales form the second factor. This impact of dissipative media is significantly more difficult to analyze. Building on recently derived relations between coherences and population derivatives, we present an analysis of the transport that allows us to account for both the effects in a rigorous manner. We demonstrate the richness hidden behind the transport even for a relatively simple system, a 4-site coarsegrained model of the Fenna-Matthews-Olson complex. The effect of the local dissipative media is highly non-trivial. We show that while the impact on the total site population may be small, there are dramatic changes to the pathway taken by the transport process. The ability to untangle the dynamics at a greater granularity opens up possibilities in terms of design of novel systems with an eye towards quantum control. I. INTRODUCTION Simulating complex chemical reactions in the condensed phase has been the holy grail of computational and theoretical chemistry. This already difficult task becomes even more arduous when the reaction involves the purely quantum mechanism of tunneling. However, this is ubiquitous in various processes like exciton transport in photosynthetic complexes, electron transfer, etc. In addition, many exciton and electron transfer processes happen in extended systems where there can be multiple pathways for the quantum "particle" to follow. A thorough understanding of the contribution of these various pathways is necessary to facilitate a more clear picture of the dynamics. The simulation of the basic dynamics of quantum particles in a condensed phase is quite challenging in and of itself. Approximations like Redfield and Förster, 1 though often used, are not universally applicable, especially in the strongly coupled regime. For numerically exact simulations of dynamics of extended systems, approaches based on tensor-networks have been gaining a lot of popularity. Most notable among them are density matrix renormalization group (DMRG) 2-4 and its time-dependent variant. 5 The family of multiconfiguration time-dependent Hartree (MCTDH) 6,7 can also be thought of as being based on tree tensor networks. However, the approaches often fail to account for the effects of (a possible continuum of) translational and vibrational degrees of freedom contributed by the solvent. a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Both authors contributed equally to this work. Methods based on simulating the reduced density matrix provide a lucrative alternative to the above-mentioned methods for simulating these systems. Of these, the quasi-adiabatic propagator path integral (QuAPI) 8,9 and hierarchical equations of motion (HEOM) [10][11][12] are the most widely used. The development of small matrix decomposition 13,14 of QuAPI has made it especially viable for simulating large systems. Additionally, tensor networks have also been shown to be exceptionally useful in increasing the efficiency of path integral methods. [15][16][17][18] These tensor network-based ideas have very recently been successfully extended to a multisite framework capable of simulating the quantum dynamics of extended systems coupled with local dissipative media. [19][20][21] Studies of population dynamics conducted with these methods, while very rich in information, are unable to provide a clear and unambiguous insight into the mechanism of the transport. Consider an extended system with a non-trivial topology allowing for long-ranged couplings between sites, and assume we are interested in the transport of an exciton. For a given initial location of the exciton, one would traditionally focus on the timedependent population of the exciton on each of the sites. We would have no further information on the route or "pathway" that the exciton took to get to a particular site. Such information, however, is crucial to optimization of materials for guided quantum transport. An extremely naïve approach to analyzing the pathways would be to track the route of the strongest couplings in the system Hamiltonian that leads from the "source" to the "sink." Such an approach would obviously miss out on the effects of the dissipative media. A different approach has been recently used to understand these pathways under a Lindbladian model Hamiltonian by evaluating the transport of the base system vis-à-vis a system with a particular chromophore dropped. 22 The idea is that dropping a chromophore that is a part of the primary pathways would lead to a large decrease of transport efficiency. Alternatively, Wu et al. 23 have used flux networks constructed on integrated flux between sites and the flux balance method 24 to analyze the pathways in the Fenna-Matthews-Olson complex (FMO). Recently, Dani and Makri 25 have shown that the instantaneous rate of change of the site population is related to the off-diagonal terms of the reduced density matrix (also called the "coherences") and rigorously derived the rate constants specific to the various state-to-state channels. While rates and kinetic models can often offer deep insights, 23,24,26 in many ultrafast chemical systems, the short-time dynamics, often called the transients, may be very important. Rate theory generally fails for such processes and ones with more than one primary timescale. In these cases, it becomes crucial to shift our attention from rates to the population transfer. "Coherence maps" are visual representations of the time evolution of the off-diagonal terms of the reduced density matrix and have recently been shown to capture important features of the structure of the Hamiltonian. 27 Building on these insights, we show how one can efficiently leverage the information in the off-diagonal terms to understand the effect of dissipative media in modulating the direct transport between sites as a function of time. The paper is organized as follows. The analysis performed in this paper and its connections with the previous works is outlined in Sec. II. Thereafter, we explore the excitation dynamics in a coarse-grained four-site model of the Fenna-Matthews-Olson complex (FMO) with a focus on how this information can be used in a directed manner to gain detailed insights into the same. Finally, some concluding remarks and future outlook are presented in Sec. IV. II. IMPORTANCE OF COHERENCES IN DIRECT UNMEDIATED POPULATION TRANSPORT Consider a system with N sites or states coupled with arbitrary harmonic baths. These baths may or may not be site-local. The Hamiltonian of such a problem is generally of the form:Ĥ =Ĥ 0 +Ĥ SB(1) whereĤ 0 is the Hamiltonian corresponding to the system andĤ SB is the Hamiltonian corresponding to the systembath coupling. (It is assumed that the system is represented in a basis that diagonalizesĤ SB .) Under Gaussian response, the harmonic baths are often obtained from a simulation of the bath response function. 28,29 Usually one simulates the time-dependent population of each of the states. Here, we define the direct "state-to-state" population transfer from state k to state j as the population transfer between them without any intermediate state, also denoted by P j←k . The objective is to be able to simulate P j←k as a function of time. Given that a "pathway" or "route" of transport is nothing but a sequence of these state-to-state population transfers, it should be possible to assemble a picture of the important pathways using them as the building blocks. If P j = Tr (ρ(t) |j j|), is the population of the jth site, it is trivial to show that the time derivative of this population can be expressed as: 25 dP j dt = Tr ρ(t)F j ,(2)whereF j = i Ĥ 0 , |j j| .(3) This commutator,F j , is exactly the same flux operator that is used for rate theory. 30,31 Though generally rate theory is formulated in terms of the equilibrium correlation functions, it has been shown that the rate for a two-state problem can be obtained as a "plateau" value of the time-dependent non-equilibrium flux, Eq. 2 and 3, as well. 32 However, here we are not interested in a rate perspective. We rather want to understand the full time dynamics with additional information about the channeldependent contributions. Following Dani and Makri 25 , we expand Eq. 2 to get dP j dt = i k j|ρ(t)|k k|Ĥ 0 |j − j|Ĥ 0 |k k|ρ(t)|j .(4) For a real symmetric time-independent system Hamiltonian, dP j dt = − 2 k j|Ĥ 0 |k Im j|ρ(t)|k ,(5) Equations 4 and 5 can be interpreted in terms of the rates along the different state-to-state channels. 25 By comparing Eq. 5 with Eq. 4, one can, for any k, interpret the term with Im k|ρ(t)|j as the rate of flow from site k into j and the term with Im j|ρ(t)|k as the rate of flow from site j to k. The time evolution of the individual coherences, j|ρ(t)|k , as captured through coherence maps, also shows very interesting features reflecting the system dynamics and equilibrium. 27 Often the full population dynamics is a fruitful interrogative tool for understanding the system. When it comes to ideas of quantum design and understanding pathways of the excitation energy transport (EET) processes, it seems to be helpful to think in terms of site-to-site population transfer. The coherences allow us a crucial ability to express the direct and unmediated transfer of population between different sites. One can directly use Eq. 4 to partition the total population change at a site into the contributions from each state-to-state channel. We define the time-dependent population flow from the kth site to the jth site as P j←k (t) = i t 0 dt j|ρ(t )|k k|Ĥ 0 |j − j|Ĥ 0 |k k|ρ(t )|j .(6) For a real symmetric time-independent system Hamiltonian, using Eq. 5, this reduces to P j←k (t) = − 2 j|Ĥ 0 |k t 0 dt Im j|ρ(t )|k . (7) Notice that the state-to-state population flow between two sites is proportional to the Hamiltonian matrix element between them. This is what we would have naïvely concluded. However, the proportionality constant is related to the time integral of the coherence. This captures the solvent effect on the system dynamics. These stateto-state populations are independent of how the simulation was done, and therefore can be applied to any level of simulation as desired. There are no further approximations over and above the ones used for simulating the time-dependent reduced density matrix of the system. Notice that Eqs. 6 and 7 uphold detailed balance in the sense that P j←k (t) = −P k←j (t) and that P j←j (t) = 0 for all j, encoding the fact that there cannot be any population transfer from a site to itself. We would like to point out that the long time limits of the time-dependent state-to-state population flow is the same as the quantum integrated population fluxes derived by Wu et al. 23 and used to construct the flux networks. Finally, the time dependent population of the jth site can be expressed as P j (t) = P j (0) + k =j P j←k (t).(8) The ability to partition the time-dependent population on a site into the components along various channels is important for understanding the effects resulting from the non-trivial interactions between specific changes in the dissipative media and the system Hamiltonian. If the system is thought of as a graph, with the sites being the vertices and the edges being the various inter-site connections, then the time-integrated coherences decompose the time-evolution of the population on a site (vertex) along all the edges that are incident on it. III. RESULTS To demonstrate the utility of this analysis of the stateto-state population transfers leveraging the information of the coherences, consider a coarse-grained system modeled on the FMO complex. FMO is a naturally occurring light-harvesting complex with eight bacteriochlorophyll monomeric sites. It is ubiquitous as a model for excitonic transport and provides a very rich set of dynamical features owing to the non-linear inter-site couplings. To enable a thorough exploration of the impact of the vibrational modes on the transfers through various state-to-state channels, we simplify the system by coarse-graining it to include the four most relevant sites. For FMO, it is known that if bacteriochlorophyll site 1 is initially excited, the primary pathway is 1 → 2 → 3 and the secondary pathway leads from 1 → 6 → 5 → 4 → 3. Thus, in our coarse-grained model we keep sites 1, 2, and 3 as is, reduce sites 4, 5, and 6 into a new renormalized 4th site and omit sites 7 and 8 entirely. This is shown in Fig. 1. Similar to the full FMO, we expect the model to have a primary pathway of 1 → 2 → 3 and a secondary pathway of 1 → 4 → 3. The coarse-grained FMO model along with its interactions with the local vibrational baths is described by the following Hamiltonian: H =Ĥ 0 +Ĥ SB ,(9)H 0 = 4 k=1 k |k k| + j =k h j,k |j k| ,(10)H SB = 4 k=1 Nosc j=1 p 2 kj 2m kj + 1 2 m kj ω 2 kj x kj − c kj |k k| m kj ω 2 kj 2 ,(11) where ω kj and c kj are the frequency and coupling of the jth harmonic mode of the bath corresponding to the kth site. The electronic excitation energies are given by k and the inter-site couplings are given by h j,k . The frequencies and couplings of the baths are charac- terized by the spectral density defined as J k (ω) = π 2 j c 2 kj m kj ω kj δ(ω kj − ω).(12) This can be calculated as the Fourier transform of the energy-gap autocorrelation function simulated using molecular dynamics. The site-dependent spectral densities and Hamiltonian for FMO have been recently calculated by Maity et al. 33 We use these parameters as the starting point for our exploration. The Hamiltonian corresponding to this coarse-grained model is given in the supplementary information. It is based on the FMO Hamiltonian obtained by Maity et al. 33 In the original work the simulation was done using QM/MM MD trajectories, TD-LC-DFTB was used for the site energies and TrESP for the couplings. 33 The average and the sitedependent spectral densities are shown in Fig. 2 for reference. In our exploration of the FMO model, we will change the spectral densities in various ways that shall be described. However, the parameters for the system Change due to site 1 (P ←1(t)). Red: Change due to site 2 (P ←2(t)). Green: Change due to site 3 (P ←3(t)). Blue: Change due to site 4 (P ←4(t)). Hamiltonian will always remain the same to ensure that the effects that we see arise solely out of the vibrational baths. Figure 3 shows the excitonic population corresponding to each of the sites for the site-specific and average spectral densities. (This information can, in principle, be calculated using many methods. Here the simulations have been conducted using the tensor network path integral method 17 based on Feynman-Vernon influence functional.) We notice that changing the average spectral density to the site-specific spectral densities has minor effects on the dynamics of bacteriochlorophyll sites 1 and 2 and negligible effects on the populations of sites 3 and 4. A key drawback of this population picture is that it washes away a lot of details. At this level, one cannot answer questions such as how does the transfer from site 1 to site 2, P 2←1 (t), change in switching between the two descriptions. Or what happens to the various contributions to the site 3 population? The analysis of the imaginary part of coherences allows us to answer these questions. In Fig. 4, we show the population dynamics of specific sites along with the individual contributions. The first thing that one immediately observes is that the primary flow of excitonic population happens along 1 → 2 → 3. To see this consider that the excitation starts on site 1. The biggest transport happens from 1 to 2 in Fig. 4 (a) (red line). Then looking at where the population goes from site 2, we see that the maximum amount goes to 3 in Fig. 4 (b) (green line). By a similar analysis, we find a secondary, slower, pathway that leads from site 1 to site 3 via site 4 (1 → 4 → 3). Additionally, one sees a non-insignificant contribution from 1 → 2 → 4 → 3. The direct transfer from site 1 to site 3 is the least important of these. While the ability to analyze the primary pathways immediately is obvious from Fig. 4, we would like to emphasize the power of the method in terms of disqualifying unimportant pathways as well. Notice that though site 1 transfers population into site 4, site 4 only transfers population into site 3. Therefore, a path like 1 → 4 → 2 → 3 is not important. We notice that with the site-specific spectral densities the excitonic flow along 1 → 2 → 3 is decreased coupled with an increased flow along the 1 → 4 → 3 pathway. As for the other two pathways, the flow along 1 → 2 → 4 → 3 increases, and the direct transfer 1 → 3 remains the same. These changes in the exitonic pathways are evidenced by the fact that in going from the average to the site specific spectral densities, the direct transfer from site 2 to site 3 (red curve in Fig. 4 (c)) shows a decrease and the transfer from 4 to 3 (blue curve in the same figure) shows an increase. Furthermore, while there is an increase in both the transfer from site 1 to site 4 (magenta curve in Fig. 4 (d)) and site 2 to site 4 (red curve in the same figure), the increase in 1 → 4 is much larger. An explanation for these changes can be made by looking at P 3←2 (t) (red curve in Fig. 4 (c)). We notice that for the site-specific spectral densities, the direct transfer from site 2 to 3 seems to be somewhat restricted causing a rerouting of the excitation through site 4. Apart from this rather broad overview of the pathways of excitation dynamics, a state-to-state analysis can uncover a wide variety of other features. For example, we can determine that the actual direct transfer from site 1 to site 2, P 2←1 (t), stops after around 0.5 ps. Thus, the red line in Fig. 4 (a) and the magenta line in Fig. 4 (b) becomes practically flat around that time. This is the case even though the populations of both sites 1 and 2 keep changing throughout the period of simulation. Additionally, on careful observation of the plots Figs. 4 (a) and (b), it is seen that around 3 ps -4 ps, there is a backflow of population from site 2 to site 1 in case of different baths. (Notice the small bump in the dashed red line in Fig. 4 (a) around that point.) Neither of these two analysis would have been evident from a rate theory perspective. To explore the effect of the site-dependence of the spectral density, we systematically change the reorganization energies on single sites using the average bath as a starting point. We scale the reorganization energies on site 3 and site 2 with factors ranging from 0.25 and 1.75 in steps of 0.25. Figure 5 shows the change in the population dynamics of site 3. Notice that in both cases, the population curve initially increases but then starts to decrease, though the reorganization energy where the maximum transfer occurs is different in the two cases. This behavior is similar to the inverted region in Marcus theory of electron transfer. 34 It is interesting that although the maximum transfer is obtained at different values, at their respective maximum reorganization energies, the two curves look remarkably similar. However, this apparent similarity hides differences in the mechanism. Let us consider the two pathways -the primary one, 1 → 2 → 3 and the secondary one, 1 → 4 → 3 separately. Figure 6 demonstrates the changes on the population dynamics along the primary pathway. In Fig. 6 (b) and (d), we see that the curves are very similar to Fig. 5 (a) and (b). This implies that the transfer 2 → 3 accounts for the main parts of the dynamics of the excitation population on site 3. The transfer from site 2 to site 3, in Fig. 6 (b) and (d), seems to hit a maximum and decrease as we go away from it. This behavior is similar to the inverted region of Marcus theory of rate of electron transfer. Interestingly, the transfer from 1 → 2 seems to plateau and vary slightly around a constant value. However, things become completely different when the reorganization energy on site 2 is scanned (Figs. 6 (c) and (d)). There is apparently very little pattern to the transfer from site 1 to site 2 as seen in Fig. 6 (c). The lack of a Marcus-like inversion region is because of the relatively large coupling between sites 1 and 2 which breaks perturbation theory. Additionally, in all these simulations, one finds that the transfer from site 1 to site 2 happens at the smallest time-scales. There are further counter-intuitive features of the state-to-state population transfer between sites 1 and 2. At a scaling of 0.75, there is an initial transfer from site 1 to site 2, but around 0.5 ps the transfer reverses and population starts moving from 2 to 1. At a scaling of 1.25, there seems to be a backflow around 1 ps from site 2 to 1, and then the regular flow resumes. For a scaling of 0.25, 0.5, 1.5 and 1.75, there are clearly two different time scales involved, a faster one upto around 0.5 ps and a slower one after that. The transfers seem to be linear rather than exponential in these two domains. These behaviors are only observable because of the ability to partition the population transfer on a site-to-site Finally, let us turn our attention to the secondary pathway of 1 → 4 → 3. The impact of the scans on the two direct transfers that make up this pathway are shown in Fig. 7. First notice that on scanning the reorganization energy on site 3 ( Fig. 7 (a) and (b)), both the direct transfers seem to decrease with the reorganization energy and then stop changing. When scanning the reorganization energy on site 2, the transfer from 1 → 4 seems to increase monotonically. However, the transfer from 4 → 3 shows a decrease followed by an increase, resulting in a minimum of amount of transfer. These changes caused in the 1 → 4 → 3 pathway is very surprising, given that the reorganization energy on site 2, which is not a part of this pathway, is being scanned. (The data corresponding to the other channels, though not explored here, has been shown in the supplementary information for completeness.) IV. CONCLUSION Many exact and approximate methods exist that can simulate dynamics in complex systems coupled with solvents and vibrational modes. However, it is a significantly different and more difficult challenge to understand the exact routes that the transport process takes. Naïve approaches of looking at the inter-site couplings fall short because of their failure to account for the nontrivial effects of the solvent modes. In this paper, we have presented a novel technique for analyzing the dynamics that yields the contribution of each channel. There has been a recent realization of the importance of the coherences or off-diagonal terms in understanding the dynamics. It has been shown that the time-derivative of the site populations can be written as a linear combination of the imaginary part of the coherences. 25 The dynamics of these coherences have also been explored quite thoroughly under different conditions. 27 Based on the relation of the time-derivatives with the imaginary parts of the coherences, we show that the change in the population of a site can be trivially decomposed into the contributions coming from different channels. Thus one can, using the coherences, study the effects of the solvent and temperature on the direct and unmediated transport between any pair of sites. Employing this insight, we can start to untangle the dynamics of systems with complex inter-site couplings. As demonstrated in the 4-site model based on the FMO, the insights uncovered can often be very non-trivial. From a fundamental perspective, just because the total population on a site has a relatively regular pattern, it is not necessary that the regularity is there in all the individual contributions. Similarly, the total population showing some strange feature does not imply the existence of the weirdness in each of the contributory dynamics. What is possibly equally important to a fundamental understanding is the fact that changing the vibrational profile on a single site not only affects the pathways involving that site, but other pathways as well. This has important implications in trying to design materials and engineer specific outcomes in complex open quantum systems. An analysis of the coherences reveals a wealth of information that lay hidden in the dynamics of the reduced density matrix. Extending the explorations in Ref. 27 , it is now possible to asssociate causes with the various changes that happen in the total population dynamics. In the near future, we will utilize these ideas in understanding other processes beyond exciton transport, especially complex reactions with multiple pathways like proton-coupled electron transfer and multi-proton transfers. Finally, the fact that the various analyses of coherence done here and earlier elsewhere 25,27 are not dependent on any single method of simulation of the time evolved reduced density matrix makes these ideas universally applicable. FIG. 1 . 1Fenna-Matthews-Olson complex with the bacteriochlorophyll units colored by the coarse-grained units used. Blue: Coarse-grained site 1. Orange: Coarse-grained site 2. Green: Coarse-grained site 3. Red: Coarse-grained site 4. Gray: Ignored. FIG. 2 . 2Site-dependent and average spectral densities for the first four bacteriochlorophyll units in FMO obtained from Maity et al.33 FIG. 3 . 3Excitonic population on different sites as a function of time. Solid line: Average spectral density. Dashed line: Different spectral densities. FIG. 4 . 4Transfer pathways of excitonic population corresponding to each site with different spectral densities. Solid line: average bath. Dashed line: Different spectral densities. Black: Total change of population of the site. Magenta: FIG. 5 . 5Total population of site 3 as a function of time on scanning the reorganization energies on site 3 and site 2 respectively. FIG. 6 . 6Population transfer along 1 → 2 and 2 → 3 for components of the primary pathway when the site-specific reorganization energies on site 3 and site 2 are scanned. FIG. 7 . 7Population transfer along 1 → 4 and 4 → 3 for components of the primary pathway when the site-specific reorganization energies on site 3 and site 2 are scanned. channel basis. ACKNOWLEDGMENTWe acknowledge Nancy Makri for useful discussions and inputs. Zwischenmolekulare Energiewanderung und Fluoreszenz. T Förster, 10.1002/andp.19484370105Annalen der Physik. 437T. 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[ "Bose-Einstein condensation for two dimensional bosons in the Gross-Pitaevskii regime", "Bose-Einstein condensation for two dimensional bosons in the Gross-Pitaevskii regime" ]
[ "Cristina Caraci ", "Serena Cenatiempo ", "Benjamin Schlein ", "\nInstitute of Mathematics\nGran Sasso Science Institute\nViale Francesco Crispi 7 67100L'AquilaItaly\n", "\nUniversity of Zurich\nWinterthurerstrasse 1908057ZurichSwitzerland\n" ]
[ "Institute of Mathematics\nGran Sasso Science Institute\nViale Francesco Crispi 7 67100L'AquilaItaly", "University of Zurich\nWinterthurerstrasse 1908057ZurichSwitzerland" ]
[]
We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.
10.1007/s10955-021-02766-6
[ "https://export.arxiv.org/pdf/2011.05962v3.pdf" ]
226,300,127
2011.05962
d98e8efd6cca0feb3ca216f87f94563f31ce3073
Bose-Einstein condensation for two dimensional bosons in the Gross-Pitaevskii regime 21 Nov 2020 January 8, 2022 Cristina Caraci Serena Cenatiempo Benjamin Schlein Institute of Mathematics Gran Sasso Science Institute Viale Francesco Crispi 7 67100L'AquilaItaly University of Zurich Winterthurerstrasse 1908057ZurichSwitzerland Bose-Einstein condensation for two dimensional bosons in the Gross-Pitaevskii regime 21 Nov 2020 January 8, 2022 We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations. Introduction We consider N ∈ N bosons trapped in the two-dimensional box Λ = [−1/2; 1/2] 2 with periodic boundary conditions. In the Gross-Pitaevskii regime, particles interact through a repulsive pair potential, with a scattering length exponentially small in N . The Hamilton operator is given by H N = N j=1 −∆ xj + N i<j e 2N V (e N (x i − x j )) (1.1) and acts on a dense subspace of L 2 s (Λ N ), the Hilbert space consisting of functions in L 2 (Λ N ) that are invariant with respect to permutations of the N particles. We assume here V ∈ L 3 (R 2 ) to be compactly supported and pointwise non-negative (i.e. V (x) ≥ 0 for almost all x ∈ R 2 ). We denote by a the scattering length of the unscaled potential V . We recall that in two dimensions and for a potential V with finite range R 0 , the scattering length is defined by 2π log(R/a) = inf φ BR |∇φ| 2 + 1 2 V |φ| 2 dx (1.2) where R > R 0 , B R is the disk of radius R centered at the origin and the infimum is taken over functions φ ∈ H 1 (B R ) with φ(x) = 1 for all x with |x| = R. The unique minimizer of the variational problem on the r.h.s. of (1.2) is non-negative, radially symmetric and satisfies the scattering equation −∆φ (R) + 1 2 V φ (R) = 0 in the sense of distributions. For R 0 < |x| ≤ R, we have φ (R) (x) = log(|x|/a) log(R/a) . By scaling, φ N (x) := φ (e N R) (e N x) is such that −∆φ N + 1 2 e 2N V (e N x)φ N = 0 We have φ N (x) = log(|x|/a N ) log(R/a N ) ∀x ∈ R 2 : e −N R 0 < |x| ≤ R , for all x ∈ R 2 with e −N R 0 < |x| ≤ R. Here a N = e −N a. The properties of trapped two dimensional bosons in the Gross-Pitaevskii regime (in the more general case where the bosons are confined by external trapping potentials) have been first studied in [10,11,12]. These results can be translated to the Hamilton operator (1.1), defined on the torus, with no external potential. They imply that the ground state energy E N of (1.1) is such that E N = 2πN 1 + O(N −1/5 ) . (1.3) Moreover, they imply Bose-Einstein condensation in the zero-momentum mode ϕ 0 (x) = 1 for all x ∈ Λ, for any approximate ground state of (1.1). More precisely, it follows from [11] that, for any sequence ψ N ∈ L 2 s (Λ N ) with ψ N = 1 and lim N →∞ 1 N ψ N , H N ψ N = 2π, (1.4) the one-particle reduced density matrix γ N = tr 2,...,N |ψ N ψ N | is such that 1 − ϕ 0 , γ N ϕ 0 ≤ CN −δ (1.5) for a sufficiently smallδ > 0. The estimate (1.5) states that, in many-body states satisfying (1.4) (approximate ground states), almost all particles are described by the one-particle orbital ϕ 0 , with at most N 1−δ ≪ N orthogonal excitations. For V ∈ L 3 (R 2 ), our main theorem improves (1.3) and (1.5) by providing more precise bounds on the ground state energy and on the number of excitations. Theorem 1.1. Let V ∈ L 3 (R 2 ) have compact support, be spherically symmetric and pointwise non-negative. Then there exists a constant C > 0 such that the ground state energy E N of (1.1) satisfies 2πN − C ≤ E N ≤ 2πN + C log N (1.6) Furthermore, consider a sequence ψ N ∈ L 2 s (Λ N ) with ψ N = 1 and such that ψ N , H N ψ N ≤ 2πN + K for a K > 0. Then the reduced density matrix γ N = tr 2,...,N |ψ N ψ N | associated with ψ N is such that 1 − ϕ 0 , γ N ϕ 0 ≤ C(1 + K) N (1.7) for all N ∈ N large enough. It is interesting to compare the Gross-Pitaevskii regime with the thermodynamic limit, where a Bose gas of N particles interacting through a fixed potential with scattering length a is confined in a box with volume L 2 , so that N, L → ∞ with the density ρ = N/L 2 kept fixed. Let b = | log(ρa 2 )| −1 . Then, in the dilute limit ρa 2 ≪ 1, the ground state energy per particle in the thermodynamic limit is expected to satisfy e 0 (ρ) = 4πρ 2 b 1 + b log b + 1/2 + 2γ + log π b + o(b) , (1.8) with γ the Euler's constant. The leading order term on the r.h.s. of (1.8) has been first derived in [17] and then rigorously established in [14], with an error rate b −1/5 . The corrections up to order b have been predicted in [1,15,16]. To date, there is no rigorous proof of (1.8). Some partial result, based on the restriction to quasi-free states, has been recently obtained in [7,Theorem 1]. Extrapolating from (1.8), in the Gross-Pitaevskii regime we expect |E N −2πN | ≤ C. While our estimate (1.6) captures the correct lower bound, the upper bound is off by a logarithmic correction. Eq. (1.7), on the other hand, is expected to be optimal (but of course, by (1.6), we need to choose K = C log N to be sure that (1.1) can be satisfied). This bound can be used as starting point to investigate the validity of Bogoliubov theory for two dimensional bosons in the Gross-Pitaevskii regime, following the strategy developed in [4] for the three dimensional case; we plan to proceed in this direction in a separate paper. The proof of Theorem 1.1 follows the strategy that has been recently introduced in [3] to prove condensation for three-dimensional bosons in the Gross-Pitaevskii limit. There are, however, additional obstacles in the two-dimensional case, requiring new ideas. To appreciate the difference between the Gross-Pitaevskii regime in twoand three-dimensions, we can compute the energy of the trivial wave function ψ N ≡ 1. The expectation of (1.1) in this state is of order N 2 . It is only through correlations that the energy can approach (1.6). Also in three dimensions, uncorrelated manybody wave functions have large energy, but in that case the difference with respect to the ground state energy is only of order N (N V (0)/2 rather than 4πa 0 N ). This observation is a sign that correlations in two-dimensions are stronger and play a more important role than in three dimensions (this creates problems in handling error terms that, in the three dimensional setting, were simply estimated in terms of the integral of the potential). The paper is organized as follows. In Sec. 2 we introduce our setting, based on a description of orthogonal excitations of the condensate on a truncated Fock space. In Sec. 3 and 4 we show how to renormalize the excitation Hamiltonian, to regularise the singular interaction. Finally, in Sec. 5, we show our main theorem. Sec. 6 and App. A contain the proofs of the results stated in 3 and 4, respectively. Finally, in App. B we establish some properties of the solution of the Neumann problem associated with the two-body potential V . Acknowledgment. We are thankful to A. Olgiati for discussions on the two dimensional scattering equation. C.C. and S.C. gratefully acknowledge the support from the GNFM Gruppo Nazionale per la Fisica Matematica -INDAM through the project "Derivation of effective theories for large quantum systems". B. S. gratefully acknowledges partial support from the NCCR SwissMAP, from the Swiss National Science Foundation through the Grant "Dynamical and energetic properties of Bose-Einstein condensates" and from the European Research Council through the ERC-AdG CLaQS. The Excitation Hamiltonian Low-energy states of (1.1) exhibit condensation in the zero-momentum mode ϕ 0 defined by ϕ 0 (x) = 1 for all x ∈ Λ = [−1/2; 1/2] 2 . Similarly as in [8,2,3], we are going to describe excitations of the condensate on the truncated bosonic Fock space F ≤N + = N k=0 L 2 ⊥ (Λ) ⊗sk constructed on the orthogonal complement L 2 ⊥ (Λ) of ϕ 0 in L 2 (Λ). To reach this goal, we define a unitary map U N : L 2 s (Λ N ) → F ≤N + by requiring that U N ψ N = {α 0 , α 1 , . . . , α N }, with α j ∈ L 2 ⊥ (Λ) ⊗sj , if ψ N = α 0 ϕ ⊗N 0 + α 1 ⊗ s ϕ ⊗(N −1) 0 + · · · + α N With the usual creation and annihilation operators, we can write and that U * N U N = 1, i.e. U N is unitary. With U N , we can define the excitation Hamiltonian L N := U N H N U * N , acting on a dense subspace of F ≤N + . To compute the operator L N , we first write the Hamiltonian (1.1) in momentum space, in terms of creation and annihilation operators a * p , a p , for momenta p ∈ Λ * = 2πZ 2 . We find U N ψ N =H N = p∈Λ * p 2 a * p a p + 1 2 p,q,r∈Λ * V (r/e N )a * p+r a * q a p a q+r (2.1) where V (k) = R 2 V (x)e −ik·x dx is the Fourier transform of V , defined for all k ∈ R 2 (in fact, (1.1) is the restriction of (2.1) to the N -particle sector of the Fock space). We can now determine L N using the following rules, describing the action of the unitary operator U N on products of a creation and an annihilation operator (products of the form a * p a q can be thought of as operators mapping L 2 s (Λ N ) to itself). For any p, q ∈ Λ * + = 2πZ 2 \{0}, we find (see [8]): U N a * 0 a 0 U * N = N − N + U N a * p a 0 U * N = a * p N − N + U N a * 0 a p U * N = N − N + a p U N a * p a q U * N = a * p a q . (2.2) where N + = p∈Λ * + a * p a p is the number of particles operator on F ≤N + . We conclude that L N = L (0) N + L (2) N + L (3) N + L (4) N (2.3) with L (0) N = 1 2 V (0)(N − 1)(N − N + ) + 1 2 V (0)N + (N − N + ) L (2) N = p∈Λ * + p 2 a * p a p + N p∈Λ * + V (p/e N ) b * p b p − 1 N a * p a p + N 2 p∈Λ * + V (p/e N ) b * p b * −p + b p b −p L (3) N = √ N p,q∈Λ * + :p+q =0 V (p/e N ) b * p+q a * −p a q + a * q a −p b p+q L (4) N = 1 2 p,q∈Λ * + ,r∈Λ * : r =−p,−q V (r/e N )a * p+r a * q a p a q+r , (2.4) where we introduced generalized creation and annihilation operators b * p = U N a * p U * N = a * p N − N + N , and b p = U N a p U * N = N − N + N a p for all p ∈ Λ * + . On states exhibiting complete Bose-Einstein condensation in the zero-momentum mode ϕ 0 , we have a 0 , a * 0 ≃ √ N and we can therefore expect that b * p ≃ a * p and that b p ≃ a p . From the canonical commutation relations for the standard creation and annihilation operators a p , a * p , we find [b p , b * q ] = 1 − N + N δ p,q − 1 N a * q a p [b p , b q ] = [b * p , b * q ] = 0 . (2.5) Furthermore, [b p , a * q a r ] = δ pq b r , [b * p , a * q a r ] = −δ pr b * q for all p, q, r ∈ Λ * + ; this implies in particular that [b p , N + ] = b p , [b * p , N + ] = −b * p . It is also useful to notice that the operators b * p , b p , like the standard creation and annihilation operators a * p , a p , can be bounded by the square root of the number of particles operators; we find b p ξ ≤ N 1/2 + ξ , b * p ξ ≤ (N + + 1) 1/2 ξ for all ξ ∈ F ≤N + . Since N + ≤ N on F ≤N + , the operators b * p , b p are bounded, with b p , b * p ≤ (N + 1) 1/2 . Quadratic renormalization From (2.4) we see that conjugation with U N extracts, from the original quartic interaction in (2.1), some large constant and quadratic contributions, collected in L N respectively. In particular, the expectation of L N on the vacuum state Ω is of order N 2 , this being an indication of the fact that there are still large contributions to the energy hidden among cubic and quartic terms in L Since U N only removes products of the zero-energy mode ϕ 0 , correlations among particles remain in the excitation vector U N ψ N . Indeed, correlations play a crucial role in the two dimensional Gross-Pitaevskii regime and carry an energy of order N 2 . To take into account the short scale correlation structure on top of the condensate, we consider the ground state f ℓ of the Neumann problem − ∆ + 1 2 V (x) f ℓ (x) = λ ℓ f ℓ (x) (3.1) on the ball |x| ≤ e N ℓ, normalized so that f ℓ (x) = 1 for |x| = e N ℓ. Here and in the following we omit the N -dependence in the notation for f ℓ and for λ ℓ . By scaling, we observe that f ℓ (e N ·) satisfies − ∆ + e 2N 2 V (e N x) f ℓ (e N x) = e 2N λ ℓ f ℓ (e N x) on the ball |x| ≤ ℓ. We choose 0 < ℓ < 1/2, so that the ball of radius ℓ is contained in the box Λ = [−1/2; 1/2] 2 . We extend then f ℓ (e N .) to Λ, by setting f N,ℓ (x) = f ℓ (e N x), if |x| ≤ ℓ and f N,ℓ (x) = 1 for x ∈ Λ, with |x| > ℓ. Then − ∆ + e 2N 2 V (e N x) f N,ℓ (x) = e 2N λ ℓ f N,ℓ (x)χ ℓ (x) ,(3.2) where χ ℓ is the characteristic function of the ball of radius ℓ. The Fourier coefficients of the function f N,ℓ are given by f N,ℓ (p) := Λ f ℓ (e N x)e −ip·x dx for all p ∈ Λ * . We introduce also the function w ℓ (x) = 1 − f ℓ (x) for |x| ≤ e N ℓ and extend it by setting w ℓ (x) = 0 for |x| > e N ℓ. Its re-scaled version is defined by w N,ℓ : Λ → R w N,ℓ (x) = w ℓ (e N x) if |x| ≤ ℓ and w N,ℓ = 0 if x ∈ Λ with |x| > ℓ. The Fourier coefficients of the re-scaled function w N,ℓ are given by w N,ℓ (p) = Λ w ℓ (e N x)e −ip·x dx = e −2N w ℓ e −N p . (3.3) We find f N,ℓ (p) = δ p,0 − e −2N w ℓ (e −N p). From the Neumann problem (3.2) we obtain − p 2 e −2N w ℓ (e −N p) + 1 2 q∈Λ * V (e −N (p − q)) f N,ℓ (q) = e 2N λ ℓ q∈Λ * χ ℓ (p − q) f N,ℓ (q). (3.4) where we used the notation χ ℓ for the Fourier coefficients of the characteristic function on the ball of radius ℓ. Note that χ ℓ (p) = ℓ 2 χ(ℓp) with χ(p) the Fourier coefficients of the characteristic function on the ball of radius one. In the next lemma, we collect some important properties of the solution of (3.1). Lemma 3.1. Let V ∈ L 3 (R 2 ) be non-negative, compactly supported (with range R 0 ) and spherically symmetric, and denote its scattering length by a. Fix 0 < ℓ < 1/2, N sufficiently large and let f ℓ denote the solution of (3.2). Then i) 0 ≤ f ℓ (x) ≤ 1 ∀ |x| ≤ e N ℓ . ii) We have λ ℓ − 2 (e N ℓ) 2 log(e N ℓ/a) ≤ C (e N ℓ) 2 log 2 (e N ℓ/a) (3.5) iii) There exist a constant C > 0 such that dx V (x)f ℓ (x) − 4π log(e N ℓ/a) ≤ C log 2 (e N ℓ/a) (3.6) iv) There exists a constant C > 0 such that |w ℓ (x)| ≤ C if |x| ≤ R 0 C log(e N ℓ/|x|) log(e N ℓ/a) if R 0 ≤ |x| ≤ e N ℓ |∇w ℓ (x)| ≤ C log(e N ℓ/a) 1 |x| + 1 for all |x| ≤ e N ℓ (3.7) v) Let w N,ℓ = 1 − f N,ℓ with f ℓ,N = f ℓ (e N x) . Then the Fourier coefficients of the function w N,ℓ defined in (3.3) are such that | w N,ℓ (p)| ≤ C p 2 log(e N ℓ/a) . (3.8) Proof. The proof of points i)-iv) is deferred in Appendix B. To prove point v) we use the scattering equation (3.4): w ℓ (e −N p) = e 2N 2p 2 q∈Λ * V (e −N (p − q)) f N,ℓ (q) − e 4N p 2 λ ℓ q∈Λ * χ ℓ (p − q) f N,ℓ (q). Using the fact that e 2N λ ℓ ≤ Cℓ −2 | ln(e N ℓ/a)| −1 and that 0 ≤ f ℓ ≤ 1, we end up with | w ℓ (e −N p)| ≤ e 2N 2p 2 ( V (e −N ·) * f N,ℓ )(p) + 2e 2N λ ℓ ( χ ℓ * f N,ℓ )(p) ≤ e 2N 2p 2 V (x)f ℓ (x)dx + Cℓ −2 | log(e N ℓ/a)| −1 χ ℓ (x)f ℓ (e N x)dx ≤ Ce 2N p 2 log(e N ℓ/a) . We now defineη : Λ → R througȟ η(x) = −N w N,ℓ (x) = −N w ℓ (e N x) . (3.9) With (3.7) we find |η(x)| ≤ CN if |x| ≤ e −N R 0 C log(ℓ/|x|) if e −N R 0 ≤ |x| ≤ ℓ (3.10) and in particular, recalling that e −N R 0 < ℓ ≤ 1/2, |η(x)| ≤ C max(N, log(ℓ/|x|)) ≤ CN (3.11) for all x ∈ Λ. Using (3.10) we find η 2 = η 2 ≤ C |x|≤ℓ | log(ℓ/|x|)| 2 d 2 x ≤ Cℓ 2 1 0 (log r) 2 rdr ≤ Cℓ 2 . In the following we choose ℓ = N −α , for some α > 0 to be fixed later, so that η ≤ CN −α . (3.12) This choice of ℓ will be crucial for our analysis, as commented below. Notice, on the other hand, that the H 1 -norms of η diverge, as N → ∞. From (3.9) and Lemma 3.1, part iv) we find η 2 H1 = |x|≤ℓ e 2N N 2 |(∇w ℓ )(e N x)| 2 d 2 x = |x|≤e N ℓ N 2 |∇w ℓ (x)| 2 d 2 x ≤ C |x|≤e N ℓ 1 (|x| + 1) 2 d 2 x ≤ CN for N ∈ N large enough. We denote with η : Λ * → R the Fourier transform ofη, or equivalently η p = −N w N,ℓ (p) = −N e −2N w ℓ (p/e N ) . (3.13) With (3.8) we can bound (since ℓ = N −α ) |η p | ≤ C |p| 2 (3.14) for all p ∈ Λ * + = 2πZ 2 \{0}, and for some constant C > 0 independent of N , if N is large enough. From (3.12) we also have η ∞ ≤ CN −α . (3.15) Moreover, (3.4) implies the relation (3.16) or equivalently, expressing also the other terms through the coefficients η p , p 2 η p + N 2 ( V (./e N ) * f N,ℓ )(p) = N e 2N λ ℓ ( χ ℓ * f N,ℓ )(p)p 2 η p + N 2 V (p/e N ) + 1 2 q∈Λ * V ((p − q)/e N )η q = N e 2N λ ℓ χ ℓ (p) + e 2N λ ℓ q∈Λ * χ ℓ (p − q)η q . (3.17) We will mostly use the coefficients η p with p = 0. Sometimes, however, it will be useful to have an estimate on η 0 (because Eq. (3.17) involves η 0 ). From (3.13) and Lemma 3.1, part iv) we find |η 0 | ≤ N |x|≤ℓ w ℓ (e N x)d 2 x ≤ C |x|≤ℓ log(ℓ/|x|)d 2 x + CN e −N ≤ Cℓ 2 . (3.18) With the coefficients (3.13) we define the antisymmetric operator B = 1 2 p∈Λ * + η p b * p b * −p −η p b p b −p (3.19) and we consider the unitary operator e B = exp   1 2 p∈Λ * + η p b * p b * −p −η p b p b −p   . (3.20) We refer to operators of the form (3.20) as generalized Bogoliubov transformations. In contrast with the standard Bogoliubov transformations e B = exp   1 2 p∈Λ * + η p a * p a * −p −η p a p a −p   (3.21) defined in terms of the standard creation and annihilation operators, operators of the form (3.20) leave the truncated Fock space F ≤N + invariant. On the other hand, while the action of standard Bogoliubov transformation on creation and annihilation operators is explicitly given by e − B a p e B = cosh(η p )a p + sinh(η p )a * −p there is no such formula describing the action of generalized Bogoliubov transformations. Conjugation with (3.20) leaves the number of particles essentially invariant, as confirmed by the following lemma. Lemma 3.2. Assume B is defined as in (3.19), with η ∈ ℓ 2 (Λ * ) and η p = η −p for all p ∈ Λ * + . Then, for every n ∈ N there exists a constant C > 0 such that, on F ≤N + , e −B (N + + 1) n e B ≤ Ce C η (N + + 1) n . (3.22) as an operator inequality on F ≤N + . The proof of (3.22) can be found in [5, Lemma 3.1] (a similar result has been previously established in [19]). With the generalized Bogoliubov transformation e B : F ≤N + → F ≤N + , we define a new, renormalized, excitation Hamiltonian G N,α : F ≤N + → F ≤N + by setting G N,α = e −B L N e B = e −B U N H N U * N e B . (3.23) In the next proposition, we collect important properties G N,α . We will use the notation K = p∈Λ * + p 2 a * p a p and V N = 1 2 p,q∈Λ * + ,r∈Λ * : r =−p,−q V (r/e N )a * p+r a * q a q+r a p (3.24) for the kinetic and potential energy operators, restricted on F ≤N + , and H N = K+V N . We also introduce a renormalized interaction potential ω N ∈ L ∞ (Λ), which is defined as the function with Fourier coefficients ω N ω N (p) := g N χ(p/N α ) , g N = 2N 1−2α e 2N λ ℓ (3.25) for any p ∈ Λ * + , and ω N (0) = g N χ(0) = πg N . (3.26) with χ(p) the Fourier coefficients of the characteristic function of the ball of radius one. From (3.5) and ℓ = N −α one has |g N | ≤ C. Note in particular that the potential ω N (p) decays on momenta of order N α , which are much smaller than e N . From Lemma 3.1 parts i) and iii) we find ω N (0) − N V f ℓ 1 ≤ C N , ω N (0) − 4π 1 + α log N N ≤ C N . (3.27) Proposition 3.3. Let V ∈ L 3 (R 2 ) be compactly supported, pointwise non-negative and spherically symmetric. Let G N,α be defined as in (3.23) and define G eff N,α := 1 2 ω N (0)(N − 1) 1 − N + N + 2N V (0) − 1 2 ω N (0) N + 1 − N + N + 1 2 p∈Λ * + ω N (p)(b p b −p + h.c. ) + √ N p,q∈Λ * + : p+q =0 V (p/e N ) b * p+q a * −p a q + h.c. + H N . (3.28) Then there exists a constant C > 0 such that E G = G N,α − G eff N,α is bounded by | ξ, E G ξ | ≤ C N 1/2−α + N −1 (log N ) 1/2 H 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 + C ξ 2 (3.29) for all α > 1, ξ ∈ F ≤N + and N ∈ N large enough. The proof of Prop. 3.3 is very similar to the proof of [4,Prop. 4.2]. For completeness, we discuss the changes in Appendix A. Cubic Renormalization Conjugation through the generalized Bogoliubov transformation (3.21) renormalizes constant and off-diagonal quadratic terms on the r.h.s. of (3.28). In order to estimate the number of excitations N + through the energy and show Bose-Einstein condensation, we still need to renormalize the diagonal quadratic term (the part proportional to N V (0)N + , on the first line of (3.28)) and the cubic term on the last line of (3.28). To this end, we conjugate G eff N,α with an additional unitary operator, given by the exponential of the anti-symmetric operator A := 1 √ N r,v∈Λ * + η r b * r+v a * −r a v − h.c. (4.1) with η p defined in (3.13). An important observation is that while conjugation with e A allows to renormalize the large terms in G N,α , it does not substantially change the number of excitations. The following proposition can be proved similarly to [3, Proposition 5.1]. Proposition 4.1. Suppose that A is defined as in (4.1). Then, for any k ∈ N there exists a constant C > 0 such that the operator inequality e −A (N + + 1) k e A ≤ C(N + + 1) k holds true on F ≤N + , for any α > 0 (recall the choice ℓ = N −α in the definition (3.13) of the coefficients η r ), and N large enough. We will also need to control the growth of the expectation of the energy H N with respect to the cubic conjugation. This is the content of the following proposition, which is proved in subsection 6.1. for all α ≥ 1, s ∈ [0; 1] and N ∈ N large enough. We use now the cubic phase e A to introduce a new excitation Hamiltonian, obtained by conjugating the main part G eff N,α of G N,α . We define R N,α := e −A G eff N,α e A (4.3) on a dense subset of F ≤N + . Conjugation with e A renormalizes both the contribution proportional to N + (in the first line in the last line on the r.h.s. of (3.28)) and the cubic term on the r.h.s. of (3.28), effectively replacing the singular potential V (p/e N ) by the renormalized potential ω N (p) defined in (3.25). This follows from the following proposition. Proposition 4.3. Let V ∈ L 3 (R 2 ) be compactly supported, pointwise non-negative and spherically symmetric. Let R N,α be defined in (4.3) and define Then for ℓ = N −α and α > 2 there exists a constant C > 0 such that 5) for N ∈ N sufficiently large. R eff N,α = 1 2 (N − 1) ω N (0)(1 − N + /N ) + 1 2 ω N (0) N + (1 − N + /N ) + ω N (0) p∈Λ * + a * p a p 1 − N + N + 1 2 p∈Λ * + ω N (p) b * p b * −p + b p b −p + 1 √ N r,v∈Λ * + : r =−v ω N (r) b * r+v a * −r a v + h.c. + H N .E R = R N,α − R eff N,α is bounded by ± E R ≤ C[N 2−α + N −1/2 (log N ) 1/2 ](H N + 1) ,(4. The proof of Proposition 4.3 will be given in Section 6. We will also need more detailed information on R eff N,α , as contained in the following proposition. Proposition 4.4. Let R eff N,α be defined in (4.4). Then, for every c > 0 there is a constant C > 0 (large enough) such that R eff N,α ≥ 2πN + ω N (0) 2 N + + c log N H N − C(log N ) 2 N 2 + N − C (4.6) for all α > 2 and N ∈ N large enough. Moreover, let f, g : R → [0; 1] be smooth, with f 2 (x) + g 2 (x) = 1 for all x ∈ R. For M ∈ N, let f M := f (N + /M ) and g M := g(N + /M ). Then there exists C > 0 such that R eff N,α = f M R eff N,α f M + g M R eff N,α g M + Θ M (4.7) with ±Θ M ≤ C log N M 2 f ′ 2 ∞ + g ′ 2 ∞ H N + 1 for all α > 2, M ∈ N and N ∈ N large enough. Proof. From (4.4), using that | ω N (0)| ≤ C we have R eff N,α ≥ N 2 ω N (0) + ω N (0) N + + 1 2 p∈Λ * + ω N (p) b * p b * −p + b p b −p + 1 √ N r,v∈Λ * + : r =−v ω N (r) b * r+v a * −r a v + h.c. + H N − C N 2 + N − C . (4.8) For the cubic term on the r.h.s. of (4.8), with p∈Λ * + | ω N (p)| 2 p 2 ≤ C log N (4.9) we can bound 1 √ N r,v∈Λ * + r =−v ω N (r) ξ, b * r+v a * −r a v ξ ≤ 1 √ N r,v∈Λ * + r =−v | ω N (r)| (N + + 1) −1/2 b r+v a −r ξ (N + + 1) 1/2 a v ξ ≤ 1 √ N r,v∈Λ * + r =−v |r| 2 (N + + 1) −1/2 b r+v a −r ξ 2 1/2 × r,v∈Λ * + r =−v | ω N (r)| 2 |r| 2 (N + + 1) 1/2 a v ξ 2 1/2 ≤ C(log N ) 1/2 √ N K 1/2 ξ (N + 1)ξ . (4.10) As for the off-diagonal quadratic term on the r.h.s of (4.8), we combine it with part of the kinetic energy to estimate. For any 0 < µ < 1, we have 1 2 p∈Λ * + ω N (p) b * p b * −p + b −p b p + (1 − µ) p∈Λ * + p 2 a * p a p = (1 − µ) p∈Λ * + p 2 b * p + ω N (p) 2(1 − µ)p 2 b −p b p + ω N (p) 2(1 − µ)p 2 b * −p − 1 4(1 − µ) p∈Λ * + | ω N (p)| 2 p 2 b p b * p + (1 − µ) p∈Λ * + p 2 a * p N + N a p since a * p a p − b * p b p = a * p (N + /N )a p . With (2.5), we conclude that 1 2 p∈Λ * + ω N (p) b * p b * −p + b −p b p + (1 − µ) p∈Λ * + p 2 a * p a p ≥ − 1 4(1 − µ) p∈Λ * + | ω N (p)| 2 p 2 a * p a p − 1 4(1 − µ) p∈Λ * + | ω N (p)| 2 p 2 . With the choice µ = C/ log N and with (4.9), we obtain 1 2 p∈Λ * + ω N (p) b * p b * −p + b −p b p + (1 − µ) p∈Λ * + p 2 a * p a p ≥ − 1 4(1 − µ) p∈Λ * + | ω N (p)| 2 p 2 a * p a p − 1 4 p∈Λ * + | ω N (p)| 2 p 2 − C . To bound the first terms on the r.h.s. of the last equation, we use the term ω N (0)N + , in (4.8). To this end, we observe that, with (3.27), | ω N (p)| 2 4(1 − µ)p 2 ≤ | ω N (0)| 2 4(1 − µ)p 2 ≤ ω N (0) 4(1 − µ)π 1 + C log N N ≤ ω N (0) 2 for every p ∈ Λ * + (notice that |p| ≥ 2π, for every p ∈ Λ * + ) and for N large enough (recall the choice µ = c/ log N ). From (4.8), we find R eff N,α ≥ N 2 ω N (0) − 1 4 p∈Λ * + | ω N (p)| 2 p 2 + ω N (0) 2 N + + c log N H N − C (log N ) 2 N N 2 + − C. (4.11) Let us now consider the second term on the r.h.s more carefully. Using that, from (3.25), ω N (p) = g N χ(p/N α ), we can bound, for any fixed K > 0, 1 4 p∈Λ * + | ω N (p)| 2 p 2 ≤ C + 1 4 p∈Λ * + : K<|p|≤N α | ω N (p)| 2 p 2 . With | ω N (p) − ω N (0)| ≤ C|p|/N α , we obtain 1 4 p∈Λ * + | ω N (p)| 2 p 2 ≤ C + | ω N (0)| 2 4 p∈Λ * + : K<|p|≤N α 1 p 2 ≤ C + 4π 2 p∈Λ * + : K<|p|≤N α 1 p 2 . (4.12) For q ∈ R 2 , let us define h(q) = 1/p 2 , if q is contained in the square of side length 2π centered at p ∈ Λ * + (with an arbitrary choice on the boundary of the squares). We can then estimate, for K large enough, 4π 2 p∈Λ * + : K<|p|≤N α 1 p 2 ≤ K/2<|q|≤N α +K h(q)dq . For q in the square centered at p ∈ Λ * + , we bound h(q) − 1 q 2 = |p 2 − q 2 | p 2 q 2 ≤ C |q| 3 . Hence 4π 2 p∈Λ * + : K<|p|≤N α 1 p 2 ≤ K/2<|q|<N α +K 1 q 2 dq + C ≤ 2πα log N + C . Inserting in (4.12), we conclude that 1 4 p∈Λ * + | ω N (p)| 2 p 2 ≤ 2πα log N + C . Combining the last bound with (3.27) (and noticing that the contribution proportional to log N cancels exactly), from (4.11) we obtain R eff N,α ≥ 2πN + ω N (0) 2 N + + c log N H N − C (log N ) 2 N N 2 + − C which proves (4.6). Next we prove (4.7). From (4.8), with the bounds (4.10) and since, by (4.9), p∈Λ * + ω N (p) ξ, b * p b * −p ξ ≤ p∈Λ * + | ω N (p) b p ξ (N + + 1) 1/2 ξ ≤   p∈Λ * + | ω N (p)| 2 p 2   1/2 (N + + 1) 1/2 ξ K 1/2 ξ ≤ C(log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ it follows that R eff N,α = 2πN + H N + θ N,α (4.13) where for arbitrary δ > 0, there exists a constant C > 0 such that ± θ N,α ≤ δH N + C(log N ) (N + + 1) . (4.14) We now note that for f : R → R smooth and bounded and θ N,α defined above, there exists a constant C > 0 such that ± [f (N + /M ), [f (N + /M ), θ N,α ]] ≤ C log N M 2 f ′ 2 ∞ (H N + 1) (4.15) for all α > 2 and N ∈ N large enough. The proof of (4.15) follows analogously to the one for (4.14), since the bounds leading to (4.14) remain true if we replace the operators b # p , # = {·, * }, and a * p a q with [f (N + /M ), [f (N + /M ), b # p ]] or [f (N + /M ), [f (N + /M ), a * p a q ] ] respectively, provided we multiply the r.h.s. by an additional factor M −2 f ′ 2 ∞ , since, for example R eff N,α = f M R eff N,α f M + g M R eff N,α g M + 1 2 [f M , [f M , R eff N,α ]] + [g M , [g M , R eff N,α ]] Writing R eff N,α as in (4.13) and using (4.15) we get Its proof makes use of localization in the number of particle and is an adaptation of the proof of [3, Proposition 6.1]. The main difference w.r.t. [3] is that here we need to localize on sectors of F ≤N where the number of particles is o(N ), in the limit N → ∞. ± [f M , [f M , R eff N,α ]] + [g M , [g M , R eff N,α ]] ≤ C log N M 2 f ′ 2 ∞ + g ′ 2 ∞ H N + 1 . Proposition 5.1. Let V ∈ L 3 (R 2 ) be compactly supported, pointwise non-negative and spherically symmetric. Let G N,α be the renormalized excitation Hamiltonian defined as in (3.23). Then, for every α ≥ 5/2, there exist constants C, c > 0 such that G N,α − 2πN ≥ c N + − C (5.1) for all N ∈ N sufficiently large. Proof. Let f, g : R → [0; 1] be smooth, with f 2 (x) + g 2 (x) = 1 for all x ∈ R. Moreover, assume that f (x) = 0 for x > 1 and f (x) = 1 for x < 1/2. For a small ε > 0, we fix M = N 1−ε and we set f M = f (N + /M ), g M = g(N + /M ). It follows from Prop. 4.4 that R eff N,α − 2πN ≥ f M R eff N,α − 2πN f M + g M R eff N,α − 2πN g M − CN 2ε−2 (log N )(H N + 1) (5.2) Let us consider the first term on the r.h.s. of (5.2). From Prop. 4.4, for all α > 2 there exist c, C > 0 such that R eff N,α − 2πN ≥ c N + − C N (log N ) 2 N 2 + − C . (5.3) On the other hand, with (4.13) and (4.14) we also find R eff N,α − 2πN ≥ cH N − C(log N ) (N + + 1) (5.4) for all α > 2 and N large enough. Moreover, due to the choice M = N 1−ε , we have (log N ) 2 N f M N 2 + f M ≤ (log N ) 2 N ε f 2 M N + . With the last bound, Eq. (5. 3) implies that f M R eff N,α − 2πN f M ≥ cf 2 M N + − C (5.5) for N large enough. Let us next consider the second term on the r.h.s. of (5.2). We claim that there exists a constant c > 0 such that g M R eff N,α − 2πN g M ≥ cN g 2 M (5.6) for all N sufficiently large. To prove (5.6) we observe that, since g(x) = 0 for all x ≤ 1/2, g M R eff N,α − 2πN g M ≥ inf ξ∈F ≤N ≥M/2 : ξ =1 1 N ξ, R eff N,α ξ − 2π N g 2 M where F ≤N ≥M/2 = {ξ ∈ F ≤N + : ξ = χ(N + ≥ M/2)ξ} is the subspace of F ≤N + where states with at least M/2 excitations are described (recall that M = N 1−ε ). To prove (5.6) it is enough to show that there exists C > 0 with inf ξ∈F ≤N ≥M/2 : ξ =1 1 N ξ, R eff N,α ξ − 2π ≥ C (5.7) for all N large enough. On the other hand, using the definitions of G N,α in (3.28), R N,α and R eff N,α in (4.4), we obtain that the ground state energy E N of the system is given by E N = inf ξ∈F ≤N + : ξ =1 ξ, e −A G N,α e A ξ = inf ξ∈F ≤N + : ξ =1 ξ, R eff N,α + E L ξ with E L = E R + e −A E G e A .±E L ≤ CN −1/2 (log N ) 1/2 (H N + 1) + e −A N −1 (H N + 1) + (N + + 1) e A + C ≤ CN −1/2 (log N ) 1/2 (H N + 1) + C With (5.4) we obtain ± E L ≤ CN −1/2 (log N ) 1/2 R eff N,α − 2πN + CN −1/2 (log N ) 3/2 N + + C , (5.8) and therefore, with N + ≤ N E N − 2πN ≤ C inf ξ∈F ≤N + : ξ =1 ξ, R eff N,α − 2πN ξ + CN 1/2 (log N ) 3/2 + C . From the result (1.3) of [10,11,12] inf ξ∈F ≤N ≥M/2 : ξ =1 1 N ξ, R eff N,α ξ − 2π ≥ inf ξ∈F ≤N + : ξ =1 1 N ξ, R eff N,α − 2πN ξ ≥ c E N N − 2π − C √ N (log N ) 3/2 − CN −1 → 0 as N → ∞. If we assume by contradiction that (5.7) does not hold true, then we can find a subsequence N j → ∞ with inf ξ∈F ≤N j ≥M j /2 : ξ =1 1 N j ξ, R eff Nj ,α ξ − 2π → 0 as j → ∞ (here we used the notation M j = N 1−ε j ). This implies that there exists a sequenceξ Nj ∈ F ≤Nj ≥Mj /2 with ξ Nj = 1 for all j ∈ N such that lim j→∞ 1 N j ξ Nj , R eff Nj ,αξ Nj = 2π . On the other hand, using the relation R eff Nj ,α = e −A G Nj ,α e A −E L,j with E L,j satisfying the bound (5.8) (with N + ≤ N j ), we obtain that there exist constants c 1 , c 2 , C > 0 such that c 1 ξ Nj , R eff N,α − 2πN j ξ Nj − CN 1/2 j (log N j ) 3/2 ≤ e Aξ Nj , G Nj ,α − 2πN j e Aξ Nj ≤ c 2 ξ Nj , R eff N,α − 2πN j ξ Nj + CN 1/2 j (log N j ) 3/2 Hence for ξ Nj = e Aξ Nj we have lim Nj →∞ 1 N j ξ Nj , G Nj ,α ξ Nj = 2π . Let now S := {N j : j ∈ N} ⊂ N and denote by ξ N a normalized minimizer of G N,α for all N ∈ N \ S. Setting ψ N = U * N e B ξ N , for all N ∈ N, we obtain that ψ N = 1 and that lim N →∞ 1 N ψ N , H N ψ N = lim N →∞ 1 N ξ N , G N,α ξ N = 2π (5.9) Eq. (5.9) shows that the sequence ψ N is an approximate ground state of H N . From (1.5), we conclude that ψ N exhibits complete Bose-Einstein condensation in the zero-momentum mode ϕ 0 , and in particular that there existδ > 0 such that |1 − ϕ 0 , γ N ϕ 0 | ≤ CN −δ . Using Lemma 3.2, Prop. 4.1 and the rules (2.2), we observe that 1 N ξ N , N + ξ N = 1 N e −B U N ψ N , N + e −B U N ψ N ≤ C N ψ N , U * N (N + + 1)U N ψ N = C N + C 1 − 1 N ψ N , a * (ϕ 0 )a(ϕ 0 )ψ N = C N + C [1 − ϕ 0 , γ N ϕ 0 ] ≤ CN −δ (5.10) as N → ∞. On the other hand, for N ∈ S = {N j : j ∈ N}, we have ξ N = χ(N + ≥ M/2)ξ N and therefore 1 N ξ N , N + ξ N ≥ M 2N = N −ε 2 . Choosing ε <δ and N large enough we get a contradiction with (5.10). This proves (5.7), (5.6) and therefore also g M R eff N,α − 2πN g M ≥ cN + g 2 M . (5.11) Inserting (5.5) and (5.11) on the r.h.s. of (5.2), we obtain that R eff N,α − 2πN ≥ cN + − C(log N )N 2ε−2 (H N + 1) − C (5.12) for N large enough. With (5.4), (5.12) implies R eff N,α − 2πN ≥ cN + − C. To conclude, we use the relation e −A G N,α e A = R eff N,α + E L and the bound (5.8). We have that for α ≥ 5/2 there exist c, C > 0 such that G N,α − 2πN ≥ ce A R eff N,α − 2πN e −A − CN −1/2 (log N ) 3/2 e A N + e A − C ≥ c e A N + e −A − C ≥ cN + − C where we used (5.12) and Prop. 4.1. We are now ready to show our main theorem. Proof of Theorem 1.1. Let E N be the ground state energy of H N . Evaluating (3.28) and (3.29) on the vacuum Ω ∈ F ≤N + and using (3.26), we obtain the upper bound E N ≤ 2πN + C log N . With Eq. (5.1) we also find the lower bound E N ≥ 2πN − C. This proves (1.6). Let now ψ N ∈ L 2 s (Λ N ) with ψ N = 1 and ψ N , H N ψ N ≤ 2πN + K . (5.13) We define the excitation vector ξ N = e −B U N ψ N . Then ξ N = 1 and, recalling that G N,α = e −B U N H N U * N e B we have, with (5.1), ψ N , (H N − 2πN )ψ N ≥ ξ N , (G N,α − 2πN )ξ N ≥ c ξ N , N + ξ N − C . (5.14) From Eqs. (5.13) and (5.14) we conclude that ξ N , N + ξ N ≤ C(1 + K) . If γ N denotes the one-particle reduced density matrix associated with ψ N , using Lemma 3.2 we obtain 1 − ϕ 0 , γ N ϕ 0 = 1 − 1 N ψ N , a * (ϕ 0 )a(ϕ 0 )ψ N = 1 − 1 N U * N e B ξ N , a * (ϕ 0 )a(ϕ 0 )U * N e B ξ N = 1 N e B ξ N , N + e B ξ N ≤ C N ξ N , N + ξ N ≤ C(1 + K) N which concludes the proof of (1.7). Analysis of the excitation Hamiltonian R N In this section, we show Prop. 4.3, where we establish a lower bound for the operator R N,α = e −A G eff N,α e A , with G eff N,α as defined in (3.28) and with A = 1 √ N r,v∈Λ * + η r b * r+v a * −r a v − h.c. . (6.1) We decompose G eff N,α = O N + K + Z N + C N + V N (6.2) with K and V N as in (3.24), and with O N = 1 2 ω N (0)(N − 1) 1 − N + N + 2N V (0) − 1 2 ω N (0) N + 1 − N + N , Z N = 1 2 p∈Λ * + ω N (p)(b p b −p + h.c. ) C N = √ N p,q∈Λ * + :p+q =0 V (p/e N ) b * p+q a * −p a q + h.c. . (6.3) We will analyze the conjugation of all terms on the r.h.s. of (6.2) in Subsections 6.2-6.6. The estimates emerging from these subsections will then be combined in Subsection 6.6 to conclude the proof of Prop. 4.3. Throughout the section, we will need Prop. 4.2 to control the growth of the expectation of the energy H N = K + V N under the action of (6.1); the proof of Prop. 4.2 is contained in Subsection 6.1. In this section, we will always assume that V ∈ L 3 (R 2 ) is compactly supported, pointwise non-negative and spherically symmetric. A priori bounds on the energy In this section, we show Prop. 4.2. To this end, we will need the following proposition. Proposition 6.1. Let V N and A be defined in (3.24) and (4.1) respectively. Then, there exists a constant C > 0 such that [V N , A] = 1 N 1/2 u,r,v∈Λ * + u =−v V ((u − r)/e N )η r b * u+v a * −u a v + h.c. + δ VN where | ξ, δ VN ξ | ≤ C(log N ) 1/2 N 1/2−α H 1/2 N ξ 2 (6.4) for any α > 0, for all ξ ∈ F ≤N + , and N ∈ N large enough. Proof. We proceed as in [ 3, Prop. 8.1], computing [a * p+u a * q a p a q+u , b * r+v a * −r a v ]. We obtain [V N , A] = 1 N 1/2 * u∈Λ * ,r,v∈Λ * + V ((u − r)/e N )η r b * u+v a * −u a v + Θ 1 + Θ 2 + Θ 3 + h.c. with Θ 1 := 1 √ N * u∈Λ * r,p,v∈Λ * + V (u/e N )η r b * p+u a * r+v−u a * −r a p a v ,Θ 2 := 1 √ N * u∈Λ * p,r,v∈Λ * + V (u/e N )η r b * r+v a * p+u a * −r−u a p a v ,Θ 3 := − 1 √ N * u∈Λ * ,p,r,v∈Λ * + V (u/e N )η r b * r+v a * −r a * p+u a p a v+u . (6.5) and with * running over all momenta, except choices for which the argument of a creation or annihilation operator vanishes. We conclude that δ VN = Θ 1 + Θ 2 + Θ 3 + h.c. . Next, we show that each error term Θ j , with j = 1, 2, 3, satisfies (6.4). To bound Θ 1 we switch to position space and apply Cauchy-Schwarz. We find | ξ, Θ 1 ξ | ≤ 1 √ N Λ 2 dxdy e 2N V (e N (x − y)) ǎ(η y )ǎ yǎx ξ ǎ yǎx ξ ≤ C η Λ 2 dxdy e 2N V (e N (x − y)) ǎ yǎx ξ 2 ≤CN −α V 1/2 N ξ 2 , for any ξ ∈ F ≤N + The term Θ 3 can be controlled similarly. We find | ξ, Θ 3 ξ | = 1 √ N Λ 2 dxdy e 2N V (e N (x − y)) ξ,b * xǎ * (η x )ǎ * yǎ xǎy ξ ≤ CN −α V 1/2 N ξ 2 . It remains to bound the term Θ 2 on the r.h.s. of (6.5). Passing to position space we obtain, by Cauchy-Schwarz, | ξ,Θ 2 ξ | = 1 √ N Λ 3 dxdydz e 2N V (e N (y − z))η(x − z) ξ,b * xǎ * yǎ * zǎ xǎy ξ ≤ CN −1/2 Λ 3 dxdydz e 2N V (e N (y − z))|η(x − z)| ǎ xǎyǎz ξ ǎ xǎy ξ ≤ CN −1/2 V 1/2 N N 1/2 + ξ Λ 3 dxdydz e 2N V (e N (y − z))|η(x − z)| 2 ǎ xǎy ξ 2 1/2 , To bound the term in the square bracket, we write it in first quantized form and, for any 2 < q < ∞, we apply Hölder inequality and the Sobolev inequality u q ≤ C √ q u H 1 to estimate (denoting by 1 < q ′ < 2 the dual index to q), N n=2 n i<j e 2N V (e N ·) * |η| 2 (x i − x j ) |ξ (n) (x 1 , . . . , x n )| 2 dx 1 . . . dx n ≤ Cq e 2N V (e N ·) * |η| 2 q ′ × N n=2 n n i=1 |∇ xi ξ (n) (x 1 , . . . , x n )| 2 + |ξ (n) (x 1 , . . . , x n )| 2 dx 1 . . . dx n ≤ Cq η 2 2q ′ (K + N + ) 1/2 N 1/2 + ξ 2 . (6.6) With the bounds (3.11), (3.12), η 2 2q ′ ≤ η 2/q ′ 2 η 2(q ′ −1)/q ′ ∞ ≤ N −2α/q ′ N 2(q ′ −1)/q ′ we conclude that | ξ, Θ 2 ξ | ≤ Cq 1/2 N −1/2 N −α/q ′ N 1/q V 1/2 N N 1/2 + ξ (K + N + ) 1/2 N 1/2 + ξ ≤ Cq 1/2 N 1/2 N −α/q ′ N 1/q V 1/2 N ξ K 1/2 ξ for any 2 < q < ∞, if 1/q + 1/q ′ = 1. Choosing q = log N , we obtain that | ξ, Θ 2 ξ | ≤ C(log N ) 1/2 N 1/2−α H 1/2 N ξ 2 . Using Prop. 6 f ξ (s) := ξ, e −sA H N e sA ξ . We compute f ′ ξ (s) = ξ, e −sA [K, A]e sA ξ + ξ, e −sA [V N , A]e sA ξ . (6.7) With Prop. 6.1, we have [V N , A] = 1 √ N u,v∈Λ * + ,u =−v ( V (·/e N ) * η)(u) b * u+v a * −u a v + h.c. + δ VN with δ VN satisfying (6.4). Switching to position space and using Prop. 4.1 we find , using (3.11) to bound η ∞ ≤ CN , 1 √ N u,v∈Λ * + ( V (·/e N ) * η)(u) ξ, e −sA b * u+v a * −u a v e sA ξ = 1 √ N Λ 2 dxdy e 2N V (e N (x − y))η(x − y) ξ, e −sAǎ * xǎ * yǎ y e sA ξ ≤ N 1/2 Λ 2 dxdy e 2N V (e N (x − y)) ǎ xǎy e sA ξ 2 1/2 × Λ 2 dxdy e 2N V (e N (x − y)) ǎ y e sA ξ 2 1/2 ≤ CN 1/2 V 1/2 N e sA ξ N 1/2 + e sA ξ (6.8) Together with (6.4) we conclude that for any α > 1/2 ξ, e −sA [V N , A]e sA ξ ≤ C ξ, e −sA H N e sA ξ + CN ξ, e −sA (N + + 1)e sA ξ (6.9) if N is large enough. Next, we analyze the first term on the r.h.s. of (6.7). We compute [K, A] = 1 √ N r,v∈Λ * + 2r 2 η r b * r+v a * −r a v + h.c. + 2 √ N r,v∈Λ * + r · v η r b * r+v a * −r a v + h.c. =: T 1 + T 2 . (6.10) With (3.17), we write T 1 = − √ N r,v∈Λ * + r =−v ( V (·/e N ) * f N,ℓ )(r) b * r+v a * −r a v + h.c. + 2 √ N r,v∈Λ * + e 2N λ ℓ ( χ ℓ * f N,ℓ )(r) b * r+v a * −r a v + h.c. =: T 11 + T 12 . (6.11) The contribution of T 11 can be estimated similarly as in (6.8); switching to position space and using (3.6), we obtain ξ 1 , T 11 ξ 2 ≤ C √ N dxdye 2N V (e N (x − y))f ℓ (e N (x − y)) ǎ xǎy ξ a y ξ ≤ C √ N dxdye 2N V (e N (x − y)) ǎ xǎy ξ 2 1/2 × dxdye 2N V (e N (x − y))f ℓ (e N (x − y)) a y ξ 2 1/2 ≤ C V 1/2 N ξ N 1/2 + ξ . (6.12) for any ξ ∈ F ≤N + . The second term in (6.11) can be controlled using that for any ξ ∈ F ≤N + and 2 ≤ q < ∞ we have N 2α Λ 2 dxdy χ(|x − y| ≤ N −α ) ǎ xǎy ξ ǎ x ξ ≤ N 2α Λ 2 dx ǎ x ξ dy χ(|x − y| ≤ N −α ) 1−1/q dy ǎ xǎy ξ q 1/q ≤ CN 2α/q q 1/2 dx ǎ x ξ 2 1/2 dxdy ǎ x ∇ yǎy ξ 2 + dxdy ǎ xǎy ξ 2 1/2 ≤ CN 2α/q q 1/2 (N + + 1) 1/2 ξ K 1/2 (N + + 1) 1/2 ξ + (N + + 1)ξ . (6.13) Hence, choosing q = log N , ξ,T 12 ξ = √ N e 2N λ ℓ Λ 2 dxdy χ(|x − y| ≤ N −α )f N,ℓ (x − y) ξ,b * xǎ * yǎx ξ ≤ CN 2α−1/2 Λ 2 dxdy χ(|x − y| ≤ N −α ) ǎ xǎy ξ ǎ x ξ ≤ C(log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ + (N + + 1) 1/2 ξ ,(6.14) With (6.12) and (6.14) we conclude that | ξ, e −A T 1 e A ξ | ≤ C(log N ) 1/2 (H N + 1) 1/2 e sA ξ (N + + 1) 1/2 e sA ξ . (6.15) for all ξ ∈ F ≤N + . As for the second term on the r.h.s. of (6.10) we have ξ, T 2 ξ ≤ C √ N r∈Λ * + |r| 2 N 1/2 + a −r ξ 2 1/2 r,v∈Λ * + |v| 2 η 2 r a v ξ 2 1/2 ≤ CN −α K 1/2 ξ 2 . (6.16) for any ξ ∈ F ≤N + . With (6.15) and Prop. 4.1, we conclude that | ξ, e −sA [K, A]e sA ξ | ≤ C ξ, e −sA H N e sA ξ + C log N ξ, e −sA N + e sA ξ . Combining with Eq. (6.9) we obtain | ξ, e −sA [H N , A]e sA ξ | ≤ C ξ, e −sA H N e sA ξ + CN ξ, e −sA N + e sA ξ . With Prop. 4.1 we obtain the differential inequality |f ′ ξ (s)| ≤ Cf ξ (s) + CN ξ, (N + + 1)ξ . By Gronwall's Lemma, we find (4.2). Analysis of e −A O N e A In this section we study the contribution to R N,α arising from the operator O N , defined in (6.3). To this end, it is convenient to use the following lemma. Lemma 6.2. Let A be defined in (4.1). Then, there exists a constant C > 0 such that p∈Λ * + F p e −A a * p a p e A = p∈Λ * + F p a * p a p + E F where | ξ 1 , E F ξ 2 | ≤ CN −α F ∞ (N + + 1) 1/2 ξ 1 (N + + 1) 1/2 ξ 2 for all α > 0, ξ 1 , ξ 2 ∈ F ≤N + , F ∈ ℓ ∞ (Λ * + ), and N ∈ N large enough. Proof. The lemma is analogous to [3,Lemma 8.6]. We estimate p∈Λ * + F p ( ξ 1 , e −A a * p a p e A ξ 2 − ξ 1 , a * p a p ξ 2 ) = 1 0 ds p∈Λ * + F p ξ 1 , e −sA [a * p a p , A]e sA ξ 2 ≤ 1 √ N 1 0 ds r,v∈Λ * + |F r+v + F −r − F v ||η r || e sA ξ 1 , b * r+v a * −r a v e sA ξ 2 | ≤ C η F ∞ (N + + 1) 1/2 ξ 1 (N + + 1) 1/2 ξ 2 . where we used Prop. 4.1. We consider now the action of e A on the operator O N , as defined in (6.3). Proposition 6.3. Let A be defined in (4.1). Then there exists a constant C > 0 such that e −A O N e A = 1 2 ω N (0)(N − 1) 1 − N + N + 2N V (0)− 1 2 ω N (0) N + (1 − N + /N )+ δ ON where ±δ ON ≤ CN 1−α (N + + 1) for all α > 0, and N ∈ N large enough. Proof. The proof is very similar to [3,Prop. 8.7]. First of all, with Lemma 6.2 we can bound ± e −A 1 2 ω N (0)(N − 1) 1 − N + N + 2N V (0) − 1 2 ω N (0) N + e A − 1 2 ω N (0)(N − 1) 1 − N + N + 2N V (0) − 1 2 ω N (0) N + ≤ CN 1−α (N + + 1) . Moreover, for the contribution quadratic in N + , we can decompose ξ, e −A N 2 + e A − N 2 + ξ = ξ 1 , e −A N + e A − N + ξ + ξ, e −A N + e A − N + ξ 2 with ξ 1 = e −A N + e A ξ and ξ 2 = N + ξ, and estimate, again with Lemma 6.2, ξ, e −A N 2 + e A − N 2 + ξ ≤ CN −α (N + + 1) 1/2 ξ (N + + 1) 1/2 ξ 1 + (N + + 1) 1/2 ξ 2 . With Prop. 4.1, we have (N + + 1) 1/2 ξ 1 ≤ C (N + + 1) 3/2 ξ . Contributions from e −A Ke A In Section 6.6 we will analyse the contributions to R N,α arising from conjugation of the kinetic energy operator K = p∈Λ * + p 2 a * p a p . To this aim we will exploit properties of the commutator [K, A], collected in the following proposition. Proposition 6.4. Let A be defined as in (4.1) and ω N (r) be defined in (3.25). Then there exists a constant C > 0 such that [K, A] = − √ N p,q∈Λ * + ,p =−q ( V (·/e N ) * f N,ℓ )(p)(b * p+q a * −p a q + h.c. ) + 1 √ N p,q∈Λ * + ,p =−q ω N (p) b * p+q a * −p a q + h.c. + δ K where ξ, δ K ξ ≤ CN −1 (log N ) 1/2 K 1/2 ξ N 1/2 + ξ + CN −α K 1/2 ξ 2 (6.17) for all α > 1, ξ ∈ F ≤N + , and N ∈ N large enough. Moreover, the operator ∆ K = 1 √ N p,q∈Λ * + ,p =−q ω N (p) b * p+q a * −p a q , A satisfies ξ, ∆ K ξ ≤ CN −α (log N ) 1/2 K 1/2 ξ 2 + CN −1 (N + + 1) 1/2 ξ 2 (6.18) for all α > 1, ξ ∈ F ≤N + , and N ∈ N large enough. Proof. To show (6.17) we recall from Eqs. (6.10), (6.11) that [K, A] = − √ N r,v∈Λ * + r =−v ( V (·/e N ) * f N,ℓ )(r) b * r+v a * −r a v + h.c. + 2 √ N r,v∈Λ * + e 2N λ ℓ ( χ ℓ * f N,ℓ )(r) b * r+v a * −r a v + h.c. + 2 √ N r,v∈Λ * + r · v η r b * r+v a * −r a v + h.c. = T 11 + T 12 + T 2 . with T 2 satisfying (6.16). Using the definition ω N (p) = 2N e 2N λ ℓ χ ℓ (p) we write T 12 = 1 √ N p,q∈Λ * + ,p =−q ω N (p) b * p+q a * −p a q + h.c. + 2 √ N e 2N λ ℓ p,q∈Λ * + ,p =−q ( χ ℓ * η)(p) b * p+q a * −p a q + h.c. = T 121 + T 122 . Hence, δ K = T 2 + T 122 . To bound T 122 we switch to position space: | ξ, T 122 ξ | ≤ CN 2α−3/2 Λ 2 χ ℓ (x − y)η(x − y) ǎ xǎy ξ ǎ x ξ ≤ CN 2α−3/2 Λ 2 χ ℓ (x − y) ǎ xǎy ξ 2 dxdy 1/2 Λ 2 |η(x − y)| 2 ǎ x ξ 2 dxdy 1/2 ≤ CN α−3/2 N 1/2 + ξ Λ 2 χ ℓ (x − y) ǎ xǎy ξ 2 dxdy 1/2 . To bound the term in the parenthesis, we proceed similarly as in (6.6). We find Λ 2 χ ℓ (x − y) ǎ xǎy ξ 2 dxdy ≤ Cq χ ℓ q ′ K 1/2 N 1/2 + ξ 2 ≤ CqN 1−2α/q ′ K 1/2 ξ 2 for any q > 2 and 1 < q ′ < 2 with 1/q + 1/q ′ = 1. Choosing q = log N , we obtain | ξ, T 122 ξ | ≤ CN −1 (log N ) 1/2 N 1/2 + ξ K 1/2 ξ With (6.16), this implies (6.17). Let us now focus on (6.18). We have 1 √ N p,q∈Λ * + ,p =−q ω N (p) b * p+q a * −p a q , A = 1 N r,p,q,v∈Λ * + , p =−q,r =−v ω N (p)η r b * p+q a * −p a q , b * r+v a * −r a v − a * v a −r b r+v . With the commutators from the proof of Prop. 8.8 in [3], we arrive at 1 √ N p,q∈Λ * + ,p =−q ω N (p) b * p+q a * −p a q , A + h.c. = 12 j=1 Υ j + h.c. where Υ 1 := − 1 N q,r,v∈Λ * + , q =v,r =−v ω N (v − q) + ω N (v) η r b * r+v b * −r a * q−v a q ,Υ 2 := 1 N q,r,v∈Λ * + , r =−v,r =−q ω N (r + q)η r (1 − N + /N )a * v a * r+q a q a r+v ,Υ 3 := 1 N r,v∈Λ * + , r =−v ω N (r + v) + ω N (r) η r (1 − N + /N )a * v a v ,Υ 4 := 1 N q,r,v∈Λ * + , q =v,r =−v ω N (r + v − q)η r (1 − N + /N )a * v a * q−r−v a −r a q , Υ 5 := − 1 N 2 p,q,r,v∈Λ * + , p =−q,r =−v ω N (p)η r a * v a * p+q a * −p a −r a r+v a q ,Υ 6 := − 1 N 2 q,r,v∈Λ * + , q =r+v ω N (r + v)η r a * v a * q−r−v a −r a q ,Υ 7 := − 1 N 2 q,r,v∈Λ * + , q =−r,r =−v ω N (r)η r a * v a * q+r a r+v a q ,Υ 8 := 1 N r,v,p∈Λ * + , p =−r−v ω N (p)η r b * p+r+v b * −p a * −r a v ,Υ 9 := 1 N p,r,v∈Λ * + , p =r,r =−v ω N (p)η r b * p−r b * r+v a * −p a v ,(6.19) and Υ 10 := 1 N q,r,v∈Λ * + , q =−r,r =−v ω N (r)η r b * q+r a * v a q b r+v , Υ 11 := − 1 N p,r,v∈Λ * + , p =−v,r =−v ω N (p)η r b * p+v a * −p a −r b r+v , Υ 12 := 1 N q,r,v∈Λ * + r =q−v,−v ω N (r + v)η r b * q−r−v a * v a −r b q . (6.20) To conclude the proof of Prop. 6.4, we show that all operators in (6.19) and (6.20) satisfy (6.18). To study all these terms it is convenient to switch to position space. We recall that ω N (p) = g N χ(ℓp) with |g N | ≤ C and ℓ = N −α . Using (6.13) we find: ξ, Υ 1 ξ ≤ CN 2α−1 Λ 2 dxdy χ ℓ (x − y) b (η x )b xǎy ξ [ ǎ x ξ + ǎ y ξ ] ≤ CN 2α−1 η Λ 2 dxdy χ ℓ (x − y) b xǎy (N + + 1) 1/2 ξ ǎ x ξ ≤ CN −α (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ . The expectation of Υ 2 is bounded following the same strategy used to show (6.13). For any 2 ≤ q < ∞ we have ξ, Υ 2 ξ ≤ CN 2α−1 Λ 3 dxdydzχ ℓ (z − y)|η(z − x)| ǎ xǎy ξ ǎ zǎx ξ ≤ CN 2α−1 Λ 2 dxdz|η(z − x)| ǎ zǎx ξ × Λ dy χ(|z − y| ≤ N −α ) 1−1/q Λ dy ǎ xǎy ξ q 1/q ≤ Cq 1/2 N 2α/q−1 η (N + + 1)ξ Λ 2 dxdy ǎ x ∇ yǎy ξ 2 + Λ 2 dxdy ǎ xǎy ξ 2 1/2 ≤ CN −α (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ , where in the last line we chose q = log N . The term Υ 3 is of lower order; using that r ω N (r)η r ≤ χ(./N α ) 2 η 2 ≤ C and Cauchy-Schwarz, we easily obtain ξ, Υ 3 ξ ≤ CN −1 (N + + 1) 1/2 ξ 2 . The term Υ 4 can be estimated as Υ 1 using (6.13): ξ, Υ 4 ξ ≤ CN 2α−1 Λ 2 dxdy χ ℓ (x − y) ǎ xǎy ξ ǎ(η y )ǎ y ξ ≤ CN 2α−1 η Λ 2 dxdy χ ℓ (x − y) ǎ xǎy ξ ǎ y (N + + 1) 1/2 ξ ≤ CN −α (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ . The term Υ 5 is bounded similarly to Υ 2 ; with q = log N we have ξ, Υ 5 ξ ≤ CN 2α−2 η Λ 3 dxdydz χ ℓ (y − z) ǎ xǎyǎz ξ N 1/2 +ǎ xǎy ξ ≤ CN 2α−3/2 η Λ 2 dxdy ǎ xǎy ξ × Λ dz χ(|y − z| ≤ N −α ) 1−1/q Λ dz ǎ xǎyǎz ξ q 1/q ≤ CN −α (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ . The terms Υ 6 and Υ 7 are of smaller order and can be bounded with Cauchy-Schwarz; we have ξ,Υ 6 ξ ≤ CN 2α−2 Λ 2 dxdydz χ ℓ (x − y) ǎ xǎy ξ ǎ(η x )ǎ y ξ ≤ CN α−3/2 Λ 2 dxdy ǎ xǎy ξ 2 1/2 Λ 2 dxdy χ(|x − y| ≤ N −α ) ǎ y ξ 2 1/2 ≤ CN −1 (N + + 1) 1/2 ξ 2 , and ξ, Υ 7 ξ ≤ CN 2α−2 Λ 3 dxdydz χ ℓ (y − z)|η(z − x)| ǎ xǎy ξ 2 ≤ CN 2α−2 Λ 3 dxdydz χ ℓ (y − z) ǎ xǎy ξ 2 1/2 × Λ 3 dxdydz |η(z − x)| 2 ǎ xǎy ξ 2 1/2 ≤ CN −1 (N + + 1) 1/2 ξ 2 . The terms Υ 8 , Υ 11 , Υ 12 are again bounded, as Υ 1 , using (6.13). We find ξ, Υ 8 + Υ 11 + Υ 12 ξ ≤ CN 2α−1 η Λ 2 dxdy χ ℓ (x − y) N 1/2 +ǎ xǎy ξ ǎ x ξ ≤ CN −α (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ . It remains to bound Υ 9 and Υ 10 . The term Υ 9 is bounded analogously to Υ 2 : ξ, Υ 9 ξ ≤ CN 2α−1 Λ 3 dxdydz χ ℓ (x − z)|η(x − y)| ǎ xǎyǎz ξ ǎ y ξ ≤ CN 2α−1 Λ 2 dxdy |η(x − y)| ǎ y ξ Λ dz χ(|y − z| ≤ N −α ) 1−1/q × Λ dz ǎ xǎyǎz ξ q 1/q ≤ Cq 1/2 N 2α/q−1 Λ 2 dxdy |η(x − y)| 2 ǎ y ξ 2 1/2 Λ 3 dxdy ǎ xǎyǎz ξ 2 L q z 1/2 ≤ CN −α (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ . As for Υ 10 , we find ξ, Υ 10 ξ ≤ CN 2α−1 Λ 3 dxdydz χ ℓ (y − z)|η(x − z)| ǎ xǎy ξ 2 Proceeding as in (6.6), we obtain ξ, Υ 10 ξ ≤ CqN 2α χ ℓ * |η| q ′ K 1/2 ξ 2 ≤ Cq η q ′ K 1/2 ξ 2 for any q > 2, and q ′ < 2 with 1/q + 1/q ′ = 1. Since, for an arbitrary q ′ < 2, η q ′ ≤ η 2 = η 2 ≤ N −α , we obtain ξ, Υ 10 ξ ≤ CN −α K 1/2 ξ 2 We conclude that for any α > 1 ξ, 12 j=1 Υ i ξ ≤ CN −α (log N ) 1/2 (K + 1) 1/2 ξ 2 + CN −1 (N + + 1) 1/2 ξ 2 . 6.4 Analysis of e −A Z N e A In this subsection, we consider contributions to R N,α arising from conjugation of Z N , as defined in (6.3). Proposition 6.5. Let A be defined in (4.1). Then, there exists a constant C > 0 such that e A Z N e −A = 1 2 p∈Λ * + ω N (p) b * p b * −p + b p b −p + δ ZN where ±δ ZN ≤ CN 1−α (H N + 1) for all α > 0, and N ∈ N large enough. Proof. We have 1 2 p∈Λ * + ω N (p) e −A b * p b * −p + b p b −p e A − b * p b * −p + b p b −p = 1 2 1 0 ds p∈Λ * + ω N (p) e −sA b * p b * −p + b p b −p , A e sA . (6.21) We compute 1 2 p∈Λ * + ω N (p) b * p b * −p , b * r+v a * −r a v − a * v a −r b r+v = − ω N (v)b * r+v b * −v b * −r + ω N (r)b * v b * r b r+v − 2 N a * r a r+v + ω N (r + v) 1 − N + N b * −r−v a * v a −r − 1 N p∈Λ * ω N (p)b * p a * −p a * v a −r a r+v . (6.22) With (6.22) we write 1 2 p∈Λ * ω N (p) b * p b * −p + b p b −p , A = with Π 1 = − 1 √ N r,v∈Λ * + r =−v ω N (v)η r b * r+v b * −v b * −r , Π 2 = 1 √ N r,v∈Λ * + : r =−v ω N (r)η r b * v b * r b r+v − 2 N a * r a r+v , Π 3 = 1 √ N r,v∈Λ * + r =−v ω N (r + v)η r 1 − N + N b * −r−v a * v a −r , Π 4 = − 1 N 3/2 r,v,p∈Λ * + : r =−v ω N (p)η r b * p a * −p a * v a −r a r+v . To bound the first term, we observe, with (4.9), | ξ, Π 1 ξ | ≤ η √ N K 1/2 N 1/2 + ξ (N + + 1) 1/2 ξ   v∈Λ * + | ω N (v)| 2 v 2   1/2 ≤ CN −α (log N ) 1/2 K 1/2 ξ (N + + 1) 1/2 ξ . The term Π 3 can be bounded similarly to Π 1 , with (4.9). We find ξ, Π 3 ξ ≤ CN −α (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ . With | ω N (r)| ≤ C, we similarly obtain | ξ, Π 2 ξ | ≤ N −1/2 η K 1/2 N 1/2 + ξ (N + + 1) 1/2 ξ ≤ CN −α K 1/2 ξ (N + + 1) 1/2 ξ . Finally, we estimate, using again (4.9), ξ, Π 4 ξ ≤ N −3/2 r,v,p∈Λ * + p 2 |η r | 2 a −p a v (N + + 1) 1/2 ξ 2 1/2 × r,v,p∈Λ * + | ω N (p)| 2 p 2 a −r a r+v ξ 2 1/2 ≤ CN −3/2 η (log N ) 1/2 K 1/2 (N + + 1)ξ (N + + 1)ξ ≤ CN −α (log N ) 1/2 K 1/2 ξ (N + + 1) 1/2 ξ . With (6.21), we conclude that 1 2 p∈Λ * ω N (p) ξ, e −A b * p b * −p + b p b −p e A ξ − ξ, b * p b * −p + b p b −p ξ ≤ CN −α (log N ) 1/2 1 0 ds K 1/2 e sA ξ (N + + 1) 1/2 e sA ξ . With Prop. 4.1, Lemma 4.2, we conclude that 1 2 p∈Λ * ω N (p) ξ, e −A b * p b * −p + b p b −p e A ξ − ξ, b * p b * −p + b p b −p ξ ≤ CN −α (log N ) 1/2 H 1/2 N ξ + N 1/2 N 1/2 + ξ (N + + 1) 1/2 ξ ≤ CN 1−α (H N + 1) 1/2 ξ 2 . 6.5 Contributions from e −A C N e A In Section 6.6 we will analyse the contributions to R N,α arising from conjugation of the cubic operator C N defined in (6.3). To this aim we will need some properties of the commutator [C N , A], as established in the following proposition. Proposition 6.6. Let A be defined in (4.1). Then, there exists a constant C > 0 such that C N , A = 2 r,v∈Λ * + V (r/e N )η r + V ((r + v)/e N )η r a * v a v 1 − N + N + δ CN where | ξ, δ CN ξ | ≤ CN 3/2−α H 1/2 N ξ (N + + 1) 1/2 ξ (6.23) for all α > 0, ξ ∈ F ≤N + , and N ∈ N large enough. Proof. We consider the commutator C N , A = p,q∈Λ * + :p+q =0 r,v∈Λ * + V (p/e N )η r b * p+q a * −p a q , b * r+v a * −r a v − a * v a −r b r+v + h.c. . As in the proof of Prop. 6.4, we use the commutators from the proof of Prop. 8.8 in [3] to conclude that C N , A = 2 r,v∈Λ * + V (r/e N )η r + V ((r + v)/e N )η r a * v a v N − N + N + 12 j=1 (Ξ j + h.c. ) where Ξ 1 := − r,v,p∈Λ * + , p =v V (p/e N )η r b * r+v b * −r a * −p a v−p , Ξ 2 := r,v,p∈Λ * + r =−p V (p/e N )η r (1 − N + /N )a * v a * −p a −r−p a r+v ,Ξ 3 := r,v,p∈Λ * + : r+v =p V (p/e N )η r (1 − N + /N )a * v a * −p a −r a r+v−p , as well as Ξ 4 := − 1 N r,v,p,q∈Λ * + :p+q =0 V (p/e N )η r a * v a * p+q a * −p a −r a r+v a q ,Ξ 5 := − 1 N r,v,q∈Λ * + : r+v =q V ((r + v)/e N )η r a * v a * q−r−v a −r a q ,Ξ 6 := − 1 N r,v,q∈Λ * + : r =−q V (r/e N )η r a * v a * q+r a r+v a q Ξ 7 := r,v,p∈Λ * + : r+v =−p V (p/e N )η r b * p+r+v b * −p a * −r a v ,Ξ 8 := r,v,p∈Λ * + : r =−p V (p/e N )η r b * p−r b * r+v a * −p a v ,Ξ 9 := − r,v,q∈Λ * + : q =v V (v/e N )η r b * q−v b * r+v a * −r a q , Ξ 10 := r,v,q∈Λ * + : r =−q V (r/e N )η r b * q+r a * v a q b r+v , Ξ 11 := − r,v,p∈Λ * + : p =−v V (p/e N )η r b * p+v a * −p a −r b r+v , Ξ 12 := r,v,q∈Λ * + : q =r+v V ((r + v)/e N )η r b * q−r−v a * v a −r b q . To prove the proposition, we have to show that all terms Ξ j , j = 1, . . . , 12, satisfy the bound (6.23). We bound Ξ 1 in position space, with Cauchy-Schwarz, by ξ, Ξ 1 ξ ≤ C Λ 3 dxdydze 2N V (e N (x − y))|η(x − z)| ǎ x ξ ǎ xǎyǎz ξ ≤ C Λ 3 dxdydz e 2N V (e N (x − y)) ǎ xǎyǎz ξ 2 1/2 × Λ 3 dxdydz e 2N V (e N (x − y))|η(x − z)| 2 ǎ x ξ 2 1/2 ≤ C η (N + + 1) 1/2 ξ V 1/2 N N 1/2 + ξ ≤ CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . We can proceed similarly to control Ξ 9 . We obtain ξ, Ξ 9 ξ ≤ CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . The expectations of the terms Ξ 3 and Ξ 12 can be bounded analogously: ξ, Ξ 3 ξ + ξ, Ξ 12 ξ ≤ C Λ 3 dxdydz e 2N V (e N (x − y))(|η(x − z)| + |η(y − z)|) ǎ xǎy ξ ǎ xǎz ξ ≤ C Λ 3 dxdydz e 2N V (e N (x − y)) ǎ xǎy ξ 2 (|η(x − z)| 2 + |η(y − z)| 2 ) 1/2 × Λ 3 dxdydz e 2N V (e N (x − y)) ǎ xǎz ξ 2 1/2 ≤ C η (N + + 1)ξ V 1/2 N ξ ≤ CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . As for Ξ 4 , we find | ξ, Ξ 4 ξ | = 1 N Λ 2 dxdydz e 2N V (e N (y − z)) ξ,ǎ * xǎ * yǎ * zǎ (η x )ǎ xǎy ξ ≤ CN −1 η Λ 2 dxdydz e 2N V (e N (y − z)) ǎ xǎyǎz ξ N 1/2 +ǎxǎy ξ ≤ CN −1 η Λ 2 dxdydz e 2N V (e N (y − z)) ǎ xǎyǎz ξ 2 1/2 × Λ 2 dxdydz e 2N V (e N (y − z)) N 1/2 +ǎ xǎy ξ 2 1/2 ≤ CN 1/2−α V 1/2 N ξ N 1/2 + ξ . The terms Ξ 5 and Ξ 6 can be bounded in momentum space, using (A.51). Hence, | ξ, Ξ 5 ξ | + | ξ, Ξ 6 ξ | ≤ CN −1 r,v,q∈Λ * + V ((v + r)/e N ) |v| |η r ||v| a v a q−r−v ξ a −r a q ξ + V (r/e N ) |r + v| |η r ||r + v| a r+q a v ξ a q a r+v ξ ≤ CN 1/2−α (N + + 1) 1/2 ξ K 1/2 ξ . Similarly we have | ξ, Ξ 2 ξ | + | ξ, Ξ 10 ξ | ≤ r,v,p∈Λ * + V (p/e N ) |p| |η r ||p| a v a −p ξ a r+v a −r−p ξ + V (r/e N ) |r + v| |η r ||r + v| a q a r+v ξ a r+q a v ξ ≤ CN 3/2−α (N + + 1) 1/2 ξ K 1/2 ξ . Next, we rewrite Ξ 7 , Ξ 8 and Ξ 11 as Ξ 7 = Λ 2 dxdy e 2N V (e N (x − y))b * xb * y a * (η x )ǎ x , Ξ 8 = Λ 2 dxdydz e 2N V (e N (x − y))η(z − x)b * xb * zǎ * yǎz , Ξ 11 = − Λ 2 dxdy e 2N V (e N (x − y))b * xǎ * yǎ (η x )b x . Thus, we obtain | ξ, Ξ 7 ξ | ≤ C η Λ 2 dxdy e 2N V (e N (x − y)) N 1/2 +ǎ xǎy ξ ǎ x ξ ≤ C η N 1/2 + V 1/2 N ξ N 1/2 + ξ ≤ CN 1/2−α V 1/2 N ξ N 1/2 + ξ , as well as | ξ, Ξ 8 ξ | ≤ C Λ 2 dxdydz e 2N V (e N (x − y))|η(x − z)| ǎ xǎyǎz ξ ǎ z ξ ≤ C Λ 2 dxdydz e 2N V (e N (x − y)) ǎ xǎyǎz ξ 2 1/2 × Λ 2 dxdydz e 2N V (e N (x − y))|η(x − z)| 2 ǎ z ξ 2 1/2 ≤ CN 1/2−α V 1/2 N ξ N 1/2 + ξ , and | ξ, Ξ 11 ξ | ≤ C η Λ 2 dxdy e 2N V (e N (x − y)) ǎ xǎy ξ N 1/2 +ǎx ξ ≤ C η V 1/2 N ξ N + ξ ≤ CN 1/2−α V 1/2 N ξ N 1/2 + ξ . Collecting all the bounds above, we arrive at (6.23). Proof of Proposition 4.3 With the results of Sections 6.1-6.5, we can now show Proposition 4.3. We assume α > 2. From Eq. (6.2), Prop. 6.3 and Prop. 6.5 we obtain that R N,α = e −A G eff N,α e A = 1 2 ω N (0)(N − 1)(1 − N + /N ) + 2N V (0) − 1 2 ω N (0) N + (1 − N + /N ) + 1 2 p∈Λ * + ω N (p) b * p b * −p + b p b −p + K + C N + V N + 1 0 ds e −sA K + C N + V N , A e sA + E (1) R with ±E (1) R ≤ CN 1−α (H N + 1) . From Prop. 6.1, Prop. 6.4 and Prop. 6.6, we can write, for N large enough, [K + C N + V N , A = 1 √ N r,v∈Λ * + ω N (r) b * r+v a * −r a v + h.c. − √ N r,v,∈Λ * + , p =−q V (r/e N ) b * r+v a * −r a v + h.c. + 2 r,v∈Λ * + V (r/e N )η r + V ((r + v)/e N )η r a * v a v (1 − N + /N ) + E (2) R where | ξ, E(2)R ξ | ≤CN 1/2−α (log N ) 1/2 H 1/2 N ξ 2 + CN 3/2−α H 1/2 N ξ (N + + 1) 1/2 ξ + CN −1 (log N ) 1/2 H 1/2 N ξ (N + + 1) 1/2 ξ .+ C N + V N , A e sA = 1 0 ds e −sA − C N + 1 √ N r,v∈Λ * + ω N (r) b * r+v a * −r a v + h.c. + 2 r,v∈Λ * + V (r/e N )η r + V ((r + v)/e N )η r a * v a v 1 − N + N e sA + E (3) R (6.24) with ±E (3) R ≤ C[N 2−α + N −1/2 (log N ) 1/2 ](H N + 1) for N ∈ N sufficiently large. We now rewrite 2 r,v∈Λ * + V (r/e N )η r + V ((r + v)/e N )η r a * v a v 1 − N + N = 4 r,v∈Λ * + V (r/e N )η r a * v a v 1 − N + N + 2 r,v∈Λ * + V ((r + v)/e N ) − V (r/e N ) η r a * v a v 1 − N + N := Q 1 + Q 2 . (6.25) With Lemma 3.1, part iii) we get 2 r∈Λ * V (r/e N )η r − 2 ω N (0) − 2N V (0) ≤ C N ,(6.26) and therefore, using Lemma 6.2 and (6.26) ± e −sA Q 1 e sA − 2 2 ω N (0) − 2N V (0) v∈Λ * + a * v a v 1 − N + N ≤ CN 1−α (N + + 1) + C N N + . (6.27) On the other hand it is easy to check that e −sA Q 2 e sA is an error term; to this aim we notice that r∈Λ * V (r/e N )η r − V ((r + v)/e N )η r ≤ CN |v|e −N . Hence with Props ± 1 √ N 1 0 ds r,v∈Λ * + ω N (r) e −sA b * r+v a * −r a v e sA − b * r+v a * −r a v + h.c. = ± 1 √ N 1 0 ds s 0 dt r,v∈Λ * + ω N (r)e −tA b * r+v a * −r a v , A e tA ≤ C 1 0 ds s 0 dt e −tA N −α (log N ) K + N −1 (N + + 1) e tA ≤ CN 1−α log N (H N + 1) . (6.29) As for the first term on the second line of (6.24), we use again Prop. 6.6. Using (6.25), (6.27) and (6.28) we have Inserting the bounds (6.27), (6.28), (6.29) and (6.30) into (6.24) we arrive at R N,α = 1 2 (N − 1) ω N (0)(1 − N + /N ) + 1 2 ω N (0) N + (1 − N + /N ) + ω N (0) p∈Λ * + a * p a p 1 − N + N + 1 2 p∈Λ * + ω N (p) b * p b * −p + b p b −p + 1 √ N r,v∈Λ * + : r =−v ω N (r) b * r+v a * −r a v + h.c. + H N + E R with ±E R ≤ C[N 2−α + N −1/2 (log N ) 1/2 ](H N + 1) for N ∈ N sufficiently large. The analysis in this section follows closely that of [3, Section 7] with some slight modifications due to the different scaling of the interaction potential and the fact that the kernel η p of e B is different from zero for all p ∈ Λ * + (in [3] η p is different from zero only for momenta larger than a sufficiently large cutoff of order one). Moreover, while in three dimensions it was sufficient to choose the function η p appearing in the generalized Bogoliubov transformation with η sufficiently small but of order one, we need here η to be of order N −α for some α > 0 large enough. As discussed in the introduction this is achieved by considering the Neumann problem for the scattering equation in (3.2) on a ball of radius ℓ = N −α ; as a consequence some terms depending on ℓ will be large, compared to the analogous terms in [3]. A.1 Generalized Bogoliubov transformations In this subsection we collect important properties about the action of unitary operators of the form e B , as defined in (3.20). As shown in [2, Lemma 2.5 and 2.6], we have, if η is sufficiently small, e −B b p e B = ∞ n=0 (−1) n n! ad (n) B (b p ) e −B b * p e B = ∞ n=0 (−1) n n! ad (n) B (b * p ) (A.1) where the series converge absolutely. To confirm the expectation that generalized Bogoliubov transformation act similarly to standard Bogoliubov transformations, on states with few excitations, we define (for η small enough) the remainder operators d q = m≥0 1 m! ad (m) −B (b q ) − η m q b ♯m αmq , d * q = m≥0 1 m! ad (m) −B (b * q ) − η m q b ♯m+1 αmq (A.2) where q ∈ Λ * + , (♯ m , α m ) = (·, +1) if m is even and (♯ m , α m ) = ( * , −1) if m is odd. It follows then from (A.1) that e −B b q e B = γ q b q + σ q b * −q + d q , e −B b * q e B = γ q b * q + σ q b −q + d * q (A.3) where we introduced the notation γ q = cosh(η q ) and σ q = sinh(η q ). It will also be useful to introduce remainder operators in position space. For x ∈ Λ, we define the operator valued distributionsď x ,ď * x through e −Bb x e B = b(γ x ) + b * (σ x ) +ď x , e −Bb * x e B = b * (γ x ) + b(σ x ) +ď * x (A.4) whereγ x (y) = q∈Λ * cosh(η q )e iq·(x−y) andσ x (y) = q∈Λ * sinh(η q )e iq·(x−y) . The next lemma is taken from [3, Lemma 3.4]. Lemma A.1. Let η ∈ ℓ 2 (Λ * + ), n ∈ Z. For p ∈ Λ * + , let d p be defined as in (A.3). If η is small enough, there exists C > 0 such that (N + + 1) n/2 d p ξ ≤ C N |η p | (N + + 1) (n+3)/2 ξ + η b p (N + + 1) (n+2)/2 ξ , (N + + 1) n/2 d * p ξ ≤ C N η (N + + 1) (n+3)/2 ξ (A.5) for all p ∈ Λ * + , ξ ∈ F ≤N + . In position space, withď x defined as in (A.4), we find (N + + 1) n/2ď x ξ ≤ C N η (N + + 1) (n+3)/2 ξ + b x (N + + 1) (n+2)/2 ξ . (A.6) Furthermore, lettingď x =ď x + (N + /N )b * (η x ), we find (N + + 1) n/2ǎ yďx ξ ≤ C N η 2 (N + + 1) (n+2)/2 ξ + η |η(x − y)| (N + 1) (n+2)/2 ξ + η ǎ x (N + + 1) (n+1)/2 ξ + η 2 ǎ y (N + + 1) (n+3)/2 ξ + η ǎ xǎy (N + 1) (n+2)/2 ξ (A.7) and, finally, (N + + 1) n/2ď xďy ξ ≤ C N 2 η 2 (N + + 1) (n+6)/2 ξ + η |η(x − y)| (N + + 1) (n+4)/2 ξ + η 2 a x (N + + 1) (n+5)/2 ξ + η 2 a y (N + + 1) (n+5)/2 ξ + η 2 a x a y (N + + 1) (n+4)/2 ξ (A.8) for all ξ ∈ F ≤n + . A first simple application of Lemma A.1 is the following bound on the growth of the expectation of N + . Lemma A.2. Assume B is defined as in (3.19), with η ∈ ℓ 2 (Λ * ) and η p = η −p for all p ∈ Λ * + . Then, there exists a constant C > 0 such that ξ, e −B N + e B − N + ξ ≤ η (N + + 1) 1/2 ξ 2 for all ξ ∈ F ≤N + . Proof. With (A.3) we write e −B N + e B − N + = 1 0 e −sB [N + , B]e sB ds = 1 0 p∈Λ * + η p e −sB (b p b −p + b * p b * −p )e sB ds = 1 0 p∈Λ * + η p (γ (s) p b p + σ (s) p b * −p + d (s) p )(γ (s) p b −p + σ (s) p b * −p + d (s) −p ) + h.c. ds with γ (s) p = cosh(sη p ), σ (s) p = sinh(sη p ). Using |γ (s) p | ≤ C and |σ (s) p | ≤ C|η p |, (A.5) in Lemma A.1 we arrive at ξ, e −B N + e B − N + ξ ≤ C (N + + 1) 1/2 ξ p∈Λ * + |η p | |η p | (N + + 1) 1/2 ξ + b p ξ ≤ C η (N + + 1) 1/2 ξ 2 A.2 Analysis of G (0) N,α = e −B L (0) N e B We define E (0) N so that G (0) N,α = e −B L (0) N e B = 1 2 V (0)(N + N + − 1)(N − N + ) + E (0) N,α . where we recall from (2.4) that We write L (0) N = 1 2 V (0)(N − 1 + N + )(N − N + ) .L (0) N = N (N − 1) 2 V (0) + N 2 V (0) q∈Λ * + b * q b q − N + . Hence, E (0) N = N 2 V (0) q∈Λ * + e −B b * q b q e B − b * q b q − N 2 V (0) e −B N + e B − N + . To bound the first term we use (A.3), |γ 2 q − 1| ≤ Cη 2 q , |σ q | ≤ C|η q |, the first bound in (A.5), Cauchy-Schwarz and the estimate η ≤ CN −α . To bound the second term, we use Lemma A.2. We conclude that | ξ, E (0) N ξ | ≤ CN 1−α (N + + 1) 1/2 ξ 2 . A.3 Analysis of G e −B Ke B = K + p∈Λ * + p 2 η p (b p b −p + b * p b * −p ) + p∈Λ * + p 2 η 2 p N − N + N N − N + − 1 N + E (K) N,α (A.9) where | ξ, E (K) N,α ξ | ≤ CN 1/2−α H 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 (A.10) for any α > 1, ξ ∈ F ≤N + and N ∈ N large enough. Proof. We proceed as in the proof of [3,Prop. 7.2]. We write e −B Ke B − K = 1 0 ds p∈Λ * + p 2 η p γ (s) p b p + σ (s) p b * −p γ (s) p b −p + σ (s) p b * p + h.c. + 1 0 ds p∈Λ * + p 2 η p γ (s) p b p + σ (s) p b * −p d (s) −p + d (s) p γ (s) p b −p + σ (s) p b * p + h.c. + 1 0 ds p∈Λ * + p 2 η p d (s) p d (s) −p + h.c. =: G 1 + G 2 + G 3 (A.11) with γ (s) p = cosh(sη p ), σ (s) p = sinh(sη p ) and where d (s) p is defined as in (A.2), with η p replaced by sη p . We find G 1 = p∈Λ * + p 2 η p b p b −p + b * −p b * p + p∈Λ * + p 2 η 2 p 1 − N + N + E K 1 with E K 1 = 2 1 0 ds p∈Λ * + p 2 η p (σ (s) p ) 2 b p b −p + b * −p b * p + 1 0 ds p∈Λ * + p 2 η p γ (s) p σ (s) p (4b * p b p − 2N −1 a * p a p ) + 2 1 0 ds p∈Λ * + p 2 η p (γ (s) p − 1)σ (s) p + (σ (s) p − sη p ) 1 − N + N . Since | (γ (s) p ) 2 − 1 | ≤ Cη 2 p , (σ (s) p ) 2 ≤ Cη 2 p , p 2 |η p | ≤ C, η ∞ ≤ N −α , we can estimate | ξ,E K 1 ξ | ≤ C p∈Λ * + p 2 |η p | 3 b p ξ (N + + 1) 1/2 ξ + C p∈Λ * + p 2 η 2 p a p ξ 2 + C p∈Λ * + p 2 η 4 p ξ 2 ≤ C η (N + + 1) 1/2 ξ 2 ≤ CN −α (N + + 1) 1/2 ξ 2 , (A.12) for any ξ ∈ F ≤N + . To bound the term G 3 in (A.11), we switch to position space: | ξ, G 3 ξ | ≤ CN 1 0 ds Λ 2 dxdy e 2N V (e N (x − y)) + N 2α−1 χ(|x − y| ≤ N −α ) × (N + + 1) −1/2ď(s) xď (s) y ξ (N + + 1) 1/2 ξ With (A.8), we obtain | ξ, G 3 ξ | ≤ CN 1−α Λ 2 dxdy e 2N V (e N (x − y)) + N 2α−1 χ(|x − y| ≤ N −α ) (N + + 1) 1/2 ξ 2 + CN −2α Λ 2 dxdy e 2N V (e N (x − y)) + N 2α−1 χ(|x − y| ≤ N −α ) (N + + 1) 1/2 ξ × ǎ x (N + + 1)ξ + ǎ y (N + + 1)ξ + ǎ xǎy (N + + 1) 1/2 ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . (A.13) Finally, we consider G 2 in (A.11). We split it as G 2 = G 21 + G 22 + G 23 + G 24 , with G 21 = 1 0 ds p∈Λ * + p 2 η p γ (s) p b p d (s) −p + h.c. , G 22 = 1 0 ds p∈Λ * + p 2 η p σ (s) p b * −p d (s) −p + h.c. G 23 = 1 0 ds p∈Λ * + p 2 η p γ (s) p d (s) p b −p + h.c. , G 24 = 1 0 ds p∈Λ * + p 2 η p σ (s) p d (s) p b * p + h.c. . (A.14) We consider G 21 first. We write G 21 = − p∈Λ * + p 2 η 2 p N + + 1 N N − N + N + E K 2 + h.c. where E K 2 = 3 j=1 E K 2j , with E K 21 = 1 2N p∈Λ * + p 2 η 2 p (N + + 1) b * p b p − 1 N a * p a p , E K 22 = 1 0 ds p∈Λ * + p 2 η p (γ (s) p − 1)b p d (s) −p , E K 23 = 1 0 ds p∈Λ * + p 2 η p b p d (s) −p . (A. 15) and where we introduced the notation d (s) −p = d (s) −p + sη p (N + /N )b * p . With (3.15), we find | ξ, E K 21 ξ | ≤ C p∈Λ * + η p a p ξ 2 ≤ CN −α N 1/2 + ξ 2 (A.16) Using |γ (s) p − 1| ≤ Cη 2 p and (A.5), we obtain | ξ, E K 22 ξ | ≤ p∈Λ * + p 2 |η p | 3 N 1/2 + ξ d (s) −p ξ ≤ CN −3α (N + + 1) 1/2 ξ 2 . (A.17) To control the third term in (A.15), we use (3.16) and we switch to position space. We find E K 23 = − N 1 0 ds Λ 2 dxdy e 2N V (e N (x − y))f N,ℓ (x − y)b xď (s) y + N 1 0 dse 2N λ ℓ Λ 2 dxdy χ ℓ (x − y)f N,ℓ (x − y)b xď (s) y = E K 231 + E K 232 . (A.18) With (A.7) and |η(x − y)| ≤ CN , we obtain | ξ, E K 231 ξ | ≤ N 1 0 ds Λ 2 dxdy e 2N V (e N (x − y)) × (N + + 1) 1/2 ξ (N + + 1) −1/2ǎ xď (s) y ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . (A.19) As for E K 232 , with (A.7) and Lemma 3.1 (recalling ℓ = N −α ), we find | ξ, E K 232 ξ | ≤ CN −α (N + + 1) 1/2 ξ 2 + Λ 2 dxdy χ(|x − y| ≤ N −α ) (N + + 1) 1/2 ξ ǎ xǎy N 1/2 + ξ (A.20) To bound the last term on the r.h.s. of (A.20) we use Hölder's and Sobolev inequality u q ≤ Cq 1/2 u H 1 , valid for any 2 ≤ q < ∞. We find Λ 2 dxdy χ(|x − y| ≤ N −α ) (N + + 1) 1/2 ξ ǎ xǎy N 1/2 + ξ ≤ C (N + + 1) 1/2 ξ Λ dx Λ dy χ(|x − y| ≤ N −α ) 1−1/q Λ dy ǎ xǎy N 1/2 + ξ q 1/q ≤ CN 2α/q−2α (N + + 1) 1/2 ξ Λ dx Λ dy ǎ xǎy N 1/2 + ξ q 1/q ≤ Cq 1/2 N 2α/q−2α (N + + 1) 1/2 ξ × Λ 2 dxdy ǎ x ∇ yǎy N 1/2 + ξ 2 + Λ 2 dxdy ǎ xǎy N 1/2 + ξ 2 1/2 ≤ Cq 1/2 N 2α/q−2α (N + + 1) 1/2 ξ K 1/2 N + ξ + N 3/2 + ξ . Choosing q = log N , we get Λ 2 dxdy χ(|x − y| ≤ N −α ) (N + + 1) 1/2 ξ ǎ xǎy (N + + 1) 1/2 ξ ≤ CN 1−2α (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ . (A.21) Therefore, for any ξ ∈ F ≤N + , | ξ, E K 232 ξ | ≤ N 1−2α (log N ) 1/2 K 1/2 ξ (N + + 1) 1/2 ξ + N −α (N + + 1) 1/2 ξ 2 . Combining the last bound with (A.16), (A.17) and (A.19), we conclude that | ξ, E K 2 ξ | ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α H 1/2 N ξ (N + + 1) 1/2 ξ . (A.22) for any α > 1, N ∈ N large enough, ξ ∈ F ≤N + . The term G 22 in (A.14) can be bounded using (A.5). We find | ξ, G 22 ξ | ≤ CN −2α (N + + 1) 1/2 ξ 2 . (A.23) We split G 23 = E K 31 + E K 32 + h.c. , with E K 31 = 1 0 ds p∈Λ * + p 2 η p γ (s) p − 1 d (s) p b −p , E K 32 = 1 0 ds p∈Λ * + p 2 η p d (s) p b −p With (A.5), we find | ξ, E K 31 ξ | ≤ C 1 0 ds p∈Λ * + p 2 |η p | 3 (d (s) p ) * ξ b −p ξ ds ≤ CN −3α (N + + 1) 1/2 ξ 2 To estimate E K 32 , we use (3.16) and we switch to position space. Proceeding as we did in (A.18), (A.19), (A.20), we obtain | ξ, E K 32 ξ | ≤ CN 1 0 ds Λ 2 dxdy e 2N V (e N (x − y)) + N 2α−1 χ(|x − y| ≤ N −α ) × (N + + 1) 1/2 ξ (N + + 1) −1/2ď(s) xby ξ . With (A.6) and (A.21) we find | ξ, E K 32 ξ | ≤ CN −α Λ 2 dxdy e 2N V (e N (x − y)) + N 2α−1 χ(|x − y| ≤ N −α ) × (N + + 1) 1/2 ξ ǎ y (N + + 1)ξ + ǎ xǎy (N + + 1) 1/2 ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ + CN 1−2α (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ . Combining the bounds for E K 31 and E K 32 , we conclude that, if α > 1, | ξ, G 23 ξ | ≤ CN 1/2−α (N + + 1) 1/2 ξ H 1/2 N ξ + CN 1−α (N + + 1) 1/2 ξ 2 (A.24) To bound G 24 in (A.14), we use (A.5), the bounds (3.14) and η 2 H1 ≤ CN , and the commutator (2.5): | ξ, G 24 ξ | ≤ C 1 0 ds p∈Λ * + p 2 η 2 p (N + + 1) 1/2 ξ (N + + 1) −1/2 d (s) p b * p ξ ≤ C (N + + 1) 1/2 ξ p∈Λ * + p 2 η 2 p |η p | (N + + 1) 1/2 ξ + N −1 η b p b * p (N + + 1) 1/2 ξ ≤ CN −α (N + + 1) 1/2 ξ 2 . Together with (A.14), (A.22), (A.23) and (A.24), this implies that In the next proposition, we consider the conjugation of the operator G 2 = − p∈Λ * + p 2 η 2 p N + + 1 N N − N + N + E K 4 with | ξ, E K 4 ξ | ≤ CN 1/2−α H 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1L (2,V ) N = N p∈Λ * + V (p/e N ) b * p b p − 1 N a * p a p + N 2 p∈Λ * + V (p/e N ) b * p b * −p + b p b −p Proposition A.5. Under the assumptions of Prop. 3.3, there is a constant C > 0 such that e −B L (2,V ) N e B = N p∈Λ * + V (p/e N )η p N − N + N N − N + − 1 N + N p∈Λ * + V (p/e N )a * p a p 1 − N + N + N 2 p∈Λ * + V (p/e N ) b p b −p + b * −p b * p + E (V ) N (A.26) where | ξ, E (V ) N ξ | ≤ CN 1/2−α H 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 . (A.27) for any α > 1, ξ ∈ F ≤N + and N ∈ N large enough. Proof. We write e −B L (2,V ) N e B = N p∈Λ * + V (p/e N )e −B b * p b p e B − p∈Λ * + V (p/e N )e −B a * p a p e B + N 2 p∈Λ * + V (p/e N )e −B b p b −p + b * p b * −p e B =: F 1 + F 2 + F 3 . (A.28) With (A.3), we find F 1 = N p∈Λ * + V (p/e N ) γ p b * p + σ p b −p γ p b p + σ p b * −p ] + N p∈Λ * + V (p/e N ) (γ p b * p + σ p b −p )d p + d * p (γ p b p + σ p b * −p ) + d * p d p where γ p = cosh η p , σ p = sinh η p and the operators d p are defined in (A.2). Using |1 − γ p | ≤ η 2 p , |σ p | ≤ C|η p | and using Lemma A.1 for the terms on the second line, we find F 1 = N p∈Λ * + V (p/e N )b * p b p + E V 1 (A.29) with ±E V 1 ≤ CN 1−α (N + + 1). Let us now consider the second contribution on the r.h.s. of (A.28). We find − F 2 = p∈Λ * + V (p/e N )a * p a p + E V 2 (A.30) with E V 2 = p∈Λ * + V (p/e N ) 1 0 e −sB (η p b −p b p + h.c. )e sB ds. With Lemma 3.2, we easily find ±E V 2 ≤ CN −α (N + + 1). Finally, we consider the last term on the r.h.s. of (A.28). With (A.3), we obtain F 3 = N 2 p∈Λ * + V (p/e N ) γ p b p + σ p b * −p γ p b −p + σ p b * p + h.c. + N 2 p∈Λ * + V (p/e N ) (γ p b p + σ p b * −p ) d −p + d p (γ p b −p + σ p b * p ) + h.c. + N 2 p∈Λ * + V (p/e N )d p d −p + h.c. =: F 31 + F 32 + F 33 . (A.31) Using |1 − γ p | ≤ Cη 2 p , |σ p | ≤ C|η p |, we obtain F 31 = N 2 p∈Λ * + V (p/e N ) b p b −p + b * −p b * p + N p∈Λ * + V (p/e N )η p N − N + N + E V 3 (A.32) with ±E V 3 ≤ CN 1−α (N + + 1) . As for F 32 in (A.31), we divide it into four parts F 32 = N 2 p∈Λ * + V (p/e N ) (γ p b p + σ p b * −p ) d −p + d p (γ p b −p + σ p b * p ) + h.c. =: F 321 + F 322 + F 323 + F 324 . (A.33) We start with F 321 , which we write as F 321 = −N p∈Λ * + V (p/e N )η p N − N + N N + + 1 N + E V 4 where E V 4 = E V 41 + E V 42 + E V 43 + h.c. , with E V 41 = N 2 p∈Λ * + V (p/e N ) (γ p − 1)b p d −p , E V 42 = N 2 p∈Λ * + V (p/e N )b p d −p E V 43 = − N 2 p∈Λ * + V (p/e N )η p N + + 1 N (b * p b p − N −1 a * p a p ) and with the notation d −p = d −p + N −1 η p N + b * p . Since |γ p − 1| ≤ Cη 2 p , η ∞ ≤ CN −α , we find easily with (A.5) that | ξ, E V 41 ξ | ≤ CN 1−3α (N + + 1) 1/2 ξ 2 . Moreover | ξ, E V 43 ξ | ≤ CN p∈Λ * + η p a p ξ 2 ≤ CN 1−α N 1/2 + ξ 2 . As for E V 42 , we switch to position space and we use (A.7). We obtain | ξ, E V 42 ξ | ≤ CN Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) 1/2 ξ (N + + 1) −1/2ǎ xďy ξ ≤ CN 1−α Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) 1/2 ξ × (N + + 1) 1/2 ξ + ǎ x ξ + ǎ y ξ + N −1/2 ǎ xǎy ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . We conclude that | ξ, E V 4 ξ | ≤ CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ + CN 1−α (N + + 1) 1/2 ξ 2 . To bound the term F 322 in (A.33), we use (A.5) and |σ p | ≤ C|η p |; we obtain | ξ, F 322 ξ | ≤ CN p∈Λ * + |η p | b −p ξ |η p | (N + + 1) 1/2 ξ + η b −p ξ ≤ CN 1−2α (N + + 1) 1/2 ξ 2 . Let us now consider the term F 323 on the r.h.s. of (A.33). We write F 323 = E V 51 + E V 52 + h.c. , with E V 51 = N 2 p∈Λ * + V (p/e N ) (γ p − 1) d p b −p , E V 52 = N 2 p∈Λ * + V (p/e N ) d p b −p . With |γ p − 1| ≤ Cη 2 p and (A.5) we obtain | ξ, E V 51 ξ | ≤ CN p∈Λ * + η 2 p d * p ξ a p ξ ≤ CN 1−3α (N + + 1) 1/2 ξ 2 . We find, switching to position space and using (A.6), | ξ, E V 52 ξ | ≤ CN Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) 1/2 ξ (N + + 1) −1/2ď xǎy ξ ≤ CN 1−α (N + + 1) 1/2 ξ Λ 2 dxdy e 2N V (e N (x − y)) ǎ y ξ + N −1/2 ǎ xǎy ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . Hence, | ξ, F 323 ξ | ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ To estimate the term F 324 in (A.33) we use (A.5) and the bound p∈Λ * + | V (p/e N )||η p | ≤ C p∈Λ * + , |p|≤e N 1 p 2 + C p∈Λ * + , |p|>e N | V (p/e N )| p 2 ≤ CN + C p∈Λ * + | V (p/e N )| 2 1/2 p∈Λ * + , |p|>e N 1 p 4 1/2 ≤ CN We find | ξ, F 324 ξ | ≤ CN p∈Λ * + V (p/e N ) |η p | (N + + 1) 1/2 ξ (N + + 1) −1/2 d p b * p ξ ≤ CN p∈Λ * + V (p/e N ) |η p | (N + + 1) 1/2 ξ × |η p | (N + + 1) 1/2 ξ + N −1 η b p b * p (N + + 1) 1/2 ξ ≤ CN p∈Λ * + V (p/e N ) |η p | (N + + 1) 1/2 ξ × |η p | (N + + 1) 1/2 ξ + N −1 η (N + + 1) 1/2 ξ + η a p ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 . Combining the last bounds, we arrive at F 32 = N p∈Λ * + V (p/e N )η p N − N + N −N + − 1 N + E V 6 with | ξ, E V 6 ξ | ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . (A.34) To control the last contribution F 33 in (A.31), we switch to position space. With (A.8) and (3.11) we obtain | ξ, F 33 ξ | ≤ CN (N + + 1) 1/2 ξ Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) −1/2ď xďy ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−2α (N + + 1) 1/2 ξ V 1/2 N ξ . The last equation, combined with (A.31), (A.32) and (A.34), implies that F 3 = N 2 p∈Λ * + V (p/e N )(b p b −p + b * −p b * p ) + N p∈Λ * + V (p/e N )η p N − N + N N − N + − 1 N + E V 7 with | ξ, E V 7 ξ | ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . Together with (A.29) and with (A.30), and recalling that b * p b p − N −1 a * p a p = a * p a p (1 − N + /N ), we obtain (A.26) with (A.27). A.4 Analysis of G (3) N,α = e −B L (3) N e B We consider here the conjugation of the cubic term L N e B = √ N p,q∈Λ * + :p+q =0 V (p/e N ) b * p+q a * −p a q + h.c. + E (3) N where | ξ, E(3)N ξ | ≤ CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ + CN 1−α (N + + 1) 1/2 ξ 2 (A.35) for any α > 1 and N ∈ N large enough. Proof. This proof is similar to the proof of [3,Prop. 7.5]. Expanding e −B a * −p a q e B , we arrive at E (3) N = √ N p,q∈Λ * + :p+q =0 V (p/e N ) (γ p+q − 1)b * p+q + σ p+q b −p−q + d * p+q a * −p a q + √ N p,q∈Λ * + ,p+q =0 V (p/e N )η p e −B b * p+q e B 1 0 ds e −sB b p b q e sB + √ N p,q∈Λ * + ,p+q =0 V (p/e N )η q e −B b * p+q e B 1 0 ds e −sB b * −p b * −q e sB + h.c. =: E (3) 1 + E (3) 2 + E (3) 3 + h.c. (A.36) where, as usual, γ p = cosh η(p), σ p = sinh η(p) and d p is as in (A.2). We consider E 1 . To this end, we write E (3) 1 = √ N p,q∈Λ * + :p+q =0 V (p/e N ) (γ p+q − 1)b * p+q + σ p+q b −p−q + d * p+q a * −p a q =: E(3) 11 + E 12 + E 13 . Since |γ p+q − 1| ≤ |η p+q | 2 and η ≤ CN −α , we find | ξ, E(3) 11 ξ | ≤ CN η 2 (N + + 1) 1/2 ξ 2 ≤ CN 1−2α (N + + 1) 1/2 ξ 2 . (A.37) As for E 12 , we commute a * −p through b −p−q (recall q = 0). With |σ p+q | ≤ C|η p+q |, we obtain | ξ, E(3)12 ξ | ≤ CN 1−α (N + + 1) 1/2 ξ 2 . (A.38) We decompose now E 13 = E (3) 131 + E (3) 132 , with E (3) 131 = √ N p,q∈Λ * + :p+q =0 V (p/e N )d * p+q a * −p a q E(3)132 = − (N + + 1) N √ N p,q∈Λ * + :p+q =0 V (p/e N )η p+q b −p−q a * −p a q .(3) where we defined d * p+q = d * p+q − (N++1) N η p+q b −p−q . The term E(3) 132 is estimated similarly to E (3) −p to the left of b −p−q ; we find ±E (3) 132 ≤ CN 1−α (N + + 1). We bound E (3) 131 in position space. We find | ξ, E (3) 131 ξ | ≤ N 1/2 Λ 2 dxdy e 2N V (e N (x − y)) ǎ x ξ ǎ yďx ξ ≤ CN 1/2−α Λ 2 dxdy e 2N V (e N (x − y)) ǎ x ξ × (N + + 1)ξ + N −1 ǎ x (N + + 1) 1/2 ξ + η ǎ y (N + + 1) 1/2 ξ + ǎ xǎy ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . With (A.37) and (A.38) we obtain | ξ, E(3)1 ξ | ≤CN 1/2−α V 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 . (A.39) Next, we focus on E E (3) 2 = √ N p,q∈Λ * + ,p+q =0 V (p/e N )η p e −B b * p+q e B × 1 0 ds γ (s) p γ (s) q b p b q + σ (s) p σ (s) q b * −p b * −q + γ (s) p σ (s) q b * −q b p + σ (s) p γ (s) q b * −p b q + √ N p,q∈Λ * + ,p+q =0 V (p/e N )η p e −B b * p+q e B 1 0 ds γ (s) p σ (s) q [b p , b * −q ] + √ N p,q∈Λ * + ,p+q =0 V (p/e N )η p e −B b * p+q e B × 1 0 ds d (s) p γ (s) q b q + σ (s) q b * −q + γ (s) p b p + σ (s) p b * −p d (s) q + d (s) p d (s) q =: E(3) 21 + E 22 + E 21 ξ | ≤ CN 1−α (N + + 1) 1/2 ξ 2 . (A.41) Since [b p , b * −q ] = −a * −q a p /N for p = −q, we find | ξ, E(3)22 ξ | ≤ CN −2α (N + + 1) 1/2 ξ 2 . (A.42)(3) As for the third term on the r.h.s. of (A.40), we switch to position space. We find E (3) 23 = √ N Λ 3 dxdydz e 2N V (e N (x − z))η(y − z) e −Bb * x e B × 1 0 ds ď (s) y b(γ (s) x ) + b * (σ (s) x ) + b(γ (s) y ) + b * (σ (s) y ) ď (s) x +ď (s) yď (s) x . Using the bounds (A.6), (A.7), (A.8) and Lemma 3.2 we arrive at | ξ, E(3)23 ξ | ≤ C √ N Λ 3 dxdydz e 2N V (e N (x − z))|η(y − z)| b x e B ξ 1 0 ds × ď (s) y b x + b(ř (s) x ) + b * (σ (s) x ) ξ + b y + b(ř (s) y ) + b * (σ (s) y ) ď (s) x ξ + ď (s) xď (s) y ξ ≤ C √ N Λ 3 dxdydz e 2N V (e N (x − z))|η(y − z)| b x e B ξ N −1 |η(x − y)| (N + + 1)ξ + η b xby ξ + η (N + + 1)ξ + η b x (N + + 1) 1/2 ξ + η b y (N + + 1) 1/2 ξ ≤ CN 1−α N 1/2 + e B ξ (N + + 1)ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 whereř indicates the function in L 2 (Λ) with Fourier coefficients r p = 1 − γ p , and the fact that η , ř , σ ≤ CN −α . Combined with (A.41) and (A.42), the last bound implies that ±E (3) 2 ≤CN 1−α (N + + 1) . (A.43) To bound the last contribution on the r.h.s. of (A.36), it is convenient to bound (in absolute value) the expectation of its adjoint E (3) * 3 = √ N p,q∈Λ * + ,p+q =0 V (p/e N )η q 1 0 ds e −sB b −q e sB × γ (s) p b −p + σ (s) p b * p + d (s) −p γ p+q b p+q + σ p+q b * −p−q + d p+q = √ N p,q∈Λ * + ,p+q =0 V (p/e N )η q 1 0 ds e −sB b −q e sB × γ (s) p γ p+q b −p b p+q + σ (s) p σ p+q b * p b * −p−q + γ (s) p σ p+q b * −p−q b −p + γ p+q σ (s) p b * p b p+q + d (s) −p γ p+q b p+q + σ p+q b * −p−q + γ (s) p b −p + σ (s) p b * p d p+q + d (s) −p d p+q + √ N p,q∈Λ * + ,p+q =0 V (p/e N )η q 1 0 ds e −sB b −q e sB γ (s) p σ p+q [b −p , b * −p−q ] =: E (3) 31 + E(3) 32 . Since q = 0, [b −p , b * −p−q ] = −a * −p−q a −p /N . Thus, we can estimate | ξ,E(3)32 ξ | ≤ CN −1/2 1 0 ds p,q∈Λ * + ,p+q =0 |η q ||η p+q | a −p−q e −sB b * −q e sB ξ a −p ξ ≤ C η 2 (N + + 1) 1/2 ξ 2 ≤ CN −2α (N + + 1) 1/2 ξ 2 . (A.44) To bound the expectation of E 31 , we switch to position space. We find | ξ, E(3)31 ξ | ≤ N 1/2 1 0 ds Λ 2 dxdy e 2N V (e N (x − y)) b * (η x )e sB ξ b xby ξ + η b x (N + + 1) 1/2 ξ + η b y (N + + 1) 1/2 ξ + N −1 |η(x − y)| (N + + 1)ξ . With Lemma 3.2, we conclude that | ξ, E(3)31 ξ | ≤CN 1/2−α V 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 . (A.45) From (A.44) and (A.45) we obtain | ξ, E(3)3 ξ | ≤CN 1/2−α V 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 .N e B = V N + 1 2 q∈Λ * + ,r∈Λ * r =−q V (r/e N )η q+r η q 1 − N + N 1 − N + + 1 N + 1 2 q∈Λ * + ,r∈Λ * : r =−q V (r/e N ) η q+r b q b −q + b * q b * −q + EN ξ | ≤CN 1/2−α V 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 (A.46)(4) for any α > 1, ξ ∈ F ≤N + and N ∈ N large enough. To show Prop. A.7, we use the following lemma, whose proof can be obtained as in [3,Lemma 7.7]. Lemma A.8. Let η ∈ ℓ 2 (Λ * ) as defined in (3.13). Then there exists a constant C > 0 such that (N + + 1) n/2 e −Bb xby e B ξ ≤ C N (N + + 1) n/2 ξ + ǎ y (N + + 1) (n+1)/2 ξ + ǎ x (N + + 1) (n+1)/2 ξ + ǎ xǎy (N + + 1) n/2 ξ for all ξ ∈ F ≤N + , n ∈ Z. Proof of Prop. A.7. We follow the proof of [3,Prop. 7.6]. We write G (4) N,α = V N + W 1 + W 2 + W 3 + W 4 with W 1 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r 1 0 ds e −sB b q b −q e sB + h.c. W 2 = p,q∈Λ * + ,r∈Λ * :r =p,−q V (r/e N ) η q+r 1 0 ds e −sB b * p+r b * q e sB a * −q−r a p + h.c. W 3 = p,q∈Λ * + ,r∈Λ * :r =−p−q V (r/e N )η q+r η p × 1 0 ds s 0 dτ e −sB b * p+r b * q e sB e −τ B b * −p b * −q−r e τ B + h.c. W 4 = p,q∈Λ * + ,r∈Λ * :r =−p−q V (r/e N ) η 2 q+r × 1 0 ds s 0 dτ e −sB b * p+r b * q e sB e −τ B b p b q+r e τ B + h.c. . (A.47) Let us first consider the term W 1 . With (A.3), we find W 1 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r 1 0 ds(γ (s) q ) 2 (b q b −q + h.c. )ds 2γ (s) q σ (s) q b * q b q + (σ (s) q ) 2 b * −q b * q + h.c. E (4) 102 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r 1 0 ds σ (s) q b * −q d (s) −q + h.c. E (4) 103 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r 1 0 ds σ (s) q d (s) q b * q + h.c. E (4) 104 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r 1 0 ds γ (s) q d (s) q b −q + h.c. E (4) 105 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r 1 0 ds d (s) q d (s) −q + h.c. . (A.50) With 1 N sup q∈Λ * + r∈Λ * + | V (r/e N )||η q+r | ≤ C < ∞ (A.51) uniformly in N ∈ N, we can estimate the first term in (A.50) by | ξ, E(4)101 ξ | ≤ CN 1−α (N + + 1) 1/2 ξ 2 . Using (A.51) and (A.5) we also find | ξ, E(4) 102 ξ | ≤ CN 1−2α (N + + 1) 1/2 ξ 2 . For the third term in (A.50) we switch to position space and use (A.6): | ξ, E(4)103 ξ | ≤ 1 2 dxdye 2N V (e N (x − y))|η(x − y)| × 1 0 ds (N + 1) −1/2ď y b * (σ (s) x )ξ (N + 1) 1/2 ξ ≤ C η ∞ η dxdye 2N V (e N (x − y)) (N + + 1) 1/2 ξ 1 0 ds × b * (σ (s) x )ξ + 1 N |η (s) (x − y)| (N + 1) 1/2 ξ + 1 √ N b * (σ (s) x )b y ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 . Consider now the fourth term in (A.50). We write E (γ (s) q − 1)d (s) q b −q E (4) 1042 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r 1 0 ds d (s) q b −q With |γ (s) q − 1| ≤ C|η q | 2 , (A.51) and d * q ξ ≤ C η (N + + 1) 1/2 ξ , we find | ξ, E(4) 1041 ξ | ≤ CN 1−3α (N + + 1) 1/2 ξ 2 As for E (4) 1042 , we switch to position space. Using (3.11) and (A.6), we obtain | ξ, E(4)1042 ξ | = 1 2 1 0 ds Λ 2 dxdy e 2N V (e N (x − y))η(x − y) ξ,ď (s) xb y ξ ≤ CN 1 0 ds Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) 1/2 ξ (N + + 1) −1/2ď(s) xby ξ ≤ CN η 1 0 ds Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) 1/2 ξ × N −1 ǎ y N + ξ + ǎ xǎy N 1/2 + ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ Let us consider the last term in (A.50). Switching to position space and using (A.8) in Lemma A.1 and again (3.11), we arrive at | ξ, E(4)105 ξ | ≤ CN Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) 1/2 ξ 1 0 ds (N + + 1) −1/2ď(s) xď (s) y ξ ≤ CN η (N + + 1) 1/2 ξ Λ 2 dxdy e 2N V (e N (x − y)) × (N + + 1) 1/2 ξ + η ǎ x ξ + η ǎ y ξ + N −1/2 η ǎ xǎy ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−2α (N + + 1) 1/2 ξ V 1/2 N ξ . Summarizing, we have shown that (A.49) can be bounded by | ξ, E(4)10 ξ | ≤CN 1/2−α V 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 (A.52) for any α > 1, ξ ∈ F ≤N + . Next, we come back to the terms W 11 , W 12 , W 13 introduced in (A.48). Using (A.51) and |γ (s) q − 1| ≤ Cη 2 q , we can write W 11 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r (b q b −q + h.c. ) + E (4) 11 , (A.53) where E(4) 11 is such that | ξ, E(4)11 ξ | ≤ CN 1−2α (N + + 1)ξ 2 . Next, we can decompose the second term in (A.48) as W 12 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r η q 1 − N + N + E (4) 12 (A.54) where ±E (4) 12 ≤ CN −α N + + N 1−3α . The third term on the r.h.s. of (A.48) can be written as W 13 = − 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r η q 1 − N + N N + + 1 N + E (4) 13 (A.55) where E (4) 13 = E(4) 131 + E 132 + E 133 + E (γ (s) q − 1)b q d (s) −q + h.c. E (4) 132 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r 1 0 ds b q d (s) −q + sη q N + N b * q + h.c. E (4) 133 = − 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r η q b * q b q N + + 1 N E (4) 134 = 1 2N q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r η q a * q a q N + + 1 N . With (A.51), we immediately find ±E (4) 133 ≤ CN 1−α (N + + 1), ±E(4) 134 ≤ CN −α (N + + 1) . With |γ 131 ξ | ≤ CN 1−3α (N + + 1) 1/2 ξ 2 .(4) Let us now consider E 132 ξ | = 1 2 1 0 ds Λ 2 dxdy e 2N V (e N (x − y))η(x − y) ξ,b xď (s) y ξ ≤ CN 1−α Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) 1/2 ξ × (N + + 1) 1/2 ξ + ǎ y ξ + ǎ x ξ + N −1 ǎ xǎy N 1/2 + ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . It follows that | ξ, E(4)13 | ≤ CN 1/2−α V 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 .(4)W 1 = 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r b q b −q + h.c. + 1 2 q∈Λ * + ,r∈Λ * :r =−q V (r/e N )η q+r η q 1 − N + N 1 − N + + 1 N + E (4) 1 (A.56) where | ξ, E(4)1 ξ | ≤ CN 1/2−α V 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 , Next, we control the term W 2 , from (A.47). In position space, we find W 2 = Λ 2 dxdy e 2N V (e N (x − y)) 1 0 ds e −sBb * xb * y e sB a * (η x )ǎ y + h.c. withη x (z) =η(x − z). By Cauchy-Schwarz, we have | ξ, W 2 ξ | ≤ Λ 2 dxdy e 2N V (e N (x − y)) 1 0 ds × (N + + 1) 1/2 e −sBb xby e sB ξ (N + + 1) −1/2 a * (η x )ǎ y ξ . With (N + + 1) −1/2 a * (η x )ǎ y ξ ≤ C η ǎ y ξ ≤ CN −α ǎ y ξ and using Lemma A.8, we obtain | ξ, W 2 ξ | ≤ CN −α Λ 2 dxdy e 2N V (e N (x − y)) ǎ y ξ × N (N + + 1) 1/2 ξ + N ǎ x ξ + N ǎ y ξ + N 1/2 ǎ xǎy ξ ≤ CN 1−α (N + + 1) 1/2 ξ 2 + CN 1/2−α (N + + 1) 1/2 ξ V 1/2 N ξ . (A.57) Also for the term W 3 in (A.47), we switch to position space. We find W 3 = Λ 2 dxdy e 2N V (e N (x − y))× (N + + 1) −1/2 e −τ B b * (η x ) b * (η y )e τ B ξ . With Lemma 3.2, we find (N + + 1) −1/2 e −τ B b * (η x ) b * (η y )e τ B ξ ≤ C η 2 (N + + 1) 1/2 ξ . Using Lemma A.8, we conclude that | ξ, W 3 ξ | ≤ C η 2 Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) 1/2 ξ × N (N + + 1) 1/2 ξ + N ǎ x ξ + N ǎ y ξ + N 1/2 ǎ xǎy ξ ≤ CN 1−2α (N + + 1) 1/2 ξ 2 + CN 1/2−2α V 1/2 N ξ (N + + 1) 1/2 ξ . (A.58) The term W 4 in (A.47) can be bounded similarly. In position space, we find W 4 = dxdy e 2N V (e N (x − y)) × 1 0 ds s 0 dτ e −sBb * xb * y e sB e −τ B b(η 2 x )b y e τ B + h.c. withη 2 the function with Fourier coefficients η 2 q , for q ∈ Λ * , and whereη 2 x (y) := η 2 (x − y). Clearly η2 x ≤ C η 2 ≤ CN −2α . With Cauchy-Schwarz and Lemma 3.2, we obtain | ξ, W 4 ξ | ≤ CN −2α 1 0 ds s 0 dτ dxdy e 2N V (e N (x − y)) × (N + + 1) 1/2b ybx e sB ξ b y e τ B ξ . Applying Lemma A.8 and then Lemma 3.2, we obtain | ξ, W 4 ξ | ≤ CN −2α 1 0 ds s 0 dτ dxdy e 2N V (e N (x − y)) b y e τ B ξ × N (N + + 1) 1/2 ξ + N ǎ x ξ + N ǎ y ξ + N 1/2 ǎ xǎy ξ ≤ CN 1−2α (N + + 1) 1/2 ξ 2 + CN 1/2−2α V 1/2 N ξ (N + + 1) 1/2 ξ .G N,α = V (0) 2 (N + N + − 1) (N − N + ) + p∈Λ * + η p p 2 η p + N V (p/e N ) + 1 2 r∈Λ * p+r =0 V (r/e N )η p+r N − N + N N − N + − 1 N + K + N p∈Λ * + V (p/e N )a * p a p 1 − N + N + p∈Λ * + p 2 η p + N 2 V (p/e N ) + 1 2 r∈Λ * : p+r =0 V (r/e N )η p+r b * p b * −p + b p b −p + √ N p,q∈Λ * + : p+q =0 V (p/e N ) b * p+q a * −p a q + h.c. + V N + E 1 (A.59) where | ξ, E 1 ξ | ≤ CN 1/2−α H 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 for any α > 1 and ξ ∈ F ≤N + . With (3.17), we find p∈Λ * + η p p 2 η p + N V (p/e N ) + 1 2 r∈Λ * : p+r =0 V (r/e N )η p+r = p∈Λ * + η p N 2 V (p/e N ) + N e 2N λ ℓ χ ℓ (p) + e 2N λ ℓ q∈Λ * χ ℓ (p − q)η q − 1 2 V (p/e N )η 0 From Lemma 3.1 and estimating χ ℓ = χ ℓ ≤ CN −α , η ≤ CN −α and χ ℓ * η = χ ℓη ≤ η ≤ CN −α , we have N e 2N λ ℓ p∈Λ * + η p χ ℓ (p) ≤ CN 2α χ ℓ η ≤ C, and e 2N λ ℓ p∈Λ * + , q∈Λ * χ ℓ (p − q)η q η p ≤ CN 2α−1 χ ℓ * η η ≤ CN −1 . Moreover, using (A.51) and the bound (3.18) we find 1 2 p∈Λ * + V (p/e N )η p η 0 ≤ CN 1−2α . We obtain p∈Λ * + η p p 2 η p + N V (p/e N ) + 1 2 r∈Λ * p+r∈Λ * + V (r/e N )η p+r N − N + N N − N + − 1 N = N 2 p∈Λ * + V (p/e N )η p N − N + N N − N + − 1 N + E 2 with ±E 2 ≤ C for all α ≥ 1/2. On the other hand, using (3.18) we have N 2 p∈Λ * + V (p/e N )η p = N 2 V (·/e N ) * η (0) − N 2 V (0)η 0 = N 2 2 dxV (x)f ℓ (x) − V (0) +Ẽ 2 with ±Ẽ 2 ≤ CN 1−2α . With the first bound in (3.27) we conclude that p∈Λ * + η p p 2 η p + N V (p/e N ) + 1 2 r∈Λ * p+r∈Λ * + V (r/e N )η p+r N − N + N N − N + − 1 N = 1 2N ω N (0) − N V (0) (N − N + − 1) (N − N + ) + E 3 (A.60) where ±E 3 ≤ C, if α ≥ 1/2. Using (3.17), we can also handle the fourth line of (A.59); we find p∈Λ * + p 2 η p + N 2 V (p/e N ) + 1 2 r∈Λ * : p+r∈Λ * + V (r/e N )η p+r b * p b * −p + b p b −p = p∈Λ * + N e 2N λ ℓ χ ℓ (p) + e 2N λ ℓ q∈Λ * χ ℓ (p − q)η q − 1 2 V (p/e N )η 0 b * p b * −p + b p b −p . (A.61) The last two terms on the right hand side of (A.61) are error terms. With (3.18) and (A.51) we have p∈Λ * + V (p/e N )η 0 b * p b * −p + b p b −p ≤ CN −2α p∈Λ * + | V (p/e N )| 2 p 2 1/2 p∈Λ * + p 2 a p ξ 2 1/2 (N + + 1) 1/2 ξ ≤ CN 1/2−2α K 1/2 ξ (N + + 1) 1/2 ξ . The second term on the right hand side of (A.61) can be bounded in position space: ξ, e 2N λ ℓ p∈Λ * + ( χ ℓ * η)(p)(b * p b * −p + b p b −p )ξ ≤ CN 2α−1 (N + + 1) 1/2 ξ Λ 2 dxdy χ ℓ (x − y)|η(x − y)| (N + + 1) −1/2b xby ξ ≤ CN α−1 (N + + 1) 1/2 ξ Λ 2 dxdy χ ℓ (x − y) (N + + 1) −1/2ǎ xǎy ξ 2 1/2 . The term in parenthesis can be bounded similarly as in (6.6). Namely, Λ 2 dxdy χ ℓ (x − y) (N + + 1) −1/2ǎ xǎy ξ 2 ≤ CqN −2α/q ′ K 1/2 ξ 2 for any q > 2 and 1 < q ′ < 2 with 1/q + 1/q ′ = 1. Choosing q = log N , we get ξ, e 2N λ ℓ p∈Λ * + ( χ ℓ * η)(p)(b * p b * −p + b p b −p )ξ ≤ CN −1 (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ , and, from (A.61), we conclude that p∈Λ * + p 2 η p + N 2 V (p/e N ) + 1 2 r∈Λ * : p+r∈Λ * + V (r/e N )η p+r b * p b * −p + b p b −p = p∈Λ * + N e 2N λ ℓ χ ℓ (p) b * p b * −p + b p b −p + E 4 , (A.62) with | ξ, E 4 ξ | ≤ CN −1 (log N ) 1/2 (N + + 1) 1/2 ξ K 1/2 ξ . if α > 1. Combining (A.59) with (A.60) and (A.62), and using the definition (3.25) we conclude that G N,α = 1 2 ω N (0)(N − 1) 1 − N + N + N V (0) − 1 2 ω N (0) N + 1 − N + N + N p∈Λ * + V (p/e N )a * p a p 1 − N + N + 1 2 p∈Λ * + ω N (p)(b p b −p + h.c. ) + √ N p,q∈Λ * + :p+q =0 V (p/e N ) b * p+q a * −p a q + h.c. + K + V N + E 5 , (A.63) with | ξ, E 5 ξ | ≤ CN 1/2−α H 1/2 N ξ (N + + 1) 1/2 ξ + CN 1−α (N + + 1) 1/2 ξ 2 + CN −1 (log N ) 1/2 K 1/2 ξ (N + + 1) 1/2 ξ + C ξ 2 , for any α > 1. Observing that | V (p/e N ) − V (0)| ≤ C|p|e −N in the second line on the r.h.s. of (A.63), we arrive at G N,α = G eff N,α + E G , with G eff N,α defined as in (3.28) and with E G that satisfies (3.29). B Properties of the Scattering Function Let V be a potential with finite range R 0 > 0 and scattering length a. For a fixed R > R 0 , we study properties of the ground state f R of the Neumann problem − ∆ + 1 2 V (x) f R (x) = λ R f R (x) (B.1) on the ball |x| ≤ R, normalized so that f R (x) = 1 for |x| = R. Lemma 3.1, parts i)-iv), follows by setting R = e N ℓ in the following lemma. Lemma B.1. Let V ∈ L 3 (R 2 ) be non-negative, compactly supported and spherically symmetric, and denote its scattering length by a. Fix R > 0 sufficiently large and denote by f R the Neumann ground state of (B.1). Set w R = 1 − f R . Then we have 0 ≤ f R (x) ≤ 1 Moreover, for R large enough there is a constant C > 0 independent of R such that λ R − 2 R 2 log(R/a) 1 + 3 4 1 log(R/a) ≤ C R 2 1 log 3 (R/a) . (B.2) and dx V (x)f R (x) − 4π log(R/a) ≤ C log 2 (R/a) . (B.3) Finally, there exists a constant C > 0 such that |w R (x)| ≤ χ(|x| ≤ R 0 ) + C log(|x|/R) log(a/R) χ(R 0 ≤ |x| ≤ R) |∇w R (x)| ≤ C log(R/a) χ(|x| ≤ R) |x| + 1 (B.4) for R large enough. To show Lemma B.1 we adapt to the two dimensional case the strategy used in [6, Lemma A.1] for the three dimensional problem. We will use some well known properties of the zero energy scattering equation in two dimensions, summarized in the following lemma. Lemma B.2. Let V ∈ L 3 (R 2 ) non-negative, with supp V ⊂ B R0 (0) for an R 0 > 0. Let a ≤ R 0 denote the scattering length of V . For R > R 0 , let φ R : R 2 → R be the radial solution of the zero energy scattering equation −∆ + 1 2 V φ R = 0 (B.5) normalized such that φ R (x) = 1 for |x| = R. Then φ R (x) = log(|x|/a) log(R/a) (B.6) for all |x| > R 0 . Moreover, |x| → φ R (x) is monotonically increasing and there exists a constant C > 0 (depending only on V ) such that φ R (x) ≥ φ R (0) ≥ C log(R/a) (B.7) for all x ∈ R 2 . Furthermore, there exists a constant C > 0 such that |∇φ R (x)| ≤ C | log(R/a)| 1 |x| + 1 (B.8) for all x ∈ R 2 . Proof. The existence of the solution of (B.5), the expression (B.6), the fact that φ R (x) ≥ 0 and the monotonicity are standard (see, for example, Theorem C.1 and Lemma C.2 in [13]). The bound (B.7) for φ R (0) follows from (B.6), comparing φ R (0) with φ R (x) at |x| = R 0 , with Harnack's inequality (see [18,Theorem C.1.3]). Finally, (B.8) follows by rewriting (B.5) in integral form φ R (x) = 1 − 1 4π R 2 log R/|x − y| V (y)φ R (y)dy . For |x| ≤ R 0 , this leads (using that φ R (y) ≤ log(R 0 /a)/ log(R/a) for all |y| ≤ R 0 and the local integrability of |.| −3/2 ) to |∇φ R (x)| ≤ C V (y)φ R (y) |x − y| dy ≤ C V 3 log(R/a) With (B.11) one finds that (B.10) ∂ r ψ R (x) |x|=R = − 1 2kR log(kae γ /2) · (kR) 4 8 log(R/a) − (kR) 2 log(R/a) − 1 2 + 2 + O((kR) 4 + (ka) 2 ) (B.12) The smallest solution of (kR) 4 Ψ R (x) := ψ k (m R (x)) = J 0 (km R (x)) − J 0 (ka) Y 0 (ka) Y 0 (km R (x)) , (B.14) with k = k(R) satisfying (B.9) and m R (x) := a exp log(R/a)φ R (x) . With this choice we have m R (x) = |x| outside the range of the potential; hence Ψ R (x) = ψ k (x) for R 0 ≤ |x| ≤ R. In particular, Ψ R satisfies Neumann boundary conditions at |x| = R. From (B.6), (B.7) and the monotonicity of φ R , we get Ca ≤ m R (x) ≤ R 0 for all 0 ≤ |x| ≤ R 0 (B.15) and for a constant C > 1, independent of R. From (B.8) we also get |∇m R (x)| ≤ C for all 0 ≤ |x| ≤ R . (B.16) With the notation h = −∆ + 1 2 V , we now evaluate Ψ R , hΨ R . To this end we note that Ψ R , hΨ R = |x|<R0 Ψ R (x)(hΨ R (x))dx + k 2 |x|≥R0 |Ψ R (x)| 2 dx . (B.17) Let us consider the region |x| < R 0 . From (B.14) and (B.11) we find, first of all, Ψ R (x) + log(m R (x)/a) log(kae γ /2) ≤ C(km R (x)) 2 , (B.18) Next, we compute −∆Ψ R (x). With J ′ 0 (r) = −J 1 (r) J ′ 1 (r) = 1 2 J 0 (r) − J 2 (r) Y ′ 0 (r) = −Y 1 (r) Y ′ 1 (r) = 1 2 Y 0 (r) − Y 2 (r) . we obtain (here, we use the notation m ′ R and m ′′ R for the radial derivatives of the radial function m R ) −∆Ψ R (x) = − ∂ 2 r Ψ R (x) − 1 |x| ∂ r Ψ R (x) = − km ′′ R (x) − J 1 (km R (x)) + J 0 (ka) Y 0 (ka) Y 1 (km R (x)) − 1 2 k 2 m ′ R (x) 2 J 2 (km R (x)) − J 0 (ka) Y 0 (ka) Y 2 (km R (x)) − 1 2 k 2 m ′ R (x) 2 − J 0 (km R (x)) + J 0 (ka) Y 0 (ka) Y 0 (km R (x)) − km ′ R (x) |x| − J 1 (km R (x)) + J 0 (ka) Y 0 (ka) Y 1 (km R (x)) . We note that, using the scattering equation (B.5), m ′′ R − (m ′ R ) 2 m R + 1 |x| m ′ R = 1 2 V m R φ R log(R/a) = 1 2 V m R log(m R /a) . (B.19) Now we write − ∆Ψ R (x) = − k m ′′ R (x) + m ′ R (x) |x| Y 1 (km R (x)) + k 2 2 (m ′ R (x)) 2 Y 2 (km R (x)) J 0 (ka) Y 0 (ka) + g R (x) (B.20) where g R (x) = 3 i=1 g (i) R (x) with g (1) R (x) = k m ′′ R (x) + m ′ R (x) |x| J 1 (km R (x)) g(2) R (x) = − 1 2 k 2 (m ′ R (x)) 2 J 2 (km R (x)) g(3) R (x) = − Next, with |J 2 (r) − r 2 /8| ≤ Cr 4 we get |g (2) R (x)| ≤ Ck 4 (m ′ R (x)) 2 (m R (x)) 2 ≤ Ck 4 . 1 2 k 2 (m ′ R (x)) 2 − J 0 (km R (x) + J 0 (ka) Y 0 (ka) Y 0 (km R (x)) = k 2 2 (m ′ R (x)) 2 Ψ R (x) . With (B.18), we can also bound |g (3) R (x)| ≤ Ck 2 (m ′ R (x)) 2 log(m R (x)/a) log(ka) ≤ Ck 2 log −1 (ka) . We conclude that |g R (r)| ≤ C(1 + V (x))k 2 for all r ≤ R 0 and R large enough. Finally, using Eq. (B.19), the expansion for Y 1 (r) in Eq. (B.11), and the bound Y 2 (r) + 4 π 1 r 2 ≤ C , we can rewrite the first term on the r.h.s. of (B.20) as − k m ′′ R (x) + m ′ R (x) |x| Y 1 (km R (x)) + k 2 2 (m ′ R (x)) 2 Y 2 (km R (x)) J 0 (ka) Y 0 (ka) = 1 π V (x) log(m R (x)/a) J 0 (ka) Y 0 (ka) + h R (x) (B.21) with |h R (x)| ≤ C(1 + V (x))k 2 for all r ≤ R 0 , R large enough. With the identities (B.20) and (B.21) we obtain − ∆Ψ R (x) − 1 π J 0 (ka) Y 0 (ka) V (x) log(m R (x)/a) ≤ C(1 + V (x))k 2 , for all |x| ≤ R 0 and for R sufficiently large. With (B.18), we conclude that, for 0 ≤ |x| ≤ R 0 , for all |x| ≤ R 0 (which follows from (B.18) and (B.15)), we get Ψ R , hΨ R ≤ k 2 Ψ R , Ψ R + Ck 2 | log(ka)| |x|≤R0 (−∆ + 1 2 V )Ψ R (x) ≤ C(1 + V (x))k 2 . (1 + V (x)) dx . On the other hand, Eq.(B.18), together with m R (x) = |x| for |x| ≥ R 0 , implies the lower bound Ψ R , Ψ R ≥ R0≤|x|≤R |Ψ R (x)| 2 dx ≥ C | log(ka)| 2 R0≤|x|≤R log 2 (|x|/a)dx ≥ CR 2 . Hence, with (B.9), we conclude that To prove the lower bound for λ R it is convenient to show some upper and lower bounds for f R . We start by considering f R outside the range of the potential. We denote ε R = √ λ R R. Keeping into account the boundary conditions at |x| = R, we find, for R 0 ≤ |x| ≤ R, λ R ≤ Ψ R , hΨ R Ψ R , Ψ R ≤ k 2 1 + C | log(ka)| R 2 ≤ 2 R 2 log(R/f R (x) = A R J 0 (ε R |x|/R) + B R Y 0 (ε R |x|/R) , with A R = J 0 (ε R ) − J 1 (ε R ) Y 0 (ε R ) Y 1 (ε R ) −1 , and B R = Y 0 (ε R ) − J 0 (ε R ) J 1 (ε R ) Y 1 (ε R ) −1 . From (B.24), we have |ε R | ≤ C | log(R/a)| −1/2 . Thus, we can expand f R for large R, using (B.11) and, for Y 0 , the improved bound Y 0 (r) − 2 π log(re γ /2) 1 − 1 4 r 2 ≤ C r 2 , we find We can also compute the radial derivative A R − 1 + ε 2 R 4 2 log(ε R e γ /2) − 1 ≤ Cε 4 R (log ε R ) 2 , B R − π 4 ε 2 R 1 − ε 2 R 8 ≤∂ r f R (x) = − ε R R A R J 1 (ε R r/R) + B R Y 1 (ε R r/R) . With the expansions (B.11) and (B.25) we conclude that for all R 0 ≤ |x| < R we have ∂ r f R (x) − ε 2 R 2|x| 1 − x 2 R 2 + ε 2 R x 2 2R 2 log(R/|x|) ≤ Cε 4 R log ε R . (B.27) The bound (B.27) shows that ∂ r f R (x) is positive, for, say, R 0 < |x| < R/2. Since ∂ r f R (x) must have its first zero at |x| = R, we conclude that f R is increasing in |x|, on R 0 ≤ |x| ≤ R. From the normalization f R (x) = 1, for |x| = R, we conclude therefore that f R (x) ≤ 1, for all R 0 ≤ |x| ≤ R. for all R * < |x| ≤ R. Finally, we show that f R (x) ≤ 1 also for |x| ≤ R 0 . First of all, we observe that, by elliptic regularity, as stated for example in [9, Theorem 11.7, part iv)], there exists 0 < α < 1 and C > 0 such that |f R (x) − f R (y)| ≤ C (V − 2λ R )f R 2 |x − y| α With V f R 2 ≤ V 3 f R 6 ≤ C f R H 1 ≤ C(1 + λ R ) f R 2 , we conclude that 0 ≤ f R (x) ≤ 1 + C f 2 for all |x| ≤ R 0 (because we know that f R (x) ≤ 1 for R 0 ≤ |x| ≤ R). To improve this bound, we go back to the differential equation (B.1), to estimate ∆f R = 1 2 V f R − λ R f R ≥ −λ R (1 + C f 2 ) (B.30) This implies that f R (x) + λ R (1 + C f 2 )x 2 /2 is subharmonic. Using (B.26), we find f R (x) ≤ 1 − Cε 2 R for |x| = R 0 . From the maximum principle, we obtain therefore that f R (x) ≤ 1 − Cε 2 R + Cλ R (1 + C f R 2 ) (B.31) for all |x| ≤ R 0 . In particular, this implies that f R 1 |x|≤R0 2 ≤ C + Cλ R f R 2 , and therefore that f R 1 R0≤|x|≤R 2 ≥ f R 2 (1 − Cλ R ) − C With f R (x) ≤ 1 for R 0 ≤ |x| ≤ R, we find, on the other hand, that f R 1 R0≤|x|≤R 2 ≤ CR. We conclude therefore that f R 2 ≤ CR and, from (B.31), that f R (x) ≤ 1 − Cε 2 R + C/R ≤ 1, for all |x| ≤ R 0 , if R is large enough. We are now ready to prove the lower bound for λ R . We use now that any function Φ satisfying Neumann boundary conditions at |x| = R can be written as Φ(x) = q(x)Ψ R (x), with Ψ R (x) the trial function used for the upper bound and q > 0 a function that satisfies Neumann boundary condition at |x| = R as well. This is in particular true for the solution f R (x) of (B.1). In the following we write f R (x) = q R (x)Ψ R (x) where q R satisfies Neumann boundary conditions at |x| = R. From (B.18), we find |Ψ R (x)| ≥ C/ log(ka). The bound f R (x) ≤ 1 implies therefore that there exists c > 0 such that q R (x) ≤ C log(ka) ∀ |x| ≤ R 0 . (B.32) From the identity hf R = (hΨ R )q R − (∆q R )Ψ R − 2∇q R ∇Ψ R we have |x|≤R dx f R hf R = |x|≤R dx |∇q R | 2 Ψ 2 R + |x|≤R dx |q R | 2 Ψ R hΨ R . From (B.22) and (B.23), we have Ψ R (x)(hΨ R )(x) − k 2 Ψ 2 R (x) ≤ C k 2 | log ka| (1 + V (x))χ(|x| ≤ R 0 ) . Hence |x|≤R dx f R hf R ≥ k 2 f R 2 − Ck 2 | log k| |x|≤R0 dx (1 + V (x))|q R (x)| 2 . (B.33) With (B.32), we obtain |x|≤R dx f R hf R ≥ k 2 f R 2 − Ck 2 log(ka) . With (B.29) (recalling that R * = max{R 0 , ea}), we bound f R 2 ≥ R * ≤|x|≤R |f R (x)| 2 dx ≥ CR 2 log 2 (R/a) and, inserting in (B.33), we conclude that λ R = f R , hf R f R , f R ≥ k 2 1 − C log 3 (R/a) R 2 ≥ 2 R 2 log(R/a) 1 + 3 4 1 log(R/a) − C log 2 (R/a) , where in the last inequality we used (B.9). To prove (B.3) we use the scattering equation (B.1) to write dx V (x)f R (x) = 2 |x|≤R dx ∆f R (x) + 2 |x|≤R dxλ R f R (x) . Passing to polar coordinates, and using that ∆f R (x) = |x| −1 ∂ r |x|∂ r f R (x), we find that the first term vanishes. Hence dx V (x)f R (x) = 2λ R dx f R (x) . With the upper bound f R (r) ≤ 1 and with (B.2), we find dx V (x)f R (x) ≤ 2πR 2 λ R ≤ 4π log(R/a) 1 + C log(R/a) . To obtain a lower bound for the same integral we use that f R (r) ≥ 0 inside the range of the potential. Outside the range of V , we use (B.26). We find dx V (x)f R (x) ≥ 4πλ R R R0 dr r (1 − Cε 2 R log(R/r)) ≥ Moreover ∂ r f R (x) = 0 if |x| = R, by construction. On the other hand, if |x| ≤ R 0 , we have w R (x) = 1 − f R (x) ≤ 1. As for the derivative, we define f R on R + through f R (r) = f R (x), if |x| = r, and we use the representation N n=0 ( 1 n=01− |ϕ 0 ϕ 0 |) ⊗n a(ϕ 0 ) N −n (N − n)! ψ N for all ψ N ∈ L 2 s (Λ N ). It is then easy to check that U * N : F ≤N + → L 2 s (Λ N ) is given by U * N {α (0) , . . . , α (N ) } = N n=0 a * (ϕ 0 ) N −n (N − n)! α (n) Proposition 4 . 2 . 42Let A be defined as in (4.1). Then there exists a constant C > 0 such that e −sA H N e sA ≤ CH N + CN (N + + 1) (4.2) [f (N + /M ), [f (N + /M ), b p ]] = f (N + /M ) − f ((N + + 1)/M ) 2 b p and f (N + /M ) − f ((N + + 1)/M ) ≤ CM −1 f ′ ∞ .With an explicit computation we obtain The next proposition combines the results of Prop. 3.3, Prop. 4.3 and Prop. 4.4. for all ξ ∈ F ≤N + . From Prop. 4.1, Prop. 4.2 and recalling the definition (6.3) of the operator C N , we deduce that 1 0 ds e −sA [K ±+ e −sA Q 2 e sA ≤ CN e −N e −sA N K 1/2 e sA ≤ CN 2 e −N (H N + 1) . (6.28) To handle the second term on the second line of (6.24), we apply Prop. 6.4 and then Prop. 4.1 and Prop. 4.2 R ≤ CN 2−α (H N + 1) + CN −1 (N + + 1). A Analysis of G N,α The aim of this section is to show Prop. 3.3. From (2.3) and (3.23), we can decompose G N,α = e −B L N e B = G To analyse G N,α we will need precise informations on the action of the generalized Bogoliubov transformation e B with B the antisymmetric operator defined in (3.19), which are summarized in subsection A.1. Then, in the subsections A.2-A.5 we prove separate bounds for the operators G Proposition A. 3 . 3Under the assumptions of Prop. 3.3, there exists a constant C > 0 such that ±E (0) N,α ≤ CN 1−α (N + + 1) for all α > 0 and N ∈ N large enough. Proof. The proof follows [3, Prop. 7.1]. We consider first conjugation of the kinetic energy operator.Proposition A.4. Under the assumptions of Prop. 3.3, there exists C > 0 such that /2 ξ 2 . 2(A.25) Combining (A.12), (A.13) and (A.25), we obtain (A.9) and (A.10). defined in (A.36). With Eq. (A.3), we find as in (A.2), with η replaced by sη. With Lemma 3.2, we get | ξ, E Together with (A.36), (A.39) and (A.43), we arrive at (A.35). . 7 . 7Under the assumptions of Prop.3.3 there exists a constant C > 0 such that | ξ, E 1| ≤ Cη 2 q , Lemma A.1 and, again, (A.51), we also obtain | ξ, E y + (N + /N )b * (η y ) and using (A.7), we obtain | ξ, E With (A.52), (A.53), (A.54), (A.55), we obtain sB e −τ B b * (η x )b * (η y )e τ B + h.c. . and thus | ξ, W 3 ξ | ≤ Λ 2 dxdy e 2N V (e N (x − y)) (N + + 1) 1/2 e −sBb xby e sB ξ From+ (A.56), (A.57), (A.58) and the last bound, we conclude that ,r∈Λ * :r =−q V (r/e N )η q+r b q b −q + h. limit of large R. Inserting in (B.12), we find that the r.h.s. changes sign around the value of k defined in (B.13). By the intermediate value theorem, we conclude that there is a k = k(R) > 0 satisfying (B.9), such that ∂ r ψ k(R) (x) = 0 if |x| = R. Now, let φ R (x) be the solution of the zero energy scattering equation (B.5), with φ R (x) = 1 for |x| = R. We set R (x) log(m R (x)/a) ≤ Ck 2 (1 + V (x)) . B.17), (B.22) and the upper bound |Ψ R (r)| ≤ C | log(ka)| (B.23) sufficiently large. Let R * = max{R 0 , ea}. Then Eq. (B.28) implies in particular that, for R large enough,f R (x) ≥ C log(R/a).(B.29) Finally, we show the bounds in (B.4). For r ∈ [R 0 , R], from (B.26) we havew R (x) − log(R/|x|) log(R/a) ≤ C log(R/a) . (B.34)As for the derivative of w R we use (B.27) to compute|∂ r f R (x)| ≤ C |x| 1 log(R/a). ′′ R (s)s + f ′ R (s) .With (B.1), we have (with V defined on R + through V (x) = V (r), if |x| = r(r) f R (r) ,By (B.34), we can estimate f R (R 0 ) ≤ C/ log(R/a). From (B.30), we also recall thatf R (r) ≤ f R (R 0 ) + CRλ R ≤ C/ log(R/a)for any r < R 0 . We conclude therefore that .1, we can now show Proposition 4.2. Proof of Prop. 4.2. The proof follows a strategy similar to [3, Lemma 8.2]. For fixed ξ ∈ F ≤N + and s ∈ [0; 1], we define With the results established in Subsections A.2 -A.5, we cam now show Prop. 3.3. Propositions A.3, A.4, A.5, A.6, A.7, imply thatA.6 Proof of Proposition 3.3 Cε 6 R .(B.25) which leads to f R (x) − 1 + ε 2 R 4 2 log(R/|x|) − 1 + x 2 R 2 − ε 4 R 16 log(R/|x|) 1 + 2x 2 R 2 ≤ Cε 4 R (log ε R ) 2 . (B.26) j=1 Π j + h.c. Combining with the bound for |x| > R 0 obtained differentiating (B.6), we obtain the desired estimate.Proof of Lemma B.1. By standard arguments (see for example[13,proof of theorem C1]), f R (x) is spherically symmetric and non-negative. We now start by proving an upper bound for λ R , consistent with (B.2). To this end, we calculate the energy of a suitable trial function. 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[ "Temperature dependence of (111) and (110) ceria surface energy", "Temperature dependence of (111) and (110) ceria surface energy" ]
[ "Anastasiia S Kholtobina *[email protected] ", "Axel Forslund ", "Andrei V Ruban ", "\nInstitute for Materials Science\nKTH Royal Institute of Technology\nBrinellvagen 23SE-100 44StockholmSweden\n", "\nUniversity of Stuttgart\n70569StuttgartGermany\n", "\nMaterials Center Leoben Forschung GmbH\nBörje Johansson Department of Physics and Astronomy, Division of Materials Theory\nKTH Royal Institute of Technology\nBrinellvagen 23, SE-100 44, Roseggerstraße 12A-8700Stockholm, LeobenSweden and, Austria\n", "\nUppsala University\n751 20UppsalaSweden\n", "\nNatalia V. Skorodumova\nKTH Royal Institute of Technology\nBrinellvagen 23, SE-100 44StockholmSweden\n", "\nKTH Royal Institute of Technology\nBrinellvagen 23, SE-100 44StockholmSweden\n" ]
[ "Institute for Materials Science\nKTH Royal Institute of Technology\nBrinellvagen 23SE-100 44StockholmSweden", "University of Stuttgart\n70569StuttgartGermany", "Materials Center Leoben Forschung GmbH\nBörje Johansson Department of Physics and Astronomy, Division of Materials Theory\nKTH Royal Institute of Technology\nBrinellvagen 23, SE-100 44, Roseggerstraße 12A-8700Stockholm, LeobenSweden and, Austria", "Uppsala University\n751 20UppsalaSweden", "Natalia V. Skorodumova\nKTH Royal Institute of Technology\nBrinellvagen 23, SE-100 44StockholmSweden", "KTH Royal Institute of Technology\nBrinellvagen 23, SE-100 44StockholmSweden" ]
[]
High temperature properties of ceria surfaces are important for many applications. Here we report the temperature dependences of surface energy for the(111)and(110)CeO2 obtained in the framework of the extended two-stage upsampled thermodynamic integration using Langevin dynamics (TU-TILD). The method was used together with machinelearning potentials called moment tensor potentials (MTPs), which were fitted to the results of the ab initio MD calculations for(111)and(110)CeO2 at different temperatures. The parameters of MTPs training and fitting were tested and the optimal algorithm for the ceria systems was proposed. We found that the temperature increases from 0 K to 2100 K led to the decrease of the Helmholtz free energy of (111) CeO2 from 0.78 J/m 2 to 0.64 J/m 2 . The energy of (110) CeO2 dropped from 1.19 J/m 2 at 0 K to 0.92 J/m 2 at 1800 K. We show that it is important to take anharmonicity into account as simple consideration of volume expansion gives wrong temperature dependences of the surface energies.I. INTRODUCTIONDue to its attractive redox, catalytic, electronic, and mechanical properties ceria (CeO2) has become a key material in many modern technologies[1][2][3][4][5][6]. In particular, the automobile exhaust gas catalysts [1,2], solid oxide fuel cells[5,6]and oxygen storage [7] largely rely on the surface properties of ceria[8]. In these applications ceria works at elevated temperatures [7] and, therefore, the knowledge of the thermodynamic properties of ceria surfaces at high temperatures is of great importance. Such information, however, is still limited, even the data about the energy of different crystallographic surfaces of ceria is scarce. At the same time, the experimental information regarding the properties of bulk ceria is available for different temperatures. For example, ceria heat capacity obtained for the temperature range of 2-900 K using the adiabatic scanning calorimetry and differential scanning calorimetry (DSC) have been reported in a number of studies[9][10][11][12][13]. The high-temperature enthalpy data for bulk ceria in the temperature range of 391-1800 K can also be found in literature[14][15][16]. Additionally, Hisashige and co-authors have measured the thermal expansion of ceria by thermomechanical analysis (TMA) in the temperature range from 100 to 800 K and the Debye temperature at room temperature by an ultrasonic pulse method[17].During the past decade a number of ab initio studies devoted to the high-temperature thermodynamic properties of bulk CeO2 were published[7,13,[18][19][20]. The self-consistent ab initio lattice dynamical (SCAILD) method[19], which includes the effects of phonon-phonon interactions using the quasi-harmonic approximation (QHA) provided a description of phonon and thermodynamic properties of bulk ceria at temperatures from 0 K up to 1500-1800 K[21]. Based on the QHA method and phonon calculations the thermodynamic properties, such as heat capacity, isothermal bulk modulus, Gibbs free energy, and coefficient of thermal expansion of CeO2 polymorphs were obtained in the temperature range of 0-1150 K [20]. Morrison et. al. [13] calculated the entropy, enthalpy, and Gibbs functions for bulk ceria at temperatures between 5 K and 400 K using the Perdew, Burke, and Ernzerhof parameterization revised for solids (PBEsol) and the simple Debye model [13]. The obtained Debye temperature (ΘD), 455 K [13], was in between the previously reported experimental data: 409 K [22], 480 K [17], and reported theoretical values of 481 K [23], 414.5-582.9 K [24]. Additionally, Klarbring et. al. [7] used the temperature dependent effective potential (TDEP) method to investigate several high temperature properties of ceria including thermal expansion [25,26]. Using PBEsol+U, Weck and Kim [23] obtained the crystalline parameters of CeO2 in good agreement with the experimental values. They also calculated the Debye temperatures within the Voigt-Reuss-Hill (VRH) approximation. Niu et. al. [24] obtained the pressure and temperature dependences of the specific heat, Debye temperature, and the thermal expansion coefficient for cubic CeO2 from the Debye-Grüneisen model.
10.1103/physrevb.107.035407
[ "https://export.arxiv.org/pdf/2301.05827v1.pdf" ]
255,877,270
2301.05827
5b8856413e6ccff0ea3cf05c2798c46331ef78bd
Temperature dependence of (111) and (110) ceria surface energy Anastasiia S Kholtobina *[email protected] Axel Forslund Andrei V Ruban Institute for Materials Science KTH Royal Institute of Technology Brinellvagen 23SE-100 44StockholmSweden University of Stuttgart 70569StuttgartGermany Materials Center Leoben Forschung GmbH Börje Johansson Department of Physics and Astronomy, Division of Materials Theory KTH Royal Institute of Technology Brinellvagen 23, SE-100 44, Roseggerstraße 12A-8700Stockholm, LeobenSweden and, Austria Uppsala University 751 20UppsalaSweden Natalia V. Skorodumova KTH Royal Institute of Technology Brinellvagen 23, SE-100 44StockholmSweden KTH Royal Institute of Technology Brinellvagen 23, SE-100 44StockholmSweden Temperature dependence of (111) and (110) ceria surface energy 1 High temperature properties of ceria surfaces are important for many applications. Here we report the temperature dependences of surface energy for the(111)and(110)CeO2 obtained in the framework of the extended two-stage upsampled thermodynamic integration using Langevin dynamics (TU-TILD). The method was used together with machinelearning potentials called moment tensor potentials (MTPs), which were fitted to the results of the ab initio MD calculations for(111)and(110)CeO2 at different temperatures. The parameters of MTPs training and fitting were tested and the optimal algorithm for the ceria systems was proposed. We found that the temperature increases from 0 K to 2100 K led to the decrease of the Helmholtz free energy of (111) CeO2 from 0.78 J/m 2 to 0.64 J/m 2 . The energy of (110) CeO2 dropped from 1.19 J/m 2 at 0 K to 0.92 J/m 2 at 1800 K. We show that it is important to take anharmonicity into account as simple consideration of volume expansion gives wrong temperature dependences of the surface energies.I. INTRODUCTIONDue to its attractive redox, catalytic, electronic, and mechanical properties ceria (CeO2) has become a key material in many modern technologies[1][2][3][4][5][6]. In particular, the automobile exhaust gas catalysts [1,2], solid oxide fuel cells[5,6]and oxygen storage [7] largely rely on the surface properties of ceria[8]. In these applications ceria works at elevated temperatures [7] and, therefore, the knowledge of the thermodynamic properties of ceria surfaces at high temperatures is of great importance. Such information, however, is still limited, even the data about the energy of different crystallographic surfaces of ceria is scarce. At the same time, the experimental information regarding the properties of bulk ceria is available for different temperatures. For example, ceria heat capacity obtained for the temperature range of 2-900 K using the adiabatic scanning calorimetry and differential scanning calorimetry (DSC) have been reported in a number of studies[9][10][11][12][13]. The high-temperature enthalpy data for bulk ceria in the temperature range of 391-1800 K can also be found in literature[14][15][16]. Additionally, Hisashige and co-authors have measured the thermal expansion of ceria by thermomechanical analysis (TMA) in the temperature range from 100 to 800 K and the Debye temperature at room temperature by an ultrasonic pulse method[17].During the past decade a number of ab initio studies devoted to the high-temperature thermodynamic properties of bulk CeO2 were published[7,13,[18][19][20]. The self-consistent ab initio lattice dynamical (SCAILD) method[19], which includes the effects of phonon-phonon interactions using the quasi-harmonic approximation (QHA) provided a description of phonon and thermodynamic properties of bulk ceria at temperatures from 0 K up to 1500-1800 K[21]. Based on the QHA method and phonon calculations the thermodynamic properties, such as heat capacity, isothermal bulk modulus, Gibbs free energy, and coefficient of thermal expansion of CeO2 polymorphs were obtained in the temperature range of 0-1150 K [20]. Morrison et. al. [13] calculated the entropy, enthalpy, and Gibbs functions for bulk ceria at temperatures between 5 K and 400 K using the Perdew, Burke, and Ernzerhof parameterization revised for solids (PBEsol) and the simple Debye model [13]. The obtained Debye temperature (ΘD), 455 K [13], was in between the previously reported experimental data: 409 K [22], 480 K [17], and reported theoretical values of 481 K [23], 414.5-582.9 K [24]. Additionally, Klarbring et. al. [7] used the temperature dependent effective potential (TDEP) method to investigate several high temperature properties of ceria including thermal expansion [25,26]. Using PBEsol+U, Weck and Kim [23] obtained the crystalline parameters of CeO2 in good agreement with the experimental values. They also calculated the Debye temperatures within the Voigt-Reuss-Hill (VRH) approximation. Niu et. al. [24] obtained the pressure and temperature dependences of the specific heat, Debye temperature, and the thermal expansion coefficient for cubic CeO2 from the Debye-Grüneisen model. The information about ceria surfaces at elevated temperatures is scarce in experimental publications and totally non-existing in theoretical reports. Zouvelou et. al. [27] described the first experimental determination of the surface energies of polycrystalline CeO2 in the argon atmosphere in the 1473-1773 K temperature range, which were measured to be 1.64-1.47 J/m 2 . Hayun et. al. [28] determined the surface energy of nanoceria for hydrated and anhydrous samples at room temperature to be 0.81 and 1.16 J/m 2 , respectively. Hayun et. al. concluded that these surface energies could be attributed to the (111) surface. This assumption was based on the systematic study of the CeO2 nanoparticles of different sizes using high-resolution transmission electron microscopy (HRTEM) [29] and the work by Vyas et. al. [30] showing that CeO2 equilibrium morphologies are dominated by the (111) facets. Here we report the calculated free energies of the (111) and (110) ceria surfaces at temperatures up to 2100 K. The values were calculated with the extended two-stage upsampled thermodynamic integration using Langevin dynamics (TU-TILD) method [31,32]. This method was previously applied in the free energy calculations of various metallic systems, including vacancy formation free energies [33], stacking fault free energies [31] as well as the surface free energy of TiN [34] and W [35]. In the current study TU-TILD was combined with machine-learning potentials, in particular, the moment tensor potentials (MTP) [36], which were trained on the results of ab initio molecular dynamics (AIMD) calculations [37]. II. METHODOLOGY The methodology of the two-stage up-sampled thermodynamic integration using Langevin dynamics was applied in this work to determine the Helmholtz free energies of the ceria bulk and surface supercells and consequently the surface free energies of (111) and (110) ceria A. Surface free energy The slab technique was used to determine the surface free energy γ as: γ(T) = F (a , T) − F (a , T) 2A ,(1) where F (a , T), and F (a , T) refer to the Helmholtz free energies of the slab and bulk calculated for the same number of formula units, respectively; a , is the lattice constant at temperature T and zero pressure, and A is the surface area of the slab. Factor 1 /2 accounts for the two surfaces of the slab. The Helmholtz free energies of both bulk and surface supercells (subscripts are omitted in the formula) can be adiabatically decomposed into the following contributions: F(a , T) = E(a ) + F (a , T),(2) where E denotes the conventional 0 K total energy of the system (either bulk or slab) and F the vibrational free energy of the lattice, obtained in the fully anharmonic form using the TU-TILD method. B. Anharmonic free energy calculations We applied the two-stage up-sampled thermodynamic integration using Langevin dynamics method, which treats the interatomic potential as an intermediate reference potential in the thermodynamic integration. The thermodynamic integration is split in two stages: first, from the harmonic to the reference potential and, secondly, from the reference potential to full DFT [38]. In the framework of this study, a modified version of the original TU-TILD method was utilized. The first modification was the usage of an optimized Einstein crystal as the analytic reference to compute the absolute free energy, instead of a quasiharmonic reference. The Einstein crystal is a simple and convenient workaround [39]. The corresponding Einstein frequency can be chosen quite arbitrarily within a reasonable interval specific for this system. Since the Einstein system is used only as an auxiliary reference for thermodynamic integration, the choice does not affect the final result obtained after the integration. The second modification of the original TU-TILD method implemented here was the usage of a machine-learning potential, namely MTP, as an efficient bridge between the analytical reference system and the DFT system. Following this formalism [40,41], the full vibrational free energy including the anharmonic is obtained as follows: F = F + F → + F → ,(3)where F → = dλ < E − E > ,(4)F → = dλ < E − E > +< ΔE > .(5) Where F is the free energy of an optimized Einstein crystal. E , E , and E are the energies of a particular atomic configuration calculated for the Einstein crystal, calculated with MTP [42] as implemented in the MLIP software [36], and calculated with low-converged DFT parameters, respectively. <…>λ denote the thermodynamic average for particular coupling constant λ, certain temperature and volume. Finally, < ΔE > is obtained within the free-energy perturbation theory and it accounts for the free energy difference between the low-and well-converged DFT calculations [32]. C. Machine learning potentials MTPs are the class of machine-learning (ML) potentials first proposed by Shapeev et. al. for single-component materials [42] and later extended to multi-component systems [43]. MTPs are efficient in combination with the TU-TILD method [1,2]. In the framework of machine learning methods, each considered model should be optimized. In order to avoid overfitting or underfitting during potentials training, the root mean square error (RMSE), calculated between the reference outputs and model predictions, were compared at the end of training process [44]. MTPs represent the energy of an atomic configuration as a sum of the contributions of the local atomic environments of each atom i: E = E ,(6) where each contribution Ei is linearly expanded via a set of basis functions, E = ξ B ,(7) where ξ = {ξα} are parameters to be found by fitting to the training set. To train MTPs, an active learning technique can be applied in the framework of the MLIP package [36]. This technique allows one to entrust training set refinement iterations to the computer, thus, completely automating the training set construction. A good training set should include all representative structures, so that the potential does not have to "extrapolate" while searching for stable phases. This goal can be achieved by treating the active learning technique as a generalization of the algorithm proposed for linearly parametrized models by Gubarev [45,46]. The full iteration of the active learning algorithm consists of five steps [47]: 1) training set is constructed from well-converged configurations obtained in DFT-AIMD calculations at the considered temperature; 2) pre-training procedure is implemented for untrained MTP with the defined MTP level and cutoff radius in order to for the first time define current MTP (MTP renewed at each iteration); 3) simulation with the current MTP is performed using the LAMMPS-MLIP interface and the extrapolative configurations are selected; the simulation is stopped when the maximum extrapolation grade [3] is exceeded, and an update of the training set is performed; 4) should new configurations be added to the training set, the total energies of these configurations are calculated with DFT-AIMD and then added to the trained set; 5) MTP is retrained using the updated training set. The whole procedure should be repeated until no new configurations appear in the third step. If only first two steps are used, we call such a procedure "passive" learning or training since the training set in this case is generated manually and MTP is not "adding" any new configurations [36]. III. COMPUTATIONAL DETAILS The DFT calculations were performed using the projector augmented wave (PAW) method [48] as implemented in the Vienna ab initio simulation package (VASP) [49]. The exchange and correlation effects were treated using the Perdew-Burke-Ernzerhof solid PBEsol [50] parametrisation of the generalized gradient approximation (GGA). Weck et. al. [51]previously demonstrated that PBEsol described the experimental crystalline parameters and properties of CeO2 and Ce2O3 with good accuracy. For our purpose, the description of cerium oxides within the DFT+U formalism is sufficient [52]. Therefore, the calculations were performed using PBEsol+U energy functional with the Hubbard parameter U of 5 eV applied to the 4f-states of ceria. The PAW potentials with the following electronic configurations were used: Ce 4s4p4f 5d6s and O 2s2p. All calculations were spin polarized with the initial ferromagnetic spin arrangement. The equilibrium lattice parameter of ceria obtained with PBEsol+U was 5.40 Å (0 K), in fair agreement with the experimental value [27]. A. Bulk free energy calculations The Helmholtz free energy of the bulk system, F (a , T) was obtained using Eq. (2). The convergence parameters were chosen to achieve the accuracy of 1 meV/atom or below. The 0 K total energy, Ebulk (a ), was calculated for the 96-atom supercell built as a 2 × 2 × 2 replication of the 12 atom cubic cell. The total energies were computed for the 12 volumes, equilibrium at different elevated temperatures [7]. For elevated temperatures we used the lattice parameters determined in our previous work [7], which were in good agreement with earlier experimental [25,26] and theoretical data [24]. The plane-wave cutoff was set to 500 eV and the k-point mesh was 2 × 2 × 2 [53]. The vibrational free energy F (a , T) was calculated by the TU-TILD method. The procedure of this calculation consisted of the following three steps: 1) ab initio molecular dynamics runs performed in VASP at 12 temperatures from 450 to 2100 K with 150 K step; 2) 12 moment tensor potentials training in the framework of the active learning algorithm; 3) two-stage up-sampled thermodynamic integration performed using Langevin dynamics with 12 trained MTPs for each considered temperature. The thermodynamic integration (step 3) includes the following sub-steps corresponding to Eqs.3-5: 3.1) calculations of MTP correction to the Einstein crystal model; 3.2) calculations of the DFT correction to the MTP free energy; 3.3) additional calculations for the up-sampling term. All 12 AIMD simulations were run with the Langevin thermostat [54] with the damping parameter of 0.01fs -1 . The van-Gunsteren-Berendsen algorithm [55] was used for the integration of the Newton's equations of motion. 1 fs time step was determined to be sufficient for AIMD runs. Each MD run was done for 6000 steps. For every temperature we trained MTP for it to reproduce the energy and forces of the 6000 atomic configurations for bulk ceria obtained from DFT-AIMD. We did it always in the framework of the active learning approach. The 16 th level of MTP and 5 Å cutoff radius were chosen. The resulting root-mean-square error (RMSE) of the energy difference between DFT and MTP was 1.6 meV/atom and RMSE of the force was 0.51 eV Å -2 demonstrating a satisfactory reproducibility of the DFT energies and forces by the fitted MTPs. Having fitted MTPs for the 12 considered temperatures, we could start the two-stage up-sampled thermodynamic integration using Langevin dynamics (Eqs. [3][4][5]. The F → correction was obtained for the 324-atom supercell built as the 3 × 3 × 3 replication of the 12 atom cubic cell. The convergence of the free-energy correction was below 1 meV/atom for all temperatures. It might be important to use a large enough supercell in this integration in order to capture the contribution of the long wavelength phonons. At every temperature, a dense set of 26 λ1 values was used for the integration in Eq. (4). For each λ1, LAMMPS MD runs up to 50 000 steps were performed to get statistically well converged results. The F → correction was obtained using the 96-atom supercell. Due to a high computational cost of the calculations at this step only five λ2 values (0, 0.25, 0.5, 0.75,1) were chosen for the integration in Eq. (5). For each λ2 one MD run with 1000 MD steps was performed. This sampling resulted in the statistical error less than 0.2 meV/atom demonstrating the excellent performance of MTP in reproducing the DFT values. Term E in Eq. (5) was calculated using the plane-wave cutoff of 500 eV and the 2 × 2 × 2 Monkhorst-Pack [53] k-point mesh for the 2x2x2 supercell. The calculations of the up-sampling term in Eq. (5), < ΔE > , were carried out with the plane-wave cutoff of 700 eV and the 4 × 4 × 4 k-point mesh. It appeared that it was enough to perform 10 up-sampling calculations at each temperature to obtain highly converged energies. B. Slab free energy calculations The Helmholtz free energy of the surface slab, F (a , T), was obtained using Eq. (2). All the parameters were converged to the accuracy of 1 meV/atom. The 0 K energy E (a ) was calculated for the two (111) surface supercells: 2 × 2 and 4 × 4 in the xy directions, both with the thickness of 9 layers, containing 36-and 144 atoms, respectively. The (110) CeO2 supercell was 2 × 1 in the xy directions with the thickness of 7 layers (42 atoms). The vacuum was 15 Å thick in all cases. The Monkhorst-Pack [53] k-point meshes of 2 × 2 × 1 and 4 × 4 × 1 were used for 4x4 and 2x2 (111) supercells, respectively. For the (110) 2x1 supercell, 4 × 6 × 1 k-point mesh was used. The plane-wave cutoff was 500 eV in all surface calculations. To calculate F (a , T), MTPs for the surface slabs were fitted in the same manner as bulk-MTPs. The initial DFT AIMD runs were performed for 12 temperatures in the ranges of 450-2100 K for the (111) slab and for 7 temperatures in the ranges of 600-1800 K for the (110) slab. For slab-MTPs, just like for bulk MTPs, the 16 th level and cutoff radius of 5 Å were chosen. At each temperature the slab-MTPs were trained to the energies and forces of the 6000 atomic configurations obtained in DFT AIMD runs. The active learning algorithm of MTP training was applied, just like in the case of bulk MTPs. The resulting root-mean-square error (RMSE) of the energy difference between DFT and MTP was 0.3 -0.5 meV/atom and the RMSE for the force was 0.04 -0.08 eV Å -2 . For the surfaces correction F IV. RESULTS AND DISCUSSION Here we present the results of the application of the described above methodology to the (111) and (110) ceria surfaces. The temperature dependence of the surface free energy for (111) CeO2 including full anharmonic vibrational contribution is shown in Fig.1.The surface energy decreases from 0.78 J/m 2 at 0 K to 0.63 J/m 2 at 2100 K. FIG. 1. The surface free energy of (111) ceria surface as the function of temperatures for different supercell sizes, type of MTP training and corrections to the free energy (Einst → MTP and MTP → DFT). Green squares: γ + γ → for the 36 atom unit cell with MTPs obtained from the "passive" learning procedure; blue circles: γ + γ → for the 144 atom unit cell with MTPs trained using "passive" learning; blue triangles: γ + γ → for the 144 atom unit cell with MTPs obtained from "active" learning; the orange triangles: γ = γ + γ → + γ → for the 144 atom unit cell using MTPs obtained "active" learning. In Fig. 1 we compare results obtained for 36 and 144 atoms supercells. It is obvious that already the 36-atom cell allows one to get reasonably accurate results up to high temperatures. Our tests done for the 144-atom cell show that the surface energy does not change whether we use the "active" or "passive" learning procedure for MTP training (Fig.1). Based on this finding, only the last variants of MTPs were used for the 36-atom cell in order to reduce computational cost. Notice that the data in Fig.1 also shows that in this case neither the computationally expensive MTP→DFT correction ( γ → including the up-sampling procedure) causes any change. This might be explained by a rather high accuracy of our initial MD calculations used to train MTPs in comparison with those used in previous works [34,35] where only the Gamma point and a rather low cutoff energy were used. Therefore, it is not surprising that the authors of Ref [38] obtained a significant MTP→DFT correction providing a noticeable contribution to the final surface free energy. Based on our tests performed for the (111) surface, we decided to use a 42-atom cell and passive training for MTPs in our calculations for the (110) surface. The MTP→DFT correction, γ → was also neglected. The results for (110) CeO2 are shown in Fig.2 A. Fig.2 B presents the surface energy dependence for (111) CeO2. The (110) surface free energy decreases from 1.19 J/m 2 at 0 K to 0.92 J/m 2 at 1800 K. Thus, the surface energy decreases in between 0 K and 1800 K is 0.27 J/m 2 for (110) CeO2 and 0.15 J/m 2 for the close packed (111) surface for the same temperature interval. We notice that it is necessary to take anharmonicity into account in order to adequately describe temperature dependence of ceria surface energy. To demonstrate this point Fig. 2 A, B show the surface energies calculated at 0 K for the volumes corresponding to the considered temperatures. Fig. 2 A, B also present the results of our transferability tests of the obtained MTPs. The following two types of the transferability tests were performed: (1) The use of the MTP trained at 1800 K for the evaluation of the surface free energy at lower temperatures. This test was applied for both ceria surfaces. MTPs trained at 1800 K ( Fig. 2 A, B) demonstrate good agreement with the results obtained with the MTPs trained at each corresponding temperature (Fig. 2 A, B) both for (111) and (110). The conclusion can be drawn that at least in the case of ceria surfaces it is enough to train MTP at a high temperature and apply it for a range of lower temperatures to estimate the surface free energy with reasonable accuracy. (2) The use of the individual MTP trained at a particular temperature for (111) ceria for the description of surface free energy of (110) ceria at the same temperature ( Fig.2 B) In this case, the best agreement can be found at lower temperatures, for example, 600 K but this approach still can be used for a rough and quick estimation of the (110) surface free energy even at high temperatures. FIG. 2. Surface energies calculated for (111) a) and (110) b) ceria surfaces as a function of the temperature. Green squares: without MTPs at 0 K using lattice parameters respective to finite temperatures. Blue circles: using the MTPs of corresponding surfaces at respective finite temperatures. Red diamonds: using the 1800 K MTPs of corresponding surfaces. Yellow stars: using the MTPs trained for (111) surface but applied for (110) surface at respective finite temperatures. Additionally, we tried to use the MTP trained for the (110) surface to describe the (111) surface. This attempt, however, showed no good results even for low temperatures. We note that the MTPs trained for the bulk neither provided any reasonable description of the (111) surface or (110) surface. Finally, we have shown that the calculation of surface free energy without vibrational contribution to Helmholtz free energy gives us quite different values than the one we obtained taking all contributions into account (green curves in Fig. 2 A, B). The calculated surface energies can be compared with the available experimental data. Hayun et. al. [28] reported the surface energy value of 0.81 J/m 2 at room temperature, which is in fair agreement with our 0K value for the (111) surface (Fig.2 B). Zouvelou et. al. [27] reported the surface energy of polycrystalline CeO2 to be 1.116 -0.998 J/m 2 in the 1473 -1773 K temperature range, thus, the surface energy decreases in this temperature interval is about 0.12 J/m 2 [27]. Note that the type of the surface was not specified in Ref. [27]. Fig.2 A, B demonstrate that for both surfaces the free energy decreases in the temperature interval of 1500 -1800 K. However, in the case of the (111) surface the modest difference of 0.03 J/m 2 was obtained whereas for the (110) surface it was 0.12 J/m 2 , in good agreement with results by Zouvelou et. al. [30]. (Fig.4) obtained from 50000 step MD runs at 1800 K. For both surfaces, as expected, the mobility of surface atoms is larger than that of the bulk atoms. Oxygens shift much further from their crystallographic sites than heavy cerium atoms. In the fluorite structure of ceria, each oxygen is placed in the middle of a tetrahedron of four cerium atoms. The oxygen atom movement is easier through the facet of the cerium tetrahedron than through its edge between two Ce atoms. Under some projection angles Figs. 3 and 4 demonstrate the characteristic triangular shape of the trajectory distributions. V. CONCLUSIONS We demonstrated that the proficient methodology for computing the fully anharmonic surface free energy from ab initio calculations based on the TU-TILD method can be successfully applied to oxide surfaces, in particular to CeO2 (111) and (110) surfaces. The optimal algorithm for the surface free energies calculation in the case of ceria systems has been proposed. It has been shown that for the considered ceria systems active training of the MTPs and utilizing of the increased supercell sizes can be excessive if the original MD calculations are performed with reasonably high accuracy. We have also found that MTP-DFT correction and up-sampling term can be neglected for ceria surfaces again if the initial MD calculations have more k-points than just Gamma point and the cutoff energy of 500 eV. The surface free energy changing from 0.78 J/m 2 at 0 K to 0.64 J/m 2 at 2100 K in the case of CeO2 (111) and from 1.19 J/m 2 at 0 K to 0.92 J/m 2 at 1800 K, in the case of the (110) surfaces. The obtained results are in reasonable agreement with the experimental data by Zouvelou et. al. [27] and Hayun et. al. [28]. It is essential to take anharmonic contributions into account to adequately describe the temperature dependence of the surface free energy of ceria. 672 atoms, respectively. All other parameters were set as for the respective bulk calculations. Correction F → for surface slabs was obtained using (111) 4 × 4 and (110) 4 × 2 supercells containing 144 and 168 atoms, respectively. The same set of Lambda values (0, 0.25, 0.5, 0.75, 1) as for the corresponding bulk calculations was used here. At each λ2, 1000 step MD runs were carried out with the following DFT parameters: the plane-wave cutoff energy of 500 eV and the 2 × 2 × 1, 4 × 4 × 1 Monkhorst-Pack [53] k-point meshes for the (111) 4 × 4 and (110) 4 × 8 supercells. For up-sampling the following parameters were used: cutoff energy of 700 eV and 6 × 6 × 1 and 4 × 4 × 1 k-point meshes for 36 and 144 atomic (111) CeO2 supercells, respectively, and the 6 × 8 × 1 k-point mesh for 42 atom (110) CeO2 cell. Finally, the surface free energy was calculated according to Eq. (1). FIG. 3 . 3Atomic trajectories plotted for the (111) surface at 1800 K. a) projection from the [-110] direction, b) projection from the [11-2] direction. Oxygen atoms are green and cerium atoms are yellow. FIG. 4. Atomic trajectories for the (110) ceria surface at 1800 K. a) projected along the in-plane [001], b) projected along the in-plane [110]. Oxygen atoms are green, cerium atoms are yellow. Figs 3 and 4 present the trajectories of oxygen and ceria atoms in (111) (Fig.3) and (110) surface slabs ACKNOWLEDGMENTSThe computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at High Performance Computing Center North and National Computer Center at Linköping University partially funded by the Swedish Research Council (VR-RFI) and through PRACE resources. We acknowledge SNIC and PRACE for awarding us access to Tetralith, Sweden,( Alternative Energy Technologies. M S Dresselhaus, I L Thomas, Nature. 414332M. S. Dresselhaus and I. L. Thomas, Alternative Energy Technologies, Nature 414, 332 (2001). CeO2 -Encapsulated Noble Metal Nanocatalysts: Enhanced Activity and Stability for Catalytic Application. S Song, X Wang, H Zhang, NPG Asia Mater. 7S. 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[ "Towards Optimally Efficient Search with Deep Learning for Large-Scale MIMO Systems", "Towards Optimally Efficient Search with Deep Learning for Large-Scale MIMO Systems", "Towards Optimally Efficient Search with Deep Learning for Large-Scale MIMO Systems", "Towards Optimally Efficient Search with Deep Learning for Large-Scale MIMO Systems", "Towards Optimally Efficient Search with Deep Learning for Large-Scale MIMO Systems", "Towards Optimally Efficient Search with Deep Learning for Large-Scale MIMO Systems" ]
[ "Le He ", "Ke He ", "Lisheng Fan ", "Xianfu Lei ", "Fellow, IEEEArumugam Nallanathan ", "Fellow, IEEEGeorge K Karagiannidis ", "Le He ", "Ke He ", "Lisheng Fan ", "Xianfu Lei ", "Fellow, IEEEArumugam Nallanathan ", "Fellow, IEEEGeorge K Karagiannidis ", "Le He ", "Ke He ", "Lisheng Fan ", "Xianfu Lei ", "Fellow, IEEEArumugam Nallanathan ", "Fellow, IEEEGeorge K Karagiannidis " ]
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This paper investigates the optimal signal detection problem with a particular interest in large-scale multiple-input multiple-output (MIMO) systems. The problem is NP-hard and can be solved optimally by searching the shortest path on the decision tree. Unfortunately, the existing optimal search algorithms often involve prohibitively high complexities, which indicates that they are infeasible in large-scale MIMO systems. To address this issue, we propose a general heuristic search algorithm, namely, hyper-accelerated tree search (HATS) algorithm. The proposed algorithm employs a deep neural network (DNN) to estimate the optimal heuristic, and then use the estimated heuristic to speed up the underlying memory-bounded search algorithm. This idea is inspired by the fact that the underlying heuristic search algorithm reaches the optimal efficiency with the optimal heuristic function. Simulation results show that the proposed algorithm reaches almost the optimal bit error rate (BER) performance in largescale systems, while the memory size can be bounded. In the meanwhile, it visits nearly the fewest tree nodes. This indicates that the proposed algorithm reaches almost the optimal efficiency in practical scenarios, and thereby it is applicable for largescale systems. Besides, the code for this paper is available at https://github.com/skypitcher/hats. Index Terms-Signal detection, integer least-squares, deep learning, maximum likelihood detection, MIMO, sphere decoding, best-first search L. He and L. Fan are both with the
10.1109/tcomm.2022.3158367.
[ "https://arxiv.org/pdf/2101.02420v6.pdf" ]
237,354,911
2101.02420
3fc853023d50605051297980ab6668819ef728b8
Towards Optimally Efficient Search with Deep Learning for Large-Scale MIMO Systems 4 Mar 2022 Le He Ke He Lisheng Fan Xianfu Lei Fellow, IEEEArumugam Nallanathan Fellow, IEEEGeorge K Karagiannidis Towards Optimally Efficient Search with Deep Learning for Large-Scale MIMO Systems 4 Mar 2022Index Terms-Signal detectioninteger least-squaresdeep learningmaximum likelihood detectionMIMOsphere decod- ingbest-first search This paper investigates the optimal signal detection problem with a particular interest in large-scale multiple-input multiple-output (MIMO) systems. The problem is NP-hard and can be solved optimally by searching the shortest path on the decision tree. Unfortunately, the existing optimal search algorithms often involve prohibitively high complexities, which indicates that they are infeasible in large-scale MIMO systems. To address this issue, we propose a general heuristic search algorithm, namely, hyper-accelerated tree search (HATS) algorithm. The proposed algorithm employs a deep neural network (DNN) to estimate the optimal heuristic, and then use the estimated heuristic to speed up the underlying memory-bounded search algorithm. This idea is inspired by the fact that the underlying heuristic search algorithm reaches the optimal efficiency with the optimal heuristic function. Simulation results show that the proposed algorithm reaches almost the optimal bit error rate (BER) performance in largescale systems, while the memory size can be bounded. In the meanwhile, it visits nearly the fewest tree nodes. This indicates that the proposed algorithm reaches almost the optimal efficiency in practical scenarios, and thereby it is applicable for largescale systems. Besides, the code for this paper is available at https://github.com/skypitcher/hats. Index Terms-Signal detection, integer least-squares, deep learning, maximum likelihood detection, MIMO, sphere decoding, best-first search L. He and L. Fan are both with the I. INTRODUCTION Given an observation vector y ∈ R n×1 and the transformation matrix H ∈ R n×m with full column rank (n ≥ m), the signal detection problem aims to recover the transmitted discrete signal x ∈ R m×1 from the linear mixing model y = Hx + w,(1) where w ∈ R n×1 denotes the additive white Gaussian noise with zero mean and unit variance. Generally, the detection problem has diverse applications which include -but it is not limited to-signal processing [1], communications [2], machine learning [3], global navigation satellite systems (GNSS) [4], and radar imaging [5]. In these areas, estimating the transmitted signal optimally in large-scale systems still remains a big challenge, and it is also of vital importance for improving the reliability of the whole system [6]. In this paper, we will henceforth concentrate on studying the optimal signal detection in large-scale multiple-input multiple-output (MIMO) wireless communication systems. However, it shall be noted that the proposed method is general, and can be applied to other applications. A large-scale MIMO system [7]- [9], which is one of the most promising wireless techniques to significantly improve the spectrum efficiency, has been widely studied during the last decades. Specifically, there are m c and n c antennas equipped at the transmitter and receiver such that n c ≥ m c ≫ 1. Moreover, H c refers to the random wireless channel state information (CSI) matrix with the (i, j)'s element h i,j ∼ CN (0, ρ) denoting the tap gain from the j-th transmit antenna to the i-th receive antenna. In further, x c denotes the transmit signal vector which is uniformly distributed over the finite set of lattice points A m . In fact, A is the alphaset depending on the underlying modulation schema, and a standard example of A would be A = {a + bj|a, b ∈ {−1, +1}} for 4quadrature amplitude modulation (4-QAM). While transmitting x c through the random wireless channel H c , it suffers from an additive white Gaussian noise w c . Therefore, the received signal can be expressed as y c = H c x c + w c .(2) In particular, we consider the following equivalent representation y = ℜ(y c ) ℑ(y c ) , H = ℜ(H c ) −ℑ(H c ) ℑ(H c ) ℜ(H c ) ,(3)x = ℜ(x c ) ℑ(x c ) , w = ℜ(w c ) ℑ(w c ) ,(4) where y c ∈ C nc×1 , H c ∈ C nc×mc , x c ∈ C mc×1 and w c ∈ C nc×1 are composed of complex values, and ℜ(·) and ℑ(·) denote the real part and imaginary part of the value, respectively. Based on this real-valued representation, (2) can be written as (1) with m = 2m c and n = 2n c . In lattice theory, H is considered to be the generator matrix [10] of the generated lattice L(H) = {Hx|x ∈ A m }, which indicates that the generated lattice is the "skewed" one of the original lattice. When H is perfectly known at the receiver, the mathematically optimal approach to solve the detection problem in terms of minimizing the average error probability, is to search the closest "skewed" lattice point Hx * to y, in terms of the Euclidean distance [2] x * = arg min x∈A m y − Hx 2 ,(5) which gives the exact optimal maximum likelihood (ML) estimate of x in (1). Since the signal vector x comprises of integer components only, (5) is also refereed as integer least-squares (ILS) problem in the literature [11]. Unfortunately, solving the ILS problem is much more challenging with comparison to the standard least-squares problem, where the latter's signal vector comprises of continuous entries rather than discrete entries, and the optimal solution can be efficiently resolved via pseudo inverse [1]. As a combinatorial optimization problem, it is known to be NP-hard because of its discrete search space. It has been proven that solving the problem optimally involves exponential complexity for all algorithms in the worst case [12], [13]. Moreover, with the deployment of 5G, the numbers of antennas m c and n c become very large to support the demand of high data rate and ultra-reliable low-latency communication (URLLC), which indicates that finding the optimal estimate becomes much more challenging as well. However, one can formulate the problem as a search in a state space, and efficiently reduce the average complexity by following a certain strategy that investigates only the necessary lattice points at each dimension [1]. In particular, the resulting state space forms a decision tree whose nodes represent decisions on symbols, and branches represent the costs of the associated decisions. Correspondingly, the strategies that decide which node should be chosen for expansion are called tree search algorithms. Once we find the shortest path, we can find the optimal solution. A. Related Works In the past decades, researchers have proposed many tree search algorithms to address the ILS problem [2], [10], [11], [14]- [22], and a comprehensive survey of these algorithms can be found in the fifty-years review of MIMO detection [6]. One common approach achieving the optimal ML detection performance is the sphere decoding (SD) algorithm, which employs a branch-and-bound (BnB) depth-first search (DFS) strategy to find the shortest path [2], [10], [11]. Besides, many variants of SD have been proposed in the literature as well. Among these variants, the Schnorr-Eucherr SD (SE-SD) achieves the optimal performance with reduced average complexity [10]. However, its average complexity still remains prohibitively high in large-scale MIMO systems with low signal-to-noise ratio (SNR) [1]. To overcome this drawback, K-best SD has been proposed to achieve a fixed and reduced complexity with a breath-first search (BrFS) strategy, whereas the performance is sacrificed [15]. More importantly, the complexity reduction can not be guaranteed in high SNR regimes [23]. Attempts that employ best-first search (BeFS) or stack algorithms to overcome the drawbacks of DFS and BrFS have been investigated in the literature [17]- [22], [24], [25]. Among these variants, A* like BeFS is the most promising variant which employs a heuristic function to predict the shortest remaining cost [17]. It guarantees to find the optimal solution with an admissible heuristic function, while it achieves the optimal efficiency with a consistent heuristic function [26]. It has been show that BeFS visits the fewest nodes among the three search strategies, but it requires exponentially increasing memory space [19]. Fortunately, several memorybounded BeFS algorithms have been proposed in the past years, and their performances are very close to the original one [21], [24], [25]. However, it is still very difficult to find a consistent heuristic function such that the optimal efficiency is achieved [20], [22]. Hence, the existing BeFS algorithms are still infeasible for large-scale MIMO systems. With the tremendous success of machine learning techniques on physical layer communication, significant improvement becomes possible for the aforementioned search strategies [27]- [32]. For example, the authors in [33] proposed a novel expectation maximization-based sparse Bayesian learning framework to learn the model parameters of the sparse virtual channel, which significantly reduces the overhead of the channel training in massive MIMO systems. To solve the crucial active antenna selection problem in massive MIMO, a deep learning (DL) based active antenna selection network was devised to utilize the probabilistic sampling theory to select the optimal location of these active antennas [34]. Besides, a preliminary theoretical analysis on DL based channel estimation was presented in [35] to understand and interpret the internal mechanism of single-input multipleoutput (SIMO) systems. The theoretical result shows that DL based channel estimation outperforms or is at least comparable with traditional channel estimation, depending on the types of channels, which encourages researchers to solve problems with promising DL based approaches. Regarding the ILS problem, researchers have proposed a DL based SD (DL-SD) algorithm which significantly improves SE-SD's performance by employing the deep neural networks (DNNs) to choose a good initial radius [36], [37]. Since the tree search can be treated as a sequential decision making process, the authors in [38] proposed a sub-optimal search algorithm named LISA with a fixed complexity, since LISA uses a DNN to make decisions with fixed steps. It was demonstrated that LISA could achieve very good performances with a fixed complexity. The success of these data-driven search algorithms certainly motivates us to improve the BeFS algorithm's performance with DL techniques. B. Contributions Consequently, we are interested in speeding up A* like BeFS algorithms with model-driven DL methods, while trying not to compromise the optimal bit error rate (BER) performance. To accomplish this goal, we propose a hyperaccelerated tree search (HATS) algorithm, which enhances the underlying efficient memory-bounded A* algorithm (SMA*) [24], through predicting the optimal heuristic with a welltrained DNN. This idea is inspired by the fact that with the optimal heuristic, A* not only becomes optimally efficient but also expands the fewest nodes [26]. In contrast, the other A* inspired search algorithms only find the admissible heuristic [20], [22], which limits the improvement of these methods. As we will show in the simulations, the proposed algorithm achieves almost the optimal BER performance in large-scale MIMO systems, while its memory size can be bounded and it reaches the lowest average complexity under practical scenarios. This suggests that the proposed algorithm is feasible for practical large-scale MIMO systems. II. MIMO SIGNAL DETECTION WITH HEURISTIC BEST-FIRST SEARCH In this section, we will take an efficient approach to interpret the optimal MIMO signal detection problem as a tree search, which enables us to decouple the vector-valued problem into a sequential decision making problem. After that, we will introduce the A* algorithm to find the shortest path, and discuss its properties accordingly. A. Tree Construction To construct a decision tree, we first perform QR decomposition on the CSI matrix H as H = Q 1 Q 2 R 0 (n−m)×m = Q 1 R,(6) where R ∈ R m×m is an upper triangular matrix, and the partitioned matrices Q 1 ∈ R n×m and Q 2 ∈ R n×(n−m) both have orthogonal columns. Then, we rewrite (1) as z = Rx + v,(7) where z Q T 1 y and v Q T 1 w. Note that the optimal MIMO signal detection problem is always squared of dimensions m after preprocessing. For convenience, we number the entries of matrices and vectors of (7) in a reverse order as    z m . . . z 1    =    r m,m . . . r m,1 . . . . . . . . . 0 . . . r 1,1       x m . . . x 1    +    v m . . . v 1    ,(8) where r i,j represents the (i, j)-th component of R after arranging from the bottom right to the upper left. After preprocessing, the squared Euclidean distance of a given candidate can be expanded as d 2 (x) = (z 1 − r 1,1 x 1 ) 2 b(x 1 ) + (z 2 − r 2,2 x 2 − r 2,1 x 1 ) 2 b(x 2 ) + · · · + (z m − r m,1 x 1 − · · · − r m,m x m ) 2 b(x m ) (9a) = m k=1 b(x k ),(9b) where the k-th incremental cost is denoted by which only depends on the partial signal vector (PSV) b(x k ) =   z k − k j=1 r k,j x j   2 ,(10)x k = [x k , x k−1 , . . . , x 1 ] T . The cumulative cost of the PSV x k is g(x k ) = k i=1 b(x i ),(11) and the successor can be computed recursively by g(x k+1 ) = b(x k+1 ) + g(x k ), ∀k = 0, 1, · · · , m − 1 (12) where the initialization g(x 0 ) ≡ 0 is set. Clearly, (12) constructs a perfect |A|-way m-level (starts from zero) decision tree. In the resulting tree, the deepest nodes represents the goal nodes associated with specific candidates. Since |A| is finite, the set of goal nodes can be enumerated as A m = {x m 1 , x m 2 , . . . , x m j , . . . , x m |A| m },(13) where each goal node x m j is associated with a signal candidate. For convenience, we denote the antecedent of a goal node x m j that locates at level k as x k j , which also represents the path leading from the root to that node. Note that we will sometimes ignore the subscript and use x k to represent an arbitrary node located at level k. We also use x j ⊆ x k (0 ≤ j ≤ k) to denote that x l is the antecedent of x k . On the contrary, x j ⊇ x k (0 ≤ k ≤ j) denotes that x j is the descendent of x k . Importantly, each node x k is associated with a branch cost b(x k ), and g(x k ) is the cumulative cost of the path leading from the root to that node. For demonstration, a simple binary 4-level decision tree is illustrated in Fig. 1, where m = 4 and A = {−1, +1}. B. Heuristic Tree Search Now it is straightforward to employ the A* algorithm [20], [22], [26], [39] to search the least-cost path on the tree. Before describing the algorithm, it is necessary to introduce the following notations, • ACTIVE is an ordered list with possibly limited space that stores nodes to be expanded, whose data structure is often a priority queue or self-balancing binary search tree (a.k.a. AVLTree) in practice. In this case, we can efficiently find the least-f -cost node with a computational complexity of O(1), which means that the algorithm is able to find the least-f-cost node at each loop with a very small fixed cost. • f (x k ) is the evaluation cost (f -cost) currently assigned to a node x k at the time. Note that f (x k ) is not static, and it may change during search. • A node is visited or generated if it has been encountered during the expansion of its parent. • A node is expanded if all of its successors have been generated during the expansion. • A node is in memory if it is inserted into ACTIVE. • The complexity of a search algorithm is defined in terms of the number of visited nodes. We are now ready to introduce the A* algorithm, and the pseudo code is presented in Algorithm 1. Throughout the paper, we shall emphasize that the tree search always starts at the dummy root x 0 . At each iteration, A* expands the least-fcost node among all the nodes that are in ACTIVE and waiting for expansion, and inserts all successors of that node into ACTIVE afterwards [22]. This process will terminate as long as a goal node is selected for expansion, and the selected goal node will become the output of the algorithm. As a heuristic BeFS algorithm, A* employs a heuristic function h(x k ) to guide the search. Specifically, the heuristic function estimates the remaining cost from x k to goal nodes, denoted by h(x k , x m ) = g(x m ) − g(x k ) (14) = m i=k+1   z i − i j=1 r i,j x j   2 , ∀x m ⊇ x k .(15) Accordingly, A* computes the evaluation cost of x k as f (x k ) = g(x k ) + h(x k ),(16) where h(x k ) represents the estimate of the minimum value of (14), and h(x m ) = 0 holds since the goal nodes have no successor at all. Therefore, f (x k ) actually estimates the cost of the shortest path of the sub-tree of x k . In particular, the optimal heuristic function always gives the minimum value of (14) as h * (x k ) = min x m ⊇x k h(x k , x m ).(17) In this case, the optimal f -cost is given by f * (x k ) = g(x k ) + h * (x k ).(18) With the optimal heuristic function, A* only expands the shortest path, as it always guides the search process towards the shortest path. C. Optimality and Optimal Efficiency We know that A* performs best with the optimal heuristic function, since only the nodes lying along the shortest path would be expanded eventually. In practical, we however often use a lower bound on (17) as the heuristic function, as finding the optimal heuristic function requires exhaustively search as well. In general, if a heuristic function h(x k ) Algorithm 1 A* Algorithm Input:z and R Output:x (estimate of the transmitted signal x) x k ← least-f -cost node in ACTIVE 7: if x k is a goal node then 8: returnx = x k with success 9: end if 10: for every successor x k+1 ⊃ x k do 11: f ′ (x k+1 ) ← g(x k+1 ) + h(x k+1 ) 12: f (x k+1 ) ← max f (x k ), f ′ (x k+1 ) 13: Insert x k+1 into ACTIVE 14: end for 15: Remove x k from ACTIVE 16: end loop never overestimates the optimal heuristic, then it is admissible [22]. In further, h(x k ) is said to be consistent if h(x k ) ≤ b(x k+1 ) + h(x k+1 ) holds for each node x k and its successors x k+1 . In other words, a heuristic function is consistent if it satisfies the triangle inequality [26]. It is clear that a consistent heuristic is also admissible, but not the vice verse. With an admissible heuristic function, A* is said to be optimal since it guarantees to find the shortest path [26]. In particular, the simplest admissible heuristic is h(x k ) = 0. Hence, the existing approaches mainly focus on finding a more meaningful admissible heuristic function based on the tighter lower bounds on (14) [20], [22]. In this case, A* is able to find the shortest path faster than the simplest heuristic. However, the performance improvements are still limited since the estimated heuristics are still far from the optimal heuristic. With a consistent heuristic function, A* is said to be optimally efficient in terms of that not only it is optimal but also no other optimal algorithm that uses the same heuristic would expand fewer nodes than it [26]. In this case, reducing the estimation error between a consistent heuristic and the optimal heuristic will only decrease the number of expanded nodes. Although A* can perform much better than other search algorithms, it has a main issue that its space complexity grows exponentially with the increasing search depth [26]. Consequently, finding a consistently good estimate of the optimal heuristic and improving the memory efficiency become the two essential approaches to improve the performance of A* in large-scale systems, which is also the intuition behind the strategy we will introduce in the following section. III. DEEP LEARNING ACCELERATED HEURISTIC BEST-FIRST SEARCH In this section, we will propose a general heuristic tree search algorithm, which significantly speeds up the heuristic BeFS algorithm by employing a DNN to estimate the optimal heuristic. After that, we will introduce the training strategy and discuss its computational complexity. A. Proposed Deep Learning Based Search Strategy Let h(x k |z, R, θ) denote the DNN parameterized by the trainable parameters θ. For convenience, we will simplify it as h θ (x k ) in the sequel. Accordingly, we define the corresponding f -cost as f θ (x k ) = g(x k ) + h θ (x k ).(19) As shown in Fig. 2, we use a fully-connected neural network (FCNN) composed of L fully-connected layers, and the number of neurons at the l-th (1 ≤ l ≤ L) layer is denoted by n l . Formally, the output of the l-th layer will be proceeded by the rectified linear unit (ReLU) based activation function, which can be expressed as p l = max 0, W l p l−1 + b l ,(20) where p l−1 ∈ R n l−1 is the output of the prior layer. Notations W l ∈ R n l ×n l−1 and b l ∈ R n l are the learnable weight matrix and bias vector, respectively. Note that the max{·, ·} operation is performed in component-wise manner. In particular, the input of the first layer is given by p 0 = z − R 0 (m−k)×1 x k .(21) In this case, the input size and output size are both fixed, which are set to n 0 = m and n L = 1, respectively. Hence, the structure of FCNN can be summarized as m, n 1 , n 2 , · · · , n l , · · · , n L−1 , 1 , and the set of trainable parameters is given by θ = W 1 , b 1 , · · · , W l , b l , · · · , W L , b L .(23) We shall emphasize that the DNN's structure is not limited to FCNN, and it should be chosen according to the problem scale and resources. In this paper, we consider FCNN since it is very simple and can be easily implemented by hardware to improve the efficiency. Besides, FCNN will be a good candidate if we consider that the signal detection problem in MIMO systems has typically much lower dimensions with comparison to computer vision tasks like image reconstruction. It should be noted that the major objective of this paper is to show the power of DNN based heuristic, and the presented DNN structure in this work is just for reference. In practice, we may choose convolutional neural networks (CNNs) to further reduce the computational complexity when the problem scale is very large. On the other hand, recursive neural networks (RNNs) will be a good candidate if the channel is correlated over time. Therefore, we need to select an appropriate network structure to approximate the heuristic according to the specific applications and available resources for the considered system. To leverage the DNN based heuristic function, we can combine it with heuristic BeFS search algorithms. Since A* may require prohibitively large memory space in large-scale problems, we will use the SMA* algorithm [24] as the underlying search strategy. As a variant of A*, SMA* is able to perform BeFS with limited memory space, and it is still equivalent to A* with enough memory size. By combining SMA* with the aforementioned deep heuristic function, we thus propose a hyper-accelerated tree search (HATS) algorithm for the optimal signal detection in large-scale MIMO problems. The pseudo code of HATS is detailed in Algorithm 2, and the associated utility functions are presented in Algorithm 3. In general, HATS works just like the A* algorithm. It keeps expanding the deepest least-f -cost node until ACTIVE is full, and the f -cost of each generated successor is computed according to (19). In particular, since the memory size is limited, HATS will forget the most unpromising node from ACTIVE, and remember its key information in its parent. Thus, the memory size can be bounded. After safely deleting the most unpromising node, the proposed algorithm is able to move forward, and recover back once there is no any other path better than the forgotten paths. Based on this strategy, HATS maintains a partially expanded sub-tree of the whole tree. Hence, the most unpromising node shall be the shallowest highest f -cost leaf node. It should be noted that a node is said to be a "leaf" node in terms of the partially expanded sub-tree. Thus, the forgotten leaf node is not necessary to be the goal node. While maintaining the partially expanded sub-tree, the proposed algorithm will recursively adjust the expanded node's cost according to the costs of its successors, and thereby the sub-tree is updated. In conclusion, at each iteration, the proposed algorithm expands the best node from memory, generates one successor and inserts the generated successor to ACTIVE at the time, and deletes the worst leaf node when the memory is full. B. Training Strategy It is clear that the training strategy is to minimize the average error between the estimated heuristic and the optimal one. To achieve this goal, we first know that with the optimal heuristic, the f -costs of the nodes on the shortest path of each sub-tree are all equal to the cumulative cost of the deepest node on that path. This can be easily proven according to the definition of the optimal heuristic. Following this point, one straightforward training strategy is to minimize the average ℓ 2 loss L(θ|D) = E D g(x m φ ) − f θ (x k φ ) 2 ,(24) where the empirical expectation is taken on the data set D, and D can be generated by traversing the shortest path at each time if x k is a goal node (k = N t ) then 8: returnx k with success 9: end if 10:x k+1 ← next not-generated valid successor or best forgotten successor ofx k 11: Insertx k+1 into its parent's generated successor list 12: ifx k+1 is not a forgotten node then 13: f if all ofx k 's successors are generated then 3:x k+1 ← least-cost successor among all generated successors and forgotten successors ofx k+1 (x k+1 ) ← max f (x k ), f θ (x k+1 ) 4: ifx k+1 's cost is finite and not equal tox k 's cost then 5: Updatex k 's cost tox k+1 's cost 6: ADJUST(x k 's parent) Remove shallowest highest-cost leaf nodex k j from ACTIVE 15: Removex k j from its parent's generated successor list 16: Rememberx k j 's cost in its parent's forgotten successor list 17: if the parent is not in ACTIVE then 18: Insert the parent into ACTIVE MAKESPACE 19: end if 20: end function slot. Specifically, the data set of T time slots can be expressed as D = S 1 1 , · · · , S m−1 1 , · · · , S 1 t , · · · , S k t , · · · , S m−1 T ,(25) where S k t denotes the k-th (1 ≤ k ≤ m − 1) sample drawn at time slot t (1 ≤ t ≤ T ). Specifically, S k t includes S k t = z, R, x k φ , x m φ ,(26) where z, R, x m φ are the associated received signal, triangular matrix and ML estimate at the time. Notation x k φ is the kth node on the shortest path x 0 → x m φ . It should be noted that the time index is only used for identifying the symbols randomly drawn from different trees. After successfully collecting enough samples from different time slots, we will further shuffle the whole data set before training the model. Hence, the time index has no impact on the final prediction. Base on the data set, we can optimize the network parameters θ by employing mini-batch gradient descent methods (e.g. the SGD optimizer [40] and ADAM optimizer [41]) to find the optimal parameters as θ * = arg min θ L(θ|D). Nevertheless, we still have to compute the ML estimate, which will slow down the sampling efficiency during the training. To tackle this problem, we can use the transmitted vector x rather than the ML estimate x m φ . This is a reasonable approximation in the sense of signal estimation, since the ML estimate is most likely to be x, especially for high SNR regimes. By using this strategy, we can train the model well very soon, and still have a very good performance. In summary, the advanced training procedure for the proposed algorithm is detailed in Algorithm 4. C. Computational Complexity The computational complexity of HATS depends on both the number of visited nodes and the visitation costs. While visiting a node x k , the computational complexity is O k + L l=1 n l n l−1 + n l ,(28) where the first term denotes the complexity of computing g(x k ) and the second term denotes the complexity of computing h θ (x k ). When the DNN estimates the optimal heuristic precisely, the algorithm visits the fewest nodes. Thus, the lower bound on the average complexity of HATS is given by O m 2 + m L l=1 n l n l−1 + n l ,(29) since only the m nodes lying on the shortest path would be eventually expanded. When the estimation is imperfect, the optimality and the optimal efficiency can not be guaranteed, and the complexity may grow exponentially with the problem scale in the worst case. However, as we have discussed in Sec. II-C, decreasing the estimation error will significantly improve the BER performance and search speed. Therefore, the performance and complexity of HATS rely on both the estimation for t = 1, 2, · · · , T do 7: Randomly collect z, R and x with random SNR 8: // Traverse the approximately shortest path of the whole tree 9: for k = 1, 2, · · · , m do 10: Select the k-th node x k on the path x Compute the average loss on D according to (24) 21: Perform a stochastic gradient decent step on the network's parameters θ 22: end for 23: Save the well-trained parameters θ quality of DNN and the system's SNR. In fact, as we will show in the simulations, the proposed algorithm achieves almost the optimal BER performance, while it reaches almost the lowest complexity in low SNR regimes. That is, with the proposed training strategy, the network is able to predict the optimal heuristic accurately. Thus, HATS can often reach the optimal efficiency under practical scenarios, which indicates that the proposed algorithm is applicable in large-scale systems. IV. SIMULATION RESULTS In this section, we will present simulation results and discussions to show the effectiveness of the proposed algorithm. Specifically, we will first introduce the environment setup and the implementation details of our model. Then, we will introduce the competing algorithms and discuss the related simulation results afterwards. A. Environment Setup In the simulations, we consider a MIMO system model where there are m c and n c antennas at the transmitter and receiver, respectively. Therefore, we have m = 2m c and n = 2n c . In addition, the signal is modulated by QPSK or 16-QAM, and the transmission experiences a random wireless channel. Moreover, the channel information can be known perfectly at the receiver. In practice, the columns of the CSI matrix may be somehow correlated. To perform simulations over correlated channels, we adopt the well-known Kroneker model introduced in [42] to generate the correlated channel matrices. Mathematically, the complex channel matrix H c in Kroneker model is given by H c = R 1 2 rHc R 1 2 t ,(30) whereH c denotes a Rayleigh flat fading channel matrix with independent and identically distributed (i.i.d.) random entries, and R r and R t are two covariance matrices at the receiver and transmitter, respectively. Without lost of generality, we assume that the correlation occurs at the transmitter, and R r and R t are set as R r = I, R t =      1 ρ · · · ρ ρ 1 · · · ρ . . . . . . . . . . . . ρ · · · ρ 1      ,(31) where ρ ∈ [0, 1] is the correlation coefficient. As to the DNN structure, we use the same DNN structure for all experiments, where there are 4 hidden layers in total and the neuron numbers of the four hidden layers are set to 128, 64, 32, and 16, respectively. According to (22), the structure of DNN can be summarized as To train the model, we employ the Adam optimizer [41] with the learning rate of 10 −6 for all experiments. The mini-batch size is set to 128 time slots, and the total batch size is 10 million in the training. The training samples are randomly generated within a range of SNRs varying from 0 dB to 30 dB. For the performance test, we will test the aforementioned algorithms with enough times to ensure a stable BER performance. B. Competing Algorithms To verify the effectiveness of the proposed algorithm, we compare the proposed algorithm with several competitive algorithms. Before discussing the results, we first introduce the following abbreviations, • DL-SD: The DL based sphere decoding algorithm introduced in [37]. • A*: The A* algorithm introduced in [24]. Note that the heuristic is set to zero constantly for comparison. • HATS(M ): The proposed algorithm in this paper, where M denotes the memory capacity. Among these algorithms, OAMP-Net2, DL-SD and the proposed HATS algorithm are DL based algorithms. Besides, SD, DL-SD, A* and HATS are search algorithms. In particular, DL-SD requires linear space with the search depth, while A* has exponential space complexity. Moreover, we will take simulations for HATS with different memory sizes. In particular, HATS(∞) denotes that the memory size is unlimited. In this case, HATS(∞) is equivalent to A* except that the heuristic is estimated by a DNN. Since SD, DL-SD and A* are optimal search algorithms, we will simply denote their BER performances as "ML" for convenience. Unlike the existing literature that measures the complexity in terms of the number of visited lattice points (goal nodes) [37], we will compare the complexities of search algorithms based on the number of visited nodes in the simulations, since it is more accurate to measure the complexity. C. BER Performance and Complexity Comparison On Uncorrelated Channels Figs. 3-5 illustrate the BER comparisons of the aforementioned algorithms in different-scale MIMO systems, where the channel is uncorrelated, and the numbers of antennas are 8, 16, 32 at both the transmitter and receiver, respectively. In addition, SNR varies from 5 dB to 15 dB. Note that algorithms have to search the closest lattice point among 2 64 candidates to reach the exact optimal BER performance in 32 × 32 MIMO, which is a very big challenge. It should be noted that all the ML performances in these figures are simulated based on the SD algorithm. In particular, the BER results of "ML" with SNR ranging from 5 dB to 9 dB are not plotted in Fig. 5, since the computational complexity of SD becomes prohibitively large in this regime. From these figures, we can find that HATS achieves almost the optimal BER performance in all three systems. When SNR = 15 dB, HATS produces only slightly 10%, 10% and 15% more error than the optimal search algorithms in 8 × 8, 16 × 16 and 32 × 32 MIMO systems, respectively. As a contrast, for the sub-optimal algorithms like MMSE and OAMP-Net2, their BER performances are significantly far from the optimal one in all three MIMO systems, especially when the problem scale becomes prohibitively large. Moreover, we can also find from these figures that the performance gaps between HATS and the other sub-optimal algorithms enlarge with the increasing SNR. In particular, when SNR = 15 dB, HATS reduces the error of MMSE and OAMP-Net2 to only about 0.0052% and 1.504% in 32 × 32 MIMO, respectively. Besides, the SNR gains of HATS over OAMP-Net2 are about 3.2 dB, 2.1 dB and 1.8 dB at the BER levels of 10 −3 , 10 −4 , and 10 −5 in 8 × 8, 16 × 16 and 32 × 32 MIMO systems, respectively. In conclusion, these results show that the proposed algorithm is robust and almost optimal even in large-scale problems. In order to show the impact of SNR and memory constraint on the computational complexity of the proposed algorithm, Figs. 6-8 are provided to show the complexity comparison under the same setup as Figs. 3-5. From the results in Figs. 6-8, we can conclude that with difference memory sizes, HATS still visits much fewer nodes than SD, DL-SD and A*. The numbers of nodes visited by SD, DL-SD and A* all grow exponentially with the decreasing SNR, which indicates that they are not applicable in large-scale problems. As a contrast, the number of nodes visited by HATS increases almost linearly in low SNR regimes. Moreover, we can find from these figures that HATS can reach the lowest complexity in a wide range of acceptable SNRs, whereas the complexities of SD, DL-SD and A* are still very high in the regime. Specifically, for the 32 × 32 MIMO where the size of search space is 2 64 and SNR = 18 dB, SD, DL-SD and A* have to visit average about 1800, 1100 and 1000 nodes to find the optimal solution, while HATS only needs to visit about average 150 nodes. In particular, we can also find that HATS is not sensitive to the memory size, and it can perform well with bounded memory size. Specifically, when the memory size is bounded, the performances of HATS(128), HATS(1024) and HATS(∞) are almost the same in all the three systems. This indicates that HATS can reach almost the optimal efficiency with bounded memory in practice. Fig. 9 illustrates the computational complexity comparison of the competing algorithms versus the problem scale, where the number of antennas varies from 8 to 24 and the associated SNR is set to 15 dB. From this figure, we can observe that the complexities of HATS(128), HATS(1024) and HATS(∞) increase almost linearly with the number of antennas, while the complexities of SD, DL-SD, A* increase exponentially with the problem scale. This implies that HATS is much more efficient and less sensitive to the problem scale by comparing to the other optimal search algorithms. As it has been shown in Figs. 3-8 that HATS can reach almost the optimal BER performance in large-scale systems, one is able to conclude that the proposed algorithm almost meets the optimal efficiency in large-scale problems, which indeed further verifies the effectiveness of the proposed algorithm. D. Further Results on Correlated Channels and Higher-Order Modulation To show the robustness of the proposed HATS in practical scenarios, we further present simulation results in Figs. 10-13, in which the channels are correlated (ρ > 0) and 16-QAM is adopted. Specifically, Fig. 10 demonstrates the average training loss of the proposed HATS on two different MIMO systems, where the numbers of transmit antennas are both 32, while the numbers of receive antennas are 32 and 160, and the channel correlation coefficients are 0.5 and 0.2, respectively. We can find from Fig. 10 that the average training losses of the proposed HATS rapidly converge to a low value in both of the two systems, which indicates that the proposed network structure and training strategy are effective for different practical systems. Besides, Fig. 11 depicts the BER performance comparison results of the aforementioned algorithms, where the correlation coefficient ρ varies in {0.1, 0.3, 0.7}, and SNR varies from 10 dB to 30 dB. From this figure, we can see that even when the size of search space grows very largely to 2 128 for a 16-QAM modulated 32 × 160 large-scale MIMO system, and the channels are slightly (ρ = 0.1), moderately (ρ = 0.3), or strongly (ρ = 0.7) correlated, HATS can still be able to reach almost the optimal ML performance in a wide range of SNRs. This indicates that the estimated heuristic is nearly admissible such that the algorithm can always successfully find the ML estimate. Moreover, Fig. 12 demonstrates the complexity comparison over SNR for the corresponding BER results of ρ = 0.1 in Fig. 11. We can find from this figure that the proposed HATS still outperforms the A* and DL-SD algorithms in a wide range of SNRs, and it visits nearly only the 512 necessary nodes connecting the shortest path for this system. This indicates that the estimated heuristic is nearly optimal, by the fact that the proposed HATS not only achieves almost the optimal BER performance, but also reaches almost the optimal efficiency. In further, we present the complexity comparison results over the correlation level in Fig. 13, where the correlation coefficient ρ ranges from 0 to 0.8. It can be seen from Fig. 13 that the proposed HATS is not sensitive to the correlation level with comparison to the original A* and DL-SD algorithms. The proposed HATS will maintain the average complexity at a low level until the correlation level grows really high to 0.8, while the other two algorithms' complexities will grow rapidly with the increasing correlation level. This indicates that the proposed model can still estimate a highquality heuristic even when the channel is strongly correlated, which indeed further verifies the robustness of the proposed algorithm. V. CONCLUSIONS AND FUTURE WORKS In this paper, we investigated the optimal signal detection problem with focus on large-scale MIMO systems. The problem can be regarded as search on the decision tree, and the optimal solution is obtained by finding the shortest path. In order to improve the search speed while does not compromise the optimality, we proposed a DL based heuristic search algorithm, namely, HATS. The proposed algorithm was shown to be almost optimally efficient, since it can reach almost the optimal BER performance while still visit almost the fewest nodes in large-scale systems. Hence, we do believe that the proposed algorithm is attractive for optimal signal detection in practical large-scale MIMO systems, especially when the problem scales largely and the optimal performance is required. Generally, this paper have focused mainly on producing hard-decision output. However, channel coding is often used in modern MIMO systems to enhance the reliability, and thereby having probabilistic soft-output will help enhance the performance of wireless transmission. Therefore, one interesting future topic of this work is to provide soft-decisions with HATS, and this could be a challenge for all tree search based algorithms. A feasible solution to address this challenge is to employ the max-log approximation to compute loglikelihood ratios (LLRs) of bits, such that we will only need to visit the nodes neighbouring to the shortest path [43]. In this way, we can easily compute LLRs of bits by designing an additional pruning strategy to neglect all the nodes which have no contributions to LLRs, which indicates that it becomes possible to provide soft-output while still visit as fewer nodes as possible. Fig. 1 . 1An example of decision tree for signal detection, where m = 4 and A = {−1, +1}. In the resulting tree, the left branch represents the symbol decision of −1, while the right branch represents the symbol decision of +1. ... ... ... ... ... ... ...... Fig. 2 . 2Network structure adopted by the proposed algorithm. 15: end for 16: // Training with random samples 17: Randomly initialize the DNN's parameters θ 18: for j = 1, 2, · · · , B do 19: Randomly select a mini-batch of samples D from D all 20: {2N t , 128, 64, 32, 16, 1} . • MMSE: The conventional minimum mean squared error estimator. • OAMP-Net2: The orthogonal approximate message passing network 2 introduced in [30]. • SD: The sphere decoding algorithm introduced in [1]. Fig. 3 . 3BER comparison versus SNR for 8 × 8 MIMO with uncorrelated channels. Fig. 4 . 4BER comparison versus SNR for 16 × 16 MIMO with uncorrelated channels. Fig. 5 . 5BER comparison versus SNR for 32 × 32 MIMO with uncorrelated channels. Fig. 6 . 6Complexity comparison versus SNR for 8×8 MIMO with uncorrelated channels. 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[ "https://github.com/skypitcher/hats." ]
[ "RobustLoc: Robust Camera Pose Regression in Challenging Driving Environments", "RobustLoc: Robust Camera Pose Regression in Challenging Driving Environments" ]
[ "Sijie Wang \nContinental-NTU Corporate Lab\nNanyang Technological University\n\n", "Qiyu Kang qiyu.kang@ \nContinental-NTU Corporate Lab\nNanyang Technological University\n\n", "Rui She rui.she@ \nContinental-NTU Corporate Lab\nNanyang Technological University\n\n", "Wee Peng Tay [email protected]@andreas.hartmannsgruber \nContinental-NTU Corporate Lab\nNanyang Technological University\n\n", "Andreas Hartmannsgruber \nContinental Automotive\nSingapore\n", "Diego Navarro Navarro [email protected] \nContinental Automotive\nSingapore\n" ]
[ "Continental-NTU Corporate Lab\nNanyang Technological University\n", "Continental-NTU Corporate Lab\nNanyang Technological University\n", "Continental-NTU Corporate Lab\nNanyang Technological University\n", "Continental-NTU Corporate Lab\nNanyang Technological University\n", "Continental Automotive\nSingapore", "Continental Automotive\nSingapore" ]
[]
Camera relocalization has various applications in autonomous driving. Previous camera pose regression models consider only ideal scenarios where there is little environmental perturbation. To deal with challenging driving environments that may have changing seasons, weather, illumination, and the presence of unstable objects, we propose RobustLoc, which derives its robustness against perturbations from neural differential equations. Our model uses a convolutional neural network to extract feature maps from multi-view images, a robust neural differential equation diffusion block module to diffuse information interactively, and a branched pose decoder with multi-layer training to estimate the vehicle poses. Experiments demonstrate that RobustLoc surpasses current stateof-the-art camera pose regression models and achieves robust performance in various environments. Our code is released at: https://github.com/sijieaaa/RobustLoc
10.48550/arxiv.2211.11238
[ "https://export.arxiv.org/pdf/2211.11238v4.pdf" ]
253,734,176
2211.11238
08c1b87c94e741c09985d37872c1ae14c6c7c2e4
RobustLoc: Robust Camera Pose Regression in Challenging Driving Environments Sijie Wang Continental-NTU Corporate Lab Nanyang Technological University Qiyu Kang qiyu.kang@ Continental-NTU Corporate Lab Nanyang Technological University Rui She rui.she@ Continental-NTU Corporate Lab Nanyang Technological University Wee Peng Tay [email protected]@andreas.hartmannsgruber Continental-NTU Corporate Lab Nanyang Technological University Andreas Hartmannsgruber Continental Automotive Singapore Diego Navarro Navarro [email protected] Continental Automotive Singapore RobustLoc: Robust Camera Pose Regression in Challenging Driving Environments Camera relocalization has various applications in autonomous driving. Previous camera pose regression models consider only ideal scenarios where there is little environmental perturbation. To deal with challenging driving environments that may have changing seasons, weather, illumination, and the presence of unstable objects, we propose RobustLoc, which derives its robustness against perturbations from neural differential equations. Our model uses a convolutional neural network to extract feature maps from multi-view images, a robust neural differential equation diffusion block module to diffuse information interactively, and a branched pose decoder with multi-layer training to estimate the vehicle poses. Experiments demonstrate that RobustLoc surpasses current stateof-the-art camera pose regression models and achieves robust performance in various environments. Our code is released at: https://github.com/sijieaaa/RobustLoc Introduction Accurate camera relocalization plays an important role in autonomous driving. Given query images, the camera relocalization task aims at estimating the camera poses for which the images are taken. In recent years, many camera relocalization approaches have been proposed. Generally speaking, they fall into two categories. 1. One solution is to treat relocalization as a matching task. This solution assumes the availability of a database or a map that stores prior information (e.g., 3D point clouds, images, or descriptors) of sample points. Given a query image, the matching model finds the best match between the query and the database based on a similarity score. The estimated camera pose is then inferred from the matched prior information in the database. 2. Another solution does not assume the availability of a database and uses only neural networks to regress the camera poses of the query images. This approach constructs an implicit relation between images and poses, which is called camera pose regression (CPR). The prerequisite of a database on the one hand can boost the accuracy of the camera relocalization by storing useful prior * These authors contributed equally. This paper has been accepted by AAAI 2023. spatially/temporally nearby frames of a single vehicle frames shared from other spatially nearby transportation agents information. On the other hand, the computation and storage requirements are proportionate to the number of sample points in the database. To decouple relocalization from the need for a database, there has been a recent surge of research interest in the second category CPR. The pioneering work PoseNet (Kendall, Grimes, and Cipolla 2015) uses a convolutional neural network (CNN) to extract features from a single image as vector embeddings, and the embeddings are directly regressed to the 6-DoF poses. To further improve the regression performance in driving scenarios, multi-view-based models extend the input from a single image to multi-view images. MapNet (Brahmbhatt et al. 2018) leverages pre-computed visual odometry to postprocess the output pose trajectory. GNNMapNet (Xue et al. 2020) integrates a graph neural network (GNN) into CNN to make image nodes interact with neighbors. The above-mentioned multi-view-based models show promising performance in benign driving environments. To operate well in challenging environments, the model must be robust to environmental perturbations (e.g., changing seasons, weather, illumination, and unstable objects), and effectively leverage neighboring information from spatially or temporally nearby frames of a single vehicle or multi-view images shared from other spatially nearby agents (e.g., using V2X communication) as shown in Fig. 1. Images sharing such neighboring information are said to be covisible. Recently, neural Ordinary Differential Equations (ODEs) (Chen et al. 2018) and Partial Differential Equations (PDEs) (Chamberlain et al. 2021b,a) have demonstrated their robustness against input perturbations (Yan et al. 2018;Kang et al. 2021). Moreover, GNNs can effectively aggregate neighborhood information. We thus propose RobustLoc that not only explores the relations between graph neighbors but also utilizes neural differential equations to improve robustness. We test our new multi-view-based model on three challenging autonomous driving datasets and verify that it outperforms existing state-of-the-art (SOTA) CPR methods. Our main contributions are summarized as follows: 1. We represent the features extracted from a CNN in a graph and apply graph neural diffusion layers at each stage. I.e., we design feature diffusion blocks at both the feature map extraction and vector embedding stages to achieve robust feature representations. Each diffusion block consists of not only cross-diffusion from node to node in a graph but also self-diffusion within each node. We also propose multi-level training with the branched decoder to better regress the target poses. 2. We conduct experiments in both ideal and challenging noisy autonomous driving datasets to demonstrate the robustness of our proposed method. The experiments verify that our method achieves better performance than the current SOTA CPR methods. 3. We conduct extensive ablation studies to provide insights into the effectiveness of our design. Related Work Camera Pose Regression Given the query images, CPR models directly regress the camera poses of these images without the need for a database. Thus, it does not depend on the scale of the database, which is definitely a born gift compared with those database methods. PoseNet (Kendall, Grimes, and Cipolla 2015) and Geo-PoseNet (Kendall and Cipolla 2017) propose the simultaneous learning for location and orientation by integrating balance parameters. MapNet (Brahmbhatt et al. 2018) uses visual odometry to serve as the post-processing technique to optimize the regressed poses. LsG (Xue et al. 2019) and LSTM-PoseNet (Walch et al. 2017) integrates the sequential information by fusing PoseNet and LSTM. AD-PoseNet and AD-MapNet (Huang et al. 2019) leverages the semantic masks to drop out the dynamic area in the image. AtLoc introduces the global attention to guide the network to learn better representation. GNNMapNet (Xue et al. 2020) expands the feature exploration from a single image to multi-view images using GNN. IRPNet (Shavit and Ferens 2021) proposes to use two branches to regress translation and orientation respectively. Coordinet (Moreau et al. 2022) uses the coordconv (Liu et al. 2018) and weighted average pooling (Hu, Wang, and Lin 2017) to capture spatial relations. There are also some models focusing on LiDARbased pose regression including PointLoc and HypLiLoc (Wang et al. 2023). Neural Differential Equations and Robustness The dynamics of a system are usually described by ordinary or partial differential equations. The paper (Chen et al. 2018) first proposes trainable neural ODEs by parameterizing the continuous dynamics of hidden units. The hidden state of the ODE network is modeled as: dy(t) dt = f θ (y(t))(1) where y(t) denotes the latent state of the trainable network f θ that is parameterized by weights θ. Recent studies (Yan et al. 2018;Kang et al. 2021) have demonstrated that neural ODEs are intrinsically more robust against input perturbations compared to vanilla CNNs. In addition, neural PDEs (Chamberlain et al. 2021b,a) have been proposed and applied to GNN, where the diffusion process is modeled on the graph. Furthermore, the stability of the heat semigroup and the heat kernel under perturbations of the Laplace operator (i.e., local perturbation of the manifold) is studied (Song et al. 2022). Proposed Model In this section, we provide a detailed description of our proposed CPR approach. We assume that the input is a set of images {I i } i∈[N ] that may be covisible (see Fig. 1). 1 Our objective is to perform CPR on the input images. RobustLoc Overview We first summarize our multi-view CPR pipeline, which can be decomposed into three different stages, as follows (see Fig. 2 and Fig. 3): 1. Given N neighboring images, a CNN extracts the feature maps of all these images. Our proposed feature map diffusion block then performs cross-self diffusion on the feature maps. 2. After feature map diffusion, a global average pooling module aggregates the feature maps as vector embeddings, which contain global representations of these images. Similarly, those vector embeddings are then diffused by cascaded diffusion blocks. 3. Based on the vector embeddings, the branched decoder module regresses the output camera poses. During training, decoding is performed on multiple levels to provide better feature constraints. Neural Diffusion for Feature Maps The input images {I i } i∈ [N ] are passed through a CNN to obtain the feature maps {m i ∈ R H×W ×C } i∈ [N ] . Here, C is the channel dimension, while H and W are the dimensions of a feature map. For each feature map m i , we denote its j-th element as m i,j ∈ R C , j ∈ [HW ]. We next describe the feature map diffusion block, where we perform crossdiffusion from node to node in a graph, and self-diffusion within each node. The two diffusion processes update the feature map by leveraging the neighboring information or only using each node's individual information, respectively. Cross-Diffusion Dynamics. To support the cross-diffusion over feature maps, we formulate the first graph in our pipeline as: G feat = (V feat , E feat ),(2) where the node set V feat = {m i,j } (i,j)∈[N ]×[HW ] contains element-wise features m i,j and the edge set E feat is defined as the complete graph edges associated with attention weights as discussed below. And the complete graph architecture is demonstrated to be an effective design shown in Table 5. To achieve robust feature interaction, we next define the cross-diffusion process as: ∂ ∂t x(t) = f cross (x(t)),(3) where f cross (x(t)) is a neural network and can be approximately viewed as a neural PDE with the partial differential operations over a manifold space replaced by the attention modules that we will introduce later. We denote the input to the feature map diffusion module as the initial state at t = t 0 as HW ] denotes the hidden state of the diffusion. The diffusion process is known to have robustness against local perturbations of the manifold (Chen et al. 1998) where the local perturbations in our CPR task include challenging weather conditions, dynamic street objects, and unexpected image noise. Therefore, we expect our module (3) is simultaneously capable of leveraging the neighboring image information and holding robustness against local perturbations. We next introduce the computation of attention weights in f cross (x(t)) for node features at time t. We first generate the embedding of each node using multi-head fully connected (FC) layers with learnable parameter matrix W k and bias b k at each head k = [K], where K is the number of heads. The output at each head k can be written as: x(t 0 ) = {m i,j } (i,j)∈[N ]×[HW ] , where x(t) = {m i,j (t)} (i,j)∈[N ]×[m FC i,j;k (t) = W k m i,j (t) + b k .(4) The attention weights are then generated by computing the dot product among all the neighboring nodes using the fea- tures m FC i,j;k (t) (i,j)∈[N ]×[HW ] . We have {a (i,j),(i ′ ,j ′ );k (t)} (i ′ ,j ′ )∈Ni,j = Softmax (i ′ ,j ′ )∈N (i,j) (m FC i,j;k (t) · m FC i ′ ,j ′ ;k (t)),(5) where N i,j denotes the set of neighbors of node m i,j . Let m weighted i,j;k (t) = (i ′ ,j ′ )∈Ni,j a (i,j),(i ′ ,j ′ );k (t)m FC i ′ ,j ′ ;k (t). (6) Finally, the updated node features are obtained by concatenating the weighted node features from all heads as f cross (x(t)) =    ∥ k∈[K] (m weighted i,j;k (t))    (i,j)∈[N ]×[HW ] .(7) Based on the above pipeline, the output of the crossdiffusion at time t = t 1 can be obtained as: x(t 1 ) = F cross (x(t 0 )),(8) where F cross (·) denotes the solution of (3) integrated from t = t 0 to t = t 1 . Self-Diffusion Dynamics. In the next step, we update each node feature independently. The node-wise feature update can be regarded as a rewiring of the complete graph to an edgeless graph, and the node-wise feature update is described as: dm i,j (t) dt = f self (m i,j (t)) = MLP(m i,j (t)).(9) And the output of self-diffusion can be obtained as: m i,j (t 2 ) = F self (m i,j (t 1 )).(10) where F self (·) denotes the solution of (9) integrated from t = t 1 to t = t 2 . As neural ODEs are robust against input perturbations (Yan et al. 2018;Kang et al. 2021), we expect the updating of each node feature according to the self-diffusion (9) to be robust against perturbations like challenging weather conditions, dynamic street objects, and image noise. Vector Embeddings and Diffusion After the feature map neural diffusion, we feed the updated feature maps into a global average pooling module to generate the vector embeddings {h i ∈ R C } i∈[N ] , where h i = Pooling(m i ).(11) Each vector embedding contains rich global representations for the input image together with the information diffused from the neighboring images. To enable diffusion for the global information, we propose to design the vector embedding graph as: G vect = (V vect , E vect ),(12) where the node set V vect = {h i } i∈[N ] contains image vector embeddings h i and the edge set E vect is also defined to be the complete graph. Based on this graph G vect , we construct the cascaded diffusion blocks, to perform global information diffusion. Within the cascaded blocks, each basic diffusion block consists of two diffusion layers: a cross-diffusion layer and a self-diffusion layer, similar to the two diffusion schemes introduced at the feature map diffusion phase. Pose Decoding In this subsection, we explain the pose decoding operations. Branched Pose Decoder. Each camera pose p = {d, r} ∈ R 6 , consists of a 3-dimensional translation d ∈ R 3 and a 3-dimensional rotation r ∈ R 3 . Thus CPR can be viewed as a multi-task learning problem. However, since the translation and rotation elements of p do not scale compatibly, the regression converges in different basins. To deal with it, previous methods consider regression for translation and rotation respectively and demonstrate it is an effective way to improve performance (Shavit and Ferens 2021). In our paper, we also follow this insight to design the decoder. Firstly, the feature embeddings {h d , h r } for translation and rotation are extracted from the feature embedding h using different non-linear MLP layers as: h d = MLP d (h),(13)h r = MLP r (h),(14) Thus, the features of translation and rotation are decoupled. Next in the second stage, the pose output can be regressed as: p = W (h d ∥ h r ) + b(15) where W , b are learnable parameters. During training, we compute the regression loss of decoded poses from multiple levels, which we will introduce below. During inference, we use the decoded pose from the last layer as the final output pose. Multi-level Pose Decoding Graph. To better regularize the whole regression pipeline, we propose to leverage the feature maps at multiple levels. As shown in Fig. 3, at the vector embedding stage, we use the vector embeddings to regress the poses, while at the feature map stage, we use the feature maps. Denoting the feature maps at layer l as {m l i ∈ R H×W ×C } i∈[N ] , the pose decoding graph at layer l can be formulated as: G pose,l = (V pose,l , E pose,l ),(16) where edge set E pose,l is defined to be connected with two spatially adjacent nodes which can be viewed as the odometry connection, while the node set V pose,l is defined depending on layers since the information used to regress poses is different: V pose,l = {h i } i∈[N ] if l = L, {m l i } i∈[N ] otherwise,(17) where L represents the last layer in our network. At the last layer where there are vector embeddings, we can directly apply the pose decoder to generate absolute pose messages. By contrast, at feature map layers, we first apply a global average pooling module on the feature maps to formulate feature vectors, and pose messages can be obtained using the pose decoder: p l i = f l decoder (h l i ) if l = L, f l decoder (Pooling(m l i )) otherwise.(18) where f l decoder (·) is the pose decoder at layer l. Using the simplified relative pose computation technique in , the relative pose messages p l i,i ′ at layer l can be generated as: p l i,i ′ = p l i ′ − p l i .(19) By leveraging multi-layer information, we expect not only the last layer but also the preceding middle-level layers can directly learn the implicit relation between images and poses, which helps to improve the robustness against perturbations. Table 1: Median and mean translation/rotation estimation error (m/ • ) on the Oxford RobotCar dataset. The best and the secondbest results in each metric are highlighted with bold and underline respectively. "-" denotes no data provided. Loss Function Following the approach in ), we use a weighted balance loss for translation and rotation predictions. For the input image I i , we denote the translation and rotation targets as d * i ∈ R 3 and r * i ∈ R 3 respectively. Then the absolute pose loss term L l i and the relative pose loss term L l i,i ′ at decoding layer l are computed as: L l i = ∥d l i − d * i ∥ exp(−α) + α + ∥r l i − r * i ∥ exp(−β) + β,(20)L l i,i ′ = ∥d l i,i ′ − d * i,i ′ ∥ exp(−γ) + γ + ∥r l i,i ′ − r * i,i ′ ∥ exp(−λ) + λ ,(21) where d l i , r l i , d l i,i ′ , r l i,i ′ are outputs at layer l, while α, β, γ, λ are all learnable parameters. Finally, the overall loss function can be obtained as: L = l∈{3,4,L} i∈[N ],i ′ ∈N l i L l i + L l i,i ′ ,(22) where N l i is the neighborhood of node i in G pose,l . We use the logarithmic form of the quaternion to represent rotation r as: r = log q = (q2,q3,q4) ∥(q2,q3,q4)∥ cos −1 q 1 if ∥(q 2 , q 3 , q 4 )∥ ̸ = 0, 0 otherwise,(23) where q = (q 1 , q 2 , q 3 , q 4 ) ∈ R 4 represents a quaternion. Experiments In this section, we first evaluate our proposed model on three large autonomous driving datasets. We next present an ablation study to demonstrate the effectiveness of our model design. 4Seasons. There are only a few existing methods designed for robust CPR in driving environments, and the experiment on the Oxford dataset is insufficient for comparison. Thus we also conduct experiments on another driving dataset to cover more driving scenarios. The 4Seasons dataset (Wenzel et al. 2020) is a comprehensive dataset for autonomous driving SLAM. It was collected in Munich, Germany, covering varying perceptual conditions. Specifically, it contains different environments including the business area, the residential area, and the town area. In addition, it consists of a wide variety of weather conditions and illuminations. In our experiments, we use 1) Business Campus (business area), 2) Neighborhood (residential area), and 3) Old Town (town area). Datasets and Implemention Perturbed RobotCar. To further evaluate the performance under challenging environments, we inject noise into the RobotCar Loop dataset and call this the Perturbed RobotCar dataset as shown in Fig. 4. We create three scenarios: 1) Medium (with fog, snow, rain, and spatter on the lens), 2) Hard (with added Gaussian noise), and 3) Hard (+ noisy training) (i.e., training with noisy augmentation). Implemention We use ResNet34 as the backbone, which is pre-trained on the ImageNet dataset. We set the maximum number of input images as 11. We resize the shorter side of each input image to 128 and set the batch size to 64. The ignore useful Figure 5: Robust features from RobustLoc. Adam optimizer with a learning rate 2 × 10 −4 and weight decay 5 × 10 −4 is used to train the network. Data augmentation techniques include random cropping and color jittering. We set the integration times t 0 = 0, t 1 = 1, and t 2 = 2. The number of attention heads is 8. We train our network for 300 epochs. All of the experiments are conducted on an NVIDIA A5000. Main Results On the Oxford RobotCar dataset, as shown in Table 1, we obtain the best performance in 10 out of 12 metrics. Using the mean error, which is easily influenced by outlier predictions, RobustLoc outperforms the baselines by a significant margin. In the most challenging route Full, to the best of our knowledge, RobustLoc is the first to achieve less than 10m mean translation error for CPR. The 4Seasons dataset consists of more varied driving scenes. As shown in Table 2, RobustLoc achieves the best performance in 11 out of 12 metrics. Again, using the mean error metric, RobustLoc outperforms the baselines by a significant margin. On the Perturbed RobotCar dataset, where the images contain more challenging weather conditions and noisy perturbations, RobustLoc achieves the best in all metrics. The superiority of RobustLoc over other baselines is more obvious in Table 3. Analysis Ablation Study. We justify our design for RobustLoc by ablating each module. From Table 4, we observe that every module in our design contributes to the final improved estimation. We see that making use of neighboring information from covisible frames and learning robust feature maps contribute to more accurate CPR. Salience Visualization. Salience maps shown in Fig. 5 suggest that in driving environments, RobustLoc pays more attention to relatively robust features such as the skyline and the road, similar to PixLoc (Sarlin et al. 2021). In addition, dynamic objects such as vehicles are implicitly suppressed in RobustLoc's regression pipeline. Diffusion and Augmentation. Using multi-level features is an effective method in dense prediction tasks such as depth estimation (Yan et al. 2021). To test if this holds in CPR, we use the feature maps from the lower stage 3 (see Fig. 3), which however does not lead to performance improvement shown in Table 4. We also utilize the multi-level concatenation strategy used in GNNMapNet. This does not lead to significant changes. These experiments demonstrate that CPR benefits more from high-level features with more semantic information than from low-level local texture features. Finally, we test the performance when training with less data augmentation, which leads to worse performance. This suggests that more extensive data augmentation can enhance the model robustness in challenging scenarios, which is consistent with the experimental results on the Perturbed Robotcar dataset in Table 3. Graph Design. We next explore the use of different graph designs for feature map diffusion and vector embedding diffusion. The grid graph stacks an image with two other spatially adjacent images as a cube, and the attention weights are formulated within the 6-neighbor area (for feature maps) or the 2-neighbor area (for vector embeddings). The self-cross graph computes attention weights first within each image and then across different images. From Table 5, we see that the complete graph has the best performance. This is because, in the complete graph, each node can interact with all other nodes, allowing the aggregation of useful information with appropriate attention weights. Rotation Representation. We compare different representations of rotation in Table 5, where the log form of the quaternion is the optimal choice. The other three representations, including the vanilla quaternion, the Lie group, and the vanilla rotation matrix, show similar performance. Trajectory Visualization. We visualize the output pose trajectories as shown in Fig. 6, where a significant gap can be seen from the comparison. RobostLoc outputs more smooth and globally accurate poses compared with the previous method, which shows the effectiveness of our design. Inference Speed. We finally test the performance using a different number of input frames. The inference speed does not drop significantly when increasing the input frames. And even the slowest one (using 11 frames) can run 50 iterations per second and achieve real-time regression. On the other hand, more frames can bring performance improvement when the input size is small, while further increasing frame size does not bring significant change. ing Initiative, as well as cash and in-kind contribution from the industry partner(s), and by the National Research Foundation, Singapore and Infocomm Media Development Authority under its Future Communications Research & Development Programme. The computational work for this article was partially performed on resources of the National Supercomputing Centre, Singapore (https://www.nscc.sg). Supplement Oxford RobotCar -Loop Baseline Models The baseline models in our comparison include: PoseNet, GeoPoseNet, MapNet, LsG, AtLoc/AtLoc+, GNNMapNet, ADPoseNet/ADMapNet, IRPNet, CoordiNet. Dataset Configuration The datasets we used in our experiments include the Oxford Robotcar dataset, the 4Seasons dataset, and the Perturbed Robotcar dataset. All of the datasets are available online at: • https://robotcar-dataset.robots.ox.ac.uk/, • https://vision.cs.tum.edu/data/datasets/4seasonsdataset/download. For each dataset and each scene, we list the corresponding data split as shown in Table 7, Table 8, and Table 9. Dataset Visualization We visualize part of the images from the datasets we used in our experiments as shonw in Fig. 7, Fig. 8, Fig. 9, Fig. 10 Figure 1 : 1Multi-view camera pose regression with neighboring information, without the need for any database. Figure 2 : 2The main architecture of RobustLoc. Feature diffusion is performed at both the feature map stage and the vector embedding stage. The branched decoder regresses the 6-DoF poses based on the vector embeddings or the pooled feature maps. The details for multi-layer decoding are shown inFig. 3. Figure 3 : 3Multi-level pose decoding. Decoding can be directly applied to vector embeddings. Feature maps are first pooled and then decoded. Figure 4 : 4Details Oxford RobotCar. The Oxford RobotCar dataset(Maddern et al. 2017) is a large autonomous driving dataset collected Visualization of the Perturbed RobotCar dataset.Medium is with fog, snow, rain, and spatter on the lens. Hard is with added Gaussian noise.by a car driving along a route in Oxford, UK. It consists of two different routes: 1) Loop with a trajectory area of 8.8×10 4 m 2 and length of 10 3 m, and 2) Full with a trajectory area of 1.2 × 10 6 m 2 and length of 9 × 10 3 m. Figure 6 : 6Trajectory visualization on the Oxford RobotCar dataset. The ground truth trajectories are shown in bold blue lines, and the estimated trajectories are shown in thin red lines. The stars mark the start of the trajectories. Figure 7 :Figure 8 : 78Visualization of the Oxford RobotCar Loop dataset. The Loop (cross-day) and the Loop (within-day) share the same images but have different data splits. Visualization of the Oxford RobotCar Full dataset. Figure 9 :Figure 10 :Figure 11 : 91011Visualization of the Perturbed RobotCar dataset. Visualization Visualization of the 4Seasons Neighborhood dataset. Figure 12 : 12, Fig. 11, and Fig. 12. Visualization Table 4 : 4Ablation study, diffusion design, and augmentation design comparison on the Oxford RobotCar dataset.We have proposed and verified the performance of a robust CPR model RobustLoc. The model's robustness derives from the use of information from covisible images and neural graph diffusion to aggregate neighboring information, which is present in challenging driving environments.Conclusion grid graph 15.67 / 2.95 self-cross graph 15.31 / 3.28 complete graph 9.37 / 2.47 Mean Error ( • ) on Business Campus quaternion 2.23 Lie group 2.20 rotation matrix 2.25 log (quaternion) 2.04 Table 5 : 5Graph design comparison on the Oxford RobotCar dataset and rotation representation comparison on the 4Seasons dataset. Mean Error (m) 5.28 5.09 4.96 4.68 4.72#frames 3 5 7 9 11 Speed (iters/s) 56 55 53 52 50 Table 6 : 6The performance using different numbers of frames on the Oxford RobotCar Loop (cross-day). Notations: In this paper, we use [N ] to denote the set of integers {1, 2, . . . , N }.We use boldfaced lowercase letters like m to denote vectors and boldface capital letters like W to denote matrices. 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IEEE. . Loop, cross-day) 2014-06-26-08-53-56 2014-06-23-15-36-04 2014-06-26-09-24-58 2014-06-23-15-41-25Loop (cross-day) 2014-06-26-08-53-56 2014-06-23-15-36-04 2014-06-26-09-24-58 2014-06-23-15-41-25 . Loop, within-day) 2014-06-26-09-24-58 2014-06-26-08-53-56 2014-06-23-15-41-25 2014-06-23-15-36-04Loop (within-day) 2014-06-26-09-24-58 2014-06-26-08-53-56 2014-06-23-15-41-25 2014-06-23-15-36-04 The Oxford RobotCar configuration. 2020-10-08-09-30-57 2021-02-25-14-16-43 2021-01-07-13-12-23Dataset Train Test Business Campus. 7Table 7: The Oxford RobotCar configuration. Dataset Train Test Business Campus 2020-10-08-09-30-57 2021-02-25-14-16-43 2021-01-07-13-12-23 Table 8: The 4Seasons configuration. Table 8: The 4Seasons configuration. . 06-26-08-53-56 2014-06-23-15-36-04 2014-06-26-09-24-58 2014-06-23-15-41-25Dataset Train Test Medium. Dataset Train Test Medium 2014-06-26-08-53-56 2014-06-23-15-36-04 2014-06-26-09-24-58 2014-06-23-15-41-25 The Perturbed RobotCar configuration. 9Table 9: The Perturbed RobotCar configuration. Codebase Our codes are developed based on the following repositories. Codebase Our codes are developed based on the following repositories: • https://github.com/BingCS/AtLoc, • https://github.com/psh01087/Vid-ODE. Our codes is released at: https://github.com/sijieaaa/RobustLoc
[ "https://github.com/sijieaaa/RobustLoc", "https://github.com/BingCS/AtLoc,", "https://github.com/psh01087/Vid-ODE.", "https://github.com/sijieaaa/RobustLoc" ]
[ "Automated design of pneumatic soft grippers through design-dependent multi-material topology optimization", "Automated design of pneumatic soft grippers through design-dependent multi-material topology optimization" ]
[ "Josh Pinskier ", "Prabhat Kumar ", "Matthijs Langelaar ", "David Howard " ]
[]
[]
Soft robotic grasping has rapidly spread through the academic robotics community in recent years and pushed into industrial applications. At the same time, multimaterial 3D printing has become widely available, enabling the monolithic manufacture of devices containing rigid and elastic sections. We propose a novel design technique that leverages both technologies and can automatically design bespoke soft robotic grippers for fruit-picking and similar applications. We demonstrate the novel topology optimisation formulation that generates multi-material soft grippers, can solve internal and external pressure boundaries, and investigate methods to produce air-tight designs. Compared to existing methods, it vastly expands the searchable design space while increasing simulation accuracy.The authors thank Prof. Krister Svanberg for providing MATLAB codes of the MMA optimizer.
10.1109/robosoft55895.2023.10122069
[ "https://export.arxiv.org/pdf/2211.13843v2.pdf" ]
254,017,402
2211.13843
11b88d4fb83e3c1675be9b95d497820c4c4cf83c
Automated design of pneumatic soft grippers through design-dependent multi-material topology optimization Josh Pinskier Prabhat Kumar Matthijs Langelaar David Howard Automated design of pneumatic soft grippers through design-dependent multi-material topology optimization Soft robotic grasping has rapidly spread through the academic robotics community in recent years and pushed into industrial applications. At the same time, multimaterial 3D printing has become widely available, enabling the monolithic manufacture of devices containing rigid and elastic sections. We propose a novel design technique that leverages both technologies and can automatically design bespoke soft robotic grippers for fruit-picking and similar applications. We demonstrate the novel topology optimisation formulation that generates multi-material soft grippers, can solve internal and external pressure boundaries, and investigate methods to produce air-tight designs. Compared to existing methods, it vastly expands the searchable design space while increasing simulation accuracy.The authors thank Prof. Krister Svanberg for providing MATLAB codes of the MMA optimizer. I. INTRODUCTION Soft robotic grasping has emerged as a safe and effective means for grasping fragile, flexible and fluctuating objects. Their inherent deformability enables them to conform to fit the objects' shape and distribute gripping force, hence gently grasping even soft objects. These soft grippers are often inspired by human hands, which are seen as the gold standard in soft and dexterous grasping. However, there is an increasing trend towards non-anthropomorphic designs, which enable diverse grasping strategies and require controllable fewer degrees of freedom (DOFs) [1]. Several mechanisms have been investigated for their actuation including pneumatic [2], tendon-driven [3], and granular (vacuum) jamming [4], [5]. Despite the diversity of grasping and actuation paradigms available in the literature, most existing grippers are handdesigned. They draw on human experience and biomimicry to navigate the complexity of designing deformable devices to generate high-quality designs [6]. The resulting generic designs emphasise universal approaches rather than bespoke designs [4]. However, real-world applications frequently require designs which are tailored to the specifics of their task. Clearly, a fruit-picking robot requires a different end effector to an assembly line robot or a human assistance robot, and an apple-picking end-effector has different requirements to a strawberry picker. 3-Material optimised soft-gripper under 50kPa pressure (5x deformation scale). Pressure is applied to the two faces on the left, causing the jaws to close on the right. Material stiffnesses are: Red -100MPa, Green -10MPa, Despite this obvious need to produce bespoke soft endeffectors, existing automated design tools are limited and underexplored. Methods including simulated and in-materio evolution have recently proven successful in designing granular jamming grippers [7], [8]. More generally, an impressive and diverse array of soft robotics have been evolved in simulation [9], [10]. However, because of the large number of evaluations required these require very cheap simulators, which are unable to capture multiphysics interactions and have a large simulation to reality gap [11]. In contrast, Topology optimization (TO) is a general purpose design tool, suitable to numerous actuation techniques and physical domains [12]. It distributes material inside a meshed (or pixelized/voxelized) space to identify the topology with the best performance, and has designed both pneumatic and tendon-driven soft grippers. [13], [6], [14]. However, the methods presented in these works require significant assumptions about the design domain and actuation, limiting both the accuracy of the simulation and the range of realisable designs. A. Topology Optimisation of Soft Grippers The current state-of-the-art in topology optimised soft grippers broadly falls into two categories. Either externally actuated grippers which use an exogenous displacement to drive their grasping behaviour (an externally routed cable or moving surface) [15], [13]; or pressure-actuated soft fingers without design dependency [16], [17]. In both cases, the actuation source is prespecified and does not form part of the optimisation problem. Whilst convenient, these assumptions do not reflect best-practice design methods which use complex pneumatic chambers and internal cable routing. To capture these features, the loading point (magnitude, direction and location) should be free to move with each iteration of the topology optimisation solver. This designdependency problem increases the solver complexity and requires auxiliary physics equations to solve and additional constraints to enforce physical limits. A small number of topology optimised soft grippers have investigated designdependent pressure optimisation, but their coarse physics approximations result in unrealisable designs, with disconnected pressurised regions [18], [19]. The above methods have been demonstrated only in single-material optimisation. However, improvements in 3D printing technology, enable the monolithic manufacture of arbitrarily complex multi-material soft robots. With two or more materials it becomes possible to strike a trade-off between the flexibility and strength of the material, and increase the overall strength of the device without compromising on its workspace. For a detailed review of soft robotic topology optimisation see [6], [20]. To the best of the authors' knowledge, there is currently no method for creating multi-material pneumatically activated soft robots using topology optimization. B. Pressure-Loaded Topology Optimsation Pressure-loaded topology optimsation is a problem that extends beyond soft robotics. It has applications in the design of pneumatically and hydraulically loaded structures like pressure vessels, dams, pumps and ships. In these problems, the fluid-solid boundary and hence the loading must move during the optimisation. In density-based topology optimisation, mesh elements are allowed to occupy a continuum between solid and void [12]. Hence, the problem is commonly approached either by attempting to explicitly identify a fluid-solid boundary, or using a mixed fluidsolid formulation [21], [22]. In contrast, the current state of the art method treats the continuous density material as a porous media, and uses the Darcy method to estimate fluid penetration as a function of density [22], [14]. This allows the boundary to be located implicitly without the need to explicitly seperately the fluid-filled and void regions. However, generating airtight in pressure-actuated compliant mechanisms remains challenging as the contiguous, closed surfaces required to hold pressure also reduce compliance. Hence, the problem and cost function design are critical to prevent leaky designs. The issue can be resolved using a material filtering scheme, which forces a solid layer between the high and low pressure regions [23], such a scheme is heavily dependent on the optimiser's initial conditions and prevents the formation of beneficial internal cavities. C. Contributions In this work, we present a novel method to design 3D multi-material pressure-actuated soft grippers using topology optimization. The method builds on our previous work into pressure-loaded topology optimization using Darcy's law [22], [14] and the extended solid-isotropic material with penalisation (SIMP) material model for the multi-material modeling [24]. An example of a soft gripper designed using this method is shown in Figure 1, it uses three materials with stiffnesses of 1 M P a, 10 M P a and 100 M P a. Using the multimaterial Darcy formulation, the solver converges to a soft gripper which clamps together using several compliant hinges. The main contributions of this work are: 1) The first presentation of a multi-material topology optimsation formulation for pneumatic soft robots with design-dependent loading conditions. 2) The development and investigation two new formulations to generate sealed pneumatic actuators, based on pressure regions and an energy penalty, respectively. 3) The design of several new multimaterial pressureactuated soft grippers. We focus on the application of this methology to soft robotic grasping, but it is generalisable to other pneumatic compliant mechanism and soft robots. II. TOPOLOGY OPTIMISATION FORMULATION In this work, we use the density based SIMP method for topology optimisation. The goal of topology optimisation is to find a discrete material layout where each region contains a unique material or is left void. To simplify the problem, SIMP allows the design variable ρ to occupy a continuum from 0 to 1, and a penalty p = 3 applied to drive the results towards a binary solution. For a single material problem this is done using the SIMP interpolation law: E i = (1 −ρ p )E min +ρ p E 1(1) where E min is a small, non-zero constant used to prevent singularities in material voids and E 1 is the elastic modulus of the material used. A. Multimaterial Modeling We apply the extended SIMP interpolation technique to model multiple materials for the gripper mechanisms [24]. In this formulation, one design variable is assigned to each material. For example, in the two-material case, the scheme with the modified SIMP formulation can be written as: E i = (1 −ρ p i1 )E min +ρ p i1 ((1 −ρ p i2 )E 1 +ρ p i2 E 2 ) (2) where E 1 and E 2 are moduli of material 1 and material 2, respectively.ρ i denotes the physical variable corresponding to design variable ρ i . {ρ i1 = 1,ρ i2 = 1} gives the second material, whereas {ρ i1 = 1,ρ i2 = 0} provides the first material. Thus,ρ i1 is called the topology variable. It decides the topology of the evolving design, whereasρ i2 decides the candidate material. Similarly the three-material case can be described by: E i = (1 −ρ p i1 )E min +ρ p i1 [((1 −ρ p i2 )E 1 + ρ p i2 ((1 −ρ p i3 )E 2 +ρ p i3 E 3 ))](3) where E 3 is the modulus of material 3. Using three materials, {ρ i1 = 1,ρ i2 = 0,ρ i3 = 0}, {ρ i1 = 1,ρ i2 = 1,ρ i3 = 0} and {ρ i1 = 1,ρ i2 = 1,ρ i3 = 1} give material 1, material 2 and material 3. To remove non-physical checkerboard patterns and intermediate (i.e non-binary) densities from the final design, we use a spacial density filter with hyperbolic projection as in [12], [25]. This takes a weighted averages of the elemental density with its neighbours, then uses a hyperbolic projection to drive towards 0/1. B. Pressure load modeling The method developed here for pneumatic soft robotic optimisations builds on our previous work into the Darcy method, a detailed description of which can be found in [22], [14]. It conceptualises the continuous design variableρ as a porous medium, and uses Darcy's law to calculate pressure losses. In it, the flux q (volumetric fliud flow rate across a unit area) is defined by the flow coefficient K(ρ i1 ) and the pressure difference ∇p as: q = − κ µ ∇p = −K(ρ i1 )∇p(4) As the topology of the multimaterial structure (whether there is a material or void) is determined byρ i1 , the flux solely depends onρ i1 , regardless of the number of materials. Hence, the flow coefficient of element i is calculated as K(ρ i ) = K v 1 − (1 − K s K v )H(ρ i1 , β κ , η κ )(5) where H(ρ i1 , β κ , η κ ) = tanh (β κ η κ ) + tanh (β κ (ρ i1 − η κ )) tanh (β κ η κ ) + tanh (β κ (1 − η κ ))(6) K s and K v are flow coefficients of solid and void phases, respectively, and η κ and β κ shape the distribution of K(ρ i ). A drainage term, Q drain , is added. It helps achieve the natural pressure field variation by draining pressure from internal cavities: Q drain = −D s H(ρ i1 , β d , η d )(ρ e )(p − p atm )(7) where Ds is drainage coefficient and p atm is the atmospheric pressure. The net flow of the system is given by the equilibrium equation: ∇ · q − Q drain = 0(8) Which is solved using the finite element method to find the equilibrium pressure distribution and transform the pressure distribution p, to a global force F to solve the mechanical equilibruim equation: Ku = F = −Tp(9) where u and K are the global displacement vector and stiffness matrix, and T tranforms elemental pressures to nodal forces. A linear system is used to facilitate a tractible and efficient solution. However, the resulting solution is accurate only for small deformations. By using two physical equation to solve for the equilibrium pressure and displacement, the formulation determines the pressure boundary at each iteration. C. Problem formulation The final optimisation problem is formulated using: min ρ − s u out (SE) 1/n such that: Ap = 0 Ku = F = −Tp nel i=1 v iρi1 ≤ (v f1 + v f2 + v f3 ) nel i=1 v i nel i=1 v iρi2 ≤ v f2 nel i=1 v i nel i=1 v iρi3 ≤ v f3 nel i=1 v i 0 ≤ρ ≤ 1                                              ,(10) where u out and SE indicate output displacement and strain energy, respectively. s is the consistent scaling parameter. A is the global flow matrix, which is found by assembling (8). We use three linear volume constraints using the definitions ρ i1 ,ρ i2 andρ i3 described above. The first constraint controls the total amount of the solid state, whereas the second and third give the material amount of phase 2 and phase 3. v f1 , v f2 and v f3 denote the volume fraction for material 1, material 2 and material 3, respectively. The cost function is selected to balance the dual requirements of maximising the deformation of the gripper, and maintaining a design which is stiff enough to grasp and hold objects. Here n = 8 was selected after some initial studies to place a soft penalty on the design's stiffness. In this work, v f1 = 0.3, v f2 = 0.2, and v f3 = 0.2 unless otherwise stated, hence the total material permitted is 70% of the volume of the design domain. Whilst it is desirable to minimise material usage, permitting more material is desirable for proof of concept. In each optimisation, the design variables are initialised with a constant densityρ in = v fn for material n. Finally, the input pressure is 50 kPa and the materials are given stiffnesses E 1 = 1 MPa, E 2 = 10 MPa, and E 3 = 100 MPa. III. SOFT GRIPPERS DESIGN To demonstrate the method and motivate the need for airtightness, this section investigates the design of pressureactuated grippers using the multimaterial Darcy formulation. The design domain of the grippers is presented in Figure 2(a). Pressure is applied from the left face, with the output direction shown on the right. To simplify the domain and reduce computation time, 2 planes of symmetry are used, reducing workspace size. The resulting design is illustrated in Figures 2(b) and 2(c), showing the design domain and undeformed configuration. The deformed configuration is show in Figure 2(d). In it, a solid face is formed on the left side, which absorbs the pressure. The internal strains are then transferred to the output face via a series of compliant hinges, one in the centre of the gripper, and four on the outer edges. Thin sections of the stiffest material E 3 are used in each hing, and joined by the softer materials E 1 and E 2 . Although quite elegant, the design illustrates two issues with existing pressure optimisation methods. The first is that the optimiser frequently falls into a local minimum in which the pressurised fluid is not allowed to penetrate deeply into the structure, preventing the formation of more complex, higher performing designs. The second is that without careful consideration of the design domain, the optimiser generates holes in the final design which spuriously increases performance by reducing stiffness in undesired locations. In this case, resealing the device is fairly trivial, but in more complex designs, doing so adversely affects performance. Hence, design methods are needed which drive closed designs. IV. AIRTIGHT DESIGN To generate closed designs, we investigate and compare three methods, and apply them to soft finger design. In soft fingers, the pressure load is often applied via a central channel in the design domain. This forces pressure deeper into the design and enhances performance but also increases its susceptibility to hole generation. Viewed from the perspective of the optimsation problem, sealed chambers reduce compliance and restrict deformation. We propose two new methods for generating sealed designs: 1) A heuristic approach, which adds material to the final design along the median pressure contour. 2) A penalty approach, which adds an energy term to the cost function and drives the optimisation to reduce pressure loss. The first approach leverages the advantages of the Darcy method, which calculates the internal pressure distribution between the inlet and outlet points. Where a face is unsealed, a smooth pressure gradient will flow from the inlet to the outlet. However, closed regions have a sharper pressure boundary. Hence by adding material along the line 0.5(P in − P atm ) we close open regions without significantly impacting regions which already have material. The second approach is more rigorous, but remains susceptible to local minima. Using the equilibrium flow from the Darcy equation, we are able to calculate the energy transferred from inlet to outlet. In a closed system, there would be no flow, hence no energy transferred. However, using the Darcy method, a small flow will always arise. We use this energy value as a penalty term in the cost function, such that we seek to minimise: min ρ − s u out E t (SE) 1/n(11) where E t is the total energy loss calculated at the boundaries and s is a constant. V. AIRTIGHT SOFT FINGERS The design domain of the soft fingers is presented in Figure 3(a). It is fixed around the edges on the left side and pressure enters via a central cavity, a single symmetry face is used to reduce the problem size. The aim is to maximise the bending on the right side. A. Heuristic Skin An example of the design of the soft bending finger is shown in Figure 3. Without any closure method, the material is distributed roughly from stiffest to softest, with the stiffest material placed around the fixed side. Bending is increased by placing holes at the top and sides of the structure. However, a closed structure is easily regenerated using the heuristic method. B. With Skin The surface can also be inserted as part of the optimisation problem by creating a non-design domain on the boundary of the optimisation region and assigning it to have stiffness E 1 . This guarantees air cannot leak, but will produce suboptimal solutions as the external boundary must bend and expand to generate deformation far from the neutral bending axis (Figure 4). In contrast, Pneunets, a state of the art design have a sinusoidal profile which localises bending in narrow sections. C. Energy Penalty Finally, the same design is presented using the energy penalty method. Here, the optimiser has reduced the overall amount of air leakage by using the low stiffness material E 1 to close sections of the chamber which contribute least to bending. As shown in Figure 5, the result is not a totally closed design, but one where the open areas have been greatly reduced. This uses the same design domain as the heuristic skin. The efficacy of this penalty can be increased by increasing the volume limit of the most elastic material E 1 . This is illustrated in Figure 6 in which the energy penalty is evaluated with V f1 = 0.2 and V f1 = 0.4. When using V f1 = 0.2, there is insufficient material to meaningfully close the design, but at V f1 = 0.4 an almost sealed chamber emerges with only a small opening around the fixed side. D. Numerical Comparison We compare the two proposed closure methods by calculating their output displacement, strain energy, mechanical work done, and energy loss across 9 different output stiff- Figure 7. A standard Pneunet design is included for comparison. It contains 7 inflatable chambers with a rectangular cross section, has total dimensions 17 mm x 15 mm x 72 mm and is made of a single material with E = E 1 = 1 MPa and a constant wall thickness of 15 mm. Unsurprisingly, the unconstrained (no skin) optimisation produces the greatest bending, strain energy, work done and energy loss. Ignoring the energy loss, the design performs extremely well. In contrast, the closed design domain performs poorly. The heuristic gives the best performance of the methods discussed in this work, with a relatively large output displacement and low strain energy and energy loss. Of the methods discussed in this work, the heuristic gives the best performance, with a relatively large output displacement and low strain energy and energy loss. Whilst the energy penalty shows promise it is impeded by minimum length scales of topology optimisation, which prevent the formation of thin skins, and tends to become trapped in suboptimal local minima. In contrast, the Pneunet design gives a relatively large displacement across the entire range of output stiffnesses and can exert a significant amount of work on the output spring, but to do so it must take up large amounts of internal strain. This inherent softness is beneficial when acting in free space or on very soft objects but detrimental when grasping stiffer ones as its output work declines at higher output stiffnesses. E. Experimental Validation To validate the concept, a heuristic skin soft finger was 3D printed using a Stratasys Connex3 Polyjet printer. It allows blending of multiple base materials to produce soft elastomers ranging from Shore-A 30 to 95 as well as rigid materials. The three optimised materials are approximated as Shore-A 30, Shore-A 60 and Shore-A 85. Figure 8 shows the resulting printed finger in its undeformed state and during inflation. Although the printed material properties are only an approximation of the optimisation materials, the qualitative behaviour of the actuator matches the optimisation, validating the modelling and design approach. In the future, the mechanical properties of the 3D printed materials will be characterised to enhance simulation accuracy further. VI. CONCLUSION Guaranteeing closure in the topology optoimisation of pneumatically actuated soft robots is a significant problem which has not been solved in existing research. We discussed two new methods for generated airtight or low-leakage soft robots. Of the two, the heuristic approach outperforms the rigorous optimisation method, but the latter approach is worthy of further investigation. In addition, this work presented a multi-material method for pneumatic topology optimisation and several new soft gripper designs. In numerical studies, the best optimised designs performed comparably to a Pneunet, a state of the art single material design. However, the topology optimisation method shows promise in generating bespoke designs for specific grasping challenges and can be generalised to any problem involving soft robotic motion. Nonlinearities including large-deformation, hyperelasticity and contact remain a challenge in topology optimisation. In future we aim to experimentally validate our designs in soft grasping and investigate the optimisation of non-linear mechanics. Fig. 1 . 1Fig. 1. 3-Material optimised soft-gripper under 50kPa pressure (5x deformation scale). Pressure is applied to the two faces on the left, causing the jaws to close on the right. Material stiffnesses are: Red -100MPa, Green -10MPa, Blue -1MPa Fig. 2 . 3 - 23Material optimised soft-gripper with stiffnesses: Red -100MPa, Green -10MPa, Blue -1MPa,(a) Design domain (b) Undeformed Side-view (c) Undeformed Top-View Fig. 3 . 3 -Fig. 4 . 3 - 3343Material optimised soft-finger with stiffnesses: Red -100MPa, Green -10MPa, Blue -1MPa,(a) Design domain (b) Undeformed (c) Deformed (5x deformation scale) (d) Optimised Pressure Distribution (Undeformed) (e) Implied pressure boundary (f) Complete design with sealed chamber Material optimised soft-finger with casing -stiffnesses: Red -100MPa, Green -10MPa, Blue -1MPa, (a) Design domain (b) Undeformed Side-view (casing not shown) (c) Undeformed top-view (casing not shown) (d) Deformed half model, showing casing (5x deformation scale) Fig. 6 . 6Energy Penalised soft-gripper with (a) V f 1 = 0.2 (b) V f 1 = 0.4nesses (springs placed at the output face) from 0.1 N m −1 to 1000 N m −1 . 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[ "Detection of Atmospheric Escape from Four Young Mini Neptunes", "Detection of Atmospheric Escape from Four Young Mini Neptunes" ]
[ "Michael Zhang \nDepartment of Astronomy\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n", "Heather A Knutson \nDivision of Geological and Planetary Sciences\nCalifornia Institute of Technology\n\n", "Fei Dai \nDivision of Geological and Planetary Sciences\nCalifornia Institute of Technology\n\n", "Lile Wang \nKavli Institute for Astronomy and Astrophysics\nPeking University\n100871BeijingChina\n", "George R Ricker \nKavli Institute for Astrophysics and Space Research\nMassachusetts Institute of Technology\n\n", "Richard P Schwarz \nCenter for Astrophysics\nHarvard & Smithsonian\n60 Garden Street02138CambridgeMAUSA\n", "Christopher Mann \nInstitute for Research on Exoplanets\nUniversity of Montreal\n\n", "Karen Collins \nCenter for Astrophysics\nHarvard University\n\n" ]
[ "Department of Astronomy\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Division of Geological and Planetary Sciences\nCalifornia Institute of Technology\n", "Division of Geological and Planetary Sciences\nCalifornia Institute of Technology\n", "Kavli Institute for Astronomy and Astrophysics\nPeking University\n100871BeijingChina", "Kavli Institute for Astrophysics and Space Research\nMassachusetts Institute of Technology\n", "Center for Astrophysics\nHarvard & Smithsonian\n60 Garden Street02138CambridgeMAUSA", "Institute for Research on Exoplanets\nUniversity of Montreal\n", "Center for Astrophysics\nHarvard University\n" ]
[]
We use Keck/NIRSPEC to survey a sample of of young (<1 Gyr), short period mini Neptunes orbiting nearby K dwargs to measure their mass loss via the metastable helium line. We detect helium absorption from all four of the targets in our initial sample. The first detection, around TOI 560b, was announced in a previous paper. We now announce three additional detections around TOI 1430.01, 2076b, and TOI 1683.01. All four planets show an average in-transit excess absorption of 0.7-1.0%. However, the outflows differ in their kinematic properties. TOI 1430b exhibits pre-ingress absorption, while TOI 2076b's outflow is exceptionally optically thick and shows significant post-egress absorption. For all four planets, the width of the measured helium absorption signal is consistent with expectations for a photoevaporative outflow (10-30 km/s, 5000-10,000 K). Unless broadening mechanisms other than thermal velocity and the bulk outflow velocity are significant, our observations disfavor core-powered mass loss models, which predict much slower (1-3 km/s) outflows. We utilize both an isothermal Parker wind model and an order-of-magnitude method to estimate the mass loss timescale, and obtain ∼ a few hundred Myr for each planet. We conclude that many, if not all, of these planets will lose their hydrogen-rich envelopes and become super Earths. Our results demonstrate that most mini Neptunes orbiting sun-like stars have primordial atmospheres, and that photoevaporation is an efficient mechanism for stripping these atmospheres and transforming these planets into super Earths.
10.3847/1538-3881/aca75b
[ "https://export.arxiv.org/pdf/2207.13099v2.pdf" ]
251,104,690
2207.13099
4974c144a07056fd21a58559323aaccfd870a229
Detection of Atmospheric Escape from Four Young Mini Neptunes January 16, 2023 Michael Zhang Department of Astronomy California Institute of Technology 91125PasadenaCAUSA Heather A Knutson Division of Geological and Planetary Sciences California Institute of Technology Fei Dai Division of Geological and Planetary Sciences California Institute of Technology Lile Wang Kavli Institute for Astronomy and Astrophysics Peking University 100871BeijingChina George R Ricker Kavli Institute for Astrophysics and Space Research Massachusetts Institute of Technology Richard P Schwarz Center for Astrophysics Harvard & Smithsonian 60 Garden Street02138CambridgeMAUSA Christopher Mann Institute for Research on Exoplanets University of Montreal Karen Collins Center for Astrophysics Harvard University Detection of Atmospheric Escape from Four Young Mini Neptunes January 16, 202310.26134/ExoFOP5)Draft version Typeset using L A T E X twocolumn style in AASTeX63 We use Keck/NIRSPEC to survey a sample of of young (<1 Gyr), short period mini Neptunes orbiting nearby K dwargs to measure their mass loss via the metastable helium line. We detect helium absorption from all four of the targets in our initial sample. The first detection, around TOI 560b, was announced in a previous paper. We now announce three additional detections around TOI 1430.01, 2076b, and TOI 1683.01. All four planets show an average in-transit excess absorption of 0.7-1.0%. However, the outflows differ in their kinematic properties. TOI 1430b exhibits pre-ingress absorption, while TOI 2076b's outflow is exceptionally optically thick and shows significant post-egress absorption. For all four planets, the width of the measured helium absorption signal is consistent with expectations for a photoevaporative outflow (10-30 km/s, 5000-10,000 K). Unless broadening mechanisms other than thermal velocity and the bulk outflow velocity are significant, our observations disfavor core-powered mass loss models, which predict much slower (1-3 km/s) outflows. We utilize both an isothermal Parker wind model and an order-of-magnitude method to estimate the mass loss timescale, and obtain ∼ a few hundred Myr for each planet. We conclude that many, if not all, of these planets will lose their hydrogen-rich envelopes and become super Earths. Our results demonstrate that most mini Neptunes orbiting sun-like stars have primordial atmospheres, and that photoevaporation is an efficient mechanism for stripping these atmospheres and transforming these planets into super Earths. ABSTRACT We use Keck/NIRSPEC to survey a sample of of young (<1 Gyr), short period mini Neptunes orbiting nearby K dwargs to measure their mass loss via the metastable helium line. We detect helium absorption from all four of the targets in our initial sample. The first detection, around TOI 560b, was announced in a previous paper. We now announce three additional detections around TOI 1430.01, 2076b, and TOI 1683.01. All four planets show an average in-transit excess absorption of 0.7-1.0%. However, the outflows differ in their kinematic properties. TOI 1430b exhibits pre-ingress absorption, while TOI 2076b's outflow is exceptionally optically thick and shows significant post-egress absorption. For all four planets, the width of the measured helium absorption signal is consistent with expectations for a photoevaporative outflow (10-30 km/s, 5000-10,000 K). Unless broadening mechanisms other than thermal velocity and the bulk outflow velocity are significant, our observations disfavor core-powered mass loss models, which predict much slower (1-3 km/s) outflows. We utilize both an isothermal Parker wind model and an order-of-magnitude method to estimate the mass loss timescale, and obtain ∼ a few hundred Myr for each planet. We conclude that many, if not all, of these planets will lose their hydrogen-rich envelopes and become super Earths. Our results demonstrate that most mini Neptunes orbiting sun-like stars have primordial atmospheres, and that photoevaporation is an efficient mechanism for stripping these atmospheres and transforming these planets into super Earths. 1. INTRODUCTION Planets between 1-4 R ⊕ are frequently found on closein orbits around nearby sun-like stars (Zhu & Dong 2021). This class of planets has no comparable analogue in the solar system, and there is considerable debate about their origins (Drazkowska et al. 2022). A now-famous "radius gap" or "Fulton gap" (Fulton et al. 2017;Fulton & Petigura 2018) separates the smaller, denser super Earths (1-1.7 R ⊕ ) from the larger, less dense mini Neptunes (2-3 R ⊕ ). The conventional explanation for this gap is that mini Neptunes have a primordial hydrogen/helium atmosphere comprising ∼1% of their total mass that significantly inflates their radii, whereas super Earths have either lost their hydrogen-rich atmospheres or never acquired them in the first place (e.g., Lee & Connors 2021;Lee et al. 2022)). If most super-Earths initially formed with hydrogen-rich envelopes, they could have been stripped away by intense X-ray and extreme UV (XUV) irradiation from the young star (photoevaporative mass loss, e.g. Owen & Wu 2017;Mills & Mazeh 2017). Alternatively, the energy powering the outflow may predominately come from the internal heat of formation ('core-powered' mass loss; Ginzburg et al. 2018;Gupta & Schlichting 2019). A third possibility is that some or most mini Neptunes have no hydrogen-rich envelopes at all, but instead formed with substantial water-rich envelopes (e.g. Mousis et al. 2020). In order to differentiate between these competing hypotheses, it is crucial to obtain observations of young mini Neptunes to ascertain whether and how quickly they are losing hydrogen/helium. Even though the study of escaping mini-Neptune atmospheres is in its infancy, observations of escaping atmospheres around larger planets have a long history, as do attempts to understand the observations with models. The first detection of Lyα absorption occurred two years after the detection of the first transiting exoplanet (Vidal-Madjar et al. 2003). Because the Lyα core is wiped out by the interstellar medium and only the highvelocity wings can be seen, Lyα observations are only logarithmically sensitive to the mass loss rate (Owen et al. 2023). Helium absorption observations hold more promise because interstellar absorption is not a problem, and because the line probes the inner regions of the outflow. The 1083 nm helium triplet is a sensitive probe of escaping atmospheres Spake et al. (2018); Oklopčić & Hirata (2018), and can be readily accessed using ground-based telescopes. This technique works best for planets with K dwarf hosts, as these stars are predicted to produce the largest metastable helium population (Oklopčić 2019). Escaping helium has been detected around numerous giant planets, including WASP-107b (Spake et al. 2018;Allart et al. 2019), WASP-69b (Nortmann et al. 2018), and HD 189733b Guilluy et al. 2020;Zhang et al. 2022a). Models capable of predicting helium absorption include the Parker wind models of Oklopčić & Hirata (2018) and Lampón et al. (2020)), multiple 1D hydrodynamic models (e.g. The PLUTO-CLOUDY Interface, or TPCI; Salz et al. 2015a), the 3D Monte Carlo simulation EVaporating Exoplanets (Bourrier & Lecavelier des Etangs 2013;Allart et al. 2018Allart et al. , 2019, and many grid-based 3D hydrodynamic codes (Wang & Dai 2018;Shaikhislamov et al. 2021;Khodachenko et al. 2021;MacLeod & Oklopčić 2022;Rumenskikh et al. 2022). Although these models have been used to reproduce the observed outflows from giant exoplanets, they are largely untested in the mini-Neptune regime. Earlier this year, we reported the first detections of ongoing mass loss from young mini Neptunes. In Zhang et al. (2022c), we detected Lyα absorption from the 400 Myr mini Neptune HD 63433c, which orbits a sun-like star, but not from the inner mini Neptune in the same system. This shows that the outer planet has an escaping hydrogen/helium atmosphere, but further interpretation is difficult because the interstellar medium blocks the core of the line, and because the non-detection from the inner planet could be due to the absence of a lightweight atmosphere, a magnetic field preventing the outflow from being accelerated to high velocities, or photoionization of escaping hydrogen (Owen et al. 2023). In Zhang et al. (2022b), we reported metastable helium ab-sorption from the outflowing atmosphere of TOI 560b, a 500 Myr mini Neptune orbiting a K dwarf. In these studies, our Microthena and TPCI models both struggled to reproduce the observations. The Microthena models could roughly match the depth of the Lyα blue wing absorption for HD 63433c, but could not explain either the red wing absorption or the nondetection around HD 63433b. For TOI 560b, both the fiducial Microthena and TPCI models overestimated the observed helium absorption, and the Microthena models predicted a blueshift instead of the observed redshift. We suspect that the poor fit could be due to large uncertainties in planetary and stellar parameters, such as the EUV flux, the atmospheric metallicity, and the stellar wind conditions. It could also be due to limitations of the model, such as the neglect of magnetic fields, which could significantly affect the outflow (e.g. Owen & Adams 2014;Khodachenko et al. 2015;Arakcheev et al. 2017). To gain a solid understanding of if and how mini Neptunes turn into super Earths, a large sample of mini Neptunes with outflow detections is required, spanning a range of radii, masses, ages, and periods. It is much easier to accumulate a large sample with helium observations than with Lyα because the latter requires Hubble Space Telescope observations of very nearby stars that have favorable radial velocities, and we have not found any additional candidate as favorable as HD 63433. Having a larger sample of helium detections will reveal whether the features we detected in TOI 560b-such as the existence of an escaping hydrogen/helium atmosphere, the redshift of the signal, the asymmetry in the light curve, or the unexpectedly low amplitude of the absorption-are common or rare. Population studies can also reveal patterns in the data that give insight into the physical processes involved. By testing our models on a population level, we can mitigate some of the large uncertainties in individual planet and stellar parameters. By focusing on young planets which nevertheless span a factor of a few in age, we catch young mini Neptunes during the most critical period of their lives-the period when they could be rapidly losing mass and transitioning into super Earths. For these reasons, we undertook a survey of escaping helium observations from eight young (< 1 Gyr) mini Neptunes orbiting K-type stars observable from the Keck telescopes. This paper presents the first four results from the survey. All of our targets are recent discoveries by the Transiting Exoplanet Survey Satellite (TESS). TESS has surveyed the entire sky, finding many more nearby transiting planets than previously known and making a survey of this type possible for the first time. Section 2 presents the observations and data reduction, section 3 presents the results, section 4 estimates the mass loss rate using two different methods and comments on the implications, and section 5 concludes. OBSERVATIONS AND DATA REDUCTION Targets The Transiting Exoplanet Survey Satellite (TESS ) has already surveyed nearly the entire sky, discovering many nearby transiting planets amenable to atmospheric characterization. The TESS Object of Interest (TOI) catalog (Guerrero et al. 2021) contains a mix of new planet candidates and confirmed planets, including previously known planets re-detected by TESS. To select targets for our Keck survey, we downloaded the catalog from ExoFOP (DOI: 10.26134/ExoFOP5) and imposed several cuts. We required planet candidates to have radii between 2 and 3 R ⊕ , periods less than 20 days, declinations greater than −20 • , and K dwarf host stars (3900 < T eff < 5300 K) with J < 9. For unpublished candidates, we required a rating of 'Validated Planet Candidate?' (VPC?) or higher by the TESS Followup Observing Program (TFOP) in order to eliminate likely false positives from our sample. This designation means that follow-up observations have at least tentatively confirmed that the transit occurs for the correct star, and that there are no Gaia DR2 stars contaminating the TESS aperture that are bright enough to cause the detection. Most of the planets and planet candidates we identified did not have well-determined ages, so we ran a Lomb-Scargle peridogram on the TESS light curve and required a robustly detected rotation period in order to limit our sample to young stars. In practice, this means that the star's rotation period must be less than 15 days. We manually examined the TESS light curve to make sure that the rotational modulations looked strong and convincing (coherent and on the order of tens of percent). We also searched for the stars in the ROSAT All-Sky Survey, an X-ray catalog. Detectable X-ray emission is also evidence of youth, and we therefore prioritized targets with detections in this catalog for our survey. In total, we are currently targeting 8 planets orbiting 7 stars, of which 3 have ROSAT detections. Although we continue to monitor the list of newly announced TESS candidates, we do not expect the survey sample to expand significantly because TESS has already completed its baseline survey and has likely discovered all of the most favorable targets. There are additional favorable candidates remaining in the southern hemisphere, but these are not accessible from Keck. During the first year of the survey, we observed four targets: TOI 560b, 1430.01, 1683.01, and 2076b. Their properties are summarized in Table 1. TOI 560b and 2076b are both confirmed planets with published discovery papers (Barragán et al. 2022;El Mufti et al. 2021 andHedges et al. 2021, respectively). They are both the inner planets of multi-planetary systems, with one and two outer mini-Neptune companions respectively. TOI 1430 and 1683 are unpublished candidates, with the former included in the ROSAT All-Sky Survey, indicating its likely youth. Neither have any detected transiting companions. Of the four targets, only TOI 560b currently has a published radial velocity mass measurement. The masses of the other planets are derived from the massradius relation of Wolfgang et al. (2016). It is possible that this relation overestimates masses for young planets. TOI 560b gives us some reason for optimism, because the RV-derived mass of 10.2 +3.4 −3.4 M ⊕ is perfectly consistent with the mass inferred from the mass-radius relation, 10 ± 2 M ⊕ . Similarly, the relation gives 8.4 ± 2 for TOI 560c, compared to an RV mass of 9.7 +1.8 −1.7 M E . We encourage RV followup of the other planets to aid the interpretation of atmospheric observations. For TOI 1430.01 and 1683.01, which do not have published discovery papers, we took additional steps to validate their planetary nature. We checked that the even transits had a consistent transit depth with the odd transits. TOI 1683.01 was verified (in part) by the Dragonfly Telephoto Array, a robotic telescope stationed in New Mexico that is comprised of 48 individual telephoto lenses acting in concert similar to a 1.0-m refractor. With Dragonfly, we detected an ontime and on-target 1.6 ppt transit event, excluding the nearby Gaia stars as potential causes of a false-positive detection by TESS. Photometric collected by the Las Cumbres Observatory have only tentatively detected the transit of TOI 1430.01, due to its shallow transit depth. We searched the Gaia DR2 catalog to ensure that no bright stars are within 21 (one TESS pixel) of the target, as this could contaminate the light curve. For TOI 1430.01, Gaia detected 6 stars within this radius, but their total flux in the Gaia G band is only 0.03% that of TOI 1430. Near TOI 1683, Gaia detected one star with a flux 0.025% that of TOI 1683. Gaia has an angular resolution of 0.4 , corresponding to 17 AU at the distance of TOI 1430 and 21 AU at the distance of TOI 1683. We therefore conclude that, when combined with our detection of helium absorption, these tests suggest that these two candidates are likely to be real planets and treat them as such in the subsequent analysis. For these two planets, we obtained transit parameters by fitting the TESS light curves using the same Dai et al. (2021). To summarize, we downloaded the TESS photometry from the Mikulski Archive for Space Telescopes (MAST). We fit each light curve with batman (Kreidberg 2015), with free parameters including the out-of-transit flux level, stellar density, quadratic limb darkening coefficients, period, transit midpoint, R p /R s , and orbital inclination. For both planets, we have multiple sectors of TESS observations spanning 2 years, giving us a very accurate ephemeris that is sufficient to predict transit times corresponding to our helium observations to within 2-3 minutes. The phased light curves and transit fits are plotted in Appendix A. We obtained age estimates for TOI 1430 and 1683 using gyrochronology. Specifically, we used Schlaufman (2010), which relates age to stellar mass and rotation period, and Mamajek & Hillenbrand (2008), which relates age to B-V color and rotation period. We obtained the rotation period from a Lomb-Scargle periodogram of the TESS SAP fluxes, and take the masses and B-V colors from the TESS Input Catalog (Stassun et al. 2019). These stellar masses are computed from the effective temperature, which in turn is obtained from spectroscopy (if available) or deredenned colors. For TOI 1430, we obtain 156 ± 30 Myr with the first method and 174 ± 8 Myr using the second; for TOI 1683, we obtain 550 ± 150 and 460 ± 110 Myr. We adopt the average of the two methods and the more conservative error bars. Keck/NIRSPEC We used Keck/NIRSPEC to observe one transit for each of the four planets. All spectra were observed in Y band with the 12 × 0.432 slit. The exposure time was 60 seconds per frame, and we used an ABBA nod to subtract background, giving us a typical observing efficiency of 77%. Table 2 shows the target-specific details for each observation. We include TOI 560b for comparison, with the full details for this target provided in Zhang et al. (2022b). All four nights of observations were affected to varying degrees by weather. During the observations of TOI 560b, 1430.01, and 1683.01, transparency was poor throughout the night. The 1683.01 half-night also suffered from poor seeing. When we began observing at the beginning of the night the seeing was 1 , but it started rising around 12:44 UTC, eventually reaching values as high as 3 . The combination of a faint host star, clouds, and slit losses from poor seeing resulted in a low SNR for this target. For TOI 2076b, a combination of telescope problems, fog, and unexpected transit timing variations (see Appendix C) prevented us from observing the first 2/3 of the transit. However, the remainder of the night had stable seeing and no detectable clouds. To analyze the data, we used the same pipeline described in Zhang et al. (2022b). Among other steps, this pipeline produces A-B subtraction images to remove the background, uses optimal extraction on the differenced images to obtain 1D spectra, fits a template to the spectra to obtain wavelength solutions, and runs molecfit to remove telluric absorption features. The pipeline calculates excess absorption during the transit as a function of time and wavelength, which can then be collapsed along either axis to give the average in-transit excess absorption spectrum or the band-integrated light curve. In addition to fitting telluric features, molecfit also fits for the full width half maximum (FWHM) of the line spread profile, but it is not stable enough to reliably fit the FWHM for every spectrum. Instead, we run it on every spectrum and take the median of the FWHMs across the night for every target except TOI 1683, which has low SNR. We obtain 3.4 pixels, corresponding to R=32,000 at the position of the helium line. We then rerun all the molecfit fits with the FWHM fixed to 3.4 pixels. This value is close to what we obtain for each individual night: 3.65 ± 0.38 (TOI 560), 3.36 ± 0.17 (TOI 1430), 3.40 ± 0.12 (TOI 2076), where the reported error is the standard deviation of the FWHM across the night. Figure 1 shows the stellar spectrum in the vicinity of the helium line, both before and after correction for tellurics by molecfit. The strongest stellar line is a Si I at a rest frame vacuum wavelength of 10830Å; the second strongest line is the metastable helium line, at 10830Å. For TOI 1430 and 2076, the helium line was far from any tellurics. For TOI 1683, telluric absorption begins to pick up only at the reddest edge of the helium line, 20 km/s from the center. As we will see, the observed planetary absorption signal is blueshifted, making the telluric absorption even less important. It is only for TOI 560 that a telluric line falls right on top of the helium line. Due to extraordinary luck, telluric absorption was minimal because this night happened to be exceptionally dry-molecfit reports a precipitable water vapor (PWV) column of 0.15 mm, compared to 0.9 mm for TOI 1430, 1.0-1.4 mm for TOI 2076, and ∼1 mm for TOI 1683. We discuss the telluric correction for TOI 560 in greater detail in Zhang et al. (2022b), and are confident that it does not greatly affect our conclusions. X-ray observations We also separately observed all of the host stars except for TOI 2076 using XMM-Newton to characterize their X-ray spectra, which are needed as inputs for photoevaporative mass loss models. XMM-Newton observes with all six instruments simultaneously, of which the most useful for our purposes are the three EPIC cameras and the Optical Monitor (OM). The OM observed with the UVW2 and UVM2 filters in order to measure the star's mid-ultraviolet flux, which destroys metastable helium. We followed the procedure outlined in Zhang et al. (2022c) to compute the star's X-ray spectrum and MUV flux from the XMM-Newton EPIC and OM observations, respectively, except that we only fit a one-temperature model for TOI 1683 due to its much lower S/N. The observation IDs, along with the X-ray spectra, light curves, and inferred model parameters for TOI 1430 and 1683 are given in Appendix B. The corresponding X-ray observations for TOI 560 are described in Zhang et al. (2022b). TOI 2076 does not have a XMM-Newton observation from our program, but it will be observed as part of PI Wheatley's program 090431 in the current cycle. TOI 2076 was also observed by the ROSAT All-Sky Survey (Boller et al. 2016). However, RASS only measured 28 photons, too few to fit the spectrum with a plasma model and obtain meaningful constraints. We instead assume that TOI 2076 has the same X-ray spectral shape as TOI 1430, a star whose estimated age and effective temperature are within 1σ of that of TOI 2076. We scale TOI 1430's X-ray flux to 17 erg s −1 cm −2 at 1 AU, derived from the L X /L Bol for TOI 2076 obtained from ROSAT data by Hedges et al. (2021). 3. RESULTS Figure 2 shows the excess absorption from each of the four planets as a function of time and wavelength, relative to the average flux outside of the white light transit. Figure 3 shows the average in-transit excess absorption spectrum in the planetary rest frame, while Figure 4 shows the band-integrated light curve in a 3Å bandpass centered on the main helium peak. Figure 1. The stellar spectra in the vicinity of the helium triplet (in red), before and after correction for tellurics. Unlike in the rest of the paper, wavelengths are in Earth's frame, not the stellar rest frame, leading telluric lines to have the same wavelengths for every target. The telluric lines do not overlap with the helium line for any target except TOI 560, for which we were lucky to have had an exceptionally dry night. 1.01 ± 0.05 −6.7 ± 1.4 < 0.40 ± 0.05, > 0.18 ± 0.05 10.0 ± 0.7 0.99 ± 0.09 * Only TOI 2076b has a visually distinguishable secondary peak. The four absorption signals share both striking similarities and striking differences. Despite having different radii, orbital periods, ages, and masses, all four planets show an average in-transit absorption depth of 0.7-1.0%. None of the planets show an evolving velocity that tracks the radial acceleration of the planet. This might cause one to suspect that stellar activity is the origin of the putative planetary signals, but the band-integrated light curves strongly argue for a planetary origin: they de-crease at the white light ingress and recover by the white light egress for all planets except TOI 2076b. One final common feature of all our observations is unexplained variations in the light curve, such as the rise prior to ingress for TOI 560b, the dip just after ingress for TOI 1430.01, the dip well before ingress for TOI 1683.01, and the decline for TOI 2076b 3 hours after egress. We attribute these features to stellar activity, because our targets are young stars and the stellar helium line is a well known tracer of chromospheric activity. It is also conceivable that some of this variability is due to absorption from the escaping planetary atmosphere, which may form extended and complex structures (see e.g. the 3D simulations of McCann et al. 2019). The differences between the planets are as interesting as the similarities. One of the most puzzling features of the TOI 560b absorption signal, and the hardest to explain with our 3D models, was the redshift (Zhang et al. 2022b). Young stars are expected to have a strong stellar wind pushing the outflow away from the star, which is toward the observer during transit, and previous helium observations of giant exoplanets-including HD 189733b Guilluy et al. 2020;Zhang et al. 2022a (Czesla et al. 2022). Our new observations show that the redshift we observed was unusual, as our other three planets all show net blueshifts. Also potentially exceptional with TOI 560b was the time evolution of the absorption; in both Figure 2 and 4, it is evident that absorption peaks after mid-transit. TOI 1430.01 does not show this characteristic, but more data is required to evaluate whether TOI 1683.01 and 2076b do. If TOI 560b has its quirks, the other planets are no less peculiar. TOI 1430.01 shows what appears to be pre-ingress absorption. Without any detected velocity evolution matching the planets orbit, it is difficult to determine whether this pre-ingress absorption might be due to stellar variability, or a combination of planetary absorption and stellar variability. Real pre-ingress absorption is not implausible, as it has been seen for HAT-P-32b (Czesla et al. 2022) and can be produced by an up-orbit stream (e.g. Lai et al. 2010;McCann et al. 2019;Matsakos et al. 2015). We are not aware of any published 3D hydrodynamic simulations that predict strong pre-ingress helium absorption, but MacLeod & Oklopčić (2022) predicts weak pre-ingress absorption in their simulation with an intermediate-strength (∼10x solar) stellar wind. Incidentally, in their simulation with a strong stellar wind (∼100x solar), the helium absorption has considerably less velocity evolution than the planet itself-reminiscent of our observations. Finally, TOI 2076b is unique among our four planets in showing extensive absorption for at least half an hour after white light egress, with the band-integrated light curve only declining 50 minutes after egress. TOI 2076b is also unique in showing both a clear secondary peak and a primary peak, both blueshifted. The peak ratio is 0.40 ± 0.05, and there is a distinct valley in between the two peaks. The peak ratio is 0.125 in the case of a perfectly optically thin outflow and 1 for a perfectly optically thick outflow; thus, the observed peak ratio indicates an effective optical depth in between these two extremes. We do not detect a secondary peak for the other three planets, indicating that their outflows are likely closer to the optically thin limit. There are relatively few detections of secondary peaks in published helium observations of giant planets (e.g. HD 189733b, HAT-P-32b), and to our knowledge, no other planet has such a prominent secondary peak separated from the main peak by such a prominent valley. This is partly due to the high SNR of our detection; with higher SNR data, HAT-P-32b (Czesla et al. 2022) may turn out to have a similar peak ratio and valley depth. Salz et al. (2018) report a similar peak ratio of 0.36 ± 0.03 for HD 189733b, corresponding to an optical depth in the main peak of 3.2, although the peaks are not as clearly separated. Coincidentally, Gaidos et al. (2022) observed the same transit of TOI 2076b from the same mountain, using the high resolution InfraRed Doppler (IRD) spectrograph on the 8.2 m Subaru telescope. They obtain lower S/N than we do, with much higher correlated noise. Their results are consistent with ours: 1% absorption in the helium line during transit, with some post-egress absorption. However, they tentatively interpret their results as stellar activity rather than planetary absorption. Given that we see planetary absorption of similar amplitude and width in every planet of our survey, we disagree with this interpretation, but more transits will be needed to conclusively disambiguate between these two possibilities. Table 3 shows the quantitative properties of the outflow: peak absorption, redshift, peak ratio, and equivalent width. Defining these properties is not trivial because the line profile is unknown. We attempted to fit two Gaussians to the excess absorption spectrum, one for each absorption peak, but the fit was not good and would have resulted in misleading values if used for quantitative purposes. Instead, we take a data-driven approach. The peak absorption is taken from the highest datapoint. The redshift is similarly taken from the position of the highest datapoint, and assigned an error of one pixel (TOI 1683.01) or half a pixel (all other planets). The peak ratio is calculated by dividing the mean of the three points closest to the secondary peak by the mean of the three points closest to the primary peak. The primary peak is identified by the point of highest absorption, while the secondary peak is assumed to be blueshifted or redshifted by the same amount as the primary peak. Because we do not know how much of the absorption at the secondary peak is actually due to highly blueshifted gas absorbing at its primary peak, these ratios are at all upper limits. For TOI 2076b, we also obtain a lower limit by subtracting off the depth of the valley. The equivalent width is calculated from the light curves in Figure 4. For each planet except TOI 2076b, we reject the two lowest in-transit points to account for outliers, and average the next four lowest intransit points. For TOI 2076b, for which we have limited in-transit coverage, we average the two lowest in-transit points. These are next to each other and are the closest points to mid-transit, giving us increased confidence that the points are not outliers. Finally, we calculate the width of the absorption by dividing the equivalent absorption by the height. The width is the most consistent metric, and is within 10% of 1.1Å for all planets. Finally, Figure 5 shows the maximum excess absorption in units of scale height and compared to the EUV flux. This way of comparing planets has been used in many previous papers (e.g. Nortmann et al. 2018;Orell-Miquel et al. 2022). The uncertainties are large because the planetary masses are not well known, but the mini Neptunes tend to be higher on this metric than the gas giants. MASS LOSS RATE Our next step is to translate the magnitude of the measured helium absorption signal into a present-day mass loss rate rate for each planet. Unfortunately, this rate is not directly derivable from the data. The observed absorption comes from a trace species-helium in the triplet ground state-comprising less than a millionth of the atoms in the outflow, and whose abundance is determined by photoionization and collisional de-excitation, among other processes, and therefore changes with distance from the planet. The properties of the outflow depend on the metallicity of the atmo- sphere (e.g. Zhang et al. 2022b), which could plausibly range from 1x to 100x solar for mini Neptunes. The stellar wind and planetary and interplanetary magnetic fields influence the outflow geometry, which in turn alters the magnitude and shape of the absorption signature. The wind and magnetic field have highly uncertain strengths, especially because the wind is highly variable. For these reasons and others, we have previously struggled to match our Lyα absorption data for HD 63433 b/c (Zhang et al. 2022c) and the helium absorption data for TOI 560b (Zhang et al. 2022b) with either the 1D hydrodynamics code The Pluto-CLOUDY Interface (Salz et al. 2015a) or the 3D hydrodynamics code Microthena (Wang & Dai 2018). Without an accurate physical model of the outflow, it is impossible to derive a precise mass loss rate. Instead, we calculate order-of-magnitude estimates of the mass loss rate using two methods. The first method, introduced in Zhang et al. (2022b), estimates the mass of metastable helium crossing in front of the star from the maximum equivalent width of the absorption, assuming the outflow is optically thin. It then converts the mass of metastable helium into total mass, assuming that 10 −6 of the helium atoms are in the metastable state, and that 10% of all atoms are helium while the other 90% are hydrogen. The 10 −6 ratio is typical of a K-type host star, as shown by our 1D and 3D simulations of HD 63433 b/c and TOI 560b, and by the independent models of Oklopčić (2019) (their Figure 2). To obtain a mass loss rate, this total mass is divided by a timescale on which the mass disappears from view, which we take to be the radius of the star divided by a typical sound speed (10 km/s). The second method we use to estimate the mass loss rate is to put the Parker wind model of Oklopčić & Hirata (2018) in a Bayesian framework and run nested sampling retrievals on our datasets. The model requires the stellar spectrum, which we reconstruct using the methodology of Zhang et al. (2022b). Importantly, the EUV reconstruction is highly uncertain because the stellar EUV is only measurable from space and no space telescopes currently have EUV capabilities (France et al. 2022). For example, France et al. (2022) compared four different methods of reconstructing the EUV spectrum for Proxima Centauri and found that they were discrepant by 3-100x, depending on the wavelength; an order of magnitude difference in EUV flux can result in a 3x difference in helium absorption depth (Oklopčić 2019). Leaving aside the EUV uncertainty, the Parker wind forward model has three free parameters: the log of the mass loss rate, the temperature, and the velocity offset, all of which are given uniform priors. The first two parameters are required to compute the Parker wind, while the third is meant to account for the moderate redshifts and blueshifts seen in the data, which could be due to a combination of the non-spherical nature of photoevaporation (only the dayside is irradiated), the stellar wind, and a small eccentricity of the planetary orbit. The Parker wind model outputs optical depth as a func-tion of impact parameter and wavelength, which we then combine with the stellar radius, stellar limb darkening coefficients, and transit parameters to derive a prediction of the excess absorption as a function of wavelength and time. The log likelihood is computed by comparing this prediction to the observed absorption and error, and the nested sampling code dynesty (Speagle 2020) repeatedly calls the log likelihood function to compute the posterior distribution. This model assumes that the outflow is isothermal, that it is spherically symmetric, and that it consists only of hydrogen and helium in a 10:1 ratio. These assumptions are of dubious validity. For TOI 560b, our fiducial 3D simulations showed an outflow that rose in temperature from 1000 K at launch to almost 8000 K at the edge of the interface with the stellar wind, while the 1D solar metallicity simulation showed an outflow that declined from 10,000 K at maximum to 3000 K at 15 R p . The TOI 560b outflow is also not symmetric, as can be seen from Figure 2 and 4; and neither is the TOI 1430.01 outflow, if the pre-ingress absorption is not in fact stellar variability. Lastly, mini Neptunes can plausibly have very high atmospheric metallicities. For all of these reasons, we do not expect the Parker wind model to be much better at estimating the mass loss rate than the order-of-magnitude method. All timescale estimates are shown in Table 4. These were obtained by assuming that 1% of the planetary mass is in the envelope, with the planetary mass being taken from Table 1. After estimating the timescale using both methods, we computed the mass loss efficiency, as defined by the η in: dm dt = η 4 R 3 XU V L XU V GM p a 2(1) To roughly calculate R XU V , we follow Wang & Dai (2018) in assuming that the EUV photosphere is at ρ = 10 −13 g cm -3 . Assuming that the atmosphere is isothermal at the equilibrium temperature, and that the white light radius corresponds to P=100 mbar, we can derive the radius of the EUV photosphere (which we assume to be similar to the XUV photosphere): ρ phot = P µ k B T eq (2) β ≡ GM c µ R p k B T eq (3) R EUV = R p 1 + β −1 ln (ρ EU V /ρ phot )(4) Adopting this procedure, we find XUV radii 28-43% higher than the white light radii. Changing the pressure of the white light photosphere by a factor of 10 only changes this result by a few percent, but violating the isothermal assumption would result in larger changes. Using Equation 1, we calculate the mass loss efficiency for each planet and report it in Table 4. Just like the mass loss timescales, these efficiencies should not be trusted to better than a factor of a few, and efficiencies greater than 1 are particularly unrealistic. However, taking into account the large uncertainties, our observational values are sensible. Caldiroli et al. (2022) used the 1D hydrodynamic code ATES to calculate the efficiency of photoevaporation from a variety of planets with different gravitational potentials and XUV irradiations. For planets similar to ours in gravitational potential and XUV irradiation, they found η ≈ 0.7, comparable to the values we computed from our observations. Our observations allow us to rule out scenarios in which the planets have a very high gravitational potential, corresponding to very low efficiencies (∼ 3 × 10 −4 ). Our detection of escaping helium from all of the first four planets in our survey strongly suggests that most mini Neptunes have hydrogen/helium atmospheres, disfavoring the water world hypothesis for their low densities. The high mass loss rates we infer, which are sufficient to strip the envelope in hundreds of Myr, show that the radius gap separating super Earths and mini Neptunes must evolve over a planet's lifetime. However, they do not necessarily disprove the primordial origin of the radius valley, as Lee & Connors (2021) find that photoevaporation would fill in a primordial radius valley instead of deepening it. Further work is necessary to investigate the consistency of our observations with the primordial radius valley hypothesis. Photoevaporation or core powered mass loss? It is worthwhile to ask why the data favors high temperatures. As many authors have pointed out (e.g. Mansfield et al. 2018;Vissapragada et al. 2020), there is a degeneracy between temperature and mass loss rate when only the equivalent width of the helium absorption is known, but this degeneracy is broken when the line shape is resolved. Specifically, the width of the line constrains the temperature (e.g. Dos . For a Parker outflow, the sound speed, the bulk outflow speed, and the thermal speed are all close to 2k B T µm H = 9.4 km/s for T=7000 K and µ = 1.3, leading to a FWHM of 2.355σ ≈ 22 km/s. By contrast, core-powered mass loss, at T ∼ 800 K, would exhibit a FWHM of only 7.5 km/s. Absorption from photoevaporation is just wide enough to be resolvable with NIRSPEC, whose line spread profile coincidentally has a FWHM of 9.4 km/s. With the line width revealing the temperature, the degeneracy is broken, and the model fits provide tight constraints on both the outflow temperature and the corresponding mass loss rate. Taken at face value, our Parker wind model appears to favor photoevaporation over core-powered mass loss as the cause of the outflows observed for all four planets. The inferred outflow temperature of several thousand Kelvin is typical of photoevaporative models (see e.g. the simulations of Salz et al. 2015b), whereas a core-powered outflow should have a temperature closer to the planet's equilibrium temperature of ∼800 K (e.g. Ginzburg et al. 2018). For these planets, it is not possible for a core-powered outflow to set the mass loss rate by going supersonic before EUV photons can penetrate the outflow (e.g. Bean et al. 2021) because the sonic radius for an outflow at T ∼ 800 K is at ∼ 20R p , whereas the helium signal's effective radius is a few R p . The inferred mass loss timescales from the observed signals are on the order of hundreds of Myr, comparable to the ages of the systems. This implies that the observed outflows can strip most or all of the envelope within the first Gyr of the planets' lives. This argument, however, rests on the assumption that the only significant broadening sources are thermal motion and the bulk velocity predicted by the Parker wind model. While natural broadening is small enough to be negligible-and is in any case included in the modelthe same may not be true for the stellar wind, especially in combination with interplanetary and planetary magnetic fields. Acting together, these mechanisms or others yet unknown are capable of accelerating a small amount of gas to >100 km/s, as evidenced by Lyα absorption detected at these velocities (e.g. Vidal-Madjar et al. 2003;Lecavelier des Etangs et al. 2012. However, Lyα can typically only probe high velocities because absorption from the interstellar medium wipes out the line core, and it is likely that the observed high-velocity gas comprises only a tiny fraction of the total outflow. Whether stellar winds substantially broaden the helium absorption, which probes the outflow properties closer to the planet (∼3 planetary radii vs. ∼12 planetary radii), is uncertain. The blueshift seen for every planet except TOI 560b, the asymmetry in the light curve of TOI 560b, and the weak velocity evolution during the transit all suggest that stellar winds and/or magnetic fields shape the outflow properties of these planets. In the planetary frame, the observed velocity evolution represents a redshift early in the transit and a blueshift late in the transit, which is expected if the outflow emerging from the planet's dayside is turned around by the stellar wind and shaped into an increasingly accelerated tail. However, the redshift seen for TOI 560b suggests that the stellar wind in this system may be weaker than expected or that magnetic fields are significant, as our 3D models including winds all predict a blueshift (Zhang et al. 2022b). The lack of post-egress absorption for all but one planet argues for similar conclusions. More 3D simulations accounting for the effects of both the stellar wind and magnetic fields, similar to e.g. Owen & Adams (2014); Khodachenko et al. (2015); Arakcheev et al. (2017), will be necessary to translate these qualitative statements into quantitative constraints on the stellar wind properties, and to explore how much the stellar wind might contributed to the observed line width. Despite these caveats, the width of the helium absorption that we observe strongly hints that the outflows are driven by photoevaporation. However, a better theoretical understanding of the mass loss process and of the properties of these particular exoplanet systems is required before we can make a definitive statement. On masses and eccentricities Masses and eccentricites are critical to the interpretation of the helium data. Under certain conditions, such as a strong background magnetic field compared to the surface magnetic field, the mass loss rate could decline exponentially with planet mass at a rate ofṁ ∝ e −b where b = GMp c 2 s Rp (Adams 2011). Under these conditions, a 2.4M ⊕ increase in the planet mass causes a factor of two decrease in the mass loss rate. Adams (2011) find, however, that when the background field is much weaker than the surface field, the mass loss rate scales asṁ ∝ b 3 e −b , so that the same change in mass only changes the mass loss rate by several percent. The eccentricity-or more precisely, the radial velocity of the planet at transit center, K sin ω-is critical to interpreting whether the outflow is blueshifted or redshifted. Taking TOI 2076b as an example, with an orbital velocity of ∼110 km/s, an eccentricity of 0.06 is enough to match the apparent blueshift of 6.7 ± 1.4 km/s. This eccentricity is entirely reasonable; Mills et al. (2019) obtained a mean eccentricity of 0.05 for systems with multiple transiting planets, and 0.21 for systems with a single transiting planet. Masses, eccentricities, and arguments of periastron can be obtained by detecting the radial velocity signature of the planets with a sufficiently high signal-tonoise ratio. This is difficult for our targets because of the youth of their host stars, but not impossible. For example, TOI 560b has a measured mass, and its measured radial velocity during transit enhances rather than diminishes the apparent redshift (Zhang et al. 2022c). We are collaborating with another team in radial velocity followup of TOI 1430.01, and they have recently obtained a tentative upper limit comparable to our assumed mass of 7 M ⊕ (private communication, Joseph Murphy). Follow-up efforts for TOI 2076 and 1683.01 are underway as part of the TESS Follow-up Program. For TOI 2076b, which is near a 3:5 period ratio with 2076c and could be in resonance, transit timing variations could also be used to constrain its mass and eccentricity. Our efforts to model the system's current sample of TTV measurements (Appendix C) were not sufficient to break the mass-eccentricity, but it is possible that future transit observations will, especially if they successfully detect the chopping signal, which our models indicate could have an amplitude of several minutes. CONCLUSION We surveyed four nearby mini Neptunes orbiting young K-type stars, and detected helium absorption from all of them. The helium absorption signals show some consistent properties across the four planets, but each planet has unique features as well. Our simple models suggest that the observed outflows velocities and temperatures appear to be consistent with expectations for photoevaporation, with an inferred mass loss rate within a factor of a few of hydrodynamical predictions. The mass loss timescales for the observed outflows are on the order of several hundred Myr. This suggests that these mini Neptunes will all likely lose their hydrogen/helium envelopes, evolving into super Earths. If we wish to develop a more detailed understanding of mass loss processes on these planets, there are several promising paths forward. On the observational side, additional helium observations of these planets are needed in order to map out the kinematic structures of their outflows in more detail, mitigate the effect of stellar variability on the measured helium signals, and characterize the magnitude of any time variability in the outflow properties. Lyα observations could be used to probe neutral hydrogen absorption far from the planet and at high velocities, providing complementary constraints on the outflow properties, although many factors could result in the non-detection of even a strong outflow. If we can measure the planetary masses for TOI 1430.01, 1683.01, and 2076b via radial velocities or transit timing variations, it would also greatly facilitate our theoretical understanding of their outflows, because it would help reduce the dimensionality of the model parameter space and minimize degeneracies with other model parameters, such as the assumed EUV flux and atmospheric composition. By expanding the sam-ple of young mini Neptunes with outflows detected in the helium line, we can also obtain a better statistical picture of their outflow properties. This will allow us to search for correlations between specific features (like redshifted/blueshifted absorption) and other planet properties. We are currently pursuing all of these approaches. For example, we have HST programs to look for Lyα absorption from TOI 560b and 560c, with plans to write more proposals if these programs return promising data. On the theoretical front, there is also much work to be done. In our previous studies of the mini-Neptunes HD 63433b/c (Zhang et al. 2022c) and TOI 560b (Zhang et al. 2022b), we modeled our observations with the 1D hydrodynamic code TPCI (Salz et al. 2015a) and the 3D hydrodynamic code Microthena (Wang & Dai 2018), and found that neither code was able to to fully reproduce the observed signals. We encourage other teams with 3D hydrodynamic codes to try modeling the four outflows reported here. It is also important to consider whether it is realistic to expect any model to match the observations, given the uncertainties in the composition of the outer atmosphere, the amount of turbulence in the outflow, the XUV flux, the planetary and interplanetary magnetic fields, the velocity of the stellar wind, and the potentially large time variability in all of these quantities due to stellar activity. If this variability turns out to be a limiting source of uncertainty, expanding the sample of mini Neptunes with helium measurements would help to mitigate against these effects by enabling broader population-level comparisons between data and model as well as an empirical search for correlations between outflow properties and planetary or stellar parameters. Software: numpy (van der Walt et al. 2011), scipy (Virtanen et al. 2020), matplotlib (Hunter 2007), dynesty (Speagle 2020 All the TESS data used in this paper can be found in MAST: 10.17909/b5pm-bb37. ACKNOWLEDGMENTS The helium data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. B. XMM-NEWTON X-RAY MEASUREMENTS As part of this paper, we analyzed the X-ray measurements of TOI 1430 and 1683 taken by XMM-Newton's three EPIC cameras. The light curves are plotted in Figure 8. The measured spectrum of TOI 1430 and the theoretical model that best explains it are plotted in Figure 7. We fit a one-component model for TOI 1683 and a two-component model for TOI 1430, because the SNR is far higher for the latter than for the former. The physical parameters we obtained from our MCMC fits to the X-ray data are shown in Table 5. We proposed for our TOI 2076b helium observations using the Hedges et al. (2021) ephemeris, but Osborn et al. (2022) subsequently discovered transit timing variations in the three-planet system. As a result, the Hedges et al. (2021) ephemeris is far less accurate than its error bars imply. Since Osborn et al. (2022), TESS has re-observed the system and captured three more transits of b, one more transit of c, and one more transit of d. The three transits of b are the three immediately following our Keck/NIRSPEC transit. Accurately predicting the transit time for our Keck observation is critical to the interpretation of the helium outflow. Ideally, we would use a Bayesian method to model the TTVs of all three planets simultaneously, and use the ensemble of TTV models consistent with the data to generate predictions of the transit time. However, Osborn et al. (2022) reports that none of their three models are robust: one relies on the dubious assumption that the planets are not in resonance, and the other two arrive at either implausibly low or implausibly high values for the planetary masses, depending on our choice of priors. To estimate the transit time, we therefore consider three possible approaches: first, we fit a linear ephemeris to the three transits in the recent TESS sector and extrapolate back one epoch; second, we fit a linear ephemeris plus a sinuisoidal TTV model to all transit times for b; third, we fit a full TTV model using the ensemble of measured transit times for all three planets. All three approaches require us to derive transit times from TESS data. We do this using the exoplanet package ) by adapting their tutorial entitled "Simultaneous Fitting of a Transit with Stellar Variability". Specifically, we download the Simple Aperture Photometry (SAP) fluxes and model the stellar variability using the quasiperiodic RotationTerm kernel from the celerite2 package (Foreman-Mackey 2018), which is a sum of two simple harmonic oscillator terms: a primary term at the stellar rotation period, and a less coherent (lower quality factor, Q) secondary term at half the stellar rotation period. Unlike in the tutorial, we set the priors on log(Q) and log(∆Q) to be Gaussians with a mean of 12 and standard deviation of 5, instead of a mean of 0 and standard deviation of 2, because visual examination of the light curve shows highly coherent rotation across the entire month for all sectors of observation. In addition, unlike in the tutorial, we fix all stellar parameters and all transit parameters except the transit times to the median values derived by Osborn et al. (2022). This is for two reasons: first, Osborn et al. (2022) includes non-TESS photometry for all planets, so their values may be more robust; second, we wanted to minimize the number of free parameters to avoid getting caught in a local minimum, which sometimes happened when we experimented with additional parameters. Our final model includes seven free parameters describing the Gaussian Processes model (mean, white noise level, rotation period, rotational variability σ rot , log(Q), log(∆Q), relative amplitude of secondary term), in addition to individual transit times for each planets (14 transits in total, 5 in the new sector). We use PyMC (Salvatier et al. 2016) to perform No-U-Turn Hamiltonian Monte Carlo sampling with 2 chains with 1500 tune-up steps and 1000 sampling steps each, and use the built-in tools to check for convergence. Below are the transit times we measure from our Gaussian Processes fit; non-TESS times from Osborn et al. (2022) are added for completeness: Figure 2 . 2Excess absorption, in percent, as a function of time and wavelength in the stellar rest frame. Clockwise from upper left: TOI 560b (first published in Zhang et al. 2022b), 1430.01, 2076b, 1683.01. The colorbars and wavelengths are matched, but the y axes are not. The red lines show the wavelengths of the helium triplet in the planet rest frame. The horizontal dashed and solid lines show the beginning of white light ingress and end of white light egress, respectively. Figure 3 . 3Average in-transit excess absorption spectrum. Note that we only have the last third of the transit for TOI 2076, making it difficult to directly compare to the other full-transit observations. Figure 4 . 4Band-integrated light curves, integrated within 1.5Å of the main peak of the helium triplet, at 10833.3Å. The wavelength range is defined with respect to the stellar frame in order to give an accurate picture of stellar variability in the helium line. The y axes are matched, but not the x axes. Dashed and solid vertical lines represent the beginning and end of the white light transit, respectively. ), WASP-107b (Allart et al. 2019; Kirk et al. 2020), WASP-69b (Nortmann et al. 2018), GJ 3470b (Palle et al. 2020), HD 209458b (Alonso-Floriano et al. 2019), and HAT-P-11b (Allart et al. 2019)-saw a blueshift. The sole exception is HAT-P-32b Figure 5 . 5The four mini Neptunes in context, with the size of each point inversely proportional to the planetary radius. The y-axis shows δRp/Heq, the ratio of the increase in apparent radius at 1083 nm to the scale height at equilibrium temperature. For all of our planets (red filled circles), the y-axis error bars are dominated by the uncertainty in mass. In particular, planets other that TOI 560 have no mass measurement. Data for published planets (black filled circles) are taken fromKasper et al. 2020, with updates from Orell- Miquel et al. (2022 (GJ 1214b, tentative) andKirk et al. (2022) (WASP-52b). Photometric detections are excluded because they are not comparable. Breaking with precedent, upper limits are excluded to reduce clutter. ), SAS, exoplanet (Foreman-Mackey et al. 2021; Foreman-Mackey et al. 2021), PyMC (Salvatier et al. 2016), theano (Theano Development Team 2016), celerite2 (Foreman-Mackey 2018) Figure 6 . 6Phase-folded TESS light curves for TOI 1430.01 (left) and TOI 1683.01 (right). Figure 7 . 7Left: X-ray spectrum of TOI 1430 (top) and TOI 1683 (bottom), measured by XMM-Newton's three EPIC cameras. Note that both the throughput and the line spread profile are heavily dependent on wavelength. Right: the theoretical model that best explains the measurements. The model consists of two components (top, TOI 1430) or one component (bottom, TOI 1683) of optically thin, collisional plasma in equilibrium. Figure 8 . 8Background-subtracted X-ray count rate as measured by XMM-Newton's EPIC cameras. Left: TOI 1430; right: TOI 1683. C. TRANSIT TIMING VARIATIONS IN TOI 2076B Figure 9 . 9TTVs computed with the above ephemeris. The shaded regions are the model predictions from the low-mass nested sampling run (left), or the high-mass nested sampling run (right).As a final step, we improved upon the ephemerides ofOsborn et al. (2022) by deriving new periods and transit midpoints. For systems with TTVs, there is no single correct definition of period. Possible definitions include: (1) the coefficient found from a linear least-squares fit of transit time vs. epoch; (2) the coefficient found from a fit of transit time vs. epoch of the form T = P E + T 0 + A sin (ω super P E) + B cos (ω super P E); (3) the period implied by the position and velocity of the planet at a given time if the other planets didn't exist. Definitions 1 and 2 are the same if the TTVs are perfectly sinuisoidal and if the observations evenly sample the superperiod. Definition 3 depends on the time chosen. For our TTV fits, we choose a time of 1743, just before the first transit of b. Below, we present the periods and mean transit times that follow from definition 1: Table 1 . 1Planet and star properties Barragán et al. (2022), b Osborn et al. (2022), c Gaia Collaboration et al. (2018), d Own fits to TESS data, f Mass-radius relation of Wolfgang et al. (2016), g Gyrochronology using TESS-derived rotation period, h Hedges et al. (2021), i TOI 2076 has TTVs. See Appendix C, j TESS Input Catalog v8.2Parameter/Planet TOI 560b a TOI 1430.01 TOI 1683.01 TOI 2076b b Rp(R⊕) 2.79 ± 0.1 2.1 ± 0.2 d 2.3 ± 0.3 d 2.52 ± 0.04 Mp(M⊕) 10.2 +3.4 −3.1 7 ± 2? f 8 ± 2? f 9? f T eff, * 4511 ± 110 5067 ± 60 c 4539 ± 100 c 5200 ± 70 R * (R ) 0.65 ± 0.02 0.784 +0.018 −0.014 0.636 +0.031 −0.017 0.762 ± 0.016 M * (M ) 0.73 ± 0.02 0.85 ± j 0.10 0.69 ± 0.09 j 0.824 +0.035 −0.037 Prot, * 12.08 ± 0.11 5.79 ± 0.15 11.3 ± 1.5 6.84 ± 0.58 J * 7.6 7.6 8.8 7.6 Age (Myr) 480 − 750 165 ± 30 g 500 ± 150 g 204 ± 50 a(AU ) 0.0604 0.0705 0.036 0.0631 FXUV (ergs −1 cm −2 ) 5000 6800 12, 000 9500 FMUV (ergs −1 cm −2 ) 3000 27, 000 4, 900 50, 000 Teq(K) 714 ± 21 813 927 797 ± 12 P (d) 6.3980420 +0.0000067 −0.0000062 7.434162 d ± 1.5 × 10 −5 3.05752578 d ± 6.3 × 10 −6 10.355183 ± 0.000065 i T0(BJDTDB − 2, 457, 000) 1517.69013 +0.00056 −0.00059 2032.74865 ± 0.0010 2232.2348 ± 0.0011 2654.9837 ± 0.0015 i a/R * 19.98 +0.61 −0.63 17.25 ± 0.54 9.99 ± 0.39 17.8 ± 0.4 i( • ) 88.37 ± 0.18 88.69 +0.30 −0.23 85.99 +0.29 −0.28 88.9 ± 0.11 h Transit duration τ14(h) 2.143 +0.029 −0.027 2.71 +0.07 −0.04 1.28 +0.06 −0.04 3.251 ± 0.03 Sources: a methodology as Table 2 . 2Keck/NIRSPEC observationsParameter/Planet TOI 560b TOI 1430.01 TOI 1683.01 TOI 2076b Date (UTC) 2021-03-18 2021-09-21 2021-10-01 2022-03-16 Time (UTC) 12:06-16:07 05:04-09:58 10:31-15:51 12:06-16:07 SNR 150 160 67 190 Pre-ingress baseline (h) 0.7 0.7 2.4 0 Transit fraction observed (%) 100 100 100 31 In-transit duration (h) 2.1 2.7 1.3 1.0 Post-egress baseline (h) 0.6 1.4 1.6 3.0 Table 3 . 3Excess absorption spectra propertiesPlanet Height (%) Redshift (km/s) Ratio* Equivalent Width (mÅ) EW/Height (Å) TOI 560b 0.72 ± 0.08 4.3 ± 1.4 < 0.27 ± 0.20 8.6 ± 0.6 1.19 ± 0.16 TOI 1430.01 0.64 ± 0.06 −4.0 ± 1.4 < 0.19 ± 0.09 6.6 ± 0.5 1.03 ± 0.12 TOI 1683.01 0.84 ± 0.17 −6.7 ± 2.8 < 0.32 ± 0.11 8.5 ± 1.6 1.01 ± 0.28 TOI 2076b Table 4 . 4Estimated timescale to lose 1% of total planet mass. Despite the naming, both methods should be considered order-of-magnitude estimates only. (see Section 4).Planet OOM (Gyr) Parker (Gyr) T Parker (kK) ηOOM η Parker TOI 560b 0.9 0.12 +0.05 −0.04 9.8 ± 0.7 0.5 3.9 TOI 1430.01 0.5 0.10 ± 0.03 6.7 ± 0.3 0.5 2.8 TOI 1683.01 0.7 0.61 +0.54 −0.27 6.7 ± 1.4 0.2 0.3 TOI 2076b 0.6 0.70 +0.16 −0.11 5.0 ± 0.4 0.4 0.3 Based on observations obtained with XMM-Newton (observation ID 0882870701 for TOI 1430, 0882870501 for TOI 1683), an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. The grant number is 80NSSC22K0742. Funding for the TESS mission is provided by NASA's Science Mission Directorate. We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This research has made use of the Exoplanet Followup Observation Program (ExoFOP; DOI: 10.26134/Ex-oFOP5) website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program APPENDIX A. TESS LIGHT CURVES0.99900 0.99925 0.99950 0.99975 1.00000 1.00025 1.00050 Relative Flux 2 1 0 1 2 Time from Mid-transit (hours) 0.00025 0.00000 0.00025 Residual Flux 0.9980 0.9985 0.9990 0.9995 1.0000 1.0005 1.0010 Relative Flux 1.0 0.5 0.0 0.5 1.0 Time from Mid-transit (hours) 0.0005 0.0000 0.0005 Residual Flux Table 5 . 5Model parameters for XMM-Newton data Flux * (erg/s/cm 2 ) 1.37 +0.15 −0.06 × 10 −13 0.32 +0.05 −0.04 × 10 −13Note- * Derived, not a fit parameter. For the range 5-100Å (0.124-2.48 keV).Parameter Value (TOI 1430) Value (TOI 1683) Metallicity 0.29 +0.03 −0.16 0.13 +0.35 −0.07 kT1 (keV) 0.16 +0.31 −0.02 0.27 +0.05 −0.03 EM1 (cm −3 ) 8.1 +6.6 −2.1 × 10 50 17 +7 −11 × 10 50 kT2 (keV) 0.68 +0.06 −0.04 EM2 (cm −3 ) 14 +8 −4 × 10 50 Table 7 . 7Linearized ephemerides for TOI 2076b planetsPlanet P (d) T0(BJDTDB − 2, 457, 000) b 10.355183 ± 0.000065 2178.6439 ± 0.0023 c 21.01544 ± 0.00027 2147.9853 ± 0.0041 d 35.12561 ± 0.00011 2289.5491 ± 0.0011 When we use only the new sector of TESS data, compute a linear ephemeris from its 3 transits of b, and extrapolate one epoch back to the Keck observation epoch, we obtain a transit time of 2654.9837 ± 0.0015 (BJD TDB -2,457,000). When we compute all transit times of all planets from all TESS data, add in the non-TESS timings reported byOsborn et al. (2022), and perform a linear least-squares fit of the b transit times to a linear ephemeris plus a sinuisoid, we obtain a transit time of 2654.9841. The sinuisoid is assumed to have a superperiod of 1/|2/P c − 1/P b | = 713.74 ± 0.14 d, consistent with the 713.1 ± 2.7 d superperiod found byOsborn et al. (2022). For our third method, we combine the N-body code TTVFast(Deck et al. 2014)with the nested sampling code dynesty (Speagle 2020) to run two full TTV models on all transit times of all planets, following the methodology ofHadden & Lithwick (2016). Briefly, we assume zero mutual inclination and have five parameters per planet (mass, ecos(ω), esin(ω), period, mean anomaly). The mean anomaly is reparameterized as the epoch 0 transit time, to minimize the degeneracy with period. In the "low-mass" TTV model, we adopt a log-uniform prior on the masses from 1-20 M ⊕ and a uniform prior on the eccentricities from 0.001-0.2; in the "high-mass" TTV model, we adopt a uniform prior on the masses and a log-uniform prior on the eccentricities, with the same bounds. After nested sampling finishes, we draw randomly from the samples, taking into account the weights, and rerun TTVFast to predict the transit time for b at the desired epoch. We obtain 2654.9830 ± 0.0024 for the low-mass model, and 2654.9837 ± 0.0019 for the high-mass model. Since all of our estimates for the transit time are within 1.6 minutes of each other, we are confident that we correctly calculated the transit time despite the TTVs. We adopt as our final answer the result of the first method, 2654.9837 ± 0.0015. Unfortunately, this timing is 51 minutes earlier than what we expected when we proposed for the Keck observations. In combination with the telescope and weather problems, this meant that we were unable to observe the first 2/3 of transit.In running the TTV models, we had hoped to compute not just the transit time corresponding to the Keck observations, but the masses of the planets. 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[ "Non-equilibrium effects in chaperone-assisted translocation of a stiff polymer", "Non-equilibrium effects in chaperone-assisted translocation of a stiff polymer" ]
[ "Haji Rouhollah \nPhysics Department\nUniversity of Science and Technology (IUST)\n16846-13114TehranIran, Iran\n", "Abdolvahab [email protected] \nPhysics Department\nUniversity of Science and Technology (IUST)\n16846-13114TehranIran, Iran\n" ]
[ "Physics Department\nUniversity of Science and Technology (IUST)\n16846-13114TehranIran, Iran", "Physics Department\nUniversity of Science and Technology (IUST)\n16846-13114TehranIran, Iran" ]
[]
Chaperone assisted biopolymer translocation is the main model proposed for translocation in vivo. A dynamical Monte Carlo method is used to simulate the translocation of a stiff homopolymer through a nanopore driven by chaperones. Chaperones are proteins that bind to the polymer near the wall and prevent its backsliding through Cis side. The important parameters include binding energy, size and the local concentration of the chaperones. The profile of these local concentrations, build up the chaperones distribution. Here we investigate the effects of binding energy, size and the exponential distribution of chaperones in their equilibration in each step of the polymer translocation needed for stable translocation time. The simulation results show that in case of chaperones with size of a monomer (λ = 1) and/or positive effective binding energy and/or uniform distribution, the chaperones binding equilibration rate/frequency is less than 5 times per monomer. However, in some special cases in exponential distribution of chaperones with size λ > 1 and negative effective binding energy the equilibration rate will diverge to more than 20 times per monomer. We show that this non-equilibrium effect results in supper diffusion, seen before. Moreover, we confirm the equilibration process theoretically.
10.1016/j.physleta.2017.11.013
[ "https://export.arxiv.org/pdf/1710.00601v1.pdf" ]
119,085,207
1710.00601
6f2535a009912758d4ebe75ee77ed870a301c8dc
Non-equilibrium effects in chaperone-assisted translocation of a stiff polymer 2 Oct 2017 Haji Rouhollah Physics Department University of Science and Technology (IUST) 16846-13114TehranIran, Iran Abdolvahab [email protected] Physics Department University of Science and Technology (IUST) 16846-13114TehranIran, Iran Non-equilibrium effects in chaperone-assisted translocation of a stiff polymer 2 Oct 2017Preprint submitted to arxiv.org January 15, 2022(Rouhollah Haji Abdolvahab)Polymer translocationFirst passage timeChaperone distributionBinding energyNanoporesupper-diffusion Chaperone assisted biopolymer translocation is the main model proposed for translocation in vivo. A dynamical Monte Carlo method is used to simulate the translocation of a stiff homopolymer through a nanopore driven by chaperones. Chaperones are proteins that bind to the polymer near the wall and prevent its backsliding through Cis side. The important parameters include binding energy, size and the local concentration of the chaperones. The profile of these local concentrations, build up the chaperones distribution. Here we investigate the effects of binding energy, size and the exponential distribution of chaperones in their equilibration in each step of the polymer translocation needed for stable translocation time. The simulation results show that in case of chaperones with size of a monomer (λ = 1) and/or positive effective binding energy and/or uniform distribution, the chaperones binding equilibration rate/frequency is less than 5 times per monomer. However, in some special cases in exponential distribution of chaperones with size λ > 1 and negative effective binding energy the equilibration rate will diverge to more than 20 times per monomer. We show that this non-equilibrium effect results in supper diffusion, seen before. Moreover, we confirm the equilibration process theoretically. Introduction Translocation of biomolecules through the nanopores Meller (2003) is one of the most important processes within biological cells. This is a ubiquitous process in cell metabolism. Protein's translocation through endoplasmic reticulum is an example. The polymer translocation also seen in proteins transport through organelles like mitochondria Alberts et al. (2002); Muthukumar (2007); Rapoport (2007). Translocation of messenger RNA through nuclear pore complexes in gene expression and in transcription through eukaryotic cells are two other biological instances Alberts et al. (2002). Translocation of DNA through protein channels covering the bacterial membrane amid phage infection is another example Bates et al. (2003); Dreiseikelmann (1994). Biotechnological examples also includes gene therapy, drug delivery and cheap rapid sequencing of the biopolymers Marzio and Kasianowicz (2003); Nakane et al. (2003); Branton et al. (2008); Cohen et al. (2012); Fanzio et al. (2012); Carson and Wanunu (2015); Liang and Zhang (2015). The experimental work of Kasianowicz et al. Kasianowicz et al. (1996) on ssRNA translocation through an α-hemolysin channel was an influential work. Hereafter, there has been many experimental and theoretical works and simulations in polymer translocation Meller (2003); Panja et al. (2013); Sun and Luo (2014); Palyulin et al. (2014). There are many different mechanisms to drive translocation of polymers. In vitro, people usually use a strong electric field to drive the translocation of highly charged biopolymers like single stranded DNA or RNA. Moreover, there could be many other parameters affecting the translocation such as crowding Gopinathan and Kim (2007); Pu et al. (2016), pressure and confinement Grayson and Molineux (2007); Panja and Molineux (2010); Molineux and Panja (2013); Palyulin et al. (2014). However, the most important model for the translocation in vivo is chaperone assisted translocation Tomkiewicz et al. (2007). This model with the name of Brownian ratchet mechanism was first proposed by Simon et al. in 1992Simon et al. (1992. In this model proteins called chaperones are bound to the polymer in the Trans side and actively pull the polymer or just prevent its backsliding through the Cis side Alberts et al. (2002); Tomkiewicz et al. (2007). Later, the experiments of Matlack et al. in 1999 highlights the problem again Liebermeister et al. (2001); Elston (2002); Zandi et al. (2003). Subsequently, many theoretical and simulations struggled to have a better understanding of different aspects of the problem Palyulin et al. (2004); Abdolvahab et al. (2008Abdolvahab et al. ( , 2011b; Luo (2011, 2014); Wei-Ping Cao and Luo (2015); Suhonen and Linna (2016); Abdolvahab (2016). There are many works on the non-equilibrium aspects of forced polymer translocation Sakaue (2007); Bhattacharya and Binder (2010); Saito and Sakaue (2013); Vollmer and de Haan (2016). This out of equilibrium property emerge as a result of force pulling the polymer through the Trans side. Here we investigate the chaperone assisted translocation which is used in vivo. Following our recent works we consider the chaperones exponential distribution effects on polymer translocation Abdolvahab (2016Abdolvahab ( , 2017. We will show that even in the case of stiff polymer the distribution may induce the non-equilibrium effects in the translocation process. In what follows we examine the chaperones binding rate effects on the translocation. Hereafter introducing our dynamical Monte Carlo simulation, we will discuss our simulation results in our main part of article in section 3. Finally we will sum up our findings in the conclusion. Simulation Theoretical model As the figure 1 shows, we simulate a stiff homopolymer consisted of M monomer with size of L = Mσ. Chaperones with the same size of λσ are distributed only in the Trans side. We suppose the λ to be an integer Abdolvahab et al. (2011b); Ambjrnsson and Metzler (2004). They have local concentration which depends on their distribution. Moreover, there is a binding energy, positive or negative, between the chaperones and the polymer. They bind (unbind) to (from) the polymer in the Trans side. The polymer always can go to the right. However, for backsliding of the polymer, its near the wall site must be unbound. The wall has no width Abdolvahab (2017). Binding of chaperones bias the translocation through the Trans side. The master equation for this process is written as: ∂P (m, t) ∂t = W + (m − 1)P (m − 1, t − 1) + W − (m + 1)P (m + 1, t + 1) − (W + (m) + W − (m))P (m, t)(1) in which W ± are the transfer rates for translocating polymer to the right and left. P (m, t) is the probability of finding polymer in time t at condition in which m monomer of it translocated to the right. Using transfer rates, W ± , and boundary conditions one could find the translocation time by calculating its mean first passage time Gardiner (2002). Describing the Monte-Carlo method We use a dynamical Monte Carlo method to simulate the translocation of a stiff polymer as follows. The polymer always can go to the right with probability of half. However, backsliding of the polymer is restricted. The polymer may come back through the Cis side only if there is not any chaperone bound to the polymer near the wall. Moreover, we use from the so called transmission boundary condition Redner (2001). It means reflective at first and absorbing at the end. As a result the polymer does not come back to the Cis when the first monomer is near the wall Abdolvahab (2017). Chaperones will try to bind/unbind in each step of the translocation by frequency f per monomer per 40. It means, for example, in the case of a polymer with m = 40 and by the frequency f = 40, the binding/unbinding process is one time. A monomer in the Trans side is selected randomly. If there is a chaperone bound to it we try to unbind it with its probability and vice versa (it will try to bind a chaperone accordingly). Due to our computational limits, we changed the frequency from 1 to 10 3 . Chaperones binding probability: There are three terms in the binding probability. Boltzmann distribution, which depend on the binding energy between chaperones and the polymer. Entropy linked to different patterns in which chaperones may distributed on the polymer and availability of the chaperones related to its local density Ambjrnsson and Metzler (2004); Abdolvahab et al. (2011b). The second term is automatically comes in the simulation. In place of the binding energy, we define effective binding energy (EBE or E ef f ) to combine the first and third term as Abdolvahab et al. (2011b) E ef f ≡ − 1 λ log [c 0 v 0 exp (−ε/k B T )] , where ε comes for the chaperone binding energy per monomer of the polymer. c 0 denotes the chaperone concentration, and v 0 stands for their volume Abdolvahab et al. (2011a); Abdolvahab (2017). Thus the binding and unbinding probabilities are written as: P bind = exp − λ i=1 E i ef f 1 + exp − λ i=1 E i ef f , P unbind = 1 1 + exp − λ i=1 E i ef f .(2) The effective binding energy is changed from −4 to 4 and the polymer length is restricted to M = 50. In order to reach to an acceptable error, we repeat the translocation process for at least 10 4 times. Moreover, we use the chaperones of different sizes of λ = 1, 2 and 6. Chaperones distributions: For simplicity we restrict the chaperones to distributed only in the right part. We consider the exponential distribution with different rates, α, for the chaperones and compare its results with the usual uniform distribution (α = 0). In order to change the chaperones distribution in our Monte Carlo simulation it is enough to change the E ef f to E ef f + αd in which d is the distance (per monomer size) between the wall and the monomer in which we need its near chaperones concentration Abdolvahab (2016Abdolvahab ( , 2017. Results and discussion Mean translocation time We simulate 1 dimensional stiff homopolymer translocation through a nanopore using a dynamic Monte Carlo method in presence of the chaperones with different sizes and different spatial distributions. There are chaperones with distinct spatial distributions and various EBEs by monomers in the Trans side(There is not any chaperone in the Cis side.). Try number, or chaperones rate of binding (its frequency denotes by f ), is an important parameter in calculating the translocation time of the polymer. In spite of this importance, there are few works on investigating its effects on the polymer translocation D' Orsogna et al. (2007). People suppose that due to interaction of the polymer with the pore and its size, the chaperones will reach to equilibrium in each step of the polymer translocation Ambjrnsson and Metzler (2004). In what follows we will show that although this is true for the case of uniform distribution of chaperones, in exponential distribution, the equilibration frequency will become large and the assumption is violated. We translocate polymers in presence of chaperones of sizes λ = 1, 2, 6 and spatial distributions of α = 0, 1, 5, 10 and EBE = −4 : +4. The figure 2 shows mean translocation time versus frequency or rate of the chaperones for different exponential chaperone distributions of α = 0, 5 and different EBEs for chaperones of size λ = 2. Different curves stand for different chaperones spatial distribution and/or different EBEs. The right figure is a zoom of the left one for faster polymers. Note that α = 0 means uniform spatial distribution of the chaperones. The simulation results show that there are different regimes based on convexity and/or equilibrium rate: • In large enough and positive EBEs the time versus rate curves are strictly ascending. They soon will reach to their equilibrium rates. • In large enough and negative EBEs the time versus rate curves are strictly descending. They quickly will reach to their equilibrium rates usually but in some special cases the equilibrium rate will become quite large. • In some intermediate energies, we will see a maximum in time versus rate curves. In large positive EBEs, the chaperones do not prefer to bind to the polymer. Hence, the probability density function of finding the monomers bound are well asymmetric to the left (see the figure 3). In asymmetric density functions of this kind, mean is always in right and close to the most probable. We will show that in this situation increasing the fluctuation, will increase the P bind (near the wall binding probability). Consequently, by increasing the rate and decreasing the fluctuation, the translocation time will be increased (see the Appendix A.1). In contrast in large negative EBEs, increasing the frequency will cause the translocation time to be decreased (see the Appendix A.2). Equilibrium rate As discussed in the previous section, an important parameter in describing the polymer translocation is the equilibrium rate (the rate from which the translocation time does not change). In this regard, the polymers have two different behaviours. All the polymers with positive E ef f , which means the chaperones prefer not to bind to the polymer, have the small equilibrium rates. In contrast in the case of negative E ef f , the equilibrium rate could be relatively large. Indeed, our simulation results show that this non-equilibrium properties comes from the chaperones exponential distribution. We did not see any divergence in the uniform distribution of the chaperones (compare the plots in figure 4). The equilibrium rate of the polymers translocation in vicinity of the chaperones with different sizes of λ = 1, 2, 6 and with chaperones spatial distribution with different exponential rates of α = 0, 1, 5, 10 are plotted against E ef f in figure 4. As it shows there is not any important divergence in the cases of uniform distribution, α = 0, and/or chaperones with the size of a monomer, λ = 1. To better understand this results one should find a more detailed description of the chaperones exponential distributions effects on the polymer translocation (see Abdolvahab (2017) It is pertinent to mention that our results paved the way for understanding the supper diffusion reported in our previous work Abdolvahab (2017). In that article we show that in some special case the scaling exponent of time vs polymer length, β (T ∼ M β ), becomes less than 1 (β < 1). It can be explained as follows. Increasing the polymer length will increase the time for chaperones to equilibrate. Consequently, the equilibration rate will be decreased. Hence, trying to calculate the scaling exponent β before equilibrium will cause the translocation time increase less than the expected amount and the exponents could even become less than 1. Mean waiting time Details of the translocation can be seen from its waiting times. Mean Waiting Time (MWT) of the translocation for chaperones of size λ = 2 and E ef f = −4, 4 for uniform chaperone's distribution and 3 different rates of f = 10, 300, 900 is shown in figure 5a. As it shows MWT for E ef f = 4 and for rates of f ≥ 300 are decreasing linearly. However, for the same E ef f but for smaller rate of f = 10, it starts from smaller amounts and it makes its translocation velocity faster. In contrast, in case of E ef f = −4, the translocation velocity increase by increasing the rate from f = 10 to f = 300, 900. As a result of the large binding probability the MWT for the rates f = 300, 900 becomes completely sawtooth shape. As we saw in figure uniform distribution and exponential distribution with α = 5, both with chaperones of size λ = 2 and E ef f = −4 in rates of f = 300 equalize to each other. MWT for these two case is plotted in figure 5b. Because of the large negative E ef f , the chaperones bind to the polymer as right as it find a free place near the wall. In the case of uniform distribution availability of the chaperones are the same through the whole polymer which cause the sawtooth shape.However, in the case of exponential distribution, the chaperones may unbind from the polymer in further sites away from the wall. Consequently, the sawtooth shape becomes smooth or disappeared. On the other hand, in this case it takes time for the system to find its equilibrium and thus we see the translocation time is decreased by rate frequency and in f = 300 it coincide with its uniform counterparts. Conclusions We simulate the translocation of stiff homopolymer through a nanopore driven by chaperones. We investigate specially the chaperones binding frequency and spatial distribution on the translocation time. Our results show that there are different patterns of equilibration in terms of chaperones size, effective binding energy and spatial distribution. In most cases the equilibrium is reaching soon (less than 10 try per monomer). However, an increase in equilibration frequency (more than 20 try per monomer) is seen in the cases of λ > 1, E ef f < 0 and α > 0 roughly in the interval of −6 < λE ef f + α < −1. In larger amounts the chaperones do not prefer to bind and in less amounts the chaperones could not unbound from further sites. This result pave the way for understanding the supper diffusion reported in Abdolvahab (2017). We also strengthen our simulation results by theoretical discussion about the effect of chaperones binding rate on translocation time. Appendix A. Fluctuation in binding probability and translocation time Based on a master equation approach and using mean first passage time theory, we provide translocation time as follows Abdolvahab et al. (2011b): T = 2τ 0 P bind N + 1 − 1 − P bind P bind 1 − (1 − P bind ) N +1 , (A.1) where τ 0 is the time takes for a bare polymer to translocate over distance of a monomer, N is the total number of the monomers and P bind is the probability of the polymer to be bound near the wall. In the small and large amount of P bind one may approximate the equation A.1 as follows: Appendix A.1 . Large binding probability In large enough binding probability P bind ≃ 1 which occurs in large negative EBE, expansion of the equation A.1 leads to: T ≃ 2τ 0 P bind (N + 1). (A.2) Fluctuation of P bind which shows itself in translocation time can be investigated as follows. Admit the change in P bind to be of order δ; P bind → P bind ± δ. We then average over this range by integration: T large ≃ 1 2δ P bind +δ P bind −δ 2τ 0 P (N + 1)dP ≃ 2τ 0 (N + 1) P bind − δ , (A.3) where T large is average of T over fluctuations of P bind for large binding probabilities. As expected from our simulation results, it shows that increasing the chaperones binding rate which decreases the fluctuation δ, will reduce the mean translocation time. Appendix A.2. Small binding probability In contrast to the previous section, in small binding probability, expansion of equation A.1 results in: T (P bind ) = 2τ 0 (N + 1) × C 0 (N) − C 1 (N)P bind + C 2 (N)P 2 bind − · · · , (A.4) C 0 (N) = 1 + N 2 , C 1 (N) = N 2 1 + N − 1 3 , C 2 (N) = N(N − 1) 3! 1 + N − 2 4 . Note that this approximation is true when NP bind ≪ 1 and the important parameter here is the Péclet number not P bind itself (see Abdolvahab et al. (2011a) for more detail). As the figure 3 shows, the binding probability distribution is exponential. Let assume its exponential determines by parameter a; P (P bind ) ∝ exp(−aP bind ). Averaging of the translocation time over fluctuations of P bind leads to: Increasing the chaperones binding rate will decrease the fluctuation over P bind , δ, and as a result the mean translocation time, T small , will be increased. It is in place to note here that the EBE in which the regimes changed, say EBE 0 , may be obtained by comparison of second and third term in equation A.4: C 1 (N)P bind > C 2 (N)P 2 bind ⇒ N 3 P bind < 1. (A.7) Presume the binding probability to be proportional to its Boltzmann distribution (P bind ∝ exp(−λEBE)) leads us to EBE 0 ≃ 1 λ ln( N 3 ). As an example in case of λ = 2 and N = 50, EBE 0 ≃ 1.4 which is compatible with the simulation results (see e.g. figure 2a). Figure 1 : 1A stiff polymer is translocating from the Cis side (left) to the Trans side (right). Chaperones of size λσ are distributed in the Trans side. The total length of the polymer is L = M σ. m monomers are no translocated to the right. Figure 2 : 2Translocation time of polymers, constructed of 50 monomers, versus try number. Chaperones size are λ = 2 and exponential rates are α = 0, 5. The dash-dotted lines are for α = 5 while the solid lines stand for α = 0. The right figure is a zoom on the left one. Figure 3 : 3Probability distribution function of finding the near the wall monomer of the polymer bound is compared for chaperones of size λ = 2 and exponential distribution with α = 5 for three different E ef f = −2, 1, 2. Figure 4 : 4Equilibrium rates are plotted against EBEs for different binding sizes λ = 1, 2, 6 and different chaperones distributions of α = 0, 1, 5and10. The dashed line shows the maximum rate in the simulation. The points over the black dashed line are not real data and just estimated (based onfigure 2). Figure 5 : 5Mean waiting times of polymer translocation in different conditions are compared. In above figure, the chaperones have uniform distribution and different EBEs of E ef f = −4, 4 and rate frequencies of f = 10, 300, 900 are compared(note that the chaperones with E ef f = −4 and rate frequencies of f = 300, 900 are exactly the same.). In the figure 5b, we compare the uniform and exponential distribution of chaperones. ⇒ P bind = − ∂ ∂a ln(Z) ≃ − ∂ ∂a ln 2δ exp(−aP bind )(1 − (aδ) 2 ) ≃ P bind + 2aδ 2 T small − T = 1 2δ P bind +δ P bind −δ T (P ) exp(−aP )dP − T (P bind ) ⇒ T small − T ≃ −C 1 (N)2aδ 2 . (A.6) ).Effective binding energy -4 -3 -2 -1 0 1 2 3 4 Equilibrium rate 0 500 1000 1500 2000 λ=1, α=0 λ=2, α=0 λ=2, α=1 λ=2, α=5 λ=2,α=10 λ=6, α=0 λ=6, α=1 λ=6, α=5 λ=6,α=10 Investigating binding particles distribution effects on polymer translocation through nanopore. R H Abdolvahab, Physics Letters A. 38010231030Abdolvahab, R. H., 2016. Investigating binding particles distribution ef- fects on polymer translocation through nanopore. Physics Letters A 380, 10231030. 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[ "Page curve from dynamical branes in JT gravity", "Page curve from dynamical branes in JT gravity" ]
[ "Kazumi Okuyama [email protected] \nDepartment of Physics\nShinshu University\n3-1-1 Asahi390-8621MatsumotoJapan\n", "Kazuhiro Sakai [email protected] \nInstitute of Physics\nMeiji Gakuin University\n1518 Kamikurata-cho, Totsuka-ku244-8539YokohamaJapan\n" ]
[ "Department of Physics\nShinshu University\n3-1-1 Asahi390-8621MatsumotoJapan", "Institute of Physics\nMeiji Gakuin University\n1518 Kamikurata-cho, Totsuka-ku244-8539YokohamaJapan" ]
[]
We study the Page curve of an evaporating black hole using a toy model given by Jackiw-Teitelboim gravity with Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) antibranes. We treat the anti-FZZT branes as dynamical objects, taking their back-reaction into account. We construct the entanglement entropy from the dual matrix model and study its behavior as a function of the 't Hooft coupling t proportional to the number of branes, which plays the role of time. By numerical computation we observe that the entropy first increases and then decreases as t grows, reproducing the well-known behavior of the Page curve of an evaporating black hole. The system finally exhibits a phase transition, which may be viewed as the end of the evaporation. We study the critical behavior of the entropy near the phase transition. We also make a conjecture about the late-time monotonically decreasing behavior of the entropy. We prove it in a certain limit as well as give an intuitive explanation by means of the dual matrix model.
10.1007/jhep02(2022)087
[ "https://arxiv.org/pdf/2111.09551v3.pdf" ]
244,345,934
2111.09551
02befa1c0f710bcb906326c453c622f2dc28d7c5
Page curve from dynamical branes in JT gravity 14 Feb 2022 Kazumi Okuyama [email protected] Department of Physics Shinshu University 3-1-1 Asahi390-8621MatsumotoJapan Kazuhiro Sakai [email protected] Institute of Physics Meiji Gakuin University 1518 Kamikurata-cho, Totsuka-ku244-8539YokohamaJapan Page curve from dynamical branes in JT gravity 14 Feb 2022Prepared for submission to JHEP We study the Page curve of an evaporating black hole using a toy model given by Jackiw-Teitelboim gravity with Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) antibranes. We treat the anti-FZZT branes as dynamical objects, taking their back-reaction into account. We construct the entanglement entropy from the dual matrix model and study its behavior as a function of the 't Hooft coupling t proportional to the number of branes, which plays the role of time. By numerical computation we observe that the entropy first increases and then decreases as t grows, reproducing the well-known behavior of the Page curve of an evaporating black hole. The system finally exhibits a phase transition, which may be viewed as the end of the evaporation. We study the critical behavior of the entropy near the phase transition. We also make a conjecture about the late-time monotonically decreasing behavior of the entropy. We prove it in a certain limit as well as give an intuitive explanation by means of the dual matrix model. Introduction The black hole information paradox has been a long-standing puzzle in the study of quantum gravity [1]. In particular, the growing behavior of the entropy of thermal radiation based on Hawking's calculation [2] apparently contradicts with the unitarity of the quantum mechanics which requires that the black hole stays in a pure state. For an evaporating black hole, the Page curve [3], a plot of the entanglement entropy of the Hawking radiation as a function of time, should show a decreasing behavior toward the end of evaporation. Recent studies revealed that the gravitational path integral receives, even semi-classically, contributions from saddle-points other than the classical black hole solution, namely the replica wormholes [4,5]. This is a key to understand how the Page curve is obtained in an expected form, which partly resolves the information paradox. The idea was refined in the form of the island formula [6], which was first derived by means of holography [4][5][6][7][8][9][10][11][12] and then consolidated by directly evaluating the gravitational path integral in quantum gravity in two dimensions [13,14]. See [15] for a recent review and references therein. In [13] the Page curve was studied by using Jackiw-Teitelboim (JT) gravity [16,17] with the end-of-the-world (EOW) branes. 1 Roughly speaking, the system is viewed as a generalization of the original Page's model [19] (see [20] for recent exact results). Page's calculation starts with a random pure state in the bipartite Hilbert space consisting of two subspaces that represent the interior and exterior of a black hole. Taking ensemble average of the state in either of the subspaces, one obtains the reduced density matrix, from which the entanglement entropy is calculated. In Page's model the ensemble is Gaussian in both subspaces. In the case of JT gravity with EOW branes, the ensemble in the interior is Gaussian whereas the average in the exterior is described by the double-scaled matrix integral of JT gravity [21]. The size of the interior subspace, which is identified with the number of branes, plays the role of time. In this paper we propose another simple toy model to understand the Page curve: JT gravity with Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) antibranes [22,23]. Our model is a simplified variant of the model of [13], with the EOW branes replaced by anti-FZZT branes. In our previous paper [24] we showed that the matrix model description of the EOW brane in [25] corresponds to that of a collection of infinitely many anti-FZZT branes with a particular set of parameters. It is therefore simpler to consider JT gravity with a single kind of anti-FZZT branes. Despite this simplification, our model captures several features of black hole entropy. Most notably, the entanglement entropy exhibits the late-time decreasing behavior which is characteristic of an evaporating black hole. 2 To reproduce this decreasing behavior, it is crucial to treat branes as dynamical objects. In the previous studies, branes are treated as either dynamical [25] or non-dynamical [13]. We will see from numerical computation that the late-time decreasing behavior of the entropy is reproduced only when we treat anti-FZZT branes as dynamical objects. In fact, we consider the 't Hooft limit, in which the back-reaction of branes is not negligible and one has to treat branes as dynamical objects. We will also study how this decreasing behavior arises from the viewpoint of the matrix model and make a conjecture about monotonicity, which we will prove in a certain limit. Our model exhibits a phase transition as "time" grows. The transition may be viewed as the end of the evaporation of black hole. We will study the critical behavior of the entropy near the transition point. One can consider JT gravity with FZZT branes and study the Page curve in the same manner. In this case, however, no phase transition occurs and the entropy continue increasing. All these results derived from the matrix model are in perfect accordance with the semi-classical analysis on the gravity side: Dilaton gravities with nontrivial dilaton potential were studied as deformations of JT gravity [28,29] and black hole solutions in these gravities were also studied [30]. JT gravity with (anti-)FZZT branes can be viewed as this type of dilaton gravity [24]. We will study its black hole solutions and see the continuous growth of the entropy in the FZZT setup and the phase transition in the anti-FZZT setup. Thus, in this paper we concentrate on the case of anti-FZZT branes. This paper is organized as follows. In section 2, we describe our model and explain the general method of computing the entropy of the Hawking radiation. In section 3, we explain how the phase transition occurs and study the critical behavior of the entropy. In section 4, we numerically study the Page curve, i.e. the evolution of the entropy as a function of the 't Hooft coupling. We also make a conjecture about the late-time monotonically decreasing behavior of the entropy. In section 5, we prove the conjecture in a certain limit. We also give an intuitive explanation of the reason why the entropy decreases. In section 6, we study black hole solutions from the viewpoint of dilaton gravity. Finally we conclude in section 7. In appendix A, we give a derivation of the Schwinger-Dyson equation (2.22) based on the saddle point method. Entropy of radiation from dynamical anti-FZZT branes In this section we will describe our model and explain the general method of computing the entropy of the Hawking radiation. In many parts of our formulation we follow the method of [13] with EOW branes being replaced by anti-FZZT branes. In our study, however, branes are treated as dynamical objects. This is along the lines of [25] and an important difference from [13]. Matrix integral and black hole microstates Let us consider general 2d topological gravity with K dynamical anti-FZZT branes. 3 It is described by the double scaling limit of the matrix integral Z = dHe − Tr V (H) det(ξ + H) −K = dHdQdQ † e − Tr V (H)−Tr Q † (ξ+H)Q . (2.1) Here H and Q are N × N hermitian and N × K complex matrices respectively. ξ is a parameter characterizing the anti-FZZT brane, which is now taken to be common to all K branes. The potential could have been normalized as 1 g s V (H), (2.2) where g s is the genus counting parameter, so that the genus expansion is manifest. In this paper we include g −1 s in V for simplicity. In the double scaling limit, N is sent to infinity and the potential turns into the effective potential. In this paper we will further take the 't Hooft limit K → ∞, g s → 0 with t ≡ g s K fixed (2.3) and evaluate quantities in the planar approximation. That is, we will ignore all higher-order corrections of expansions in g s and K −1 . The matrices H, Q are often denoted by their components H ab , Q ai , where a, b = 1, . . . , N are "color" indices and i, j = 1, . . . , K are "flavor" indices. The color degrees of freedom are used for describing bulk gravity while the flavor degrees of freedom are thought of as describing the interior partners of the early Hawking radiation. One can regard the matrix element H ab as H ab = a|H|b , (2.4) where H is a Hamiltonian operator and {|a } N a=1 form an orthonormal basis of the corresponding N dimensional Hilbert space a|b = δ ab , 1 = a |a a|. (2.5) For ith random vector variable Q ai we consider the (canonical) thermal pure quantum state [31,32] |ψ i = a e − 1 2 βH |a Q ai = a,b |b (e − 1 2 βH ) ba Q ai . (2.6) Here β is the inverse temperature, which is identified with the (renormalized) length of an asymptotic boundary in 2d gravity. |ψ i play the role of the black hole microstates. Ensemble average To study the entropy, we will compute the average of overlaps such as ψ i |ψ j . We define the average of O by O = dHdQdQ † e − Tr V (H)−Tr Q † (ξ+H)Q O. (2.7) Here the angle brackets represent averaging over the color degrees of freedom while the overline represents averaging over the flavor degrees of freedom. It is convenient to change the variable as Q = (ξ + H) − 1 2 C, (2.8) so that the new random variable C obeys the Gaussian distribution O = dHdCdC † det(ξ + H) −K e − Tr V (H)−Tr C † C O. (2.9) Thus, in terms of C the flavor average becomes nothing but the Gaussian average. Note that the determinant factor is recovered from the integration measure. On the other hand, the thermal pure quantum state (2.6) becomes (see also appendix D of [13]) |ψ i = a,b |b e − 1 2 βH (ξ + H) − 1 2 ba C ai . (2.10) For our discussion it is convenient to express (2.10) as |ψ i = a,b |b ( √ A) ba C ai (2.11) with A(H) = e −βH ξ + H . (2.12) We then consider the overlaps such as W ij ≡ ψ i |ψ j = a,b A ab C * ai C bj , W ij W ji = | ψ i |ψ j | 2 = a,b,a ,b A ab A b a C * ai C bj C a i C * b j . (2.13) Recall that the Gaussian average of C can be computed by the Wick contraction C * ai C bj = δ ab δ ij , C * ai C bj C a i C * b j = δ ij δ ab δ a b + δ aa δ bb . (2.14) By using these formulas, the average of the overlaps (2.13) are given by ψ i |ψ j = δ ij Tr A, | ψ i |ψ j | 2 = δ ij (Tr A) 2 + Tr A 2 . (2.15) As discussed in [13], one can visualize the above computation (2.15) by drawing diagrams. For instance, ψ i |ψ j in (2.13) can be represented by the following diagram ψ i |ψ j = (C † AC) ij = i j (C † ) ia A ab C bj . (2.16) The black thick curve labeled by the color matrix A ab corresponds to the asymptotic boundary of 2d spacetime while the dashed lines correspond to the flavor degrees of freedom C, C † . The gravitational path integral in the presence of branes is given by the matrix integral (2.9). One can easily see that the gravitational computations in eq. (2.10) and Figure 3 of [13] agree with the first and the second lines of (2.15), respectively. Reduced density matrix of radiation As explained in [13], the reduced density matrix of radiation is represented by the ensemble average of ij = W ij K i=1 W ii = W Tr W ij . (2.17) This is normalized as Tr = 1. Let us first consider the "purity" Tr 2 as an example. In the planar approximation, we can take the average of the numerator and the denominator of Tr 2 independently Tr 2 ≈ Tr W 2 Tr W 2 = K(Tr A) 2 + K 2 Tr A 2 (K Tr A) 2 = 1 K + Tr A 2 (Tr A) 2 . (2.18) Similarly, the average of Tr n is approximated as Tr n ≈ Tr W n (Tr W ) n = Tr W n (K Tr A) n . (2. 19) Tr W n in the numerator can be computed by using the Wick contraction of C and C † . In the planar approximation one obtains Tr W = K Tr A, Tr W 2 = K(Tr A) 2 + K 2 Tr A 2 , Tr W 3 = K(Tr A) 3 + 3K 2 Tr A 2 Tr A + K 3 Tr A 3 , Tr W 4 = K(Tr A) 4 + 6K 2 Tr A 2 (Tr A) 2 + 2K 3 (Tr A 2 ) 2 + 4K 3 Tr A 3 Tr A + K 4 Tr A 4 . (2.20) In fact, Tr W n can be computed efficiently by means of the generating function R(λ) = Tr 1 λ − = ∞ n=0 Tr n λ n+1 = K λ + ∞ n=1 Tr W n λ n+1 (K Tr A) n . (2.21) In the planar approximation R(λ) satisfies λR(λ) = K + ∞ n=1 R(λ) n Tr A n (K Tr A) n . (2.22) This equation was derived diagrammatically in [13]. We give an alternative derivation based on the saddle point method in appendix A. By plugging (2.21) into (2.22), Tr W n can be obtained recursively. In this way, in the planar approximation we obtain Tr n ≈ 1 K n−1 + n(n − 1) 2K n−2 Tr A 2 Tr A 2 + · · · + n K Tr A n−1 Tr A n−1 + Tr A n Tr A n . (2.23) Thus, to compute Tr n we need to evaluate Tr A n = dHe − Tr V (H) det(ξ + H) −K Tr A n . (2.24) We evaluate it in the double scaling limit. In the planar approximation we have only to consider the genus zero part. It can be expressed in terms of the leading-order density ρ 0 (E) of the eigenvalues of H. As we studied in [24], for Witten-Kontsevich topological gravity with general couplings {t k } (k ∈ Z ≥0 ), ρ 0 (E) is given by ρ 0 (E) = 1 √ 2πg s E E 0 dv √ E − v ∂f (−v) ∂(−v) (2.25) with f (u) := ∞ k=0 (δ k,1 − t k ) u k k! . (2.26) The threshold energy E 0 is determined by the condition (the genus-zero string equation) f (−E 0 ) = 0. (2.27) In this paper we consider JT gravity, which corresponds to a particular background t k = γ k with [33-35] 4 γ 0 = γ 1 = 0, γ k = (−1) k (k − 1)! (k ≥ 2). (2.28) As we studied in [24], the effect of anti-FZZT branes, i.e. the insertion of det(ξ + H) −K amounts to shifting the couplings t k of topological gravity as 5 t k = γ k + t(2k − 1)!!(2ξ) −k− 1 2 . (2.29) This is valid as long as Re ξ > 0. Here t is the 't Hooft coupling in (2.3). Thus, (2.24) is evaluated as Tr A n ≈ Tr A n g=0 = ∞ E 0 dEρ 0 (E)A(E) n , ≡ Z n ,(2.30) where ρ 0 (E) is now evaluated in the background (2.29). In this background, (2.26) becomes f (u = −v) ≡ f (v, t) = √ vI 1 (2 √ v) + t √ 2ξ + 2v , (2.31) where we have changed the variable as v = −u for convenience and I k (z) denotes the modified Bessel function of the first kind. (2.25) then becomes ρ 0 (E) = 1 √ 2πg s E E 0 dv I 0 (2 √ v) √ E − v − t E + ξ E − E 0 2(E 0 + ξ) . (2.32) Note that in [24] we calculated ρ 0 (E) for JT gravity in the presence of K FZZT branes. The above ρ 0 (E) for anti-FZZT branes is essentially identical to this except the sign of the 't Hooft coupling t. Note also that this expression of ρ 0 (E) is valid as long as t is not greater than the critical value t c . We will explain this in section 3.1. We emphasize that we have treated anti-FZZT branes as dynamical objects. More specifically, in (2.24) the color average is evaluated in the presence of the determinant factor det(ξ + H) −K and as a consequence the deformed eigenvalue density (2.32) is used in (2.30). This is the main difference from the approach of [13], which is based on the probe brane approximation at genus-zero Tr A n probe g=0 = dHe − Tr V (H) Tr A n g=0 = ∞ 0 dEρ JT 0 (E)A(E) n ,(2.33) where the original JT gravity density of state ρ JT 0 (E) is given by ρ JT 0 (E) = sinh(2 √ E) √ 2πg s . (2.34) However, in [13] the same 't Hooft limit as ours (2.3) is used. As we argued in [24], in this limit the back-reaction of (anti-)FZZT branes cannot be ignored and the couplings t k are shifted due to the insertion of branes. As a consequence, the eigenvalue density is deformed from ρ JT 0 (E) in (2.34) to ρ 0 (E) in (2.32). Thus we have to use the dynamical brane picture in this limit. Resolvent of reduced density matrix and entropy We saw in the last subsection that the ensemble averages of in the planar approximation are expressed in terms of Z n in (2.30). On the other hand, the general expression (2.23) of Tr n is rather complicated as a function of n and it is difficult to apply the replica trick directly to (2.23) to calculate the entropy. Instead, as detailed in [13], we can study the entropy using the resolvent R(λ) for in (2.21). By substituting (2.30), the Schwinger-Dyson equation (2.22) for R(λ) becomes λR(λ) = K + ∞ n=1 R(λ) n (KZ 1 ) n ∞ E 0 dEρ 0 (E)A(E) n = K + ∞ E 0 dEρ 0 (E) w(E)R(λ) K − w(E)R(λ) ,(2.35) where we have defined w(E) = A(E) Z 1 . (2.36) Following [13], we divide the integral in (2.35) into two pieces λR(λ) = K + E K E 0 dEρ 0 (E) w(E)R(λ) K − w(E)R(λ) + ∞ E K dEρ 0 (E) w(E)R(λ) K − w(E)R(λ) ≈ K + E K E 0 dEρ 0 (E) w(E)R(λ) K − w(E)R(λ) + λ 0 R(λ),(2.37) where λ 0 and E K are defined by λ 0 = 1 K ∞ E K dEρ 0 (E)w(E), K = E K E 0 dEρ 0 (E). (2.38) By rewriting (2.37) as R(λ) = K λ − λ 0 + 1 λ − λ 0 E K E 0 dEρ 0 (E) w(E)R(λ) K − w(E)R(λ) , (2.39) we can solve R(λ) by the iteration starting from R(λ) = K/(λ − λ 0 ). As discussed in [13], the second order iteration gives R(λ) ≈ K λ − λ 0 + 1 λ − λ 0 E K E 0 dEρ 0 (E) w(E)K(λ − λ 0 ) −1 K − w(E)K(λ − λ 0 ) −1 = K λ − λ 0 + 1 λ − λ 0 E K E 0 dEρ 0 (E) w(E) λ − λ 0 − w(E) . (2.40) Using (2.38), we find R(λ) = 1 λ − λ 0 E K E 0 dEρ 0 (E) + E K E 0 dEρ 0 (E) w(E) λ − λ 0 − w(E) = E K E 0 dEρ 0 (E) 1 λ − λ 0 − w(E) . (2.41) The eigenvalue density D(λ) of the density matrix ij is obtained from the discontinuity of R(λ) D(λ) = R(λ − i0) − R(λ + i0) 2πi = E K E 0 dEρ 0 (E)δ λ − λ 0 − w(E) . (2.42) Finally, the von Neumann entropy is given by S = − dλD(λ)λ log λ = − E K E 0 dEρ 0 (E)(λ 0 + w(E)) log λ 0 + w(E) . (2.43) We will use this expression to study the Page curve numerically in section 4. In the rest of this section let us make several comments on the above approximation. We can check that D(λ) in (2.42) is normalized correctly Tr 0 = dλD(λ) · 1 = E K E 0 dEρ 0 (E) = K, (2.44) where we have used (2.38). We also find Tr = dλD(λ) · λ = E K E 0 dEρ 0 (E)(λ 0 + w(E)) = Kλ 0 + E K E 0 dEρ 0 (E)w(E) = ∞ E K dEρ 0 (E)w(E) + E K E 0 dEρ 0 (E)w(E) = ∞ E 0 dEρ 0 (E)w(E) = 1 Z 1 ∞ E 0 dEρ 0 (E)A(E) = 1 Z 1 Z 1 = 1, (2.45) where we have used (2.38). λ 0 in (2.38) can be written as λ 0 = 1 K ∞ E 0 dEρ 0 (E)w(E) − 1 K E K E 0 dEρ 0 (E)w(E) = 1 K − w K ,(2.46) where we have used (2.45) and defined w K ≡ 1 K E K E 0 dEρ 0 (E)w(E) = E K E 0 dEρ 0 (E)w(E) E K E 0 dEρ 0 (E) . (2.47) That is, w K is the average of w(E) in the "post Page" subspace E < E K . Thus the resolvent is written as R(λ) = Tr 1 λ − = E K E 0 dEρ 0 (E) 1 λ − λ(E) , (2.48) where λ(E) = 1 K + w(E) − w K . (2.49) λ(E) behaves as λ(E) ≈    1 K (K g −1 s ), w(E) − w K (K g −1 s ). (2.50) This corresponds to Figure 6 in [13]. Note that the density matrix ij is originally a matrix in the flavor space, but after taking the average the spectrum λ(E) of ij is effectively written in terms of energy eigenvalues in the "color" space. We have to project quantities onto the "post Page" subspace E < E K to ensure that the number of total state is K = E<E K dEρ 0 (E). Phase transition An interesting feature of the anti-FZZT brane background in JT gravity is that the system exhibits a phase transition as the 't Hooft coupling t varies. In this section we discuss this phase transition and study the critical behavior of the entropy. Threshold energy Let us first clarify the definition of the threshold energy E 0 . In the last section we saw that E 0 is determined by the threshold energy condition (2.27). For JT gravity with anti-FZZT branes, f is given by (2.31) and the condition (2.27) is written explicitly as Figure 1. We see that √ EI 1 (2 √ E) oscillates for E < 0 and increases monotonically for E > 0 starting from the origin, while −t/ 2(E + ξ) is always negative. Therefore E 0 has to be negative if it exists. However, the number of real solutions of (3.1) varies depending on the values of t and ξ. In particular, (3.1) has no real solution when t is very large, whereas it has multiple real solutions when ξ is large and t is small. 6 On the other hand, one can easily see that (3.1) always has at least one real solution for sufficiently small t. We define E 0 as the largest real solution of (3.1) (i.e. the solution with the smallest absolute value), so that it is continuously deformed from E 0 = 0 for the original JT gravity case t = 0. E 0 I 1 (2 E 0 ) = − t 2(E 0 + ξ) . (3.1) As we see from Figure 1, |E 0 | is small for small t. If we increase t, |E 0 | also increases. Then there exists a critical point t = t c beyond which E 0 no longer continues as a real solution. Thus, we expect a phase transition. This transition is qualitatively very similar to the one discussed in [25] in the case of EOW branes. If one continuously increases t beyond the critical point, E 0 and the second largest root turn into a pair of complex roots. It is therefore very likely that the saddle point of the matrix integral is described by an eigenvalue density with "Y" shaped support, similar to the one studied in [25]. It would be interesting to study the model in this "Y" shaped phase further. In this paper we view this phase transition as the end of the black hole evaporation and focus on the physics before the phase transition. Behavior of threshold energy near t = t c At the critical value t = t c , the equation (3.1) has a double root E c 0 . Thus t c is determined by the condition f (E c 0 , t c ) = 0, ∂ v f (E c 0 , t c ) = 0 (3.2) with f (v, t) given in (2.31). Expanding the equation f (E 0 , t) = 0 around (v, t) = (E c 0 , t c ), we find 0 = f (E c 0 , t c ) + ∂ v f (E c 0 , t c )(E 0 − E c 0 ) + 1 2 ∂ 2 v f (E c 0 , t c )(E 0 − E c 0 ) 2 + ∂ t f (E c 0 , t c )(t − t c ) + · · · . (3. 3) The first two terms vanish due to (3.2). Thus, near t = t c we find E 0 − E c 0 ≈ C √ t c − t (t < t c ), (3.4) where C is given by C = 2∂ t f (E c 0 , t c ) ∂ 2 v f (E c 0 , t c ) . (3.5) Effective zero-temperature entropy and von Neumann entropy near t = t c In [25], the effective zero-temperature entropy S eff was introduced. It is defined by the behavior of ρ 0 (E) near E = E 0 : ρ 0 (E) ∼ e S eff E − E 0 . (3.6) From (2.25) we find e S eff = √ 2 πg s ∂ v f (E 0 , t) = √ 2 πg s I 0 (2 E 0 ) − t (2ξ + 2E 0 ) 3 2 . (3.7) In Figure 2, we show the plot of S eff . We see that S eff is a monotonically decreasing function of t. Near t = t c , using (3.2) and (3.4) we find e S eff ∼ ∂ v f (E 0 , t) = ∂ v f (E c 0 , t c ) + ∂ 2 v f (E c 0 , t c )(E 0 − E c 0 ) + ∂ t ∂ v f (E c 0 , t c )(t − t c ) + · · · ∼ √ t c − t. (3.8) Using the above results, we can evaluate the critical behavior of the von Neumann entropy (2.43). It turns out that the critical behavior of the von Neumann entropy S(t) in (2.43) is determined by that of the eigenvalue density near E = E 0 One can show that the contribution of the E-integral (2.43) away from the edge E = E 0 is finite at t = t c . Subtracting this finite contribution and using (3.9) near E = E 0 in (2.43), we find ρ 0 (E) ∼ e S eff E − E 0 ∼ √ t c − t E − E 0 .(3.S(t) − S(t c ) ∼ √ t c − t. (3.10) In the next section we will confirm this behavior numerically. Numerical study of Page curve In section 2 we saw how to calculate the entropy. In this section we will numerically study the Page curve, i.e. the time evolution of the entropy. In Page's original calculation log K is regarded as "time" [19]. Since we take the 't Hooft limit (2.3), we will regard log t as "time." We will plot the entropy as a function of t rather than log t, which is convenient for seeing the critical behavior discussed in section 3. Von Neumann entropy We consider the von Neumann entropy (2.43) in JT gravity in the presence of K anti-FZZT branes. As discussed in section 2 we compute the entropy for dynamical branes, but it is interesting to compute it in the probe brane approximation as well for the sake of comparison. In the probe brane approximation, we have E 0 = 0 and the eigenvalue density is given by ρ JT 0 (E) in (2.34). We show the plot of the von Neumann entropy S in Figure 3. We see that the entropy for the dynamical brane (solid blue curve) starts to decrease relative to the probe brane case (dashed orange curve). This is very similar to the Page curve of an evaporating black hole. As we saw in the last section, we observe that the entropy exhibits the critical behavior (3.10). Rényi entropy It is also interesting to consider the Rényi entropy. The nth Rényi entropy S n is defined by Tr n = e −(n−1)Sn . (4.1) For large K, Tr n is dominated by the last term of (2.23). Thus as t grows, S n approaches S n := − 1 n − 1 log Z n (Z 1 ) n . (4.2) Let us first consider the second Rényi entropy S 2 = − log g s t + Z 2 Z 2 1 . (4.3) In Figure 4, we show the plot of S 2 as a function of t. We can see that S 2 first increases and then decreases. Near t = t c , S 2 exhibits a critical behavior S 2 (t) − S 2 (t c ) ∼ √ t c − t. (4.4) This behavior can be derived in the same way as in the case of the von Neumann entropy. We can see that the plot of S 2 has a similar behavior with the Page curve of Hawking radiation from an evaporating black hole (see e.g. Figure 7 in [15]). We regard S 2 as an analogue of the thermodynamic entropy S BH of an evaporating black hole. From Figure 4, we see that S 2 is a monotonically decreasing function of t (represented by the dashed orange curve). Let us next consider the third Rényi entropy As t grows, this approaches e −2S 3 = 1 K 2 + 3Z 2 KZ 2 1 + Z 3 Z 3 1 .e −2 S 3 = Z 3 Z 3 1 . (4.6) In Figure 5 we show the plot of S 3 for anti-FZZT branes. We see that this is qualitatively very similar to the S 2 case. In particular, S 3 is bounded from above by S 3 , which monoton-ically decreases. From Figure 4 and 5, it is natural to regard S n in (4.2) as an analogue of the thermodynamic entropy S BH of black hole, since S BH decreases monotonically during the evaporation process as well S n ↔ S BH . (4.7) Based on the above numerical results, we conjecture that S n defined in (4.2) is a monotonically decreasing function of t. More specifically, we conjecture that ∂ t S n < 0 for n > 1, 0 < t < t c . (4.8) We will study this monotonic behavior in the next section. Monotonicity of S n In this section we study the monotonically decreasing behavior of S n . We will prove our conjecture (4.8) in the large ξ limit. We will also discuss how to understand intuitively this monotonically decreasing behavior. Leading-order eigenvalue density and its derivative In this subsection let us derive some useful formulas about the leading-order eigenvalue density ρ 0 (E) in (2.25) for Witten-Kontsevich gravity with general couplings {t k }, which we will use in the next subsection. To study general Witten-Kontsevich gravity it is convenient to introduce the Itzykson-Zuber variables [40] I n (u) ≡ I n (u, {t k }) = ∞ m=0 t n+m u m m! (n ≥ 0). (5.1) In terms of I n , f (u) in (2.26) is written as f (u) = u − I 0 (u) (5.2) and the leading-order density ρ 0 (E) in (2.25) becomes ρ 0 (E) = 1 √ 2πg s E E 0 dv 1 − I 1 (−v) √ E − v . (5.3) Let us now consider the anti-FZZT brane background (2.29). We are interested in how the entropy evolves as the 't Hooft coupling t in (2.3) grows. Note that t is implicitly related to E 0 by the string equation (2.27), from which one finds 0 = ∂ t f (−E 0 ) = (∂ t E 0 )(I 1 (−E 0 ) − 1) − ∂ t I 0 (u) u=−E 0 = (∂ t E 0 )(I 1 (−E 0 ) − 1) − 1 √ 2ξ + 2E 0 . (5.4) By using this relation, the t-derivative of ρ 0 (E) is calculated as ∂ t ρ 0 (E) = (∂ t E 0 )∂ E 0 ρ 0 (E) − 1 √ 2πg s E E 0 dv ∂ t I 1 (−v) √ E − v = ∂ t E 0 √ 2πg s I 1 (−E 0 ) − 1 √ E − E 0 − 1 √ 2πg s E E 0 dv(E − v) − 1 2 (2ξ + 2v) − 3 2 = 1 2πg s (E − E 0 )(ξ + E 0 ) − 1 2πg s (E + ξ) E − E 0 ξ + E 0 = 1 2πg s (E + ξ) ξ + E 0 E − E 0 . (5.5) Note that the background (2.29) is written for JT gravity, but we have never used the specific values of γ k in the above derivation. Therefore, (5.5) is in fact valid for the anti-FZZT brane background of other gravities as well. Proof in the large ξ limit In the last section we conjectured that S n defined in (4.2) is a monotonically decreasing function of t. In this subsection let us prove this conjecture (4.8) at large ξ. From the expression (4.2), we see that (4.8) is equivalent to ∂ t Z n nZ n > ∂ t Z 1 Z 1 . (5.6) By using the property ρ 0 (E 0 ) = 0 and the expression (5.5), the t-derivative of Z n in (2.30) is calculated as ∂ t Z n = ∞ E 0 dE∂ t ρ 0 (E)A(E) n = ∞ E 0 dE 1 2πg s ξ + E 0 E − E 0 e −nβE (E + ξ) n+1 = √ ξ + E 0 2πg s e −nβE 0 ∞ 0 d E E − 1 2 ( E + E 0 + ξ) −n−1 e −nβ E , (5.7) where we have set E = E − E 0 . For large ξ, (5.7) is evaluated as ∂ t Z n = e −nβE 0 2 √ nπβg s ξ −n− 1 2 + O ξ −n− 3 2 . (5.8) On the other hand, by plugging (2.32) into (2.30), Z n is written as Z n = ∞ E 0 ρ 0 (E)A(E) n = 1 √ 2πg s ∞ E 0 dvI 0 (2 √ v) ∞ v dE(E − v) − 1 2 (E + ξ) −n e −nβE − t 2πg s (E 0 + ξ) 1 2 ∞ E 0 dE(E − E 0 ) 1 2 (E + ξ) −n−1 e −nβE . (5.9) In the same way as in (5.7)-(5.8), the above integrals at large ξ are evaluated as Z n = 1 √ 2nπβg s ∞ E 0 dvI 0 (2 √ v)(v + ξ) −n e −nβv + O ξ −n−1 − te −nβE 0 4 √ πg s (nβ) 3 2 ξ −n− 3 2 + O ξ −n− 5 2 = 1 √ 2nπβg s ξ n ∞ E 0 dvI 0 (2 √ v)e −nβv + O ξ −n−1 . (5.10) Here we see that in the leading-order of the large-ξ approximation the first integral in (5.9) is dominant and the second integral does not contribute. Indeed, from (5.10) one can reproduce (5.8) by using the relation ∂ t E 0 = − √ 2ξI 0 (2 √ E 0 ) −1 + O(ξ −3/2 ), which follows from (3.1) or (5.4). Thus we obtain ∂ t Z n nZ n = e −nβE 0 n √ 2ξ ∞ E 0 dvI 0 (2 √ v)e −nβv + O ξ − 3 2 . (5.11) To prove (5.6) at large ξ, it is sufficient to show that (5.11) monotonically increases as n grows: ∂ n ∂ t Z n nZ n > 0. (5.12) Since ∂ t Z n /nZ n > 0, 7 this is equivalent to showing that ∂ n log ∂ t Z n nZ n > 0. (5.13) The l.h.s. of (5.13) is rewritten as ∂ n log ∂ t Z n nZ n = −βE 0 − 1 n + ∞ E 0 dvI 0 (2 √ v)βve −nβv ∞ E 0 dvI 0 (2 √ v)e −nβv = ∞ E 0 dvI 0 (2 √ v) [nβ(v − E 0 ) − 1] e −nβv n ∞ E 0 dvI 0 (2 √ v)e −nβv = ∞ 0 dṽI 0 (2 √ṽ + E 0 ) (nβṽ − 1) e −nβṽ ne nβE 0 ∞ E 0 dvI 0 (2 √ v)e −nβv ,(5.14) where we have setṽ = v − E 0 . The denominator of the last expression in (5.14) is positive (see footnote 7). By renamingṽ as v, the numerator is evaluated as Since the integrand is positive for any E 0 satisfying E c 0 < E 0 < 0, 8 (5.15) is positive. Thus we have proved (5.13). Hence (4.8) has been proved at large ξ. ∞ 0 dvI 0 (2 v + E 0 )nβve −nβv − ∞ 0 dvI 0 (2 v + E 0 )e −nβv = −I 0 (2 v + E 0 )ve −nβv ∞ 0 + ∞ 0 dv I 0 (2 v + E 0 ) + I 1 (2 √ v + E 0 ) √ v + E 0 v e −nβv − ∞ 0 dvI 0 (2 v + E 0 )e −nβv = ∞ 0 dv I 1 (2 √ v + E 0 ) √ v + E 0 ve −nβv . Intuitive understanding of the decreasing behavior of S n Beyond the large ξ approximation, it does not seem easy to find a simple analytic proof of the monotonically decreasing behavior of S n . Alternatively, in this subsection we will explain how to understand intuitively the monotonically decreasing behavior of S n . In contrast to the proof in the last subsection, the idea we will describe does not depend on the details of the JT gravity background and thus it can be generalized to the other gravity cases as well. The replica index n is sometimes identified as an analogue of the inverse temperature (see e.g. [41]). Here we will pursue this analogy. To do this, let us consider the change of variable from E to E given by 9 e −E = A(E) = e −βE ξ + E . (5.18) Then we find Z n = ∞ E 0 dEρ 0 (E)A(E) n = ∞ E 0 dED(E)e −nE ,(5.19) where E 0 = − log A(E 0 ) and D(E) = ∂E ∂E −1 ρ 0 (E) = ξ + E β(ξ + E) + 1 ρ 0 (E). (5.20) Z n in (5.19) takes the form of the canonical partition function with inverse temperature n and density of states D(E). In this picture the "thermodynamic entropy" is expressed as [41,42] S therm = (1 − n∂ n ) log Z n . (5.21) On the other hand, as we saw in the last subsection ∂ t S n < 0 is equivalent to (5.12), which is written as −∂ n ∂ t Z n nZ n = 1 n 2 ∂ t S therm < 0. (5.22) Therefore, the monotonically decreasing behavior of S n is interpreted as that of the thermodynamic entropy S therm . Let us list some useful properties of S therm : 8 This is easily seen from the graph of 1. The threshold energy E 0 does not contribute to S therm : If we define Z n by √ EI1(2 √ E) = I1(2 √ E)/ √ E × E inZ n = e −nE 0 Z n , Z n = ∞ 0 dED(E + E 0 )e −nE ,(5.23) then we find S therm = (1 − n∂ n )(−nE 0 + log Z n ) = (1 − n∂ n ) log Z n . (5.24) 2. The overall scale of Z n does contribute to S therm : If we define Z n = e S(t) Z n ,(5.25) where S(t) is n-independent, then we find S therm = (1 − n∂ n )(S(t) + log Z n ) = S(t) + (1 − n∂ n ) log Z n . (5.26) If we further assume that Z n is t-independent, then the monotonically decreasing behavior of S(t) implies that of S therm . 3. S therm is written as S therm = log Z n + n E , (5.27) where E is given by E = −∂ n log Z n = 1 Z n ∞ 0 dED(E + E 0 )e −nE E. (5.28) Let us now focus on the case of JT gravity with anti-FZZT branes. In Figure 6 we plot ρ 0 (E) in (2.32) for several different values of t. Due to property 1 of S therm , it is convenient to plot ρ 0 (E) against E − E 0 (see Figure 6b) to consider the behavior of the entropy. Then we observe that the overall scale of ρ 0 clearly decreases as t grows. 10 By crude approximation, the overall scale of ρ 0 gives that of Z n and thus this implies the decreasing behavior of S therm , as explained in property 2. More precisely, as described in property 3, S therm is related to Z n by (5.27). We found numerically that each individual terms log Z n and n E are not necessarily monotonically decreasing functions of t for generic values of ξ, but the sum of them is always monotonically decreasing. To summarize, we have seen that the monotonically decreasing behavior of S n is equivalent to that of the "thermodynamic entropy" S therm if we regard the replica index n as the inverse temperature. Its decreasing behavior is intuitively understood from that of the overall scale of ρ 0 (E). It is well known that the entropy of the Hawking radiation is bounded from above by the thermodynamic entropy S BH of the black hole, which is given by the area of horizon in the semi-classical approximation. For an evaporating black hole, the area of horizon decreases as time passes and this explains the decreasing behavior of the Page curve. As we saw in (4.7), S n (not S therm ) corresponds to S BH for n > 1. In general, S n and S therm are different quantities. However, one can easily see that S n becomes equal to S therm in the limit n → 1 lim n→1 S n = lim n→1 S therm . (5.29) Thus the thermodynamic entropy S BH of black hole literally corresponds to the "thermodynamic entropy" S therm in the limit n → 1. Dilaton gravity Recently, dilaton gravities with nontrivial dilaton potential were studied as deformations of JT gravity [28,29]. Black hole solutions in these gravities were also discussed in [30]. JT gravity with (anti-)FZZT branes can be viewed as this type of dilaton gravity [24]. In this section we will study black hole solutions from the viewpoint of dilaton gravity. 11 The action of dilaton gravity is written as [30] I = − 1 2 d 2 x √ g(φR + W (φ)). (6.1) We derived that in the case of JT gravity with K (anti-)FZZT branes the dilaton potential is given by [24] W (φ) =        2φ + t √ 2ξ ξ + π 2 φ 2 e −2πφ (anti-FZZT), 2φ − t √ 2ξ ξ + π 2 φ 2 e −2πφ (FZZT). (6.2) The general Euclidean black hole solution is given by ds 2 = A(r)dt 2 + dr 2 A(r) , φ(r) = r, A(r) = r r h dφW (φ),(6.3) where r = r h is the horizon at which A(r) vanishes. This is a one-parameter family of solutions parametrized by r h = φ h . The value of φ h is not fixed by the equation of motion. 4πT The entropy of this solution is given by W (φ) t < t c t = t c t > t c φ hS = 2πφ h + S 0 . (6.4) The physical condition is A(r) > 0 for r > r h . For a fixed temperature T , φ h is determined by the condition W (φ h ) 4π = T. (6.5) In Figure 7 we show the plot of W (φ). For the FZZT branes in Figure 7a, (6.5) has a unique solution φ h for a given value of T . As t increases, φ h also increases. Thus the entropy increases as a function of t. On the other hand, from Figure 7b for the anti-FZZT branes, one can see that there are two solutions of (6.5) if t is not too large. As discussed in [30], the stable solution with minimal free energy corresponds to the largest root φ h of (6.5) (see Figure 8). The largest root φ h of (6.5) decreases as t increases. This explains the decreasing behavior of the entropy (6.4). Beyond some critical value t = t c , there is no solution of (6.5) for a given temperature. This might be interpreted that for t > t c there is no stable black hole solutions; at t = t c the stable black hole disappears. This suggests that, to model the black hole evaporation process, the anti-FZZT brane setup is more suitable than the FZZT brane setup. Conclusion and outlook In this paper we studied the entanglement entropy in the matrix model of JT gravity with anti-FZZT branes, which serves as a toy model of an evaporating black hole. The entanglement entropy is defined between the color and flavor sectors, which correspond respectively to bulk gravity and to the interior partners of the early Hawking radiation. We computed the entropy in the planar approximation as well as in the 't Hooft limit. The 't Hooft coupling t, which is proportional to the number of branes, plays the role of time. We computed numerically the von Neumann and Rényi entropies as functions of t. In both cases, the entropy first increases and then decreases, which is peculiar to the Page curve of an evaporating black hole. We stress that we treated the anti-FZZT branes as dynamical objects and this was crucial to reproduce the late-time decreasing behavior of the entropy, because otherwise the entropy approaches a constant value at late time in the probe brane approximation [13], as we saw in Figure 3. We saw that the system exhibits a phase transition at t = t c . This may be viewed as the end of the evaporation of the black hole. We studied the critical behavior of the entropy and derived that it scales as in (3.8). As t grows toward t = t c , the Rényi entropy becomes dominated by S n in (4.2). We conjectured that S n monotonically decreases and proved this conjecture in the large ξ limit. We also gave an intuitive explanation of this decreasing behavior. We studied black hole solutions of dilaton gravity that describes JT gravity with (anti-)FZZT branes and saw the continuous growth of the entropy in the FZZT setup as well as a phase transition in the anti-FZZT setup. This suggests that the anti-FZZT brane setup is more suitable to model an evaporating black hole. There are many interesting open questions. We have seen that our model of dynamical branes in JT gravity serves as a good toy model for an evaporating black hole. We hope that the behavior of our model beyond the phase transition t > t c would shed light on the deep question of the unitarity in black hole evaporation, e.g. the final state proposal in [44]. It would be interesting to study our matrix model beyond the phase transition t > t c along the lines of [25]. Our analysis of the Page curve was limited to the planar approximation. It would be interesting to compute the higher genus corrections to the Page curve. More ambitiously, it would be very interesting if we can compute the Page curve of our model non-perturbatively in g s . We leave this as an interesting future problem. We can also repeat the analysis of the Petz map in [13] using our setup of dynamical branes. It would be interesting to study how the entanglement wedge reconstruction is modified from the result of [13] if we take account of the back-reaction of branes. In our calculation of the Page curve, the decreasing behavior of entropy comes from the last term of (2.23), which is interpreted as a contribution of replica wormholes [13,14]. The appearance of the replica wormhole is closely related to the ensemble average on the boundary side of the AdS/CFT correspondence. The role of the ensemble average in the gravitational path integral is still not well-understood and there are many conceptual issues related to this problem, such as the factorization puzzle (see e.g. [45][46][47][48][49][50] and references therein). It is believed that the Rényi entropy is a self-averaging quantity [13]. Nonetheless, it would be interesting to see how the Page curve of our model would look like if we pick a certain member of the ensemble and do not take an average over the random matrix (see e.g. [51] for a study in this direction). Acknowledgments This work was supported in part by JSPS KAKENHI Grant Nos. 19K03845, 19K03856, 21H05187 and JSPS Japan-Russia Research Cooperative Program. A preliminary result of this work was presented by one of the authors (KO) in the KMI colloquium at Nagoya University on October 13, 2021. A Schwinger-Dyson equation from saddle point approximation In this appendix we will derive the Schwinger-Dyson equation (2.22) based on the saddle point method. Let us consider the integral K i=1 dφ † i dφ i e − ij φ † i (λδ ij − ij )φ j . (A.1) Then the two point function φ i φ † j is equal to the resolvent φ i φ † j = (λ − ) −1 ij . (A.2) We can rewrite the integral as The density matrix ij is given by dφ † dφe −φ † (λ− )φ = dφ † dφdG ij δ(G ij − φ i φ † j )e − Tr(λ− )G = dφ † dφdGdΣe Σ ji (G ij −φ i φ †ij = C † i AC j KZ 1 = C † i AC j , A = A KZ 1 . (A.4) After integrating out C † , C we have In the planar approximation, the G-and Σ-integrals can be evaluated by the saddle point approximation. The saddle point equations read ∂I ∂Σ ij = −G ij + (Σ −1 ) ij = 0, ∂I ∂G ij = −Σ ij + λδ ij − δ ij Tr A 1 − AG = 0. (A.7) Multiplying the second equation of (A.7) by G ij and summing over i, j, we find −K + λ Tr G − Tr G Tr A 1 − AG = 0. (A.8) This is rewritten as λ Tr G = K + Tr G Tr A 1 − AG = K + Tr G ∞ n=1 Tr A n Tr G n−1 . (A.9) In the planar approximation we have Tr G ≈ i φ i φ † i = i (λ − ) −1 ii = R(λ). (A.10) We also find Tr G n = φ i 1 φ † i 2 φ i 2 φ † i 3 · · · φ in φ † i 1 ≈ φ † i 1 φ i 1 · φ † i 2 φ i 2 · · · φ † in φ in = (Tr G) n = R(λ) n . Figure 1 : 1The threshold energy E 0 is determined by the equation (3.1). The red curve represents the graph of √ EI 1 (2 √ E) while the blue and green curves are the graphs of −t/ 2(E + ξ) with t = 3 and t = t c ≈ 4.46 respectively, where we set ξ = 27. As we see in this example, (3.1) could have multiple real solutions. The threshold energy E 0 is determined as the largest real solution, as indicated by a dot. The horizontal location of the green dot gives the critical value E c 0 .The threshold energy E 0 is determined as a real solution of this equation. Here, t > 0 by definition and we take ξ > 0 in order for the shift of the couplings (2.29) to be valid. We show the plots of both sides of the equation (3.1) in Figure 2 : 2Plot of the effective entropy S eff . We set ξ = 18, g s = 1/100 in this figure. Figure 3 : 3Plot of the von Neumann entropy S in (2.43) as a function of t = g s K. We set ξ = 18, β = 4, g s = 1/100 in this figure. The solid blue curve is the dynamical brane case while the dashed orange curve is the probe brane case. Figure 4 :Figure 5 : 45Plot of the second Rényi entropy S 2 in (4.3) as a function of t = g s K. We set ξ = 18, β = 4, g s = 1/100 in this figure. The solid blue curve is S 2 in (4.3). The dashed orange curve represents S 2 in (4.2) while the green dashed curve is S 2 without taking account of the back-reaction. Plot of the third Rényi entropy S 3 in (4.5) as a function of t = g s K. We set ξ = 18, β = 4, g s = 1/100 in this figure. The solid blue curve is S 3 in (4.5). The dashed orange curve represents S 3 in (4.6). 7 This follows from (5.11) with I0(2 √ v) = ∂v[ √ vI1(2 √ v)] > 0for v > E c 0 , as we can see fromFigure 1. Figure 1 .Figure 6 : 169 (5.18) is rewritten as β(ξ + E)e β(ξ+E) = βe E+βξ . (5.16) This is solved by the Lambert function W (z)e W (z) = z as β(ξ + E) = W βe E+βξ . Plot of ρ 0 (E) against (a) E and (b) E − E 0 for anti-FZZT branes. We set ξ = 18 in this plot. Figure 7 : 7Plot of W (φ) in (6.2) for (a) FZZT branes and (b) anti-FZZT branes. We set ξ = 18 in this plot. Figure 8 : 8Stability of black hole solutions in the anti-FZZT brane setup. The largest root φ h of (6.5) corresponds to the stable solution. The solution no longer exists for t > t c . = dGdΣe Tr ΣG−Tr log Σ−Tr(λ− )G . (A.3) dC † dCe − Tr C † C+Tr G = e −Tr log(1− AG) , (A.5) where Tr log(1 − AG) should be understood as the trace of both color and flavor indices. Then (A.3) becomes dGdΣe −I where the action I is given by I = − Tr ΣG + Tr log Σ + λ Tr G + Tr log(1 − AG). (A.6) A n R(λ) n . (A.12) A classification of branes in JT gravity is found in[18].2 The Page curve of an evaporating black hole in JT gravity was studied in different approaches[26,27]. In this paper we will eventually restrict ourselves to the JT gravity case, but most parts of our formalism can be applied to other 2d gravities as well. Another way to obtain JT gravity is to take the p → ∞ limit of the (2, p) minimal string[21,36].Entanglement entropy in this context was studied recently in[37].5 This shift of couplings was first recognized in the theory of soliton equations[38] and appears in various contexts of matrix models and related subjects. For more details, see[24] and references therein. See also[39] for a similar problem in JT gravity with conical defects. As we see in(5.20), D(E) is proportional to ρ0(E) up to a prefactor. Since this prefactor is independent of t, the graph of D(E) against E − E0 also decreases as t grows, in a similar way as ρ0(E) inFigure 6b. See also[43] for recent related studies. Thus we have re-derived the Schwinger-Dyson equation (2.22), which was originally derived by means of diagrams in[13]. The above saddle point method can be generalized to the Grassmann-odd integral (i.e. to the case of FZZT branes). Breakdown of Predictability in Gravitational Collapse. 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[ "Transition from Susceptible-Infected to Susceptible-Infected-Recovered Dynamics in a Susceptible-Cleric-Zombie-Recovered Active Matter Model", "Transition from Susceptible-Infected to Susceptible-Infected-Recovered Dynamics in a Susceptible-Cleric-Zombie-Recovered Active Matter Model" ]
[ "A Libál \nMathematics and Computer Science Department\nBabeş-Bolyai University\n400084Cluj-NapocaRomania\n", "1ÁP Forgács \nMathematics and Computer Science Department\nBabeş-Bolyai University\n400084Cluj-NapocaRomania\n", "Néda \nMathematics and Computer Science Department\nBabeş-Bolyai University\n400084Cluj-NapocaRomania\n", "C Reichhardt \nTheoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "N Hengartner \nTheoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "C J O Reichhardt \nTheoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n" ]
[ "Mathematics and Computer Science Department\nBabeş-Bolyai University\n400084Cluj-NapocaRomania", "Mathematics and Computer Science Department\nBabeş-Bolyai University\n400084Cluj-NapocaRomania", "Mathematics and Computer Science Department\nBabeş-Bolyai University\n400084Cluj-NapocaRomania", "Theoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Theoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Theoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA" ]
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The Susceptible-Infected (SI) and Susceptible-Infected-Recovered (SIR) models provide two distinct representations of epidemic evolution, distinguished by the lack of spontaneous recovery in the SI model. Here we introduce a new active matter epidemic model, the "Susceptible-Cleric-Zombie-Recovered" (SCZR) model, in which spontaneous recovery is absent but zombies can recover with probability γ via interaction with a cleric. Upon interacting with a zombie, both susceptibles and clerics can enter the zombie state with probability β and α, respectively. By changing the intial fraction of clerics or their healing ability rate γ, we can tune the SCZR model between SI dynamics, in which no susceptibles or clerics remain at long times, and SIR dynamics, in which no zombies remain at long times. The model is relevant to certain real world diseases such as HIV where spontaneous recovery is impossible but where medical interventions by a limited number of caregivers can reduce or eliminate the spread of infection. arXiv:2210.07310v1 [cond-mat.soft]
10.1103/physreve.107.024604
[ "https://export.arxiv.org/pdf/2210.07310v1.pdf" ]
252,907,723
2210.07310
b8cec67a8dce0b4a70f1141d6cb330c004a63481
Transition from Susceptible-Infected to Susceptible-Infected-Recovered Dynamics in a Susceptible-Cleric-Zombie-Recovered Active Matter Model A Libál Mathematics and Computer Science Department Babeş-Bolyai University 400084Cluj-NapocaRomania 1ÁP Forgács Mathematics and Computer Science Department Babeş-Bolyai University 400084Cluj-NapocaRomania Néda Mathematics and Computer Science Department Babeş-Bolyai University 400084Cluj-NapocaRomania C Reichhardt Theoretical Division and Center for Nonlinear Studies Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA N Hengartner Theoretical Division and Center for Nonlinear Studies Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA C J O Reichhardt Theoretical Division and Center for Nonlinear Studies Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Transition from Susceptible-Infected to Susceptible-Infected-Recovered Dynamics in a Susceptible-Cleric-Zombie-Recovered Active Matter Model (Dated: October 17, 2022) The Susceptible-Infected (SI) and Susceptible-Infected-Recovered (SIR) models provide two distinct representations of epidemic evolution, distinguished by the lack of spontaneous recovery in the SI model. Here we introduce a new active matter epidemic model, the "Susceptible-Cleric-Zombie-Recovered" (SCZR) model, in which spontaneous recovery is absent but zombies can recover with probability γ via interaction with a cleric. Upon interacting with a zombie, both susceptibles and clerics can enter the zombie state with probability β and α, respectively. By changing the intial fraction of clerics or their healing ability rate γ, we can tune the SCZR model between SI dynamics, in which no susceptibles or clerics remain at long times, and SIR dynamics, in which no zombies remain at long times. The model is relevant to certain real world diseases such as HIV where spontaneous recovery is impossible but where medical interventions by a limited number of caregivers can reduce or eliminate the spread of infection. arXiv:2210.07310v1 [cond-mat.soft] I. INTRODUCTION Understanding the propagation of infectious diseases is an intensely studied issue, and a variety of different epidemic models and methods to simulate the spread of disease have been developed [1][2][3][4]. Two of the most widely used disease propagation models are the Susceptible-Infected (SI) and Susceptible-Infected-Recovered (SIR) models [1][2][3][4]. In the SI model, illustrated in Fig. 1(a), there are only susceptibles (S) and infectives (I) present. There is no spontaneous recovery, and the model contains only a single probability β for an S to transform to an I. As shown in Fig. 1(b), the SIR model adds a spontaneous recovery process with rate µ for an I to become recovered (R). A key difference between the SI and SIR models is that in the SI model the amount of S present drops to zero at long times, but in the SIR model the amount of I present drops to zero. A wide range of diseases can be described using these two models. Diseases with lifelong transmittivity and no recovery are captured by the SI model, while situations where reinfection is impossible but spontaneous recovery occurs can be represented with the SIR model. Numerous variations of the SI and SIR models have been considered over the years [2][3][4][5], including epidemic spreading on networks [6], memory effects [7], adding vaccination [8], spatial heterogeneity [9,10], social distancing [11], diffusion [12], and models that include details on mobility patterns in attempts to more accurately portray real world epidemics [13,14]. Despite the large number of models that have been explored, we did not find any descriptions of a model in which a transition from SI to SIR behavior naturally emerges. Such transitions could arise for certain types of infectious disease where spontaneous recovery does not occur but where direct medical intervention can result in recovery or a reduced rate of infectiousness. Ref. [15], a zombie (Z, green) interacting with S recovers with probability κ and turns the S into Z with probability β. (d) In our SCZR model, we divide the susceptible population into S and clerics (C, brown). Z can only recover when in contact with C with probability γ, but interaction with Z causes S to turn into Z with probability β and C to turn into Z with probability α. ample, in the human immunodeficiency virus (HIV), an untreated patient remains contagious, but when appropriate medical interventions are applied, the patient becomes effectively cured and has a rate of infectiousness that drops dramatically or even reaches zero. In such cases, if there is an insufficient supply of resources or treating agents (doctors), the course of the epidemic will follow the SI model, but if there are ample resources or treating agents, the epidemic will instead fall in the SIR regime. Standard SI and SIR type models assume homogeneous mixing of infectious and susceptible individuals, either across the entire population or within strata. For many diseases, that assumption is known to fail and in Refs. [16,17], the impact of the failure of the homogeneity assumption is studied. In our previous work [18], we showed that a run-and-tumble active matter model combined with SIR dynamics produces different regimes of behavior when quenched disorder is introduced, due to the lack of homogeneous mixing in the system. For low infection rates, the quenched disorder strongly affects the duration of the epidemic as well as the final epidemic size or fraction of S that survive to the end of the epidemic. When the infection rate is high, the quenched disorder has little impact and the epidemic propagates as waves through the system. The term "active matter" encompasses self driven systems such as an assembly of self-motile particles that undergo contact interactions with each other [19,20]. In our previous work [18], we considered run-and-tumble particles moving in two dimensions and subjected to rules of how an infection spreads when a contact interaction occurs between an S and an I particle. Active matter systems are attractive for epidemic modeling since they allow real world effects such as spatial heterogeneity to be incorporated easily because density heterogeneities arise naturally from the interactions among the particles, and there have now been several studies in which active matter is used to study epidemics [21][22][23]. There have also been several experimental realizations of active matter systems that can mimic social dynamics through the activity and tracking of individual active particles, so the type of active matter epidemic systems we consider here should be feasible to create experimentally [24,25]. Here we introduce a new model for epidemic spreading featuring multiple susceptible species and no spontaneous recovery, and show that in this model, an easily tunable transition between SI and SIR behavior occurs. We specifically consider a modification of the Susceptible-Zombie-Removed (SZR) model proposed by Alemi et al. [15]. Figure 1(c) shows the dynamics of the SZR model. Unlike the SIR model, the SZR model has no spontaneous recovery. Instead, when an S and a zombie (Z) interact, the Z transitions to recovered (R) with probability κ, while the S transitions to Z with probability β. In our modification of the model, there is again no spontaneous recovery, but we break the susceptible population into two portions: susceptibles (S) and clerics (C). As illustrated in Fig. 1(d), when an S interacts with a Z, the S becomes a Z with probability β, as in the SZR model; however, the S cannot cause the Z to recover. Instead, only an interaction between a C and a Z can cause the Z to recover with probability γ, while with probability α, the C becomes a Z. We call this the Susceptible-Cleric-Zombie-Removed or "SCZR" model. Although, as in Ref. [15], we have placed the model in a zombie framework, the model can be rephrased in terms of certain real world diseases such as HIV which, if left untreated, confer a lifelong ability to infect; however, under medical treatment from a health care provider, the infection rate can be reduced or dropped to zero, resulting in an effectively recovered individual. In this case, the zombie class would be simply be labeled as infected (I) while the cleric class would represent some form of health care provider or medical resources. As we show below, the SCZR model exhibits SI behavior when the initial fraction of C or the healing rate γ is low, since in this case the Z wipe out both the C and the S so that a finite fraction of Z remain at the end of the epidemic. In contrast, when the initial fraction of C or the healing rate γ is high enough, the C are able to eliminate the Z so that a finite fraction of S and C remain at the end of the epidemic, which is behavior associated with an SIR model. II. MODELING AND CHARACTERIZATION OF THE SCZR DYNAMICS We consider a two-dimensional assembly of N = 5000 run-and-tumble active particles in a system of size L × L where L = 200.0 and where there are periodic boundary conditions in both the x and y directions. The motion of the particles is obtained by integrating the following overdamped equation of motion in discrete time: α d v i = F dd i + F m i .(1) Here v i = dr i /dt is the velocity and r i is the position of particle i, and the damping constant α d = 1.0. The interaction between two particles, each of radius r a = 1.0, is modeled with a harmonic repulsive potential F dd i = N i =j k(2r a −|r ij |)Θ(|r ij |−2r a )r ij , where Θ is the Heaviside step function, r ij = r i − r j ,r ij = r ij /|r ij |, and the repulsive spring force constant is k = 20.0. Each particle is subjected to an active motor force F m i = F Mmi of magnitude F M applied in a randomly chosen directionm i during a continuous run time of τ l ∈ [1.5×10 4 , 3.0×10 4 ] before instantaneously changing to a new randomly chosen direction. This type of runand-tumble dynamics of active particles has been used extensively to model active matter systems [19,20,26], active ratchets [27], active jamming [28] and motility induced phase separation [26,29]. In another version of active matter, the particles undergo driven diffusion; however, many of the generic phases are the same for both run-and-tumble and driven diffusive active matter [26,30], so we expect that our results will also be relevant to driven diffusive systems. For sufficiently large density or activity, both run-and-tumble and driven diffusive active particles begin to exhibit self-clustering, leading to what is known as motility-induced phase separation (MIPS) [19,20,26,[31][32][33][34]. We select the run length range and motor force value such that the system is in the MIPS regime, and thus creates large connected active clusters similar to those employed in our previous active matter epidemic model [18]. Each particle tracks which one of the four possible states, S, Z, C or R, it is currently occupying. These states are linked together by the following equations: dS= −βSZ (2) dZ= αCZ + βSZ − γCZ (3) dC= −αCZ (4) dR= γCZ .(5) According to these equations, when an S particle encounters a Z particle, it changes its label to Z with rate β. More interestingly, when a C and Z particle come in contact, a change in state occurs with rate α + γ. For interactions in which a state change occurs, with probability α/(α + γ) the C particle becomes a Z, and with probability γ/(α + γ), the Z morphs into R. Our simulation discretizes time in ∆-sized steps, and in the above dynamic, rates are changed into probabilities. Specifically, the probability that an S particle in contact with a Z particle morphs into a Z particle is 1−e −∆β . Similarly, the probability that a change occurs during a Z and C particle encounter is 1 − e −∆(α−γ) . The probability of transitions from C to Z and Z to R remains unchanged. If at a given time step an S particle is in contact with multiple Z particles, or a Z particle is in contact with multiple C or S particles, every possible pair interaction is computed independently using the unmodified states of all particles, and the state of each particle is updated simultaneously at the end of the computation when we apply all S → Z, Z → R, and C → Z transitions. There are no concurrency issues since each type of particle can undergo only one type of transition. The R state is absorbing since the R particles experience no further state transitions, but there is no mechanism to replenish the initial pool of either S or C particles. The epidemic ends when either there are no more S and C particles or there are no more Z particles. Therefore, there are only two possible types of final state for the SCZR model: an SI-like situation in which all S and C particles have been transformed into Z and R particles (indicating that the zombies or the clinical cases prevail), and an SIR-like situation in which all Z particles have been extinguished by becoming R particles (indicating that the medical community prevails and no zombies or clinical cases remain). While the time t d to reach the final state is finite, we observe in simulations that t d can become very long because, in order for the epidemic to come to a conclusion, it is necessary for the remaining S and C or the remaining Z particles to come into contact with Z or C particles, respectively. We initialize the system by randomly placing the particles at non-overlapping positions in the sample. Initially all of the particles are set to the S state. We allow the system to evolve for 5 × 10 5 simulation time steps until a large MIPS cluster emerges, and we define this state to be the t = 0 condition. We then randomly select five particles and change their state to Z. We choose five particles rather than one particle in order to lower the probability of a failed outbreak. We also randomly select a fraction ranging from 10% to 100% of the S to change into C. The system continues to evolve under both the motion of the particles and the reactions between states S, C, Z, and R until there are either no S or C particles or there are no Z particles, indicating that further epidemiological change is impossible. We consider different values of α, β, and γ in addition to varying the fraction of C in the initial population. III. RESULTS In Figure 2 we illustrate the spatial evolution of our system under the SCZR model at fixed α = 5 × 10 −6 , β = 1 × 10 −5 and γ = 1.9 × 10 −5 . For Fig. 2(a,b,c), the initial fraction of C is c 0 ≡ C(t = 0)/N = 0.2, and over time we find an SI-like behavior in which the zombie outbreak prevails and the populations of S and C drop to zero. When c 0 is raised to c 0 = 0.4, Fig. 2(d,e,f) shows an SIR-like behavior in which recovery prevails and the population of Z drops to zero. The initial condition of the MIPS cluster is identical for the two cases, and the motion of the particles is not influenced by their epidemiological state. The peak of the zombie outbreak is shown in Figs. 2(b) and 2(e), and the particle positions are different for the two cases only because the peak in Fig. 2(e) occurs at a later time of t = 9.67 × 10 5 compared to the peak in Fig. 2(b), which falls at t = 4.85×10 5 . In general we find that the progression of an SIR-like epidemic is significantly slower than that of an SI-like epidemic. The end state of the epidemic is illustrated in Fig. 2(c) when the last C is eliminated after a time of t = 1.606 × 10 6 , and in Fig. 2(e) when the last Z is eliminated after a time of t = 2.277 × 10 6 . In the well-mixed mean field limit, when β > α we would expect that all of the S are eliminated prior to the elimination of the last C for the c 0 = 0.2 system. In practice, due to the heterogeneity of our system, we found that out of all the SI simulations we considered, the S were eliminated prior to the C 78% of the time, and the C were eliminated prior to the S 22% of the time. In Fig. 3(a) we plot the epidemic curves s = S/N , c = C/N , z = Z/N , and r = R/N versus simulation time for the c 0 = 0.2 system in the SI regime from Fig. 2(a,b,c). At first, r and z increase at roughly the same rate until z passes through a local peak. Meanwhile, since β > α, s decreases more rapidly than c, and at longer times z undergoes a modest decrease from its peak value so that, at the end of the epidemic, s = 0, c = 0, z = 0.25, and r = 0.75. Figure 3(b) shows the epidemic curves for the SIR regime with c 0 = 0.4 from Fig. 2(d,e,f). Here the evolution to the final state occurs much more slowly, and in order to show the behavior of z clearly we plot z on a separate y axis scale, which is why the curve has a noisy appearance. Both s and c decrease with time, but after passing through a peak, z drops to z = 0 at the end of the epidemic while the values of s, c, and r all remain finite. At late times during the epidemic in Fig. 3(b), where all of the epidemic curves become relatively flat, a strongly stochastic process occurs in which the surviving C and Z need to come into contact with each other in order to end the epidemic. Since the motion of both C and Z is diffusive in nature, this slows the progression of the epidemic and introduces more stochasticity. For late times in Fig. 3(a), as the surviving Z transform the remaining C into Z, z increases with each transformation and so there is a higher probability of making contact with the remaining C, shortening Fig. 2(a,b,c). Here, s = c = 0 in the final state. (b) SIR-like progression at c0 = 0.4 corresponding to the system in Fig. 2(d,e,f). The value of z is plotted on a separate y axis for better visibility. In the final state, z = 0. the epidemic. In contrast, for late times in Fig. 3(b), the surviving C transform the remaining Z into R, which are epidemiologically inert, so there is no increase in c with each transformation and the total duration t d of the epidemic is longer. y (a) (d) y (b) (e) x y (c) x (f) We next consider how changing the values of the model parameters c 0 , α, β, and γ affects the epidemic outcomes. To characterize the outcome of a given simulation, we introduce the quantity υ = (s f + c f )/(s 0 + c 0 ),(6) where s 0 = S(t = 0)/N is the initial fraction of susceptibles, s f = S(t = t d )/N is the final fraction of susceptibles at time t = t d equal to the duration of the epidemic, and c f = C(t = t d )/N is the final fraction of clerics. Using υ we can determine what fraction of the initial population of S and C survive the epidemic. In the SI-like regime, υ = 0, and in the SIR-like regime, υ remains finite. From an epidemiological point of view, υ gives an indication of how effective the medical intervention by the clerics is at suppressing the epidemic. High values of υ are desirable since this indicates that a smaller fraction of the population caught the disease. For any individual simulation with a given set of parameters, it is possible to have either SI or SIR behavior emerge due to the stochasticity, so we average υ over an ensemble of 50 runs for each parameter choice, where each run has a different random seed for the initial particle positions and placement of Z and C particles. When υ remains high, the SIR behavior is dominant and the Z are usually eliminated from the system, while when υ becomes small, the SI behavior is dominant and the S and C are usually eliminated from the system so that the zombies prevail. In Fig. 4 we plot phase diagrams of υ as a function of c 0 , the initial cleric fraction, versus γ, the probability of the transition C + Z → C + R. Each diagram contains 160 points, and each point is averaged over 50 different initial realizations. In the blue region, υ is high and we find SIR-like behavior where S and C survive while Z are eliminated, while in the green region, υ is low and the system is SI-like, with Z persisting to the end of the epidemic and all of the S and C vanishing. Figure 4(a) shows the phase diagram for samples with α = 5 × 10 −6 and β = 1 × 10 −5 , as in Figs. 2 and 3. At higher γ, the zombies are more effectively healed by the clerics, and the initial fraction c 0 of C needed to produce SIR-like behavior drops to lower values, as shown by the solid line which is a fit of the SI-SIR transition to the form c 0 ∝ a(γ + b) −1 . For a simple way to understand the general form of this curve, consider the early time behavior of an individual Z particle. As it moves, the Z encounters a C with probability c 0 and an S with probability 1 − c 0 . The Z always survives an encounter with S, but it only survives an encounter with C with probability 1 − γ. Thus, the probability that the Z survives is Z survive = (1 − γ)c 0 + (1 − c 0 ) and the probability that the Z is destroyed by turning into an R is Z destroy = γc 0 . At the SI-SIR transition, we have Z survive = Z destroy , meaning that the transition line is expected to fall at c 0 = 0.5(γ) −1 . The actual location of the SI-SIR transition line is affected by the values of α and β because these control the way in which the populations of S, C, Z, and R evolve over time. If we cut the probability α of the C + Z → Z + Z transition in half to α = 2.5 × 10 −6 , the phase diagram in Fig. 4(b) indicates that the SI-SIR transition line shifts to lower values of c 0 since it becomes more difficult for the Z to eliminate all of the C. If we instead double α to α = 1 × 10 −5 , as in Fig. 4(c), we reach the limit in which α = β and the S and C particles are both equally likely to be infected upon encountering a Z. Here, not only does the SI-SIR transition line shift to higher c 0 , but for small values of γ only SI behavior can occur even if the entire population apart from the zombie index cases is initialized to state C. If we leave α unchanged but double β, the probability of the S + Z → Z + Z transition, to β = 2 × 10 −5 , Fig. 4(d) shows that at low γ, the location of the SI-SIR transition does not change very much, but at higher γ, it shifts to higher c 0 . In order to illustrate some representative averaged epidemic curves, in Fig. 5(a) we reproduce the phase diagram of Fig. 4(a) for α = 5 × 10 −6 and β = 1 × 10 −5 with a black line indicating the location of a horizontal cut. Figure 5(b) shows υ versus γ at the cut location of c 0 = 0.5. When γ < 9 × 10 −6 , there are no realizations in which SIR behavior occurs; instead, the Z always wipe out all of the S and C. Similarly, for γ > 1.1×10 −5 , there are no realizations in which SI behavior occurs, and the Z are always fully eliminated. The kink in the curve marks the transition to fully SIR behavior. The value of υ indicates how effective the clerics are at suppressing the epidemic. When υ increases, it means that a greater fraction of the population was never infected by the disease. For γ just above the transition into fully SIR behavior, over 75% of the population still becomes infected before the zombies are eliminated, whereas for higher γ, the majority of the population is able to avoid becoming infected. For the three points highlighted in black in Fig. 5(b), we show averaged epidemic curves with s, c, z, and r plotted as a function of normalized timet = t/t d in Figs. 5(c,d,e). For γ = 9×10 −6 in Fig. 5(c), we are still in the SI dominated regime and the z curve is higher than the s and c curves. Although in any individual run we either have z = 0 or s = c = 0, for the ensemble average s and c are finite since SIR behavior emerges 10% of the time. Since we are working at c 0 = 0.5, we have s = c at the beginning of the epidemic, and although s drops more rapidly than c as the epidemic progresses, by the end of 5.0×10 6 1. the epidemic s ≈ c, due in large part to the many SI runs for which s = c = 0. In Fig. 5(d) at γ = 1.2 × 10 −5 , all 50 simulations are in the SIR regime so that z = 0 at the end of the epidemic, while the final value of r ≈ 0.5 shows that on average half of the population becomes infected before the zombies are extinguished. Since we have β = 2α, the value of s drops approximately twice as fast as the value of c at early times in the epidemic, but as the supply of Z is depleted through healing by the clerics, both s and c reach a plateau, and in the final state c > s. For γ = 1.9 × 10 −5 in Fig. 5(e), well within the SIR regime, z remains quite small throughout the epidemic. Although we still find c > s at the end of the epidemic, both quantities have dropped only slightly from the original levels and are not very different from each other, and 90% of the population is able to avoid becoming infected. As shown in Fig. 6(a), we next consider a vertical cut at γ = 1 × 10 −5 from the phase diagram in Fig. 4(a) for α = 5 × 10 −6 and β = 1 × 10 −5 . In Fig. 6(b) we plot υ versus c 0 along this cut. For c 0 < 0.5, all of the realizations are in the SI regime and the Z prevail, while for c 0 ≥ 0.6, all of the realizations are in the SIR regime and there are no Z remaining at the end of the epidemic. The black points in Fig. 6(b) correspond to the values of c 0 at which the averaged epidemic curves in Figs. 6(c,d,e) were obtained. At c 0 = 0.4 in the SI regime, Fig. 6(c) shows that at the end of the epidemic, s = c = 0 and the average fraction of zombies is z = 0.28. When c 0 = 0.5 in Fig. 6(d), the system is in the SI regime 36% of the time, so that the final value of z is greater than zero. Although c and s approach each other toward the end of the epidemic, we find that c > s by a small amount since the s = c = 0 behavior of the SI regime is no longer dominant. In Fig. 6(e), for c 0 = 0.6 the system is fully in the SIR regime, and throughout the epidemic we find not only that c > s but that the difference between c and s remains constant. This is an indication of the importance of the stochastic diffusive process that occurs in our model in order to permit Z to come into contact with S or C. For c 0 = 0.4 in Fig. 6(c), at early times in the epidemic a Z encounters an S 60% of the time but a C only 40% of the time. Since S are twice as likely as C to be infected, s drops much more rapidly than c in this regime. When c 0 is increased to c 0 = 0.5 in Fig. 6(d), a Z is equally likely to encounter an S or a C at early times, and we see that the doubled infection probability causes s to drop about twice as fast as c, as also shown in Fig. 5(c,d,e). Further increasing c 0 to c 0 = 0.6 in Fig. 6(e) means that at early times a Z encounters a C 60% of the time and an S only 40% of the time. Since the C are more resistant to infection, the relative fraction 5.0×10 6 1. of C and S in the population remains nearly constant. Increasing c 0 even further produces many short-lived epidemics in which s and c do not change very much from their initial values. We can analytically evaluate υ for well mixed systems whose dynamics is described through Equations (2)-(5). Using a standard argument (see [35]) and some algebra, we can show that s s 0 α = c c 0 β This provides us with the opportunity to compute a target for υ: υ = (s f + c 0 (s f /s 0 ) α/β )/(s 0 + c 0 ) . Failure to hit that target in simulations is an indication that the homogeneous mixing assumption failed. From the data in Figs. 5(c,d,e) and 6(c,d,e), we find that the predicted value of υ is higher than the actual value of υ, but that the agreement between predicted and actual improves as we move deeper into the SIR regime. This could be an indication that the SIR regime is better mixed than the SI regime, possibly due to the faster dynamics that tend to occur for SI behavior. In Fig. 7 we plot the distribution P (t d ) of the duration t d of the individual epidemics for the runs in all of the phase diagrams in Fig. 4. The data is split into two distributions, with the first for simulations that ended in the SI regime with a finite number of Z remaining, and the second for simulations that ended in the SIR regime with no Z remaining. For the SI case, there are no epidemics of short duration. This is because all C and S must be eliminated in the SI regime, and the elimination process requires a minimum amount of time to occur. In the inset we show the same data on a log-linear scale, indicating that some of the SI epidemics last for an extremely long time before reaching a final state. These lengthy epidemics occur for values of c 0 and γ at which the behavior is evenly split between SI and SIR on average. There is also a peak in P (t d ) near t d = 1.5 × 10 5 simulation time steps. In the SIR regime, there is a large peak in P (t d ) at small t d corresponding to failed outbreaks in which the C can rapidly encounter and cure the small number of Z present at early times before the epidemic gets going. This is followed by a gap similar to what we observed previously in SIR simulations [18], and then by a second peak representing epidemics that involve a substantial portion of the population. Here we find that if the epidemic in the SIR regime is able to become established, it lasts longer than the typical epidemic in the SI regime, but that there is a high probability for the SIR epidemic to be extinguished before it can become established. IV. DISCUSSION As we noted earlier, although we have cast our SCZR model in terms of zombies and clerics, it could also be rephrased so that the zombies are disease-spreading individuals that cannot spontaneously recover from the disease they have caught, and the clerics are medical care providers who can cure the infected individuals or at least render them non-infectious. In this picture, when we take α < β but α > 0, this would mean that the medical care providers are more careful than the general population and take more precautions against becoming infected, but that they are not immune from becoming infected. The transition between SI and SIR behavior is significant because it indicates that by introducing a larger number of medical care providers (increasing c 0 ) or giving the medical care providers more effective treatment protocols (increasing γ), the disease can be prevented from entering the SI regime in which the entire population winds up getting infected eventually, and can instead be held in the SIR regime, ideally in the limit where t d is short and the epidemic never becomes established in the population. Some of the next steps for our SCZR model would be to consider the effect of adding fixed spatial heterogeneity such as quenched disorder. For example, the C might be confined to only certain regions of the system, as in real world scenarios where impassable terrain or military blockades are present. Other situations include considering the case where the R are not epidemiologically inert but can produce infection at greatly reduced rates β β and α α, to represent situations in which the medical care givers only reduce the infec-tiousness rather than fully eliminating it. Active matter models in general also readily allow other effects to be captured, such as introducing a small fraction of very active particles with increased motor force F M embedded in a population of reduced mobility or much smaller F M in order to represent different types of mobility patterns in social systems. Another question that could be explored with the SCZR model is what is the nature of the transition from the SI to the SIR regime. Although the transition is somewhat sharp in our phase diagrams, it may be only a crossover. Note that in the limit c 0 = 1, the SCZR model becomes equivalent to the SZR model of Ref. [15]. In this limit, Fig. 4 shows that for certain parameter regimes there is still a transition from SI to SIR behavior; however, it is much more intuitive from a medical intervention point of view to tune between the two regimes using the c 0 and γ parameters of the SCZR model than by using the parameter α (which is written as β in the SZR model). Epidemic models show various types of critical phenomena associated with directed percolation transitions [36,37]; however, such transitions can be screened or modified by the introduction of quenched disorder [38], so we expect that there could be various types of critical behavior in our system. V. SUMMARY We have introduced a model for epidemics that we call the Susceptible-Cleric-Zombie-Removed or SCZR model, and we demonstrate the use of this model with active matter run-and-tumble particles. In the SCZR model, the infectious agents are the zombies, and there is no spontaneous recovery. There is an initial population of susceptibles and clerics. With probability α for clerics and β for susceptibles, interaction with a zombie causes infection into the zombie state, while with probability γ, a cleric interacting with a zombie causes the zombie to enter an epidemiologically inert recovered state. We show that by varying the initial density of clerics or their healing rate γ, we can tune the SCZR model between SI and SIR regimes. If the initial cleric density or the healing rate γ is low, the zombies eliminate all of the clerics and susceptibles to give SI behavior, while if the initial cleric density or healing rate γ is high enough, the clerics are able to heal all of the zombies and SIR behavior emerges. Our model has implications for real world diseases where infections are lifelong and spontaneous recovery does not occur, but where medical intervention can produce recovery or at least drive the rate of infectiousness to zero. One example of this type of disease is the human immunodeficiency virus (HIV). In this case, the zombies would be infected persons and the clerics would represent medical caregivers that can provide treatment. The SCZR model could provide a good staring point for creating new types of epidemic models where treatment is needed for recovery and there are finite or limited treatment resources available. FIG. 1 . 1(a) In the SI model, there is no spontaneous recovery, and susceptibles (S, yellow) that come into contact with infectives (I, red) become infected with probability β. (b) The SIR model adds a spontaneous recovery process in which an I transitions to recovered (R, blue open circle) at a rate µ. (c) In the Living-Zombie-Recovered model introduced in FIG. 2 . 2Snapshots of the time evolution of the SCZR system for α = 5 × 10 −6 , β = 1 × 10 −5 , and γ = 1.9 × 10 −5 . Yellow disks are susceptibles (S), brown disks are clerics (C), green disks are zombies (Z), and open blue circles are recovered (R). (a,b,c) are for an initial cleric fraction of c0 = 0.2, and (d,e,f) are for c0 = 0.4. (a,d) The t = 0 moment where the MIPS cluster is present. (b,e) The peak of the zombie outbreak, which occurs at t = 4.85 × 10 5 in (b) and at t = 9.67 × 10 5 in (e). (c,f) The final state, which is reached at t = 1.606 × 10 6 in (c) and t = 2.277 × 10 6 in (f). (a,b,c) show an SI-like evolution in which all S and C are eliminated in the final state, while (d,e,f) show an SIR-like evolution in which all Z are eliminated in the final state. FIG. 3 . 3Epidemic curves for the individual runs illustrated in Fig. 2 with α = 5 × 10 −6 , β = 1 × 10 −5 and γ = 1.9 × 10 −5 showing the fractions of susceptible s (yellow), cleric c (brown), zombie z (green), and recovered r (blue) particles versus time t. (a) SI-like progression at c0 = 0.2 corresponding to the system in showing heat maps of υ , the average fraction of the initial population of S and C that survive the epidemic, as a function of initial cleric fraction c0 vs the probability γ of the transition C + Z → C + R. Blue indicates SIR behavior in which Z are eliminated, and green indicates SI behavior in which S and C are eliminated. In general, as γ increases, the SIR behavior emerges at a lower value of c0.(a) Samples of the type shown in Figs. 1 to 3 with α = 5 × 10 −6 and β = 1 × 10 −5 . (b) Samples with the same β = 1 × 10 −5 where α, the probability for C+Z → Z+Z, has been halved to α = 2.5×10 −6 . (c) Samples with the same β = 1 × 10 −5 in which α has been doubled to α = 1 × 10 −5 . (d) Samples with the same α = 5 × 10 −6 in which β, the probability for S + Z → Z + Z, is doubled to β = 2 × 10 −5 . The solid lines in the figures are fits of the form c0 ∝ a(γ + b) −1 where (a) a = 7.77 × 10 −6 and b = 3.781 × 10 −6 , (b) a = 6.912 × 10 −6 and b = 5.696 × 10 −6 , (c) a = 9.056 × 10 −6 and b = 9.498 × 10 −7 , and (d) a = 1.212 × 10 −5 and b = 8.381 × 10 −6 . The phase diagram with a heat map of υ as a function of c0 vs γ from Fig. 4(a) with α = 5 × 10 −6 and β = 1 × 10 −5 . (b) A horizontal slice of υ vs γ taken at c0 = 0.5 along the black line in panel (a). (c,d,e) Epidemic curves averaged over 50 runs taken at the black points in panel (b) showing s (yellow), c (orange), z (green), and r (blue) vs the rescaled timet = t/t d . (c) At γ = 9 × 10 −6 , SI behavior occurs 90% of the time, so the averaged values of s and c do not reach zero but are lower than the averaged value of z. (d) At γ = 1.2 × 10 −5 , all runs are in the SIR regime and on average 50% of the population is never infected. (e) At γ = 1.4 × 10 −5 , the clerics become more effective at reducing the impact of the epidemic, and on average 90% of the population is never infected. FIG. 6 . 6The phase diagram with a heat map of υ as a function of c0 vs γ from Fig. 4(a) with α = 5 × 10 −6 and β = 1 × 10 −5 . (b) A vertical slice of υ vs c0 taken at γ = 1 × 10 −5 along the black line in panel (a). (c,d,e) Epidemic curves averaged over 50 runs taken at the black points in panel (b) showing s (yellow), c (orange), z (green), and r (blue) vst. (c) At c0 = 0.4, only SI behavior occurs. (d) At c0 = 0.5, we find mixed behavior, with an SI response occurring 36% of the time and an SIR response appearing in the remaining 64% of runs. (e) At c0 = 0.6, all runs are in the SIR regime. FIG. 7. The distribution P (t d ) of epidemic durations for the runs presented in the phase diagrams of Fig. 4. Blue: runs in which the final state was in the SIR regime with all of the Z eliminated. Green: runs in which the final state was in the SI regime with a finite population of Z surviving at the end of the epidemic. Inset: The same data plotted on a log-linear scale.0 1×10 6 2×10 6 3×10 6 4×10 6 5×10 6 t d 0 1×10 7 2×10 7 3×10 7 4×10 7 5×10 7 6×10 7 7×10 7 8×10 7 P(t d ) (a) SIR phase SI phase 0.0 5.0×10 6 1.0×10 7 t d 10 8 10 7 10 6 P(t d ) (b) ACKNOWLEDGMENTSThis work was supported by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U. S. Department of Energy (Contract No. 892333218NCA000001). NH benefited from resources provided by the Center for Nonlinear Studies (CNLS). PF and AL were supported by a grant of the Romanian Ministry of Education and Research, CNCS -UEFISCDI, project number PN-III-P4-ID-PCE-2020-1301, within PNCDI III. A contribution to the mathematical theory of epidemics. 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[ "ON THE SUPPORT OF A HYPOELLIPTIC DIFFUSION ON THE HEISENBERG GROUP", "ON THE SUPPORT OF A HYPOELLIPTIC DIFFUSION ON THE HEISENBERG GROUP", "ON THE SUPPORT OF A HYPOELLIPTIC DIFFUSION ON THE HEISENBERG GROUP", "ON THE SUPPORT OF A HYPOELLIPTIC DIFFUSION ON THE HEISENBERG GROUP" ]
[ "Marco Carfagnini ", "Marco Carfagnini " ]
[]
[]
We provide an elementary proof of the support of the law of a hypoelliptic Brownian motion on the Heisenberg group H. We consider a control norm associated to left-invariant vector fields on H, and describe the support in terms of the space of finite energy horizontal curves.2020 Mathematics Subject Classification. Primary 58J65, 60H10; Secondary 60J60, 60H05 .
10.30757/alea.v20-26
[ "https://export.arxiv.org/pdf/2101.11045v2.pdf" ]
257,804,506
2101.11045
346812a43ab46128a4b2cc74226c8b07cc282029
ON THE SUPPORT OF A HYPOELLIPTIC DIFFUSION ON THE HEISENBERG GROUP 28 Mar 2023 Marco Carfagnini ON THE SUPPORT OF A HYPOELLIPTIC DIFFUSION ON THE HEISENBERG GROUP 28 Mar 2023arXiv:2101.11045v2 [math.PR] We provide an elementary proof of the support of the law of a hypoelliptic Brownian motion on the Heisenberg group H. We consider a control norm associated to left-invariant vector fields on H, and describe the support in terms of the space of finite energy horizontal curves.2020 Mathematics Subject Classification. Primary 58J65, 60H10; Secondary 60J60, 60H05 . Introduction The purpose of this paper is to describe the support of the law of a hypoellptic diffusion g t on the Heisenberg group H. The novelty of this paper is the norm on the path space W 0 (H) of H-valued continuous curves starting at the identity that is used for the support. The group H is the simplest example of a Carnot group and it comes with a natural left-invariant distance, the Carnot-Carathéodory distance d cc . This is the control distance associated to the left-invariant vector fields on H, see Definition 2.5. The corresponding control norm d c is defined as d c (x) := d cc (x, e), where e ∈ H is the identity. Our main result is to prove a support theorem for the hypoelliptic Brownian motion g t with respect to the norm max 0 t 1 d c (γ(t)) for γ ∈ W 0 (H). As pointed out in Remark 2.16 it is sufficient to describe the support with respect to some equivalent norms. The support of a diffusion was first studied by Stroock and Varadhan [18], which we now describe briefly. Let X t be an R d -valued diffusion which is solution to the stochastic differential equation (1.1) dX t = σ (t, X t ) • dW t + b (t, X t ) dt, X 0 = 0, where σ(t, x) is a d×ℓ matrix whose entries are functions of (t, x) ∈ [0, 1]×R d , and b(t, x) is a vector in R d , and W t is an ℓ-dimensional Brownian motion, and •d denotes the stochastic differential in Stratonovich's sense. We can view the process {X t } 0 t 1 as a W 0 (R d )-valued random variable, where W 0 (R d ) is the space of R d -valued continuous paths starting at zero. Let µ be the law of {X t } 0 t 1 and S µ its support. If H denotes the subset of W 0 (R d ) consisting of absolutely continuous paths, then to any φ ∈ H one can associate a deterministic path x φ as being the solution to the ordinary differential equation x ′ φ (t) = σ (t, x φ (t)) φ ′ (t)dt + b (t, x φ (t)) dt, (1.2) x φ (0) = 0. We follow [14] and refer to solutions to (1.2) as controlled systems. Then (1.3) S µ = {x φ , φ ∈ H} ∞ , where the closure is taken in the uniform topology in W 0 (R d ). Note that the hypoelliptic Brownian motion g t can be viewed as an R 3valued stochastic process. This is not a Gaussian process and it satisfies the stochastic differential equation   dg 1 (t) dg 2 (t) dg 3 (t)   =   1 0 0 1 − 1 2 B 2 (t) 1 2 B 1 (t)   dB 1 (t) dB 2 (t) , g 0 = (0, 0, 0). Stroock-Varadhan original support theorem (1.3) was proven under the assumption that σ is C 2 in space and C 1 in time, bounded together with its partial derivatives of order one and two, and b is globally Lipschitz and bounded. In a series of papers by Gyöngy [5,6,8], and by Gyöngy-Pröhle [9] (1.3) is proven for processes driven by continuous semi-martingales under milder assumptions on σ and b. In particular, (1.3) for the law of {g t } 0 t 1 with respect to the uniform norm max 0 t 1 |γ(t)| R 3 on W 0 (R 3 ) follows from [9, Theorem 3.1]. Moreover, (1.3) for the hypoelliptic Brownian motion can also be proven by rough path theory and continuity of the Lyons-Itô map, that is, the solution map of a rough differential equation [3, Section 10.3, Section 13.7]. In the current paper we prove (1.3) for the hypoelliptic Brownian motion on the Heisenberg group. Differently from [9], we replace the Euclidean norm in R 3 by the control norm d c , which is a more natural norm and it is consistent with the geometry of the Heisenberg group. Our proof does not rely on rough path theory and it is based on a time change argument. We mention that (1.3) for diffusion processes on Hilbert spaces is proven in [1,7]. We also mention that in [15] a rough paths approach is used, and a support theorem in the p-variational topology is proven. One can ask under what condition the closure in (1.3) coincides with the whole path space W 0 (R d ). This question has been addressed in [14], where the author gives nearly necessary and sufficient conditions for (1.4) W 0 (R d ) = {x φ , φ ∈(1.5) W 0 (H) = H(H) dc = S µ , where µ is the law of {g t } 0 t 1 and the closure is taken with respect to the norm max 0 t 1 d c (γ(t)) for γ ∈ W 0 (H). First, we show that S µ ⊂ H(H) dc by constructing a family of stochastic processes {g δ } δ>0 that approximates g in the sense that the law µ δ of g δ converges weakly to the law µ of g. We further study relations between the measures µ δ and µ, and prove that they are singular. In particular we show that the space H(H) of finite energy horizontal curves has µ-measure zero. The space H(H) can then be viewed as a Cameron-Martin space in a non-Gaussian setting by (1.5) and since µ(H(H)) = 0. We then prove that H(H) dc ⊂ S µ using Theorem 3.6 and the explicit form of the process g t . Namely, g t = (B t , A t ), where B t is a two-dimensional standard Brownian motion and A t is the corresponding Lévy's stochastic area. Our proof relies on the classical identity A t = b τ (t) , where b t is a one-dimensional standard Brownian motion independent of B t , and τ (t) is a stopping time. The paper is organized as follows. In Section 2 we describe the Heisenberg group H, and the corresponding control norm and hypoelliptic Brownian motion, and state the main result of the paper Theorem 2.15. Section 3 then contains the proof of Theorem 2.15. The setting and the main result 2.1. Heisenberg group as Lie group. The Heisenberg group H as a set is R 3 ∼ = R 2 × R with the group multiplication given by (v 1 , z 1 ) · (v 2 , z 2 ) := x 1 + x 2 , y 1 + y 2 , z 1 + z 2 + 1 2 ω (v 1 , v 2 ) , where v 1 = (x 1 , y 1 ) , v 2 = (x 2 , y 2 ) ∈ R 2 , and ω : R 2 × R 2 −→ R, ω (v 1 , v 2 ) := x 1 y 2 − x 2 y 1 . The identity in H is e = (0, 0, 0) and the inverse is given by (v, z) −1 = (−v, −z). The Lie algebra of H can be identified with the space R 3 ∼ = R 2 ×R with the Lie bracket defined by [(a 1 , c 1 ) , (a 2 , c 2 )] = (0, ω (a 1 , a 2 )) . The set R 3 ∼ = R 2 × R with this Lie algebra structure will be denoted by h. Let us now recall some basic notation for Lie groups. Suppose G is a Lie group, then the left and right multiplication by an element k ∈ G are denoted by L k : G −→ G, g −→ k −1 g, R k : G −→ G, g −→ gk. Recall that the tangent space T e G can be identified with the Lie algebra g of left-invariant vector fields on G, that is, vector fields X on G such that dL k • X = X • L k , where dL k is the differential of L k . More precisely, if A is a vector in T e G, then we denote byà ∈ g the (unique) left-invariant vector field such thatÃ(e) = A. A left-invariant vector field is determined by its value at the identity, namely,à (k) = dL k •Ã (e). For the Heisenberg group the differential of left and right multiplication can be described explicitly as follows. Proposition 2.1. Let k = (k 1 , k 2 , k 3 ) = (k, k 3 ) and g = (g 1 , g 2 , g 3 ) = (g, g 3 ) be two elements in H. Then, for every v = (v 1 , v 2 , v 3 ) = (v, v 3 ) in T g H, the differentials of the left and right multiplication are given by dL k : T g H −→ T k −1 g H, dR k : T g H −→ T gk H, dL k (v) = v 1 , v 2 , v 3 + 1 2 ω(v, k) , dR k (v) = v 1 , v 2 , v 3 + 1 2 ω(v, k) . (2.1) 2.2. Heisenberg group as a sub-Riemannian manifold. The Heisenberg group H is the simplest non-trivial example of a sub-Riemannian manifold. We define X, Y and Z as the unique left-invariant vector fields satisfying X e = ∂ x , Y e = ∂ y and Z e = ∂ z , that is, X = ∂ x − 1 2 y∂ z , Y = ∂ y + 1 2 x∂ z , Z = ∂ z . Note that the only non-zero Lie bracket for these left-invariant vector fields is [X, Y ] = Z, so the vector fields {X, Y } satisfy Hörmander's condition. We define the horizontal distribution as H := span {X, Y } fiberwise, thus making H a sub-bundle in the tangent bundle T H. To finish the description of the Heisenberg group as a sub-Riemannian manifold we need to equip the horizontal distribution H with an inner product. For any p ∈ H we define the inner product ·, · Hp on H p so that {X (p) , Y (p)} is an orthonormal (horizontal) frame at any p ∈ H. Vectors in H p will be called horizontal, and the corresponding norm will be denoted by · Hp . In addition, Hörmander's condition ensures that a natural sub-Laplacian on the Heisenberg group (2.2) ∆ H = X 2 + Y 2 is a hypoelliptic operator by [10]. We recall now another important object in sub-Riemannian geometry, namely, horizontal curves. Notation 2.2. A curve γ(t) = (x (t) , y (t) , z (t) ) in H will be denoted by (x (t) , z (t)), and its corresponding tangent vector γ ′ (t) in T H γ(t) will be denoted by γ ′ (t) = x ′ (t) , y ′ (t) , z ′ (t) = x ′ (t) , z ′ (t) . Definition 2.3. An absolutely continuous path t −→ γ(t) ∈ H, for a.e. t ∈ [0, 1] is said to be horizontal if γ ′ (t) ∈ H γ(t) for a.e. t, that is, the tangent vector to γ (t) at every point γ (t) is horizontal. Equivalently we can say that γ is horizontal if c γ (t) := dL γ(t) (γ ′ (t)) ∈ H e for a.e. t. Note that for γ(t) = (x (t) , z (t)) we have c γ (t) = dL γ(t) γ ′ (t) = x ′ (t) , z ′ (t) − 1 2 ω(x (t) , x ′ (t)) , (2.3) where we used Proposition 2.1. Equation (2.3) can be used to characterize horizontal curves in terms of the components as follows. The curve γ is horizontal if and only if, for a.e. 0 t 1 (2.4) z ′ (t) − 1 2 ω(x (t) , x ′ (t))) = 0. Definition 2.4. We say that a horizontal curve t −→ γ(t) ∈ H, 0 t 1 has finite energy if (2.5) γ 2 H(H) := 1 0 |c γ (s) | 2 He ds = 1 0 |dL γ(s) γ ′ (s) | 2 He ds < ∞. We denote by H (H) the space of finite energy horizontal curves starting at the identity. The inner product corresponding to the norm · H(H) is denoted by ·, · H(H) . Note that the Heisenberg group as a sub-Riemannian manifold comes with a natural left-invariant distance. Definition 2.5. For any x, y ∈ H the Carnot-Carathéodory distance is defined as d cc (x, y) := inf 1 0 |c γ (s) | He ds, γ : [0, 1] −→ H, γ(0) = x, γ(1) = y, γ is horizontal} . Another consequence of Hörmander's condition for left-invariant vector fields X, Y and Z is that we can apply the Chow-Rashevskii theorem. As a result, given two points in H there exists a horizontal curve connecting them, and therefore the Carnot-Carathéodory distance is finite on H. The Carnot-Carathéodory distance in Definition 2.5 is the control distance associated to the vector fields X, Y , and Z on H [2, Definition 5.2.2]. The control norm d c : H → R is then defined as d c (x) = d cc (x, e). Note that (2.6) d c (y −1 x) = d cc (x, y), by left-invariance of X, Y , and Z, and the definition of d cc . The control norm is an example of a homogeneous norm. Definition 2.6. Let ρ : H → [0, ∞) be a continuous function with respect to the Euclidean topology. Then ρ is a homogeneous norm if it satisfies the following properties ρ(δ λ (x)) = λρ(x), for every λ > 0, and x ∈ H, ρ(x) = 0 if and only if x = e, where δ λ (x) := λx 1 , λx 2 , λ 2 x 3 . If ρ 1 and ρ 2 are two homogeneous norms, then there exists a constant c > 0 such that (2.7) c −1 ρ 1 (x) ρ 2 (x) cρ 1 (x), for every x ∈ H, [2, Proposition 5.1.4]. We consider the following homogeneous norm (2.8) |x| := (x 2 1 + x 2 2 ) 2 + x 2 3 1 4 , for every x = (x 1 , x 2 , x 3 ) ∈ H. By (2.6) and (2.7) it follows that c −1 d cc (x, y) |y −1 x| cd cc (x, y), for any x, y ∈ H. Finally, we need to describe a hypoelliptic Brownian motion with values in H. Definition 2.7. An H-valued Markov process g t is called a hypoelliptic Brownian motion if its generator is the sub-Laplacian 1 2 ∆ H defined by Equation (2.2). One can write a stochastic differential equation for g t . This form is the standard stochastic differential equation for a Lie group-valued Brownian motion, namely, dL gt (dg t ) = (dB 1 (t), dB 2 (t), 0), g 0 = e, where B t = (B 1 (t), B 2 (t)) is a standard two-dimensional Brownian motion. An explicit solution is given by (2.9) g t := (B t , A t ) , where A t := 1 2 t 0 ω (B s , dB s ) is the Levy's stochastic area. Note that we used the Itô integral rather than the Stratonovich integral. However, these two integrals are equal since the symplectic form ω is skew-symmetric, and therefore Lévy's stochastic area functional is the same for both integrals. Notation 2.8. Throughout the paper we fix a filtered probability space (Ω, F, F t , P). We denote the expectation under P by E. 2.3. The Wiener meaure. We recall here the definition of Wiener measure, and collect some notations that will be used throughout the paper. Notation 2.9 (Topology on H). Let | · | be the homogeneous norm in (2.8). We consider the topology on H whose open balls centered at the identity are {x ∈ H, |x| < r}. Note that by (2.7) all homogeneous norms induce the same topology. Once a homogeneous norm ρ on H is fixed, one can introduce a topology on W 0 (H) in the following way. We endow W 0 (H) with the following norm h W 0 (R n ) := max 0 t 1 |h(t)| R n , h ∈ W 0 (R n ) , and the associated distance d W 0 (R n ) (h, k) = max 0 t 1 |h(t) − k(t)| R n , where | · | R n is the Euclidean norm.η ρ := max 0 t 1 ρ(η(t)), η ∈ W 0 (H) , and the associated distance is max 0 t 1 ρ(η(t) −1 γ(t)) for any η, γ ∈ W 0 (H). Definition 2.12. Let W 0 (H) be the Wiener space over H, and {g t } 0 t 1 be the hypoelliptic Brownian motion defined by (2.9). We call its law the horizontal Wiener measure and we denote it by µ. The process g t can be viewed as a W 0 (H)-valued random variable, that is, g : Ω −→ W 0 (H) , ω → {t → g t (ω)} . The measure µ is then given by µ(E) = P g −1 (E) = P (g ∈ E) for any Borel set E in W 0 (H). We denote the support of µ by S µ , that is, S µ is the smallest closed subset of W 0 (H) having µ-measure one. Remark 2.13. Note that even though the hypoelliptic Brownian motion g t is an R 3 -valued stochastic process, it is not a Gaussian process, and its law µ is not a Gaussian measure on W 0 (H). Moreover, contrary to the Euclidean case, the space W 0 (H) is not a Banach space. It is easy to see that the space W 0 (H) is closed under the norm max 0 t 1 ρ(γ(t)) for γ ∈ W 0 (H), where ρ is a homogeneous norm on H, but W 0 (H) is not a linear space. Let us denote by π the projection (2.10) π : W 0 (H) −→ W 0 (R 2 ), π(γ) = (γ 1 , γ 2 ), for any γ = (γ 1 , γ 2 , γ 3 ) ∈ W 0 (H). Remark 2.14. Let φ = (φ 1 , φ 2 , φ 3 ) ∈ H (H) be a finite energy horizontal curve as in Definition 2.4. Then π(φ) is in the Cameron-Martin space on R 2 , that is, π(φ) is an absolutely continuous R 2 -valued curve starting at zero such that 1 0 |π(φ) ′ (s)| 2 R 2 ds < ∞.S µ = H (H) dc = W 0 (H) , where the closure is taken with respect to the norm max 0 t 1 d c (γ(t)), for γ ∈ W 0 (H), and d c is the control norm induced by the Carnot-Carathéodory distance. Remark 2.16. It is enough to prove Theorem 2.15 for the homogeneous norm | · | given by (2.8). Indeed, if ρ 1 and ρ 2 are two homogeneous norms on H and · ρ 1 , · ρ 2 denote the corresponding norms on W 0 (H), that is, γ ρ i := max 0 t 1 ρ i (γ(t)), γ ∈ W 0 (H) . Then H (H) ρ 1 = H (H) ρ 2 , since · ρ 1 , · ρ 2 are equivalent by (2.7), and hence Theorem 2.15 holds for any homogeneous norm as soon as it holds for one norm. Let us denote by H (H) the closure of H (H) with respect to the norm max 0 t 1 ρ(γ(t)), γ ∈ W 0 (H), and for ρ a homogeneous norm on H. By Remark 2.16, H (H) is independent of ρ. Proof of Theorem 2.15 We will divide the proof of Theorem 2.15 in two steps. First, we introduce a family of processes that approximates {g t } 0 t 1 . This is used in Corollary 3.4 to show that the support S µ is contained in H (H). The reverse inclusion is proven in Corollary 3.7 which follows from Theorem 3.6. In Proposition 3.10 we prove that H (H) = W 0 (H), which concludes the proof of Theorem 2.15. 3.1. Approximation of the hypoelliptic Brownian motion. The aim of this step is to show that the support S µ of the law of {g t } 0 t 1 is contained in H (H). This will be accomplished by constructing a horizontal piecewise approximation g δ (t) of g t such that µ δ → µ weakly, where µ δ is the law of g δ (t). Different approximations of a Brownian motion have been extensively studied over the decades, see for example Wong-Zakai [19], Kunita [13], Nakao-Yamamoto [17], Ikeda-Nakao-Yamato [11], and Ikeda-Watanabe [12, Chapter 6, Section 7] for more details. We are not able to refer to all the vast literature on the subject, but we mentioned some results which are closer and more relevant to the techniques we use in this paper. Let {B δ } δ>0 be an approximation of a two-dimensional Brownian motion, that is, (3.1) E max 0 t 1 |B δ (t) − B t | 2 R 2 −→ 0, as δ → 0, such that (3.2) E max 0 t 1 A δ (t) − A t | 2 R −→ 0 as δ → 0,B i,δ (t) := B i (kδ) + f i t − kδ δ (B i (kδ + δ) − B i (kδ)) ,g δ (t) = (g 1,δ (t), g 2,δ (t), g 3,δ (t)) , where g 1,δ (t) = B 1,δ (t) g 2,δ (t) = B 2,δ (t) (3.5) g 3,δ (t) = A δ (t). Let C 2 p R 2 be the space of piecewise continuously twice differentiable curves in R 2 starting at zero, and set H p (H) := γ : [0, 1] −→ H, π(γ) ∈ C 2 p (R 2 ), γ 3 (t) = 1 2 t 0 ω π(γ)(s), π(γ) ′ (s) ds , where γ = (π(γ), γ 3 ), that is, H p (H) is the set of piecewise continuously twice differentiable horizontal curves. Clearly we have that H (H) = H p (H). We can view g δ as a H p (H)-valued random variable, that is, g δ : Ω −→ H p (H) , ω → {t → g δ (t, ω)} , (3.6) and hence we can induce a probability measure µ δ on W 0 (H) by µ δ (E) := P g −1 δ (E ∩ H p (H)) for any Borel set E in W 0 (H). Proposition 3.2. Let S µ δ be the support of the measure µ δ . Then S µ δ ⊂ H p (H) = H (H). Proof. By 3.6 we have that g δ (Ω) ⊂ H p (H) and hence (H p (H)) ⊂ Ω. Therefore by the definition of µ δ it follows that Ω ⊂ g −1 δ g δ (Ω) ⊂ g −1 δ1 = P g −1 δ (H p (H)) = µ δ (H p (H)) µ δ H p (H) 1, and the proof is complete since S µ δ is the smallest closed subset of W 0 (H) having µ δ -measure one. We can now state and prove the main result of this section, that is, that the family {g δ } δ>0 is an approximation of the hypoelliptic Brownian motion g in the sense that E max 0 t 1 ρ(g δ (t) −1 g t ) 2 −→ 0 as δ → 0 for any homogeneous norm ρ. As a consequence, the support of the measure µ is contained in H (H). Theorem 3.3. Let {g δ } δ>0 be the sequence defined by (3.5), and ρ be a homogeneous norm on H. Then (3.7) lim δ→0 E max 0 t 1 ρ(g δ (t) −1 g t ) 2 = 0. Proof. By (2.7) and (2.8) we have that max 0 t 1 ρ(g δ (t) −1 g t ) 4 C max 0 t 1 |g −1 δ (t)g t | 4 C max 0 t 1 |B t − B δ (t)| 4 + C max 0 t 1 A t − A δ (t) − 1 2 ω (B δ (t), B t ) 2 C max 0 t 1 |B t − B δ (t)| 2 + max 0 t 1 A t − A δ (t) − 1 2 ω (B δ (t), B t ) 2 , and hence E max 0 t 1 ρ(g δ (t) −1 g t ) 2 CE max 0 t 1 |B t − B δ (t)| 2 + CE max 0 t 1 |A t − A δ (t)| + CE max 0 t 1 1 2 ω (B δ (t), B t ) CE max 0 t 1 |B t − B δ (t)| 2 + CE max 0 t 1 |A t − A δ (t)| 2 1 2 + CE max 0 t 1 1 2 ω (B δ (t), B t ) , for some constant C (which varies from line to line). By (3.1) and (3.2), we only need to show that E max 0 t 1 1 2 ω (B δ (t), B t ) −→ 0, as δ → 0. Since B i is independent of B j,δ − B j when i = j, and max 0 t 1 1 2 ω (B δ (t), B t ) 1 2 max 0 t 1 |B 1 (t)| max 0 t 1 |B 2,δ (t) − B 2 (t)| + 1 2 max 0 t 1 |B 2 (t)| max 0 t 1 |B 1,δ (t) − B 1 (t)|, the proof is complete. Corollary 3.4. We have that µ δ → µ weakly. In particular (3.8) S µ ⊂ H (H). Proof. Let us first show that {g δ } δ>0 converges to g in probability in W 0 (H). For any fixed ε > 0 we have that P max 0 t 1 ρ(g δ (t) −1 g t ) > ε 1 ε 2 E max 0 t 1 ρ(g δ (t) −1 g t ) 2 which goes to zero by Theorem 3.3. Therefore {g δ } δ>0 converges to g in distribution, and hence µ δ converges weakly to µ in W 0 (H). Thus, for any closed set F in W 0 (H) we have that µ(F ) lim sup δ→0 µ δ (F ). In particular, for F = H p (H) and by Proposition 3.2 it follows that µ H p (H) lim sup δ→0 µ δ (H p (H)) = 1. Since S µ is the smallest closed subset having µ-measure one, we have that S µ ⊂ H p (H) = H (H). We conclude this section showing that for each fixed δ, the measures µ and µ δ are singular. Proof. From the proof of Proposition 3.2 we know that µ δ (H (H)) = 1. It is then enough to show that µ (H (H)) = 0. Let us denote by ν the law of a two-dimensional standard Brownian motion. By definition of g t and π (2.10), the following diagram commutes Ω W 0 (H) W 0 R 2 , g B π and for any Borel set E in W 0 R 2 we have that ν(E) := P B −1 (E) = P g −1 • π −1 (E) = µ π −1 (E) . Moreover, from Remark 2.14 we know that π (H (H)) is the Cameron-Martin space over R 2 , which is known to have ν-measure zero, see [4] for more details. Therefore we can conclude that µ (H (H)) µ π −1 π (H (H)) = ν (π (H (H))) = 0. 3.2. Support of the Wiener measure. The goal of this section is to prove that H (H) ⊂ S µ which will follow from Theorem 3.6. Moreover, in Proposition 3.10 we show that H (H) = W 0 (H). Theorem 3.6. Let φ = (π(φ), φ 3 ) =∈ H p (H). For δ > 0 let us denote by E δ,φ the event E δ,φ := max 0 t 1 |B t − π(φ)(t)| R 2 < δ . Then for any ε > 0 lim δ→0 P max 0 t 1 ρ(φ(t) −1 g t ) > ε | E δ,φ = 0, where ρ is a homogeneous norm on H. Proof. By (2.7) it is enough to prove it for the homogeneous norm given by (2.8). For φ ∈ H p (H) we have that max 0 t 1 |φ(t) −1 g t | 4 max 0 t 1 |B t − π(φ)(t)| 4 R 2 + max 0 t 1 1 2 t 0 ω B s − π(φ)(s), dB s − π(φ) ′ (s)ds + t 0 ω B s − π(φ)(s), π(φ) ′ (s) ds 2 max 0 t 1 |B t − π(φ)(t)| 2 R 2 + max 0 t 1 1 2 t 0 ω B s − π(φ)(s), dB s − π(φ) ′ (s)ds + t 0 ω B s − π(φ)(s), π(φ) ′ (s) ds 2 . Therefore on the event E δ,φ we have that max 0 t 1 |φ(t) −1 g t | 2 max 0 t 1 |B t − π(φ)(t)| 2 R 2 + max 0 t 1 1 2 t 0 ω B s − π(φ)(s), dB s − π(φ) ′ (s)ds + t 0 ω B s − π(φ)(s), π(φ) ′ (s) ds δ 2 + max 0 t 1 t 0 ω B s − π(φ)(s), π(φ) ′ (s) ds + max 0 t 1 1 2 t 0 ω B s − π(φ)(s), dB s − π(φ) ′ (s)ds δ 2 + δC φ + max 0 t 1 1 2 t 0 ω B s − π(φ)(s), dB s − π(φ) ′ (s)ds , where C φ := 1 0 |φ ′ 1 (s)| + |φ ′ 2 (s)|ds. It then follows that P max 0 t 1 |φ(t) −1 g t | > ε |E δ,φ P max 0 t 1 1 2 t 0 ω B s − π(φ)(s), dB s − π(φ) ′ (s)ds > ε 2 − δC φ − δ 2 | E δ,φ . Note that this last expression only depends on the process B t − π(φ)(t). Since φ = (π(φ), φ 3 ) ∈ H p (H), by Remark 2.14 we know that π(φ) belongs to the Cameron-Martin space over R 2 . Therefore from the Cameron-Martin-Girsanov Theorem there exists a probability measure Q φ such that the process B φ t := B t + π(φ)(t) is a Brownian motion under Q φ . More precisely there exists an exponential martingale E φ such that Q φ (A) = E E φ ½ A ∀A ∈ F, where E φ = exp − 1 0 π(φ) ′ (s), dB s R 2 ds − 1 2 1 0 |π(φ) ′ (s)| 2 R 2 ds . Note that d (B t − π(φ)(t)) = dB t − π(φ) ′ (t)dt, and dB t = dB φ t − π(φ) ′ (t)dt, that is, the law of B t − π(φ)(t) under P is the same as the law of B t under Q φ . Therefore we can write P (E δ,φ ) = P max 0 t 1 |B t − π(φ)(t)| R 2 < δ = Q φ max 0 t 1 |B t | R 2 < δ = E E φ ½ E δ = E E φ |E δ P (E δ ) , where we set E δ := {max 0 t 1 |B t | R 2 < δ}. Similarly we have that P max 0 t 1 1 2 t 0 ω B s − π(φ)(s), dB s − π(φ) ′ (s)ds > ε 2 − δC φ − δ 2 , E δ,φ = E E φ |F ε δ,φ ∩ E δ P F ε δ,φ ∩ E δ , where F ε δ,φ := max 0 t 1 1 2 t 0 ω (B s , dB s ) > ε 2 − δC φ − δ 2 . Therefore it follows that P max 0 t 1 |φ(t) −1 g t | > ε |E δ,φ P max 0 t 1 1 2 t 0 ω B s − π(φ)(s), dB s − π(φ) ′ (s)ds > ε 2 − δC φ − δ 2 |E δ,φ = P F ε δ,φ ∩ E δ E E φ |F ε δ,φ ∩ E δ P (E δ ) E [E φ |E δ ] = P F ε δ,φ | E δ × E E φ |F ε δ,φ ∩ E δ E [E φ |E δ ] (3.9) We will show later in the paper, see Lemma 3.9, that for any ε > 0 and any φ ∈ H p (H) we have that (3.10) lim δ→0 E E φ | F ε δ,φ ∩ E δ E [E φ | E δ ] = 1. In light of 3.9 and 3.10, the proof will be completed once we show that lim δ→0 P F ε δ,φ | E δ := lim δ→0 P max 0 t 1 1 2 t 0 ω (B s , dB s ) > ε 2 − δC φ − δ 2 max 0 t 1 |B t | R 2 < δ = 0. The process A t := 1 2 t 0 ω (B s , dB s ) is a square integrable martingale with zero mean, and therefore there exists a one dimensional Brownian motion b t such that b τ (t) = 1 2 t 0 ω (B s , dB s ) , where τ (t) = 1 4 t 0 B 1 (s) 2 + B 2 (s) 2 ds. Moreover it is known that b t is inde- pendent of B t [P max 0 t 1 |b τ (t) | > ε 2 − δC φ − δ 2 |, max 0 t 1 |B t | R 2 < δ P max 0 t 1 4 δ 2 |b t | > ε 2 − δC φ − δ 2 |, max 0 t 1 |B t | R 2 < δ = P max 0 t 1 4 δ 2 |b t | > ε 2 − δC φ − δ 2 = P max 0 t 1 |b t | > 2( ε 2 δ − C φ − δ) , which goes to zero as δ goes to zero. Proof. Let us first prove that for any φ ∈ H p (H) and ε > 0 we have that µ (B ε (φ)) > 0, where B ε (φ) := γ ∈ W 0 (H) , max 0 t 1 |φ −1 (t)γ(t)| < ε Indeed, for any φ ∈ H p (H) and ε > 0 we have that µ (B ε (φ)) := P (g ∈ B ε (φ)) = P max 0 t 1 |φ(t) −1 g t | < ε P max 0 t 1 |φ(t) −1 g t | < ε | E δ,φ P (E δ,φ ) , where E δ,φ := {max 0 t 1 |B t − π(φ)(t)| R 2 < δ} . From Theorem (3.6) there exists a δ 0 such that for every δ ∈ (0, δ 0 ) P max 0 t 1 |φ(t) −1 g t | < ε | E δ,φ 1 2 , for any ε > 0. Combining everything together we have that µ (B ε (φ)) 1 2 P sup 0 t 1 |B t − π(φ)(t)| R 2 < δ 0 2 ,O ⊂ H p (H) c , that is, S µ := F closed µ(F )=1 F ⊃ H p (H) , and since S µ is closed, we have that S µ ⊃ H p (H) = H (H). The proof of Theorem 3.6 will be completed once we show (3.10). Before proceeding to the proof of (3.10), we need the following lemma whose proof can be found in [12, pp. 536-537]. Lemma 3.8 (pp. 536-537 in [12]). Let I 1 , . . . , I n be n random variables on a probability space (Ω, F, P). Let {A δ } 0<δ<1 be a family of events in F and a 1 , . . . , a n be n numbers. If for every real number c and every 1 i n lim sup δ→0 E [exp(c I i ) |A δ ] exp(c a i ), then lim δ→0 E exp n i=1 I i |A δ = exp n i=1 a i . Lemma 3.9. Let E δ and F ε δ,φ be given as in the proof of Theorem 3.6. Then lim δ→0 E E φ | F ε δ,φ ∩ E δ E [E φ | E δ ] = 1. Proof. Let us first prove that (3.11) lim δ→0 E E φ | E δ = exp − 1 2 1 0 |π(φ) ′ (s)| 2 R 2 ds . Since E φ = exp − 1 0 π(φ) ′ (s), dB s R 2 ds − 1 2 1 0 |π(φ) ′ (s)| 2 R 2 ds , by Lemma 3.8 and the definition of E δ , it is enough to show that for any real number c and i = 1, 2 lim sup δ→0 E exp −c 1 0 φ ′ i (s)dB i (s) max 0 t 1 |B t | R 2 < δ 1. For φ ∈ H p (H) we can write In a similar way it can be shown that 1 0 φ ′ i (s)dB i (s) = φ ′ i (1)B i (1) − 1 0 φ ′′ i (s)B i (lim δ→0 E E φ | F ε δ,φ ∩ E δ = exp − 1 2 1 0 |π(φ) ′ (s)| 2 R 2 ds , and the proof is completed. The following Proposition completes the proof of Theorem 2.15. Proof. Any element in W 0 (H) can be approximated with piecewise linear curves in the uniform topology. It is then enough to prove that for any piecewise linear curve ξ there exists a sequence of horizontal finite energy curves φ ξ n n∈N such that d W 0 (H) φ ξ n , ξ → 0. Let us first explain the geometric construction through the following example. Consider the curve t → ξ(t) = (0, 0, t) ∈ H for t ∈ [0, 1], which is the prototype of a nonhorizontal curve. Let us define a family of finite energy horizontal curves φ n by φ n (s) := 2 n cos n 2 s , 1 n sin n 2 s , s . Geometrically, the curves φ n are helics that shrink around the ξ as n goes to infinity. Indeed, −→ 0, as n goes to infinity. Now, let ξ(t) = (a 1 t, a 2 t, a 3 t) be a linear curve in H, where a 1 , a 2 , a 3 ∈ R. Then set φ n (s) := a 1 s + 2 n cos n 2 a 3 s , a 2 s + 1 n sin n 2 a 3 s , a 3 s − a 2 s n cos n 2 a 3 s + a 1 s 2n sin n 2 a 3 s + 1 n s 0 2a 2 cos n 2 a 3 u − a 1 sin n 2 a 3 u du . It is easy to check that for any n ∈ N, φ n is a finite energy horizontal curve such that (φ −1 n ξ)(s) = − 2 n cos n 2 a 3 s , − 1 n sin n 2 a 3 s , 1 n s 0 a 1 sin n 2 a 3 u − 2a 2 cos n 2 a 3 u du , which implies that d W 0 (H) (φ n , ξ) → 0 as n goes to infinity. Notation 2. 10 ( 10Standard Wiener space). We denote by W 0 (R n ) the space of R n -valued continuous functions on [0, 1] starting at 0. This space comes with the norm Definition 2.11 (Wiener space over H). The Wiener space over H, denoted by W 0 (H), is the space of H-valued continuous functions on [0, 1] starting at identity in H. B 1,δ (s)B ′ 2,δ (s) − B 2,δ (s)B ′ 1,δ (s) ds.Let f 1 and f 2 be differentiable functions on [0, 1] such that f i Proposition 3 . 5 . 35For each δ the measures µ and µ δ are singular. Corollary 3. 7 . 7H (H) ⊂ S µ . Proposition 3 . 10 . 310We have that H (H) = W 0 (H). H} to hold. We prove (1.4) for the hypoelliptic Brownian motion. Our proof is explicit and it relies on the group structure of H. The main results of this paper are contained in Theorem 2.15, where we prove (1.3) and (1.4) for the hypoelliptic Brownian motion on the Heisenberg group. More precisely, if H (H) denotes the set of finite energy horizontal curves, then we prove that∞ 2.4.Main result. Now we have all the ingredients needed to state the main result of this paper, that is, we describe the support of the Wiener measure for the hypoelliptic Brownian motion g t in terms of horizontal paths.Theorem 2.15. Let W 0 (H) be the Wiener space over H, and µ be the horizontal Wiener measure on W 0 (H), and H (H) be the space of horizontal curves with finite energy. Then 12, Chapter 6 p. 470]. Hence we have that and the latter is positive since π(φ) is in the Cameron-Martin space over R 2 . Therefore, if O is any open set in W 0 (H) with µ(O) = 0 then O ⊂ H p (H) c , and henceO open µ(O)=0 s)ds, and hence on the event E δ we have that exp −c for some finite constant k φ only depending on φ. Therefore we have that|B t | R 2 < δ E lim sup δ→0 e −ck φ δ |E δ 1.1 0 φ ′ i (s)dB i (s) exp (−ck φ δ) , lim sup δ→0 E exp −c 1 0 φ ′ i (s)dB i (s) max 0 t 1 Acknowledgement. The author wishes to thank M. 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[ "INTRODUCING A NEW INTRINSIC METRIC", "INTRODUCING A NEW INTRINSIC METRIC" ]
[ "Oona Rainio ", "Matti Vuorinen " ]
[]
[]
A new intrinsic metric called t-metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains G R n . The behaviour of the new metric is also studied under a few examples of conformal and quasiconformal mappings, and the differences between the balls drawn with all the metrics considered are compared by both graphical and analytical means.
10.1007/s00025-021-01592-2
[ "https://export.arxiv.org/pdf/2010.01984v1.pdf" ]
222,134,186
2010.01984
af80166467f5f0fcdf5e748b30ca21b0a8910f65
INTRODUCING A NEW INTRINSIC METRIC Oona Rainio Matti Vuorinen INTRODUCING A NEW INTRINSIC METRIC A new intrinsic metric called t-metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains G R n . The behaviour of the new metric is also studied under a few examples of conformal and quasiconformal mappings, and the differences between the balls drawn with all the metrics considered are compared by both graphical and analytical means. Introduction In geometric function theory, one of the topics studied deals with the variation of geometric entities such as distances, ratios of distances, local geometry and measures of sets under different mappings. For such studies, we need an appropriate notion of distance that is compatible with the class of mappings studied. In classical function theory of the complex plane, one of the key concepts is the hyperbolic distance, which measures not only how close the points are to each other but also how they are located inside the domain with respect to its boundary. The hyperbolic distance also serves as a model when we need generalisations to subdomains G of arbitrary metric spaces X. These generalized distances behave like the hyperbolic metric in the aspect that they define the Euclidean topology and, in particular, we can cover compact subsets of G using balls of the generalized metrics. Thus, the boundary of the domain has a strong influence on the inner geometry of the domain defined by some chosen metric. Since the classical hyperbolic geometry acts as a model, some of its key features are inherited by the generalizations but not all. For instance, it is desirable to study local behaviour of functions and we need to have a metric that is locally comparable with the Euclidean geometry. Such a metric is here called an intrinsic metric. Note that there is no established definition for this concept and it is sometimes required, for instance, that the closures of the balls defined with an intrinsic metric never intersect the boundary of the domain. An example of an intrinsic metric is the following new metric, on which this work focuses. File: main.tex, printed: 2021-12-3, 18.46 Definition 1.1. Let G be some non-empty, open, proper and connected subset of a metric space X. Choose some metric η G defined in the closure of G and denote η G (x) = η G (x, ∂G) = inf{η G (x, z) | z ∈ ∂G} for all x ∈ G. The t-metric for a metric η G in a domain G is a function t G : G × G → [0, 1], t G (x, y) = η G (x, y) η G (x, y) + η G (x) + η G (y) , for all x, y ∈ G. Here, we mostly focus on the special case where G R n and η G is the Euclidean distance. Our work in this paper is motivated by the research of several other mathematicians. During the past thirty years, many intrinsic metrics have been introduced and studied [3,4,7,10,11]. It is noteworthy that each metric might be used to discover some intricate features of mappings not detected by other metrics. Since our new metric differs slightly from other intrinsic metrics and has a relatively simple definition, it could potentially be a great help for new discoveries about intrinsic geometry of domains. For instance, there is one inequality that is an open question for the triangular ratio metric, but could potentially be proved for the t-metric, see Conjecture 4.13 and Remark 4.14. Unlike several other hyperbolic type metrics, such as the triangular ratio metric or the hyperbolic metric itself, the t-metric does not have the property about the closed balls never intersecting with the boundary, see Theorem 5.5. This is an interesting aspect for this metric clearly fulfills most of the others, if not all, properties of a hyperbolic type metrics listed in [7, p. 79]. Consequently, we have found an intrinsic metric that does not have one of the common properties of hyperbolic type metrics. In this paper, we will study this new metric and its connection to other metrics. In Section 3, we prove that the function of Definition 1.1 is really a metric and find the sharp inequalities between this metric and several hyperbolic type metrics, including also the hyperbolic metric, in different domains. In Section 4, we show how the t-metric behaves under certain quasiconformal mappings and find the Lipschitz constants for Möbius maps between balls and half-spaces. Finally, in Section 5, we draw t-metric disks and compare their certain properties to those of other metric disks. Acknowledgements. The research of the first author was supported by Finnish Concordia Fund. Preliminaries In this section, we will introduce the definitions of a few different metrics and metric balls that will be necessary later on but, first, let us recall the definition of a metric. Definition 2.1. For any non-empty space G, a metric is a function η G : G × G → [0, ∞) that fulfills the following three conditions for all x, y, z ∈ G: (1) Positivity: η G (x, y) ≥ 0, and η G (x, y) = 0 if and only if x = y, (2) Symmetry: η G (x, y) = η G (y, x), (3) Triangle inequality: η G (x, y) ≤ η G (x, z) + η G (z, y). Let η G be now some arbitrary metric. An open ball defined with it is B η (x, r) = {y ∈ G | η G (x, y) < r} and the corresponding closed ball is B η (x, r) = {y ∈ G | η G (x, y) ≤ r}. Denote the sphere of these balls by S η (x, r). For Euclidean metric, these notations are B n (x, r), B n (x, r) and S n−1 (x, r), respectively, where n is the dimension. In this paper, the unit ball B n = B n (0, 1), the upper half-plane H n = {(x 1 , ..., x n ) ∈ R n | x n > 0} and the open sector S θ = {x ∈ C | 0 < arg(x) < θ} with an angle θ ∈ (0, 2π) will be commonly used as domains G. Note also that the unit vectors will be denoted by {e 1 , ..., e n }. Let us now define the metrics needed for a domain G R n . Denote the Euclidean distance between the points x, y by |x − y| and let d G (x) = inf{|x − z| | z ∈ ∂G}. Suppose that the t-metric is defined with the Euclidean distance so that t G (x, y) = |x − y| |x − y| + d G (x) + d G (y) for all x, y ∈ G, if not otherwise specified. The following hyperbolic type metrics will be considered: The triangular ratio metric: s G : G × G → [0, 1], s G (x, y) = |x − y| inf z∈∂G (|x − z| + |z − y|) , the j * G -metric: j * G : G × G → [0, 1], j * G (x, y) = |x − y| |x − y| + 2 min{d G (x), d G (y)} , and the point pair function: p G : G × G → [0, 1], p G (x, y) = |x − y| |x − y| 2 + 4d G (x)d G (y) . Out of these hyperbolic type metrics, the triangular ratio metric was studied by P. Hästö in 2002 [9], and the two other metrics are more recent. As pointed out in [8], the j * G -metric is derived from the distance ratio metric found by F.W. Gehring and B.G. Osgood in [6]. Note that there are proper domains G in which the point pair function is not a metric [3, Rmk 3.1 p. 689]. Define also the hyperbolic metric as chρ H n (x, y) = 1 + |x − y| 2 2d H n (x)d H n (y) , x, y ∈ H n , sh 2 ρ B n (x, y) 2 = |x − y| 2 (1 − |x| 2 )(1 − |y| 2 ) , x, y ∈ B n in the upper half-plane H n and in the Poincaré unit disk B n [7, (4.8), p. 52 & (4.14), p. 55]. In the two-dimensional space, th ρ H 2 (x, y) 2 = th 1 2 log |x − y| + |x − y| |x − y| − |x − y| = x − y x − y , th ρ B 2 (x, y) 2 = th 1 2 log |1 − xy| + |x − y| |1 − xy| − |x − y| = x − y 1 − xy = |x − y| A[x, y] ,(x, y) ≤ s G (x, y) ≤ 2j * G (x, y) holds for all x, y ∈ G. Lemma 2.4. [7, p. 460] For all x, y ∈ G ∈ {H n , B n },(1)th ρ H n (x, y) 4 ≤ j * H n (x, y) ≤ s H n (x, y) = p H n (x, y) = th ρ H n (x, y) 2 ≤ 2th ρ H n (x, y) 4 ,(2)th ρ B n (x, y) 4 ≤ j * B n (x, y) ≤ s B n (x, y) ≤ p B n (x, y) ≤ th ρ B n (x, y) 2 ≤ 2th ρ B n (x, y) 4 . t-Metric and Its Bounds Now, we will prove that our new metric is truly a metric in the general case. Theorem 3.1. For any metric space X, a domain G X and a metric η G defined in G, the function t G is a metric. Proof. The function t G is a metric if it fulfills all the three conditions of Definition 2.1. Trivially, the first two conditions hold. Consider now a function f : [0, ∞) → [0, ∞), f (x) = x/(x + k) where k > 0 is a constant. Since f is increasing in its whole domain [0, ∞), x ≤ y ⇔ x x + k ≤ y y + k . Because η G is a metric, η G (x, y) ≤ η G (x, z) + η G (z, y) for all x, y, z ∈ G. Furthermore, η G (z) ≤ min{η G (x, z) + η G (x), η G (z, y) + η G (y)}. From these results, it follows that t G (x, y) = η G (x, y) η G (x, y) + η G (x) + η G (y) ≤ η G (x, z) + η G (z, y) η G (x, z) + η G (z, y) + η G (x) + η G (y) = η G (x, z) η G (x, z) + η G (z, y) + η G (x) + η G (y) + η G (z, y) η G (x, z) + η G (z, y) + η G (x) + η G (y) ≤ η G (x, z) η G (x, z) + η G (x) + η G (z) + η G (z, y) η G (z, y) + η G (y) + η G (z) = t G (x, z) + t G (z, y) for all x, y, z ∈ G. Thus, t G fulfills the triangle inequality. We now show that the method of proof of Theorem 3.1 can be used to prove that several other functions are metrics, too. Theorem 3.2. If G is a proper subset of a metric space X, η G some metric defined in the closure of G and c G : G × G → [0, ∞) some symmetric function such that, for all x, y, z ∈ G, c G (x, z) ≤ η G (z, y) + c G (x, y), (3.3) then any function φ G : G × G → [0, 1], defined as φ G (x, x) = 0, φ G (x, y) = η G (x, y) η G (x, y) + c G (x, y) if x = y for all x, y ∈ G, is a metric in the domain G. Proof. Since η G is a metric and c G is both symmetric and non-negative, the function φ G trivially fulfills the first two conditions of Definition 2.1. Note that, by the triangle inequality of the metric η G and the inequality (3.3), the inequalities η G (x, y) ≤ η G (x, z) + η G (z, y), c G (x, z) ≤ η G (z, y) + c G (x, y), c G (z, y) ≤ η G (x, z) + c G (x, y), hold for all x, y, z ∈ G. Now, φ G (x, y) = η G (x, y) η G (x, y) + c G (x, y) ≤ η G (x, z) + η G (z, y) η G (x, z) + η G (z, y) + c G (x, y) = η G (x, z) η G (x, z) + η G (z, y) + c G (x, y) + η G (z, y) η G (x, z) + η G (z, y) + c G (x, y) ≤ η G (x, z) η G (x, z) + c G (x, z) + η G (z, y) η G (z, y) + c G (z, y) = φ G (x, z) + φ G (z, y), so the function φ G fulfills the triangle inequality and it must be a metric. Remark 3.4. (1) If the function c G of Theorem 3.2 is strictly positive, the condition φ G (x, x) = 0 does not need to be separately specified. Namely, this condition follows directly from the fact that η G (x, x) = 0 for a metric η G . Note also that if c G is a null function, the function φ G becomes the discrete metric. (2) If η G is a metric, then η α G is a metric, too, for 0 < α ≤ 1, but this is not true for α > 1 [7, Ex. 5.24, p. 80]. Corollary 3.5. The function ψ : B n × B n → [0, 1], defined as ψ(x, x) = 0, ψ(x, y) = |x − y| |x − y| + c|x||y| if x = y, for all x, y ∈ B n with a constant 0 < c ≤ 1, is a metric on the unit ball. Proof. Since now c|x|(|z| − |y|) ≤ |z| − |y| ≤ |z − y| ⇒ c|x||z| ≤ |z − y| + c|x||y|, for all x, y, z ∈ B n , the result follows from Theorem 3.2. Corollary 3.6. If G is a proper subset of a metric space X and η G is some metric defined in the closure of G such that η G (x) = inf{η G (x, u) | u ∈ ∂G} ≤ 1 for all x ∈ G, then a function υ G : G × G → [0, 1], defined as υ G (x, y) = η G (x, y) η G (x, y) + c (1 + η G (x))(1 + η G (y)) with a constant 0 < c ≤ √ 2 is a metric in the domain G. Proof. Fix c G (x, y) = c (1 + η G (x))(1 + η G (y)). Now, c G (x, z) − c G (x, y) = c (1 + η G (x))(1 + η G (z)) − c (1 + η G (x))(1 + η G (y)) = c · (1 + η G (x))(1 + η G (z)) − (1 + η G (x))(1 + η G (y)) (1 + η G (x))(1 + η G (z)) + (1 + η G (x))(1 + η G (y)) = c 1 + η G (x)(η G (z) − η G (y)) 1 + η G (z) + 1 + η G (y) ≤ √ 2c(η G (z) − η G (y)) 1 + 1 ≤ η G (z) − η G (y) ≤ η G (z, y), so the inequality (3.3) holds for all x, y, z ∈ G and the result follows from Theorem 3.2. Corollary 3.7. The function χ : B n × B n → [0, 1], defined as χ(x, y) = |x − y| |x − y| + c (2 − |x|)(2 − |y|) for all x, y ∈ B n with a constant 0 < c ≤ √ 2, is a metric on the unit ball. Proof. Follows from Corollary 3.6. Let us focus again on the t-metric. Since the result of Theorem 3.1 holds for any metric η G , the t-metric is trivially a metric also when defined for the Euclidean metric. Below, we will consider the t-metric in this special case only. Let us next prove the inequalities between the t-metric and the three hyperbolic type metrics defined earlier. Theorem 3.8. For all domains G R n and all points x, y ∈ G, the following inequalities hold: (1) j * G (x, y)/2 ≤ t G (x, y) ≤ j * G (x, y), (2) p G (x, y)/2 ≤ t G (x, y) ≤ p G (x, y), (3) s G (x, y)/2 ≤ t G (x, y) ≤ s G (x, y). Furthermore, in each case the constants are sharp for some domain G. Proof. (1) t G (x, y) ≤ j * G (x, y) follows trivially from the definitions of these metrics. If d G (x) ≤ d G (y), then d G (y) ≤ |x − y| + d G (x) and d G (x) + d G (y) ≤ |x − y| + 2d G (x). In the same way, if d G (y) ≤ d G (x), then d G (x) + d G (y) ≤ |x − y| + 2d G (y). It follows from this that d G (x) + d G (y) ≤ |x − y| + 2 min{d G (x), d G (y)}. With this information, we can write j * G (x, y) = |x − y| |x − y| + 2 min{d G (x), d G (y)} = 2|x − y| |x − y| + 2 min{d G (x), d G (y)} + |x − y| + 2 min{d G (x), d G (y)} ≤ 2|x − y| |x − y| + 2 min{d G (x), d G (y)} + d G (x) + d G (y) ≤ 2|x − y| |x − y| + d G (x) + d G (y) = 2t G (x, y). The equality t G (x, y) = j * G (x, y) holds always when d G (x) = d G (y). For x = i and y = ki, lim k→0 + (t H 2 (x, y)/j * H 2 (x, y)) = 1/2. Thus, the first inequality of the theorem and its sharpness follow. (2) From Lemma 2.2 and Theorem 3. 8(1), it follows that t G (x, y) ≤ p G (x, y). Let us now prove that p G (x, y)/2 ≤ t G (x, y). This is clearly equivalent to |x − y| + d G (x) + d G (y) ≤ 2 |x − y| 2 + 4d G (x)d G (y). (3.9) Fix u = |x − y|, v = min{d G (x), d G (y)} and k = |d G (x) − d G (y)|. The inequality (3.9) is now u + 2v + k ≤ 2 u 2 + 4v(v + k) ⇔ k 2 + 4uv + 2uk − 3u 2 − 12v 2 − 12kv ≤ 0. Define a function f (k) = k 2 + 4uv + 2uk − 3u 2 − 12v 2 − 12kv. Since the inequality above is equivalent to f (k) ≤ 0, we need to find out the greatest value of this function. There is no upper limit for u ≥ 0 or v ≥ 0 but 0 ≤ k ≤ u. We can solve that f (k) = 2k + 2u − 12v = 0 ⇔ k = 6v − u. Since f (0) = −3u 2 + 4uv − 12v 2 ≤ −2u 2 − 8v 2 ≤ 0, f (6v − u) = −4u 2 + 16uv − 42v 2 ≤ −26v 2 ≤ 0, f (u) = −8uv − 12v 2 ≤ 0, f (k) is always non-positive on the closed interval k ∈ [0, u] and, consequently, the in- equality p G (x, y)/2 ≤ t G (x, y) follows. For x = ki and y = i, lim k→0 + (t H 2 (x, y)/p H 2 (x, y)) = 1/2 and, for x = ki and y = 1+ki, lim k→0 + (t H 2 (x, y)/p H 2 (x, y)) = 1. (3) By the triangle inequality and Lemma 2.3 and Theorem 3.8(1), y). For x = ki and y = i, lim k→0 + (t H 2 (x, y)/s H 2 (x, y)) = 1/2 and, for x = ki and y = 1 + ki, lim k→0 + (t H 2 (x, y)/s H 2 (x, y)) = 1. Proof. If G is a proper subdomain, there must exist some ball B n (x, r) ⊂ G with S n−1 (x, r)∩ ∂G = ∅ where r > 0. Fix z ∈ S n−1 (x, r) ∩ ∂G and y ∈ [x, z] so that |y − z| = kr with k ∈ (0, 1). Clearly, d G (x) = r, d G (y) = kr, |x−y| = (1−k)r and inf z∈∂G (|x−z|+|z −y|) = 1 + k. Consequently, s G (x, y) 2 = |x − y| inf z∈∂G (|x − z| + |z − y|) + inf z∈∂G (|x − z| + |z − y|) ≤ |x − y| inf z∈∂G (|x − z| + |z − y|) + d G (x) + d G (y) ≤ t G (x, y) ≤ j * G (x, y) ≤ s G (x,t G (x, y) = (1 − k)r (1 − k)r + r + kr = 1 − k 2 , j * G (x, y) = p G (x, y) = s G (x, y) = 1 − k 1 + k . It follows that lim k→0 + t G (x, y) j * G (x, y) = lim k→0 + t G (x, y) j * G (x, y) = lim k→0 + t G (x, y) j * G (x, y) = lim k→0 + 1 + k 2 = 1 2 , lim k→1 − t G (x, y) j * G (x, y) = lim k→1 − t G (x, y) j * G (x, y) = lim k→1 − t G (x, y) j * G (x, y) = lim k→1 − 1 + k 2 = 1. Thus, regardless of how G is chosen, the inequalities of Theorem 3.8 are sharp. Next, we will study the connection between the t-metric and the hyperbolic metric. Theorem 3.11. For all x, y ∈ G ∈ {H n , B n }, the inequality Proof. In the case G = H n , the result follows directly from Lemma 2.4(1) and Theorem 3.8 (3). 1 2 th ρ G (x, y) 2 ≤ t G (x, y) ≤ th ρ G (x, Suppose now that G = B n . From Lemma 2.4(2) and Theorem 3.8(1), it follows that t B n (x, y) ≤ j * B n (x, y) ≤ th ρ B n (x, y) 2 . Thus, the inequality t B n (x, y) ≤ th(ρ B n (x, y)/2) holds for all x, y ∈ B n and the sharpness follows because x = ke 1 and y = −ke 1 fulfill lim k→1 − (t B n (x, y)/th(ρ B n (x, y)/2)) = 1. Since the values of the t-metrics and the hyperbolic metric in the domain B n only depend on how the points x, y are located on the two-dimensional plane fixed by these two points and the origin, we can assume without loss of generality that n = 2. Consider now the quotient t B 2 (x, y) th(ρ B 2 (x, y)/2) = A[x, y] |x − y| + 1 − |x| + 1 − |y| .t B 2 (x, y) th(ρ B 2 (x, y)/2) ≥ |x − y| + 1 − |x| − |y| + |x||y| |x − y| + 1 − |x| + 1 − |y| ≥ |x − y| − |x| − |y| + 1 |x − y| − |x| − |y| + 2 ≥ 1 2 , which proves that th(ρ B 2 (x, y)/2)/2 ≤ t B 2 (x, y). By the observation above, this also holds in the more general case where n is not fixed. The inequality is sharp, too: For x = ke 1 and y = −ke 1 , lim k→0 + (t B n (x, y)/th(ρ B n (x, y)/2)) = 1/2. Theorem 3.12. For a fixed angle θ ∈ (0, 2π) and for all x, y ∈ S θ , the following results hold: (1) t S θ (x, y) ≤ th(ρ S θ (x, y)/2) ≤ 2(π/θ) sin(θ/2)t S θ (x, y) if θ ∈ (0, π), (2) t S θ (x, y) ≤ th(ρ S θ (x, y)/2) ≤ 2t S θ (x, y) if θ = π,(3)(π/θ)t S θ (x, y) ≤ th(ρ S θ (x, y)/2) ≤ 2t S θ (x, y) if θ ∈ (π, 2π). Proof. Follows from Theorems 3.8(3) and 3.11, and [12, Cor. 4.9, p. 9]. Quasiconformal Mappings and Lipschitz Constants In this section, we will study the behaviour of the t-metric under different conformal and quasiconformal mappings in order to demonstrate how this metric works. Remark 4.1. The t-metric is invariant under all similarity maps. In particular, the tmetric defined in a sector S θ is invariant under a reflection over the bisector of the sector and a stretching x → r · x with any r > 0. Consequently, this allows us to make certain assumptions when choosing the points x, y ∈ S θ . First, let us study how the t-metric behaves under a certain conformal mapping between two sectors with angles at most π. Lemma 4.2. If α, β ∈ (0, π] and f : S α → S β , f (z) = z (β/α) , then for all x, y ∈ S α t Sα (x, y) 2 ≤ t S β (f (x), f (y)) ≤ β sin(α/2) α sin(β/2) t Sα (x, y) if α ≤ β, β sin(α/2) α sin(β/2) t Sα (x, y) ≤ t S β (f (x), f (y)) ≤ 2t Sα (x, y) otherwise. Proof. If α ≤ β, by Theorem 3.8(3) and [12, Lemma 5.11, p. 13], t S β (f (x), f (y)) ≥ s S β (f (x), f (y)) 2 ≥ s Sα (x, y) 2 ≥ t Sα (x, y) 2 . (4.3) Suppose that α ≤ β still. Fix x = e hi and y = re ki , where 0 < h ≤ k < α and r ≥ 1 without loss of generality. Consider the quotient t S β (f (x), f (y)) t Sα (x, y) = |1 − (re (k−h)i ) β α |(|1 − re (k−h)i | + sin(γ) + r sin(µ)) |1 − re (k−h)i |(|1 − (re (k−h)i ) β α | + sin( β α γ) + r β α sin( β α µ)) , (4.4) where µ = min{h, α − h} and γ = min{k, α − k}. This quotient is strictly decreasing with respect to r and, since r ≥ 1, it attains its maximum value when r = 1. Consequently, the quotient (4.4) has an upper limit of sin( β 2α (k − h))(2 sin( k−h 2 ) + sin(γ) + sin(µ)) sin( k−h 2 )(2 sin( β 2α (k − h)) + sin( β α γ) + sin( β α µ)) . (4.5) The value of the quotient above is at greatest, when k − h is at minimum and both γ and µ are at maximum. This happens when h < α/2 and k = α − h. Now, γ = µ = h and the quotient (4.5) is sin( β 2α (α − 2h))(sin( α 2 − h) + sin(h)) sin( α 2 − h)(sin( β 2α (α − 2h)) + sin( β α h)) . Since the expression above is strictly increasing with respect to h and h < α/2, the maximum value of the quotient (4.4) is lim h→ α 2 − sin( β 2α (α − 2h))(sin( α 2 − h) + sin(h)) sin( α 2 − h)(sin( β 2α (α − 2h)) + sin( β α h)) = β sin(α/2) α sin(β/2) , (4.6) which, together with the inequality (4.3), proves the first part of our theorem. Suppose next that α > β instead. It can be now proved that the minimum value of the quotient (4.4) is the same limit value (4.6) and, by Theorem 3.8(3) and [12, Lemma 5.11, p. 13], t S β (f (x), f (y)) ≤ 2t Sα (x, y). Thus, the theorem follows. Let us now consider a more general result than the one above. Namely, instead of studying a conformal power mapping, we can assume that, for domains G 1 , G 2 ⊂ R 2 , the mapping f : G 1 → G 2 = f (G 1 ) is a K-quasiconformal homeomorphism, see [13,Ch. 2]. Let c(K) be as in [7,Thm 16.39,p. 313]. Now, c(K) ≥ K and c(K) → 1 whenever K → 1. See also the book [5] by F.W. Gehring and K. Hag. Theorem 4.7. If α, β ∈ (0, 2π) and f : S α → S β = f (S α ) is a K-quasiconformal homeomorphism, the following inequalities hold for all x, y ∈ S α . (1) β 2c(K) K π sin(β/2) t Sα (x, y) K ≤ t S β (f (x), f (y)) ≤ c(K)( π α sin( α 2 )) 1/K t Sα (x, y) 1/K if α, β ∈ (0, π],(2)1 2c(K) K t Sα (x, y) K ≤ t S β (f (x), f (y)) ≤ c(K)π β ( π α sin( α 2 )) 1/K t Sα (x, y) 1/K if α ∈ (0, π) and β ∈ (π, 2π), (3) 1 2 ( α c(K)π ) K t Sα (x, y) K ≤ t S β (f (x), f (y)) ≤ c(K)π β t Sα (x, y) 1/K if α, β ∈ [π, 2π). Proof. Follows from Theorem 3.8(3) and [12,Cor. 5.7,p. 12]. Next, we will focus on the radial mapping, which is another example of a quasiconformal mapping, see [13, 16.2, p. 49]. Theorem 4.8. If f : G → G with G = B 2 \{0} is the radial mapping defined as f (z) = |z| a−1 z for some 0 < a < 1, then for all x, y ∈ G such that |x| = |y|, the sharp inequality t B 2 \{0} (x, y) ≤ t B 2 \{0} (f (x), f (y)) ≤ 1 2 a − 1 t B 2 \{0} (x, y) holds. Proof. Fix x = re ki and y = re −ki with 0 < r < 1 and 0 < k < π/2. Now, f (x) = r a e ki and f (y) = r a e −ki . Consider the quotient t G (f (x), f (y)) t G (x, y) = |f (x) − f (y)|(|x − y| + d G (x) + d G (y)) |x − y|(|f (x) − f (y)| + d G (f (x)) + d G (f (y))) , (4.9) where |x − y| = 2r sin(k), |f (x) − f (y)| = 2r a sin(k), d G (x) = d G (y) = min{r, 1 − r}, d G (f (x)) = d G (f (y)) = min{r a , 1 − r a }. If 0 < r < r a < 1/2, the quotient (4.9) is t G (f (x), f (y)) t G (x, y) = r a (r sin(k) + r) r(r a sin(k) + r a ) = 1. If 0 < r ≤ 1/2 < r a < 1, the quotient (4.9) is t G (f (x), f (y)) t G (x, y) = r a (r sin(k) + r) r(r a sin(k) + 1 − r a ) , (4.10) which is decreasing with respect to k. Since lim k→0 + r a (r sin(k) + r) r(r a sin(k) + 1 − r a ) = r 1+a r(1 − r a ) = r a 1 − r a is increasing with respect to r and r ≤ 1/2, the maximum value of the quotient (4.10) is 1/(2 a − 1). The other limit value lim k→1 − r a (r sin(k) + r) r(r a sin(k) + 1 − r a ) = r a (r + r) r(r a + 1 − r a ) = 2r a is increasing with respect to r a and r a > 1/2, so the quotient (4.10) is always more than 1. If 1/2 < r < r a < 1, the quotient (4.9) is t B 2 \{0} (f (x), f (y)) t B 2 \{0} (x, y) = r a (r sin(k) + 1 − r) r(r a sin(k) + 1 − r a ) , (4.11) which is decreasing with respect to k. Since r > 1/2 and lim k→0 + r a (r sin(k) + 1 − r) r(r a sin(k) + 1 − r a ) = r a (1 − r) r(1 − r a ) , is decreasing with respect to r, the quotient (4.10) is less than 1/(2 a − 1). The other limit value is lim k→1 − r a (r sin(k) + 1 − r) r(r a sin(k) + 1 − r a ) = r a−1 , which is clearly more than 1. Thus, the minimum value of the quotient (4.9) is 1 and the maximum value 1/(2 a − 1), so the theorem follows. Let us now find Lipschitz constants of a few different mappings for the t-metric. G 1 → G 2 = f (G 1 ) with G 1 , G 2 ∈ {H n , B n }, the inequality 1 2 t G 1 (x, y) ≤ t G 2 (f (x), f (y)) ≤ 2t G 1 (x, y) holds for all x, y ∈ G 1 . Proof. By Theorem 3.11 and the conformal invariance of the hyperbolic metric, 1 2 t G 1 (x, y) ≤ 1 2 th ρ G 1 (x, y) 2 = 1 2 th ρ G 2 (f (x), f (y)) 2 ≤ t G 2 (f (x), f (y)) ≤ th ρ G 2 (f (x), f (y)) 2 = th ρ G 1 (x, y) 2 ≤ 2t G 1 (x, y). It follows from Theorem 4.12 that the Lipschitz constant Lip(f |G 1 ) for the t-metric in any conformal mapping f : G 1 → G 2 = f (G 1 ), G 1 , G 2 ∈ {H n , B n }, is at most 2. Suppose now that h is the Möbius transformation h : B 2 → H 2 , h(z) = (1 − z)i/(1 + z). Since, for x = 0 and y = 1−k k+1 with 0 < k < 1, lim k→1 − t H n (h(x) , h(y)) t B n (x, y) = lim k→1 − (k + 1) = 2, the Lipschitz constant Lip(h|B 2 ) is equal to 2. However, for certain Möbius transformations, there might be a better constant than 2. For instance, the following conjecture is supported by several numerical tests. In the next few results, we will study a mapping f * : S θ → S θ , f * (x) = x/|x| 2 defined in some open sector S θ , and find its Lipschitz constants for the t-metric. Theorem 4.15. If θ ∈ (0, π] and f * is the mapping f * : S θ → S θ , f * (x) = x/|x| 2 , the Lipschitz constant Lip(f * |S θ ) for the t-metric is 1 + sin(θ/2). Proof. Without loss of generality, we can fix x = e hi and y = re ki with 0 < h ≤ π/2, h ≤ k < θ and r > 0. Since x * = e hi and y * = (1/r)e ki , it follows that t S θ (x * , y * ) t S θ (x, y) = 1 + r 2 − 2r cos(k − h) + r sin(h) + sin(min{k, θ − k}) 1 + r 2 − 2r cos(k − h) + sin(h) + r sin(min{k, θ − k}) . To maximize this, we clearly need to choose k = θ/2 and make r and h as small as possible. If k = θ/2, lim h→0 + , r→0 + t S θ (x * , y * ) t S θ (x, y) = 1 + sin(θ/2), so the theorem follows. Theorem 4.16. If x * = x/|x| 2 and y * = y/|y| 2 , the equality s S θ (x, y) = s S θ (x * , y * ) holds in an open sector S θ with θ ∈ (0, 2π). Proof. Fix x = e hi and y = re ki where r > 0 and 0 < h ≤ k < θ. Clearly, x * = x = e hi and y * = (1/r)e ki . Suppose first that θ ≤ π. By the known solution to Heron's problem, the infimum inf z∈∂S θ (|x − z| + |z − y|) is min{|x − y|, |x − y |}, where y is the point y reflected over the left side of the sector θ. Clearly, |x − y| ≤ |x − y | ⇔ |e −hi − re ki | ≤ |e hi − re (2θ−k)i | ⇔ |1 − re (h+k)i | ≤ |1 − re (2θ−h−k)i | ⇔ h + k ≤ 2θ − h − k ⇔ (h + k)/2 ≤ θ/2. By symmetry, we can suppose that (h + k)/2 ≤ θ/2 without loss of generality. Note that it follows from above that not only inf z∈∂S θ (|x − z| + |z − y|) = |x − y| but also inf z∈∂S θ (|x * − z| + |z − y * |) = |x * − y * |. Now, s S θ (x, y) = |x − y| |x − y| = |1 − re (k−h)i | |1 − re (k+h)i | = |1 − re (h−k)i | |1 − re −(k+h)i | = |r − e (k−h)i | |r − e (k+h)i | = |1 − (1/r)e (k−h)i | |1 − (1/r)e (k+h)i | = |x * − y * | |x * − y * | = s S θ (x * , y * ). Consider now the case where θ > π. If k − h ≥ π, then s S θ (x, y) = 1 = s S θ (x * , y * ) always so suppose that k − h < π instead. This leaves us three possible options. If (h + k)/2 ≤ π/2, then (h + k)/2 < θ/2 and s S θ (x, y) = |x − y| |x − y| = |x * − y * | |x * − y * | = s S θ (x * , y * ), just like above. By symmetry, s S θ (x, y) = s S θ (x * , y * ) also if (k + h)/2 ≥ θ − π/2. If π/2 < (k + h)/2 < θ − π/2 instead, then s S θ (x, y) = |x − y| |x| + |y| = |1 − re (k−h)i | 1 + r = |r − e (k−h)i | 1 + r = |1 − (1/r)e (k−h)i | 1 + 1/r = |x * − y * | |x * | + |y * | = s S θ (x * , y * ). Theorem 4.17. If θ ∈ [π, 2π) and f * is the mapping f * : S θ → S θ , f * (x) = x/|x| 2 , the Lipschitz constant Lip(f * |S θ ) for the t-metric is 2. Proof. It follows from Theorems 4.16 and 3.8(3) that t S θ (x, y) 2 ≤ t S θ (x * , y * ) ≤ 2t S θ (x, y) for all x, y ∈ S θ . Since for x = e hi and y = re πi/2 with h < π/2 and r > 0, lim h,r→0 + t S θ (x * , y * ) t S θ (x, y) = lim h,r→0 + 1 + r 2 − 2r cos(π/2 − h) + r sin(h) + 1 1 + r 2 − 2r cos(π/2 − h) + sin(h) + r = 2, and it follows that sup{ t S θ (x * , y * ) t S θ (x, y) | x, y ∈ S θ , x = y, θ ∈ [π, 2π)} = 2. Comparison of Metric Balls Next, we will graphically demonstrate the differences and similarities between the various metrics considered in this paper by drawing for each metric several circles centered at the same point but with different radii. In all of the figures of this section, the domain G ⊂ R 2 is a regular five-pointed star and the circles have a radius of r = 1/10, ..., 9/10. The center of these circles is in the center of G in the first figures, and then off the center in the rest of the figures. All the figures in this section were drawn by using the contour plot function contour in R-Studio and choosing a grid of the size 1,000×1,000 test points. While we graphically only inspect circles and disks, we will also prove some properties for the n-dimensional metric balls. For several hyperbolic type metrics, the metric balls of small radii resemble Euclidean balls, but the geometric structure of the boundary of the domain begins to affect the shape of these balls when their radii grow large enough, see [7,Ch. 13,. By analysing this phenomenon more carefully, we can observe, for instance, that the balls are convex with radii less than some fixed r 0 > 0 in the case of some other metrics, see [7,Thm 13.6,p. 241; Thm 13.41 p. 256; Thm 13.44, p. 258]. From the figures of this section, we see that the four metrics studied here share this same property. In particular, we notice that, while the metric disks with small radii are convex and round like Euclidean disks, the metric circles with larger radii are non-convex and have corner points. By a corner point, we mean here such a point on the circle arc that has many possible tangents. In the following theorem, we will prove a property that can be seen from Figures 1c, 1d, 2c and 2d. Theorem 5.1. If the domain G is a polygon, then the corner points of the circles S p (x, r) and S t (x, r) are located on the the angle bisectors of G. Proof. Suppose G has sides l 0 and l 1 that have a common endpoint k. Fix x ∈ G and choose some point y ∈ G so that k is the vertex of G that is closest to y and there is no other side closer to y than l 0 and l 1 . Thus, d G (y) = min{d(y, l 0 ), d(y, l 1 )} and, for a fixed distance |x − y|, d G (y) is at maximum when d(y, l 0 ) = d(y, l 1 ). The condition d(y, l 0 ) = d(y, l 1 ) is clearly fulfilled when y is on the bisector of ∠(l 0 , l 1 ) and, the greater the d G (y), the smaller the distances p G (x, y) and t G (x, y) are now. Consequently, if the (a) s G -metric circles. (b) j * G -metric circles. (c) p G -metric circles. circle S p (x, r) or S t (x, r) has a corner point, it must be located on an angle bisector of G. However, it can been seen from Figures 2a and 2b that the circles with s G -and j * Gmetrics can have corner points also elsewhere than on the angle bisectors of the domain (a) s G -metric circles. (b) j * G -metric circles. (c) p G -metric circles. G. We also notice that the circles in Figure 2b clearly differ those in Figures 2a and 2c. This can be described with the concept of starlikeness, which is a looser form of convexity. Namely, a set K is starlike with respect to a point x ∈ K if and only if the segment [x, y] belongs to K fully for every y ∈ K. In particular, the five-pointed star domain is starlike with respect to its center. The disks by the j * G -and t G -metrics (Figures 2a and 2c) are clearly not starlike and, even if it cannot be clearly seen from Figure 2d, there are disks drawn with the point pair function p G that are not starlike. Lemma 5.2. There exist disks B j * (x, r), B p (x, r), and B t (x, r) that are not starlike with respect to their center. Proof. Consider a domain G = H 2 ∪ {z ∈ C | − 1 < Re(z) < 1, −3 < Im(z) ≤ 0}. Fix x = −2i and y = 3 + i. Clearly, d G (x) = d G (y) = 1 and |x − y| = 3 √ 2. Consequently, j * G (x, y) = t G (x, y) = 3 3 + √ 2 < 0.7, p G (x, y) = 3 √ 11 < 0.91. The segment [x, y] does not clearly belong to G fully and no disk in G can contain this segment. However, its end point y is clearly included in the disks B j * (x, 0.7), B t (x, 0.7) and B p (x, 0.91). Thus, we have found examples of non-starlike disks. There are no disks or balls like this for the triangular ratio metric. For several common hyperbolic type metrics η G , the closed ball B η (x, M ) with M = η G (x, y) and x, y ∈ G is always a compact subset of the domain G, see [7, p. 79]. For instance, the hyperbolic metric ρ G has this property [7, p. 192]. As can be seen from the figures, the j * -metric, the triangular ratio metric and the point pair function share this property, too. Proof. If the ball B η (x, r), η G ∈ {j * G , p G , s G }, touches the boundary of G, then there is some point y ∈ S η (x, r) with d G (y) = 0 and j * G (x, y) = p G (x, y) = s G (x, y) = 1. Thus, we need to just prove that the balls with radius 1 always touch the boundary. Consider first the balls B j * (x, r) and B p (x, r), with a radius r = 1. Since, for all the points y on their boundary, j * G (x, y) = 1 ⇔ 2 min{d G (x), d G (y)} = 0 ⇔ d G (x) = 0 or d G (y) = 0, p G (x, y) = 1 ⇔ 4d G (x)d G (y) = 0 ⇔ d G (x) = 0 or d G (y) = 0, the balls B j * (x, 1) and B p (x, 1) touch the boundary of G. Consider yet the triangular ratio metric. Because only balls with radius r = 1 can touch the boundary, B s (x, 1) ∩ ∂G = ∅. However, if s G (x, y) = 1, there is some point z ∈ ∂G such that |x − y| = |x − z| + |z − y|. This means that z is on a line segment [x, y] and, since z / ∈ B s (x, 1), z must be arbitrarily close to the point y. Thus, d G (y) = 0 and the ball B s (x, 1) touches the boundary. However, the t-metric differs from the hyperbolic type metrics in this aspect: the closure of a t-metric ball is a compact set, if and only if the radius of the ball is less than 1/2. Theorem 5.5. The balls B t (x, r) touch the boundary of the domain G R n if and only if r ≥ 1 2 . Proof. If B t (x, r) touches the boundary, there must be some y ∈ S t (x, r) such that d G (y) = 0. Since d G (x) ≤ |x − y| + d G (y), it follows that r = t G (x, y) = |x − y| |x − y| + d G (x) + d G (y) ≥ |x − y| |x − y| + |x − y| + 0 + 0 = 1 2 . Thus, only balls B t (x, r) with a radius r ≥ 1 2 can touch the boundary of G. Let us yet prove that the balls B t (x, 1 2 ) always touch the boundary of G. For any point y ∈ S t (x, 1 2 ), it holds that |x − y| = 1/2 and t G (x, y) = |x − y| |x − y| + d G (x) + d G (y) = 1 2 ⇔ d G (y) = |x − y| − d G (x). Since only balls B t (x, r) with r ≥ 1 2 can touch the boundary of G, B t (x, 1 2 ) ∩ ∂G = ∅ and d G (x) ≥ 1/2. Thus, d G (y) = 1/2 − d G (x) ≤ 0 and, since the distances cannot be negative, d G (y) = 0 and the ball B t (x, 1 2 ) truly touches the boundary of G. The result above is visualized in Figures 1d and 2d. Proposition 3 . 10 . 310For any fixed domain G R n , the inequalities of Theorem 3.8 are sharp. y) 2 2holds and the constants here are sharp. Theorem 4 . 12 . 412For all conformal mappings f : Conjecture 4 . 13 . 413For all a, x, y ∈ B 2 , the Möbius transformation T a :B 2 → B 2 , T a (z) = (z − a)/(1 − az) fulfills the inequality t B 2 (T a (x), T a (y)) ≤ (1 + |a|)t B 2 (x, y). Remark 4 . 14 . 414It is also an open question whether the inequality of Conjecture 4.13 holds for the triangular ratio metric or so called Barrlund metric, but numerical tests suggest so, see [3, Conj. 1.6, p. 684] and [4, Conj. 4.3, p. 25]. (d) t G -metric circles. Figure 1 . 1Circles in a five-pointed star domain with different metrics. (d) t G -metric circles. Figure 2 . 2Circles in a five-pointed star domain with different metrics. Lemma 5. 3 . 3[7, p. 206] The balls B s (x, r) in any domain G R n are always starlike with respect to their center x. Lemma 5 . 4 . 54The balls B j * (x, r), B p (x, r) and B s (x, r) touch the boundary of the domain G R n if and only if r = 1. G Anderson, M Vamanamurthy, M Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley-InterscienceG. Anderson, M. Vamanamurthy and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley-Interscience, 1997. The hyperbolic metric and geometric function theory. A F Beardon, D Minda, Proc. International Workshop on Quasiconformal Mappings and their Applications (IWQCMA05). S. Ponnusamy, T. Sugawa and M. VuorinenInternational Workshop on Quasiconformal Mappings and their Applications (IWQCMA05)A.F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Proc. In- ternational Workshop on Quasiconformal Mappings and their Applications (IWQCMA05), eds. S. Ponnusamy, T. Sugawa and M. Vuorinen (2006), 9-56. Lipschitz conditions, triangular ratio metric, and quasiconformal maps. J Chen, P Hariri, R Klén, M Vuorinen, Ann. Acad. Sci. Fenn. Math. 40J. Chen, P. Hariri, R. Klén and M. Vuorinen, Lipschitz conditions, triangular ratio metric, and quasiconformal maps. Ann. Acad. Sci. Fenn. Math., 40 (2015), 683-709. Barrlund's distance function and quasiconformal maps. M Fujimura, M Mocanu, M Vuorinen, Complex Var. Elliptic Equ. M. Fujimura, M. Mocanu and M. Vuorinen, Barrlund's distance function and quasiconformal maps, Complex Var. Elliptic Equ. (2020), 1-31. The ubiquitous quasidisk. F W Gehring, K Hag, With contributions by Ole Jacob Broch. Mathematical Surveys and Monographs. 184American Mathematical SocietyF.W. Gehring and K. Hag, The ubiquitous quasidisk. With contributions by Ole Jacob Broch. Mathematical Surveys and Monographs, 184. American Mathematical Society, Providence, RI, 2012. Uniform domains and the quasi-hyperbolic metric. F W Gehring, B G Osgood, J. Analyse Math. 36F.W. Gehring and B.G. Osgood, Uniform domains and the quasi-hyperbolic metric, J. Analyse Math., 36 (1979), 50-74. P Hariri, R Klén, M Vuorinen, Conformally Invariant Metrics and Quasiconformal Mappings. SpringerP. Hariri, R. Klén and M. Vuorinen, Conformally Invariant Metrics and Quasiconformal Map- pings. Springer, 2020. Inequalities and Bilipschitz Conditions for Triangular Ratio Metric. P Hariri, M Vuorinen, X Zhang, Rocky Mountain J. Math. 47P. Hariri, M. Vuorinen and X. Zhang, Inequalities and Bilipschitz Conditions for Triangular Ratio Metric. Rocky Mountain J. Math., 47, 4 (2017), 1121-1148. A new weighted metric, the relative metric I. P Hästö, J. Math. Anal. Appl. 274P. Hästö, A new weighted metric, the relative metric I. J. Math. Anal. Appl., 274 (2002), 38-58. Geometry of the Cassinian metric and its inner metric. Z Ibragimov, M Mohapatra, S Sahoo, X Zhang, Bull. Malays. Math. Sci. Soc. 401Z. Ibragimov, M. Mohapatra, S. Sahoo and X. Zhang, Geometry of the Cassinian metric and its inner metric. Bull. Malays. Math. Sci. Soc., 40 (2017), no. 1, 361-372. A Gromov hyperbolic metric vs the hyperbolic and other related metrics. M Mohapatra, S Sahoo, Comput. Methods Funct. Theory. 183English summaryM. Mohapatra and S. Sahoo, A Gromov hyperbolic metric vs the hyperbolic and other related metrics. (English summary) Comput. Methods Funct. Theory, 18 (2018), no. 3, 473-493. O Rainio, M Vuorinen, Triangular Ratio Metric Under Quasiconformal Mappings In Sector Domains. Arxiv. O. Rainio and M. Vuorinen, Triangular Ratio Metric Under Quasiconformal Mappings In Sector Domains. Arxiv, 2005.11990. Lectures on n-dimensional quasiconformal mappings. J Väisälä, Lecture Notes in Math. 229J. Väisälä, Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Math. Vol. 229,
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[ "CLUSTER CATEGORIES AND RATIONAL CURVES", "CLUSTER CATEGORIES AND RATIONAL CURVES" ]
[ "Zheng Hua ", "Bernhard Keller " ]
[]
[]
We study rational curves on smooth complex Calabi-Yau threefolds via noncommutative algebra. By the general theory of derived noncommutative deformations due to Efimov, Lunts and Orlov, the structure sheaf of a rational curve in a smooth CY 3-fold Y is pro-represented by a nonpositively graded dg algebra Γ. The curve is called nc rigid if H 0 Γ is finite dimensional. When C is contractible, H 0 Γ is isomorphic to the contraction algebra defined by Donovan and Wemyss. More generally, one can show that there exists a Γ pro-representing the (derived) multipointed deformation (defined by Kawamata) of a collection of rational curvesThe collection is called nc rigid if H 0 Γ is finite dimensional. We prove that Γ is a homologically smooth bimodule 3CY algebra. As a consequence, we define a (2CY) cluster category C Γ for such a collection of rational curves in Y . It has finite-dimensional morphism spaces iff the collection is nc rigid. When t i=1 C i is (formally) contractible by a morphism Y → X, then C Γ is equivalent to the singularity category of X and thus categorifies the contraction algebra of Donovan and Wemyss. The Calabi-Yau structure on Y determines a canonical class [w] (defined up to right equivalence) in the zeroth Hochschild homology of H 0 Γ. Using our previous work on the noncommutative Mather-Yau theorem and singular Hochschild cohomology, we prove that the singularities underlying a 3-dimensional smooth flopping contraction are classified by the derived equivalence class of the pair (H 0 Γ, [w]). We also give a new necessary condition for contractibility of rational curves in terms of Γ.
null
[ "https://export.arxiv.org/pdf/1810.00749v6.pdf" ]
119,179,328
1810.00749
3d12c67d0f61c8167cb70ebf653ae9a810981c8f
CLUSTER CATEGORIES AND RATIONAL CURVES Zheng Hua Bernhard Keller CLUSTER CATEGORIES AND RATIONAL CURVES We study rational curves on smooth complex Calabi-Yau threefolds via noncommutative algebra. By the general theory of derived noncommutative deformations due to Efimov, Lunts and Orlov, the structure sheaf of a rational curve in a smooth CY 3-fold Y is pro-represented by a nonpositively graded dg algebra Γ. The curve is called nc rigid if H 0 Γ is finite dimensional. When C is contractible, H 0 Γ is isomorphic to the contraction algebra defined by Donovan and Wemyss. More generally, one can show that there exists a Γ pro-representing the (derived) multipointed deformation (defined by Kawamata) of a collection of rational curvesThe collection is called nc rigid if H 0 Γ is finite dimensional. We prove that Γ is a homologically smooth bimodule 3CY algebra. As a consequence, we define a (2CY) cluster category C Γ for such a collection of rational curves in Y . It has finite-dimensional morphism spaces iff the collection is nc rigid. When t i=1 C i is (formally) contractible by a morphism Y → X, then C Γ is equivalent to the singularity category of X and thus categorifies the contraction algebra of Donovan and Wemyss. The Calabi-Yau structure on Y determines a canonical class [w] (defined up to right equivalence) in the zeroth Hochschild homology of H 0 Γ. Using our previous work on the noncommutative Mather-Yau theorem and singular Hochschild cohomology, we prove that the singularities underlying a 3-dimensional smooth flopping contraction are classified by the derived equivalence class of the pair (H 0 Γ, [w]). We also give a new necessary condition for contractibility of rational curves in terms of Γ. Introduction The study of rational curves in algebraic varieties lies at the core of birational geometry. A smooth rational curve C in a quasi-projective variety Y is called rigid if the component of the Hilbert scheme of curves containing C is a finite scheme. Note that this is weaker than the notion of infinitesimally rigid, which says that Ext 1 Y (O C , O C ) = 0. If a curve is not rigid then we call it movable. When Y is a smooth projective surface, a smooth rational curve C ⊂ Y is rigid if and only if its normal bundle N C/Y is negative. And if C is rigid then it is contractible, i.e. for the formal completion Y of Y along C there exists a birational morphism f : Y → X to a normal surface X that contracts C. The definition of contractibility in general can be found in Definition 2.9. In this article, we will focus on the case when Y is a smooth complex Calabi-Yau threefold, i.e. ω Y is trivial. The situation is much more complicated than the surface case. We call a rational curve C ⊂ Y of type (a, b) if it has normal bundle O(a)⊕O(b). By the adjunction formula, we have a + b = −2. A (−1, −1)-curve is contractible. The underlying singular variety X is equivalent to the singular hypersurface x 2 + y 2 + u 2 + v 2 = 0. There exists a different resolution Y + → X and the birational map Y Y + is called the Atiyah flop. In [68], Reid proves that a (0, −2)-curve is either contractible or movable. The contractible case corresponds to the Pagoda flops. Laufer proves that a contractible curve is of the types (−1, −1), (0, −2) or (1, −3) (c.f. [72]). Katz and Morrison show that any simple flopping contraction (see definition in Section 2.5) can be constructed as base change of a universal contraction [41]. In general, it is not true that all rigid curves are contractible. A counter example was constructed by Clemens [16]. We study the contractibility of rational curves in Calabi-Yau 3-folds via noncommutative methods. In general, given a rational curve C ⊂ Y the problem is two-fold: (1) Find infinitesimal criteria for the contractibility of C. (2) If C is contractible, determine the underlying singularity of the contraction. Our research is motivated by a remarkable paper of Donovan and Wemyss. In [18], Donovan and Wemyss considered the algebra Λ that represents the noncommutative deformation functor of O C for a contractible rational curve C ⊂ Y . They prove that Λ is finite dimensional and call it the contraction algebra. Indeed, the contraction algebra can be defined in a more general context where Y may be neither CY nor smooth, and the birational morphism may contract a divisor containing C. However we will focus on the special case when Y is a smooth CY 3-fold and the contraction is not divisorial. Donovan and Wemyss conjectured that the 3-dimensional simple flops are classified by the isomorphism types of the contraction algebras (cf. Conjecture 5.10). In order to deal with the case of general flops where the exceptional fiber can have multiple irreducible components, Kawamata proposes to study the multi-pointed noncommutative deformation of a semi-simple collection (see definition in Section 2) 1 of sheaves E 1 , . . . , E t . A case of special interest is when the collection is O C1 , . . . , O Ct where C i are irreducible components of the reduced exceptional fiber of a contraction (see Example 6.5 of [42]). We consider the derived noncommutative deformation theory of E := t i=1 E i for a semi-simple collection of sheaves {E i } t i=1 on a smooth CY 3-fold Y . By a result of Efimov, Lunts and Orlov (cf. Theorem 2.2), such deformation functor is pro-represented by a nonpositively graded dg algebra Γ. We call Γ the derived deformation algebra of the semi-simple collection {E i } t i=1 . We call a semi-simple collection {E i } t i=1 nc rigid ("nc" stands for noncommutative) if H 0 Γ is finite dimensional. Given a collection of smooth rational curve C 1 , . . . , C t such that {O Ci } t i=1 form a semi-simple collection, we call {C i } t i=1 a nc-rigid collection of rational curves if H 0 Γ is finite dimesional. If t = 1 and C = C 1 is nc rigid rational curve then the abelianization of H 0 Γ represents the commutative deformation functor of O C . Therefore, a nc rigid curve is in particular rigid. Our first result is: Theorem A. (Corollary 3.9) Let C 1 , . . . , C t be a collection of rational curves in a smooth quasiprojective Calabi-Yau 3-fold Y such that {O Ci } t i=1 form a semi-simple collection. The derived deformation algebra Γ of t i=1 O Ci is a non positive pseudo-compact dg algebra that is (1) homologically smooth; (2) bimodule 3CY. Moreover, Γ is exact 3CY in either one of the following cases (a) Y is projective; (b) there is a (formal) contraction f : Y → X such that Ex( f ) = t i=1 C i , where Ex( f ) stands for the reduced exceptional fiber of f . This theorem establishes a link between birational geometry and the theory of cluster categories. We consider the triangle quotient C Γ := per(Γ)/D f d (Γ) (see Section 3.3). It is Hom-finite if and only if {C i } t i=1 is nc rigid. By a result of Amiot [3], it is then a 2CY category. When C is contractible by a morphism Y → X, then H 0 Γ is isomorphic to the contraction algebra Λ defined in [18] and C Γ is equivalent to the singularity category of X and thus categorifies the contraction algebra of Donovan and Wemyss. If Γ is exact 3CY, Van den Bergh proved that it is quasi-isomorphic to a (complete) Ginzburg algebra D(Q, w) for some finite quiver Q and a potential w. If we fix the CY structure on Y , then there is a canonical class [w], defined up to right equivalence, in the zeroth Hochschild homology of H 0 Γ (see Proposition 4.8). The class [w] can be viewed as the "classical shadow" of the Calabi-Yau structure on Y . Our second result is: given by a dg bimodule Z such that HH 0 (Z) (as defined in [47]) takes [w] to [w ] in HH 0 (H 0 Γ ) = HH 0 (Γ ). Then X is isomorphic to X . In particular, s is equal to t. For general (non-simple) flopping contractions, there exist derived equivalent algebras H 0 Γ and H 0 Γ that are non-isomorphic. August proves that the isomorphism classes of such algebras in a fixed derived equivalence class of H 0 Γ are precisely the contraction algebras for the iterated flops of Y (see Theorem 1.4 of [4]). Different contraction algebras in the same derived equivalence class are related by the iterated mutations of the tilting objects. The mutations are the homological counterpart of flops between different minimal models. We refer to [87,88] for the general framework of the homological minimal model program. Theorem B says that the underlying singularity type of the smooth minimal models is determined by the derived equivalence class of the pair (H 0 Γ, [w]). We sketch the idea of the proof. From 3-dimensional birational geometry we know that the underlying (isolated) singularity of a smooth flopping contraction is a hypersurface (see Section 2.5). It is a classical theorem of Mather and Yau that up to isomorphism, a germ of isolated hypersurface singularity is determined by its Tyurina algebra (see [67] for the analytic case and [30] for the formal case). Next we prove that the derived equivalence class of H 0 Γ together with the canonical class [w] recovers the Tyurina algebra of the singularity. We solve this problem in two steps. First, we prove that the Tyurina algebra, therefore the isomorphism class of the hypersurface singularity, can be recovered from the (Z-graded dg enhanced) cluster category C Γ . This result, proved in Section 5, should have independent interest. Secondly, we show that the isomorphism class of the Ginzburg algebra D(Q, w) that is quasi-isomorphic to Γ can be recovered from the data (H 0 Γ, [w]). The proof uses a result of the first author and Gui-song Zhou in noncommutative differential calculus of potentials with finitedimensional Jacobi-algebras [35]. Finally, we prove (in Section 4.4 and 4.5) that any derived Morita equivalence D(H 0 Γ) D(H 0 Γ ) preserving the canonical class yields a derived Morita equivalence per(Γ) per(Γ ). Note that in [33], the first author and Toda gave an alternative definition of the contraction algebra associated to a flopping contraction using the category of matrix factorizations. In this definition, the contraction algebra carries an additional (compared with the definition in [18]) Z/2graded A ∞ -structure. In [34], the first author proved that the Tyurina algebra of the singularity can be recovered from the Z/2-graded A ∞ -structure. Our proof of Theorem B shows that the Z/2graded A ∞ -structure on the contraction algebra can be recovered from the class [w]. Theorem B without the condition on the preservation of the canonical class is precisely the generalization of the conjecture by Donovan and Wemyss stated by August in Conjecture 1.3 of [4]. See Conjecture 5.10 for the original conjecture of Donovan and Wemyss, which is for simple flopping contractions. The generalized form of the conjecture has recently been proved in Appendix A of [40], cf. also [39], by combining the derived Auslander-Iyama correspondence of [loc. cit.] with our Theorem 5.8. The geometric meaning of the class [w] remains to be understood. It is believed that the vanishing of [w] is closely related to the condition that X is quasi-homogeneous. Our third result is a necessary condition on the contractibility of a nc rigid rational curve in a smooth CY 3-fold. Let u be a variable of degree 2. Theorem C. (Proposition 6.9) Let C be a nc rigid rational curve in a smooth CY 3-fold Y . If C is contractible, then its derived deformation algebra Γ is k[u −1 ]-enhanced (see definition in Section 6). Moreover, H 0 Γ is a symmetric Frobenius algebra. We conjecture that a nc rigid rational curve C is contractible if and only if Γ is k[u −1 ]-enhanced (see Conjecture 6.8). The paper is organized as follows. In Section 2 we review basics on derived noncommutative deformation theory, noncommutative crepant resolutions and flopping contractions. Concerning derived deformations, we complement the results of [25] by explaining the link to classical deformations in abelian categories in subsection 2.3. In Section 3, we discuss various notions of Calabi-Yau structures in geometry and algebra and prove Theorem A. The notion of cluster category is introduced in Section 3.3. When the curve is contractible, there are two cluster categories associated to it: one via derived deformation and the other via the NCCR. We prove that these two cluster categories are algebraically equivalent. In Section 4, we recall the definition of Ginzburg algebras and several results in noncommutative differential calculus including the noncommutative Mather-Yau theorem and noncommutative Saito theorem. We further show that for a contractible curve in a CY 3-fold, there exists a Ginzburg algebra weakly equivalent to the derived deformation algebra whose potential is canonically defined up to right equivalence. Then we establish a relation between the silting theory of a non positive dg algebra and the silting theory of its zeroth homology. In Section 5, we study the relation between the cluster category associated to the contractible curves and their underlying singularities via Hochschild cohomology. In particular, Theorem B is proved. In Section 6, we introduce the notion of k[u −1 ]-enhancement of dg algebras. For derived deformation algebra, we establish a link between the existence of k[u −1 ]-enhancement and contractibility of rational curve and prove Theorem C. don Katz and Gui-song Zhou for inspiring discussions, and Aron Heleodoro for the help on understanding [27]. The second author thanks Akishi Ikeda for kindly explaining the details of Section 3.5 of [31] and Gustavo Jasso for help with section 7.2. He is grateful to Zhengfang Wang for inspiring discussions on singular Hochschild cohomology and for providing reference [29]. Both authors want to thank Michael Wemyss for many valuable comments and suggestions, in particular for drawing our attention to the work of August [4,5]. They are greatly indebted to an anonymous referee for reading previous versions of the manuscript with great care, pointing out numerous local errors and inaccuracies and helping to improve the readability of the paper. The research of the first author was supported by RGC General Research Fund no. 17330316, no. 17308017. Preliminaries 2.1. Notation and conventions. Throughout k will be a ground field unless stated otherwise. Unadorned tensor products are over k. Let V be a k-vector space. We denote its dual vector space by DV . When V is graded, DV is understood as the the dual in the category of graded vector spaces. For a subspace V of a complete topological vector space V , we denote the closure of V in V by (V ) c . By definition a pseudo-compact k-vector space is a linear topological vector space which is complete and whose topology is generated by subspaces of finite codimension. Following [84] we will denote the corresponding category by P C(k). We have inverse dualities D : M od(k) → P C(k) op : V → Hom k (V, k) D : P C(k) → M od(k) op : W → Hom P C(k) (W, k) where we recall that for V ∈ M od(k) the topology on DV is generated by the kernels of DV → DV where V runs through the finite dimensional subspaces of V . Similarly, if V is graded then D is understood in the graded sense. For the definition of Hom-space and tensor product in P C(k), we refer to Section 4 of [84]. Using the tensor product in P C(k), we define the pseudo-compact dg algebras, modules and bimodules to be the corresponding objects in the category of graded objects of P C(k). Let A be pseudo-compact dg k-algebra. Denote by P C(A e ) the category of pseudocompact A-bimodules. We will sometimes take a finite dimensional separable k-algebra l to be the ground ring. The definition of the duality functor D on P C(l e ) requires some extra care due to the noncommutativity of l. We refer to Section 5 of [84] for the detailed discussion. Denote by PCAlgc(l) the category of augmented pseudo-compact dg algebras A whose underlying graded algebras have their augmentation ideal equal to their Jacobson radical (cf. Proposition 4.3 and section 6 of [84]). Our main interest is in the case when l ∼ = ke 1 × ke 2 × . . . × ke n for central orthogonal idempotents (e i ) i . For an object A ∈ PCAlgc(l), we use Hom-spaces and tensor products in P C(l e ) to define the Hochschild and cyclic (co)homology. For details, we refer to Section 7 and Appendix A of [84]. If A is an l-algebra in PCAlgc(l), we use HH * (A), HH * (A), HC * (A) to denote the continuous Hochschild homology, cohomology and cyclic homology of A. Because for a pseudocompact dg algebra, we will only consider continuous Hochschild homology, cohomology and cyclic homology, there is no risk of confusion. By an abuse of notations, for A ∈ PCAlgc(l) we denote by D(A) the pseudo-compact derived category of A. Its subcategories per(A) and D f d (A) are defined as the thick subcategory generated by the free A-module A and as the full subcategory of all objects with homology of finite total dimension. Similar to the algebraic case, the notion of homological smoothness can be defined in the pseudo-compact setting. We refer to the Appendix of [57] for a careful treatment. For the bar-cobar formalism and Koszul duality of pseudo-compact dg algebras, we refer to Appendix A and D of [83]. 2.2. Derived deformation theory. We briefly recall the setup of derived noncommutative deformation theory of Efimov, Lunts and Orlov. In this section, we fix a field k. We refer to [50] for foundational material on dg categories. For a dg category A, we denote by D(A) the derived category of right dg A-modules. Fix a positive integer n and let l be the separable k-algebra ke 1 × . . . × ke n . An l-algebra A is a k-algebra together with a morphism of k-algebras l → A (note that l is not necessarily central in A). An equivalent datum is that of the k-category with n objects 1, . . . , n, whose morphism space from i to j is given by e j Ae i . An l-augmented (dg) algebra is a (dg) l-algebra R together with an l-algebra morphism R → l such that the composition l → R → l is the identity morphism. Its augmentation ideal is the kernel of the augmentation morphism R → l. An artinian l-algebra is an augmented l-algebra whose augmentation ideal is finite-dimensional and nilpotent. A dg l-algebra is artinian if it is an augmented dg l-algebra whose augmentation ideal is finite-dimensional and nilpotent. Denote by Art l and cArt l the categories of artinian l-algebras and of commutative artinian l-algebras. Denote by dgArt l the category of artinian dg algebras and by dgArt − l the subcategory of dgArt l consisting of dg algebras concentrated in nonpositive degrees. Fix a dg category A and a dg A-module E with a decomposition E = E 1 ⊕· · ·⊕E n . We view E as an l op ⊗A-module in the natural way. The dg endomorphism l-algebra of E is the dg endomorphism algebra over A of the sum E viewed as an l-algebra in the natural way. We are going to define a pseudo-functor Def(E) from dgArt l to the category Gpd of groupoids. This pseudo-functor assigns to an artinian dg l-algebra R the groupoid Def R (E) of R-deformations of E in the derived category D(A). We will mostly follow the notations of [24] and identify R with the dg category with n objects 1, . . . , n, where the morphism complex from i to j is given by e j Re i . Denote the dg category R op ⊗A by A R . The augmentation ε : R → l yields the functor of extension of scalars ε * taking a dg R-module S to the dg l op ⊗A-module ε * (S) = l L ⊗ R S.σ : ε * (S) → E is an isomorphism in D(l op ⊗A). A morphism f : (S, σ) → (T, τ ) between two R-deformations of E is an isomorphism f : S → T in D(A R ) such that τ • ε * (f ) = σ. This defines the groupoid Def R (E). A homomorphism of augmented artinian dg l-algebras φ : R → Q induces the functor φ * : Def R (E) → Def Q (E) given by Q L ⊗ R ?. Thus we obtain a pseudo-functor Def(E) : dgArt l → Gpd. We call Def(E) the pseudo-functor of derived deformations of E. We denote by Def − (E) the restrictions of the pseudo-functor Def(E) to the subcategory dgArt − l . The category of augmented dg l-algebras can be naturally enhanced to a weak 2-category. We refer to Definition 11.1 of [25] for the precise definition of the 2-category structure. In particular, we denote the corresponding 2-categorical enhancements of dgArt l , dgArt − l and Art l by 2-dgArt l , 2-dgArt − l and 2-Art l (in [25], they are denoted by 2 -dgArt l etc.). By Proposition 11.4 of [25], there exists a pseudo-functor DEF(E) from 2-dgArt l to Gpd and which is an extension to 2-dgArt l of the pseudo-functor Def(E). Similarly, there exists a pseudo-functor DEF − (E) extending Def − (E). The main theorem of [25] is: . Let E be the direct sum of the E i and C the extension algebra Ext * A (E, E) considered as a graded l-algebra. Assume that we have (a) C p = 0 for all p < 0; (b) C 0 = l; (c) dim k C p < ∞ for all p and dim k C p = 0 for all p 0. Denote by C the dg endomorphism l-algebra of E. Let A be a strictly unital minimal model of C. Then the pseudo-functor DEF − (E) is pro-representable by the dg l-algebra Γ = DBA, where B denotes the bar construction. That is, there exists an equivalence of pseudo-functors DEF − (E) h Γ from 2-dgArt − l to Gpd, where h Γ denotes the groupoid of 1-morphisms 1-Hom(Γ, ?). In the case where the dg category A is given by an algebra A concentrated in degree 0 and E is a one-dimensional A-module, Booth [12, Theorem 3.5.9] obtains an analogous prorepresentability result for the set-valued framed deformation functor Def fr,≤0 A (E) without having to impose the finiteness condition (c). Let Y be a smooth algebraic variety. A collection of compactly supported coherent sheaves E 1 , . . . , E t on Y is called semi-simple if Hom Y (E i , E i ) ∼ = k for all i and Hom Y (E i , E j ) = 0 for all i = j. The finiteness assumption in Theorem 2.2 is satisfied by any semi-simple collection. Let E be the direct sum of such a collection of coherent sheaves on Y . We may denote the completion of Y along the support of E by Y . Corollary 2.3. Given a semi-simple collection {E i } t i=1 in D b (coh Y ) , denote by C the dg endomorphism l-algebra of E := t i=1 E i . Let A be a strictly unital minimal model of C and Γ = DBA. Then there is an equivalence DEF − (E) h Γ . We call Γ the derived deformation algebra of the collection {E i } t i=1 in Y . When we want to emphasize the dependence on Y and {E i } t i=1 , Γ is replaced by Γ Y E . The semi-simple collection {E i } t i=1 is called nc rigid if dim k H 0 (Γ Y E ) < ∞. In this paper, we are mainly interested in the case when {E i } t i=1 is (the structure sheaves) a collection of smooth rational curves C 1 , . . . , C t that satisfies the condition that Hom Y (O Ci , O Cj ) = 0 for i = j. For such a collection of rational curves C := {C i } n i=1 where we write Γ Y C for Γ Y {O C i } t i=1 . In the context of classical noncommutative deformation theory, the representability of noncommutative deformations of contractible rational curves was proved by Donovan and Wemyss. Theorem 2.4. (Proposition 3.1, Corollary 3.3 [18]) Let f : Y → X be a simple flopping contraction of 3-folds (see definition in Section 2.5) and let C be the reduced exceptional fiber of f . The functor π 0 (Def cl (O C )) : Art k → Set is representable. The artinian algebra Λ representing it is called the contraction algebra associated to f : Y → X. The definition of the classical deformation functor Def cl is recalled in the next section (cf. Section 2 of [18]). If Γ is the derived deformation algebra of C (with t = 1), it follows from the above Theorem and Theorem 2.5 below that the contraction algebra Λ is isomorphic to H 0 Γ. Indeed, they both represent the same deformation functor and this determines them up to (non unique) isomorphism (cf. the proof of Theorem 2.14 in [78]). 2.3. Link to classical deformations. Let A be a dg category. Let H ⊂ D(A) be the heart of a t-structure on D(A). We assume that H is faithful, i.e. the higher extension groups computed in H are canonically isomorphic to those computed in D(A). Let R be an augmented artinian l-algebra. By an R-module in H, we mean an object M of H endowed with an algebra homomorphism R → End(M ). Given such an R-module, we denote by ?⊗ R M the unique right exact functor mod R → H extending the obvious additive functor proj R → H taking R to M . Here we denote by proj R the category of finitely generated projective (right) R-modules and by mod R the category of finitely generated R-modules. Notice that ?⊗ R is a left Kan extension and thus unique up to unique isomorphism and functorial in M . It can be computed using projective resolutions. It is obvious how to define morphisms of R-modules in H. Let E be the direct sum of a collection of n objects E 1 , . . . , E n of H. We view E as an l-module in H in the natural way. For an augmented artinian l-algebra R, we define the groupoid Def cl R (E) of classical deformations of E as follows: Its objects are pairs (M, µ) where M is an R-module in H such that the functor ?⊗ R M is exact and µ : l⊗ R M ∼ → E is an isomorphism of l-modules in H. A morphism (L, λ) → (M, µ) is an isomorphism f : L → M of R-modules in H such that µ • (l⊗ R f ) = λ. For an augmented l-algebra A and an augmented artinian l-algebra R, we define G(A, R) to be the groupoid whose objects are the morphisms A → R of augmented l-algebras and whose morphisms φ 1 → φ 2 are the invertible elements r of R such that φ 2 (a) = rφ 1 (a)r −1 for all a in A. Theorem 2.5. Suppose that in addition to the above assumptions, E satisfies the hypotheses of Theorem 2.2. Let Γ be the pseudo-compact dg l-algebra defined there. Let R be an augmented artinian l-algebra. Then H 0 Γ represents the classical deformations of E in the sense that there is an equivalence of groupoids Def cl R (E) ∼ → G(H 0 Γ, R). Proof. By Theorem 2.2, we have an equivalence of groupoids DEF − (E)(R) ∼ → 1-Hom(Γ, R), where 1-Hom denotes the groupoid of 1-morphisms in 2-dgArt. We will show that DEF − (E)(R) is equivalent to Def cl R (E) and 1-Hom(Γ, R) is equivalent to G(H 0 Γ, R). We start with the second equivalence. By Definition 11.1 of [25], an object of 1-Hom(Γ, R) is a pair (M, θ) consisting of • a dg bimodule M in D(Γ op ⊗R) such that the restriction to R of M is isomorphic to R in D(R) and • an isomorphism θ : M L ⊗ R l → l in D(Γ op ). A 2-morphism f : (M 1 , θ 1 ) → (M 2 , θ 2 ) is an isomorphism f : M 1 → M 2 in D(Γ op ⊗R) such that θ 2 •(f L ⊗ R l) = θ 1 . We define a functor F : G(H 0 Γ, R) → 1-Hom(Γ, R) as follows: Let φ : H 0 (Γ) → R be a morphism of augmented l-algebras. Since Γ is concentrated in degrees ≤ 0, we have a canonical algebra morphism Γ → H 0 (Γ). By composing it with φ we get a morphism of augmented dg lalgebras Γ → R. It defines a structure of dg bimodule M on R. We put F φ = (M, θ), where θ : R L ⊗ R l → l is the canonical isomorphism. Now let φ 1 and φ 2 be two morphisms of augmented algebras H 0 (Γ) → R. Put (M i , θ i ) = F φ i , i = 1, 2. Let r an be an invertible element of R such that φ 2 (a) = rφ 1 (a)r −1 for all a in H 0 (Γ). Then it is clear that the left multiplication with r defines an isomorphism of bimodules M 1 → M 2 compatible with the θ i . Recall that Γ op ⊗R is concentrated in degrees ≤ 0 so that its derived category has a canonical t-structure. Since the M i live in the heart of this t-structure on D(Γ op ⊗R), it is also clear that F is fully faithful. It remains to be checked that F is essentially surjective. So let (M, θ) be given. Since M is quasi-isomorphic to R when restricted to R, its homology is concentrated in degree 0. We can therefore replace M with H 0 (M ), which is an ordinary H 0 (Γ)-R-bimodule isomorphic to R as a right R-module (we also consider it as a left Γ-module via the canonical morphism Γ → H 0 (Γ)). In particular, M is right projective and so M L ⊗ R ? = M ⊗ R ?. We choose an isomorphism f : M ∼ → R of right R-modules. After multiplying f with an invertible element of l, we may assume that f ⊗ R l = θ. The left Γ-module structure on M yields an algebra morphism φ : H 0 (Γ) → End R (M ) ∼ → End R (R) = R. It is clear that f yields an isomorphism between (M, θ) and F φ. We now construct an equivalence from DEF − (E)(R) to Def cl R (E). Recall from Proposition 11.4 of [25] that the groupoid DEF − (E)(R) equals the groupoid Def R (E) of Definition 2.1 (but DEF − has enhanced 2-functoriality). Let P → E be a cofibrant resolution of E. Since the graded algebra Ext * (E, E) has vanishing components in degree −1 and in all sufficiently high degrees, we can apply Theorem 11.8 of [25] to conclude that the groupoid Def R (E) is equivalent to the groupoid Def h R (P ) of homotopy deformations of Definition 4.1 of [24]. We now construct an equivalence F from Def h R (P ) to Def cl R (E). Let (S, σ) be an object of Def h R (P ). We may assume that S = R⊗ l P as a graded bimodule and that σ is the canonical isomorphism l⊗ R (R⊗P ) ∼ → P . Let I denote the augmentation ideal of R. Then S has a finite filtration by the dg submodules I p S, p ≥ 0, and each subquotient is isomorphic to a summand of a finite sum of copies of l⊗ R S = P . Thus, the underlying dg A-module M of S is isomorphic in D(A) to a finite iterated extension of objects of add(E), the subcategory of direct factors of finite direct sums of copies of E. Therefore, M still lies in the heart H. Note that as shown in the proof of Theorem 11.8 of [25], S is cofibrant over R op ⊗A. Therefore, M is cofibrant over A. The left R-module structure on S yields an algebra homomorphism R → End(M ). Since each object of mod R is a finite iterated extension of one-dimensional l-modules, the functor ?⊗ R S : D(R) → D(A) takes mod R to H. Since ?⊗ R S is a triangle functor, the induced functor mod R → H is exact. Clearly it restricts to the natural functor proj R → H and is therefore isomorphic to ?⊗ R M : mod R → H. Finally, the isomorphism l⊗ R S ∼ → E yields an isomorphism µ : l⊗ R M ∼ → E. In this way, to an object (S, σ) of Def h R (P ), we have associated an object F (S, σ) = (M, µ) of Def cl R (E). Notice that by what we have just shown, we may also describe M ∈ H as the zeroth homology H 0 H (S) with respect to the t-structure associated with H, that we have an isomorphism l⊗ R M ∼ → H 0 H (l L ⊗ R S) and that the isomorphism µ : l⊗ R M ∼ → E is induced by σ : l L ⊗ R S ∼ → E. Recall that a morphism (S 1 , σ 1 ) → (S 2 , σ 2 ) of Def h R (P ) is a class of isomorphisms S 1 → S 2 of dg R op ⊗A-modules compatible with the σ i modulo homotopies compatible with the σ i . Since the functor Def h R (P ) → Def R (E) is an equivalence, these morphisms are in bijection with the isomorphisms S 1 → S 2 of D(R op ⊗A) compatible with the σ i . Clearly each such morphism induces an isomorphism (M 1 , λ 1 ) → (M 2 , λ 2 ), where (M i , λ i ) = F (S i , σ i ), i = 1, 2. It follows from Lemma 2.6 below that this assignment is a bijection. It remains to be shown that F : Def h R (P ) → Def cl R (E) is essentially surjective. Since we have an equivalence Def h R (P ) ∼ → Def R (E), it suffices to lift a given object (M, µ) of Def cl R (E) to an object (S, σ) of Def R (E). Let A denote the dg endomorphism l-algebra RHom A (M, M ). Then M becomes canonically an object of D(A op ⊗A). Now since M is in the heart of a t-structure, its negative self-extension groups vanish and we have a quasi-isomorphism τ ≤0 A ∼ → End H (M ). Thus, in the homotopy category of dg l-algebras, we have a morphism R → End H (M ) ∼ → τ ≤0 A → A. By tensoring with A we obtain a morphism Since H is a faithful heart, one obtains that this triangle functor is fully faithful. We claim that we have a square of triangle functors commutative up to isomorphism D b (mod R) ?⊗ R M / / D b (H) ?⊗ H R D(R) ? L ⊗ A S / / D(A). To check this, one has to check that the bimodules S and M ⊗ H R are isomorphic in D(R op ⊗A). This is easy using Lemma 2.6 below. Since ?⊗ R M : mod R → H is exact, the given isomorphism l⊗ R M ∼ → E yields an isomorphism l L ⊗ R M ∼ → E in D b( H) and thus an isomorphism σ : l L ⊗ R S ∼ → E in D(A). It is now clear that we can recover M as H 0 R (S) and that µ : l⊗ R M ∼ → E is the morphism induced by σ : l L ⊗ R S ∼ → E in H 0 R . By the description of F given above, this shows that we do have F (S, σ) ∼ → (M, µ). Let A be a dg category and B an ordinary k-algebra. Let X and Y be objects of D(B op ⊗A) and let res(X) be the restriction of X to A. The left action of B on X defines an algebra morphism α X : B → End D(A) (res(X)). Let M(X, Y ) be the space of all morphisms f : res(X) → res(Y ) in D(A) such that f • α X (b) = α Y (b) • f for all b ∈ B. The restriction functor induces a natural map for all n > 0, the map Φ is bijective. Φ : Hom D(B op ⊗A) (X, Y ) → M(X, Y ). Proof. We adapt the argument of Section 5 of [49]. We may suppose that X is cofibrant over B op ⊗A and in particular cofibrant over A. Then the sum total dg module of the bar resolution . . . / / B⊗B ⊗p ⊗X / / . . . / / B⊗B⊗X / / B⊗X / / 0 is still cofibrant over B op ⊗A and quasi-isomorpic to X. We use it to compute Hom D(B op ⊗A) (X, Y ). By applying Hom B op ⊗A (?, Y ) to the bar resolution, we get a double complex D of the form Hom A (X, Y ) / / Hom A (B⊗X, Y ) / / . . . / / Hom A (B ⊗p ⊗X, Y ) / / . . . We have to compute H 0 of the product total complex Tot Π D. Let D ≥0 be the double complex obtained by applying the intelligent truncation functor τ ≥0 to each column of D. Let D <0 be the kernel of D → D ≥0 . We claim that the product total complex of D <0 is acyclic. Indeed, the homology of the pth column of D <0 in degree −q is isomorphic to Hom D(A) (B ⊗p ⊗X, Σ −q Y ). It vanishes for −q < 0 by our assumption. To show that the product total complex of D <0 is acyclic, we consider the column filtration F p D <0 . Then D <0 is the inverse limit of the F p D <0 . By induction on p, each F p D <0 has an acyclic total complex. Moreover, the transition maps F p+1 D <0 → F p D <0 induce componentwise surjections in the total complexes. It follows that the inverse limit of the total complexes of the F p D <0 is still acyclic and this inverse limit is the product total complex of D <0 . So it is enough to compute H 0 of Tot Π D ≥0 . For this, let us denote by Hom 0 A (X, Y ) the quotient of Hom 0 A (X, Y ) by the nullhomotopic morphisms and similarly for Hom 0 A (B⊗X, Y ). The space we have to compute is the homology in degree 0 of the total complex of the double complex Hom 1 A (X, Y ) / / Hom 1 A (B⊗X, Y ) Hom 0 A (X, Y ) O O / / Hom 0 A (B⊗X, Y ) O O , where the lower left corner is in bidegree (0, 0). This equals the intersection of the kernels of the maps from bidegree (0, 0) to bidegrees (0, 1) and (1, 0). The kernels of the vertical maps are respectively Hom D(A) (X, Y ) and Hom D(A) (B⊗X, Y ) and the space we have to compute thus identifies with the kernel of the map Hom D(A) (X, Y ) → Hom D(A) (B⊗X, Y ) which is easily seen to take f : X → Y to f • α X − α Y • f .. Let N 0 = R, N 1 , N 2 , . . . , N t be pairwise non-isomorphic indecomposables in CM R and A := End R ( t i=0 N i ). We call A a noncommutative resolution (NCR) of R if it has finite global dimension. A NCR is called a noncommutative crepant resolution (NCCR) if A further satisfies that (a) A ∈ CM R ; (b) gldim(A) = n. If A is a NCCR, we call ⊕ t i=0 N i a tilting module. Under the above conditions, Iyama shows that ⊕ t i=1 N i is a cluster tilting object (see definition in Section 3.3) in CM R . Denote l for A/radA and e 0 for the idempotent given by the projection R⊕ t i=1 N i → R. Let S 0 , S 1 , . . . , S t be the simple A-modules with S 0 corresponding to the summand R of R⊕ t i=1 N i . DeT l V, d) ∼ → A for A, where T l V is the graded completion of the ordinary tensor algebra T l V with respect to the two sided ideal generated by V . Put Γ = T l V / T l V e 0 T l V . Then one has CM R ∼ = per(Γ)/thick(S 1 , . . . , S t ) and furthermore Γ has the following properties (1) Γ has finite dimensional cohomology in each degree; (2) Γ is concentrated in negative degrees; (3) H 0 Γ = A/Ae 0 A; (4) As a graded algebra Γ is of the form T l V 0 for V 0 = (1 − e 0 )V (1 − e 0 ) with l := l/ke 0 . 2.5. Flopping contraction. Definition 2.9. A smooth rational curve C in a normal variety Y is called contractible if there exists an open subscheme Y • ⊂ Y containing C and a proper birational morphism f • : Y • → X • such that (1) X • is normal, (2) the exceptional locus Ex(f • ) contains C, (3) f • is an isomorphism in codimension one. The above definition of contractibility is more restrictive than the standard one since it rules out the divisorial contraction (by the last condition). If Y is a 3-fold (which is our main interest), then Ex(f ) must have dimension one by condition (3). However, it may contain other components besides C. If C is a contractible curve in Y , denote by X the formal completion of X • along the exceptional subscheme, i.e. where f • is not an isomorphism. Consider the Cartesian diagram Y f / / Y • f • X / / X • where Y is the fiber product. We call f : Y → X the formal contraction associate to the contraction f • : Y • → X • . The following definition is a special case of Definition 6.10 of [59]. Definition 2.10. Let Y be a normal variety of dimension 3. A flopping contraction is a proper birational morphism f : Y → X to a normal variety Y such that f is an isomorphism in codimension one, and K Y is f -trivial. If Y is smooth, then we call f a smooth flopping contraction. In this paper, we only consider smooth flopping contractions unless stated otherwise. Given a 3-dimensional flopping contraction f : Y → X, let D be a divisor on Y such that −(K Y + D) is f -ample. By Theorem 6.14 of [59], there exists a D-flop f + : Y + → X. To be more precise, f + is a proper birational morphism that is an isomorphism in codimension one, and K Y + + D + is f + -ample where D + is the birational transform of D on Y + . In particular, X is Gorenstein terminal. Without loss of generality, we may work locally near the exceptional fiber of f . By the classification theorem of 3-dimensional terminal singularities, X has an isolated cDV singularity (see Corollary 5.38 of [59]). Recall that a 3-fold singularity (X, 0) is called cDV if a generic hypersurface section 0 ∈ H ⊂ X is a Du Val singularity. Because H has embedded dimension 3, X has embedded dimension 4, i.e. X is a hypersurface. Denote by Ex(f ) the reduced exceptional fiber of f . It is well known that Ex(f ) is a tree of rational curves Ex(f ) = t i=1 C i with normal crossings such that C i ∼ = P 1 (c.f. Lemma 3.4.1 of [82]). We call a 3-dimensional flopping contraction f : Y → X simple if Ex(f ) ∼ = P 1 . Let p be the singular point of X. By the remark above, R := O X,p is a complete local ring of the form k[[x, y, u, v]]/(g). Now we set X := Spec R and Y := X × X Y and denote by f the base change of f . We call the triple ( Y , f , R) the formal contraction associated to the flopping contraction f : Y → X. Note that Y is Calabi-Yau. Let ( Y , f , R) be a formal flopping contraction. Now we consider the NCCR associated to a three dimensional flopping contraction f : Y → X constructed as follows. For i = 1, . . . , t, let L i be a line bundle on Y such that deg Cj L i = δ ij . Define N i to be given by the maximal extension (2.1) 0 / / L −1 i / / N i / / O ⊕ri Y / / 0 associated to a minimal set of r i generators of H 1 ( Y , L −1 i ). Set N i := Rf * N i = f * N i for i = 1, . . . , t. We set A := End Y (O Y ⊕ N 1 ⊕ . . . N t ) ∼ = End R (R ⊕ N 1 ⊕ . . . ⊕N t ). (2.2) The second isomorphism can be proved as follows. It is clear that the natural morphism from A to End R (R ⊕ N 1 ⊕ . . . ⊕N t ) is an isomorphism away from the exceptional locus, which has codimension 3. Since both are reflexive R-modules, it must be an isomorphism. For simplicity, we denote by N the direct sum t i=1 N i and denote by N the direct sum t i=1 N i . Theorem 2.11. ([82, Corollary 3.2.10]) The functor F := RHom Y (O Y ⊕ N , −) defines a triangle equivalence between D b (coh Y ) and D b (Mod A), with quasi-inverse F −1 := (−)⊗ A (O Y ⊕ N ). In addition, A is itself Cohen-Macaulay. It follows that A is a NCCR. Corollary 2.12. Let ( Y , f , R) be a 3-dimensional formal flopping contraction. Then (1) The structure sheaves of the irreducible components {C i } t i=1 of Ex( f ) form a semi-simple collection in coh( Y ). (2) The derived deformation algebra Γ Y C of the collection C := {C i } t i=1 is linked by quasi- isomorphisms to Γ in Theorem 2.8. (3) For any i = 1, . . . , t, C i is nc rigid. Proof. To prove that Hom Y (O Ci , O Cj ) = 0 for i = j, we simply need to use the condition that Ex( f ) is a tree of rational curves with normal crossings. For any i = 1, . . . , t, it is easy to check that S i ∼ = F (ΣO Ci (−1)). Note that ( t i=1 O Ci )⊗( t i=1 L −1 i ) ∼ = t i=1 O Ci (−1). By Theorem 2.11, F induces an isomorphism of A ∞ algebras (2.3) Ext * Y ( t i=1 O Ci , t i=1 O Ci ) ∼ = Ext * A ( t i=1 S i , t i=1 S i ). Let l = A/rad(A) and l = l/ke 0 . Then there is a natural isomorphism of A-modules l ∼ = t i=0 S i , where S 0 is the simple A-module that corresponds to the summand R of R⊕ t i=1 N i . By Lemma 4.1 of [81], the vector space V in Theorem 2.8 can be chosen as D ΣExt ≥1 A (l, l) . Therefore, Γ := T l V / T l V e 0 T l V ∼ = DB(Ext * A (l, l)) represents the noncommutative deformations of semi-simple collection {S i } t i=1 ∈ D b (mod A). Part (2) follows from the isomorphism 2.3. Denote by Γ Y Ci the derived deformation algebra of O Ci . We have Γ Y Ci ∼ = Γ/ j =i Γe j Γ, and H 0 Γ Y Ci ∼ = H 0 Γ/ j =i H 0 Γe j H 0 Γ. Then part (3) follows from (1) of Theorem 2.8. Calabi-Yau structure and cluster category In this section, we first review several notions of Calabi-Yau property for triangulated categories, for homologically smooth dg algebras and for proper dg algebras. Then we recall geometric versions of the Calabi-Yau property and translate them into algebraic notions for endomorphism algebras of generators respectively for derived deformation algebras. Finally, we classify Calabi-Yau structures for 3-dimensional flopping contractions and review the cluster category. CY structures. 3.1.1. CY triangulated categories. Let T be a Hom-finite k-linear triangulated category. Definition 3.1. A right Serre functor for T is a triangle functor S : T → T such that there are bifunctorial isomorphisms T (Y, SX) → DT (X, Y ) for all X, Y ∈ T . It is a Serre functor if it is an autoequivalence. One can show that a right Serre functor exists if and only if for each object X of T , the functor DT (X, ?) is representable and in this case, the right Serre functor is unique up to canonical isomorphism of triangle functors [9,83]. Let d be an integer. The triangulated category T is d-Calabi-Yau if it admits a Serre functor isomorphic to Σ d . CY smooth dg algebras. A dg-k-algebra Γ is called homologically smooth if Γ is perfect in D(Γ e ). Then one checks that D f d (Γ), the subcategory of D(Γ) consisting of the modules whose homology is of finite total dimension, is contained in the perfect derived category per(Γ). Put Θ Γ = RHom Γ e (Γ, Γ e ). Then we have a canonical isomorphism HH d (Γ) ∼ → Hom D(Γ e ) (Θ Γ , Σ −d Γ). Definition 3.2. The dg algebra Γ is called bimodule dCY if it is homologically smooth and there is an isomorphism in D(Γ e ) η : Θ Γ ∼ → Σ −d Γ. A class η ∈ HH d (Γ) is called a dCY structure if the corresponding morphism η : Θ Γ → Σ −d Γ is an isomorphism in D(Γ e ). A dCY structure η is called exact if there exists a class ξ ∈ HC d−1 (Γ) such that Bξ = η, where B is the Connes morphism. A choice of ξ will be called an exact lifting of the dCY structure η. We call a bimodule dCY algebra Γ an exact dCY algebra if the dCY structure is exact in addition. Definition 3.3. The dg algebra Γ is said to satisfy the relative dCY property if for L ∈ D f d (Γ) and M ∈ per(Γ), we have a bifunctorial isomorphism DRHom D(Γ) (L, M ) RHom D(Γ) (M, Σ d L). Remark 3.4. If Γ is a topologically homologically smooth pseudo-compact dg algebra in PCAlgc(l), we call Γ a bimodule dCY if η is an isomorphism in the pseudo-compact derived category of bimodules. The isomorphism η represents a class in the continuous Hochschild homology HH d (Γ). Exactness is defined similarly by taking the continuous cyclic homology. We call a bimodule dCY pseudo-compact dg algebra Γ in PCAlgc(l) an exact dCY algebra if the dCY structure is exact in addition. Similarly, for a pseudo-compact algebra Γ in PCAlgc(l), the relative dCY property is defined by replacing D(Γ), per(Γ) and D f d (Γ) with their pseudo-dompact counter-parts. Given a homologically smooth dg algebra Γ, it follows from Lemma 3.4 in [52] that we have the implications Γ is bimodule dCY ⇒ Γ satisfies the relative dCY property ⇒ D f d (Γ) is a Hom-finite dCY triangulated category. A similar chain of implications holds in the pseudo-compact case. 3.1.3. CY proper dg algebras. Let A be a dg algebra. Suppose that A is proper, i.e. that its homology is of finite total dimension. Then the category per(A) is Hom-finite. The proper dg algebra A is called perfectly dCY if there is an isomorphism DA ∼ → Σ d A in D(A e ) . By the following lemma, the triangulated category per(A) is then dCY. Definition 3.5. Let k be a field of characteristic zero. Given a finite-dimensional A ∞ -algebra A, a cyclic A ∞ -structure of degree d on A is a non degenerate symmetric bilinear form (?, ?) : A × A → Σ d A of degree d such that (m n (a 1 , . . . , a n ), a n+1 ) = (−1) n (−1) |a1|(|a2|+...+|an+1|) (m n (a 2 , . . . , a n+1 ), a 1 ) In this case, we have in particular an isomorphism DA ∼ → Σ d A in the derived category of Abimodules. Thus, if a dg algebra is quasi-isomorphic to an A ∞ -algebra admitting a cyclic A ∞structure of degree d, then it is perfectly dCY. Lemma 3.6. Let T ≥ 1 be an integer, l a product of t copies of k and A a dg algebra augmented over l and such that A belongs to the triangulated subcategory of D(A⊗A op ) generated by l⊗l op (in particular, the dg algebra A is proper). Let Γ be the pseudo-compact dg algebra DBA, where B denotes the bar construction over l. Then Γ is topologically homologically smooth. Moreover, if A is perfectly d-Calabi-Yau, then Γ is bimodule d-Calabi-Yau. Proof. We refer to [46] for a brief summary of the bar-cobar formalism that we are going to use and to [62] and Appendix A of [84] for in-depth treatments. Let C be the augmented dg coalgebra BA and τ : C → A the canonical twisting cochain. Since τ is acyclic, the canonical morphism C → C⊗ τ A⊗ τ C is a weak equivalence of dg C-bicomodules (cf. [46]). Recall (for example from Appendix A of [84]) that a dg comodule is fibrant if its underlying graded comodule is cofree. Clearly, this holds for C⊗ τ A⊗ τ C. Since A belongs to the triangulated subcategory of the derived category of A⊗A op generated by l⊗l, the dg bicomodule C⊗ τ A⊗ τ C belongs to the triangulated subcategory of the coderived category of C⊗C op generated by C⊗C op . By applying the duality D we obtain that the morphism (3.1) Γ⊗ τ DA⊗ τ Γ → Γ is a weak equivalence and that the object on the left is cofibrant. Moreover, we see that Γ belongs to the perfect derived category of Γ⊗Γ op . This means that Γ is topologically homologically smooth. Now suppose that A is perfectly d-Calabi-Yau. Since A is proper, it is weakly equivalent to its pseudo-compact completion A. By the Calabi-Yau property, we have an isomorphism DA ∼ → Σ d A in the pseudo-compact derived category of dg A-bimodules. Now we compute the inverse dualizing complex of Γ using the resolution 3.1. We have isomorphisms in the pseudo-compact derived category of Γ-bimodules RHom Γ e (Γ, Γ e ) = Hom Γ e (Γ⊗ τ DA⊗ τ Γ, Γ e ) = Hom τ,τ psc (DA, Γ e ) = Hom τ,τ psc (Σ d A, Γ e ) = Σ −d Γ⊗ τ DA⊗ τ Γ = Σ −d Γ. Here Hom τ,τ psc denotes the space of morphisms in the category of pseudo-compact vector spaces twisted twice by τ . This shows that Γ is topologically homologically bimodule d-Calabi-Yau. 3.1.4. CY structures in geometry, algebraic consequences. We let k = C be the field of complex numbers unless specified otherwise. Definition 3.7. Let Y be a d-dimensional smooth quasi-projective C-variety. We call Y a d- dimensional Calabi-Yau variety if there is an isomorphism ω Y := Ω d Y ∼ = O Y , i.e. there exists a nowhere vanishing d-form. We call a nowhere vanishing section η : O Y → ω Y a dCY structure on Y . We call the dCY structure exact if the the d-form η is exact, i.e. there exists a (d − 1)-form ξ ∈ Ω d−1 Y such that dξ = η. A choice of ξ will be called an exact lifting of the dCY structure η. If Y is a smooth formal scheme, we may define dCY structure in a similar way by considering de Rham complex of formal scheme Ω * Y . Given a smooth quasi-projective variety Y of dimension d and a bounded complex of coherent sheaves L on Y whose cohomology has proper support, the derived endomorphism algebra A := RHom D(Qcoh Y ) (L, L) is a proper dga. We will show that A is perfectly dCY if Y is d-Calabi-Yau. Lemma 3.8. Let Y be a quasi-projective smooth CY d-fold with a fixed CY structure. Let L ∈ D b c (cohY ) be a bounded complex of coherent sheaves with proper support. Then A = RHom D(Qcoh Y ) (L, L) is perfectly d-Calabi-Yau. Proof. Since we work over a field of characteristic zero, there exists Y , a smooth projective variety that compactifies Y . Denote by i : Y → Y the canonical embedding. Since i is an open immersion and L has proper support on Y , RHom D(QcohY ) (i * L, i * L) is quasi-isomorphic to A as dg algebras. From now on, we simply assume that A = RHom D(QcohY ) (i * L, i * L). We adopt the notation of the Appendix to write Qcoh(Y ) for the dg category of (fibrant replacements of) unbounded complexes of quasi-coherent sheaves and coh(Y ) for its full dg subcategory of complexes with coherent cohomology and bounded cohomological amplitude. By Theorem 7.1 there is a bifunctorial quasi-isomorphism DHom dg coh(Y ) (M, N ) Hom dg coh(Y ) (N, M ⊗ ω Y [d]). Now let M = N = i * L. Then M ⊗ ω Y ∼ = i * (L⊗ω Y ) ∼ = i * L using the Calabi-Yau structure on Y . By the bifunctoriality this is an isomorphism in D(A e ). Proposition 3.9. Let {E i } t i=1 be a semi-simple collection of compactly supported sheaves in a smooth quasi-projective CY d-fold Y . Write Γ := Γ Y E for the derived deformation algebra of E := t i=1 E i . Then Γ is topologically homologically smooth and bimodule d-Calabi-Yau. Proof. Let A be the derived endomorphism algebra of E. Since Y is smooth and E has compact support, it is proper. Moreover, it can be chosen augmented. Clearly, it has its homology concentrated in non negative degrees and H 0 (A) is isomorphic to a product of t copies of k, which is l. By Corollary 4.1 of [56], the dg algebra A belongs to the triangulated subcategory of its derived category generated by the semi-simple object l. We know that Γ is quasi-isomorphic to DBA. Thus, by Lemma 3.6, Γ is topologically homologically smooth. Since Y is smooth and d-Calabi-Yau, A is perfectly d-Calabi-Yau by Lemma 3.8. Hence Γ is bimodule d-Calabi-Yau by Lemma 3.6. Remark 3.10. If we assume that Y is smooth and projective, then one can show that Ext * Y (E, E) is a cyclic A ∞ -algebra. This can be proved by reducing to the analytic case and applying the holomorphic Chern-Simons theory (see Example 10.2.7 [60]). An algebraic proof of the case when Y is a projective Gorenstein CY curve can be found in [76]. Proposition 3.11. Let {E i } t i=1 be a semi-simple collection of sheaves in a smooth projective CY d-fold Y . Write Γ := Γ Y E for the derived deformation algebra of E := t i=1 E i . Then Γ is an exact d-CY-algebra. Proof. As in the proof of Lemma 3.9, we see that Γ is topologically homologically smooth and bimodule d-CY. The exactness of the CY-structure follows from Theorem 12.1 of [84] and the remark above. Theorem 3.12. (Iyama-Reiten) Let R be an equi-codimensional Gorenstein normal domain of dimension d over an algebraically closed field k, and let A be an NCCR. Then A satisfies the relative dCY property. Moreover, if R is complete local then A is bimodule dCY. Proof. The fact that A satisfies the relative dCY property is proved in Theorem 4.23 of [86]. We have A = End R (R ⊕ N ) for the Cohen-Macaulay module N = t i=1 N i with N 1 , . . . , N t indecomposable. Denote by l the algebra k t+1 . By Theorem 1.1 of [81], the algebra A is quasi-isomorphic to a pseudo-compact dg algebra Γ := ( T l (V ), d) in PCAlgc(l) for a finite-dimensional graded l-bimodule V concentrated in degrees ≤ 0 and a differential d taking V into the square of the kernel of the augmentation T l (V ) → l. Since A is of finite global dimension, Γ is homologically smooth and by the first part, we have bifunctorial isomorphisms Hom D( Γ) (M, P ) = DHom D( Γ) (P, Σ d M ) for M in D f d ( Γ) and P in per( Γ). Let Θ = RHom Γ e ( Γ, Γ e ) be the inverse dualizing complex of Γ. By Lemma 4.1 of [51], we have bifunctorial isomorphisms Hom D( Γ) (L L ⊗ Γ Θ, M ) = DHom D( Γ) (M, L) for M in D f d ( Γ) and an arbitrary object L of D( Γ). By combining these with the previous isomorphisms we find Hom D( Γ) (P L ⊗ Γ Θ, M ) = Hom D( Γ) (Σ −d P, M ) for P in per( Γ) and M in D f d ( Γ). Since an object L of D( Γ) is perfect if and only if Hom D( Γ) (L, M ) is finite-dimensional for each M in D f d ( Γ) , we see that P L ⊗ Γ Θ is perfect for each perfect P . Now by taking P = Γ and M = Σ n S i , where n ∈ Z and S i is one of the t + 1 simple Γ-modules, we see that as a right Γ-module, Θ is quasi-isomorphic to Σ −d Γ. For the rest of the argument, let us replace Γ by the quasi-isomorphic pseudocompact algebra G = H 0 ( Γ), which is isomorphic to the original algebra A. Since G and Γ have canonically equivalent derived categories and derived categories of bimodules, we can view Θ as a G-bimodule complex concentrated in degree d. After replacing it with a quasi-isomorphic bimodule complex, we may assume that it is an actual G-bimodule concentrated in degree d. Moreover, we know that as a right module, it is isomorphic to G. Thus, there is an automorphism σ of G such that Σ d Θ is isomorphic to σ G as a bimodule. Since each object L of D f d (G) is perfect, we have Hom D(G (L σ , M ) = Hom D(G) (L, M ) for all L and M in D f d (G), which shows that there is a functorial isomorphism L ∼ → L σ for each L in D f d (G). In particular, for L, we can take the finite-dimensional quotients of G. We deduce that in each finite-dimensional quotient of G, the automorphism σ induces an inner automorphism. Thus, σ itself is inner and σ G is isomorphic to G as a bimodule. This shows that Θ is quasi-isomorphic to Σ −d Γ as a bimodule. Corollary 3.13. Let R be a local complete equi-codimensional Gorenstein normal domain of dimension d over an algebraically closed field k of characteristic zero, and let A be an NCCR. Let Γ be the dg algebra constructed in Theorem 2.8. Then Γ is topologically homologically smooth and bimodule dCY. Classification of CY structures for 3-dimensional flopping contractions. If ( Y , f , R) is a 3-dimensional formal flopping contraction, then R is a hypersurface. The natural isomorphism f * ω Y ∼ = ω R identifies a CY structure η on Y with a nonzero section f (η) of ω R . By the Gorenstein property, f (η) defines an isomorphism R ∼ = ω R . Theorem 3.14. Let ( Y , f , R) be a 3-dimensional formal flopping contraction. The space of 3CY structures can be identified with R × . Moreover, every Calabi-Yau structure on Y is exact. The space of all exact liftings of a 3CY structure can be identified with the cohomology group H 1 ( Y , Ω 1 Y ). Proof. Assume that C := Ex(f ) has t irreducible components C 1 , . . . , C t . Because R has rational singularities, H 0 ( Y , Ω 3 Y ) ∼ = H 0 ( Y , O Y ) ∼ = R. Then the first claim follows. The Hodge-to-de Rham spectral sequence with E 1 term E pq 1 = H q ( Y , Ω p Y ), converges to H p+q DR ( Y , C). We claim that H 1 (Ω 1 Y ) ∼ = C t . Because the first Chern classes of L i for i = 1, . . . , t are linearly independent, dim C H 1 (Ω 1 Y ) ≥ t. We write X := Spec R. By the Leray spectral sequence H p ( X, R q f * Ω 1 ) ⇒ H p+q ( Y , Ω 1 ), we have an exact sequence 0 → H 0 ( X, R 1 f * Ω 1 ) → H 1 ( Y , Ω 1 ) → H 1 ( X, f * Ω 1 ) → 0 The right most term vanishes since X is affine. Since Y is a small resolution of 3-dimensional Gorenstein singularities, the normal bundle of C i is O(a)⊕O(b) with (a, b) = {(−1, −1), (0, −2), (1, −3)} ([72, Theorem 4]). By the exact sequence 0 → O(−a)⊕O(−b) → Ω 1 | Ci → O(−2) → 0 we have H 1 (C i , Ω 1 | Ci ) ∼ = H 1 (C i , O(−2)) = C. By the normal crossing condition, there is a short exact sequence of sheaves 0 / / Ω 1 | C → Ω 1 | C1 ⊕Ω 1 | ∪ t i=2 Ci / / Ω 1 |p / / 0 where p = C 1 ∩ (∪ t i=2 C i ). Then we get a surjection H 1 (C, Ω 1 | C ) → H 1 (C 1 , Ω 1 | C1 )⊕H 1 (∪ t i=2 C i , Ω 1 ∪ t i=2 Ci ). By induction, dim C H 1 (C, Ω 1 | C ) ≤ t. Then the conclusion follows from the formal function theorem and the Lerray spectral sequence. The term E 30 2 of the Hodge-to-de Rham spectral sequence is the quotient H 0 ( Y , Ω 3 )/dH 0 ( Y , Ω 2 ). Recall that H 1 ( Y , Ω 1 ) admits a C-basis by c 1 of the line bundles L i with i = 1, . . . , t. Because (1, 1) classes are d-closed, the differential H 1 ( Y , Ω 1 ) → H 0 ( Y , Ω 3 )/dH 0 ( Y , Ω 2 ) is zero. Moreover since H 2 (O Y ) = 0, we have E 30 r = E 30 2 for r ≥ 2. Because H 3 DR ( Y , C) ∼ = H 3 DR (C, C) = 0, E 30 2 must vanish. Therefore, all 3-forms are exact. Denote by Ω ≤i Y the stupid truncation σ ≤i Ω * Y of the de Rham complex. There is a long exact sequence of hypercohomology . . . B / / H i−n ( Y , Ω i Y ) I / / H 2i−n ( Y , Ω ≤i Y ) S / / H 2i−n ( Y , Ω ≤i−1 Y ) B / / H i−n+1 ( Y , Ω i Y ) / / . . .2 ( Y , Ω ≤3 Y ) = H 2 DR ( Y ) ∼ = H 1 ( Y , Ω 1 Y ) . So the last claim is proved. Proof. This is an immediate consequence of Theorem 3.12, Theorem 2.11 and Corollary 9.3 of [84]. The following proposition follows immediately from the Hochschild-Kostant-Rosenberg theorem. Proposition 3.16. Let ( Y , f , R) be a 3-dimensional formal flopping contraction. Let A = End R (R⊕N ) be the corresponding NCCR. Then there is a bijective correspondence between the space of 3CY structures (resp. exact 3CY structures) on Y and that of A. 3.3. Cluster category. Let Γ be a dg k-algebra. Suppose that Γ has the following properties: (1) Γ is homologically smooth, i.e. Γ is a perfect Γ e -module; (2) for each p > 0, the space H p Γ vanishes; (3) H 0 Γ is finite dimensional; (4) Γ satisfies the relative 3CY property. By property (1), D f d (Γ) is a subcategory of the perfect derived category per(Γ). The generalized cluster category C Γ is defined to be the triangle quotient per(Γ)/D f d (Γ). We denote by π the canonical projection functor π : per(Γ) → C Γ . For simplicity, we will omit the adjective 'generalized' and call C Γ the cluster category associated to Γ. An object T ∈ C Γ is called a cluster-tilting object if (1) Ext 1 CΓ (T, T ) = 0; (2) For any object X such that Ext 1 CΓ (T, X) = 0, one has X ∈ add(T ). Amiot has proved that π(Γ) is a cluster-tilting object in C Γ (Theorem 2.1 [3]). We call H 0 Γ the CY tilted algebra associated to the cluster category C Γ , cf. [69]. Theorem 3.19. [3, Theorem 2.1] 2 Let Γ be a dg k-algebra with the above properties. Then the cluster category C Γ is Hom-finite and 2CY as a triangulated category. Moreover, the object π(Γ) is a cluster tilting object. Its endomorphism algebra is isomorphic to H 0 Γ. Definition 3.20. Let {C i } t i=1 be a collection of smooth rational curves in a smooth quasi-projective CY 3-fold Y with fixed CY-structure η : O Y ∼ → ω Y , such that {O Ci } form a semi-simple collection. Denote by C(Y, {C i } t i=1 , η) for the (topological) cluster category associated to the derived deformation algebra of t i=1 O Ci . We call C(Y, {C i } t i=1 , η) the cluster category associated to the triple (Y, η, {C i } t i=1 ) . Definition 3.21. Let R be a complete local equicodimensional Gorenstein normal domain of dimension 3 over an algebraically closed field k of characteristic zero, and let A be the NCCR associated to the collection of indecomposables R, N 1 , . . . , N t . Fix a 3CY structure η ∈ HH 3 (A, A). Denote by C(R, {N i } t i=1 , η) the cluster category associated to the dg algebra Γ constructed in Theorem 2.8, and call it the cluster category associated to the triple (R, {N i } t i=1 , η). 2 In the original statement of [3], the author assumed that Γ is bimodule 3CY. However, the proof is still valid under the weaker assumption that Γ satisfies the relative 3CY property. A priori, the dg algebra Γ constructed in Theorem 2.8 is pseudo-compact. However, if D(Γ) denotes the ordinary derived category and D pc (Γ) the pseudo-compact derived category, then the natural functor D pc (Γ) → D(Γ) induces equivalences in the perfect derived categories and in the subcategories of objects with finite-dimensional total homology. Therefore, the two candidates for the cluster category are equivalent. The following result is an immediate consequence of Corollary 2.12. Corollary 3.22. Let ( Y , f , R) be a 3-dimensional formal flopping contraction, and let A be the NCCR associated to the collection of indecomposables R, N 1 , . . . , N t constructed in Section 2.5. Fix a 3CY structure η on Y and denote its counter part on A by the same symbol. Then there is a triangle equivalence bewteen the cluster categories C(Y, {C i } t i=1 , η) C(R, {N i } t i=1 , η). Ginzburg algebras In this section we introduce the notion of Ginzburg (dg) algebra and prove several properties of it. The cluster category can be defined via the Ginzburg algebra, which provides an effective tool to do computations. 4.1. Definitions. Fix a commutative ring k. Let Q be a finite quiver, possibly with loops and 2-cycles. Denote by Q 0 and Q 1 the set of nodes and arrows of Q respectively. Denote by kQ the path algebra and by kQ the complete path algebra with respect to the two-sided ideal generated by arrows. For each a ∈ Q 1 , we define the cyclic derivative D a with respect to a as the unique linear map (Ginzburg) Let Q be a finite quiver and w a potential on Q. Let Q be the graded quiver with the same vertices as Q and whose arrows are • the arrows of Q (of degree 0); • an arrow a * : j → i of degree −1 for each arrow a : i → j of Q; • a loop t i : i → i of degree −2 for each vertex i of Q. The (complete) Ginzburg (dg)-algebra D(Q, w) is the dg k-algebra whose underlying graded algebra is the completion (in the category of graded vector spaces) of the graded path algebra kQ with respect to the two-sided ideal generated by the arrows of Q. Its differential is the unique linear endomorphism homogeneous of degree 1 satisfying the Leibniz rule, and which takes the following values on the arrows of Q: • da = 0 for a ∈ Q 1 ; • d(a * ) = D a w for a ∈ Q 1 ; • d(t i ) = e i ( a∈Q1 [a, a * ])e i for i ∈ Q 0 where e i is the idempotent associated to i. Denote by l the product i∈Q0 ke i . Then kQ is isomorphic to the complete tensor algebra T l V with V being the vector space spanned by arrows of Q. Remark 4.2. In most references, the above definition corresponds to the complete Ginzburg algebra while the algebra without taking the graded completion is called Ginzburg algebra. The complete Ginzburg algebra D(Q, w) is considered as an object of PCAlgc(l). Because we only consider complete Ginzburg algebra in this paper, we will call it the Ginzburg algebra for simplicity. Definition 4.3. Let Q be a finite quiver and w a potential on Q. The Jacobi algebra Λ(Q, w) is the zeroth homology of D(Q, w), which is the quotient algebra kQ/((D a w, a ∈ Q 1 )) c where ((D a w, a ∈ Q 1 )) c is the closed two-sided ideal generated by D a w. A Ginzburg algebra D(Q, w) is called Jacobi-finite if dim k Λ(Q, w) < ∞. Van den Bergh showed the following result. The above theorem was first proved by Van den Bergh in the algebraic setting in [52]. But the same proof can be adapted to the pseudo-compact case (cf. [84]). Given a Jacobi-finite Ginzburg algebra Γ := D(Q, w), there is an associated cluster category C Γ := per(Γ)/D f d (Γ). For a Ginzburg algebra Γ = D(Q, w), denote Λ(Q, w) by Λ for short. The image of w under the canonical map HH 0 ( kQ, kQ) = kQ cyc → HH 0 (Λ, Λ) = Λ cyc , denoted by [w], is a canonical class associated to the Ginzburg algebra D(Q, w). Therefore, we see that starting from a Ginzburg algebra Γ = D(Q, w) we get not only a triangulated category C Γ but an additional piece of information that is a canonical class [w] in the 0-th Hochschild homology of the CY tilted algebra. We will show in the next section that this class is determined by the CY structure up to right equivalences. 4.2. Existence and uniqueness of potential. The definition of Ginzburg algebra is not homotopically invariant. It is important to know when a bimodule 3CY dg algebra admits a model given by a Ginzburg algebra. The following theorem is due to Van den Bergh. Let k be a field and l be a finite dimensional commutative separable k-algebra. Assume that Γ is a pseudo-compact dg l-algebra in PCAlgc(l) concentrated in nonpositive degrees. Then the following are equivalent (1) Γ is exact 3CY. (2) Γ is weakly equivalent to a Ginzburg algebra D(Q, w) for some finite quiver Q with w contains only cubic terms and higher. The following result of Van den Bergh provides a lot of examples of dg algebras whose 3CY structures can be lifted to exact ones. By putting the above two theorems together, we see that if Γ is a pseudo-compact dg l-algebra in PCAlgc(l) concentrated in degree zero that is bimodule 3CY then it is quasi-isomorphic to a Ginzburg algebra D(Q, w) for some finite quiver Q and potential w. Let Γ be a pseudo-compact 3CY dg l-algebra in PCAlgc(l) with a fixed exact 3CY structure τ ∈ HH 3 (Γ, Γ). Suppose there are two pairs (Q, w) and (Q , w ) such that Γ is quasi-isomorphic to D(Q, w) and D(Q , w ) respectively. Assume that under these quasi-isomorphisms we have Bχ and Bχ identified with τ where χ, χ are the canonical classes in HC 2 defined in Remark 4.5. Then Q = Q and w is right equivalent to w . In particular, the classes [w], [w ] ∈ H 0 Γ/[H 0 Γ, H 0 Γ] c are right equivalent. Proof. In Theorem 11.2.1 of [83], it is proved that there is a weak equivalence ( T l V, d) → Γ where V = Σ −1 (DExt * Γ (l, l)) ≤0 such that (1) V = V c + lz with V c = Σ −1 (DExt 1 Γ (l, l) ⊕ DExt 2 Γ (l, l)) and z being an l-central element of degree −2. (2) dz = σ ησ with η ∈ (V c ⊗ l V c ) l being a non-degenerate and anti-symmetric element. Here σ ⊗σ ∈ l⊗ k l is the Casimir element corresponds to the trace l → k (see Section 5 of [83]). Since Q (resp. Q ) depends only on l-bimodule structure on Ext 1 Γ (l, l) (see [82,Section 10.3], we have Q = Q . Note that the perfect pairing on V c is determined by the bimodule 3CY structure τ but does not depend on the exact lifting. Using the perfect pairing, any non-degenerate and antisymmetric element in (V c ⊗ l V c ) l can be reduced to a canonical form by choosing appropriate basis on Ext 1 Γ (l, l). The element η defines a bisymplectic form ω η of degree −1 on T l V (see definition in Section 10.1 of [83]). By Lemma 11.3.1 of [83] there exists a potential w ∈ T l V /[ T l V, T l V ] of degree zero such that for any f ∈ T l V df = {w, f } ωη where {−, −} ωη is the Poisson bracket associate to the bisymplectic form ω η . Since w does not have constant terms, it is uniquely determined by the derivation d = {w, −} ωη . In other words, any two potentials w and w without constant terms satisfying the above equation differ by an automorphism T l V → T l V . Moreover, since w and w are of degree zero, therefore does not involve variables in Ext ≥2 Γ (l, l), this automorphism is precisely a formal change of variables on Ext 1 Γ (l, l). Such a formal change of variable induces an isomorphism from D(Q, w) to D(Q, w ) (see [35,Theorem 4.3]), therefore an automorphism of the complete path algebra of Q and an automorphism of H 0 Γ. Remark 4.9. Theorem 11.2.1 of [83], can be viewed as a Darboux-Weinstein theorem in noncommutative formal symplectic geometry. On Ext * Γ (l, l), the cyclic A ∞ -structure can be interpreted as a symplectic structure. The symplectic structure restricts to the truncation Ext 1 Γ (l, l)⊕Ext 2 Γ (l, l) so that Ext 1 Γ (l, l) is a (graded) Lagrangian. Then Theorem 11.2.1 of [83] says that there exists a coordinate on Ext 1 Γ (l, l) under which the symplectic form can be normalized so that it has constant coefficients, which is in particular exact. The differential d of Γ can be interpreted as a homological vector field of degree 1. Then the contraction of the normalized symplectic form by d is the exterior derivation of a potential w of degree 0. Note that a different choice of Darboux coordinates can only differ by a change of variables on Ext 1 Γ (l, l), which leads to the above proposition. (f ) = t i=1 C i , and let A = End R ( t i=1 N i ⊕R) be the NCCR associated to it. Fix a 3CY structure η on Y , and therefore on A. Denote the CY tilted algebra of C(Y, {C i } t i=1 , η) C(R, {N i } t i=1 , η) by Λ. Then there exists a canonical class, defined up to right equivalence, on HH 0 (Λ) = Λ/[Λ, Λ] c represented by a potential. The canonical class [w] in the 0-th Hochschild homology of H 0 Γ is part of the "classical shadow" of the CY structure. The class plays a crucial role in the geometric applications. When Γ is weakly equivalent to a Jacobi-finite Ginzburg algebra D( F , w) for a complete free algebra F = k x 1 , . . . , x n , then this class vanishes if and only if w is right equivalent to a weighted homogeneous noncommutative polynomial (see Theorem 4.16). Therefore, the quasi-homogeneity of a potential is indeed a homotopy invariant of the CY algebra. This motivates the following definition. Definition 4.11. Let k be a field and l be a finite dimensional commutative separable k-algebra, and let Γ be a pseudo-compact dg l-algebra in PCAlgc(l) concentrated in nonpositive degrees. Assume that Γ is exact 3CY. Then Γ is called quasi-homogeneous if the canonical class [w] is (right equivalent to) zero. The notion of quasi-homogeneity is expected to be independent of choices of CY structure. In the case of simple flopping contractions, the first author and Gui-song Zhou have conjectured that this notion of quasi-homogeneity is indeed equivalent to the quasi-homogeneity of the underlying hypersurface singularity R (see Conjecture 4.18 [35]). 4.3. Properties of Jacobi-finite Ginzburg algebras. In this section, we collect several results about Jacobi-finite Ginzburg algebras. We take k to be the field of complex numbers, though some of the results are valid more generally. Theorem 4.12. [35, Theorem 3.16] Let Q be a finite quiver and w be a potential in CQ cyc . Assume that the Jacobi algebra Λ(Q, w) is finite dimensional. Then w is right equivalent to a formal series with only finitely many nonzero terms. As a consequence, we may assume the potential is a noncommutative polynomial to begin with if the Jacobi algebra is known to be finite dimensional. Let Q be a finite quiver and let w, w ∈ CQ cyc be two potentials with only cubic terms and higher. Suppose that the Jacobi algebras Λ(Q, w) and Λ(Q, w ) are both finite dimensional. Then the following two statements are equivalent: (1) There is an algebra isomorphism γ : Λ(Q, w) ∼ = Λ(Q, w ) so that γ * ([w]) = [w ] in Λ(Q, w ) cyc . (2) w and w are right equivalent in CQ cyc . The noncommutative Mather-Yau theorem has an immediate application to Ginzburg algebras. such that β(t i ) = t i for any i ∈ Q 0 . Definition 4.15. Fix F to be the complete free associative algebra C x 1 , . . . , x n . Let (r 1 , . . . , r n ) be a tuple of rational numbers. A potential w ∈ F cyc := F /[ F , F ] c is said to be weighted-homogeneous of type (r 1 , . . . , r n ) if it has a representative which is a linear combination of monomials x i1 x i2 · · · x ip such that r i1 + r i2 + . . . r ip = 1. Let w ∈ F cyc be a potential with only cubic terms and higher such that the Jacobi algebra associated to w is finite dimensional. Then [w] = 0 if and only if w is right equivalent to a weighted-homogenous potential of type (r 1 , . . . , r n ) for some rational numbers r 1 , . . . , r n which lie strictly between 0 and 1/2. Moreover, in this case, all such types (r 1 , . . . , r n ) agree with each other up to permutations on the indices 1, . . . , n. Recall that C Γ is constructed as the Verdier quotient of the perfect derived category of Γ by its full subcategory of dg modules whose homology is of finite total dimension. The category of perfect dg Γ-modules has a canonical dg enhancement and we obtain a natural dg enhancement (C Γ ) dg for C Γ using the Drinfeld quotient of the dg category of strictly perfect dg Γ-modules by its full subcategory on the dg modules whose homology is of finite total dimension. Theorem 4.17. Let Q be a finite quiver and w a Jacobi-finite potential on Q. Let Γ be the complete Ginzburg algebra associated with (Q, w). Denote by Λ dg the dg endomorphism algebra of Γ in the canonical dg enhancement of the cluster category C Γ . Then there is a canonical isomorphism in the homotopy category of dg algebras Γ ∼ / / τ ≤0 Λ dg . Proof. There is a canonical morphism Γ = RHom Γ (Γ, Γ) → (C Γ ) dg (Γ, Γ) = Λ dg where the right hand side denotes the dg endomorphism algebra of Γ in the canonical dg enhancement (C Γ ) dg of C Γ . It suffices to show that the canonical map H −p (Γ) = Hom per(Γ) (Γ, Σ −p Γ) → Hom CΓ (Γ, Σ −p Γ) is invertible for p ≥ 0. By Proposition 2.8 of [3], we have Hom CΓ (Γ, Σ −p Γ) = colim n Hom per(Γ) (τ ≤n Γ, τ ≤n (Σ −p Γ)). We have Hom per(Γ) (τ ≤n Γ, τ ≤n (Σ −p Γ)) = Hom per(Γ) (τ ≤n Γ, Σ −p Γ). Consider the canonical triangle τ ≤n Γ / / Γ / / τ >n Γ / / Σ(τ ≤n Γ). Recall that by Lemma 2.5 of [3], the spaces H p (Γ) are finite-dimensional for all p ∈ Z. Therefore, the object τ >n Γ belongs to D f d (Γ) and Σ −p Γ belongs to per(Γ). Corollary 4.18. Let Q be a quiver with one node and arbitrary number of loops and Γ = D(Q, w) a Jacobi-finite Ginzburg algebra. Then H 0 Γ is self-injective and there is an isomorphism Σ 2 Γ ∼ / / τ ≤−1 Γ in the derived category of dg Γ-modules. In particular, we have H i (Γ) = 0 for odd i and H i (Γ) ∼ = H 0 (Γ) for even i ≤ 0. Proof. By [3], the cluster category C Γ is a Hom-finite 2-Calabi-Yau category and the image T of Γ in C Γ is a cluster-tilting object in C Γ . By Theorem 4.1 of [1], the cluster-tilting objects of C Γ are in bijection with the support τ -tilting modules over End(T ). Since End(T ) = H 0 (Γ) is local, the only support τ -tilting modules over End(T ) are 0 and End(T ), by example 6.1 of [loc. cit.]. Thus, the only cluster-tilting objects of C Γ are T and ΣT . In particular, Σ 2 T has to be isomorphic to T (since Hom(ΣT, Σ 2 T ) = 0 and Σ 2 T must be a cluster-tilting object). This implies that H 0 Γ = End(T ) is self-injective, since, by the 2-Calabi-Yau property, we have an isomorphism of right End(T )-modules DHom(T, T ) = Hom(T, Σ 2 T ) = Hom(T, T ). Let φ : Σ 2 Γ → Γ be a lift of an isomorphism Σ 2 T → T in C Γ . Let p ≥ 2. In the commutative square Hom per(Γ) (Σ p Γ, Σ 2 Γ) / / Hom CΓ (Σ p T, Σ 2 T ) φ * Hom per(Γ) (Σ p Γ, Γ) / / Hom CΓ (Σ p T, T ) the horizontal arrows are isomorphisms by Theorem 4.17 and the right vertical arrow φ * is an isomorphism. Thus, the morphism φ : Σ 2 Γ → Γ induces isomorphisms in H i for i ≤ −2. Moreover, we have H −1 (Γ) = Hom(T, Σ −1 T ) = Hom(T, ΣT ) = 0 since Σ −1 T is isomorphic to ΣT . It follows that φ induces an isomorphism Σ 2 Γ ∼ → τ ≤−1 Γ. Remark 4.19. For the pair (Q, w) associated to 3-dimensional flopping contractions, one can show that H 0 Γ is indeed symmetric (see Theorem 6.5). In the context of general contraction with one dimensional fiber, Kawamata has proved that the classical (multi-pointed) deformation algebra of the reduced exceptional fiber is always self-injective (see Proposition 6.3 [42]). So in particular it is Gorenstein. This result overlaps with the above corollary in the case of simple flopping contractions. For a general finite quiver Q, the 0-th homology of a Jacobi-finite Ginzburg algebra D(Q, w) is not self-injective. Moreover, Kawamata proves that the deformation algebra is always isomorphic to its opposite algebra (see Corollary 6.3 of [43]). Proof. a) By Theorem 4.17, the space H −1 Γ is isomorphic to C Γ (T, Σ −1 T ) and H 0 Γ is isomorphic to the endomorphism algebra of T . By Proposition 3.6 of [38], the endomorphism algebra is selfinjective if and only if C Γ (T, Σ −1 T ) vanishes if and only if T is isomorphic to Σ 2 T in C Γ . b) By combining the functorial isomorphism from T to Σ 2 T with the Calabi-Yau property we get an isomorphism of bimodules over the endomorphism algebra of T C Γ (T, T ) ∼ → C Γ (T, Σ 2 T ) ∼ → DC Γ (T, T ). Since H 0 Γ is in particular selfinjective, the space H −1 Γ vanishes by a). We get an isomorphism of graded algebras k[u, u −1 ]⊗ k C Γ (T, T ) ∼ → p∈Z C Γ (T, Σ p T ) where u is of degree 2 by sending u to the functorial isomorphism T ∼ → Σ 2 T . Thanks to Theorem 4.17, by truncation, we get an isomorphism of graded algebras k[u −1 ]⊗ k H 0 Γ ∼ → H * Γ. 4.4. Silting theory for a non positive dg algebra and its zeroth homology. Let T be a triangulated category. Recall that a tilting object for T is a classical generator T of T such that T (T, Σ p T ) vanishes for all p = 0. A silting object [55] for T is a classical generator T of T such that T (T, Σ p T ) vanishes for all p > 0. The advantage of silting objects over tilting objects is that (under suitable finiteness assumptions) they are stable under mutation [2]. We recall fundamental definitions and results from [2]. Assume from now on that T is k-linear, Hom-finite and has split idempotents. In particular, it is a Krull-Schmidt category, i.e. indecomposables have local endomorphism rings and each object is a finite direct sum of indecomposables (which are then unique up to isomorphism and permutation). An object of T is basic if it is a direct sum of pairwise non isomorphic indecomposables. If X is an object of T and U a full additive subcategory stable under retracts, a left U-approximation of X is a morphism f : X → U to an object of U such that each morphism X → V to an object of U factors through f : X → U . It is minimal if each endomorphism g : U → U such that g • f = f is an isomorphism. Notice that the morphism f : X → U is a minimal left U-approximation iff the morphism f * : U(U, ?) → T (X, ?)| U is a projective cover in the category of left U-modules. In particular, minimal left approximations are unique up to non unique isomorphism when they exist. Existence is automatic if U has finitely many indecomposables U 1 , . . . , U n (which is the case in our applications) because then the functor T (X, ?)| U corresponds to a finite-dimensional left module over the finite-dimensional endomorphism algebra of the sum of the U i . A (minimal) right U-approximation is defined dually. For an object X of T , we denote by addX the full subcategory formed by all direct factors of finite direct sums of copies of X. Let M be a basic silting object of T and X an indecomposable direct summand of M . Denote by M/X the object such that M ∼ = X ⊕ M/X. By definition, the left mutation µ X (M ) of M at X is the silting object M/X ⊕ Y , where Y is defined by a triangle . Conversely, if we start from a dg algebra Γ whose homologies are finite-dimensional and vanish in degrees > 0 and from an algebraic triangle equivalence D(Γ ) ∼ → D(Γ), then the image M of Γ in per(Γ) is a silting object. In any case, the dg algebra Γ is an exact bimodule 3-Calabi-Yau and has its homology concentrated in degrees ≤ 0. By Van den Bergh's theorem [84], the dg algebra Γ is again a Jacobi-finite Ginzburg algebra (up to weak equivalence). In particular, for M we can take the mutation M = µ X Γ, where X = e i Γ for a vertex i of the quiver of Γ. We define the associated Ginzburg algebra Γ to be the left mutation of Γ at i. Notice that by construction, we have a canonical derived equivalence from Γ to Γ. In the same way, we can define the right mutation Γ of Γ at i using the right mutation M = µ − X (Γ) of Γ at X. The right mutation Γ turns out to be quasi-isomorphic to the left mutation Γ . Indeed, by Theorem 4.17, these algebras are the τ ≤0truncations of the derived endomorphism algebras of the images π(M ) and π(M ) in the cluster category C Γ . Now we have π(M ) ∼ = π(M ) because they are the left resp. right mutation in the sense of Iyama-Yoshino [37] of the cluster-tilting object π(Γ) at π(X) and for cluster-tilting objects in 2-Calabi-Yau triangulated categories, right and left mutation coincide up to isomorphism. Now let A be a dg k-algebra whose homologies H p A vanish in all degrees p > 0. An object of per(A) is called 2-term if it is the cone over a morphism of add(A). We write 2-per(A) for the full subcategory of per(A) formed by the 2-term objects. We write 2silt(A) for the set of isomorphism classes of 2-term silting objects. Our aim is to compare 2silt(A) with 2silt(H 0 A). Note that by our assumption on A, we have a canonical morphism A → H 0 A in the homotopy category of dg algebras. We write F : per(A) → per(H 0 A) for the derived tensor product over A with H 0 A. Part a) of the following theorem is due to Brüstle-Yang [14]. In particular, for each 2-term object T , the functor F induces an isomorphism X / / E / / Y / / ΣX and X → EEnd A (T ) ∼ → End H 0 A (F T ). Proof. Part a) is Proposition A.3 of [14]. For part b), using the assumption and the 5-lemma, we check successively that F induces the following bijections: 1) For P, Q ∈ add(A) and p ≥ −1 Hom(P, Σ p Q) ∼ → Hom(F P, Σ p F Q). Now let A be a pseudocompact dg algebra in PCAlgc(l) strictly concentrated in degrees ≤ 0. Let e be an idempotent of H 0 A and A the derived endomorphism algebra of the image of A in the Verdier quotient of per(A) by the thick subcategory generated by eA. Then A is concentrated in degrees ≤ 0 and we have a canonical morphism A → A in the homotopy category of PCAlgc(l). If A is of the form ( T l (V ), d) for a pseudocompact l-bimodule V concentrated in degrees ≤ 0, where T l (V ) is the completed tensor algebra, then A is quasi-isomorphic to the quotient of A by the twosided closed ideal generated by e (cf. [13]). Put A 0 = H 0 A so that we have a canonical morphism p : A → A 0 . Let B and B 0 be pseudocompact dg algebras in PCAlgc(l), X ∈ D(A op ⊗B) such that X B is perfect and Q ∈ D(B op ⊗B 0 ) such that Q B0 is perfect. 2) For X L ⊗ B Q ∼ → A 0 L ⊗ A0 Y in D(A op ⊗B 0 ). Thus, the square D(A) A0 X / / D(B) Q D(A 0 ) Y / / D(B 0 ) is commutative up to isomorphism, where we write dg bimodules instead of derived tensor products by dg bimodules. Remark 4.24. In our applications in this article, the idempotent e will be 0. We state and prove the proposition in the general case because it provides an alternative approach to the problem of relating the tilting theory of maximal modification algebras [87] to that of the associated contraction algebras as treated by August in [5]. Let R be a complete local cDV singularity and M a maximal basic rigid object in the category of Cohen-Macaulay modules over R containing R as a direct summand. We can take A = End R (M ) and e the idempotent corresponding to the projection on R. Then A 0 = H 0 A is isomorphic to the stable endomorphism algebra of M , i.e. the contraction algebra associated with M . Let N be another maximal basic rigid object containing R as a direct summand, B its endomorphism algebra and B 0 the associated contraction algebra. Then Since by assumption RHom B0 (Z, Z) is concentrated in degrees ≥ 0 and A in degrees ≤ 0, this morphism factors uniquely through a morphism A 0 = H 0 A → RHom B0 (Z, Z) in the homotopy category of dg algebras. Let us show how to refine this argument so as to obtain an object Y of D(A op 0 ⊗B 0 ) which restricts to Z ∈ D(A op ⊗B 0 ). We may and will assume that A is cofibrant and strictly concentrated in degrees ≤ 0. We may and will also assume that Z is cofibrant as a dg A -B 0 -bimodule. The left A -module structure on Z then yields a morphism of dg algebras A −→ Hom B0 (Z, Z). Since A is strictly concentrated in non positive degrees, it factors uniquely through a morphism of dg algebras A −→ τ ≤0 Hom B0 (Z, Z). Since Z B0 has no negative selfextensions, we have a surjective quasi-isomorphism of dg algebras τ ≤0 Hom B0 (Z, Z) −→ H 0 Hom B0 (Z, Z). The composition A −→ τ ≤0 Hom B0 (Z, Z) −→ H 0 Hom B0 (Z, Z) uniquely factors through an algebra morphism A 0 = H 0 (A ) → H 0 Hom B0 (Z, Z). We thus obtain a commutative square of dg algebra morphisms A / / ( ( Hom B0 (Z, Z) τ ≤0 Hom B0 (Z, Z) O O A 0 = H 0 (A ) / / H 0 Hom B0 (Z, Z). We factor the morphism A → A 0 as the composition A → A 0 → A 0 of an acyclic fibration with a cofibration. We consider the diagram A / / ( ( Hom B0 (Z, Z) A 0 / / τ ≤0 Hom B0 (Z, Z) O O A 0 = H 0 (A ) / / H 0 Hom B0 (Z, Z). Here the morphism represented by a dotted arrow exists so that the diagram becomes commutative because A → A 0 is a cofibration and τ ≤0 Hom B0 (Z, Z) → H 0 Hom B0 (Z, Z) is an acyclic fibration. Thus, we obtain a structure of dg A 0 -B 0 -bimodule on Z wich restricts to the given structure of dg A -B 0 -bimodule. Since we have the quasi-isomorphism A 0 → A 0 , we can find a bimodule Y in D(A op 0 ⊗B 0 ) unique up to isomorphism and which restricts (up to isomorphism) to Z in D(A op ⊗B 0 ). Now suppose we have a second object Y in D(A op 0 ⊗B 0 ) which becomes isomorphic to Z in D(A op ⊗H 0 B). We have a chain of isomorphisms Y ). Clearly, the composition Hom D(A op ⊗B0) (Y, Y ) = Hom D(A e ) (A , RHom B0 (Y, Y )) = Hom D(A e ) (H 0 A , H 0 RHom B0 (Y, Y )) = Hom H 0 (A ) e (H 0 A , H 0 RHom B0 (Y, Y )) = Hom D(A e 0 ) (A 0 , H 0 RHom B0 (Y, Y )) = Hom D(A e 0 ) (A 0 , RHom B0 (Y, Y )) = Hom D(A op 0 ⊗B0) (Y,Hom D(A op 0 ⊗B0) (Y, Y ) ∼ → Hom D(A op ⊗B0) (Y, Y ) of these isomorphisms is given by the restriction of scalars functor D(A 0 ⊗H 0 B) → D(A ⊗H 0 B). Now any restriction of scalars functor reflects isomorphisms because it is compatible with the forgetful functors to the derived category of vector spaces. Thus, isomorphisms are preserved. This shows the uniqueness. Let B be a dg k-algebra whose homologies are finite-dimensional and vanish in degrees > 0. Let C be a finite-dimensional basic k-algebra (i.e. C is basic as a right module over itself) and let Z be an object of D(C op ⊗H 0 B) such that ? L ⊗ C Z : D(C) → D(H 0 B) is an equivalence. Notice that Z H 0 B is a tilting object in per(H 0 B) and in particular a silting object, which is basic by our assumption on C. Theorem 4.25. Assume that Z H 0 B is a 2-term silting object and we have H −1 (B) = 0. Then there is a dg algebra A whose homologies H p A are finite-dimensional and whose components vanish in degrees p > 0, a derived equivalence ? L ⊗ A X : D(A) ∼ → D(B), an isomorphism of algebras φ : H 0 A ∼ → C and an isomorphism φ Z ∼ → X L ⊗ B H 0 B in D(A op ⊗H 0 B), where the left A-module structure on φ Z is defined via the composition A → H 0 A → C. In particular, we have a diagram, commutative up to isomorphism D(A) H 0 A z z X / / D(B) H 0 B D(H 0 A) φ C / / D(C) Z / / D(H 0 B) , where we write dg bimodules instead of derived tensor products by dg bimodules. Proof. By part a) of Proposition 4.22, there is a 2-term silting object M of per(B) such that M L ⊗ B H 0 B is isomorphic to Z H 0 B . Since M is silting, its derived endomorphism algebra has its homology concentrated in non positive degrees and we define A = τ ≤0 RHom(M, M ). We let X ∈ D(A op ⊗B) be the dg bimodule given by M with its canonical left A-action. Since X L ⊗ B H 0 B is isomorphic to the tilting object Z H 0 B , it has no self-extensions in degree < 0. Therefore, Proposition 4.23 yields an object Y of D(H 0 (A) op ⊗H 0 (B)) and an isomorphism ψ : X L ⊗ B H 0 B ∼ → Y | A op ⊗H 0 B in D(A op ⊗H 0 B),H 0 A ∼ → End D(H 0 B) (X L ⊗ B H 0 B). Via ψ, we get an isomorphism H 0 A ∼ → End D(H 0 B) (Y ) given by the left action of H 0 A on Y . We choose an isomorphism Y H 0 B ∼ → Z H 0 B in D(H 0 B) and define φ : H 0 A ∼ → C so as to make the following square commutative H 0 A / / φ End(Y H 0 B ) C / / End(Z H 0 B ). By Lemma 2.6, the chosen isomorphism Y H 0 B ∼ → Z H 0 B lifts to an isomorphism Y ∼ → φ Z in D(H 0 (A) op ⊗H 0 (B)). Whence a composed isomorphism X L ⊗ B H 0 B ∼ → Y | A op ⊗H 0 B ∼ → φ Z| A op ⊗H 0 B 4.5. Cyclic homology and preservation of the canonical class. Let k be a field of characteristic 0 and l a finite product of copies of k. Let V be a pseudocompact l-bimodule and d a continous differential on the completed tensor algebra T l (V ). Put A = ( T l (V ), d). We define Ω l A by the short exact sequence 0 / / Ω 1 A / / A⊗ l A µ / / A / / 0 , where µ is the multiplication of A. Then the morphism A⊗ l V ⊗ l A → Ω 1 A taking a⊗v⊗b to av⊗b − a⊗vb is an isomorphism of graded l-bimodules, cf. Example 3.10 of [77]. We can describe the induced differential on A⊗ l V ⊗ l A as follows (cf. Proposition 3.7 of [52]): Let D : A → A⊗ l V ⊗ l A be the unique continuous bimodule derivation which restricts to the map v → 1⊗v⊗1 on V . We have D(v 1 . . . v n ) = 1⊗v 1 ⊗(v 2 . . . v n ) + n−1 i=2 v 1 . . . v i−1 ⊗v i ⊗v i+1 . . . v n + (v 1 . . . v n−1 )⊗v n ⊗1. Then the induced differential on A⊗ l V ⊗ l A sends a⊗v⊗b to (−1) |a| aD(dv)b + (da)⊗v⊗b + (−1) (|v|+|a|) a⊗v⊗(db). (A⊗ l V ⊗ l A) ∼ → (V ⊗ l A) l taking a⊗v⊗b to (−1) |a|(|v|+|b|) v⊗ba. The induced differential on the right hand side is given as follows: If D(dv) = i a i ⊗v i ⊗b i , then d(v⊗a) = (−1) |v| v⊗(da) + i (−1) |ai|(|vi|+|bi|+|a|) v i ⊗b i aa i . Following section 3 of [77] we define morphisms of complexes ∂ 1 : (V ⊗ l A) l → A l and ∂ 0 : A l → (V ⊗ l A) l as follows: ∂ 1 sends v⊗a to va − (−1) |v||a| av and ∂ 0 sends v 1 . . . v n to i ±v i ⊗v i+1 . . . v n v 1 . . . v i−1 , where the sign is determined by the Koszul sign rule. We then have ∂ 0 ∂ 1 = 0 = ∂ 1 ∂ 0 . The (continuous) Hochschild homology of A is computed by the total complex of of the form ( T l (V ), d), where V is the l-bimodule with basis given by the arrows α, α * and t i . Let t be the sum of the t i . By definition, we have d(t) = α [α, α * ]. Thus, t defines an element in HC 2 (A). Proof. We compute S(t) using the above description of S. We need to lift t to an element of the total complex computing cyclic homology. We have d(t) = ∂ 1 ( α α⊗α * ). We have d( Proof. We know that the class [t ] ∈ HC 2 (Γ ) is non degenerate in the sense that B[t ] ∈ HH 3 (Γ ) defines an isomorphism Σ 3 Θ Γ ∼ → Γ in D(Γ e ), where Θ Γ is the inverse dualizing complex. Thus the image τ of [t ] under HC 2 (X) −1 is a non degenerate element of HC 2 (A). The proof of Theorem 10.2.2 in [84] then shows that there is a quiver Q , a potential w and a weak equivalence s : Γ(Q , w ) → A which takes [t ] to τ . Thus, the composition HC 2 (X) • HC 2 (s) = HC 2 ( s X) takes [t ] to [t ] and the isomorphism HC 0 ( s X) takes [w ] = S[t ] to [w ] = S[t ]. CY tilted algebras and singularities 5.1. Basics on Hochschild cohomology. Let k be a commutative ring and A be a unital k-algebra projective over k. Denote by A the quotient A/k · 1. Define the normalized bar complex associated to A to be the complex B k A := A⊗ k T ΣA⊗ k A with differential n−1 i=0 (−1) i b i : A⊗A ⊗n−1 ⊗A → A⊗A ⊗n−2 ⊗A b i (a 0 , . . . , a n ) = (a 0 , . . . , a i a i+1 , . . . , a n ). Let A be an augmented dg k-algebra. Denote by A the kernel of the augmentation. Then the bar complex B k A is equipped with a second differential induced from the differential d A on A. Given an A-bimodule M , the Hochschild cochain complex C * (A, M ) is equipped with a second differential d induced by d A and the internal differential d M on M. The ith Hochschild cohomology of the dg algebra A with coefficients in the bimodule M is defined to be HH i (A, M ) : = H i (C * (A, M ), d + δ). It is well-known that HH i (A, M ) is isomorphic to Ext i A e (A, M ). When A is a smooth commutative k-algebra, HH * (A, A) is isomorphic to the polyvector fields on Spec A by the Hochschild-Kostant-Rosenberg theorem. For non-smooth algebras, there exist different variants of Hochschild cohomology. Let A be an associative k-algebra projective over k. Define the module of Kähler differentials Ω A to be the kernel of the multiplication map µ : A⊗A → A. Clearly, Ω A inherits a bimodule structure from A⊗A. It is easy to show that Ω A is generated as a bimodule by the elements of the form xdy := xy⊗1 − x⊗y. The left and right module structure are given by a(xdy) = (ax)dy, (xdy)a = xd(ya) − xyda. Define the module of n-forms to be the n-fold tensor product Ω n A := Ω A ⊗ A Ω A ⊗ . . . ⊗ A Ω A . Using the above identities, one can check that Ω n A is generated as a bimodule by the elements of the form a 0 da 1 da 2 . . . da n . There is an isomorphism of bimodules Ω n A ∼ = A⊗ k A ⊗n defined by a 0 da 1 da 2 . . . da n → a 0 ⊗a 1 ⊗ . . . ⊗a n . Set Ω 0 A = A and Ω 1 A = Ω A . Write Ω * A for n≥0 Ω n A . The bimodule structure on Ω A naturally extends to an associative algebra structure on Ω * A . The obvious differential D : a 0 da 1 da 2 . . . da n → da 0 da 1 da 2 . . . da n makes Ω * A into a differential graded algebra. For m ∈ Z, C m (A, Σ n Ω n A ) = Hom k (A ⊗(n+m) , A⊗ k A ⊗n ). Consider the chain maps θ n : C * (A, Σ n Ω n A ) → C * (A, Σ n+1 Ω n+1 A ) between the Hochschild cochain complexes defined by f → f ⊗Id ΣA . Definition 5.1. Let A be an associative k-algebra. Then the singular Hochschild cochain complex of A, denoted by C * sg (A, A), is defined as the colimit of the inductive system in the category of cochain complexes of k-modules, such that the following diagram commutes 0 / / C * (A, A) θ0 / / C * (A, ΣΩ 1 A ) θ1 / / . . . / / C * (A, Σ n Ω n A ) θn / / . . . .Ext * A e (A, A) / / Ext * A e (A, A) HH * (A, A) ∼ = O O / / HH * sg (A, A) ∼ = O O From Wang's result, we see that the singular Hochschild cohomology admits a structure of graded commutative algebra. Notice that this is not immediate from generalities about monoidal triangulated categories because the singularities category of A e does not have any obvious monoidal structure. 5.2. Hochschild cohomology of Gorenstein algebras. A (not necessarily commutative) Noetherian ring A is called Gorenstein if it has finite injective dimension both as a left and right Amodule. As in the commutative case, we denote by CM A the category of maximal Cohen-Macaulay (left) A-modules and denote by CM A its stable category. Buchweitz proved that if A is Gorenstein, then CM A is equipped with a structure of triangulated category and CM A ∼ = D sg (A). We recall a fundamental result on extension groups in the stable category over Gorenstein rings due to Buchweitz. Proof. By Definition 5.1, we have a morphism HH i (R, R) → HH i sg (R, R) for all i. In order to apply Proposition 5.3 to show that it is an isomorphism, we need to check that R⊗R is Noetherian. This follows from Theorem 1.6 of [80]. A commutative local complete Gorenstein k-algebraR is called a hypersurface algebra ifR ∼ = k[[x 1 , . . . , x n ]]/(g). We say thatR is a hypersurface algebra with isolated singularities if g has an isolated critical point. Theorem 5.5. (Theorem 3.2.7 [29]) Let R = k[x 1 , . . . , x n ]/(g) be a hypersurface algebra with isolated singularities. Denote by M g the Milnor algebra k[x 1 , . . . , x n ]/( ∂g ∂x1 , . . . , ∂g ∂xn ), and by K g and T g the kernel and cokernel of the endomorphism of M g defined by multiplication with g. Then for r ≥ n, there is an isomorphism of R-modules HH r (R, R) ∼ = T g r is even, HH r (R, R) ∼ = K g r is odd. Proof. The proof in [29] shows that in degrees r ≥ n, the Hochschild cohomology HH r (R, R) is isomorphic to the homology in degree r of the complex k[u]⊗K(R, ∂g ∂x 1 , . . . , ∂g ∂x n ), where u is of degree 2 and K denotes the Koszul complex. Put P = k[x 1 , . . . , x n ]. Since R is quasi-isomorphic to K(P, g) and the (∂g)/(∂x i ) form a regular sequence in P , the Koszul complex is quasi-isomorphic to K(M g , g). Note that T g is the Tyurina algebra k[x 1 , . . . , x n ]/(g, ∂g ∂x1 , . . . , ∂g ∂xn ). Since K g is the kernel of the multiplication map g : M g → M g it is a module over T g . Lemma 5.6. Let A be a commutative k-algebra such that A and A e are noetherian. Let S ⊂ A be a multiplicative subset. If M is a finitely generated A-module and L an A-module, we have a canonical isomorphism RHom A e (L, M ) S ∼ → RHom A e S (L S , M S ). Proof. Since L is finitely generated over A, it is finitely generated over A e . Since A e is noetherian, we have a projective resolution P → L with finitely generated components. This implies that we have isomorphisms RHom A e (L, M ) S = Hom A e (P, M ) S = Hom A e (P, M S ). Since A S ⊗ A P ⊗ A A S → L S is a projective resolution over A e S , we find Hom A e (P, M S ) = Hom A e S (A S ⊗ A P ⊗ A A S , M S ) = RHom A e S (L S , M S ). Remark 5.7. In the setting of Theorem 5.5, assume that g has isolated singularities and that the origin is a singular point of the vanishing locus of g. If we denote by m the maximal ideal (x 1 , . . . , x n ) ⊂ k[x 1 , . . . , x n ], then g ∈ m. Denote by M g,m , T g,m and K g,m the localizations of M g , T g and K g . It follows from the lemma that Theorem 5.5 still holds if one replaces R by R m := k[x 1 , . . . , x n ] m /(g) and replaces T g and K g by T g,m and K g,m . For a Noetherian k-algebra A, the derived category of singularities D sg (A) is equipped with a canonical dg enhancement, obtained from its construction as a Verdier quotient of two canonically enhanced triangulated categories [48,20]. Instead of HH * sg (A, A), one may also consider the Hochschild cohomology of the dg category D sg (A), which we will denote by HH * (D sg (A)). Moreover, ifR = k[[x 1 , . . . , x n ]]/(g ) is another hypersurface algebra with isolated singularity such that D sg (R ) is quasi-equivalent with D sg (R) as dg-categories, thenR is isomorphic toR. Proof. Because g has an isolated critical point, we may assume that g is a polynomial without loss of generality. Denote by R the algebra k[x 1 , . . . , x n ] m /(g). Notice that R has an isolated singularity at the origin and that its completion identifies withR. By Theorem 5.7 of [22], the triangulated category D sg (R) is the Karoubi envelope of D sg (R). Therefore, the two dg categories have equivalent derived categories and there is a natural isomorphism HH * (D sg (R)) ∼ → HH * (D sg (R)). Orlov proved in [70] that D sg (R) is triangle equivalent with the homotopy category of matrix factorizations MF(k[x 1 , . . . , x n ] m , g). The triangle equivalence is lifted to an equivalence of dg categories by the work of Blanc, Robalo, Toën and Vezzosi [8]. Therefore, the dg category D sg (R) is 2-periodic and so is its Hochschild cohomology. So there exists a natural isomorphism ofR-modules HH 0 (D sg (R)) ∼ → HH 2r (D sg (R)) for all r ∈ Z. By Theorem B of [26], the bounded dg derived category D b dg (mod R) is smooth. Thus, by Theorem 5.8, we have HH 2r (D sg (R)) ∼ → HH 2r sg (R, R). By Corollary 5.4, for r 0, we have HH 2r (R, R) ∼ → HH 2r sg (R, R) and by Theorem 5.5 and Remark 5.7, we have HH 2r (R, R) ∼ → T g . Because g has an isolated critical point, there is an isomorphism T g ∼ = T g . Then the first claim follows. The second claim folllows from the formal version of the Mather-Yau theorem (see Theorem 1.1 [30]). Classification of 3-dimensional smooth flops. Let ( Y , f , R) be a 3-dimensional formal flopping contraction with Ex(f ) = t i=1 C i , and let A = End R ( t i=1 N i ⊕R) be the NCCR associated to it. We have associated to it an exact 3CY algebra: the derived deformation algebra Γ of the semisimple collection O C1 , . . . , O Ct , and the cluster category C Γ . There are two relaxations of the above context. First we take Y a CY 3-fold with a semi-simple collection of rational curves O C1 , . . . , O Ct . We may still define the derived deformation algebra Γ and the cluster category C Γ . However, in general Γ is only bimodule CY and C Γ may not be Hom-finite. If we assume that the Y is projective then Γ will be exact. However, most Y are not expected to have CY compactifications. The second relaxation is to take a 3-dimensional hypersurface ring R with isolated singularities. Associate to it is the derived category of singularity D sg (R). This is a Hom-finite CY category. One may ask when is it possible to express it as a cluster category of a CY algebra. In the case of flopping contractions, these two relaxations are related by Theorem 2.8. By [82], Y admits a tilting bundle, Γ is exact and has finite dimensional cohomology. By Corollary 2.12, Theorem 4.6 and 4.7, Γ is weakly equivalent to a Ginzburg algebra D(Q, w) with t nodes. On the other hand if R admits a NCCR A then D sg (R) is equivalent to C Γ for the CY algebra Γ. By Theorem 2.8, A admits a minimal model ( [81]). Since A is Calabi-Yau, d can be derived from a potential, i.e. ( T l V, d) is a Ginzburg algebra (see [81,Proposition 1.2]). By the derived equivalence theorem 2.11 of Van den Bergh, the derived deformation algebra of the semi-simple colleciton O C1 , . . . , O Ct is isomorphic to the quotient T l V / T l V e 0 T l V in Theorem 2.8. It is natural to expect that the deformation theory of the exceptional curves and the singularity theory of R should determine each other since both are governed by the CY algebra Γ. T l V, d) with the dual space of Σ −1 Ext ≥1 A ( t i=0 S i , t i=0 S i ) (see section 4 of Recall that the CY tilted algebra Λ := H 0 Γ is isomorphic to End CΓ (π(Γ)) (c.f. Theorem 3.19). Donovan and Wemyss conjectured that Λ alone can already determine the analytic type of R. Conjecture 5.10. Let ( Y , f , R) and ( Y , f , R ) be two 3-dimensional simple formal flopping contractions with associated CY tilted algebras Λ and Λ . Then the following are equivalent (1) R is isomorphic to R . (2) Λ is isomorphic to Λ . Donovan and Wemyss have extended this conjecture to the case of not necessarily simple formal flopping contractions by replacing (2) with (2') Λ is derived equivalent to Λ , cf. Conjecture 1.3 in [4]. In this situation, the implication from (1) to (2) is known to be true by iterating a construction of Dugas [21]. The implication from (2') to (1) is one of the main open problems in the homological minimal model program for 3-folds. In this section, we will prove a slightly weaker version of this implication. The exactness of Γ poses a strong constraint on Λ, i.e. the relations of Λ can be written as cyclic derivative of a potential w by Theorem 4.6. If we fix the exact CY structure then w is uniquely determined up to right equivalences (Proposition 4.8). The proof of this theorem will take up the rest of the section. Here we highlight major components of the proof. First we prove that the cluster category C Γ is dg equivalent to the D sg (R) with its canonical Z-graded dg structure (Lemma 5.12). The second step is to establish the result that analytic type of isolated hypersurface singularity (with fixed embedded dimension) is determined by its Z-graded dg category of singularities (Theorem 5.9). In the last step, we prove that Γ can be reconstructed from the CY tilted algebra Λ together with the class [w] ∈ HH 0 (Λ) represented by the potential. Let (R, m) be a complete commutative Noetherian local Gorenstein k-algebra of Krull dimension n with isolated singularity and with residue field k. Suppose that R admits a NCCR, then D sg (R) has another dg model via the triangle equivalence D sg (R) C Γ (c.f. Theorem 2.8). We first prove that these two models are dg quasi-equivalent. Lemma 5.12. In the homotopy category of dg categories, there is an isomorphism between C Γ = per(Γ)/D f d (Γ) and the category of singularities D sg (R) = D b (R)/K b (proj(R)), both equipped with their canonical dg enhancements. Proof. Let A and B be two pretriangulated dg categories. We call a triangle functor F : H 0 (A) → H 0 (B) algebraic if there is a dg A-B-bimodule X such that we have a square of triangle functors, commutative up to isomorphism H 0 (A) F / / H 0 (B) D(A) ? L ⊗ A X / / D(B), where the vertical arrows are induced by the Yoneda functors. We know from [79] that morphisms A → B in the homotopy category of dg categories are in bijection with isomorphism classes of right quasi-representable A-B-bimodules in the derived category of bimodules. Thus, it suffices to show that the triangle equivalence C Γ ∼ → D sg (R) is algebraic. We use the notation of subsection 2.4 and put N = N 1 ⊕ . . . ⊕ N t . Let F denote the thick triangulated subcategory of per(A) generated by the simples S 1 , . . . , S t . Let us recall from Proposition 3 of [71] that we have a diagram of triangle functors, commutative up to isomorphism and whose rows and columns are exact sequences of triangulated categories 0 0 where the categories add(R) and add(R ⊕ N ) are full subcategories of CM R and the last equality denotes the equivalence given by the functor Hom(R ⊕ N, ?). We endow per(A)/per(R) with the dg enhancement given by the dg quotient [48,20]. It is then clear that the triangle functors of the middle row and of the middle column are algebraic. Let us show that the functor D b (CM R ) → CM R is algebraic. The canonical dg enhancement of CM R is given by the triangle equivalence from the homotopy category H 0 (A) of the dg category A of acyclic complexes over proj(R) to CM R taking an acyclic complex P to its zero cycles Z 0 (P ). Let B be the dg enhanced derived category D b dg (CM R ). We define a B-A-bimodule X by putting which shows that X(?, M ) is quasi-representable by a complete resolution of M . By dévissage, it follows that X(?, M ) is quasi-representable for any bounded complex M and one checks easily that the (derived=non derived) tensor product with X induces the canonical triangle functor D b (CM R ) → CM R . It follows that, at the level of dg categories, CM R identifies with the dg quotient of D b (CM R ) by per(R). In other words, the canonical equivalence 0 / / F / / per(A)/per(R) O O / / CM R O O / / 0 0 / / F / / per(A) O O / / D b (CM R ) O O / / 0 per(R) O O per(R) O O 0 O O 0 O O Here the category D b (CM R ) isD b (CM R )/per(R) ∼ → CM R is algebraic. Therefore, the induced functor per(A)/per(R) → CM R is algebraic. Thus, the whole diagram is made up of algebraic functors. Now notice that the inclusion CM R ⊂ mod R induces algebraic equivalences D b (CM R ) ∼ → D b (mod R) so that we get algebraic equivalences CM R ∼ ← − D b (CM R )/per(R) ∼ → D b (mod R)/per(R) To construct the above square, we may assume that Z is a 2-term silting object since by part (3) of Theorem 7.2 of [5], the standard derived equivalence given by Z is the composition of equivalences given by 2-term tilting complexes and their inverses. Let A be the dg algebra obtained by applying Theorem 4.25 to B = Γ , H 0 (B) = Λ , C = Λ and Z = Z. Its homologies H p (A) are finitedimensional and vanish for p > 0. Thus, it is quasi-isomorphic to a dg algebra of the form ( T l V, d), where V is a graded bimodule whose components vanish in degrees > 0 and are finite-dimensional in all degrees ≤ 0. So we may assume that A is in PCAlgc(l). By Theorem 4.25, there is moreover a dg A-Γ -bimodule X yielding an equivalence ? Proof. We may assume that Γ itself is a differential graded k[u −1 ]-algebra. The multiplication with u −1 yields a functorial morphism Id → Σ −2 of triangle functors D(Γ) → D(Γ) and per(Γ) → per(Γ). This induces a functorial morphism of triangle functors Id → Σ −2 in the cluster category C Γ . To check that it is invertible, it is enough to check that its action on the cluster-tilting object Γ ∈ C Γ is invertible (since C Γ equals its thick subcategory generated by Γ). Now by our assumption, the morphism u −1 : Σ 2 Γ → Γ induces isomorphisms in H n for n ≤ −2 and the 0-map for n ≥ −1. Thus, it induces a quasi-isomorphism Σ 2 Γ → τ ≤−2 Γ. We claim that the canonical morphism τ ≤−2 Γ → Γ becomes invertible in the cluster category. Indeed, the homology of its cone is of finite total dimension since H p Γ is of finite dimension for all integers p by Lemma 2.5 of [3]. It follows that u induces an isomorphism Σ 2 Γ ∼ → Γ in C Γ . The rest follows because we have isomorphisms of H 0 (Γ)-bimodules The simplest case is when Γ = k[t] with zero differential. It is k[u −1 ]-enhanced by setting u −1 = t. This is the derived deformation algebra for O C of a (−1, −1)-curve, which is always contractible. Proposition 6.6. Let F = k x be the complete free algebra of one generator and w ∈ F be an element with no constant term. Then Γ : H 0 (Γ) ∼ → C Γ (Γ, Γ) ∼ → C Γ (Γ, Σ 2 Γ) ∼ → DC Γ (Γ, Γ= D(F, w) is k[u −1 ]-enhanced. Proof. A general element w ∈ F is of the form: w = x n+1 + higher order terms. When n = 1 we are in the case of (−1, −1) curve. We assume that n ≥ 2. The Jacobi algebra of D(F, w) is isomorphic to k[[x]]/(x n ). It is always finite dimensional. Because w = x n+1 · u for some unit u ∈ k[[x]], [w] = 0 in k[[x]]/(x n ). By Theorem 4.16, w is right equivalent to x n+1 . Without loss of generality, we may assume w = x n+1 n+1 to begin with. Then the Ginzburg algebra D(F, w) is isomorphic to k x, θ, t with dt = [x, θ] and dθ = x n . It is easy to check that the two-sided differential ideal (t, [x, θ]) is acyclic. As a consequence, the quotient morphism Γ = (k x, θ, t , d) → Γ := (k x, θ, t /(t, [x, θ]), d) is a quasi-isomorphism of dg algebras. Note that Γ is isomorphic to the complex . . . k[[x]]θ 3 d / / k[[x]]θ 2 d / / k[[x]]θ d / / k[[x]] / / 0 where d(θ 2k ) = 0, d(θ 2k+1 ) = x n θ 2k . Define the action of u −1 on Γ by multiplication by θ 2 . It is easy to check it makes Γ a dg k[u −1 ]algebra. From the above proposition, we see that the Ginzburg algebras associated to the "one-loop quiver" are essentially classified by the dimension of their Jacobi algebras. Moreover, they all admit k[u −1 ]enhancements. If dim k (H 0 Γ) = n for n > 1, then Γ is equivalent to the derived deformation algebra of a floppable (0, −2) curve of width n (see [68] for the geometric definition of width). The following corollary can be viewed as a noncommutative counter part of the classification theorem of Reid [68]. Corollary 6.7. Let C be a rational curve in a quasi-projective smooth CY 3-fold Y of normal bundle O C ⊕O C (−2). Denote its derived deformation algebra by Γ. Then (1) C is movable if and only if Γ has infinite dimensional Jacobi algebra; (2) If C is rigid then it is contractible. The dimension of H 0 Γ is equal to n for n > 1 if and only if the underlying singularity is isomorphic to the germ of hypersurface x 2 + y 2 + u 2 + v 2n = 0 at the origin. It is proved by Laufer (c.f. [72]) that a contractible rational curve in a CY 3-fold must have normal bundle of types (−1, −1), (0, −2) or (1, −3). Donovan and Wemyss give an example of a rigid rational curve of type (1, −3) that is not nc rigid (see Example 6.4 in [19]). In their example, there exists a birational morphism that contracts a divisor containing the (1, −3) curve. Kawamata asked whether it is true that C is contractible if it is nc rigid (see Question 6.6 of [43]). We formulate a conjecture in terms of the derived deformation algebra. Conjecture 6.8. Let C ⊂ Y be a nc rigid rational curve in a smooth quasi-projective CY 3-fold. Denote its associated derived deformation algebra by Γ Y C . Then C is contractible if and only if Γ Y C is k[u −1 ]-enhanced. Note that one direction of the conjecture follows from our Theorem 4.17. Proposition 6.9. Let C ⊂ Y be a contractible rational curve in a smooth quasi-projective CY 3-fold. Then Γ Y C is k[u −1 ]-enhanced. Proof. Denote R for the ring of formal functions on the singularity underlying the contraction. For simplicity, we denote the derived deformation algebra Γ Y C by Γ. By Proposition 5.12, C Γ is quasiequivalent to D sg (R) as dg categories. Under the equivalence, the projection image of Γ is identified with the Cohen-Macaulay module N ∈ D sg (R) ∼ = CM R . By Theorem 4.17, Γ is isomorphic to τ ≤0 Λ dg where Λ dg is the dg endomorphism algebra of N in CM R . Because R is a hypersurface ring, the dg category CM R carries a canonical Z/2-graded structure (equivalently k[u, u −1 ]-structure) by Eisenbud's theorem [23]. Therefore, Γ is k[u −1 ]-enhanced. quasi-equivalent to the dg enhanced derived category of the (essentially) small dg category coh. There is a natural functor Ψ : IndCoh → Qcoh which commutes with coproducts and, restricted to coh, becomes the inclusion coh → Qcoh. If X is smooth then Ψ X : IndCoh(X) → Qcoh(X) is an equivalence [27, Chapter 4, Lemma 1.1.3]. Let f : X → Y be a proper morphism. There exists a continuous pushforward functor f IndCoh * : IndCoh(X) → IndCoh(Y ) with a commutative diagram IndCoh(X) f IndCoh * / / Ψ X IndCoh(Y ) Ψ Y Qcoh(X) f * / / Qcoh(Y )IndCoh(X) + Ψ X / / Qcoh(X) + IndCoh(Y ) + f ! O O Ψ Y / / Qcoh(Y ) + f Qcoh,! O O (7.1) where the superscript + refers to the subcategory consisting of objects whose cohomological amplitude is bounded below. Note that the similar diagram with + removed is not commutative (see [27,Ch. 4 such that IndCoh(X) ∨∨ IndCoh(X) ∨ IndCoh(X). The dual category C ∨ of a dualizable dg category C can be identified with the category of continuous dg functors from C to dg k where dg k is the category of dg k-modules. ) as dg functors from coh(X × X) to dg k and show there is a natural isomorphism between them. By [27, Chapter 3, Proposition 3.1.7], we have an equivalence Qcoh(X × X) Qcoh(X)⊗Qcoh(X). Therefore we obtain a coh(X) op ⊗coh(X)-module structure by letting M and N vary in coh(X). Let f : X → X × X be the diagonal map and p : X → Spec k be the counit map. By [27,Chapter 5,Theorem 4.2.5] there is a commutative diagram of dg functors (both the upper and the lower square commute, c.f. [28, 9.2.3].) IndCoh(k) ∨ D k / / IndCoh(k) IndCoh(X) ∨ D X / / (p ! ) ∨ O O IndCoh(X) p * O O IndCoh(X × X) ∨ D X×X / / f ∨ * O O IndCoh(X × X) f ! O O By [28, Corollary 9.5.9], if we restrict to coh(X) (resp. coh(k) and coh(X ×X)) we get a commutative diagram coh(k) op D k / / coh(k) coh(X) op D X / / (p * ) op O O coh(X) p * O O coh(X × X) op D X×X / / (f * ) op O O coh(X × X) f ! O O Then we have a natural isomorphism p * • f ! • D X×X D k • (p * ) op • (f * ) op . Since X is smooth coherent sheaves are dualizable. In this case D X (E) = E ∨ ⊗ω X [d] for E ∈ coh(X) ([28, Lemma 9.5.5]). 3 Given an object M ∨ N in coh(X × X), we may compute p * • f ! • D X×X (M ∨ N ) ∼ = p * (f ! ((M ⊗ω X [d]) (N ∨ ⊗ω X [d]))) ∼ = p * (f * (M N ∨ )⊗ω X [d]) ∼ = Hom dg coh(X) (N, M ⊗ω X [d]) In the last step, we use the condition that X is smooth and proper. On the other hand 7.2. Inverse dualizing sheaves and bimodules, after Ben-Zvi-Francis-Nadler. The following proposition is a consequence of Theorem 4.7 and Corollary 4.8 of Ben-Zvi-Francis-Nadler's paper [7] combined with Grothendieck duality. The proof we give is an elaboration on Example 2.7 of [58]. Remark 7.3. One could make further use of the results of [7] to show more precisely that the derived endomorphism algebra of G G ∨ in D(Y × Y ) is quasi-isomorphic to B e and that the canonical equivalence of the theorem is given by RHom(G G ∨ ) : D(Y × Y ) ∼ → D(B e ). Proof. Let k = C. We mostly work in the ∞-category St of k-linear stable presentable ∞-categories whose 1-morphisms are cocontinuous k-linear exact ∞-functors (equivalently: left adjoints of k-linear exact ∞-functors). We recommend section 2 of [58] for a concise but readable introduction to this setting. Each (large) pretriangulated dg k-category A such that H 0 (A) has arbitrary (set-indexed) coproducts and is compactly generated gives rise to an object of St; each dg functor F : A → B between two such categories such that H 0 (F ) commutes with arbitrary coproducts gives rise to a 1-morphism in St and this 1-morphism is an equivalence iff H 0 (F ) is an equivalence. To make these statements more precise, let us denote by dgcat pretr,⊕ k the category -whose objects are the pretriangulated dg categories A such that H 0 (A) has arbitrary coproducts and is compactly generated and -whose morphisms are the dg functors F such that H 0 (F ) commutes with arbitrary coproducts. We then have a canonical ∞-functor (where on the left, we write the ordinary category instead of its nerve) can : dgcat pretr,⊕ k −→ St. Each object X of St has an underlying ∞-category X ∞ (obtained by forgetting the k-linear structure). It is a stable presentable ∞-category and its 1-categorical truncation τ (X ∞ ) is naturally a triangulated category with arbitrary coproducts, cf. section 1.4.4 of [65]. Similarly, if f : X → Y is a 1-morphism of St, then its underlying ∞-functor f ∞ : X ∞ → Y ∞ is exact and cocontinuous and its 1-categorical truncation τ (f ∞ ) : τ (X ∞ ) → τ (Y ∞ ) is naturally a triangulated functor and commutes with arbitrary coproducts. To state these facts more precisely, let us denote by Tria ⊕ the category whose objects are the triangulated categories with arbitrary coproducts and whose morphisms are the triangle functors which commute with arbitary coproducts; let us denote by Pr L st Lurie's ∞-category of presentable stable ∞-categories whose 1-morphisms are the exact left adjoint ∞-functors. Then we have ∞-functors The ∞-category Pr L st is by definition a subcategory of the ∞-category Pr st of presentable stable ∞-categories whose 1-morphisms are all exact ∞-functors. We will use the following facts: a) If f : X → Y is a 1-morphism of Pr st , then the values of f and of τ (f ) on objects are equal in τ (Y). b) A 1-morphism f : X → Y of Pr st belongs to Pr L st iff its truncation τ (f ) : τ (X ) → τ (Y) is a (triangle) functor which commutes with arbitrary coproducts, cf. Prop. 1.4.4.1 (2) of [65]. c) A 1-morphism f : X → Y of Pr st admits a left adjoint f λ : Y → X in Pr st iff the (triangle) functor τ (f ) admits a left adjoint τ (f ) λ . By definition, in this case, the 1-morphism f λ belongs to Pr L st . Moreover, the functor τ (f λ ) is isomorphic to τ (f ) λ since truncation preserves adjunctions. For a dg algebra A, we denote by D(A) the object of St corresponding to the dg category of K-projective dg A-modules (i.e. dg A-modules which are homotopy equivalent to cofibrant dg Amodules). Notice that D(A) denotes an object of St whereas D(A) denotes the ordinary derived category (with its triangulated structure). The ordinary derived category D(A) is naturally equivalent to the truncation τ (D(A) ∞ ). Let X be a quasi-projective variety (or more generally, a quasi-projective separated scheme). Let D(X) denote the object of St corresponding to the dg category of fibrant complexes of quasi-coherent sheaves on X. Then the truncation τ (D(X) ∞ ) identifies with the derived category D(Qcoh(X)). Let T be a compact generator of the unbounded derived category of quasi-coherent sheaves on X. Suppose that T is a fibrant complex of quasi-coherent sheaves and let A = Hom(T, T ) be its dg endomorphism algebra. The homotopy category of fibrant complexes of quasi-coherent sheaves on X is compactly generated by T and so the dg A-module Hom(T, I) is K-projective for all fibrant complexes I of quasi-coherent sheaves on X. Moreover, the dg functor Hom(T, ?) induces a quasiequivalence between the homotopy categories of fibrant complexes on X and K-projective dg Amodules. Thus, the dg functor Hom(T, ?) induces a 1-morphism D(X) → D(A) in St. Its truncation τ (Hom(T, ?) ∞ ) identifies with RHom(T, ?) : D(Qcoh(X)) → D(Qcoh(Y )), which is an equivalence by our assumption on T . Thus, we have an equivalence D(X) → D(A) in St. As detailed in section 4.8 of [65] and section 2.2 of [58], the ∞-category St is endowed with a symmetric monoidal structure. The unit of the monoidal structure on St is the k-linear (symmetric monoidal) ∞-category D(k). If A 1 and A 2 are two dg algebras, then, by part (2) Hence, by fact c) above, the ∞-functor Ev A admits a left adjoint Ev A λ and τ (Ev A λ ) is isomorphic to ? L ⊗ A e Θ. In particular, the left adjoint Ev A λ takes k to Θ (fact a) above). Let X 1 and X 2 be quasi-projective varieties (or more generally, quasi-compact, separated schemes). They are in particular perfect stacks in the sense of [7]. By Theorem 4.7 of [loc. cit.], we have a canonical equivalence D(X 1 )⊗D(X 2 ) ∼ → D(X 1 × X 2 ). Fix a quasi-projective variety X. By Corollary 4.8 of [7] and the first three lines of its proof, the object D(X) becomes its own dual and the evaluation morphism is given by a composition admits the left adjoint ∆ * (π * (?)⊗Σ −d ω −1 Y ) which takes k to ∆ * (Σ −d ω −1 Y ). As above, it follows that the ∞-functor Ev Y = ev Y ∞ admits a left adjoint Ev Y λ and that this left adjoint takes k to ∆ * (Σ −d ω −1 Y ). As we have seen above, under our hypotheses, the dg functor taking a fibrant complex of quasicoherent sheaves C to Hom(G, C) induces an equivalence D(Y ) ∼ → D(B). We deduce an equivalence D(Y × Y ) D(Y ) ⊗ D(Y ) Theorem B. (Theorem 5.11) Let f : Y → X and f : Y → X be two formal flopping contractions with reduced exceptional fibers Ex( f ) = t i=1 C i and Ex( f ) = s i=1 C i . Denote respectively by Γ and Γ the derived deformation algebras of t i=1 O Ci and s i=1 O C i , and by [w] ∈ HH 0 (H 0 Γ) and [w ] ∈ HH 0 (H 0 Γ ) the canonical classes. Suppose there is a triangle equivalence ? L ⊗ H 0 Γ Z : D(H 0 Γ) ∼ → D(H 0 Γ ) an artinian augmented dg l-algebra R. An object of the groupoid Def R (E) is a pair (S, σ), where S is an object of D(A R ) and R op ⊗A → A op ⊗A in the homotopy category of dg categories. The associated restriction functor D(A op ⊗A) → D(R op ⊗A) sends M to an object S of D(R op ⊗A). By construction, the restriction of S to A is isomorphic to M in D(A) and the left action of R on S induces the given algebra morphism R → End H (M ). Since H is the heart of a t-structure, we have a canonical realization functor D b (H) → D(A) extending the inclusion H → D(A), cf. Section 3.1.10 of [6] or Section 3.2 of [54]. Moreover, since H is faithful, the realization functor is fully faithful. Since we only know how to compare tensor functors, we use a different construction to extend the inclusion H → D(A) to a triangle functor D b (H) → D(A). Let H dg be the full subcategory of the dg category of right A-modules formed by cofibrant resolutions of the objects of H. We have an equivalence of k-categories H ∼ → H 0 (H dg ). Since H is the heart of a t-structure, the homology of the dg category H dg is concentrated in degrees ≥ 0. Thus, we have quasi-equivalences τ ≤0 H dg ∼ → H 0 (H dg ) ∼ → H. Therefore, in the homotopy category of dg categories, we obtain a morphism H → τ ≤0 (H dg ) → H dg → D dg (A) where D dg (A) denotes the dg category of cofibrant dg A-modules. It gives rise to an H-A-bimodule R. Let H b (H) denote the category modulo homotopy of bounded complexes of objects of H. Using the fact that short exact sequences of H give rise to triangles in D(A), one checks that the induced functor ?⊗ H R : H b (H) → D(A) vanishes on the bounded acyclic complexes and therefore induces a triangle functor D b (H) → D(A) still denoted by ?⊗ H R. Lemma 2 . 6 . 26If we have Hom D(A) (res(X), Σ −n res(Y )) = 0 Thanhoffer de Völcsey and Van den Bergh prove that CM R admits an explicit dg model in this case. Theorem 2.8. ([81, Theorem 1.1]) There exists a finite dimensional graded l-bimodule V and a minimal model ( Take i = 3 and n = 4 . 4Let leftmost term vanishes and H Corollary 3 . 15 . 315Let R be a complete local equi-codimensional Gorenstein normal domain of dimension d over an algebraically closed field of characteristic zero, and let A be an NCCR. Then every dCY structure on A is exact. Remark 3 . 17 . 317If Γ is a pseudo-compact dg l-algebra in PCAlgc(l), we may define a continuous version of cluster category. Condition (1) is replaced by (1 ) Γ is topologically homologically smooth, and the topological cluster category C Γ is defined to be the triangle quotient per(Γ)/D f d (Γ) where per(Γ) and D f d (Γ) are considered as subcategories of the pseudo-compact derived category. We refer to the Appendix of[57] for the details. Remark 3 . 18 . 318If we drop the assumption that H 0 Γ is finite dimensional, the quotient category C Γ = per(Γ)/D f d (Γ) is no longer Hom-finite. The Calabi-Yau property only holds when one restricts to suitable subcategories, cf. Proposition 2.16 of[74]. D a : kQ/[kQ, kQ] → kQ which takes the class of a path p to the sum p=uav vu taken over all decompositions of the path p. The definition can be extended to kQ cyc := kQ/[ kQ, kQ] c where the superscript c stands for the completion with respect to the adic topology defined above. An element w in kQ/[ kQ, kQ] c is called a potential on Q. It is given by a (possibly infinite) linear combination of cycles in Q. Theorem 4. 4 . 4(Van den Bergh)[52, Appendix] Let Q be finite quiver and w be a potential. Then D(Q, w) is topologically homologically smooth and bimodule 3CY. Remark 4. 5 . 5There exists a canonical exact CY structure on Γ = D(Q, w). We follow the notation of[84] to write M l := M/[l, M ] for a l-bimodule M . Because the reduced cyclic homology of Γ is equal to the homology of (Γ/l + [Γ, Γ]) l (see Proof of Theorem 11.2.1 of [83]), a class of HC 2 (Γ, Γ) is represented by a degree −2 element χ of Γ such that dχ ∈ l + [Γ, Γ]. By the definition of d of Γ, χ := i∈Q0 t i represents a class in HC red 2 (Γ, Γ). Because Γ is cofibrant, by Proposition 7.2.1 of [83] the Hochschild chain complex of Γ is quasi-isomorphic to the mapping cone of Ω 1 l Γ/[Γ, Ω 1 l Γ] ∂1 / / Γ/[l, Γ] with differential defined by ∂ 1 (aDb) = [a, b], where Db = 1⊗b − b⊗1. In other words, a class in HH 3 (Γ, Γ) is represented by a pair of elements (ω, a) of degree (−2, −3) satisfying ∂ 1 (ω) = da and dω = 0. Because d and D commute, (Dχ, 0) represents a class in HH 3 (Γ, Γ), which is the image of χ under the Connes map. In [83], a class in HH d (Γ, Γ) is called non-degenerate if the corresponding mormphism Θ Γ → Σ −d Γ is an isomorphism. By Lemma 11.1.2 of [83], (Dχ, 0) is non-degenerate. Assume that k has characteristic zero and let Γ be a pseudocompact dg algebra in PCAlgc(l) concentrated in degree zero. Then Γ is bimodule dCY if and only if it is exact dCY. Given a pseudo-compact l-algebra A, two classes [w] and [w ] in A/[A, A] c are called right equivalent if there exists an automorphism γ of A such that γ * [w] = [w ]. Now we assume that a bimodule 3CY dg algebra Γ in PCAlgc(l) is exact. So it admits a model given by D(Q, w). Note that being bimodule CY and exact CY are homotopically invariant properties. The next proposition shows that the right equivalence class of [w] in HH 0 (Λ(Q, w)) for such a dg algebra is indeed a homotopy invariant. The proof is implicitly contained in Van den Bergh's proof of Theorem 4.6 (cf. proof of Theorem 11.2.1 of[84]). We recall it for completeness. See Remark 4.9 for a conceptional explanation of Van den Bergh's result. Proposition 4.8. (Van den Bergh) Let k be a field of characteristic zero and l = ke 1 × . . . × ke n . Corollary 4 . 10 . 410Let ( Y , f , R) be a 3-dimensional formal flopping contraction with reduced exceptional fiber Ex Theorem 4 . 413. (Noncommutative Mather-Yau theorem)[35, Theorem 3.5] Corollary 4 . 414. [35, Theorem 4.3] Fix a finite quiver Q. Let w, w ∈ CQ cyc be two potentials with only cubic terms and higher, such that the Jacobi algebras Λ(Q, w) and Λ(Q, w ) are both finite dimensional. Assume there is an algebra isomorphism γ : Λ(Q, w) → Λ(Q, w ) so that γ * ([w]) = [w ]. Then there exists a dg algebra isomorphism β : D(Q, w) ∼ = / / D(Q, w ) Theorem 4 . 416. (Noncommutative Saito theorem)[36, Theorem 1.2] By the 3 - 3Calabi-Yau property, we have Hom per(Γ) (Σ −1 τ >n Γ, Σ −p Γ) = DHom per(Γ) (Γ, Σ p+2 τ >n Γ)) which vanishes because τ >n Γ has no homology in degrees > 0. Similarly, we have Hom per(Γ) (τ >n Γ, Σ −p Γ) = DHom per(Γ) (Γ, Σ p+3 (τ >n Γ)) which vanishes for the same reason. Thus we have Hom per(Γ) (τ ≤n Γ, Σ −p Γ) = Hom per(Γ) (Γ, Σ −p Γ). Corollary 4 . 20 . 420Let Γ be the Ginzburg algebra of a Jacobi-finite quiver with potential. Let T be the image of Γ in the cluster category C Γ . a) H 0 Γ is selfinjective if and only if H −1 Γ vanishes if and only if T is isomorphic to Σ 2 T in C Γ . b) If the identity functor of C Γ is isomorphic to Σ 2 , then H 0 Γ is symmetric and there is an isomorphism of graded algebras H 0 (Γ)⊗k[u −1 ] ∼ → H * (Γ), where u is of degree 2. is a minimal left add(M/X)-approximation. It is not hard to show that then E → Y is a minimal right add(M/X)-approximation, which implies that Y is indecomposable. Indeed, let us recall the argument: Let M = M/X. Since M is silting, we have an exact sequence of End(M )-modulesHom(M , E) → Hom(M , Y ) → 0 ,where Hom(M , E) is projective over End(M ). Saying that E → Y is a minimal right add(M/X)approximation is equivalent to saying that Hom(M , E) → Hom(M , Y ) is a projective cover. If Y was decomposable, the morphism Hom(M , E) → Hom(M , Y ) would therefore split into a direct sum of two surjective morphisms and this splitting would be induced by a splitting of the morphism E → Y as a direct sum of two non trivial morphisms E → Y and E → Y . But then X would be decomposable, a contradiction. The right mutation µ − X (M ) is defined dually. The right mutation of µ X (M ) at Y is isomorphic to M . Example 4 . 21 . 421Suppose that Γ is a Jacobi-finite Ginzburg algebra associated with a finite quiver and a potential not containing cycles of length ≤ 2. Then A = Γ satisfies our assumptions and Γ is a basic silting object in per(Γ). Let M be a silting object in per(Γ) and Γ the derived endomorphism algebra of M . Then the homologies H p (Γ ) are finite-dimensional and vanish in degrees p > 0. Since M generates per(Γ), the Γ -Γ-bimodule M yields an algebraic triangle equivalence D(Γ ) ∼ → D(Γ) The functor F induces a bijection 2silt(A) ∼ → 2silt(H 0 A). b) Suppose that we have H −1 (A) = 0. Then the functor F restricts to an equivalence 2-per(A) ∼ → 2-per(H 0 A). P ∈ add(A), M ∈ 2-per(A) and p ∈ {−1, 0} Hom(P, Σ p M ) ∼ → Hom(F P, Σ p F M ). 3) For M, M ∈ 2-per(A) Hom(M, M ) ∼ → Hom(F M, F M ). Proposition 4 . 23 . 423Suppose that eX L ⊗ B Q vanishes and that the object X L ⊗ B Q of D(B 0 ) has no selfextensions in degrees p < 0. Then there is an object Y of D(A op 0 ⊗B 0 ), unique up to isomorphism, such that we have an isomorphism X = Hom R (N, M ) yields a derived equivalence ? L ⊗ A X : D(A) ∼ → D(B) taking eA to eB. Moreover, the complex X L ⊗ B B 0 is a 2-term silting object of per(B 0 ) (as it follows from silting reduction [2] combined with part a) of Proposition 4.22) and hence a tilting object since B 0 is symmetric. Thus, the hypotheses of the proposition hold and there is a canonical two-sided tilting complex Y in D(A op 0 ⊗B 0 ). Clearly, the construction is compatible with compositions via derived tensor products. Proof of the Proposition. Put U = X L ⊗ B Q viewed as an object in D(A op ⊗B 0 ). The morphism A → A induces the Verdier quotient per(A) → per(A ) = per(A)/thick(eA)and is therefore a dg quotient. By the universal property of the dg quotient, there is an object Z in D(A op ⊗B 0 ) unique up to isomorphism such that the restriction of Z along A op ⊗B 0 → A op ⊗B 0 is isomorphic to U in D(A op ⊗B 0 ). We have a canonical morphism in the homotopy category of dg algebras A / / RHom B0 (Z, Z). where the left A-module structure on Y comes from the canonical morphismA → H 0 (A). By part b) of Proposition 4.22, we have an isomorphism End(M ) ∼ → End(M L ⊗ B H 0 B). By construction, we have an isomorphism H 0 A ∼ → End(M ) or equivalently H 0 A ∼ → End(X B ). Thus, the composition of ? L ⊗ B H 0 B with ? L ⊗ A X induces an isomorphism For an l-bimodule M , we write M l for the coinvariant module M/[l, M ]. For an A-bimodule M , we let M be the coinvariant module M/[A, M ]. We have an isomorphism of graded A-modules Lemma 4. 26 . 26The image of the class of t under S : HC 2 (A) → HC 0 (A) is the canonical class [w], i.e. the image of w under the projection HC 0 (T l V ) → HC 0 (A). * ) = ∂ 0 (w) and S(t) is the image of w in H 0 (A/([A, A] + l) = HC 0 (Γ). Notice that BS(t) = B(w)is indeed a boundary in the Hochschild complex: It is the differential of α α⊗α * − t. Corollary 4 . 27 . 427Let Γ = Γ(Q , w ) be a Ginzburg algebra and A a pseudo-compact dg algebra inPCAlgc(l) concentrated in degrees ≤ 0. Let X be a dg A-Γ -bimodule such that ? L ⊗ A X : D(A) → D(Γ )is an equivalence. Then there is a quiver with potential (Q , w ) and a weak equivalence s : Γ(Q , w ) → A such that for the restriction s X along s, the isomorphism HC 0 ( s X) takes the class [w ] to [w ]. It is a projective bimodule resolution of A. Let M be an A-bimodule. The Hochschild cochain complex with coefficients in the bimodule M is defined to be the complex C * (A, M ) := Hom A e (B k (A), M ) with differential δ(f ) = −(−1) n f • b for f : A⊗A ⊗n ⊗A → M . The i-th Hochschild cohomology of the algebra A with coefficients in the bimodule M is defined to be HH i (A, M ) := H i (C * (A, M ), δ). Namely, C * sg (A, A) := colim n C * (A, Σ n Ω n A ). Its cohomology groups are denoted by HH * sg (A, A). By construction, we have a natural chain morphism from C * (A, A) to C * sg (A, A), which induces a natural morphism from HH * (A, A) to HH * sg (A, A).Let A be a Noetherian k-algebra. Define D sg (A) to be the Verdier quotient of D b (A) by the subcategory per(A). We denote the extension group in D sg (A) by Ext i A (?, ?). The singular Hochschild cohomology groups are related to the extension groups in D sg (A e ). Proposition 5.2. (Theorem 3.6 [85]) Let A be a Noetherian k-algebra. Then there exists a natural isomorphism HH * sg (A, A) ∼ → Ext * A e (A, A), Proposition 5.3. (Corollary 6.3.4 of [15]) Let A be a Gorenstein ring and let X and Y be objects in D b (A). There exists a positive integer m depending on A, X and Y such that the natural morphism Ext i A (X, Y ) → Ext i A (X, Y ) is surjective for i = m and is an isomorphism for i > m. Combining Proposition 5.2 and Proposition 5.3, we obtain the following result. Corollary 5 . 4 . 54Let R be a commutative noetherian Gorenstein k-algebra. If R⊗R is noetherian, there exists a positive integer m such that for i > m, the natural morphismHH i (R, R) → HH i sg (R, R) is an isomorphism. Theorem 5.8.(Keller [53]) There is a canonical morphism of graded algebrasHH * sg (A, A) → HH * (D sg (A)). It is invertible if D b dg (mod A)is smooth. Now we establish the main result of the subsection. Theorem 5. 9 . 9LetR = k[[x 1 , . . . , x n ]]/(g) be a hypersurface algebra with isolated singularity. Denote byT g the Tyurina algebra of g. Then there is an isomorphism of k-algebras HH 0 (D sg (R)) ∼ =Tg. Theorem 5 . 11 . 511Let ( Y , f , R) and ( Y , f , R ) be formal flopping contractions. Given exact CY structures η and η on Y and Y respectively, denote by (Λ, [w]) and (Λ , [w ]) the associated CY tilted algebras and the canonical classes of potentials. If there exists a derived equivalence from Λ to Λ given by a bimodule complex Z such that the induced map HH 0 (Z) takes [w] to [w ], then R is isomorphic to R . the bounded derived category of the exact category CM R , the functor per(R) → D b (CM R ) is induced by the inclusion proj(R) → CM R , the functor per(A) → D b (CM R ) is induced by ? L ⊗ A (R ⊕ N ) and per(R) → per(A) is induced by the inclusion add(R) → add(R ⊕ N ) = proj(A) , X(P, M ) = Hom(P, ΣM ) where P is an acyclic complex of finitely generated projective R-modules and M a bounded complex over CM R . For M ∈ B, denote by M ∧ the representable dg functor B(?, M ). We have M ∧ ⊗ B X = X(?, M ). If M is concentrated in degree 0, we have canonical isomorphisms H 0 X(P, M ) = Hom CM R (Z 0 (P ), M ) and H p X(P, M ) = Hom CM R (Z 0 (P ), Σ p M ) , L⊗ A X : D(A) ∼ → D(Γ ) and an isomorphism φ : H 0 (A) → Λ such that we have an isomorphism φ Z ∼ → X L ⊗ Γ Λ in the derived category D(A op ⊗Λ ) and in particular a square, commutative up to isomorphism Theorem 7. 1 . 1Let X be a smooth and proper k-scheme of dimension d for k = C. Let M, N be objects of coh(X). There is a bifunctorial quasi-isomorphismDHom dg coh(X) (M, N ) Hom dg coh(X) (N, M ⊗ω X [d]). Proof. We will interpret M ∨ N → DHom dg coh(X) (M, N ) and M ∨ N → Hom dg coh(X) (N, M ⊗ω Y [d] D k • (p * ) op • (f * ) op (M ∨ N ) ∼ = D k Hom dg coh(X) (M, N ) ∼ = DHom dg coh(X) (M, N ) Proposition 7. 2 ( 2). Let Y be a smooth quasi-projective variety of dimension d. Let G be a perfect generator of D(QcohY ) and B = RHom(G, G). Then there is a canonical equivalenceD(Qcoh(Y × Y )) → D(B⊗B op ) taking ∆ * (Σ −d ω −1 Y ) to the inverse dualizing complex Θ B = Hom B e (B, B e ) of B and ∆ * (O Y ) to the identity bimodule B. In particular, if Y is d-Calabi-Yau, then B is bimodule d-Calabi-Yau. / Tria ⊕ . / / D(A 1 ⊗A 2 ) induced by the dg functor taking a pair (L, M ) of cofibrant dg modules to the cofibrant A 1 ⊗A 2module L⊗M . For any dg algebra A, the object D(A) is dualizable in the monoidal ∞-category St and its dual is equivalent to D(A op ), cf. Prop. 4.3 (3) of [7]. More precisely, the duality between D(A) and D(A op ) is given by the evaluation morphism D(A op )⊗D(A) → D(k) and the co-evaluation morphism D(k) → D(A)⊗D(A op ). The composition of the co-evaluation morphism with the canonical morphism D(A)⊗D(A op ) ∼ → D(A⊗A op ) is the morphism D(k) → D(A⊗A op ) induced by the dg functor taking a complex V to V ⊗A, where A is the identity bimodule. The evaluation morphism is obtained as the composition D(A op )⊗D(A) ∼ / / D(A op ⊗A) ev A / / D(k), where the second morphism is induced by the dg functor taking a cofibrant dg bimodule M to M ⊗ A e A. Let us abbreviate ev A ∞ by Ev A . Then the truncation τ (Ev A ) identifies with the derived functor ? L ⊗ A e A : D(A op ⊗A) → D(k). When A is smooth, this functor is isomorphic to RHom A e (Θ, ?), where Θ = Θ A = RHom A e (A, A e ) is the inverse dualizing bimodule. Thus, it admits the left adjoint ?⊗Θ : D(k) → D(A op ⊗A). / / D(X × X) ev X / / D(k) ,where the truncation τ (ev X ∞ ) identifies with the composition of derived functorsD(X × X) ∆ * / / D(X) π * / / D(k).The co-evaluation morphism is the compositionD(k) / / D(X × X) D(X)⊗D(X) ∼ o o ,where the first morphism is induced by the dg functor taking a complex V to V ⊗I, where I is an injective resolution of ∆ * (O X ). Now consider the smooth quasiprojective variety Y of the claim. In this case, by Grothendieck duality, the functor ∆ * : D(Y × Y ) → D(Y ) admits the left adjoint ∆ * (?)⊗Σ −d ω −1 Y . Thus, the composed functor D(Y × Y ) ∆ * / / D(Y ) π * / / D(k). / / D(Y ) ⊗ D(Y ) ∨ ∼ / / D(B) ⊗ D(B op ) ∼ / / D(B ⊗ B op ) .The intrinsic descriptions above show that under the 1-categorical truncation of this equivalence, the object ∆ * (Σ −d ω −1 Y ) corresponds to the inverse dualizing complex Θ B = Hom B e (B, B e ) of B. Moreover, the object ∆ * (O Y ) corresponds to the identity bimodule B ∈ D(B e ) as we see by examining the co-evaluation morphisms. Thus, an isomorphism ω Y ∼ → O Y yields an isomorphism Θ ∼ → Σ −d B in D(B e ). Theorem 2.2. [25, Theorem 15.1, 15.2] Let E 1 , . . . , E n be a collection of objects in D(A) ).6.2. k[u −1 ]-structures on Ginzburg algebras associated to contractible curves. Recall that for a Jacobi-finite Ginzburg algebra Γ := D(Q, w) associate to quiver Q with only one vertex (and multiple loops), H 0 (Γ) is self-injective. On the other hand, if Γ is non-degenerately k[u −1 ]-enhanced then H 0 (Γ) is symmetric. See [27, Ch. 4, Prop. 2.1.2]. Moreover, since f is proper f IndCoh * sends coh(X) to coh(Y ). The above commutative diagram is compatible with tensor products by quasi-coherent complexes ([27, Ch. 4, Prop. 2.1.4]). The pushforward functor f IndCoh * admits a continuous right adjoint functor f ! : IndCoh(Y ) → IndCoh(X) ([27, Ch. 4, 5.1.5]) and it is compatible with the tensor product with quasi-coherent complexes ([27, Ch. 4, 5.1.7]). To distinguish f ! from the right adjoint of f * in Qcoh, we denote the latter by f Qcoh,! . We have the following comparison theorem between these two functors ([27, Ch. 4, Lemma 5.1.9]): There is a commutative diagram , Remark 5.1.10]). This result shows that we can compute f ! E via f Qcoh,! E for E ∈ coh(Y ). The left adjoint of f * , denoted by f IndCoh, * exists. It is compatible with tensor products by quasicoherent complexes and satisfies a similar commutative diagram as diagram 7.1 with + removed ([27, Ch. 4, Prop. 3.1.6]). For a scheme X of finite type, by [27, Ch. 5, Th. 4.2.5] there is a canonical equivalence D X : IndCoh(X) ∨ IndCoh(X) Kawamata calls it a simple collection. In[28], Gaitsgory denotes by ω X the !-pullback p ! k for p : X → k, which differs from the the standard notion of dualizing complex by[d]. Acknowledgments. The first author wishes to thank Yujiro Kawamata, Mikhail Kapranov, Shel-We have already seen that in (−1, −1) and (0, −2) cases, the k[u −1 ]-structure on Γ can be computed explicitly. However, we don't have any explicit construction for the (1, −3) case even though we know it must exist. We do have an explicit formula for the symmetric Frobenius structure on the CY tilted algebra H 0 Γ in term of the residue map of matrix factorizations (see[33]).Appendix: Serre duality for sheaves and modulesIn this Appendix, we give two proofs of the link the between the inverse dualizing sheaf on a smooth quasi-projective variety Y and the inverse dualizing bimodule for the derived endomorphism algebra of any perfect generator of D(Qcoh(Y )).The proof in section 7.1 is based on the work of Gaitsgory[28]and Gaitsgory-Rosenblyum[27]. The proof in section 7.2 is essentially taken from Example 2.7 of[58]. It combines Grothendieck duality with the results of Ben-Zvi-Francis-Nadler's article[7]. Both proofs rely on the foundational work of Toën, Joyal, Lurie and many others. 7.1. Dg Serre duality, after Gaitsgory. We recall a result of Gaitsgory and Gaitsgory-Rosenblyum which could be viewed as Serre duality for the dg category of coherent sheaves. They use the category of ind coherent sheaves which behaves better on singular spaces. Though we only need the smooth case, we recall the basics of ind coherent sheaves and their properties. We will follow[27]and[28].We consider (quasi-)coherent sheaves on quasi-compact separated Noetherian schemes. We write Qcoh for the dg category of (fibrant replacements of) unbounded complexes of quasi-coherent sheaves and coh for its full dg subcategory of complexes with coherent cohomology and bounded cohomological amplitude. Let IndCoh be the ind-completion of coh. Thus, the dg category IndCoh isNotice that the first three rows form a short exact sequence and that the composition of the last vertical morphism with the second last vertical morphism is only homotopic to zero. Now let Q be a finite quiver, l the product over the ke i , where i runs through the vertices of Q, w a potential on Q and A = Γ the associated complete Ginzburg algebra with generators the arrows α of Q in degree 0, the reversed arrows α * in degree −1 and the loops t i in degree −2. Then A is Thus, it will suffice to prove that the equivalence between the cluster category of Γ and the stable category CM R is algebraic. Now using the notations of Theorem 2.8 put Γ = ( T l V, d) so that we have a quasi-isomorphism Γ ∼ → A. It induces an algebraic equivalence per( Γ) ∼ → per(A). This equivalence induces an algebraic equivalence tria(e 0 Γ) → per(R), where tria(e 0Γ ) is the triangulated subcategory generated by the Γ-module e 0 Γ. The quotient map Γ → Γ induces an algebraic triangle functor per( Γ) → per(Γ) and we know from Lemma 7.2 of[44]that it is a localization with kernel tria(e 0 Γ). We obtain a diagram of triangle functors, commutative up to isomorphism, whose vertical arrows are equivalences and whose rows are exact.By passage to the dg quotient, the rightmost vertical arrow is an algebraic triangle equivalence. By composing the algebraic inclusion F → per(A)/per(R) with a quasi-inverse of the algebraic equivalence per(Γ) ∼ → per(A)/per(R), we obtain an algebraic inclusion F → per(Γ) whose image identifies with the thick subcategory of per(Γ) generated by the simple H 0 (Γ)-modules. This subcategory equals D f g (Γ). Indeed, clearly it is contained in D f d (Γ) and conversely, it contains the. This yields the first row of the following diagram with exact rows whose vertical arrows are equivalences.Again by passage to the dg quotient, the rightmost vertical arrow is an algebraic triangle equivalence.Proof of Theorem 5.11. Let (Q, w) and (Q , w ) be the quivers with potential constructed from the formal flopping contractions and Γ and Γ the associated Ginzburg dg algebras. Let Λ = H 0 (Γ) and Λ = H 0 (Γ ) be the associated contraction algebras. Recall that these algebras are symmetric so that tilting objects coincide with silting objects in their derived categories. We will construct a quiver with potential (Q , w ) with associated Ginzburg algebra Γ , a dg Γ -Γ -bimoduleZ and an isomorphism ψ from Λ = H 0 (Γ ) to C such that the squareis commutative and the isomorphism HC 0 (Z) takes the class [w ] to [w ]. Here we write dg bimodules instead of derived tensor products, algebras instead of their derived categories and the top and bottom arrows are equivalences. Notice that by[47], Hochschild homology is functorial with respect to right perfect dg bimodules so that the notation HC 0 (Z) does make sense.By Corollary 4.27, there is a quiver with potential (Q , w ) and a weak equivalence s : Γ → A from the associated Ginzburg algebra Γ to A such that the isomorphism HC 0 ( s X) takes the class [w ] to[w ]. We defineZ = s X and ψ = φ • H 0 (s) to obtain the diagramSo by construction, the isomorphism HH 0 ( s X) = HH 0 ( ψ Z) takes [w ] to[w ]. We may assume all potentials to contain no cycles of length ≤ 2 and then it follows that ψ = φ • H 0 (s) induces an isomorphism of quivers Q ∼ → Q. Indeed, it induces an isomorphism in the Jacobian algebras and the vertices i of the quiver Q are in bijection with the isomorphism classes of simple modules S i of the pseudocompact algebra Λ = H 0 (Γ) and the number of arrows from i to j equals the dimension of the space of extensions Ext 1 Λ (S j , S i ). By Corollary 4.14, there is an isomorphism β : Γ → Γ. The dg bimodule sβ −1 X yields an algebraic triangle equivalence per(Γ) ∼ → per(Γ ). Such an equivalence induces an equivalence between the subcategories of dg modules with finite-dimensional homologies because their objects M are characterized as those for which Hom(P, M ) is finite-dimensional for any object P . Thus, the algebraic triangle equivalence per(Γ) ∼ → per(Γ ) induces an algebraic triangle equivalence in the cluster categories. Hence, by Lemma 5.12, it induces an algebraic triangle equivalence D sg (R) ∼ → D sg (R ) and therefore an algebra isomorphism HH 0 (D sg (R)) ∼ = HH 0 (D sg (R )). By Theorem 5.9, we get an isomorphism R ∼ = R .6. Contractibility of rational curve 6.1. dg k[u −1 ]-algebras. In this section, we define dg-k[u −1 ]-algebras and study their properties. All the definitions and results can be adapted to the pseudo-compact case, with notations appropriately replaced by their pseudo-compact counterparts. Proof. Since A is homologically smooth, D f d (A) is contained in per(A) and clearly it consists of torsion modules. Conversely, we know from the proof of Lemma 6.3 that per(A) u −1 -tor is the thick subcategory of D(A) generated by the cone P over the morphism A → Σ −2 A given by the multiplication by u −1 . Since A is non degenerate, the object P lies in D f d (A).Proposition 6.5. Let Q be a finite quiver with potential w such that the associated Ginzburg algebra Γ = D(Q, w) has finite-dimensional Jacobi algebra. Assume that Γ is equipped with a nondegenerate k[u −1 ]-enhancement. Then C Γ is a Z/2-graded 0CY triangulated category, equivalent with the category of perfect modules over Γ⊗ k[u −1 ] k[u, u −1 ] as k[u, u −1 ]-enhanced triangulated categories. In particular, the Jacobi algebra H 0 (Γ) is a symmetric Frobenius algebra. S a commutative dg k-algebra and Γ a dg S-algebra. 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