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Gait recognition is one of the most critical long-distance identification technologies and increasingly gains popularity in both research and industry communities. Despite the significant progress made in indoor datasets, much evidence shows that gait recognition techniques perform poorly in the wild. More importantly, we also find that some conclusions drawn from indoor datasets cannot be generalized to real applications. Therefore, the primary goal of this paper is to present a comprehensive benchmark study for better practicality rather than only a particular model for better performance. To this end, we first develop a flexible and efficient gait recognition codebase named OpenGait. Based on OpenGait, we deeply revisit the recent development of gait recognition by re-conducting the ablative experiments. Encouragingly,we detect some unperfect parts of certain prior woks, as well as new insights. Inspired by these discoveries, we develop a structurally simple, empirically powerful, and practically robust baseline model, GaitBase. Experimentally, we comprehensively compare GaitBase with many current gait recognition methods on multiple public datasets, and the results reflect that GaitBase achieves significantly strong performance in most cases regardless of indoor or outdoor situations. Code is available at //github.com/ShiqiYu/OpenGait.

This paper studies the third-order characteristic of nonsingular discrete memoryless channels and the Gaussian channel with a maximal power constraint. The third-order term in our expansions employs a new quantity here called the \emph{channel skewness}, which affects the approximation accuracy more significantly as the error probability decreases. For the Gaussian channel, evaluating Shannon's (1959) random coding and sphere-packing bounds in the central limit theorem (CLT) regime enables exact computation of the channel skewness. For discrete memoryless channels, this work generalizes Moulin's (2017) bounds on the asymptotic expansion of the maximum achievable message set size for nonsingular channels from the CLT regime to include the moderate deviations (MD) regime, thereby refining Altu\u{g} and Wagner's (2014) MD result. For an example binary symmetric channel and most practically important $(n, \epsilon)$ pairs, including $n \in [100, 500]$ and $\epsilon \in [10^{-10}, 10^{-1}]$, an approximation up to the channel skewness is the most accurate among several expansions in the literature. A derivation of the third-order term in the type-II error exponent of binary hypothesis testing in the MD regime is also included; the resulting third-order term is similar to the channel skewness.

Causal multiteam semantics is a framework where probabilistic dependencies arising from data and causation between variables can be together formalized and studied logically. We consider several logics in the setting of causal multiteam semantics that can express probability comparisons concerning formulae and constants, and encompass interventionist counterfactuals and selective implications that describe consequences of actions and consequences of learning from observations, respectively. We discover complete characterizations of expressivity of the logics in terms of families of linear equations that define the corresponding classes of causal multiteams (together with some closure conditions). The characterizations yield a strict hierarchy of expressive power. Finally, we present some undefinability results based on the characterizations.

Deep reinforcement learning has recently emerged as an appealing alternative for legged locomotion over multiple terrains by training a policy in physical simulation and then transferring it to the real world (i.e., sim-to-real transfer). Despite considerable progress, the capacity and scalability of traditional neural networks are still limited, which may hinder their applications in more complex environments. In contrast, the Transformer architecture has shown its superiority in a wide range of large-scale sequence modeling tasks, including natural language processing and decision-making problems. In this paper, we propose Terrain Transformer (TERT), a high-capacity Transformer model for quadrupedal locomotion control on various terrains. Furthermore, to better leverage Transformer in sim-to-real scenarios, we present a novel two-stage training framework consisting of an offline pretraining stage and an online correction stage, which can naturally integrate Transformer with privileged training. Extensive experiments in simulation demonstrate that TERT outperforms state-of-the-art baselines on different terrains in terms of return, energy consumption and control smoothness. In further real-world validation, TERT successfully traverses nine challenging terrains, including sand pit and stair down, which can not be accomplished by strong baselines.

In this short note, I derive the Bell-CHSH inequalities as an elementary result in the present-day theory of statistical causality based on graphical models or Bayes' nets, defined in terms of DAGs (Directed Acyclic Graphs) representing direct statistical causal influences between a number of observed and unobserved random variables. I show how spatio-temporal constraints in loophole-free Bell experiments, and natural classical statistical causality considerations, lead to Bell's notion of local hidden variables, and thence to the CHSH inequalities. The word "local" applies to the way that the chosen settings influence the observed outcomes. The case of contextual setting-dependent hidden variables (thought of as being located in the measurement devices and dependent on the measurement settings) is automatically covered, despite recent claims that Bell's conclusions can be circumvented in this way.

The Variational Monte Carlo (VMC) is a promising approach for computing the ground state energy of many-body quantum problems and attracts more and more interests due to the development of machine learning. The recent paradigms in VMC construct neural networks as trial wave functions, sample quantum configurations using Markov chain Monte Carlo (MCMC) and train neural networks with stochastic gradient descent (SGD) method. However, the theoretical convergence of VMC is still unknown when SGD interacts with MCMC sampling given a well-designed trial wave function. Since MCMC reduces the difficulty of estimating gradients, it has inevitable bias in practice. Moreover, the local energy may be unbounded, which makes it harder to analyze the error of MCMC sampling. Therefore, we assume that the local energy is sub-exponential and use the Bernstein inequality for non-stationary Markov chains to derive error bounds of the MCMC estimator. Consequently, VMC is proven to have a first order convergence rate $O(\log K/\sqrt{n K})$ with $K$ iterations and a sample size $n$. It partially explains how MCMC influences the behavior of SGD. Furthermore, we verify the so-called correlated negative curvature condition and relate it to the zero-variance phenomena in solving eigenvalue functions. It is shown that VMC escapes from saddle points and reaches $(\epsilon,\epsilon^{1/4})$ -approximate second order stationary points or $\epsilon^{1/2}$-variance points in at least $O(\epsilon^{-11/2}\log^{2}(1/\epsilon) )$ steps with high probability. Our analysis enriches the understanding of how VMC converges efficiently and can be applied to general variational methods in physics and statistics.

We describe a family of decidable propositional dynamic logics, where atomic modalities satisfy some extra conditions (for example, given by axioms of the logics K5, S5, or K45 for different atomic modalities). It follows from recent results (Kikot, Shapirovsky, Zolin, 2014; 2020) that if a modal logic $L$ admits a special type of filtration (so-called definable filtration), then its enrichments with modalities for the transitive closure and converse relations also admit definable filtration. We use these results to show that if logics $L_1, \ldots , L_n$ admit definable filtration, then the propositional dynamic logic with converse extended by the fusion $L_1*\ldots * L_n$ has the finite model property.

Knowledge graph embedding (KGE) is a increasingly popular technique that aims to represent entities and relations of knowledge graphs into low-dimensional semantic spaces for a wide spectrum of applications such as link prediction, knowledge reasoning and knowledge completion. In this paper, we provide a systematic review of existing KGE techniques based on representation spaces. Particularly, we build a fine-grained classification to categorise the models based on three mathematical perspectives of the representation spaces: (1) Algebraic perspective, (2) Geometric perspective, and (3) Analytical perspective. We introduce the rigorous definitions of fundamental mathematical spaces before diving into KGE models and their mathematical properties. We further discuss different KGE methods over the three categories, as well as summarise how spatial advantages work over different embedding needs. By collating the experimental results from downstream tasks, we also explore the advantages of mathematical space in different scenarios and the reasons behind them. We further state some promising research directions from a representation space perspective, with which we hope to inspire researchers to design their KGE models as well as their related applications with more consideration of their mathematical space properties.

We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.