We study an independence test based on distance correlation for random fields $(X,Y)$. We consider the situations when $(X,Y)$ is observed on a lattice with equidistant grid sizes and when $(X,Y)$ is observed at random locations. We provide \asy\ theory for the sample distance correlation in both situations and show bootstrap consistency. The latter fact allows one to build a test for independence of $X$ and $Y$ based on the considered discretizations of these fields. We illustrate the performance of the bootstrap test in a simulation study involving fractional Brownian and infinite variance stable fields. The independence test is applied to Japanese meteorological data, which are observed over the entire area of Japan.
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We consider the goodness-of fit testing problem for H\"older smooth densities over $\mathbb{R}^d$: given $n$ iid observations with unknown density $p$ and given a known density $p_0$, we investigate how large $\rho$ should be to distinguish, with high probability, the case $p=p_0$ from the composite alternative of all H\"older-smooth densities $p$ such that $\|p-p_0\|_t \geq \rho$ where $t \in [1,2]$. The densities are assumed to be defined over $\mathbb{R}^d$ and to have H\"older smoothness parameter $\alpha>0$. In the present work, we solve the case $\alpha \leq 1$ and handle the case $\alpha>1$ using an additional technical restriction on the densities. We identify matching upper and lower bounds on the local minimax rates of testing, given explicitly in terms of $p_0$. We propose novel test statistics which we believe could be of independent interest. We also establish the first definition of an explicit cutoff $u_B$ allowing us to split $\mathbb{R}^d$ into a bulk part (defined as the subset of $\mathbb{R}^d$ where $p_0$ takes only values greater than or equal to $u_B$) and a tail part (defined as the complementary of the bulk), each part involving fundamentally different contributions to the local minimax rates of testing.
The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well as its Laplace approximation. In particular, the critical radius of the tube is generalized for evaluation of the asymptotic approximation error. As an example, we discuss the approximation of the largest eigenvalue distribution of the Wishart matrix with a non-identity matrix parameter. The Bonferroni method is the tube method when the index set is a finite set. We provide the formula for the asymptotic approximation error for the Bonferroni method when the variance is not constant.
We study iterated vector fields and investigate whether they are conservative, in the sense that they are the gradient of some scalar-valued function. We analyze the conservatism of various iterated vector fields, including gradient vector fields associated to loss functions of generalized linear models. We relate this study to optimization and derive novel convergence results for federated learning algorithms. In particular, we show that for certain classes of functions (including non-convex functions), federated averaging is equivalent to gradient descent on a surrogate loss function. Finally, we discuss a variety of open questions spanning topics in geometry, dynamical systems, and optimization.
We define wedge-lifted codes, a variant of lifted codes, and we study their locality properties. We show that (taking the trace of) wedge-lifted codes yields binary codes with the $t$-disjoint repair property ($t$-DRGP). When $t = N^{1/2d}$, where $N$ is the block length of the code and $d \geq 2$ is any integer, our codes give improved trade-offs between redundancy and locality among binary codes.
The aim of this paper is to study the asymptotic behavior of a particular multivariate risk measure, the Covariate-Conditional-Tail-Expectation (CCTE), based on a multivariate statistical depth function. Depth functions have become increasingly powerful tools in nonparametric inference for multivariate data, as they measure a degree of centrality of a point with respect to a distribution. A multivariate risks scenario is then represented by a depth-based lower level set of the risk factors, meaning that we consider a non-compact setting. More precisely, given a multivariate depth function D associated to a fixed probability measure, we are interested in the lower level set based on D. First, we present a plug-in approach in order to estimate the depth-based level set. In a second part, we provide a consistent estimator of our CCTE for a general depth function with a rate of convergence, and we consider the particular case of the Mahalanobis depth. A simulation study complements the performances of our estimator.
The list-decodable code has been an active topic in theoretical computer science since the seminal papers of M. Sudan and V. Guruswami in 1997-1998. There are general result about the Johnson radius and the list-decoding capacity theorem for random codes. However few results about general constraints on rates, list-decodable radius and list sizes for list-decodable codes have been obtained. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new simple but strong upper bounds for list-decodable codes based on various covering codes. Then any good upper bound on the covering radius imply a good upper bound on the size of list-decodable codes. Hence the list-decodablity of codes is a strong constraint from the view of covering codes. Our covering code upper bounds for $(d,1)$ list decodable codes give highly non-trivial upper bounds on the sizes of codes with the given minimum Hamming distances. Our results give exponential improvements on the recent generalized Singleton upper bound of Shangguan and Tamo in STOC 2020, when the code lengths are very large. The asymptotic forms of covering code bounds can partially recover the list-decoding capacity theorem, the Blinovsky bound and the combinatorial bound of Guruswami-H{\aa}stad-Sudan-Zuckerman. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial list-decodable codes.
Model-based approaches to the verification of non-terminating Cyber-Physical Systems (CPSs) usually rely on numerical simulation of the System Under Verification (SUV) model under input scenarios of possibly varying duration, chosen among those satisfying given constraints. Such constraints typically stem from requirements (or assumptions) on the SUV inputs and its operational environment as well as from the enforcement of additional conditions aiming at, e.g., prioritising the (often extremely long) verification activity, by, e.g., focusing on scenarios explicitly exercising selected requirements, or avoiding vacuity in their satisfaction. In this setting, the possibility to efficiently sample at random (with a known distribution, e.g., uniformly) within, or to efficiently enumerate (possibly in a uniformly random order) scenarios among those satisfying the given constraints is a key enabler for the viability of the verification process, e.g., via simulation-based statistical model checking. Unfortunately, in case of non-trivial combinations of constraints, iterative approaches like Markovian random walks in the space of sequences of inputs in general fail in extracting scenarios according to a given distribution, and can be very inefficient to produce legal scenarios of interest. We show how, given a set of constraints on the input scenarios succinctly defined by finite memory monitors, a data structure (scenario generator) can be synthesised, from which any-horizon scenarios satisfying the input constraints can be efficiently extracted by (possibly uniform) random sampling or (randomised) enumeration. Our approach enables seamless support to virtually all simulation-based approaches to CPS verification, ranging from simple random testing to statistical model checking and formal (i.e., exhaustive) verification.
Given a random matrix $X= (x_1,\ldots, x_n)\in \mathcal M_{p,n}$ with independent columns and satisfying concentration of measure hypotheses and a parameter $z$ whose distance to the spectrum of $\frac{1}{n} XX^T$ should not depend on $p,n$, it was previously shown that the functionals $\text{tr}(AR(z))$, for $R(z) = (\frac{1}{n}XX^T- zI_p)^{-1}$ and $A\in \mathcal M_{p}$ deterministic, have a standard deviation of order $O(\|A\|_* / \sqrt n)$. Here, we show that $\|\mathbb E[R(z)] - \tilde R(z)\|_F \leq O(1/\sqrt n)$, where $\tilde R(z)$ is a deterministic matrix depending only on $z$ and on the means and covariances of the column vectors $x_1,\ldots, x_n$ (that do not have to be identically distributed). This estimation is key to providing accurate fluctuation rates of functionals of $X$ of interest (mostly related to its spectral properties) and is proved thanks to the introduction of a semi-metric $d_s$ defined on the set $\mathcal D_n(\mathbb H)$ of diagonal matrices with complex entries and positive imaginary part and satisfying, for all $D,D' \in \mathcal D_n(\mathbb H)$: $d_s(D,D') = \max_{i\in[n]} |D_i - D_i'|/ (\Im(D_i) \Im(D_i'))^{1/2}$. Possibly most importantly, the underlying concentration of measure assumption on the columns of $X$ finds an extremely natural ground for application in modern statistical machine learning algorithms where non-linear Lipschitz mappings and high number of classes form the base ingredients.
Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale datasets. Among the possible strategies to alleviate this issue, practitioners can rely on computing estimates of these distances over subsets of data, {\em i.e.} minibatches. While computationally appealing, we highlight in this paper some limits of this strategy, arguing it can lead to undesirable smoothing effects. As an alternative, we suggest that the same minibatch strategy coupled with unbalanced optimal transport can yield more robust behavior. We discuss the associated theoretical properties, such as unbiased estimators, existence of gradients and concentration bounds. Our experimental study shows that in challenging problems associated to domain adaptation, the use of unbalanced optimal transport leads to significantly better results, competing with or surpassing recent baselines.
Fine-grained action segmentation and recognition is an important yet challenging task. Given a long, untrimmed sequence of kinematic data, the task is to classify the action at each time frame and segment the time series into the correct sequence of actions. In this paper, we propose a novel framework that combines a temporal Conditional Random Field (CRF) model with a powerful frame-level representation based on discriminative sparse coding. We introduce an end-to-end algorithm for jointly learning the weights of the CRF model, which include action classification and action transition costs, as well as an overcomplete dictionary of mid-level action primitives. This results in a CRF model that is driven by sparse coding features obtained using a discriminative dictionary that is shared among different actions and adapted to the task of structured output learning. We evaluate our method on three surgical tasks using kinematic data from the JIGSAWS dataset, as well as on a food preparation task using accelerometer data from the 50 Salads dataset. Our results show that the proposed method performs on par or better than state-of-the-art methods.
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