Let $\mathbf{X} = (X_i)_{1\leq i \leq n}$ be an i.i.d. sample of square-integrable variables in $\mathbb{R}^d$, \GB{with common expectation $\mu$ and covariance matrix $\Sigma$, both unknown.} We consider the problem of testing if $\mu$ is $\eta$-close to zero, i.e. $\|\mu\| \leq \eta $ against $\|\mu\| \geq (\eta + \delta)$; we also tackle the more general two-sample mean closeness (also known as {\em relevant difference}) testing problem. The aim of this paper is to obtain nonasymptotic upper and lower bounds on the minimal separation distance $\delta$ such that we can control both the Type I and Type II errors at a given level. The main technical tools are concentration inequalities, first for a suitable estimator of $\|\mu\|^2$ used a test statistic, and secondly for estimating the operator and Frobenius norms of $\Sigma$ coming into the quantiles of said test statistic. These properties are obtained for Gaussian and bounded distributions. A particular attention is given to the dependence in the pseudo-dimension $d_*$ of the distribution, defined as $d_* := \|\Sigma\|_2^2/\|\Sigma\|_\infty^2$. In particular, for $\eta=0$, the minimum separation distance is ${\Theta}( d_*^{\frac{1}{4}}\sqrt{\|\Sigma\|_\infty/n})$, in contrast with the minimax estimation distance for $\mu$, which is ${\Theta}(d_e^{\frac{1}{2}}\sqrt{\|\Sigma\|_\infty/n})$ (where $d_e:=\|\Sigma\|_1/\|\Sigma\|_\infty$). This generalizes a phenomenon spelled out in particular by Baraud (2002).
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We merge computational mechanics' definition of causal states (predictively-equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely-applicable method that infers causal structure directly from observations of a system's behaviors whether they are over discrete or continuous events or time. A structural representation -- a finite- or infinite-state kernel $\epsilon$-machine -- is extracted by a reduced-dimension transform that gives an efficient representation of causal states and their topology. In this way, the system dynamics are represented by a stochastic (ordinary or partial) differential equation that acts on causal states. We introduce an algorithm to estimate the associated evolution operator. Paralleling the Fokker-Plank equation, it efficiently evolves causal-state distributions and makes predictions in the original data space via an RKHS functional mapping. We demonstrate these techniques, together with their predictive abilities, on discrete-time, discrete-value infinite Markov-order processes generated by finite-state hidden Markov models with (i) finite or (ii) uncountably-infinite causal states and (iii) continuous-time, continuous-value processes generated by thermally-driven chaotic flows. The method robustly estimates causal structure in the presence of varying external and measurement noise levels and for very high dimensional data.
We consider an optimal control problem for the steady-state Kirchhoff equation, a prototype for nonlocal partial differential equations, different from fractional powers of closed operators. Existence and uniqueness of solutions of the state equation, existence of global optimal solutions, differentiability of the control-to-state map and first-order necessary optimality conditions are established. The aforementioned results require the controls to be functions in $H^1$ and subject to pointwise upper and lower bounds. In order to obtain the Newton differentiability of the optimality conditions, we employ a Moreau-Yosida-type penalty approach to treat the control constraints and study its convergence. The first-order optimality conditions of the regularized problems are shown to be Newton diffentiable, and a generalized Newton method is detailed. A discretization of the optimal control problem by piecewise linear finite elements is proposed and numerical results are presented.
We consider the problem of testing for two Gibbs probabilities $\mu_0$ and $\mu_1$ defined for a dynamical system $(\Omega,T)$. Due to the fact that in general full orbits are not observable or computable, one needs to restrict to subclasses of tests defined by a finite time series $h(x_0), h(x_1)=h(T(x_0)),..., h(x_n)=h(T^n(x_0))$, $x_0\in \Omega$, $n\ge 0$, where $h:\Omega\to\mathbb R$ denotes a suitable measurable function. We determine in each class the Neyman-Pearson tests, the minimax tests, and the Bayes solutions and show the asymptotic decay of their risk functions as $n\to\infty$. In the case of $\Omega$ being a symbolic space, for each $n\in \mathbb{N}$, these optimal tests rely on the information of the measures for cylinder sets of size $n$.
Defining multivariate generalizations of the classical univariate ranks has been a long-standing open problem in statistics. Optimal transport has been shown to offer a solution by transporting data points to grid approximating a reference measure (Chernozhukov et al., 2017; Hallin, 2017; Hallin et al., 2021a). We take up this new perspective to develop and study multivariate analogues of popular correlations measures including the sign covariance, Kendall's tau and Spearman's rho. Our tests are genuinely distribution-free, hence valid irrespective of the actual (absolutely continuous) distributions of the observations. We present asymptotic distribution theory for these new statistics, providing asymptotic approximations to critical values to be used for testing independence as well as an analysis of power of the resulting tests. Interestingly, we are able to establish a multivariate elliptical Chernoff-Savage property, which guarantees that, under ellipticity, our nonparametric tests of independence when compared to Gaussian procedures enjoy an asymptotic relative efficiency of one or larger. Hence, the nonparametric tests constitute a safe replacement for procedures based on multivariate Gaussianity.
The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series $(\y_n)_{n \in \mathbb{Z}}$ with independent components is studied under the asymptotic regime where the sample size $N$ converges towards $+\infty$ while the dimension $M$ of $\y$ and the smoothing span of the estimator grow to infinity at the same rate in such a way that $\frac{M}{N} \rightarrow 0$. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically $\mathcal{N}_{\mathbb{C}}(0,\I_M)$ distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of $\frac{1}{M} + \frac{\sqrt{M}}{N}+ (\frac{M}{N})^{3}$ where $N$ is the sample size. Numerical simulations supports our results.
In Chen and Zhou 2021, they consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other is bounded by $(ts)^{H-1}$ with $H\in (\frac12,\,1)$, up to a constant factor. In this paper, we investigate the same problem but with the assumption of $H\in (0,\,\frac12)$. The starting point of this paper is a new relationship between the inner product of $\mathfrak{H}$ and that of the Hilbert space $\mathfrak{H}_1$ associated with the fractional Brownian motion $(B^{H}_t)_{t\ge 0}$. Based on this relationship and some known estimation of the inner product of $\mathfrak{H}_1$, we prove the strong consistency with $H\in (0, \frac12)$, and the asymptotic normality and the Berry-Ess\'{e}en bounds with $H\in (0,\frac38)$ for both the least squares estimator and the moment estimator of the drift parameter constructed from the continuous observations.
Conditional gradient methods (CGM) are widely used in modern machine learning. CGM's overall running time usually consists of two parts: the number of iterations and the cost of each iteration. Most efforts focus on reducing the number of iterations as a means to reduce the overall running time. In this work, we focus on improving the per iteration cost of CGM. The bottleneck step in most CGM is maximum inner product search (MaxIP), which requires a linear scan over the parameters. In practice, approximate MaxIP data-structures are found to be helpful heuristics. However, theoretically, nothing is known about the combination of approximate MaxIP data-structures and CGM. In this work, we answer this question positively by providing a formal framework to combine the locality sensitive hashing type approximate MaxIP data-structures with CGM algorithms. As a result, we show the first algorithm, where the cost per iteration is sublinear in the number of parameters, for many fundamental optimization algorithms, e.g., Frank-Wolfe, Herding algorithm, and policy gradient.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.
We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.
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