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This paper addresses the problem of estimating entropy-regularized optimal transport (EOT) maps with squared-Euclidean cost between source and target measures that are subGaussian. In the case that the target measure is compactly supported or strongly log-concave, we show that for a recently proposed in-sample estimator, the expected squared $L^2$-error decays at least as fast as $O(n^{-1/3})$ where $n$ is the sample size. For the general subGaussian case we show that the expected $L^1$-error decays at least as fast as $O(n^{-1/6})$, and in both cases we have polynomial dependence on the regularization parameter. While these results are suboptimal compared to known results in the case of compactness of both the source and target measures (squared $L^2$-error converging at a rate $O(n^{-1})$) and for when the source is subGaussian while the target is compactly supported (squared $L^2$-error converging at a rate $O(n^{-1/2})$), their importance lie in eliminating the compact support requirements. The proof technique makes use of a bias-variance decomposition where the variance is controlled using standard concentration of measure results and the bias is handled by T1-transport inequalities along with sample complexity results in estimation of EOT cost under subGaussian assumptions. Our experimental results point to a looseness in controlling the variance terms and we conclude by posing several open problems., Comment: 30 pages, 7 figures

This paper considers the use of recently proposed optimal transport-based multivariate test statistics, namely rank energy and its variant the soft rank energy derived from entropically regularized optimal transport, for the unsupervised nonparametric change point detection (CPD) problem. We show that the soft rank energy enjoys both fast rates of statistical convergence and robust continuity properties which lead to strong performance on real datasets. Our theoretical analyses remove the need for resampling and out-of-sample extensions previously required to obtain such rates. In contrast the rank energy suffers from the curse of dimensionality in statistical estimation and moreover can signal a change point from arbitrarily small perturbations, which leads to a high rate of false alarms in CPD. Additionally, under mild regularity conditions, we quantify the discrepancy between soft rank energy and rank energy in terms of the regularization parameter. Finally, we show our approach performs favorably in numerical experiments compared to several other optimal transport-based methods as well as maximum mean discrepancy., Comment: 36 pages, 5 figures

This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycentric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing., Comment: ICML 2022

The framework of optimal transport has been leveraged to extend the notion of rank to the multivariate setting while preserving desirable properties of the resulting goodness-of-fit (GoF) statistics. In particular, the rank energy (RE) and rank maximum mean discrepancy (RMMD) are distribution-free under the null, exhibit high power in statistical testing, and are robust to outliers. In this paper, we point to and alleviate some of the practical shortcomings of these proposed GoF statistics, namely their high computational cost, high statistical sample complexity, and lack of differentiability with respect to the data. We show that all these practically important issues are addressed by considering entropy-regularized optimal transport maps in place of the rank map, which we refer to as the soft rank. We consequently propose two new statistics, the soft rank energy (sRE) and soft rank maximum mean discrepancy (sRMMD), which exhibit several desirable properties. Given $n$ sample data points, we provide non-asymptotic convergence rates for the sample estimate of the entropic transport map to its population version that are essentially of the order $n^{-1/2}$ when the starting measure is subgaussian and the target measure has compact support. This result is novel compared to existing results which achieve a rate of $n^{-1}$ but crucially rely on both measures having compact support. We leverage this result to demonstrate fast convergence of sample sRE and sRMMD to their population version making them useful for high-dimensional GoF testing. Our statistics are differentiable and amenable to popular machine learning frameworks that rely on gradient methods. We leverage these properties towards showcasing the utility of the proposed statistics for generative modeling on two important problems: image generation and generating valid knockoffs for controlled feature selection., Comment: 46 pages, 10 figures. Replacement note: Substantial revision over V2: Title change, first authors contribution change, new improved theoretical results relaxing compactness assumptions