Minimum intrinsic dimension scaling for entropic optimal transport

Motivated by the manifold hypothesis, which states that data with a high extrinsic dimension may yet have a low intrinsic dimension, we develop refined statistical bounds for entropic optimal transport that are sensitive to the intrinsic dimension of the data. Our bounds involve a robust notion of intrinsic dimension, measured at only a single distance scale depending on the regularization parameter, and show that it is only the minimum of these single-scale intrinsic dimensions which governs the rate of convergence. We call this the Minimum Intrinsic Dimension scaling (MID scaling) phenomenon, and establish MID scaling with no assumptions on the data distributions so long as the cost is bounded and Lipschitz, and for various entropic optimal transport quantities beyond just values, with stronger analogs when one distribution is supported on a manifold. Our results significantly advance the theoretical state of the art by showing that MID scaling is a generic phenomenon, and provide the first rigorous interpretation of the statistical effect of entropic regularization as a distance scale.Comment: 53 page

Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space

This work establishes fast rates of convergence for empirical barycenters over a large class of geodesic spaces with curvature bounds in the sense of Alexandrov. More specifically, we show that parametric rates of convergence are achievable under natural conditions that characterize the bi-extendibility of geodesics emanating from a barycenter. These results largely advance the state-of-the-art on the subject both in terms of rates of convergence and the variety of spaces covered. In particular, our results apply to infinite-dimensional spaces such as the 2-Wasserstein space, where bi-extendibility of geodesics translates into regularity of Kantorovich potentials

Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent

We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric. Although the objective is geodesically non-convex, Riemannian GD empirically converges rapidly, in fact faster than off-the-shelf methods such as Euclidean GD and SDP solvers. This stands in stark contrast to the best-known theoretical results for Riemannian GD, which depend exponentially on the dimension. In this work, we prove new geodesic convexity results which provide stronger control of the iterates, yielding a dimension-free convergence rate. Our techniques also enable the analysis of two related notions of averaging, the entropically-regularized barycenter and the geometric median, providing the first convergence guarantees for Riemannian GD for these problems.Comment: 48 pages, 8 figure

Frog Model Wakeup Time on the Complete Graph

The frog model is a system of random walks where active particles set sleeping particles in motion. On the complete graph with n vertices it is equivalent to a well-understood rumor spreading model. We given an alternate and elementary proof that the wakeup time, that is, the expected time for every particle to be activated, is &Theta(log n). Additionally, we give an explicit distributional equation for the wakeup time as a mixture of geometric random variables

Statistical aspects of optimal transport

Optimal transport (OT) is a flexible framework for contrasting and interpolating probability measures which has recently been applied throughout science, including in machine learning, statistics, graphics, economics, biology, and more. In this thesis, we study several statistical problems at the forefront of applied optimal transport, prioritizing statistically and computationally practical results. We begin by considering one of the most popular applications of OT in practice, the barycenter problem, providing dimension-free rates of statistical estimation. In the Gaussian case, we analyze first-order methods for computing barycenters, and develop global, dimension-free rates of convergence despite the non-convexity of the problem. Extending beyond the Gaussian case, however, is challenging due to the fundamental curse of dimensionality for OT, which motivates the study of a regularized, and in fact more computationally feasible, form of optimal transport, dubbed entropic optimal transport (entropic OT). Recent work has suggested that entropic OT may escape the curse of dimensionality of un-regularized OT, and in this thesis we develop a refined theory of the statistical behavior of entropic OT by showing that entropic OT does attain truly dimension-free rates of convergence in the large regularization regime, as well as automatically adapts to the intrinsic dimension of the data in the small regularization regime. We also consider the rate of approximation of entropic OT in the semi-discrete case, and complement these results by considering the problem of trajectory reconstruction, proposing two practical methods based off both un-regularized and entropic OT.Ph.D