9 research outputs found
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
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