Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings

Abstract

This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is {\it strictly weaker} than convergence in the exact 2-Wasserstein distance. Specifically, a sequence of centered Gaussian measures converges in the 2-Sinkhorn divergence if the corresponding covariance operators converge in the Hilbert-Schmidt norm. This is in contrast to the previous known result that a sequence of centered Gaussian measures converges in the exact 2-Wasserstein distance if and only if the covariance operators converge in the trace class norm. In the reproducing kernel Hilbert space (RKHS) setting, the {\it kernel Gaussian-Sinkhorn divergence}, which is the Sinkhorn divergence between Gaussian measures defined on an RKHS, defines a semi-metric on the set of Borel probability measures on a Polish space, given a characteristic kernel on that space. With the Hilbert-Schmidt norm convergence, we obtain {\it dimension-independent} convergence rates for finite sample approximations of the kernel Gaussian-Sinkhorn divergence, with the same order as the Maximum Mean Discrepancy. These convergence rates apply in particular to Sinkhorn divergence between Gaussian measures on Euclidean and infinite-dimensional Hilbert spaces. The sample complexity for the 2-Wasserstein distance between Gaussian measures on Euclidean space, while dimension-dependent and larger than that of the Sinkhorn divergence, is exponentially faster than the worst case scenario in the literature.

arXiv comment51 pages, minor revision, references added

References in corpus (9)

• Stochastic Optimization for Large-scale Optimal Transport
• Sample Complexity of Sinkhorn divergences
• Interpolating between Optimal Transport and MMD using Sinkhorn Divergences
• Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form
• How Well Do WGANs Estimate the Wasserstein Metric?
• The statistical effect of entropic regularization in optimal transportation
• Entropic regularization of Wasserstein distance between infinite-dimensional Gaussian measures and Gaussian processes
• Kinetic correlation functionals from the entropic regularisation of the strictly-correlated electrons problem
• Alpha Procrustes metrics between positive definite operators: a unifying formulation for the Bures-Wasserstein and Log-Euclidean/Log-Hilbert-Schmidt metrics