Metrizing Weak Convergence with Maximum Mean Discrepancies
Abstract
This paper characterizes the maximum mean discrepancies (MMD) that metrize the weak convergence of probability measures for a wide class of kernels. More precisely, we prove that, on a locally compact, non-compact, Hausdorff space, the MMD of a bounded continuous Borel measurable kernel k, whose reproducing kernel Hilbert space (RKHS) functions vanish at infinity, metrizes the weak convergence of probability measures if and only if k is continuous and integrally strictly positive definite (i.s.p.d.) over all signed, finite, regular Borel measures. We also correct a prior result of Simon-Gabriel & Sch\"olkopf (JMLR, 2018, Thm.12) by showing that there exist both bounded continuous i.s.p.d. kernels that do not metrize weak convergence and bounded continuous non-i.s.p.d. kernels that do metrize it.
Keywords
Cite
@article{arxiv.2006.09268,
title = {Metrizing Weak Convergence with Maximum Mean Discrepancies},
author = {Carl-Johann Simon-Gabriel and Alessandro Barp and Bernhard Schölkopf and Lester Mackey},
journal= {arXiv preprint arXiv:2006.09268},
year = {2021}
}
Comments
14 pages. Corrects in particular Thm.12 of Simon-Gabriel and Sch\"olkopf, JMLR, 19(44):1-29, 2018. See http://jmlr.org/papers/v19/16-291.html