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arXiv · 2006.09268

Metrizing Weak Convergence with Maximum Mean Discrepancies

Carl-Johann Simon-Gabriel·Alessandro Barp·Bernhard Schölkopf·Lester Mackey
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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 & Schölkopf (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.

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BibTeXRIS

Carl-Johann Simon-Gabriel, Alessandro Barp, Bernhard Schölkopf, Lester Mackey. 2021-09-03. Metrizing Weak Convergence with Maximum Mean Discrepancies. https://arxiv.org/abs/2006.09268

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