Metrizing Weak Convergence with Maximum Mean Discrepancies

• URL: http://arxiv.org/abs/2006.09268v3
• Date: Fri, 3 Sep 2021 13:03:54 GMT
• Title: Metrizing Weak Convergence with Maximum Mean Discrepancies
• Authors: Carl-Johann Simon-Gabriel and Alessandro Barp and Bernhard Sch\"olkopf and Lester Mackey
• Abstract summary: This paper characterizes the maximum mean discrepancies (MMD) that metrize the weak convergence of probability measures for a wide class of kernels. We prove that, on a locally compact, non-compact, Hausdorff space, the MMD of a bounded continuous Borel measurable kernel k, metrizes the weak convergence of probability measures if and only if k is continuous.
• Score: 88.54422104669078
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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