# Metrizing Weak Convergence with Maximum Mean Discrepancies

arXiv: https://arxiv.org/abs/2006.09268 · HTML: https://mlsanity.com/paper/2006.09268/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2020.

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## Abstract (arXiv)

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.