Metrizing Weak Convergence with Maximum Mean Discrepancies
Authors: Carl-Johann Simon-Gabriel, Alessandro Barp, Bernhard Schölkopf, Lester Mackey; Volume 24, Issue 184, Pages 1-20, 2023.
Abstract
This paper investigates the maximum mean discrepancies (MMD) that can be used to measure the weak convergence of probability measures for a broad range of kernels. Specifically, the authors prove that on a locally compact, non-compact, Hausdorff space, the MMD of a bounded continuous Borel measurable kernel $k$, which has RKHS-functions that vanish at infinity (i.e., $H_k \subset C_0$), can be used to metrize the weak convergence of probability measures if and only if $k$ is continuous and integrally strictly positive definite ($\int$s.p.d.) over all signed, finite, regular Borel measures. Additionally, the authors correct a previous result by Simon-Gabriel and Schölkopf (JMLR 2018, Thm. 12) by showing that there are bounded continuous $\int$s.p.d. kernels that do not metrize weak convergence, as well as bounded continuous non-$\int$s.p.d. kernels that do metrize it.
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