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 & 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.

14 pages. Corrects in particular Thm.12 of Simon-Gabriel and Schölkopf, JMLR, 19(44):1-29, 2018. See http://jmlr.org/papers/v19/16-291.html

References in corpus (6)

• Training generative neural networks via Maximum Mean Discrepancy optimization
• Statistical Inference for Generative Models with Maximum Mean Discrepancy
• Learning to Draw Samples with Amortized Stein Variational Gradient Descent
• Stein Point Markov Chain Monte Carlo
• The reproducing Stein kernel approach for post-hoc corrected sampling
• Asymptotically Optimal One- and Two-Sample Testing with Kernels

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• Nonparametric approximation of conditional expectation operators
• A Note on Optimizing Distributions using Kernel Mean Embeddings
• MMD-based Variable Importance for Distributional Random Forest
• A Kernel Two-sample Test for Dynamical Systems
• Minimum Discrepancy Methods in Uncertainty Quantification