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 RKHS-functions vanish at infinity (i.e., Hk ⊂ C0), metrizes the weak convergence of probability measures if and only if k is continuous and integrally strictly positive definite (∫ s.p.d.) over all signed, finite, regular Borel measures. We also correct a prior result of Simon-Gabriel and Schölkopf (JMLR 2018, Thm. 12) by showing that there exist both bounded continuous ∫ s.p.d. kernels that do not metrize weak convergence and bounded continuous non-∫ s.p.d. kernels that do metrize it
| Author(s): | Simon-Gabriel, C.-J. and Barp, A. and Schölkopf, B. and Mackey, L. |
| Journal: | Journal of Machine Learning Research |
| Volume: | 24 |
| Number (issue): | 184 |
| Year: | 2023 |
| Project(s): | |
| Bibtex Type: | Article (article) |
| State: | Published |
| URL: | https://www.jmlr.org/papers/volume24/21-0599/21-0599.pdf |
| Article Number: | 599 |
| Electronic Archiving: | grant_archive |
| Links: | |
BibTex
@article{SimBarSchMac21,
title = {Metrizing Weak Convergence with Maximum Mean Discrepancies},
journal = {Journal of Machine Learning Research},
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 RKHS-functions vanish at infinity (i.e., Hk ⊂ C0), metrizes
the weak convergence of probability measures if and only if k is continuous and integrally
strictly positive definite (∫ s.p.d.) over all signed, finite, regular Borel measures. We also
correct a prior result of Simon-Gabriel and Schölkopf (JMLR 2018, Thm. 12) by showing that
there exist both bounded continuous ∫ s.p.d. kernels that do not metrize weak convergence
and bounded continuous non-∫ s.p.d. kernels that do metrize it},
volume = {24},
number = {184},
year = {2023},
slug = {simbarschmac21},
author = {Simon-Gabriel, C.-J. and Barp, A. and Sch{\"o}lkopf, B. and Mackey, L.},
url = {https://www.jmlr.org/papers/volume24/21-0599/21-0599.pdf}
}