用最大平均离散度量化弱收敛
Metrizing Weak Convergence with Maximum Mean Discrepancies
摘要：
本文描述了最大平均差异（MMD），度量的概率测度的弱收敛的一个广泛的类的核。更精确地说，我们证明了在局部紧的非紧Hausdorff空间上，有界连续Borel可测核k的MMD度量概率测度的弱收敛当且仅当k是连续的且积分严格正定的（i.s.p.d.）有限的正则Borel测度我们还通过证明存在两个有界连续i.s.p.d.，纠正了Simon-Gabriel & Schölkopf（JMLR，2018，Thm.12）的先前结果。不度量弱收敛和有界连续非I.S.P.D.的核。使其量化的核。
英文摘要：
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.
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