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Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions

Abstract

Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures μ\mu from some set MM to functions in a reproducing kernel Hilbert space (RKHS) with kernel kk. The RKHS distance of two mapped measures is a semi-metric dkd_k over MM. We study three questions. (I) For a given kernel, what sets MM can be embedded? (II) When is the embedding injective over MM (in which case dkd_k is a metric)? (III) How does the dkd_k-induced topology compare to other topologies on MM? The existing machine learning literature has addressed these questions in cases where MM is (a subset of) the finite regular Borel measures. We unify, improve and generalise those results. Our approach naturally leads to continuous and possibly even injective embeddings of (Schwartz-) distributions, i.e., generalised measures, but the reader is free to focus on measures only. In particular, we systemise and extend various (partly known) equivalences between different notions of universal, characteristic and strictly positive definite kernels, and show that on an underlying locally compact Hausdorff space, dkd_k metrises the weak convergence of probability measures if and only if kk is continuous and characteristic.

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