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

By Carl-Johann Simon-Gabriel, Bernhard Schölkopf. 2016

99 citations recorded by the source corpus.

Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures $\mu$ from some set $M$ to functions in a reproducing kernel Hilbert space (RKHS) with kernel $k$. The RKHS distance of two mapped measures is a semi-metric $d_k$ over $M$. We study three questions. (I) For a given kernel, what sets $M$ can be embedded? (II) When is the embedding injective over $M$ (in which case $d_k$ is a metric)? (III) How does the $d_k$-induced topology compare to other topologies on $M$? The existing machine learning literature has addressed these questions in cases where $M$ 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, $d_k$ metrises the weak convergence of probability measures if and only if $k$ is continuous and characteristic.

Subfields: Artificial Intelligence, Computer Vision and Pattern Recognition

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Citations in the corpus

Consistent Kernel Mean Estimation for Functions of Random Variables (2016, methodology): “Using Corollary 52 of Simon-Gabriel and Schölkopf (2016) we conclude that: µ̂kxX → µ kx X implies that P̂ converges weakly to P .”
Generalized Energy Based Models (2020, background): “This will be the case if the kernel is characteristic Simon-Gabriel and Scholkopf (2018).”