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Kernel mean embeddings have recently attracted the attention of the machinelearning community. They map measures $\mu$ from some set $M$ to functions in areproducing kernel Hilbert space (RKHS) with kernel $k$. The RKHS distance oftwo 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 theembedding injective over $M$ (in which case $d_k$ is a metric)? (III) How doesthe $d_k$-induced topology compare to other topologies on $M$? The existingmachine learning literature has addressed these questions in cases where $M$ is(a subset of) the finite regular Borel measures. We unify, improve andgeneralise those results. Our approach naturally leads to continuous andpossibly even injective embeddings of (Schwartz-) distributions, i.e.,generalised measures, but the reader is free to focus on measures only. Inparticular, we systemise and extend various (partly known) equivalences betweendifferent notions of universal, characteristic and strictly positive definitekernels, and show that on an underlying locally compact Hausdorff space, $d_k$metrises the weak convergence of probability measures if and only if $k$ iscontinuous and characteristic.

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