Hilbert space embeddings and metrics on probability measures

Abstract

A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings: we denote this as _k, indexed by the kernel function k that defines the inner product in the RKHS. We present three theoretical properties of _k. First, we consider the question of determining the conditions on the kernel k for which _k is a metric: such k are denoted characteristic kernels. Unlike pseudometrics, a metric is zero only when two distributions coincide, thus ensuring the RKHS embedding maps all distributions uniquely (i.e., the embedding is injective). While previously published conditions may apply only in restricted circumstances (e.g. on compact domains), and are difficult to check, our conditions are straightforward and intuitive: bounded continuous strictly positive definite kernels are characteristic. Alternatively, if a bounded continuous kernel is translation-invariant on R^d, then it is characteristic if and only if the support of its Fourier transform is the entire R^d. Second, we show that there exist distinct distributions that are arbitrarily close in _k. Third, to understand the nature of the topology induced by _k, we relate _k to other popular metrics on probability measures, and present conditions on the kernel k under which _k metrizes the weak topology.

Tasks

Reproductions