Hilbert Space Embeddings and Metrics on Probability Measures

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 gamma(k), indexed by the kernel function k that defines the inner product in the RKHS. We present three theoretical properties of gamma(k). First, we consider the question of determining the conditions on the kernel k for which gamma(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: integrally 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 the distance between distributions under gamma(k) results from an interplay between the properties of the kernel and the distributions, by demonstrating that distributions are close in the embedding space when their differences occur at higher frequencies. Third, to understand the nature of the topology induced by g gamma(k), we relate g gamma(k) to other popular metrics on probability measures, and present conditions on the kernel k under which gamma(k) metrizes the weak topology.