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 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 diff...

• Type: Journal
• Year: 2010
• Where: JMLR
• Authors: Bharath K. Sriperumbudur, Arthur Gretton, Kenji Fukumizu, Bernhard Schölkopf, Gert R. G. Lanckriet
• Keywords: Embedding, Hilbert Space, JMLR 2010, Probability Measures