Injective Hilbert Space Embeddings of 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). The embedding function has been proven to be injective when the reproducing kernel is universal. In this case, the embedding induces a metric on the space of probability distributions defined on compact metric spaces. In the present work, we consider more broadly the problem of specifying characteristic kernels, defined as kernels for which the RKHS embedding of probability measures is injective. In particular, characteristic kernels can include non-universal kernels. We restrict ourselves to translation-invariant kernels on Euclidean space, and define the associated metric on probability measures in terms of the Fourier spectrum of the kernel and characteristic functions of these measures. The su...

• Type: Conference
• Year: 2008
• Where: COLT
• Authors: Bharath K. Sriperumbudur, Arthur Gretton, Kenji Fukumizu, Gert R. G. Lanckriet, Bernhard Shoelkopf
• Keywords: Characteristic Kernels, COLT 2008, Embedding, Machine Learning, Probability Measures