UCL logo

UCL Discovery

UCL home » Library Services » Electronic resources » UCL Discovery

Hilbert space embeddings and metrics on probability measures

Sriperumbudur, BK; Gretton, A; Fukumizu, K; Schölkopf, B; Lanckriet, GRG; (2009) Hilbert space embeddings and metrics on probability measures.

Full text not available from this repository.


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 {\em 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 $\bb{R}^d$, then it is characteristic if and only if the support of its Fourier transform is the entire $\bb{R}^d$. Second, we show that there exist distinct distributions that are arbitrarily close in $\gamma_k$. Third, to understand the nature of the topology induced by $\gamma_k$, we relate $\gamma_k$ to other popular metrics on probability measures, and present conditions on the kernel $k$ under which $\gamma_k$ metrizes the weak topology.

Type: Article
Title: Hilbert space embeddings and metrics on probability measures
Additional information: 48 pages
Keywords: stat.ML, stat.ML, math.ST, stat.TH
UCL classification: UCL > School of Life and Medical Sciences > Faculty of Life Sciences > Gatsby Computational Neuroscience Unit
URI: http://discovery.ucl.ac.uk/id/eprint/1334295
Downloads since deposit
Download activity - last month
Download activity - last 12 months
Downloads by country - last 12 months

Archive Staff Only

View Item View Item