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#### Injective Hilbert Space Embeddings of Probability Measures

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##### Citation

Sriperumbudur, B., Gretton, A., Fukumizu, K., Lanckriet, G., & Schölkopf, B. (2008).
Injective Hilbert Space Embeddings of Probability Measures.* Proceedings of the 21st Annual Conference
on Learning Theory (COLT 2008),* 111-122.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-C83D-C

##### 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).
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 support of the
kernel spectrum is important in finding whether a
kernel is characteristic: in particular, the embedding
is injective if and only if the kernel spectrum
has the entire domain as its support. Characteristic
kernels may nonetheless have difficulty in distinguishing
certain distributions on the basis of finite
samples, again due to the interaction of the kernel
spectrum and the characteristic functions of the
measures.