Injective hilbert space embeddings of probability measures
Year: 2008
Type: article
Abstract: A Hilbert space embedding for probability measures\nhas recently been proposed, with applications\nincluding dimensionality reduction, homogeneity\ntesting and independence testing. This embedding\nrepresents any probability measure as a mean element\nin a reproducing kernel Hilbert space (RKHS).\nThe embedding function has been proven to be injective\nwhen the reproducing kernel is universal.\nIn this case, the embedding induces a metric on the\nspace of probability distributions defined on compact\nmetric spaces.\nIn the present work, we consider more broadly the\nproblem of specifying characteristic kernels, defined\nas kernels for which the RKHS embedding\nof probability measures is injective. In particular,\ncharacteristic kernels can include non-universal kernels.\nWe restrict ourselves to translation-invariant\nkernels on Euclidean space, and define the associated\nmetric on probability measures in terms of\nthe Fourier spectrum of the kernel and characteristic\nfunctions of these measures. The support of the\nkernel spectrum is important in finding whether a\nkernel is characteristic: in particular, the embedding\nis injective if and only if the kernel spectrum\nhas the entire domain as its support. Characteristic\nkernels may nonetheless have difficulty in distinguishing\ncertain distributions on the basis of finite\nsamples, again due to the interaction of the kernel\nspectrum and the characteristic functions of the\nmeasures.
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1515072779