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Universality, Characteristic Kernels and RKHS Embedding of Measures | | Bharath K. Sriperumbudur
; Kenji Fukumizu
; Gert R. G. Lanckriet
; | | Date: |
3 Mar 2010 | | Abstract: | A Hilbert space embedding for probability measures has recently been
proposed, wherein any probability measure is represented as a mean element in a
reproducing kernel Hilbert space (RKHS). Such an embedding has found
applications in homogeneity testing, independence testing, dimensionality
reduction, etc., with the requirement that the reproducing kernel is
characteristic, i.e., the embedding is injective.
In this paper, we generalize this embedding to finite signed Borel measures,
wherein any finite signed Borel measure is represented as a mean element in an
RKHS. We show that the proposed embedding is injective if and only if the
kernel is universal. This therefore, provides a novel characterization of
universal kernels, which are proposed in the context of achieving the Bayes
risk by kernel-based classification/regression algorithms. By exploiting this
relation between universality and the embedding of finite signed Borel measures
into an RKHS, we establish the relation between universal and characteristic
kernels. | | Source: | arXiv, 1003.0887 | | Services: | Forum | Review | PDF | Favorites | |
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