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Abstract: A Hilbert space embedding for probability measures has recently beenproposed, wherein any probability measure is represented as a mean element in areproducing kernel Hilbert space RKHS. Such an embedding has foundapplications in homogeneity testing, independence testing, dimensionalityreduction, etc., with the requirement that the reproducing kernel ischaracteristic, 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 anRKHS. We show that the proposed embedding is injective if and only if thekernel is universal. This therefore, provides a novel characterization ofuniversal kernels, which are proposed in the context of achieving the Bayesrisk by kernel-based classification-regression algorithms. By exploiting thisrelation between universality and the embedding of finite signed Borel measuresinto an RKHS, we establish the relation between universal and characteristickernels.



Autor: Bharath K. Sriperumbudur, Kenji Fukumizu, Gert R. G. Lanckriet

Fuente: https://arxiv.org/



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