On the relation between universality, characteristic kernels and RKHS embedding of measures

Bharath Sriperumbudur, Kenji Fukumizu, Gert Lanckriet
Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, PMLR 9:773-780, 2010.

Abstract

Universal kernels have been shown to play an important role in the achievability of the Bayes risk by many kernel-based algorithms that include binary classification, regression, etc. In this paper, we propose a notion of universality that generalizes the notions introduced by Steinwart and Micchelli et al. and study the necessary and sufficient conditions for a kernel to be universal. We show that all these notions of universality are closely linked to the injective embedding of a certain class of Borel measures into a reproducing kernel Hilbert space (RKHS). By exploiting this relation between universality and the embedding of Borel measures into an RKHS, we establish the relation between universal and characteristic kernels. The latter have been proposed in the context of the RKHS embedding of probability measures, used in statistical applications like homogeneity testing, independence testing, etc.

Cite this Paper

BibTeX
@InProceedings{pmlr-v9-sriperumbudur10a, title = {On the relation between universality, characteristic kernels and RKHS embedding of measures}, author = {Sriperumbudur, Bharath and Fukumizu, Kenji and Lanckriet, Gert}, booktitle = {Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics}, pages = {773--780}, year = {2010}, editor = {Teh, Yee Whye and Titterington, Mike}, volume = {9}, series = {Proceedings of Machine Learning Research}, address = {Chia Laguna Resort, Sardinia, Italy}, month = {13--15 May}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v9/sriperumbudur10a/sriperumbudur10a.pdf}, url = {https://proceedings.mlr.press/v9/sriperumbudur10a.html}, abstract = {Universal kernels have been shown to play an important role in the achievability of the Bayes risk by many kernel-based algorithms that include binary classification, regression, etc. In this paper, we propose a notion of universality that generalizes the notions introduced by Steinwart and Micchelli et al. and study the necessary and sufficient conditions for a kernel to be universal. We show that all these notions of universality are closely linked to the injective embedding of a certain class of Borel measures into a reproducing kernel Hilbert space (RKHS). By exploiting this relation between universality and the embedding of Borel measures into an RKHS, we establish the relation between universal and characteristic kernels. The latter have been proposed in the context of the RKHS embedding of probability measures, used in statistical applications like homogeneity testing, independence testing, etc.} }
Endnote
%0 Conference Paper %T On the relation between universality, characteristic kernels and RKHS embedding of measures %A Bharath Sriperumbudur %A Kenji Fukumizu %A Gert Lanckriet %B Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2010 %E Yee Whye Teh %E Mike Titterington %F pmlr-v9-sriperumbudur10a %I PMLR %P 773--780 %U https://proceedings.mlr.press/v9/sriperumbudur10a.html %V 9 %X Universal kernels have been shown to play an important role in the achievability of the Bayes risk by many kernel-based algorithms that include binary classification, regression, etc. In this paper, we propose a notion of universality that generalizes the notions introduced by Steinwart and Micchelli et al. and study the necessary and sufficient conditions for a kernel to be universal. We show that all these notions of universality are closely linked to the injective embedding of a certain class of Borel measures into a reproducing kernel Hilbert space (RKHS). By exploiting this relation between universality and the embedding of Borel measures into an RKHS, we establish the relation between universal and characteristic kernels. The latter have been proposed in the context of the RKHS embedding of probability measures, used in statistical applications like homogeneity testing, independence testing, etc.
RIS
TY - CPAPER TI - On the relation between universality, characteristic kernels and RKHS embedding of measures AU - Bharath Sriperumbudur AU - Kenji Fukumizu AU - Gert Lanckriet BT - Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics DA - 2010/03/31 ED - Yee Whye Teh ED - Mike Titterington ID - pmlr-v9-sriperumbudur10a PB - PMLR DP - Proceedings of Machine Learning Research VL - 9 SP - 773 EP - 780 L1 - http://proceedings.mlr.press/v9/sriperumbudur10a/sriperumbudur10a.pdf UR - https://proceedings.mlr.press/v9/sriperumbudur10a.html AB - Universal kernels have been shown to play an important role in the achievability of the Bayes risk by many kernel-based algorithms that include binary classification, regression, etc. In this paper, we propose a notion of universality that generalizes the notions introduced by Steinwart and Micchelli et al. and study the necessary and sufficient conditions for a kernel to be universal. We show that all these notions of universality are closely linked to the injective embedding of a certain class of Borel measures into a reproducing kernel Hilbert space (RKHS). By exploiting this relation between universality and the embedding of Borel measures into an RKHS, we establish the relation between universal and characteristic kernels. The latter have been proposed in the context of the RKHS embedding of probability measures, used in statistical applications like homogeneity testing, independence testing, etc. ER -
APA
Sriperumbudur, B., Fukumizu, K. & Lanckriet, G.. (2010). On the relation between universality, characteristic kernels and RKHS embedding of measures. Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 9:773-780 Available from https://proceedings.mlr.press/v9/sriperumbudur10a.html.

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