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Universality, Characteristic Kernels and RKHS Embedding of Measures

Authors: Bharath K. Sriperumbudur, Kenji Fukumizu, Gert R. G. Lanckriet - Journal of Machine Learning Research, JMLR 2011 cited by 264

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.

10.1162/15324430276018525210.1162/153244302760185252A Kernel Method for theTwo-Sample ProblemA Kernel Method for the Two-Sample ProblemUniversal KernelsUniversal KernelsKernel Measures ofConditional DependenceKernel Measures of Conditional DependenceA Kernel StatisticalTest of IndependenceA Kernel Statistical Test of IndependenceInjective Hilbert SpaceEmbeddings of…Injective Hilbert Space Embeddings of Probability MeasuresCharacteristic Kernelson Groups and SemigroupsCharacteristic Kernels on Groups and SemigroupsKernel Choice andClassifiability for RKH…Kernel Choice and Classifiability for RKHS Embeddings of Probability DistributionsKernel dimensionreduction in regressionKernel dimension reduction in regressionHilbert Space Embeddingsand Metrics on…Hilbert Space Embeddings and Metrics on Probability MeasuresOn the relation betweenuniversality…On the relation between universality, characteristic kernels and RKHS embedding of measuresVECTOR VALUEDREPRODUCING KERNEL…VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES AND UNIVERSALITYA Kernel Two-Sample TestA Kernel Two-Sample TestEquivalence ofdistance-based and…Equivalence of distance-based and RKHS-based statistics in hypothesis testingKernel Bayes' rule:Bayesian inference with…Kernel Bayes' rule: Bayesian inference with positive definite kernelsA Primer on ReproducingKernel Hilbert SpacesA Primer on Reproducing Kernel Hilbert SpacesKernel Mean ShrinkageEstimatorsKernel Mean Shrinkage EstimatorsDensity Estimation inInfinite Dimensional…Density Estimation in Infinite Dimensional Exponential FamiliesGaussian Processes andKernel Methods: A Revie…Gaussian Processes and Kernel Methods: A Review on Connections and EquivalencesKernel DistributionEmbeddings: Universal…Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on DistributionsA Kernel MultipleChange-point Algorithm…A Kernel Multiple Change-point Algorithm via Model SelectionEstimating Rényi'sα-Cross-Entropies in a…Estimating Rényi's α-Cross-Entropies in a Matrix-Based WayA Kernel Two-Sample Testfor Functional DataA Kernel Two-Sample Test for Functional DataKernel PartialCorrelation Coefficient…Kernel Partial Correlation Coefficient - a Measure of Conditional DependenceUniversality,Characteristic Kernels…Universality, Characteristic Kernels and RKHS Embedding of MeasuresEarlier referencesFocus paperCiting papersOlderNewer

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