Universality, Characteristic Kernels and RKHS Embedding of Measures

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.

30 pages, 1 figure

References in corpus (1)

• A Kernel Method for the Two-Sample Problem

Cited by in corpus (15)

• Equivalence of distance-based and RKHS-based statistics in hypothesis testing
• The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
• Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques
• On the optimal estimation of probability measures in weak and strong topologies
• Gaussian RBF Centered Kernel Alignment (CKA) in the Large Bandwidth Limit
• Supervised Learning and Model Analysis with Compositional Data
• MMD Aggregated Two-Sample Test
• A Kernel Stein Test for Comparing Latent Variable Models
• Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents
• Data-Driven Observability Analysis for Nonlinear Stochastic Systems
• Handling Hard Affine SDP Shape Constraints in RKHSs
• A uniform kernel trick for high-dimensional two-sample problems
• Graph Neural Networks with Composite Kernels
• Conditional expectation using compactification operators
• Deep Unified Representation for Heterogeneous Recommendation