Unbounded Characteristic and Universal Kernels

Key points

• Kernel methods are among the most powerful tools in machine learning and statistics, with a large number of successful applications.1
• Their immense success stems from the flexible function class associated to each kernel---its reproducing kernel Hilbert space (RKHS)---which facilitates statistical analysis, as well as from their computational tractability and applicability to many domains.1
• Even though unbounded kernels have received significant attention over the past decade (for instance, in the construction of kernel-based discrepancy and dependence measures such as the maximum mean discrepancy, the Hilbert-Schmidt independence criterion, and the kernel Stein discrepancy), surprisingly little is known about the relations of these notions in the unbounded case.1
• In the present paper we tackle this severe bottleneck, establishing their relations under mild assumptions.1

Sources (1)

• [1]Unbounded Characteristic and Universal Kernels
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 7, 09:25 AM

  Multiple notions (such as characteristic, $L_p$-universal, and integrally strictly positive definite) capture the expressivity of kernels and their RKHSs and play a key role in understanding the statistical properties of kernel methods; these concepts and their relations are well-understood for bounded kernels.

  Kernel methods are among the most powerful tools in machine learning and statistics, with a large number of successful applications.

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