🤖 AI Summary
This study addresses the theoretical bottleneck wherein the relationships among expressiveness concepts—such as characteristicity and universality—remain unknown for unbounded kernels. By leveraging reproducing kernel Hilbert space (RKHS) theory and functional analysis techniques, we conduct rigorous mathematical derivations. As the first work to systematically elucidate the intrinsic connections among various kernel expressiveness notions in the unbounded setting, this project clarifies the statistical properties of unbounded kernels and their associated RKHSs under mild assumptions, thereby establishing fundamental theoretical links between key concepts. This research fills a critical gap in the expressiveness theory of unbounded kernels and strengthens the theoretical foundations of kernel methods.
📝 Abstract
Kernel methods are among the most powerful tools in machine learning and statistics, with a large number of successful applications. 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. 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. 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. In the present paper we tackle this severe bottleneck, establishing their relations under mild assumptions.