Unbounded Characteristic and Universal Kernels
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, Lp-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.
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BibTeX
@misc{cribeiroramallo2026unbounded,
title = {{Unbounded Characteristic and Universal Kernels}},
author = {Jose Cribeiro-Ramallo and Florian Kalinke and Zoltán Szabó},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09731},
eprint = {2610.09731},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09731},
}
RIS
TY - UNPB
TI - Unbounded Characteristic and Universal Kernels
AU - Jose Cribeiro-Ramallo
AU - Florian Kalinke
AU - Zoltán Szabó
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09731
UR - https://arxiv.org/abs/2610.09731
AB - 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.
ER -