math.STOct 7, 2026

Unbounded Characteristic and Universal Kernels

Authors: Jose Cribeiro-Ramallo, Florian Kalinke, Zoltán Szabó

Organizations: Karlsruhe Institute of Technology · London School of Economics

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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, LpL_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.

Figures & tables

StatementContentConstraint
Theorem 2(X,ρK)(\mathcal{X},\rho_{K}) has F\mathscr{F} -s.n.t. ⇔\Leftrightarrow KK is F\mathscr{F} -i.s.p.d.F⊆[Pb2(K;X)]0\mathscr{F}\subseteq[\mathcal{P}_{b}^{2}(K;\mathcal{X})]^{0}
Theorem 3KK is F\mathscr{F} -characteristic ⇒\Rightarrow KK is F\mathscr{F} -i.s.p.d.0∈F⊆Pb(K;X)0\in\mathscr{F}\subseteq\mathcal{P}_{b}(K;\mathcal{X})
Theorem 3KK is F\mathscr{F} -characteristic ⇐\Leftarrow KK is F\mathscr{F} -i.s.p.d.(F,+)⊆Pb(K;X)(\mathscr{F},+)\subseteq\mathcal{P}_{b}(K;\mathcal{X}) is a group
Theorem 5KK is P1p(K;X)\mathcal{P}_{1}^{p}(K;\mathcal{X}) -characteristic ⇔\Leftrightarrow KK is [Pbp(K;X)]0[\mathcal{P}_{b}^{p}(K;\mathcal{X})]^{0} -characteristicp∈[1,∞)p\in[1,\infty)
Theorem 7KK is LpL_{p} -universal ⇔\Leftrightarrow KK is Ap(K;X)\mathcal{A}^{p}(K;\mathcal{X}) -i.s.p.d.p∈[1,∞)p\in[1,\infty) , HK\mathscr{H}_{K} : separable
Theorem 8Ap(K;X)=Pb(K;X)\mathcal{A}^{p}(K;\mathcal{X})=\mathcal{P}_{b}(K;\mathcal{X})p∈[1,∞)p\in[1,\infty)
Table 1: Summary of our main results.
Figure 1: Relations between the definitions used with classical notions in the literature. The items are ordered by the year of the introduction of the concepts on the r.h.s.
Figure 2: Implications of our results for a continuous kernel KK on a separable topological space X\mathcal{X} . Def. indicates that it follows from the definition. Char., Lem., Pr., and Th. stand for characteristic, lemma, proposition, and theorem, respectively.
Figure 3: Relationships of the different sets relevant to our results. The set Pb∞(K;X)\mathcal{P}_{b}^{\infty}(K;\mathcal{X}) is defined in ( 27 ) and discussed afterwards.
StatementTypeContent
Lemma B.1ownunbounded f≥0f\geq 0 admits a P\mathbb{P} with f∈L1(X,P)f\in L_{1}(\mathcal{X},\mathbb{P}) but f∉Lq(X,P)f\not\in L_{q}(\mathcal{X},\mathbb{P}) for q>1q>1
Lemma B.2owna measure in [Pbp(K;X)]0[\mathcal{P}_{b}^{p}(K;\mathcal{X})]^{0} has a decomposition in P1p(K;X)\mathcal{P}_{1}^{p}(K;\mathcal{X})
Lemma B.3owna measure in [Pbp(ρ;X)]0[\mathcal{P}_{b}^{p}(\rho;\mathcal{X})]^{0} has a decomposition in P1p(ρ;X)\mathcal{P}_{1}^{p}(\rho;\mathcal{X})
Lemma B.4owninjectivity of the barycenter implies s.n.t.
Lemma B.5own\Absp(P)\Abs_{p}(\mathbb{P}) and Lp(X,P)L_{p}(\mathcal{X},\mathbb{P}) are isometrically isomorphic
Corollary B.6own\Absp′(P)\Abs_{p^{\prime}}(\mathbb{P}) is a group
Table 2: Summary of own and external results in the supplement.
Figure 4: Summary of the dependencies of our results and remarks. R1←R2R_{1}\leftarrow R_{2} means that “result R1R_{1} depends on R2R_{2} ”. Results and remarks from the main text are written in black, our auxiliary results are in blue, and external results are indicated by green.

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