Unbounded Characteristic and Universal Kernels

1 INTRODUCTION

• These expressivity notions help determine the validity of kernel-based discrepancy and dependence measures, including MMD and HSIC.Source: §1 · pp. 1–2
• The paper establishes relations among characteristic, Lp-universal, i.s.p.d., and s.n.t. properties for general domains and unbounded kernels.The introduction identifies these relations as open for unbounded kernels on topological domains.Source: §1 · p. 2
• For subgroups of finite signed measures, i.s.p.d. and characteristic properties are equivalent, while no Stein kernel is Lp-universal for p∈[1,∞).Source: §1 · p. 2
• Equivalence among Lp-universality for all p∈[1,∞) persists for unbounded kernels under mild conditions.The cited earlier result concerned c0-kernels on locally compact second-countable spaces.Source: §1 · p. 2
• The analysis uses sets of finite signed measures absolutely continuous with respect to kernel-dependent p-integrable measures satisfying 1−1/p integrability conditions.Source: §1 · p. 2

2 NOTATIONS

• The section sets general conventions for spaces, products, measures, and function spaces used throughout the paper.Source: §2 · pp. 2–3
• It defines F-characteristic, F-i.s.p.d., and Lp-universality for measure families, with bounded-kernel cases recovering classical notions.The section also introduces F-s.n.t. and summarizes how the extended notions specialize to classical ones.Source: §2 · pp. 2, 4
• Semimetric terminology includes negative type and F-strong negative type, with kernel-induced semimetrics connecting distance and kernel formulations.Source: §2 · pp. 4–5

3 RESULTS

• The results connect characteristic, integrally strictly positive definite, strong negative type, and Lp-universality properties for general kernels, including unbounded ones.The section summarizes relationships across several measure families and kernel settings.Source: §3 · p. 5
• Finite p-th moment measure sets are natural domains for kernel statistics, including MMD, HSIC, and KSD.MMD requires both probability measures to belong to P1(K; X); the stated sufficient HSIC condition requires second moments for its component measures.Source: §3 · p. 5

3.1 Characteristic, I.s.p.d., and S.n.t. Properties

• A semimetric space induced by K has F-s.n.t. if and only if K is F-i.s.p.d., for F contained in the specified finite-second-moment measure set.The theorem states the equivalence for F ⊆ [P2_b(K; X)]0.Source: §3.1 · p. 7
• F-characteristic implies F-i.s.p.d. when 0 belongs to F, and the converse holds when F is also closed under negation and addition.The result extends a known result from Banach spaces to groups with addition.Source: §3.1 · p. 7
• Pp(K; X)-characteristic is equivalent to [Pp_b(K; X)]0-characteristic and to [Pp_b(K; X)]0-i.s.p.d., extending prior results to unbounded kernels with finite p-th moment.The cited prior results are Sriperumbudur et al. (2010, Lemma 8) and Simon-Gabriel and Schölkopf (2018, Theorem 8(i)–(ii)).Source: §3.1 · p. 7
• For integer p ≥ 2, Pp(K; X)-characteristic is equivalent to the induced semimetric space having [Pp_b(K; X)]0-s.n.t.For p = 2, this reduces to the result of Sejdinovic et al. (2013b, Proposition 29).Source: §3.1 · p. 7

3.2 Lp-universality

• Lp-universality is equivalent to Ap(K; X)-i.s.p.d., and Ap(K; X) equals Pb(K; X), so Lp-universality holds for every finite p whenever it holds for one.These results apply to separable RKHSs for Theorem 7 and arbitrary RKHSs for Theorem 8.Source: §3.2 · p. 8
• The results establish Lp-universality for an unbounded kernel on Rd and for shifted fractional Brownian kernels, while ruling it out for Stein kernels.The shifted fractional Brownian kernel is Lp-universal for any p ∈ [1, ∞); the Stein-kernel conclusion applies to any p ≥ 1.Source: §3.2 · p. 8

• Table 2 summarizes results from the supplement, including both the authors’ results and external results.Source: Table 2 · p. 11
• Figure 4 maps dependencies among results and remarks from the main text and supplement.An arrow R1 ← R2 means R1 depends on R2.Source: Figure 4 · p. 11, Unbounded Characteristic and Universal… · p. 11
• Figure 4 distinguishes main-text results and remarks in black, auxiliary results in blue, and external results in green.Source: Figure 4 · p. 11

• For q ≥ p, the proof establishes P1(K; X) ⊆ Pq1(K; X) when K is bounded.It bounds the integrand using boundedness of K and finiteness of the measure.Source: §A.1 · p. 12
• For an unbounded K and q > p, the proof constructs a probability measure witnessing strictness of the inclusion between the corresponding moment classes.It uses unboundedness to choose a measure belonging to one class but not the other.Source: §A.1 · p. 12
• The constructed measure also establishes the claim for finite signed measures.Source: §A.1 · p. 12

A.2 Proof of Theorem 2

• Assuming F-s.n.t., the proof derives strict positivity of the kernel integral; conversely, F-i.s.p.d. yields negative type and strict negativity for the induced semimetric integral.Source: §A.2 · pp. 12–13

A.3 Proof of Theorem 3

• F-characteristic implies F-i.s.p.d. by comparing embeddings of a measure and zero.Source: §A.3 · p. 13
• If F is closed under negation and addition, F-i.s.p.d. implies F-characteristic by applying strict positivity to the difference of two distinct measures.Source: §A.3 · p. 13

A.4 Proof of Theorem 5

• Pp1-characteristic implies Pp_b-characteristic by applying the embedding-norm argument to the difference of distinct finite measures.The proof uses the stated homogeneity for a positive scaling factor M.Source: §A.4 · p. 13
• The proof links each characteristic property to integral strict positive definiteness on the corresponding zero-mass measure class.It invokes Theorem 3 to obtain the required strict positivity.Source: §A.4 · pp. 13–14
• For distinct P,Q in Pp1(K;X), the proof sets F=P−Q≠0 and uses Ppb(K;X)-characteristic to infer integral strict positive definiteness by Theorem 3, toward establishing Pp1-characteristic.Source: §A.4 · p. 14

A.5 Proof of Theorem 7

• The diagram combines the mean embedding restricted to Absp′(P), the map iP, and SK; their injectivity is used to prove density and strict positivity.The proof identifies these as its two goals.Source: §A.5 · p. 14
• Lp-universality is used to establish integral strict positive definiteness on Absp′(P), yielding the required injectivity in the diagram.The argument invokes Theorem C.3 and Theorem 3.Source: §A.5 · p. 14
• If K is Ap(K; X)-integrally strictly positive definite, the proof derives injectivity of the maps and concludes HK = Lp(X, P) for all P ∈ Pp1(K; X).It passes through Absp′(P)-characteristicness and injectivity of SK.Source: §A.5 · p. 14

A.6 Proof of Theorem 8

• For p = 1, Lemma B.8 gives the inclusion; for p > 1, the proof takes an arbitrary nonzero F ∈ Pb(K; X) and constructs a measure ν.The construction is intended to ensure ν ∈ Pp_b(K; X), F ≪ ν, and fK ∈ Lp′(X, ν).Source: §A.6 · pp. 14–15
• The proof defines a positive measurable function fK and shows it is integrable with respect to |F| before using it to define ν.The passage states fK ∈ L1(X, |F|) and ν(X) > 0.Source: §A.6 · p. 15
• The argument verifies F ≪ ν and fK ∈ Lp′(X, ν) using the change-of-measure property and the assumed membership F ∈ Pb(K; X).It also establishes the needed integrability condition for ν.Source: §A.6 · pp. 15–16
• Normalizing ν by its positive total mass gives the probability measure P := ν/ν(X) used to complete the proof.The construction ensures ν(X) > 0.Source: §A.6 · pp. 15–16

Lemma B.1

• Every nonnegative, Borel-measurable unbounded function is integrable under some probability measure but not in Lq for any q > 1.Source: B AUXILIARY RESULTS · p. 16

Lemma B.4

• Lemma B.4 relates injectivity of the barycenter to strong negative type for semimetric spaces of negative type when p ≥ 2.Source: B AUXILIARY RESULTS · pp. 16, 18

Lemmas B.2–B.3

• In both kernel and semimetric settings, each nonzero zero-mass signed measure F in the stated P_b class equals M(P − Q) for distinct probability measures P,Q in the corresponding P_1 class, with M > 0.Source: B AUXILIARY RESULTS · p. 17

Lemma B.5

• For P ∈ P_1(K; X) and p ∈ (1,∞], Absp(P) is normed and isometrically isomorphic to Lp(X,P).Source: B AUXILIARY RESULTS · p. 18

Corollary B.6

• For p′ ∈ (1,∞] and P ∈ P_1(K; X), (Absp′(P), +) is a group.Source: B AUXILIARY RESULTS · p. 19

Lemma B.7

• For p ∈ [1,∞) and P ∈ P_1(K; X), Absp′(P) ⊆ P_b(K; X), which implies Ap(K; X) ⊆ P_b(K; X).Source: B AUXILIARY RESULTS · p. 19

Lemma B.8

• For p ∈ [1,∞), P_b(K; X) ⊆ Ap(K; X).Source: B AUXILIARY RESULTS · p. 19

Lemma B.9

• The exponential kernel K(x,y) = e^(a⟨x,y⟩), for a > 0 on R^d, is P_b(K; R^d)-integrally strictly positive definite.Source: B AUXILIARY RESULTS · p. 20

B AUXILIARY RESULTS

• The Gaussian kernel is M_1(R^d)-characteristic and therefore M_b(R^d)-i.s.p.d., establishing claim (g).The stated verification assumes f > 0 and F ≠ 0.Source: B AUXILIARY RESULTS · p. 21
• For a P_p^1(K; X)-characteristic kernel K, adding c > 0 preserves P_p^b(K; X) and makes K + c P_p^b(K; X)-i.s.p.d., extending a known result to unbounded kernels.The result is stated as an extension of Simon-Gabriel and Schölkopf (2018, Theorem 8(i)–(iii)).Source: B AUXILIARY RESULTS · p. 21

C EXTERNAL RESULTS

• Proposition C.1 relates the classes of measures for which a kernel and its generated semimetric are integrable.It is attributed to Sejdinovic et al. (2013b, Proposition 20).Source: C EXTERNAL RESULTS · p. 22
• Fubini–Tonelli is stated for nonnegative functions on products of σ-finite measure spaces.The appendix cites Folland (1999, Theorem 2.37.b).Source: C EXTERNAL RESULTS · p. 22
• For a separable RKHS, the SK operator is injective exactly when the RKHS is dense in Lp(X, P).The result also states HK ⊆ Lp(X, P) under the listed assumptions.Source: C EXTERNAL RESULTS · p. 22
• The appendix includes a generalized Hölder inequality for finite measures and exponents pm ∈ [1, ∞].It is attributed to Folland (1999, Exercise 6.31).Source: C EXTERNAL RESULTS · p. 22