[논문 리뷰] Reproducing Kernel Hilbert Spaces on Banach Completions of Virtual Persistence Diagram Groups

Charles Fanning, Mehmet Emin Aktas|arXiv (Cornell University)|2026. 02. 16.

Topological and Geometric Data Analysis인용 수 0

한 줄 요약

본 연구는 translation-invariant Gaussian kernels와 Banach-space 모델의 비-이산 가상 지속 다이어그램 그룹에 대한 RKHS 경계(RKHS bounds)를 제시하여 Lipschitz 제어와 random feature 스킴을 가능하게 한다.

ABSTRACT

Persistent homology maps a simplicial complex filtered by elements in $\mathbb R$ to finite formal sums of elements of $\mathbb R_{\leq}^{2} = \{ (b,d) \in \mathbb R^2 \cup \{ \infty \} \mid b < d \}$ called (finite) persistence diagrams. This map is stable with respect to the $p$--Wasserstein distance for all $p \in \left[1, + \infty ight]$. Bubenik and Elchesen extend the free translation-invariant commutative Lipschitz monoid of finite persistence diagrams $D(X,A) = D(X)/D(A)$ on arbitrary metric pairs $(X,d,A)$ with $A \subset X$ onto the free translation-invariant abelian Lipschitz group of virtual persistence diagrams $K(X,A) = K(X)/K(A)$ as an isometric embedding $D(X,A) \hookrightarrow K(X,A)$ via the Grothendieck group completion. They prove that the $p$-Wasserstein distance is translation invariant on $D(X,A)$ if and only if $p=1$ and define the unique translation-invariant embedding of $W_1[d]$ into $K(X,A)$ as $ρ.$ When $K(X,A)$ is locally compact abelian, translation-invariant kernels can be constructed via positive-definite functions and Bochner's theorem on the Pontryagin dual. We prove that, for the metric topology induced by $ρ$, the group $(K(X,A),ρ)$ is locally compact if and only if it is discrete, equivalently when the pointed metric space $(X/A,d_1,[A])$ is uniformly discrete, and hence this approach fails outside that case. Assuming instead that $(X/A,d_1,[A])$ is separable and not uniformly discrete, we develop a translation-invariant kernel theory for non--locally compact virtual persistence diagram groups. The group $K(X,A)$ embeds isometrically into its canonical Banach-space linearization $B=\widehat V(X,A)\cong\mathcal F(X/A,d_1)$, and each bounded symmetric positive operator $Q\colon B o B^\ast$ determines a translation-invariant Gaussian kernel $k(x,y)=\exp\!\left(- frac12\,\langle Q(x-y),x-y angle_{B,B^\ast} ight).$

연구 동기 및 목표

• Motivate and formalize virtual persistence diagrams and their Grothendieck/Banach-space completion.
• Construct translation-invariant Gaussian kernels on the Banach model to study W1-geometry of diagrams.
• Provide Lipschitz, mass, and covering-number bounds for RKHSs on non-locally compact diagram groups.
• Develop random Fourier feature approximations with probabilistic error control.
• Demonstrate the pipeline on graph-based filtrations with non-uniform label spaces.

제안 방법

• Model virtual persistence diagrams via the Grothendieck completion K(X,A) and its Banach completion B ≅ F(X/A, d1).
• Define translation-invariant Gaussian kernels on B using Gaussian measures with covariance operators, yielding kJ,Σ,t(x,y)=exp(-t/2 ||Σ1/2 J(x−y)||^2).
• Obtain Lipschitz bounds for RKHS functions restricted to K(X,A) in terms of covariance data (Theorem 4.4).
• Relate kernel geometry to diagram mass and offer a Rayleigh-quotient criterion for seminorm equivalence (Theorem 4.7).
• Embed B into ℓ2 via J with a Gaussian measure γΣ to realize random Fourier features and provide concentration and covering-number results (Section 4.5).
• Use Lipschitz-free space interpretations to connect non-locally compact cases to Banach-space kernels (Definitions 3–4).

실험 결과

연구 질문

• RQ1 How can translation-invariant kernels be constructed on the Banach completion of virtual persistence diagram groups?
• RQ2 What are the Lipschitz and metric-entropy implications of translating W1 geometry into RKHS settings for virtual diagrams?
• RQ3 When and how do Gaussian kernels on the Banach model yield controllable finite-sample approximations via random Fourier features?
• RQ4 How does the non-uniform discreteness of X/A affect local compactness and the availability of Pontryagin-dual kernel constructions?
• RQ5 Can the developed kernels provide practical bounds on diagram mass and bi-Lipschitz equivalence across covariance choices?

주요 결과

• A family of explicit Lipschitz bounds for RKHSs associated with translation-invariant Gaussian kernels on the Banach completion B is established (Theorem 4.4).
• Covering-number bounds for subsets of K(X,A) are derived via the feature metric and the Wasserstein geometry (Theorem 4.6).
• A kernel value bound kJ,Σ,t(g,0) yields an explicit upper bound on the diagrammatic mass M(g) (Theorem 4.5).
• A Rayleigh-quotient criterion characterizes when two covariance operators induce bi-Lipschitz equivalent seminorms and thus equivalent Gaussian kernels (Theorem 4.7).
• Random Fourier feature maps are developed with uniform concentration bounds and Lipschitz control, enabling finite-sample kernel approximations (Section 4.5; Lemmas/Corollaries 4.8–4.11).
• The analytic pipeline is demonstrated on virtual diagrams from lower-star clique filtrations with various label spaces, isolating label-space geometry effects on Lipschitz constants and robustness.