Reproducing Kernel Hilbert Spaces on Banach Completions of Virtual Persistence Diagram Groups
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| Natura: | Preprint |
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2026
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| _version_ | 1866908844322455552 |
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| author | Fanning, Charles Aktas, Mehmet |
| author_facet | Fanning, Charles Aktas, Mehmet |
| contents | 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 $p \in \left[1, + \infty \right]$. 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 $Q\colon B\to B^\ast$ determines a translation-invariant Gaussian kernel $k(x,y)=\exp\!\left(-\tfrac12\,\langle Q(x-y),x-y\rangle_{B,B^\ast}\right).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15153 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Reproducing Kernel Hilbert Spaces on Banach Completions of Virtual Persistence Diagram Groups Fanning, Charles Aktas, Mehmet Functional Analysis Algebraic Topology 46E22, 22A05, 55N31 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 $p \in \left[1, + \infty \right]$. 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 $Q\colon B\to B^\ast$ determines a translation-invariant Gaussian kernel $k(x,y)=\exp\!\left(-\tfrac12\,\langle Q(x-y),x-y\rangle_{B,B^\ast}\right).$ |
| title | Reproducing Kernel Hilbert Spaces on Banach Completions of Virtual Persistence Diagram Groups |
| topic | Functional Analysis Algebraic Topology 46E22, 22A05, 55N31 |
| url | https://arxiv.org/abs/2602.15153 |