Reproducing Kernel Hilbert Spaces on Banach Completions of Virtual Persistence Diagram Groups

TL;DR

This paper develops a mathematical framework for reproducing kernel Hilbert spaces on virtual persistence diagram groups, extending stability and kernel construction methods to non-locally compact cases in topological data analysis.

Contribution

It introduces a translation-invariant kernel theory for virtual persistence diagram groups beyond locally compact cases, using Banach space linearization and positive operators.

Findings

Characterizes when the group is locally compact and discrete.

Develops translation-invariant kernels via positive-definite functions.

Embeds the group into a Banach space for kernel construction.

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 $p \in \left[1, + \infty \right]$. 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 $\rho.$ 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 $\rho$, the group $(K(X,A),\rho)$ 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 $k(x,y)=\exp\!\left(-\tfrac12\,\langle Q(x-y),x-y\rangle_{B,B^\ast}\right).$