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

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 $k(x,y)=\exp!\left(-\tfrac12,\langle Q(x-y),x-y\rangle_{B,B^\ast}\right).$

💡 Research Summary

This paper addresses the fundamental problem of constructing translation‑invariant reproducing kernel Hilbert spaces (RKHS) for virtual persistence diagram groups, denoted K(X,A), in the setting where the group is not locally compact. Persistent homology assigns to a filtered simplicial complex a persistence diagram, a finite multiset of birth–death points in ℝ²₍≤₎. The authors build on the work of Bubenik and Elchesen, who introduced the free commutative monoid of finite diagrams D(X,A) and its Grothendieck group K(X,A). They showed that the 1‑Wasserstein distance W₁ extends to a translation‑invariant metric ρ on K(X,A), while for p>1 translation invariance fails in general.

The first major contribution is a rigorous analysis of the topological nature of (K(X,A),ρ). By invoking a rigidity theorem for locally compact abelian groups, the authors prove that (K(X,A),ρ) is locally compact if and only if it is discrete, which in turn holds precisely when the pointed metric space (X/A, d₁,