Reproducing Kernel Hilbert Spaces for Virtual Persistence Diagrams

A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Persistence diagrams do not carry intrinsic spectral or kernel structures, so applications typically use auxiliary vectorizations of diagrams. Virtual persistence diagrams, given by the Grothendieck completion of finite diagrams with the $W_1$ metric, yield a group structure with additive cancellation and a translation-invariant metric. For a finite metric pair $(X,d,A)$ we use the identification $K(X,A)\cong \mathbb Z^{|X\setminus A|}$ and parametrize its Pontryagin dual torus. The Lipschitz seminorms of characters in the $W_1$ geometry are expressed in terms of edgewise phase differences on the quotient $X/A$. A weighted graph Laplacian on $X/A$ determines a Dirichlet symbol $\lambda(\theta)$, and the corresponding heat spectral multipliers induce translation-invariant kernels and their reproducing-kernel Hilbert spaces. We obtain explicit global $W_1$-Lipschitz bounds for all functions in these spaces. Random Fourier feature maps are constructed by sampling from the heat measures; they are unbiased kernel approximations and satisfy asymptotic Lipschitz estimates based on the same spectral quantities. We apply these kernels and their finite-dimensional approximations in a synthetic segmentation experiment that compares baseline, Wasserstein, and Reproducing Kernel Hilbert Space (RKHS)-based losses.