Random Walks on Virtual Persistence Diagrams

TL;DR

This paper constructs a translation-invariant heat semigroup on virtual persistence diagram groups, leading to new kernels and metrics that facilitate analysis of topological data through random walks.

Contribution

It introduces a novel heat semigroup on virtual persistence diagram groups with explicit Fourier exponents, enabling the development of new kernels and metrics for topological data analysis.

Findings

Constructed a translation-invariant heat semigroup on $K(X,A)$.

Derived explicit Fourier exponents for the kernels.

Established monotonicity properties via convex orders on mixing measures.

Abstract

In the uniformly discrete case of virtual persistence diagram groups $K(X,A)$, we construct a translation-invariant heat semigroup. The kernels are supported on a countable subgroup $H$, and the restriction to $H$ has Fourier exponent $\lambda_H$ satisfying $\lambda_H(\theta)=\sum_{\kappa\in H\setminus\{0\}}\bigl(1-\Re\theta(\kappa)\bigr)\nu(\kappa),$ for a symmetric $\nu\in\ell^1(H\setminus\{0\})$. This gives a symmetric jump process on $H$. The exponent $\lambda_H$ determines heat kernels, which define reproducing kernel Hilbert spaces and their associated semimetrics. Convex orders on the mixing measures give monotonicity for the kernels, Hilbert spaces, and semimetrics.