Random Walks on Virtual Persistence Diagrams

Abstract

Persistence diagrams, with the bottleneck and Wasserstein metrics, represent the interval decomposition of persistence modules constructed from filtered topological data. Finite persistence diagrams on a metric pair $(X,A)$ form the free translation-invariant commutative Lipschitz monoid $D(X,A)$. The Grothendieck group $K(X,A)$ is the free translation-invariant Abelian Lipschitz group of virtual persistence diagrams, and the canonical embedding $D(X,A) \hookrightarrow K(X,A)$ is isometric for the Wasserstein-1 distance with translation-invariant metric $ρ$. When the pointed metric space $(X/A,\overline d_1,[A])$ is uniformly discrete, $(K(X,A),ρ)$ is a discrete locally compact abelian group, but may be uncountable. We construct a symmetric, translation-invariant Markov semigroup $(P_t)_{t\ge0}$ on $\ell^2(K(X,A))$ as the projective limit of semigroups defined on the finitely generated subgroups $K(X_F,A)$, induced by the VPD metric $ρ$. Its convolution kernels $(p_t)_{t\ge0}$ define a random walk on $K(X,A)$, and for each $t\ge0$ the support of $p_t$ is contained in a countable subgroup $H\le K(X,A)$. On $H$, the semigroup $(p_t)_{t\ge0}$ has a Lévy-Khintchine representation $\widehat p_t(θ)=\exp(-tλ_H(θ))$ for $θ\in\widehat H$, and the kernels $k_t(x,y)=p_t(x-y)$ define reproducing kernel Hilbert spaces $\mathcal H_t$ with dense truncated subspaces. We show that a small collection of scalar random-walk invariants determined by the Lévy-Khintchine exponent -- including return probabilities, collision probabilities, and diagonal resolvent values -- controls global regularity properties of diagram functionals.

Figures (6)

• Figure 1: (a) A finite virtual persistence diagram on the Cayley graph of $S_3$ with its word metric, with basepoint $0\in K(X,A)$ and element $(123)\in S_3$ at distance $\rho(0,(123))=2$. (b) An infinite uniformly discrete virtual persistence diagram modeled on the Cayley graph of the free group $F_2=\langle a,b\rangle$, with element $ab\in F_2$ at distance $\rho(0,ab)=2$.
• Figure 2: The generators of the virtual persistence diagram group $K(X,A)$ induced by the Cayley graph of $S_3$ with its word metric.
• Figure 3: Weighted graphs used to generate the persistence diagrams.
• Figure 4: The persistence diagrams of the weighted graphs in Figure \ref{['fig:graphs']}.
• Figure 5: The virtual persistence diagram associated to the two persistence diagrams in Figure \ref{['fig:pds']}.

Theorems & Definitions (36)

• Lemma 1: fanning2025reproducingkernelhilbertspaces
• Lemma 2: fanning2025reproducingkernelhilbertspaces
• Corollary 3: Spectral form fanning2025reproducingkernelhilbertspaces
• Definition 1
• Lemma 4
• proof
• Theorem 5
• proof
• Lemma 6
• proof