Convergence of persistence diagram in the sparse regime

Abstract

The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Čech filtration over a scaled random sample rn−1Xn={rn−1X1,…,rn−1Xn}, such that rn→0 as n→∞. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: nrnd→0, n→∞. In this setting, we show that the asymptotics of the kth persistence diagram depends on the limit value of the sequence nk+2rnd(k+1). If nk+2rnd(k+1)→∞, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If rn decays faster so that nk+2rnd(k+1)→c∈(0,∞), the persistence diagram weakly converges to a limiting point process without normalization. Finally, if nk+2rnd(k+1)→0, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the M0-topology.