Convergence of persistence diagram in the sparse regime

2021/03/24 by Owada, Takashi · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2103.12943

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 r_n^-1\mathcal X_n = \ r_n^-1X₁,…, r_n^-1X_n \, such that r_n→ 0 as n→∞. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: nr_n^d→0, n→∞. In this setting, we show that the asymptotics of the kth persistence diagram depends on the limit value of the sequence n^k+2r_n^d(k+1). If n^k+2r_n^d(k+1) → ∞, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If r_n decays faster so that n^k+2r_n^d(k+1) → c∈ (0,∞), the persistence diagram weakly converges to a limiting point process without normalization. Finally, if n^k+2r_n^d(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 \mathcal M₀-topology.

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• Limit Theorems for Verbose Persistence Diagrams

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