Large deviation principle for persistence diagrams of random cubical filtrations

TL;DR

This paper establishes a large deviation principle for the persistence diagrams of random cubical filtrations, extending the understanding of their asymptotic behavior as the window size grows infinitely large.

Contribution

It introduces a large deviation principle for persistence diagrams of random cubical filtrations, with a novel method for lifting large deviation principles from Betti numbers to diagrams.

Findings

Proved a strong law of large numbers for persistence diagrams.

Established a large deviation principle with a Fenchel–Legendre rate function.

Developed a general method for lifting large deviation principles to persistence diagrams.

Abstract

The objective of this article is to investigate the asymptotic behavior of the persistence diagrams of a random cubical filtration as the window size tends to infinity. Here, a random cubical filtration is an increasing family of random cubical sets, which are the union of randomly generated higher-dimensional unit cubes with integer coordinates in a Euclidean space. We first prove the strong law of large numbers for the persistence diagrams, inspired by the work of Hiraoka, Shirai, and Trinh, where the persistence diagram of a filtration of random geometric complexes is considered. As opposed to prior papers treating limit theorems for persistence diagrams, the present article aims to further study the large deviation behavior of persistence diagrams. We prove a large deviation principle for the persistence diagrams of a class of random cubical filtrations, and show that the rate function is given as the Fenchel--Legendre transform of the limiting logarithmic moment generating function. In the proof, we also establish a general method of lifting a large deviation principle for the tuples of persistent Betti numbers to persistence diagrams for broad application.