Probabilistic Analysis of Multiparameter Persistence Decompositions

Abstract

Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical applications and their computational efficiency. Empirical observations indicate that modules that decompose into only intervals are rare. To support this observation, we show that for numerous common multiparameter constructions, such as density- or degree-Rips bifiltrations, and across a general category of point samples, the probability of the homology-induced persistence module decomposing into intervals goes to zero as the sample size goes to infinity.

Figures (11)

• Figure 1: Left: A point set $S$ of $5$ points. Right: A Poisson point sample with a subcube that contains an isolated scaled copy of $S$.
• Figure 2: Two non-thin indecomposable persistence modules over finite posets. Both posets are subposets of $\mathbb{R}_+^2$.
• Figure 3: The construction of \ref{['lem:existence_cech_non_interval']} for $\mathbb{R}^{2}$: two perturbed equilateral triangles glued along $\sigma$.
• Figure 4: The Čech and offset bifiltration of \ref{['lem:existence_cech_non_interval']} restricted to the subposet $P\subset\mathbb{R}_+^{2}$.
• Figure 5: The setup of the proof of \ref{['thm:process_estimation']}.

Theorems & Definitions (26)

• Theorem 2.1
• proof
• Corollary 2.2
• Lemma 3.1
• Lemma 3.2
• Theorem 4.1
• Lemma 4.2
• proof
• Remark 4.3
• proof : Proof of \ref{['thm:non_interval_cech_random']}