Probabilistic Analysis of Multiparameter Persistence Decompositions
CoRR(2024)
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.
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