On Interval Decomposability of 2D Persistence Modules

TL;DR

This paper investigates the structure of 2D persistence modules, introducing new algebraic concepts and providing algorithms to determine interval decomposability efficiently, which advances understanding in multidimensional persistent homology.

Contribution

It defines pre-interval representations, establishes their equivalence with interval modules over certain grids, and develops algorithms with heuristics for decomposability testing.

Findings

Equivalence of pre-interval, interval, and thin representations over equioriented 2D grids.

A new criterion for decomposability without explicit decomposition.

An algorithm with complexity analysis and heuristics for 2D persistence modules.

Abstract

In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a wild problem. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more natural algebraic definition, and study the relationships between pre-interval, interval, and indecomposable thin representations. We show that over the ``equioriented'' commutative $2$D grid, these concepts are equivalent. Moreover, we provide a criterion for determining whether or not an $n$D persistence module is interval/pre-interval/thin-decomposable without having to explicitly compute decompositions. For $2$D persistence modules, we provide an algorithm together with a worst-case complexity analysis that uses the total number of intervals in an equioriented commutative $2$D grid. We also propose several heuristics to speed up the computation.