Morse-based Fibering of the Persistence Rank Invariant

TL;DR

This paper introduces a Morse-theoretic approach to compute and interpret the rank invariant in multi-parameter persistent homology, facilitating visualization and analysis for any number of parameters.

Contribution

It demonstrates how discrete Morse theory can determine the rank invariant from critical points, enabling simplified computation and interpretation across multiple parameters.

Findings

Rank invariant is fully determined by critical points in Morse theory.

Critical points partition parameter space into classes with equivalent rank invariants.

Persistence diagrams for all lines in a class can be derived from a single representative.

Abstract

Although there is no doubt that multi-parameter persistent homology is a useful tool to analyse multi-variate data, efficient ways to compute these modules are still lacking in the available topological data analysis toolboxes. Other issues such as interpretation and visualization of the output remain difficult to solve. Software visualizing multi-parameter persistence diagrams is currently only available for 2-dimensional persistence modules. One of the simplest invariants for a multi-parameter persistence module is its rank invariant, defined as the function that counts the number of linearly independent homology classes that live in the filtration through a given pair of values of the multi-parameter. We propose a step towards interpretation and visualization of the rank invariant for persistence modules for any given number of parameters. We show how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points whose coordinates are critical with respect to a discrete Morse gradient vector field. These critical points partition the set of all lines of positive slope in the parameter space into equivalence classes, such that the rank invariant along lines in the same class are also equivalent. We show that we can deduce all persistence diagrams of the restrictions to the lines in a given class from the persistence diagram of the restriction to a representative in that class.