Stratifying multiparameter persistent homology

TL;DR

This paper introduces new algebraic invariants for multiparameter persistent homology in topological data analysis, extending the well-understood one-parameter theory to handle multiple parameters using multigraded algebra techniques.

Contribution

It proposes multigraded Hilbert series, associated primes, and local cohomology as novel invariants for analyzing multiparameter persistent homology, generalizing existing one-parameter invariants.

Findings

Provides a stratification of the support region for multiparameter modules.

Measures the size of components supported on different strata.

Extends the algebraic framework of persistent homology to multiple parameters.

Abstract

A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.