Multidimensional Persistence and Noise

TL;DR

This paper develops a framework for analyzing multidimensional persistence modules using tame functors and noise systems, introducing a feature counting invariant that generalizes classical barcodes and is stable under noise.

Contribution

It introduces noise systems and a feature counting invariant for multidimensional persistence, extending classical barcode methods to higher dimensions with stability guarantees.

Findings

The feature counting invariant is 1-Lipschitz continuous.

Noise systems induce a pseudo-metric topology on tame functors.

The invariant generalizes classical barcodes for 1D persistence.

Abstract

In this paper we study multidimensional persistence modules [5,13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.