Exterior Critical Series of Persistence Modules

TL;DR

This paper introduces the exterior critical series, a new discrete invariant for persistence modules that extends the concept of the barcode to multi-dimensional cases and captures features beyond the rank invariant.

Contribution

The paper proposes the exterior critical series as a novel invariant that is complete for 1D modules and applicable to multi-dimensional modules, detecting additional features.

Findings

Complete invariant for 1D persistence modules

Applicable to multi-dimensional modules

Detects features not captured by the rank invariant

Abstract

The persistence barcode is a well-established complete discrete invariant for finitely generated persistence modules [5] [1]. Its definition, however, does not extend to multi- dimensional persistence modules. In this paper, we introduce a new discrete invariant: the exterior critical series. This invariant is complete in the one-dimensional case and can be defined for multi-dimensional persistence modules, like the rank invariant [2]. However, the exterior critical series can detect some features that are not captured by the rank invariant.