Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions

TL;DR

This paper introduces the signed barcode, a new visual tool for understanding the structure of multi-parameter persistence modules, based on rank decompositions and minimal rank-exact resolutions, with theoretical foundations and experimental illustrations.

Contribution

It develops the theory of signed barcodes for multi-parameter persistence, establishing conditions for their existence and uniqueness, and links them to minimal rank-exact resolutions.

Findings

Signed barcodes decompose the rank invariant as a linear combination of indicator modules.

Existence and uniqueness conditions for signed barcodes are established.

Experimental results demonstrate the utility of signed barcodes in analyzing persistence modules.

Abstract

In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode decomposes the rank invariant as a $\Z$-linear combination of rank invariants of indicator modules supported on segments in the poset. We develop the theory behind these decompositions, both for the usual rank invariant and for its generalizations, showing under what conditions they exist and are unique. We also show that, like its unsigned counterpart, the signed barcode reflects in part the algebraic structure of the module: specifically, it derives from the terms in the minimal rank-exact resolution of the module, i.e., its minimal projective resolution relative to the class of short exact sequences on which the rank invariant is additive. To complete the picture, we show some experimental results that illustrate the contribution of the signed barcode in the exploration of multi-parameter persistence modules.