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

arXiv: https://arxiv.org/abs/2107.06800 · HTML: https://mlsanity.com/paper/2107.06800/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2021.

## Built on by (most cited first, 6 of 6)

- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions](https://mlsanity.com/paper/2306.11170/) (2023)
- [EMP: Effective Multidimensional Persistence for Graph Representation Learning](https://mlsanity.com/paper/2401.13713/) (2024)
- [Stability for Inference with Persistent Homology Rank Functions](https://mlsanity.com/paper/2307.02904/) (2023)
- [COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules](https://mlsanity.com/paper/2609.22012/) (2026)

## Abstract (arXiv)

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.