# Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and its Applications

arXiv: https://arxiv.org/abs/2111.15058 · HTML: https://mlsanity.com/paper/2111.15058/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2021.

## Built on by (most cited first, 4 of 4)

- [GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning](https://mlsanity.com/paper/2304.04970/) (2023)
- [Persistent Topological Features in Large Language Models](https://mlsanity.com/paper/2410.11042/) (2024)
- [D-GRIL: End-to-End Topological Learning with 2-parameter Persistence](https://mlsanity.com/paper/2406.07100/) (2024)
- [Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data](https://mlsanity.com/paper/2502.16049/) (2025)

## Abstract (arXiv)

The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval I of a mathbbZ2-indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank over I by computing the barcode of the zigzag module obtained by restricting the bifiltration inducing M to that path. If the bifiltration and I have at most t simplices and points respectively, this computation takes O(tω) time where ω∈[2,2.373) is the exponent of matrix multiplication. Among others, we apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M, determine whether M is interval decomposable and, if so, compute all intervals supporting its summands.