# Rectangular Approximation and Stability of 2-parameter Persistence Modules

arXiv: https://arxiv.org/abs/2108.07429 · HTML: https://mlsanity.com/paper/2108.07429/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Aug 2021.

## Built on by (most cited first, 1 of 1)

- [TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration](https://mlsanity.com/paper/2510.05102/) (2025)

## Abstract (arXiv)

One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of 1-parameter persistence modules are stable in the sense that the bottleneck distance between two diagrams equals the interleaving distance between their generating modules. However, in multi-parameter setting this property breaks down in general. A simple special case of persistence modules called rectangle decomposable modules is known to admit a weaker stability property. Using this fact, we derive a stability-like property for 2-parameter persistence modules. For this, first we consider interval decomposable modules and their optimal approximations with rectangle decomposable modules with respect to the bottleneck distance. We provide a polynomial time algorithm to exactly compute this optimal approximation which, together with the polynomial-time computable bottleneck distance among interval decomposable modules, provides a lower bound on the interleaving distance. Next, we leverage this result to derive a polynomial-time computable distance for general multi-parameter persistence modules which enjoys similar stability-like property. This distance can be viewed as a generalization of the matching distance defined in the literature.