# Induced Matchings and the Algebraic Stability of Persistence Barcodes

arXiv: https://arxiv.org/abs/1311.3681 · HTML: https://mlsanity.com/paper/1311.3681/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Nov 2013.

## Built on by (most cited first, 7 of 7)

- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [Efficient Betti Matching Enables Topology-Aware 3D Segmentation via Persistent Homology](https://mlsanity.com/paper/2407.04683/) (2024)
- [Fast Topological Signal Identification and Persistent Cohomological Cycle Matching](https://mlsanity.com/paper/2209.15446/) (2022)
- [Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds](https://mlsanity.com/paper/2402.15058/) (2024)
- [Unreduced Persistence Diagrams for Topological Machine Learning](https://mlsanity.com/paper/2507.07156/) (2025)
- [Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability](https://mlsanity.com/paper/2603.14169/) (2026)
- [Persistent Magnitude Homology for Quantitative Equational Theories](https://mlsanity.com/paper/2608.21479/) (2026)

## Abstract (arXiv)

We define a simple, explicit map sending a morphism f:M rightarrow N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and mathopcoker f. As an immediate corollary, we obtain a new proof of the algebraic stability of persistence, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules, and yields a novel "single-morphism" characterization of the interleaving relation on persistence modules.