Induced Matchings and the Algebraic Stability of Persistence Barcodes

Abstract

We define a simple, explicit map sending a morphism $f:M\to N$ 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 $\mathrm{coker}f$. 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 $\delta$-interleaving morphism between two persistence modules induces a $\delta$-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.

Expanded journal version, to appear in Journal of Computational Geometry. Includes a proof that no definition of induced matching can be fully functorial (Proposition 5.10), and an extension of our single-morphism characterization of the interleaving relation to multidimensional persistence modules (Remark 6.7). Exposition is improved throughout. 11 Figures added

References in corpus (3)

• Confidence sets for persistence diagrams
• Multidimensional Interleavings and Applications to Topological Inference
• Persistence modules: Algebra and algorithms

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• Lifespan Functors and Natural Dualities in Persistent Homology
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• Persistent homology for functionals
• Persistent transcendental Bézout theorems