Categorification of Persistent Homology

Abstract

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are (R, ≤)-indexed diagrams in some target category. A set of such diagrams has an interleaving distance, which we show generalizes the previously-studied bottleneck distance. To illustrate the utility of this approach, we generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of ε-interleavings of (R, ≤)-indexed diagrams in some target category, and show that if the target category is abelian, so is this category of interleavings.2010 Mathematics Subject Classification. 55N99, 68W30, 18A25, 18E10, 54E35. Key words and phrases. Applied topology, persistent topology, topological persistence, diagrams indexed by the poset of real numbers, interleaving distance.The first author gratefully acknowledges support from AFOSR grant # FA9550-13-1-0115.1 Two foundational papers in this subject are [ELZ02] and [ZC05]. In the first, Edelsbrunner, Letscher and Zomorodian define persistent homology for (Z + , ≤)-indexed diagrams of finite dimensional vector spaces, that are obtained from filtered finite simplicial complexes by taking simplicial homology with coefficients in a field. In the second, Zomorodian and Carlsson take a purely algebraic point of view. They define persistent homology for tame (Z + , ≤)-indexed diagrams of finite-dimensional vector spaces, and prove a bijection between isomorphism classes of such tame diagrams and finite barcodes whose endpoints lie in Z + ∪ {∞}. (Z + , ≤)-indexed diagrams of finite dimensional vector spaces are called persistence modules.These papers are rounded out by [CSEH07], where Cohen-Steiner, Edelsbrunner and Harer prove that persistent homology is useful in applications by showing that it is stable in the following sense. Let f, g : X → R be continuous functions on a triangulable space. Define an (R, ≤)-indexed diagram of topological spaces, F , by setting F (a) = f −1 (−∞, a] and letting F (a ≤ b) be given by inclusion. Define G similarly using g. Let H be the singular homology functor with coefficients in a field. Assume that HF and HG are diagrams of finite dimensional vector spaces and that they are tame. Then the bottleneck distance between HF and HG is bounded by the supremum norm between f and g.This stability result is significantly strengthened by Chazal, Cohen-Steiner, Glisse, Guibas and Oudot in [CCSG + 09]. They drop the assumptions that X be triangulable, that f, g be continuous, and that HF and HG be tame. Their approach is crucial to this paper. They explicitly work with (R, ≤)-indexed diagrams, though they consider them from an algebraic, not categorical, point of view. They define the interleaving distance, d, between such diagrams, and define the bottleneck distance, d B , between such diagrams using limits of discretizations, and show that d B ≤ d.The basic idea of persistent homology has been extended in numerous ways. Here we focus o...

Higher Interpolation and Extension for Persistence Modules

“…1 Similarly, let Cat denote the category of small categories and functors. Theorem 2.3 (see [6]). For each category C and functor H :…”

“…3 Moreover, these diagrams admit a bottleneck distance d Bot and the following isometry theorem establishes that the assignment dgm which sends a (tame) persistence module to its corresponding diagram preserves distances. Theorem 2.6 (see [6,10,19]). The equality…”

“…Introduction. The combination of rigorously developed foundations [6,10], efficient computability [21,25], and stability properties [8,11] has resulted in the widespread adoption of topological persistence [15,25] as a technique for the analysis of large and complex datasets [7,16,22]. The output of this process is a collection of persistent homology groups, which are typically represented via a barcode or a persistence diagram.…”

“…For a small category C, we will denote the category of functors from C to D by D C . Although we will survey some relevant definitions and results here, the reader is invited to consult [5,6,8,10,23] for detailed background material on the categorical and metric aspects of persistence modules.…”

“…We therefore follow [6] and work with C R for an arbitrary category C, keeping in mind that this functor category specializes to Mod whenever C = Vect. We call C R the category of persistence modules with values in C.…”