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Categorification of persistent homology

Peter Bubenik, Jonathan A. Scott

arXiv:1205.3669v3math.ATcs.CGmath.CT

TL;DR

The paper redevelops persistent homology categorically by studying real-indexed diagrams and their interleaving distance. It generalizes persistence results, including stability, and establishes structural properties of interleavings in abelian target categories.

  • Problem

    Persistent homology is redeveloped from an abstract categorical point of view to clarify its ideas and enable broader generalization.

  • Method

    The paper studies diagrams indexed by the real numbers in target categories, defining persistent homology through images and organizing diagrams with interleavings and their distance.

  • Results

    The interleaving distance is an extended pseudometric, finite type diagrams are exactly the tame diagrams, and finite barcodes embed isometrically under bottleneck and interleaving distances.

  • Takeaways & Limitations

    The categorical framework generalizes stability results for persistence, extended persistence, and kernel, image, and cokernel persistence, while yielding an abelian category of interleavings for abelian targets.

  • Takeaways & Limitations

    The Stability Theorem applies to persistence modules rather than persistence diagrams, so hard stability for diagrams requires additional detailed analysis.

Abstract

from arXiv · show

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The 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 greatly generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of interleavings of these diagrams, and show that if the target category is abelian, so is this category of interleavings.

1. Introduction

The paper redevelops persistent homology categorically, treating real-indexed diagrams as primary objects and using interleavings to unify and generalize persistence stability results. It also constructs an abelian category of interleavings and extends the framework to extended, kernel, image, and cokernel persistence.

  • 1.3. Our contributions.: Persistent homology is redeveloped categorically by taking diagrams indexed by (R, ≤) in a fixed target category as the main objects.These diagrams may take values in categories such as topological spaces or finite-dimensional vector spaces.
  • 1.3. Our contributions.: The interleaving distance is defined for real-indexed diagrams and shown to generalize the previously established bottleneck-distance framework.For finite barcodes, the bottleneck metric embeds isometrically into the interleaving metric space of diagrams.
  • 1.3. Our contributions.: For any topological space X, functions f and g, and functor H, the interleaving distance between HF and HG is bounded above by the supremum norm between f and g.This removes several assumptions from earlier stability results, including triangulability, continuity, and tameness in the stated setting.
  • 1.3. Our contributions.: The same supremum-norm stability bound is proved for generalized extended persistence diagrams of pairs of spaces, including not-necessarily-continuous maps into (−∞, M].The construction uses any functor on pairs of spaces.
  • 1.3. Our contributions.: A category of interleavings is defined, and it is abelian whenever the base category is abelian, yielding direct sums, kernels, images, and cokernels.This framework also generalizes prior stability results for kernel, image, and cokernel persistence and includes an extended-persistence version.
  • 1.4. Comparison with other recent work.: The categorical stability theorem is formal and broadly applicable, but it concerns persistence modules rather than their corresponding persistence diagrams.Hard stability for persistence diagrams requires additional detailed analysis, such as an isometry theorem.

2. Background

The background introduces the category-theoretic language used throughout the paper and explains how standard persistent-homology constructions fit within the categorical approach.

  • 2. Background: The paper begins by presenting the basic category-theory definitions used throughout its development.Later sections use abelian-category and algebraic definitions for specialized arguments.
  • 2. Background: A category consists of objects together with sets of morphisms between every pair of objects, equipped with composition and identity structure.The paper writes f : X → Y when f is a morphism from X to Y.

2.1. Categorical terminology.

This section introduces categories, functors, diagrams, natural transformations, and the categorical structures used to formulate persistent homology.

  • Categories: A category consists of objects, morphisms, associative composition, and identity morphisms.Composition and identities satisfy the usual compatibility laws.
  • Examples: Top and Vec are categories whose morphisms are continuous maps and linear transformations, respectively.Vec contains finite-dimensional vector spaces over a fixed field.
  • Indexing categories: A poset is identified with a category having one morphism exactly when its ordering relation holds.The real numbers, integers, nonnegative integers, and [n] provide indexing posets.
  • Functors: A functor maps objects and morphisms between categories while preserving composition and identities.Singular homology defines functors from Top to graded vector spaces and, degreewise, to Vec.
  • Natural transformations: Natural transformations assign compatible morphisms between functors, and objectwise isomorphisms form natural isomorphisms.Translations on (R, ≤) provide a natural transformation from the identity to translation by ε.
  • Diagrams: A diagram is a functor indexed by a category, and diagrams with natural transformations form a category.For (R, ≤), a diagram assigns objects F(a) and morphisms F(a) → F(b) whenever a ≤b.

2.2. Categorical persistent homology.

Persistent homology is represented categorically by diagrams of topological spaces or vector spaces indexed by ordered parameter values. Persistent groups are images of homology maps across a specified parameter interval.

  • Filtered complexes: A filtered simplicial complex produces an [n]-indexed diagram of topological spaces through inclusions.Each index carries a filtration subcomplex, and each order relation induces an inclusion.
  • Filtered complexes: Applying degree-k homology to the filtration yields an [n]-indexed diagram of finite-dimensional vector spaces.Maps are induced by the inclusions between filtration stages.
  • Filtered complexes: Summing homology over all degrees gives a Vec[n]-valued diagram.The value at each index is the direct sum of the homology groups of the corresponding filtration subcomplex.
  • Sublevel-set filtrations: A real-valued function defines sublevel sets that form an (R, ≤)-indexed diagram of topological spaces.For a ≤b, the sublevel-set inclusion is a continuous map.
  • Sublevel-set filtrations: Applying singular homology to a sublevel-set diagram produces an indexed diagram of vector spaces, finite-dimensional under the stated homological finiteness condition.The all-degree construction similarly lies in Vec(R,≤) when the total homology is finite-dimensional at every parameter.
  • Indexing variants: Diagrams indexed by [n], (Z+, ≤), and (Z, ≤) can be treated as real-indexed diagrams using retraction functors.For [n], the retraction rounds interior real values down and truncates outside the endpoints.
  • Persistent homology: The p-persistent kth homology group at a is the image of the map from HkF(a) to HkF(a + p).Diagrams in Vec[n], Vec(Z+,≤), and Vec(R,≤) are called persistence modules.

2.3. Abelian categories.

This section develops categorical notions of zero objects, kernels, cokernels, products, coproducts, pull-backs, and push-outs, culminating in abelian categories and their vector-space example.

  • Basic categorical structures: Initial and terminal objects admit unique maps in opposite directions, while an object that is both is a zero object.A zero object induces a zero morphism between every pair of objects.
  • Kernels and cokernels: Kernels are universal morphisms into a source object whose composite with the given morphism is zero.The kernel morphism is a monomorphism and represents a subobject.
  • Kernels and cokernels: Cokernels are universal morphisms out of a target object that annihilate the given morphism.The cokernel morphism is an epimorphism and represents a quotient object.
  • Limits and colimits: Products and coproducts are characterized by universal properties for jointly mapping into or out of two objects.They are unique up to canonical isomorphism when they exist.
  • Limits and colimits: Pull-backs and push-outs are universal constructions completing compatible diagrams through a unique mediating morphism.Both constructions are unique up to canonical isomorphism.
  • Abelian categories: An abelian category has zero objects, products, coproducts, kernels, and cokernels, with monomorphisms as kernels and epimorphisms as cokernels.Such categories are preadditive, so morphism sets are abelian groups and composition is bilinear.
  • Example: vector spaces: In finite-dimensional vector spaces, kernels, images, and cokernels have their standard linear-algebraic meanings.Monomorphisms are injective maps, epimorphisms are surjective maps, and coker f = W/f(V).

2.4. Algebra.

The algebraic background introduces graded rings and graded F[t]-modules, then states the structure theorem for finitely generated modules over a principal ideal domain.

  • Graded algebra: A non-negatively graded ring decomposes as R = ⊕∞_n=0 R_n with multiplication preserving degree.The polynomial ring F[t] is graded by degree.
  • Graded algebra: A graded F[t]-module decomposes into graded pieces with multiplication by t increasing degree appropriately.Finite type means that every graded component is finite-dimensional over F.
  • Structure theorem: Every finitely generated module over a principal ideal domain is a direct sum of a finite-rank free submodule and finitely many cyclic torsion modules.The free rank and torsion ideals are uniquely determined up to ordering.

3. Interleavings of diagrams

The section defines categorical ε-interleavings for real-indexed diagrams and uses them to construct an extended pseudometric. Interleaving distance is monotone under functors and becomes an extended metric after identifying distance-zero diagrams.

  • Definition: For real-indexed diagrams, Tb shifts an index a to a + b, with ηb providing the natural transformation from the identity functor to Tb.These shifts satisfy TbTc = Tb+c and ηbηc = ηb+c.
  • Definition: An ε-interleaving consists of natural transformations ϕ: F ⇒ GTε and ψ: G ⇒ FTε satisfying two compatibility identities.The identities require (ψTε)ϕ = Fη2ε and (ϕTε)ψ = Gη2ε.
  • Distance: The interleaving distance d(F, G) is the infimum of ε values for which F and G are ε-interleaved, and is ∞ when no such ε exists.If F and G are ε-interleaved, they are also ε′-interleaved for every ε′ ≥ ε, so the admissible values form a ray.
  • Metric properties: The distance is an extended pseudometric: it satisfies the metric axioms except that it may be ∞ and distance zero need not imply isomorphism.The triangle inequality follows by composing interleavings, yielding d(F, H) ≤ d(F, G) + d(G, H).
  • Metric properties: After identifying diagrams at interleaving distance zero, d becomes an extended metric on the resulting equivalence classes.The relation F ~ G defined by d(F, G) = 0 is an equivalence relation.
  • Functoriality: Applying a functor to ε-interleaved diagrams preserves ε-interleaving and cannot increase their interleaving distance.For H: D → E, the induced diagrams satisfy d(HF, HG) ≤ d(F, G).

4. Diagrams of vector spaces

The paper treats persistent homology as diagrams in Vec(R,≤), characterizes tame diagrams through finite decompositions, and identifies their barcode structure. This categorical framework also recovers the bottleneck metric via interleavings.

  • Persistent homology calculations are modeled as (R,≤)-indexed diagrams of finite-dimensional vector spaces over a fixed field.
  • Finite type diagrams are direct sums of interval diagrams χI, whose interval endpoints determine their critical values.
  • Interval diagrams χI are indecomposable, providing the basic building blocks for finite type diagrams.
  • A diagram in Vec(R,≤) is tame if and only if it has finite type.
  • Finite type diagrams have essentially unique direct-sum decompositions, yielding a Krull–Schmidt theorem and a bijection with finite barcodes.
  • The interval-diagram mapping gives an isometric embedding of finite barcodes with bottleneck distance into diagrams with interleaving distance.

5. Stability

The stability theorem compares diagrams induced by arbitrary real-valued functions through the interleaving distance. Their distance is bounded by the supremum norm of the difference between the functions.

  • The framework generalizes earlier stability results for persistent homology by allowing arbitrary topological spaces and functions that need not be continuous.
  • Given functions f,g:X→R, sublevel-set diagrams F and G are constructed using inclusions between their sublevel sets.
  • Applying any functor H, including singular homology or rational homotopy groups, preserves the interleaving bound for HF and HG.
  • If ε=∥f−g∥∞, then F(a)⊆G(a+ε) and G(a)⊆F(a+ε), so F and G are ε-interleaved.

6. Extended persistence

The paper generalizes extended persistence by encoding upward and downward filtrations as real-indexed diagrams of pairs, then proves stability under arbitrary functors on pairs.

  • Construction: Extended persistence is generalized from finite simplicial filtrations to real-indexed diagrams of pairs for maps f:X→R bounded above by M.The construction allows f to be noncontinuous and uses an arbitrary spacing parameter s>0.
  • Construction: For c<M+s, the diagram uses (f^-1(-∞,c],∅), while for c≥M+s it uses (X,f^-1[2M+s−c,∞)).The transition includes F(c)=(X,∅) for M≤c<M+s and F(M+s)=(X,f^-1(M)).
  • Construction: The structure maps F(c≤d) are inclusions across the upward, downward, and transition regimes.These inclusions establish the real-indexed diagram of pairs.
  • Construction: If f is also bounded below by m, the construction begins at (∅,∅) below m and eventually reaches (X,X).The terminal regime starts at c≥2M+s−m.
  • Stability: For maps f,g:X→(−∞,M] with ε=∥f−g∥∞, the induced extended persistence diagrams are ε-interleaved.The proof establishes ε-interleavings of the pair diagrams and transfers them through any functor H:Pair→D.

7. Abelian structure of interleavings

The paper constructs a category of ε-interleavings and proves that it preserves abelian structure, yielding direct sums and stability results for kernels, images, and cokernels.

  • Category of interleavings: The category Intε(D) consists of ε-interleavings of real-indexed diagrams and is functorial in both ε and the base category D.Functoriality in ε enlarges interleavings coherently, while functoriality in D follows by composition with a functor.
  • Abelian structure: The interleaving category Intε(A) is preadditive, with morphism pairs inheriting abelian-group structure from A(R,≤).Composition remains bilinear because it is restricted from the additive diagram category.
  • Abelian structure: Every monomorphism in Intε(A) is a kernel, and every epimorphism is a cokernel.Together with the existence of zero objects, finite products and coproducts, kernels, and cokernels, this proves abelianness.
  • Abelian structure: When A is abelian, Intε(A) has a zero object, finite products and coproducts, kernels, and cokernels.The zero object is the trivial interleaving of the constant zero diagram; pull-backs and push-outs are formed componentwise in A(R,≤).
  • Consequences: ε-interleavings are preserved under direct sums and under taking kernels, images, and cokernels of morphisms.These closure properties support corresponding persistence constructions inside the abelian interleaving category.
  • Consequences: The framework yields stability bounds for kernels, images, and cokernels in extended persistence, including d(ker Hα,ker Hβ), d(im Hα,im Hβ), d(coker Hα,coker Hβ)≤∥f−f′∥∞.The theorem applies to maps induced by a continuous h:Y→X and functions that need not be continuous or tame.

8. Future work

The paper leaves extensions beyond real-line indexing and a categorical bottleneck-distance theory for future work.

  • Scope: The framework is developed for diagrams indexed by (R,≤), while multidimensional, circle-valued, and zig-zag persistence require more general indexing categories.The authors state that this generalization will be presented in [BdSS13].
  • Open problems: A categorical definition of bottleneck distance for arbitrary (R,≤)-indexed vector-space diagrams, together with an Isometry Theorem, remains desirable.
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