PaperlayerView exact revision on arXiv

Source-linked AI summary

Generalized Persistence Diagrams

Amit Patel

arXiv:1601.03107v5math.AT

TL;DR

Persistence diagrams are well established for vector-space modules, but integer coefficients can yield indecomposables that do not resemble intervals. This paper defines type A and type B diagrams for modules valued in broader categories and proves stability results, while leaving bottleneck stability for future investigation.

  • Problem

    With integer coefficients, indecomposable persistence modules need not resemble intervals and can be hard to interpret, especially under perturbations.

    Source: §1 · p. 2

  • Method

    The paper defines type A diagrams for constructible modules in symmetric monoidal categories with images and type B diagrams when the category is also abelian.

    Source: §1 · p. 2

  • Results

    Type B diagram erosion distance is at most the modules’ interleaving distance, while type A has a weaker one-way semicontinuity result; both are stable to sufficiently small perturbations.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Type B offers a stronger stability statement than type A, although it may contain less information.

    Source: §1 · p. 2

    Whether bottleneck stability extends to type B diagrams is left for future investigation.

    Source: §1 · p. 3

Abstract

from arXiv · show

We generalize the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the setting of constructible persistence modules valued in a symmetric monoidal category. We call this the type A persistence diagram of a persistence module. If the category is also abelian, then we define a second type B persistence diagram. In addition, we show that both diagrams are stable to all sufficiently small perturbations of the module.

1 Introduction

Classical persistence diagrams and barcodes describe vector-space modules through interval data, but integer coefficients can produce indecomposables that do not resemble intervals. This paper defines type A and type B diagrams for broader categories and proves stability results, while leaving bottleneck stability for future work.

1 Introduction

  • For vector-space modules, the barcode lists interval indecomposables, and plotting their endpoints gives a persistence diagram equivalent to the barcode.The diagram can also be defined through ranks of maps between persistence-module values.Source: §1 · pp. 1–2
  • With integer coefficients, indecomposable modules need not resemble interval modules, making them hard to interpret, especially under perturbations.This contrasts with vector-space modules, whose indecomposables are interval modules.Source: §1 · pp. 1–2
  • Type A diagrams generalize the Cohen-Steiner, Edelsbrunner, and Harer diagram to constructible modules in symmetric monoidal categories with images; abelian categories also admit type B diagrams.Type B may contain less information than type A, but has a stronger stability statement; depending on the category, either diagram may fail to be a complete invariant.Source: §1 · p. 2
  • Interleaved modules have stable diagrams: type B has erosion distance at most the interleaving distance, while type A satisfies a weaker one-way semicontinuity statement.The paper calls these continuity of type B and semicontinuity of type A; both hold for sufficiently small perturbations.Source: §1 · p. 2
  • The authors leave open whether bottleneck stability extends to type B persistence diagrams.They note that a one-parameter family of interleaved modules may be constructible, but do not pursue this stability question.Source: §1 · pp. 2–3

2 Persistence Modules

A persistence module is a functor from the ordered real numbers into a category, with constructibility restricting its behavior to finitely many changes. The paper uses several symmetric monoidal categories to illustrate how the choice of category changes the module structure.

2 Persistence Modules

  • A persistence module is a functor from the real numbers ordered by ≤ into a category C.Constructible persistence modules are the objects considered in the paper.Source: §2 · pp. 3, 5
  • A module is S-constructible when its behavior follows the specified finite set S of real values, and it is constructible if some finite S suffices.A module constructible with respect to S and T is also constructible with respect to their union.Source: §2 · pp. 3–4
  • The paper considers categories including finite sets, vector spaces, finitely generated abelian groups, finite abelian groups, and representations of the natural numbers.Finite sets under disjoint union yield merge trees; representations of the natural numbers correspond to vector-space endomorphisms.Source: §2 · pp. 4–5
  • The category determines which decompositions and exact-sequence behaviors are available: vector-space short exact sequences split, but those in finitely generated abelian groups need not.For representations of the natural numbers, indecomposables are Jordan blocks, and short exact sequences need not split.Source: §2 · pp. 4–5

3 Interleaving Distance

Interleaving distance measures how much two persistence modules must be shifted to admit compatible maps between them. The section defines this distance and connects it to perturbations of functions and their induced modules.

3 Interleaving Distance

  • An ε-shift sends each parameter r to r + ε, and an ε-interleaving consists of maps in both directions between the shifted modules.The maps must satisfy the interleaving compatibility condition.Source: §3 · p. 6
  • The interleaving distance is the minimum ε ≥ 0 for which two modules are ε-interleaved, and it is infinite if they are not interleaved.For constructible modules, a minimum interleaving exists whenever an interleaving exists.Source: §3 · p. 6
  • If ε exceeds the maximum difference between two Morse functions, their sublevel-set filtrations induce ε-interleaved persistence modules.The inclusions between the sublevel sets give the interleaving by functoriality of homology.Source: §3 · p. 6
  • Interleavings appeared in earlier work and have since been abstracted to other settings.The passage cites for the idea and for naming interleavings.Source: §3 · p. 6

4 Persistence Diagrams

The section frames persistence diagrams as finite maps on a poset of intervals, extending the classical diagram through Möbius inversion. Constructibility ensures the resulting diagram has finite support on intervals determined by a finite set of parameter values.

4 Persistence Diagrams

  • Constructibility of a map means its value is unchanged under interval containment when the intervals intersect S identically.The persistence-diagram construction is presented as an instance of Möbius inversion for constructible maps on a poset.Source: §4 · pp. 7–8
  • A persistence diagram is a finite map from the interval poset Dgm to an abelian group G.Dgm consists of half-open and half-infinite intervals, ordered by containment.Source: §4 · pp. 6–7
  • For an S-constructible map, Möbius inversion yields an S-finite map satisfying the inversion formula.An S-finite map can be nonzero only on intervals [s_i, s_j) or [s_i, ∞) for values in S.Source: §4 · p. 7
  • The Cohen-Steiner, Edelsbrunner, and Harer persistence diagram is the Möbius inversion of the rank function.The generalized construction uses the same inversion framework for constructible maps into an abelian group.Source: §4 · p. 8

5 Erosion Distance

The erosion distance compares persistence diagrams by allowing each diagram to be eroded toward the diagonal. It is the least erosion scale permitting maps in both directions, and erosion also commutes with the Möbius inversion formula.

5 Erosion Distance

  • ε-erosion shifts diagram points toward the diagonal, with points close to the diagonal disappearing into it.The figure depicts each point moving to the lower-right corner of a square of side length 2ε centered at that point.Source: Figure 1 · p. 9, §5 · p. 9
  • An ε-erosion between diagrams consists of maps from each diagram’s ε-erosion to the other diagram.The erosion distance is the minimum ε for which such maps exist, or infinity if none exists.Source: §5 · p. 9
  • Growε commutes with the Möbius inversion formula for the constructible map and finite map described in Proposition 5.1.This proposition connects the growth operation used in defining erosion with Möbius inversion.Source: §5 · p. 9

6 Grothendieck Groups

The section develops Grothendieck groups for symmetric monoidal and abelian categories. These constructions encode isomorphism classes and, in the abelian case, identify classes related by short exact sequences.

6 Grothendieck Groups

  • The section considers Grothendieck groups for essentially small symmetric monoidal categories and, when the category is abelian, for abelian categories.Every abelian category is symmetric monoidal under direct sum.Source: §6 · p. 10

6.1 Symmetric Monoidal Category

  • For an essentially small symmetric monoidal category, the Grothendieck group A(C) is the group completion of the monoid of isomorphism classes.For additive Krull-Schmidt categories, A(C) is freely generated by indecomposable isomorphism classes and has a translation-invariant partial order.Source: §6.1 · p. 10

6.2 Abelian Category

  • For an abelian category, B(C) quotients A(C) by relations from short exact sequences, and the quotient can discard torsion information.For finitely generated abelian groups, the quotient map forgets the torsion part.Source: §6.2 · pp. 11–12

7 Diagram of a Module

The paper assigns type A diagrams to persistence modules valued in symmetric monoidal categories with images, and type B diagrams when the category is also abelian. The diagrams are constructed through Möbius inversion and have examples across several categories.

7 Diagram of a Module

  • The construction assigns a type A persistence diagram to modules in symmetric monoidal categories with images and a type B diagram when the category is abelian.Both diagrams are S-finite when the module is S-constructible.Source: §7 · pp. 12–13
  • The construction forms dFI, maps it into A(C), and, for abelian C, composes with the quotient to B(C) before taking Möbius inversions.The type A and type B diagrams are defined as the respective Möbius inversions.Source: §7 · p. 13
  • The FinSet example has a type A diagram but no type B diagram because FinSet is not abelian.Figure 2 presents the persistence module and its type A diagram.Source: §7 · p. 14, Figure 2 · p. 15
  • For vector-space modules, the type A and type B diagrams coincide because the quotient map A(Vec) → B(Vec) is an isomorphism.The paper also gives an example of a finitely generated abelian-group module whose type B diagram is zero because the quotient forgets torsion.Source: §7 · p. 14
  • The paper also presents an example of a finitely generated abelian-group module with its type A diagram and a type B diagram that is zero.The zero type B diagram follows in this example because the quotient map forgets torsion.Source: §7 · p. 14
  • The FinAb example presents both type A and type B persistence diagrams.The passages identify the example but do not state a specific comparison between those diagrams.Source: §7 · p. 15

8 Stability

Section 8 relates interleaving distance between persistence modules to erosion distance between their diagrams and discusses stability under perturbations. Its arguments include a concrete-category assumption for diagram chasing and a bound on the perturbation size.

8 Stability

  • The section relates interleaving distance between persistence modules to erosion distance between their persistence diagrams.The supplied passage does not state the resulting bound.Source: §8 · p. 15
  • The first theorem’s proof assumes a concrete category with concrete images to make diagram chasing possible.All essentially small abelian categories satisfy this criterion by the Freyd–Mitchell embedding theorem.Source: §8 · p. 15
  • The semicontinuity discussion requires ε < ρ, where ρ is half the injectivity radius of S in R.It also identifies as a fundamental limitation that not all short exact sequences in C split.Source: §8 · p. 19
  • Theorem 8.2 is introduced as a continuity result for persistence modules in an essentially small, concrete, abelian category.The supplied theorem passage does not include the inequality that completes its statement.Source: §8 · p. 19

9 Concluding Remarks

The concluding remarks outline possible uses of the framework for torsion and time-series data, and pose an open question about extending diagrams to non-constructible modules.

Torsion in data

  • With integer coefficients, type A diagrams measure torsion and satisfy semicontinuity, while type B diagrams forget torsion; finite abelian coefficients yield type B diagrams encoding simple torsion.The authors suggest this as an approach to studying torsion in data.Source: §9 · p. 20

Time series

  • For time series, forward shifts should induce dynamics on the shape at each scale, but the appropriate algebraic object remains unclear and may require richer structure than vector spaces or abelian groups.The authors suggest the flexibility in choosing the category may allow more structure to be encoded.Source: §9 · p. 20

Non-constructible modules

  • For an infinite point set, the induced persistence module is not constructible, raising the question of whether it has a persistence diagram.The question is addressed for vector spaces by a rectangular-measure construction.Source: §9 · p. 21
  • For abelian categories, the authors propose assigning B(C) elements to rectangles without constructibility, but leave open whether this assignment is a rectangular measure.The proposed argument is similar to the proof of Proposition 7.1.Source: §9 · p. 21

A Krull-Schmidt

This appendix gives a compact treatment of Krull–Schmidt categories, where objects decompose into indecomposables and such decompositions are unique up to permutation and isomorphism.

A Krull-Schmidt

  • A Krull–Schmidt category is additive, every object is a finite direct sum of objects with local endomorphism rings, and an indecomposable object is nonzero and not a sum of two nonzero objects.An additive category has abelian hom-sets, bilinear composition, and finite products equal to finite coproducts.Source: A Krull-Schmidt · p. 23
  • In an additive category, an object is indecomposable exactly when its endomorphism ring is local.The appendix notes that each summand in a Krull–Schmidt decomposition is indecomposable.Source: A Krull-Schmidt · p. 23
  • Every object in a Krull–Schmidt category is isomorphic to a finite direct sum of indecomposables.Source: A Krull-Schmidt · p. 23
  • Any two finite decompositions of an object into indecomposables have the same number of summands, which match up by a permutation and isomorphism.The proof uses locality to identify a summand in one decomposition with a summand in the other, then repeats on the quotient.Source: A Krull-Schmidt · p. 23
Loading 1601.03107v5…