Stability of higher-dimensional interval decomposable persistence modules

Bjerkevik, Håvard Bakke

doi:10.48550/arxiv.1609.02086

Cited by 11 publications

(32 citation statements)

References 8 publications

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“…As a first step towards a general study of stability properties of the rank invariant that we introduce, we verify that the rank invariant is stable to perturbation of the zigzag modules valued in the category of finite dimensional vector spaces or in the category of finite sets (Section 6). This stability straightforwardly follows from the equivalence between our persistence diagrams and standard ones (Sections 3 and 5), together with stability results by Botnan, Lesnick and Bjerkevik [8,10].…”

Section: Introductionsupporting

confidence: 67%

“…We obtain Theorem 6.6 below as a corollary of Theorem 5.2. We emphasize that the statement of this theorem is not the same as that of [10,Theorem 4.13] nor its strengthened version due to Bjerkevik [8]: a priori, dgm ZZ set (M ) is defined in a purely set-theoretical setting (Definitions 4.3 and 4.4), without invoking the notions of homology or Reeb graph considered in [10]. Our geometric interpretation of set-valued zigzag modules as Reeb graphs is used for proving Theorem 6.6 as follows: By functoriality of the linearization functor L F (Definition 5.1), d vec I (L F (M ), L F (N )) ≤ d set I (M , N ), (which was already discussed in [10]).…”

Section: Erosion and Bottleneck Stability For Set-valued Zigzag Modulesmentioning

confidence: 96%

“…Let µ : Con op (P) × Con op (P) → Z be the Möbius function of the poset Con(P) op . Then, for I ∈ Con op (P) 8 More precisely, the codomain of µ Q is the multiple of 1 in a specified base ring rather than Z. Remark 7.19.…”

Section: The Möbius Function µmentioning

confidence: 99%

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Generalized Persistence Diagrams for Persistence Modules over Posets

Kim,

Memoli

2018

Preprint

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When a category C satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors F : P → C from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules F : P → vec of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset P of F : P → C in defining Patel's generalized persistence diagram of F .By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type A persistence diagram to Lipschitz continuity theorem for the category of sets.

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Section: Introductionsupporting

confidence: 67%

Section: Erosion and Bottleneck Stability For Set-valued Zigzag Modulesmentioning

confidence: 96%

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Generalized Persistence Diagrams for Persistence Modules over Posets

Kim,

Memoli

2018

Preprint

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“…The general strategy for the proof of stability, particularly in Lemma 6.27 and Theorem 6.28 below, is similar to that used by Bjerkevik [3] to study algebraic stability in the context of multi-parameter persistence modules. The arguments needed for p-sheaves, however, are quite distinct.…”

Section: Persistence Diagramsmentioning

confidence: 99%

“…As pointed out earlier, the general strategy for the proof of stability is similar to that used by Bjerkevik in the study of multi-parameter persistence [3], although the arguments for p-sheaves differ substantially. A key matching result needed in the proof of the stability theorem, we employ the following classic result in combinatorics and graph theory.…”

Section: 2mentioning

confidence: 99%

Correspondence Modules and Persistence Sheaves: A Unifying Perspective on One-Parameter Persistent Homology

Hang¹,

Mio²

2020

Preprint

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We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of one-parameter persistence is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions of persistence sheaves. Applications include: (a) a Mayer-Vietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a real-valued function and (b) the construction of slices of 2-parameter persistence modules along negatively sloped lines.

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Analysis of Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence

2020

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When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. In a similar vein, studying the dynamics of social networks leads to the problem of characterizing groups/communities as they form and disperse throughout time.Motivated by this, we study the problem of obtaining persistent homology based summaries of timedependent data. Given a finite dynamic graph (DG), we first construct a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DG under a suitable distance between DGs that we identify.More precisely, our stability theorem can be interpreted as providing a lower bound for the distance between DGs. Since it relies on barcodes, and their bottleneck distance, this lower bound can be computed in polynomial time from the DG inputs.Since DGs can be given rise by applying the Rips functor (with a fixed threshold) to dynamic metric spaces, we are also able to derive related stable invariants for these richer class of dynamic objects.Along the way, we propose a summarization of dynamic graphs that captures their time-dependent clustering features which we call formigrams. These set-valued functions generalize the notion of dendrogram, a prevalent tool for hierarchical clustering. In order to elucidate the relationship between our distance between two DGs and the bottleneck distance between their associated barcodes, we exploit recent advances in the stability of zigzag persistence due to Botnan and Lesnick, and to Bjerkevik.

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Stability of higher-dimensional interval decomposable persistence modules

Cited by 11 publications

References 8 publications

Generalized Persistence Diagrams for Persistence Modules over Posets

Generalized Persistence Diagrams for Persistence Modules over Posets

Correspondence Modules and Persistence Sheaves: A Unifying Perspective on One-Parameter Persistent Homology

Analysis of Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence

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