Generalized persistence diagrams

Patel, Amit

doi:10.1007/s41468-018-0012-6

Cited by 60 publications

(95 citation statements)

References 18 publications

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“…Since Z op + is a thin category, given another functor G : R d → Z op + , the interleaving distance (Definition 3.2) between F and G can be written as Remark 3.4. The metric d I is closely related to the erosion distance [58]. See Remark 6.3.…”

Section: The Interleaving Distance Between Integer-valued Functions mentioning

confidence: 99%

“…Remark 6.3. In order to compare the rank invariants, the author of [59] makes use of a generalization of the erosion distance in [58], which is denoted by d E (see Section C). It can be deduced that for the LHS of inequality (14) coincides with d E (rk(M ), rk(N )).…”

Section: Given Anymentioning

confidence: 99%

“…been studied in various contexts and with various names such as Betti curve, feature counting function, etc, [2,28,34,35,42,62]. The rank function rk(M X ) of M X has also been extensively considered [17,19,51,58,59]. We observe that both of these functions (1) can themselves be computed in polynomial time, (2) can be compared to each other via the interleaving distance d Z I for integer-valued functions (see Section 3.2) and (3) are stable to perturbations of (X , d X (·)) under d dyn (Theorems 4.…”

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Spatiotemporal Persistent Homology for Dynamic Metric Spaces

Kim

Mémoli

2020

Discrete Comput Geom

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Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS).In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter "spatiotemporal" filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov-Hausdorff distance which permits metrizing the collection of all DMSs.On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well known interleaving distance. In either case, the metric d can be computed in polynomial time. IntroductionStability and tractability of TDA for studying metric spaces. Finite point clouds or finite metric spaces are amongst the most common data representations considered in topological data analysis (TDA) [13,29,33]. In particular, the stability of the Single Linkage Hierarchical Clustering (SLHC) method [16] or the stability of the persistent homology of filtered Rips complexes built on metric spaces [22,23] motivates adopting these constructions when studying metric spaces arising in applications.Whereas there has been extensive applications of TDA to static metric data (thanks to the aforementioned theoretical underpinnings), there is not much study of dynamic metric data from the TDA perspective. Our motivation for considering dynamic metric data stems from the study and characterization of flocking/swarming behaviors of animals [5,36,37,39,53,57,63,69], convoys [41], moving clusters [43], or mobile groups [40,70]. In this paper, by extending ideas from [16,22,23,47,46], we aim at establishing a TDA framework for the study of dynamic metric spaces (DMSs) which comes together with stability theorems. We begin by describing and comparing relevant work with ours. Lack of an adequate metric for DMSs. In [55], Munch considers vineyards -a certain notion of time-varying persistence diagrams introduced by Cohen-Steiner et al. [25] -as signatures for dynamic point clouds. Munch, in particular, shows that vineyards are stable 1 [24] under perturbations of the input dynamic point cloud ...

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Section: The Interleaving Distance Between Integer-valued Functions mentioning

confidence: 99%

Section: Given Anymentioning

confidence: 99%

mentioning

confidence: 99%

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Spatiotemporal Persistent Homology for Dynamic Metric Spaces

Kim

Mémoli

2020

Discrete Comput Geom

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“…The persistence diagram of [CSEH07] easily generalizes [Pat18] to the setting of constructible persistence modules valued in any skeletally small abelian category C. The rank function of such a persistence module records the image of each F(r s) as an element of the Grothendieck group of C. Here we are using the Grothendieck group of an abelian category: this is the abelian group with one generator for each isomorphism class of objects and one relation for each short exact sequence. The persistence diagram of F is then the Möbius inversion of this rank function.…”

Section: Introductionmentioning

confidence: 99%

“…Proposition 4.6 (Positivity[Pat18]): Let F be a constructible persistence module valued in a skeletally small abelian category C. Then for any I ∈ Dgm, we have [0] F (I).…”

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Bottleneck stability for generalized persistence diagrams

McCleary

Patel

2020

Proc. Amer. Math. Soc.

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On the Stability of Interval Decomposable Persistence Modules

Bjerkevik

2021

Discrete Comput Geom

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The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for n-dimensional rectangle decomposable persistence modules up to a constant $$2n-1$$ 2 n - 1 that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved for $$n=2$$ n = 2 . We then apply the technique to prove stability for block decomposable modules, from which novel results for zigzag modules and Reeb graphs follow. These results are improvements on weaker bounds in previous work, and the bounds we obtain are optimal.

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Generalized persistence diagrams

Cited by 60 publications

References 18 publications

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

Bottleneck stability for generalized persistence diagrams

On the Stability of Interval Decomposable Persistence Modules

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