Computational Complexity of the Interleaving Distance

Bjerkevik, Håvard Bakke; Botnan, Magnus Bakke

doi:10.48550/arxiv.1712.04281

Cited by 2 publications

(5 citation statements)

References 26 publications

(37 reference statements)

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“…Next, in regard to item (b) above, we address the issue of computability of the metric between invariants of DMSs. In [7,8], Bjerkevik and Botnan show that computing the interleaving distance d I [52] between multidimensional persistence modules can in general be NP-hard. Also, since we are not guaranteed to have interval decomposability [9,17] of the 3-dimensional modules considered in this paper, we are not in a position to utilize the bottleneck distance and relevant algorithms developed by Dey and Xin [28] instead of d I .…”

Section: Contributionsmentioning

confidence: 99%

“…Given δ > 0, deciding whether d Vec I (M , N ) ≤ δ is in general NP-hard [7,8]. In Theorem 6.2, substituting the comparison of M and N with that of rk(M ) and rk(N ) results in doubling of the underlying dimension of the interleaving distance.…”

Section: Interleaving Stability Of Rank Invariants and Dimension Func...mentioning

confidence: 99%

“…This implies that rk k (γ X ) is an enriched version of the k-th CROCKER plot C k (γ X ) of γ X . 8 Therefore, Theorem 4.4 can be interpreted somehow as establishing the stability of the CROCKER plots of a DMS.…”

Section: B Relationship Between the Rank Invariant And Crocker-plotmentioning

confidence: 99%

“…In order to obtain a lower bound for d dyn between two DMSs, computing the distance between the Betti-0 functions of the DMSs (the LHS of the inequaliy in (8)) is better than computing the distance between their 0-th rank invariants (the LHS of the inequaliy in ( 7)) : Proposition 4.7. For any two DMSs γ…”

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confidence: 99%

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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: Contributionsmentioning

confidence: 99%

Section: Interleaving Stability Of Rank Invariants and Dimension Func...mentioning

confidence: 99%

Section: B Relationship Between the Rank Invariant And Crocker-plotmentioning

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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“…Computing the interleaving distance both on Reeb graphs and merge trees is NP-hard [34,35]. However if we restrict to labeling the vertices on merge trees, e.g.…”

Section: Introductionmentioning

confidence: 99%

The ℓ ∞-Cophenetic Metric for Phylogenetic Trees As an Interleaving Distance

Munch

Stefanou

2019

Association for Women in Mathematics Series

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There are many metrics available to compare phylogenetic trees since this is a fundamental task in computational biology. In this paper, we focus on one such metric, the ∞ -cophenetic metric introduced by Cardona et al. This metric works by representing a phylogenetic tree with n labeled leaves as a point in R n(n+1)/2 known as the cophenetic vector, then comparing the two resulting Euclidean points using the ∞ distance. Meanwhile, the interleaving distance is a formal categorical construction generalized from the definition of Chazal et al., originally introduced to compare persistence modules arising from the field of topological data analysis. We show that the ∞ -cophenetic metric is an example of an interleaving distance. To do this, we define phylogenetic trees as a category of merge trees with some additional structure; namely labelings on the leaves plus a requirement that morphisms respect these labels. Then we can use the definition of a flow on this category to give an interleaving distance. Finally, we show that, because of the additional structure given by the categories defined, the map sending a labeled merge tree to the cophenetic vector is, in fact, an isometric embedding, thus proving that the ∞ -cophenetic metric is, in fact, an interleaving distance.

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Computational Complexity of the Interleaving Distance

Cited by 2 publications

References 26 publications

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

The ℓ ∞-Cophenetic Metric for Phylogenetic Trees As an Interleaving Distance

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