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

Munch, Elizabeth; Stefanou, Anastasios

doi:10.1007/978-3-030-11566-1_5

Cited by 19 publications

(18 citation statements)

References 28 publications

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“…Moreover, we can use the notion of interleavings to define an extended pseudo-metric on ordered merge trees, see [18] for the application of the interleaving distance to (un-ordered) merge trees, [3] for the simplest construction of the interleaving distance for persistence modules valued in arbitrary categories C, and [12] for a more subtle variant of the interleaving distance. It should be noted that defining metrics using ordered or labeled merge trees is not entirely new and was developed in part by Elizabeth Munch and Anastasios Stefanou in [19].…”

Section: Maps Of Ordered and Chiral Merge Treesmentioning

confidence: 99%

The fiber of the persistence map for functions on the interval

Curry

2018

J Appl. and Comput. Topology

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In this paper we study functions on the interval that have the same persistent homology, which is what we mean by the fiber of the persistence map. By imposing an equivalence relation called graph-equivalence, the fiber of the persistence map becomes finite and a precise enumeration is given. Graph-equivalence classes are indexed by chiral merge trees, which are binary merge trees where a left-right ordering of the children of each vertex is given. Enumeration of merge trees and chiral merge trees with the same persistence makes essential use of the Elder Rule, which is given its first detailed proof in this paper. arXiv:1706.06059v2 [math.AT]

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Section: Maps Of Ordered and Chiral Merge Treesmentioning

confidence: 99%

The fiber of the persistence map for functions on the interval

Curry

2018

J Appl. and Comput. Topology

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“…The connectedness of branches—irrespective of deformation or bending—is topology, and it is useful for describing variation in plant architecture ( Li et al, 2017 ). Describing branching patterns is relevant to describing phylogenetic trees, too, to which Topological Data Analysis approaches have been applied ( Munch and Stefanou, 2018 ). We converted shape into a topological feature space to comprehensively describe leaf shape diversity where other methods have failed.…”

Section: Discussionmentioning

confidence: 99%

Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace

Angelovici

et al. 2018

Front. Plant Sci.

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Current morphometric methods that comprehensively measure shape cannot compare the disparate leaf shapes found in seed plants and are sensitive to processing artifacts. We explore the use of persistent homology, a topological method applied as a filtration across simplicial complexes (or more simply, a method to measure topological features of spaces across different spatial resolutions), to overcome these limitations. The described method isolates subsets of shape features and measures the spatial relationship of neighboring pixel densities in a shape. We apply the method to the analysis of 182,707 leaves, both published and unpublished, representing 141 plant families collected from 75 sites throughout the world. By measuring leaves from throughout the seed plants using persistent homology, a defined morphospace comparing all leaves is demarcated. Clear differences in shape between major phylogenetic groups are detected and estimates of leaf shape diversity within plant families are made. The approach predicts plant family above chance. The application of a persistent homology method, using topological features, to measure leaf shape allows for a unified morphometric framework to measure plant form, including shapes, textures, patterns, and branching architectures.

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“…Most important for comparison to our work is the p-cophenetic distance [14], which is reviewed in Definition 33. In [39,31] it was shown that the ∞-cophenetic distance is equal to the interleaving distance. Consequently, by Theorem 4, the ∞-cophenetic distance and the ∞-presentation distance are the same.…”

Section: Other Metrics On Merge Treesmentioning

confidence: 99%

The Universal $\ell^p$-Metric on Merge Trees

Cardona¹,

Curry²,

Lam³

et al. 2021

Preprint

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Adapting a definition given by Bjerkevik and Lesnick [9] for multiparameter persistence modules, we introduce an p -type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the p-Wasserstein distance between the associated barcodes. For each p ∈ [1, ∞], we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the p = ∞ case, this gives a novel proof of universality for the interleaving distance on merge trees. ACM Subject ClassificationMathematics of computing → Algebraic topology; Theory of computation → Unsupervised learning and clustering; Theory of computation → Computational geometry Keywords and phrases merge trees, hierarchical clustering, persistent homology, Wasserstein distances, interleavings Funding Justin Curry: Supported by NSF CCF-1850052 and NASA 80GRC020C0016Acknowledgements While Håvard Bjerkevik was not directly involved in this project, he has had a major influence on it, via his collaboration with ML on presentation distances for multiparameter persistence modules [9]. In particular, Håvard kindly agreed to share an early draft of [9] with our group in July 2020, which inspired many of the ideas in our paper.

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

Cited by 19 publications

References 28 publications

The fiber of the persistence map for functions on the interval

The fiber of the persistence map for functions on the interval

Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace

The Universal $\ell^p$-Metric on Merge Trees

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