Decorated Merge Trees for Persistent Topology

Curry, Justin; Hang, Haibin; Mio, Washington; Needham, Tom; Okutan, Osman Berat

doi:10.48550/arxiv.2103.15804

Cited by 4 publications

(6 citation statements)

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“…Both the bottleneck distance d B and the merge tree interleaving distance d I turn out to be instances of a general categorical definition of interleaving distances introduced by Bubenik and Scott [13]; this is shown for d B in [6] and for d I in [23,Proposition 3.11]. As such, the two distances are closely related.…”

Section: Overviewmentioning

confidence: 92%

“…Following [23], we define the geometric realization of a constructible persistent set; this relates our categorical definition of a merge tree to the topological and graph-theoretic definitions that one typically sees in the literature. Our definition is equivalent to that of [23,Definition A.3] in the constructible setting, though slightly different in the details.…”

Section: Example 7 Letting πmentioning

confidence: 99%

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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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Section: Overviewmentioning

confidence: 92%

Section: Example 7 Letting πmentioning

confidence: 99%

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

Cardona¹,

Curry²,

Lam³

et al. 2021

Preprint

Self Cite

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show abstract

“…Example 2.21. Consider the strict barcode B = [0, ∞), [1,8), [2,7), [3,6), [4,5) . According to the formula in Proposition 2.20,…”

Section: Combinatorial Trees Merge Treesmentioning

confidence: 99%

“…We note that the study of the (non-) injectivity of certain topological transforms is also an aspect of topological inverse problems, see [23,13,6,21,25] for a sampling of these articles and [24] for a recent survey. Better understanding the precise failure of injectivity of certain TDA invariants led to the development of enriched topological summaries (ETS) that remediate these failures, opening a promising line of research; see [2] and [5] for some examples of these ETS.…”

Section: Introductionmentioning

confidence: 99%

From Trees to Barcodes and Back Again II: Combinatorial and Probabilistic Aspects of a Topological Inverse Problem

Curry¹,

DeSha²,

Garin³

et al. 2021

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In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. The first important outcome of our study is a clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees. Generic BHV trees on n + 1 leaf nodes fall into (2n − 1)!! distinct strata, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., (n+1)!n!2 −n . The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data and opens the door to doing more precise science.

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“…The answer, in general, is no since many different Reeb graphs can have the same persistence diagram. Despite difficulties, consideration of these inverse problems are increasingly of interest in topological data analysis as a whole [14,15,27], as they can provide insight into how much information is lost in the process of computing such a topological signature, be it a persistence diagram or a Reeb graph. However, they are notoriously difficult, as key information is inevitably lost when going from a space to such a signature.…”

Section: Introductionmentioning

confidence: 99%

Realizable piecewise linear paths of persistence diagrams with Reeb graphs

Alharbi¹,

Chambers²,

Munch³

2021

Preprint

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Reeb graphs are widely used in a range of fields for the purposes of analyzing and comparing complex spaces via a simpler combinatorial object. Further, they are closely related to extended persistence diagrams, which largely but not completely encode the information of the Reeb graph. In this paper, we investigate the effect on the persistence diagram of a particular continuous operation on Reeb graphs; namely the (truncated) smoothing operation. This construction arises in the context of the Reeb graph interleaving distance, but separately from that viewpoint provides a simplification of the Reeb graph which continuously shrinks small loops. We then use this characterization to initiate the study of inverse problems for Reeb graphs using smoothing by showing which paths in persistence diagram space (commonly known as vineyards) can be realized by a path in the space of Reeb graphs via these simple operations. This allows us to solve the inverse problem on a certain family of piecewise linear vineyards when fixing an initial Reeb graph.

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Decorated Merge Trees for Persistent Topology

Cited by 4 publications

References 0 publications

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

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

From Trees to Barcodes and Back Again II: Combinatorial and Probabilistic Aspects of a Topological Inverse Problem

Realizable piecewise linear paths of persistence diagrams with Reeb graphs

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