Persistent Homology and Euler Integral Transforms

Levanger, Rachel; Mai, Huy

doi:10.48550/arxiv.1804.04740

Cited by 3 publications

(8 citation statements)

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“…Beyond the setting of [DSW15] considered here, there have been few attempts so far at solving inverse problems in persistence theory. [TMB14], [CMT18], and [GLM18] investigated the injectivity of the persistent homology transform, an invariant that assigns to a subanalytic complex X ⊂ R d the set of persistence diagrams arising from the functions •, v for v ∈ S d−1 . Whereas the approach of that paper is to employ extrinisically defined filtrations, our approach in this paper is intrinsic and local.…”

Section: Introductionmentioning

confidence: 99%

Barcode Embeddings for Metric Graphs

Oudot,

Solomon

2017

Preprint

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Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we study a rich homology-based invariant first defined by Dey, Shi, and Wang in [DSW15], which we think of as embedding a metric graph in the barcode space. We prove that this invariant is locally injective on the space of metric graphs and globally injective on a GHdense subset. Moreover, we define a new topology on MGraphs, which we call the fibered topology, for which the barcode transform is injective on a generic (open and dense) subset. Contents 1. Introduction Our Contributions Prior Work 2. Background 2.1. Combinatorial and Metric Graphs 2.2. Pointed Metric Graphs 2.3. Metric-Measure Graphs 2.4. Morse-Type Functions 2.5. Extended Persistent Homology 2.6. Reeb Graphs 3. The Barcode Embedding and the Persistence Distortion Pseudometric 4. Stability 5. Inverse Problem 6. Overview of the Proofs of Theorems 5.4, 5.7, and 5.9, and Proposition 5.5 7. Proof of Proposition 6.2: Ψ G is a local isometry 7.1. Proof of Proposition 6.2 8. Proof of Theorem 6.1: Recovering G up to isometry when Ψ G is injective 9. Proposition 5.5: Gromov-Hausdorff Density of Ψ G -injective Graphs 10. Proposition 6.3: Z-linear independence of edge weights implies Ψ G injective for G ∈ MGraphs * 11. The Case of Self-Loops and Few Vertices of Valence not Equal to Two 12. Theorem 5.7: Local Injectivity 13. Conclusion & Discussion 14. Acknowledgements A. Rips filtrations on geodesic trees References

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Section: Introductionmentioning

confidence: 99%

Barcode Embeddings for Metric Graphs

Oudot,

Solomon

2017

Preprint

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“…does not preserve all relevant information of a shape (Crawford et al, 2020). Using the Euler calculus, Ghrist et al (2018) (Corollary 6 therein) showed that the persistent homology transform (PHT) (Turner et al, 2014b), motivated by integral geometry and differential topology, concisely summarizes information Turner et al (2014b)). Since D is not a vector space and the distances on D (e.g., the p-th Wasserstein and bottleneck distances (Cohen-Steiner et al, 2007)) are very abstract, many fundamental concepts in classical statistics are not easy to implement with summaries resulting from the PHT.…”

Section: Overview Of Topological Data Analysismentioning

confidence: 99%

Randomness and Statistical Inference of Shapes via the Smooth Euler Characteristic Transform

Meng¹,

Crawford²,

Eloyan³

2022

Preprint

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In this paper, we provide the mathematical foundations for the randomness of shapes and the distributions of smooth Euler characteristic transform. Based on these foundations, we propose an approach for testing hypotheses on random shapes. Simulation studies are provided to support our mathematical derivations and show the performance of our proposed hypothesis testing framework. Our discussions connect the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, statistical inference, and medical imaging.

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“…Recent work of Ghrist et al [GLM18] and, independently, of Curry et al [CMT18], using ideas of Schapira [Sch95], demonstrates the injectivity of the ECT in all dimensions, and for the larger class of subanalytic compact sets. Because the Euler Characteristic curve of the functions f v can be derived from their persistence module, this, in turn, implies the injectivity of the PHT.…”

Section: Extrinsic Persistent Homology Transformsmentioning

confidence: 99%

“…In particular, if χ 1 = χ 2 then the scaling term in (R S • R s ) is constant and nonzero. To take advantage of this theorem, [GLM18] define a Radon transform that can be computed using the ECT, and then find an appropriate "inverse" Radon transform.…”

Section: Extrinsic Persistent Homology Transformsmentioning

confidence: 99%

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Inverse Problems in Topological Persistence

Oudot¹,

Solomon²

2018

Preprint

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In this survey, we review the literature on inverse problems in topological persistence theory. The first half of the survey is concerned with the question of surjectivity, i.e. the existence of right inverses, and the second half focuses on injectivity, i.e. left inverses. Throughout, we highlight the tools and theorems that underlie these advances, and direct the reader's attention to open problems, both theoretical and applied.

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Persistent Homology and Euler Integral Transforms

Cited by 3 publications

References 4 publications

Barcode Embeddings for Metric Graphs

Barcode Embeddings for Metric Graphs

Randomness and Statistical Inference of Shapes via the Smooth Euler Characteristic Transform

Inverse Problems in Topological Persistence

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