The Gudhi Library: Simplicial Complexes and Persistent Homology

Maria, Clément; Boissonnat, Jean‐Daniel; Glisse, Marc; Yvinec, Mariette

doi:10.1007/978-3-662-44199-2_28

Cited by 188 publications

(145 citation statements)

References 14 publications

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“…The compressed annotation matrix implementation of persistent cohomology (denoted by CAM in the following) is part of the Gudhi library [14,15].…”

Section: Methodsmentioning

confidence: 99%

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

Boissonnat

Dey²,

Maria³

2015

Algorithmica

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International audiencePersistent homology with coefficients in a field coincides with the samefor cohomology because of duality. We propose an implementation of a recently intro-duced algorithm for persistent cohomology that attaches annotation vectors with thesimplices. We separate the representation of the simplicial complex from the represen-tation of the cohomology groups, and introduce a new data structure for maintainingthe annotation matrix, which is more compact and reduces substantially the amountof matrix operations. In addition, we propose a heuristic to simplify further the repre-sentation of the cohomology groups and improve both time and space complexities.The paper provides a theoretical analysis, as well as a detailed experimental studyof our implementation and comparison with state-of-the-art software for persistenthomology and cohomology

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“…The compressed annotation matrix implementation of persistent cohomology (denoted by CAM in the following) is part of the Gudhi library [14,15].…”

Section: Methodsmentioning

confidence: 99%

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

Boissonnat

Dey²,

Maria³

2015

Algorithmica

Self Cite

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“…Further, the maximum level of its graph filtration is m + 1, and the form is unique (see Theorem 2 in ). The finiteness and uniqueness of the filtration levels over finite graphs have been already applied in some software packages such as JavaPlex (Adams et al, 2014) and Gudhi (Maria, Boissonnat, Glisse, & Yvinec, 2014). It should be noticed that a set of unique positive weights can be obtained by removing any multiplicative weights when identical edge weights exist.…”

Section: Appendix A: Graph Filtrationmentioning

confidence: 99%

A concise and persistent feature to study brain resting‐state network dynamics: Findings from the Alzheimer's Disease Neuroimaging Initiative

Kuang

Chen

Caselli

et al. 2018

Human Brain Mapping

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Alzheimer’s disease (AD) is the most common type of dementia in the elderly with no effective treatment currently. Recent studies of non-invasive neuroimaging, resting-state functional magnetic resonance imaging (rs-fMRI) with graph theoretical analysis have shown that patients with AD and mild cognitive impairment (MCI) exhibit disrupted topological organization in large-scale brain networks. In previous work, it is a common practice to threshold such networks. However, it is not only difficult to make a principled choice of threshold values, but worse is the discard of potential important information. To address this issue, we propose a threshold-free feature by integrating a prior persistent homology-based topological feature (the zeroth Betti number) and a newly defined connected component aggregation cost feature to model brain networks over all possible scales. We show that the induced topological feature (Integrated Persistent Feature – IPF) follows a monotonically decreasing convergence function and further propose to use its slope as a concise and persistent brain network topological measure. We apply this measure to study rs-fMRI data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) and compare our approach with five other widely used graph measures across five parcellation schemes ranging from 90 to 1024 region-of-interests (ROIs). The experimental results demonstrate that the proposed network measure shows more statistical power and stronger robustness in group difference studies in that the absolute values of the proposed measure of AD are lower than MCI’s, and much lower than normal control’s, providing empirical evidence for decreased functional integration in AD dementia and MCI.

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“…There are numerous libraries that compute persistent homology and that are aimed at different public. Some of the most recent ones are the TDA package in R [10] intended for statisticians, DIPHA [11] and GUDHI [12] that are state-of-the-art approaches from the computational topology community or HomCloud [13] which aims at a more experimentalist public with additional tools and graphical output. This list is non-exhaustive and many more exist.…”

Section: Computationmentioning

confidence: 99%

Persistent Homology and Materials Informatics

2018

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This paper provides an introduction to persistent homology and a survey of its applications to materials science. Mathematical prerequisites are limited to elementary linear algebra. Important concepts in topological data analysis such as persistent homology and persistence diagram are explained in a selfcontained manner with several examples. These tools are applied to glass structural analysis, crystallization of granular systems, and craze formation of polymers.

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The Gudhi Library: Simplicial Complexes and Persistent Homology

Cited by 188 publications

References 14 publications

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

A concise and persistent feature to study brain resting‐state network dynamics: Findings from the Alzheimer's Disease Neuroimaging Initiative

Persistent Homology and Materials Informatics

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