Bayesian Topological Learning for Classifying the Structure of Biological Networks

Maroulas, Vasileios; Micucci, Cassie Putman; Nasrin, Farzana

doi:10.1214/21-ba1270

Cited by 5 publications

(2 citation statements)

References 43 publications

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“…Persistent homology is one of the central tools of topological data analysis and in recent years has found a number of applications to a diverse range of fields such as neuroscience [9,6], biology and biochemistry [8,20], materials science [23,17,21], and the study of sensor networks [10]. In order to apply persistent homology to a finite and discrete data set X, one must be able to convert X into a filtered simplicial complex for which the persistent homology groups can then be computed.…”

Section: Introductionmentioning

confidence: 99%

The homotopy category of acyclic complexes of pure-projective modules

Gillespie

2023

Forum Mathematicum

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Let R be any ring with identity. We show that the homotopy category of all acyclic chain complexes of pure-projective R-modules is a compactly generated triangulated category. We do this by constructing abelian model structures that put this homotopy category into a recollement with two other compactly generated triangulated categories: The usual derived category of R and the pure derived category of R.

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Section: Introductionmentioning

confidence: 99%

The homotopy category of acyclic complexes of pure-projective modules

Gillespie

2023

Forum Mathematicum

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“…In recent years, topology and geometry blended with statistical methods have seen increasing application to the study of data analysis, visualization, and dimensionality reduction [1][2][3][4][5][6][7][8][9]. These applications range from classification and clustering in fields such as action recognition [10], handwriting analysis [11], and biology [12][13][14][15][16], to classification of high entropy alloy [17] and gas separation [18], to the analysis of complex biological networks [19], and other complex dynamical systems [20,21] and sensor networks [22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Quantum Persistent Homology

Bernardo¹,

Maroulas²,

Siopsis³

2022

Preprint

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Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent topological features while one adjusts the resolution. However, the computation of such topological features is often a rather formidable task necessitating the sub-sampling the underlying data. To remedy this, we develop an efficient quantum computation of persistent Betti numbers, which track topological features of data across different scales. Our approach employs a persistent Dirac operator whose square yields the persistent combinatorial Laplacian, and in turn the underlying persistent Betti numbers which capture the persistent features of data. We also test our algorithm on point cloud data.

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Vietoris thickenings and complexes are weakly homotopy equivalent

Gillespie

2023

J Appl. and Comput. Topology

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Bayesian Topological Learning for Classifying the Structure of Biological Networks

Cited by 5 publications

References 43 publications

The homotopy category of acyclic complexes of pure-projective modules

The homotopy category of acyclic complexes of pure-projective modules

Quantum Persistent Homology

Vietoris thickenings and complexes are weakly homotopy equivalent

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