Efficient and robust persistent homology for measures

Buchet, Mickaël; Chazal, Frédéric; Oudot, Steve; Sheehy, Donald R.

doi:10.1016/j.comgeo.2016.07.001

Cited by 66 publications

(79 citation statements)

References 9 publications

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“…Remark 4.3. For simplicity, theorem 4.1 is stated for the Euclidean distance, but the proof of theorem 4.1 can be extended to the case of the power distance [6,57]. To define this distance, assign non-negative weights w(x) to points in X and f (X).…”

Section: Enclosing Ballsmentioning

confidence: 99%

Persistent homology for low-complexity models

Lotz

2019

Proc. R. Soc. A.

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We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is determined by the Gaussian width of a structure associated to the data set, rather than its size, and such a reduction can be computed efficiently. We further relate the Gaussian width to the doubling dimension of a finite metric space, which appears in the study of the complexity of other methods for approximating persistent homology. This allows to literally replace the ambient dimension by an intrinsic notion of dimension related to the structure of the data.

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Section: Enclosing Ballsmentioning

confidence: 99%

Persistent homology for low-complexity models

Lotz

2019

Proc. R. Soc. A.

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“…The introduction of the DTM has motivated further works and applications in various directions such as topological data analysis ( Buchet et al, 2015a ), GPS trace analysis ( Chazal et al, 2011a ), density estimation ( Biau et al, 2011 ), hypothesis testing Brécheteau (2019) , and clustering ( Chazal et al, 2013 ), just to name a few. Approximations, generalizations, and variants of the DTM have also been considered ( Guibas et al, 2013 ; Phillips et al, 2014 ; Buchet et al, 2015b ; Brécheteau and Levrard, 2020 ).…”

Section: Geometric Reconstruction and Homology Inferencementioning

confidence: 99%

An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

Chazal

Michel

2021

Front. Artif. Intell.

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256

148

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With the recent explosion in the amount, the variety, and the dimensionality of available data, identifying, extracting, and exploiting their underlying structure has become a problem of fundamental importance for data analysis and statistical learning. Topological data analysis (tda) is a recent and fast-growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. It proposes new well-founded mathematical theories and computational tools that can be used independently or in combination with other data analysis and statistical learning techniques. This article is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of tda for nonexperts.

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“…For each simplex σ with |σ| > 2 we add it at the maximum time of addition of its edges. This is an analytical solution for adding simplices to the Weighted Vietoris-Rips construction, studied in [6], for the semi metric space M (of genes) with weights on X, the original data matrix. We are essentially assigning to each gene its expression associated to the subject s. In the actual implementation, we multiplied the weights by a scaling term, since all distances are bound by 1 above but the weights themselves can be higher.…”

Section: Tda Workflowmentioning

confidence: 99%

A Topological Data Analysis Approach on Predicting Phenotypes from Gene Expression Data

Mandal

Guzmán-Sáenz

Haiminen

et al. 2020

Algorithms for Computational Biology

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The goal of this study was to investigate if gene expression measured from RNA sequencing contains enough signal to separate healthy and afflicted individuals in the context of phenotype prediction. We observed that standard machine learning methods alone performed somewhat poorly on the disease phenotype prediction task; therefore we devised an approach augmenting machine learning with topological data analysis.We describe a framework for predicting phenotype values by utilizing gene expression data transformed into sample-specific topological signatures by employing feature subsampling and persistent homology. The topological data analysis approach developed in this work yielded improved results on Parkinson's disease phenotype prediction when measured against standard machine learning methods.This study confirms that gene expression can be a useful indicator of the presence or absence of a condition, and the subtle signal contained in this high dimensional data reveals itself when considering the intricate topological connections between expressed genes.

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Efficient and robust persistent homology for measures

Cited by 66 publications

References 9 publications

Persistent homology for low-complexity models

Persistent homology for low-complexity models

An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

A Topological Data Analysis Approach on Predicting Phenotypes from Gene Expression Data

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