A roadmap for the computation of persistent homology

Otter, Nina; Porter, Mason A.; Tillmann, Ulrike; Grindrod, Peter; Harrington, Heather A.

doi:10.1140/epjds/s13688-017-0109-5

Cited by 548 publications

(553 citation statements)

References 140 publications

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“…This includes, for instance, critical points, integral lines, separating surfaces, voids, and so forth. Moreover, Persistent Homology (PH) brings an appealing framework for measuring the salience of topological features in the data, which is well established both at a theoretical and practical level . In practice, it allows users to discriminate important features from nonsignificant configurations and provides the necessary basis for multiscale data analysis .…”

Section: Introductionmentioning

confidence: 99%

A Topological Data Analysis perspective on noncovalent interactions in relativistic calculations

Olejniczak

Gomes

Tierny

2019

Int J of Quantum Chemistry

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Topological Data Analysis (TDA) is a powerful mathematical theory, largely unexplored in theoretical chemistry. In this work we demonstrate how TDA provides new insights into topological features of electron densities and reduced density gradients, by investigating the effects of relativity on the bonding of the Au4‐S‐C6H4‐S′‐Au′4 molecule. Whereas recent analyses of this species carried out with the Quantum Theory of Atoms‐In‐Molecules (a previous study) concluded, from the emergence of new topological features in the electron density, that relativistic effects yielded noncovalent interactions between gold and hydrogen atoms, we show from their low persistence values (which decrease with increased basis set size) these features are not significant. Further analysis of the reduced density gradient confirms no relativity‐induced noncovalent interactions in Au4‐S‐C6H4‐S′‐Au′4. We argue TDA should be integrated into electronic structure analysis methods, and be considered as a basis for the development of new topology‐based approaches.

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Section: Introductionmentioning

confidence: 99%

A Topological Data Analysis perspective on noncovalent interactions in relativistic calculations

Olejniczak

Gomes

Tierny

2019

Int J of Quantum Chemistry

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“…Readers looking for an easy way to test out some of these ideas and computations should visit the website (Tralie, 2016), which does computation of persistent homology in the browser for example point clouds. This is not by any means an exhaustive list; see Otter, Porter, Tillmann, Grindrod, and Harrington (2015) for a good survey on software packages.…”

Section: Available Softwarementioning

confidence: 99%

A User’s Guide to Topological Data Analysis

Munch

2017

Learning Analytics

114

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ABSTRACT. Topological data analysis (TDA) is a collection of powerful tools that can quantify shape and structure in data in order to answer questions from the data's domain. This is done by representing some aspect of the structure of the data in a simplified topological signature. In this article, we introduce two of the most commonly used topological signatures. First, the persistence diagram represents loops and holes in the space by considering connectivity of the data points for a continuum of values rather than a single fixed value. The second topological signature, the mapper graph, returns a 1-dimensional structure representing the shape of the data, and is particularly good for exploration and visualization of the data. While these techniques are based on very sophisticated mathematics, the current ubiquity of available software means that these tools are more accessible than ever to be applied to data by researchers in education and learning, as well as all domain scientists.

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“…Persistent Homology [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55] is a means of topological data analysis. Now let us use an example to show how the topological data analysis methods can overcome the limitations of geometrical methods.…”

Section: Persistent Homologymentioning

confidence: 99%

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

2018

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Atom probe tomography (APT) represents a revolutionary characterization tool for materials that combine atomic imaging with a time-of-flight (TOF) mass spectrometer to provide direct space three-dimensional, atomic scale resolution images of materials with the chemical identities of hundreds of millions of atoms. It involves the controlled removal of atoms from a specimen's surface by field evaporation and then sequentially analyzing them with a position sensitive detector and TOF mass spectrometer. A paradox in APT is that while on the one hand, it provides an unprecedented level of imaging resolution in three dimensions, it is very difficult to obtain an accurate perspective of morphology or shape outlined by atoms of similar chemistry and microstructure. The origins of this problem are numerous, including incomplete detection of atoms and the complexity of the evaporation fields of atoms at or near interfaces. Hence, unlike scattering techniques such as electron microscopy, interfaces appear diffused, not sharp. This, in turn, makes it challenging to visualize and quantitatively interpret the microstructure at the "meso" scale, where one is interested in the shape and form of the interfaces and their associated chemical gradients. It is here that the application of informatics at the nanoscale and statistical learning methods plays a critical role in both defining the level of uncertainty and helping to make quantitative, statistically objective interpretations where heuristics often dominate. In this chapter, we show how the tools of Topological Data Analysis provide a new and powerful tool in the field of nanoinformatics for materials characterization.

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A roadmap for the computation of persistent homology

Cited by 548 publications

References 140 publications

A Topological Data Analysis perspective on noncovalent interactions in relativistic calculations

A Topological Data Analysis perspective on noncovalent interactions in relativistic calculations

A User’s Guide to Topological Data Analysis

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

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