Maximally persistent cycles in random geometric complexes

Bobrowski, Omer; Kahle, Matthew; Škraba, Primož

doi:10.1214/16-aap1232

Cited by 56 publications

(71 citation statements)

References 39 publications

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“…Briefly, the persistent homology of aČech or a Rips complex tracks the evolution of the homology of the complex as the radius r changes from zero to infinity. In this section we will review some recent work related to the persistent homology of random geometric complexes [10,22].…”

Section: Persistent Homologymentioning

confidence: 99%

Topology of random geometric complexes: a survey

Bobrowski

Kahle

2018

J Appl. and Comput. Topology

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106

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In this expository article, we survey the rapidly emerging area of random geometric simplicial complexes. Random geometric complexes may be viewed as higher-dimensional generalizations of random geometric graphs, where vertices are generated by a random point process, and edges are placed based on proximity. Extending the notion of connected components and cycles in graphs, the main object of study has been the homology of these complexes. We review the results known to date about the probabilistic behavior of the homology (and related structures) generated by these random complexes.

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Section: Persistent Homologymentioning

confidence: 99%

Topology of random geometric complexes: a survey

Bobrowski

Kahle

2018

J Appl. and Comput. Topology

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106

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“…(These are common phenomena for barcodes, and have been addressed theoretically in a number of studies (e.g. [8]). ) As a consequence, the RST procedure will not work for the H 1 points in this particular example.…”

Section: Example 1: Two Circlesmentioning

confidence: 99%

Modeling and replicating statistical topology and evidence for CMB nonhomogeneity

Adler

Agami

Pranav

2017

Proc. Natl. Acad. Sci. U.S.A.

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Under the banner of "big data," the detection and classification of structure in extremely large, high-dimensional, data sets are two of the central statistical challenges of our times. Among the most intriguing new approaches to this challenge is "TDA," or "topological data analysis," one of the primary aims of which is providing nonmetric, but topologically informative, preanalyses of data which make later, more quantitative, analyses feasible. While TDA rests on strong mathematical foundations from topology, in applications, it has faced challenges due to difficulties in handling issues of statistical reliability and robustness, often leading to an inability to make scientific claims with verifiable levels of statistical confidence. We propose a methodology for the parametric representation, estimation, and replication of persistence diagrams, the main diagnostic tool of TDA. The power of the methodology lies in the fact that even if only one persistence diagram is available for analysis-the typical case for big data applications-the replications permit conventional statistical hypothesis testing. The methodology is conceptually simple and computationally practical, and provides a broadly effective statistical framework for persistence diagram TDA analysis. We demonstrate the basic ideas on a toy example, and the power of the parametric approach to TDA modeling in an analysis of cosmic microwave background (CMB) nonhomogeneity.

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“…The main idea is to use topological features (eg, homology, Euler characteristic, persistent homology) as a "signature" for various types of complex high-dimensional data.Geometric complexes are used often in TDA to translate data points into a combinatorial-topological space, which in turn can be fed into a software algorithm that calculates its relevant topological properties. It is therefore desired to develop a solid statistical theory for geometric complexes (see eg, [2,6,10,35]), and an imperative part of this effort is to develop its probabilistic foundations (see eg, [1,4,12,37]). Most of the results on random geometric complexes and graphs to date have been studied for point processes in a Euclidean space.…”

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confidence: 99%

Random Čech complexes on Riemannian manifolds

Bobrowski

Oliveira

2018

Random Struct Algorithms

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In this paper we study the homology of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.

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Maximally persistent cycles in random geometric complexes

Cited by 56 publications

References 39 publications

Topology of random geometric complexes: a survey

Topology of random geometric complexes: a survey

Modeling and replicating statistical topology and evidence for CMB nonhomogeneity

Random Čech complexes on Riemannian manifolds

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