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JournalISSN: 0179-5376

Discrete and Computational Geometry 

Springer Science+Business Media
About: Discrete and Computational Geometry is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Polytope & Convex hull. It has an ISSN identifier of 0179-5376. Over the lifetime, 2568 publications have been published receiving 78044 citations. The journal is also known as: Geometry (New York. Print) & Discrete and computational geometry (Print).


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Journal ArticleDOI
TL;DR: Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
Abstract: We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility.

1,671 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology over a polynomial ring of a particular graded module.
Abstract: We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of S3 and Z2 coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domain in any dimension.

1,528 citations

Journal ArticleDOI
TL;DR: The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane and it is proved that under mild assumptions on the function, the persistence diagram is stable.
Abstract: The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes.

1,078 citations

Journal ArticleDOI
TL;DR: The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.
Abstract: We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ . These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.

799 citations

Journal ArticleDOI
TL;DR: This work considers the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space and shows how to “learn” the homology of the sub manifold with high confidence.
Abstract: Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces. We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question.

680 citations

Performance
Metrics
No. of papers from the Journal in previous years
YearPapers
202327
202278
2021154
202095
201976
201882