Quantifying Transversality by Measuring the Robustness of Intersections
TLDR
Persistent homology assigns to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then it is proved that the robustness is stable.Abstract:Ā
By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then we prove that the robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.read more
Citations
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Journal Article
Statistical topological data analysis using persistence landscapes
TL;DR: In this article, the persistence landscape is defined as a topological summary for data that is easy to combine with tools from statistics and machine learning, in contrast to the standard topological summaries.
Journal ArticleDOI
Computing Robustness and Persistence for Images
TL;DR: A fast hierarchical algorithm is given using the dual complexes of oct-tree approximations of the function to study 3-dimensional images of plant root systems and the structure of the homology classes and their robustness, over all level and interlevel sets, can be visualized by a triangular diagram of dots.
Journal ArticleDOI
Homology and robustness of level and interlevel sets
TL;DR: In this paper, the robustness of the homology classes under perturbations of a continuous function was quantified using well groups, and the ranks of these groups from the same extended persistence diagram were shown.
Book ChapterDOI
The Stability of the Apparent Contour of an Orientable 2-Manifold
Herbert Edelsbrunner,Herbert Edelsbrunner,Herbert Edelsbrunner,Dmitriy Morozov,Amit Patel,Amit Patel +5 more
TL;DR: The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f :š¯•„ā†’ā„¯2, is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent.
Proceedings ArticleDOI
2D Vector Field Simplification Based on Robustness
TL;DR: A novel simplification scheme derived from the recently introduced topological notion of robustness is proposed, which provides a complementary view on flow structure compared to the traditional topological-skeleton-based approaches.
References
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Book
The Logic of Scientific Discovery
TL;DR: The Open Society and Its Enemies as discussed by the authors is regarded as one of Popper's most enduring books and contains insights and arguments that demand to be read to this day, as well as many of the ideas in the book.
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Stable mappings and their singularities
TL;DR: In this article, the Whitney C? topology is used to classify singularities on 2-manifolds. But the Thom-Boardman invariants are not included in this classification.
Journal ArticleDOI
Topological Persistence and Simplification
Edelsbrunner,Letscher,Zomorodian +2 more
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.
Book
Computational Topology: An Introduction
TL;DR: In this article, the authors present an introduction to the field of computational topology, combining concepts from topology and algorithms, and the main approach is the discovery of topology through algorithms.