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Proximity of persistence modules and their diagrams
TLDR
This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.
The Structure and Stability of Persistence Modules
This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such
The Gudhi Library: Simplicial Complexes and Persistent Homology
TLDR
The main algorithmic and design choices that have been made to represent complexes and compute persistent homology in the Gudhi library are presented and benchmarks for the code are provided.
Metric graph reconstruction from noisy data
TLDR
This work presents a novel algorithm that takes as an input such a data set, and outputs the underlying metric graph with guarantees, and implements the algorithm, and evaluates its performance on a variety of real world data sets.
Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra
TLDR
It is proved that the set of lines tangent to four possibly intersecting convex polyhedra in $\mathbb{R}^3$ with a total of $n$ edges consists of $\Theta(n^2)$ connected components in the worst case.
Convergence rates for persistence diagram estimation in topological data analysis
TLDR
It is shown that the use of persistent homology can be naturally considered in general statistical frameworks and established convergence rates of persistence diagrams associated to data randomly sampled from any compact metric space to a well defined limit diagram encoding the topological features of the support of the measure from which the data have been sampled.
Optimizing persistent homology based functions
TLDR
This work proposes a general framework that allows us to define and compute gradients for persistence-based functions in a very simple way, and provides a simple, explicit and sufficient condition for convergence of stochastic subgradient methods for such functions.
Farthest-polygon Voronoi diagrams
Persistence-sensitive simplication of functions on surfaces in linear time
Persistence provides a way of grading the importance of homological features in the sublevel sets of a real-valued function. Following the definition given by Edelsbrunner, Morozov and Pascucci, an
Cost-Optimal Trees for Ray Shooting
TLDR
A cost measure for space decompositions like kd-trees and octrees is developed, and proved it to reliably predict the average cost of ray shooting.
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