Proximity of Persistence Modules and their Diagrams

被引:191
作者
Chazal, Frederic [1 ]
Cohen-Steiner, David
Glisse, Marc
Guibas, Leonidas J.
Oudot, Steve Y. [1 ]
机构
[1] INRIA Saclay Ile France, F-91893 Orsay, France
来源
PROCEEDINGS OF THE TWENTY-FIFTH ANNUAL SYMPOSIUM ON COMPUTATIONAL GEOMETRY (SCG'09) | 2009年
基金
美国国家科学基金会;
关键词
Topological persistence; Stability; Persistence diagram; Discretization; Topological Data Analysis;
D O I
10.1145/1542362.1542407
中图分类号
O29 [应用数学];
学科分类号
070104 [应用数学];
摘要
Topological persistence has proven to be a key concept for the study of real-valued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable spaces. In this paper, we present new stability results that do not suffer from the above restrictions. Furthermore, by working at an algebraic level directly, we make 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. Along the way, we extend the definition of persistence diagram to a larger setting, introduce the notions of discretization of a persistence module and associated pixelization map, define a proximity measure between persistence modules, and show how to interpolate between persistence modules, thereby lending a more analytic character to this otherwise algebraic setting. We believe these new theoretical concepts and tools shed new light on the theory of persistence, in addition to simplifying proofs and enabling new applications.
引用
收藏
页码:237 / 246
页数:10
相关论文
共 21 条
[1]
[Anonymous], 2001, ALGEBRAIC TOPOLOGY
[2]
ATTALI D, WORKSH TOPOIN VIS 09
[3]
Inferring local homology from sampled stratified spaces [J].
Bendich, Paul ;
Cohen-Steiner, David ;
Edelsbrunner, Herbert ;
Harer, John ;
Morozov, Dmitriy .
48TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE, PROCEEDINGS, 2007, :536-546
[4]
Multidimensional size functions for shape comparison [J].
Biasotti, S. ;
Cerri, A. ;
Frosini, P. ;
Giorgi, D. ;
Landi, C. .
JOURNAL OF MATHEMATICAL IMAGING AND VISION, 2008, 32 (02) :161-179
[5]
CAGLIARI F, 2007, TITLE ONE DIMENSIONA
[6]
Carlsson G., 2005, INT J SHAPE MODELING, V11, P149, DOI [DOI 10.1145/1057432.1057449, DOI 10.1142/S0218654305000761, 10.1142/S0218654305000761]
[7]
CARLSSON G, 2007, P 23 ACM S COMP GEOM, P184, DOI DOI 10.1145/1247069.1247105
[8]
CHAZAL F, 2007, GEOMETRIC INFE UNPUB
[9]
CHAZAL F, 2008, 6568 INRIA
[10]
Chazal F, 2009, PROCEEDINGS OF THE TWENTIETH ANNUAL ACM-SIAM SYMPOSIUM ON DISCRETE ALGORITHMS, P1021