Persistent Homology under Non-uniform Error

Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbations vary in their magnitude across the domain.

[1]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[2]  A. Robinson.  I. Introduction , 1991 .

[3]  Moo K. Chung,et al.  Persistence Diagrams of Cortical Surface Data , 2009, IPMI.

[4]  R. Ho.  Algebraic Topology , 2022 .

[5]  Herbert Edelsbrunner,et al.  Topological Persistence and Simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[6]  Vin de Silva,et al.  On the Local Behavior of Spaces of Natural Images , 2007, International Journal of Computer Vision.

[7]  Eduardo A. Fierro.  Structures , 2003, Composite Architecture.

[8]  Daniela Giorgi,et al.  Retrieval of trademark images by means of size functions , 2006, Graph. Model..

[9]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2007, Discret. Comput. Geom..

[10]  Herbert Edelsbrunner,et al.  Quantifying Transversality by Measuring the Robustness of Intersections , 2009, Found. Comput. Math..

[11]  Earl F. Glynn,et al.  Comparison of Pattern Detection Methods in Microarray Time Series of the Segmentation Clock , 2008, PloS one.

[12]  David Cohen-Steiner,et al.  Extending Persistence Using Poincaré and Lefschetz Duality , 2009, Found. Comput. Math..

[13]  Alexander Russell,et al.  Computational topology: ambient isotopic approximation of 2-manifolds , 2003, Theor. Comput. Sci..

[14]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2005, Discret. Comput. Geom..

[15]  Herbert Edelsbrunner,et al.  Homology and Robustness of Level and Interlevel Sets , 2011, ArXiv.

[16]  Vin de Silva,et al.  Coverage in sensor networks via persistent homology , 2007 .

[17]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[18]  Gunnar E. Carlsson,et al.  Zigzag Persistence , 2008, Found. Comput. Math..