A Brief History of Persistence

Persistent homology is currently one of the more widely known tools from computational topology and topological data analysis. We present in this note a brief survey on the evolution of the subject. The goal is to highlight the main ideas, starting from the subject's computational inception more than 20 years ago, to the more modern categorical and representation-theoretic point of view.

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

[2]  W. Crawley-Boevey.  Decomposition of pointwise finite-dimensional persistence modules , 2012, 1210.0819.

[3]  Patrizio Frosini,et al.  Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions , 2008, ArXiv.

[4]  Herbert Edelsbrunner,et al.  An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere , 1995, Comput. Aided Geom. Des..

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

[6]  Steve Oudot,et al.  Towards persistence-based reconstruction in euclidean spaces , 2007, SCG '08.

[7]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[8]  Gunnar Carlsson,et al.  Numeric invariants from multidimensional persistence , 2017, J. Appl. Comput. Topol..

[9]  Herbert Edelsbrunner,et al.  Topological persistence and simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[10]  Emerson G. Escolar,et al.  Persistence Modules on Commutative Ladders of Finite Type , 2014, Discret. Comput. Geom..

[11]  Heather A. Harrington,et al.  Stratifying Multiparameter Persistent Homology , 2017, SIAM J. Appl. Algebra Geom..

[12]  Jean-Guillaume Dumas,et al.  Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms , 2003, Algebra, Geometry, and Software Systems.

[13]  Mikael Vejdemo-Johansson,et al.  javaPlex: A Research Software Package for Persistent (Co)Homology , 2014, ICMS.

[14]  Leonidas J. Guibas,et al.  Persistence Barcodes for Shapes , 2005, Int. J. Shape Model..

[15]  Mason A. Porter,et al.  A roadmap for the computation of persistent homology , 2015, EPJ Data Science.

[16]  Robert Ghrist,et al.  Elementary Applied Topology , 2014 .

[17]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[18]  Cary Webb.  Decomposition of graded modules , 1985 .

[19]  Steve Oudot,et al.  The Structure and Stability of Persistence Modules , 2012, Springer Briefs in Mathematics.

[20]  Primoz Skraba,et al.  Zigzag persistent homology in matrix multiplication time , 2011, SoCG '11.

[21]  Michal Adamaszek,et al.  On Vietoris–Rips complexes of ellipses , 2019, Journal of Topology and Analysis.

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

[23]  Vin de Silva,et al.  Metrics for Generalized Persistence Modules , 2013, Found. Comput. Math..

[24]  Afra Zomorodian,et al.  Computing Persistent Homology , 2004, SCG '04.

[25]  Leonidas J. Guibas,et al.  Proximity of persistence modules and their diagrams , 2009, SCG '09.

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

[27]  Michael Lesnick,et al.  The Theory of the Interleaving Distance on Multidimensional Persistence Modules , 2011, Found. Comput. Math..

[28]  J. Latschev.  Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold , 2001 .

[29]  P. Gabriel.  Unzerlegbare Darstellungen I , 1972 .

[30]  Michal Adamaszek,et al.  The Vietoris-Rips complexes of a circle , 2015, ArXiv.

[31]  Wojciech Chachólski,et al.  Multidimensional Persistence and Noise , 2015, Foundations of Computational Mathematics.

[32]  L. Vietoris.  Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen , 1927 .

[33]  Frédéric Chazal,et al.  Weak feature size and persistent homology: computing homology of solids in Rn from noisy data samples , 2005, SCG.

[34]  Patrizio Frosini,et al.  Measuring shapes by size functions , 1992, Other Conferences.

[35]  Michael Lesnick,et al.  Interactive Visualization of 2-D Persistence Modules , 2015, ArXiv.

[36]  Afra Zomorodian,et al.  Computing Multidimensional Persistence , 2010, J. Comput. Geom..

[37]  Kevin P. Knudson.  A refinement of multi-dimensional persistence , 2007, 0706.2608.

[38]  Konstantin Mischaikow,et al.  Morse Theory for Filtrations and Efficient Computation of Persistent Homology , 2013, Discret. Comput. Geom..

[39]  Michael Lesnick,et al.  Algebraic Stability of Zigzag Persistence Modules , 2016, Algebraic & Geometric Topology.

[40]  Virginia Vassilevska Williams,et al.  Multiplying matrices faster than coppersmith-winograd , 2012, STOC '12.

[41]  Steve Oudot,et al.  Persistence Theory - From Quiver Representations to Data Analysis , 2015, Mathematical surveys and monographs.

[42]  Afra Zomorodian,et al.  The Theory of Multidimensional Persistence , 2007, SCG '07.

[43]  Dmitriy Morozov,et al.  Dualities in persistent (co)homology , 2011, ArXiv.

[44]  P. Frosini,et al.  Size homotopy groups for computation of natural size distances , 1999 .

[45]  Vin de Silva,et al.  The observable structure of persistence modules , 2014, 1405.5644.

[46]  Peter Bubenik,et al.  Categorification of Persistent Homology , 2012, Discret. Comput. Geom..