Persistent homology for random fields and complexes

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices and, in most detail, the algebraic topology of the excursion sets of random elds.

[1]  Leopold Flatto,et al.  Random coverings , 1977 .

[2]  Oleg Viro,et al.  Some integral calculus based on Euler characteristic , 1988 .

[3]  B. Okun,et al.  Euler characteristic in percolation theory , 1990 .

[4]  Klaus Mecke,et al.  Euler characteristic and related measures for random geometric sets , 1991 .

[5]  Andrew Ranicki,et al.  Ends of Complexes , 1996 .

[6]  Étale Groupoids,et al.  Homology Theory , 2009 .

[7]  Robert Ghrist,et al.  Finding Topology in a Factory: Configuration Spaces , 2002, Am. Math. Mon..

[8]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

[9]  Robert J. Adler,et al.  Euler characteristics for Gaussian fields on manifolds , 2003 .

[10]  Westone,et al.  Home Page , 2004, 2022 2nd International Conference on Intelligent Cybernetics Technology & Applications (ICICyTA).

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

[12]  Vin de Silva,et al.  HOMOLOGICAL SENSOR NETWORKS , 2005 .

[13]  Peter Bubenik,et al.  A statistical approach to persistent homology , 2006, math/0607634.

[14]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[15]  R. Ghrist.  Barcodes: The persistent topology of data , 2007 .

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

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

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

[19]  Yuliy Baryshnikov,et al.  Target Enumeration via Euler Characteristic Integrals , 2009, SIAM J. Appl. Math..

[20]  G. Carlsson,et al.  Statistical topology via Morse theory, persistence and nonparametric estimation , 2009, 0908.3668.

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

[22]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[23]  Yuliy Baryshnikov,et al.  Euler integration over definable functions , 2009, Proceedings of the National Academy of Sciences.

[24]  Matthew Kahle,et al.  Random Geometric Complexes , 2009, Discret. Comput. Geom..

[25]  Frédéric Chazal,et al.  Geometric Inference for Measures based on Distance Functions , 2011 .

[26]  Moo K. Chung,et al.  Topological Data Analysis , 2012 .

[27]  Stephen Smale,et al.  A Topological View of Unsupervised Learning from Noisy Data , 2011, SIAM J. Comput..

[28]  Frédéric Chazal,et al.  Geometric Inference for Probability Measures , 2011, Found. Comput. Math..

[29]  Omer Bobrowski,et al.  EULER INTEGRATION OF GAUSSIAN RANDOM FIELDS AND PERSISTENT HOMOLOGY , 2010, 1003.5175.

[30]  H. Edelsbrunner,et al.  Persistent Homology — a Survey , 2022 .

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