Matroid Filtrations and Computational Persistent Homology

This technical report introduces a novel approach to efficient computation in homological algebra over fields, with particular emphasis on computing the persistent homology of a filtered topological cell complex. The algorithms here presented rely on a novel relationship between discrete Morse theory, matroid theory, and classical matrix factorizations. We provide background, detail the algorithms, and benchmark the software implementation in the Eirene package.

[1]  Michael Robinson,et al.  Topological Signal Processing , 2014 .

[2]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[3]  Ulrich Bauer,et al.  Persistence in discrete Morse theory , 2011 .

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

[5]  Ulrich Bauer,et al.  Distributed Computation of Persistent Homology , 2014, ALENEX.

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

[7]  Günter M. Ziegler,et al.  Oriented Matroids , 2017, Handbook of Discrete and Computational Geometry, 2nd Ed..

[8]  B. Sturmfels.  Oriented Matroids , 1993 .

[9]  R. Forman.  Morse Theory for Cell Complexes , 1998 .

[10]  Jim Lawrence,et al.  Oriented matroids , 1978, J. Comb. Theory B.

[11]  Emil Sköldberg,et al.  Morse theory from an algebraic viewpoint , 2005 .

[12]  Dmitry N. Kozlov,et al.  Discrete Morse Theory for free chain complexes , 2005, ArXiv.

[13]  Robert Ghrist,et al.  Discrete Morse Theory for Computing Cellular Sheaf Cohomology , 2013, Foundations of Computational Mathematics.

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

[15]  E. Pastalkova,et al.  Clique topology reveals intrinsic geometric structure in neural correlations , 2015, Proceedings of the National Academy of Sciences.

[16]  Dmitry N. Kozlov,et al.  Combinatorial Algebraic Topology , 2007, Algorithms and computation in mathematics.

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

[18]  Klaus Truemper,et al.  Matroid decomposition , 1992 .

[19]  Vidit Nanda,et al.  Discrete Morse theory and classifying spaces , 2016, Advances in Mathematics.

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

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

[22]  Pawel Dlotko,et al.  Coreduction Homology Algorithm for Regular CW-Complexes , 2011, Discret. Comput. Geom..

[23]  Michelle L. Wachs,et al.  Torsion in the matching complex and chessboard complex , 2004 .

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