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Matroids and Canonical Forms: Theory and Applications University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2017 Matroids And Canonical Forms: Theory And Applications Gregory Henselman University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Applied Mathematics Commons, and the Mathematics Commons Recommended Citation Henselman, Gregory, "Matroids And Canonical Forms: Theory And Applications" (2017). Publicly Accessible Penn Dissertations. 2332. https://repository.upenn.edu/edissertations/2332 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/2332 For more information, please contact [email protected]. Matroids And Canonical Forms: Theory And Applications Abstract This document introduces a combinatorial theory of homology, a topological descriptor of shape. The past fifteen years have seen a steady advance in the use of techniques and principles from algebraic topology to address problems in the data sciences. This new subfield of opologicalT Data Analysis [TDA] seeks to extract robust qualitative features from large, noisy data sets. A primary tool in this new approach is the homological persistence module, which leverages the categorical structure of homological data to generate and relate shape descriptors across scales of measurement. We define a combinatorial analog to this structure in terms of matroid canonical forms. Our principle application is a novel algorithm to compute persistent homology, which improves time and memory performance by up to several orders of magnitude over current state of the art. Additional applications include new theorems in discrete, spectral, and algebraic Morse theory, which treats the geometry and topology of abstract space through the analysis of critical points, and a novel paradigm for matroid representation, via abelian categories. Our principle tool is elementary exchange, a combinatorial notion that relates linear and categorical duality with matroid complementarity. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Electrical & Systems Engineering First Advisor Robert W. Ghrist Keywords Combinatorial Optimization, Jordan Canonical Form, Matrix Factorization, Matroid Representation, Morse Theory, Persistent Homology Subject Categories Applied Mathematics | Mathematics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/2332 MATROIDS AND CANONICAL FORMS: THEORY AND APPLICATIONS Gregory F Henselman A DISSERTATION in Electrical and Systems Engineering Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2017 Supervisor of Dissertation Robert W Ghrist, Andrea Mitchell University Professor of Mathematics and Electri- cal and Systems Engineering Graduate Group Chairperson Alejandro Ribeiro, Rosenbluth Associate Professor of Electrical and Systems Engi- neering Dissertation Committee: Chair: Alejandro Ribeiro, Rosenbluth Associate Professor of Electrical and Systems Engineering Supervisor: Robert W Ghrist, Andrea Mitchell University Professor of Mathematics and Electrical and Systems Engineering Member: Rakesh Vohra, George A. Weiss and Lydia Bravo Weiss University Professor of Economics and Electrical and Systems Engineering MATROIDS AND CANONICAL FORMS: THEORY AND APPLICATIONS COPYRIGHT 2017 Gregory F Henselman For Linda iii DEDICATION GIATHGIAGIAMOUPOUMOUEDW SETOMUALOGIATONPAPPOUM OUPOUMOUEDWSETOPNEUMA TOUGIATHMHTERAMOUPOUMO UEDWSETHAGAPHTHSGIATONP ATERAMOUPOUMOUEDWSETH NELPIDATOUGIATHNADELFHM OUPOUKRATAEITONDROMOMA SKAIGIATONJAKOOPOIOSFWT IZEITAMATIAMASGIAKAITLINT OUOPOIOUHFILIAEINAIHKALU TERHMOUSUMBOULHGIAJANE THSOPOIASTOERGOEINAIAUTO iv ACKNOWLEDGEMENTS I would like to thank my advisor, Robert Ghrist, whose encouragement and support made this work possible. His passion for this field is inescapable, and his mentorship gave purpose to what would otherwise have become a host of burdens. A tremendous debt of gratitude is owed to Chad Giusti, whose sound judgement has been a guiding light for the past nine years, and whose mathematical insights first led me to consider research in computation. A similar debt is owed to the colleagues who shared in the challenges of the doctoral program: Jaree Hudson, Kevin Donahue, Nathan Perlmutter, Adam Burns, Eusebio Gardella, Justin Hilburn, Justin Curry, Elaine So, Tyler Kelly, Ryan Rogers, Yiqing Cai, Shiying Dong, Sarah Costrell, Iris Yoon, and Weiyu Huang. Thank you for your support in at the high points and the low. This work has benefitted enormously from interaction with mathematicians who took it upon themselves to help my endeavors. To Brendan Fong, David Lipsky, Michael Robinson, Matthew Wright, Radmila Sazdanovic, Sanjeevi Krishnan, PawelDlotko, Michael Lesnick, Sara Kalisnik, Amit Patel, and Vidit Nanda, thank you. Many faculty have made significant contributions to the content of this text, whether directly or indirectly. Invaluable help in the preparation of this text came from Ortwin Knorr, whose instruction is a constant presence in my writing. To Sergey Yuzvinsky, Charles Curtis, Hal Sadofsky, Christopher Phillips, Dev Sinha, Nicholas Proudfoot, Boris Botvinnik, David Levin, Peng Lu, Alexander Kleshchev, Yuan Xu, James Isenberg, Christopher Sinclair, Peter Gilkey, Santosh Venkatesh, Matthew Kahle, Mikael Vejdemo- v Johansson, Ulrich Bauer, Michael Kerber, Heather Harrington, Nina Otter, Vladimir Itskov, Carina Curto, Jonathan Cohen, Randall Kamien, Robert MacPherson, and Mark Goresky, thank you. Special thanks to Klaus Truemper, whose text opened the world of matroid decomposition to my imagination. Special thanks are due, also, to my dissertation committee, whose technical insights continue to excite new ideas for the possibilities of topology in complex systems, and whose coursework led directly to my work in combinatorics. Thank you to Mary Bachvarova, whose advice was decisive in my early graduate education, and whose friendship offered shelter from many storms. Finally, this research was made possible by the encouragement, some ten years ago, of my advisor Erin McNicholas, and of my friend and mentor Inga Johnson. Thank you for bringing mathematics to life, and for showing me the best of what it could be. Your knowledge put my feet on this path, and your faith was the reason I imagined following it. vi ABSTRACT MATROIDS AND CANONICAL FORMS: THEORY AND APPLICATIONS Gregory F Henselman Robert W Ghrist This document introduces a combinatorial theory of homology, a topological descriptor of shape. The past fifteen years have seen a steady advance in the use of techniques and principles from algebraic topology to address problems in the data sciences. This new subfield of Topological Data Analysis [TDA] seeks to extract robust qualitative features from large, noisy data sets. A primary tool in this new approach is the homological persistence module, which leverages the categorical structure of homological data to generate and relate shape descriptors across scales of measurement. We define a combinatorial analog to this structure in terms of matroid canonical forms. Our principle application is a novel algorithm to compute persistent homology, which improves time and memory performance by up to several orders of magnitude over current state of the art. Additional applications include new theorems in discrete, spectral, and algebraic Morse theory, which treats the geometry and topology of abstract space through the analysis of critical points, and a novel paradigm for matroid representation, via abelian categories. Our principle tool is elementary exchange, a combinatorial notion that relates linear and categorical duality with matroid complementarity. vii Contents 1 Introduction 1 2 Notation 14 I Canonical Forms 18 3 Background: Matroids 19 3.1 Independence, rank, and closure . 19 3.2 Circuits . 22 3.3 Minors . 25 3.4 Modularity . 27 3.5 Basis Exchange . 28 3.6 Duality . 32 4 Modularity 33 4.1 Generators . 33 4.2 Minimal Bases . 38 viii 5 Canonical Forms 40 5.1 Modular Filtrations . 41 5.2 Nilpotent Canonical Forms . 43 5.3 Graded Canonical Forms . 49 5.4 Generalized Canonical Forms . 54 II Algorithms 61 6 Algebraic Foundations 62 6.1 Biproducts . 62 6.2 Idempotents . 64 6.3 Arrays . 68 6.4 Kernels . 70 6.4.1 The Splitting Lemma . 72 6.4.2 The Exchange Lemma . 74 6.4.3 Idempotents . 81 6.4.4 Exchange . 82 6.5 The Schur Complement . 84 6.5.1 Diagramatic Complements . 89 6.6 M¨obiusInversion . 90 7 Exchange Formulae 95 7.1 Relations . 95 7.2 Formulae . 98 ix 8 Exchange Algorithms 105 8.1 LU Decomposition . 105 8.2 Jordan Decomposition . 110 8.3 Filtered Exchange . 117 8.4 Block exchange . 121 III Applications 124 9 Efficient Homology Computation 125 9.1 The linear complex . 125 9.2 The linear persistence module . 129 9.3 Homological Persistence . 131 9.4 Optimizations . 133 9.4.1 Related work . 133 9.4.2 Basis selection . 135 9.4.3 Input reduction . 139 9.5 Benchmarks . 142 10 Morse Theory 145 10.1 Smooth Morse Theory . 147 10.2 Discrete and Algebraic Morse Theory . 151 10.3 Results . 157 11 Abelian Matroids 176 11.1 Linear Matroids . 177 x 11.2 Covariant Matroids . 181 xi List of Tables 9.1 Wall-time in seconds. Comparison results for eleg, Klein, HIV, drag 2, and random are reported for computation on the cluster. Results for fract r are reported for computation on the shared memory system. 143 9.2 Max Heap in GB. Comparison results for eleg, Klein, HIV, drag 2, and random
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