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Tensors: Geometry and Applications Tensors: Geometry and Applications J. M. Landsberg Graduate Studies in Mathematics Volume 128 American Mathematical Society http://dx.doi.org/10.1090/gsm/128 Tensors: Geometry and Applications Tensors: Geometry and Applications J. M. Landsberg Graduate Studies in Mathematics Volume 128 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Rafe Mazzeo Martin Scharlemann Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 15–01, 15A69, 68Q17, 14M17, 94A12, 94A13, 20G05, 62E10, 14N05. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-128 Library of Congress Cataloging-in-Publication Data Landsberg,J.M. Tensors : geometry and applications / J. M. Landsberg. p. cm. — (Graduate studies in mathematics ; v. 128) Includes bibliographical references and index. ISBN 978-0-8218-6907-9 (alk. paper) 1. Multilinear algebra. 2. Tensor products. 3. Matrices. 4. Signal theory (Telecommunica- tion) I. Title. QA199.5.L36 2011 512.5—dc23 2011021758 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 Contents Preface xi §0.1. Usage xi §0.2. Overview xii §0.3. Clash of cultures xvii §0.4. Further reading xviii §0.5. Conventions, acknowledgments xix Part 1. Motivation from applications, multilinear algebra, and elementary results Chapter 1. Introduction 3 §1.1. The complexity of matrix multiplication 5 §1.2. Definitions from multilinear algebra 7 §1.3. Tensor decomposition 11 §1.4. P v. NP and algebraic variants 18 §1.5. Algebraic statistics and tensor networks 22 §1.6. Geometry and representation theory 25 Chapter 2. Multilinear algebra 27 §2.1. Rust removal exercises 28 §2.2. Groups and representations 30 §2.3. Tensor products 32 §2.4. The rank and border rank of a tensor 35 §2.5. Examples of invariant tensors 39 v vi Contents §2.6. Symmetric and skew-symmetric tensors 40 §2.7. Polynomials on the space of matrices 48 §2.8. Decomposition of V ⊗3 52 §2.9. Appendix: Basic definitions from algebra 55 §2.10. Appendix: Jordan and rational canonical form 57 §2.11. Appendix: Wiring diagrams 58 Chapter 3. Elementary results on rank and border rank 67 §3.1. Ranks of tensors 68 §3.2. Symmetric rank 70 §3.3. Uniqueness of CP decompositions 72 §3.4. First tests of border rank: flattenings 74 §3.5. Symmetric border rank 76 §3.6. Partially symmetric tensor rank and border rank 78 §3.7. Two useful techniques for determining border rank 79 §3.8. Strassen’s equations and variants 81 §3.9. Equations for small secant varieties 86 §3.10. Equations for symmetric border rank 88 §3.11. Tensors in C2⊗Cb⊗Cc 91 Part 2. Geometry and representation theory Chapter 4. Algebraic geometry for spaces of tensors 97 §4.1. Diagnostic test for those familiar with algebraic geometry 98 §4.2. First definitions 98 §4.3. Examples of algebraic varieties 101 §4.4. Defining equations of Veronese re-embeddings 105 §4.5. Grassmannians 106 §4.6. Tangent and cotangent spaces to varieties 107 §4.7. G-varieties and homogeneous varieties 110 §4.8. Exercises on Jordan normal form and geometry 111 §4.9. Further information regarding algebraic varieties 111 Chapter 5. Secant varieties 117 §5.1. Joins and secant varieties 118 §5.2. Geometry of rank and border rank 120 §5.3. Terracini’s lemma and first consequences 122 Contents vii §5.4. The polynomial Waring problem 125 §5.5. Dimensions of secant varieties of Segre varieties 127 §5.6. Ideas of proofs of dimensions of secant varieties for triple Segre products 130 §5.7. BRPP and conjectures of Strassen and Comon 132 Chapter 6. Exploiting symmetry: Representation theory for spaces of tensors 137 §6.1. Schur’s lemma 138 §6.2. Finite groups 139 §6.3. Representations of the permutation group Sd 140 ⊗d §6.4. Decomposing V as a GL(V )-module with the aid of Sd 144 d §6.5. Decomposing S (A1⊗···⊗An)asaG = GL(A1) ×···× GL(An)-module 149 §6.6. Characters 151 §6.7. The Littlewood-Richardson rule 153 §6.8. Weights and weight spaces: a generalization of eigenvalues and eigenspaces 158 §6.9. Homogeneous varieties 164 §6.10. Ideals of homogeneous varieties 167 §6.11. Symmetric functions 170 Chapter 7. Tests for border rank: Equations for secant varieties 173 §7.1. Subspace varieties and multilinear rank 174 §7.2. Additional auxiliary varieties 177 §7.3. Flattenings 179 §7.4. Inheritance 184 §7.5. Prolongation and multiprolongation 186 §7.6. Strassen’s equations, applications and generalizations 192 §7.7. Equations for σ4(Seg(PA × PB × PC)) 199 §7.8. Young flattenings 202 Chapter 8. Additional varieties useful for spaces of tensors 207 §8.1. Tangential varieties 208 §8.2. Dual varieties 211 §8.3. The Pascal determinant 214 §8.4. Differential invariants of projective varieties 215 §8.5. Stratifications of PV ∗ via dual varieties 219 viii Contents §8.6. The Chow variety of zero cycles and its equations 221 §8.7. The Fano variety of linear spaces on a variety 226 Chapter 9. Rank 229 §9.1. Remarks on rank for arbitrary varieties 229 §9.2. Bounds on symmetric rank 231 §9.3. Examples of classes of polynomials and their ranks 235 Chapter 10. Normal forms for small tensors 243 §10.1. Vector spaces with a finite number of orbits 244 §10.2. Vector spaces where the orbits can be explicitly parametrized246 §10.3. Points in C2⊗Cb⊗Cc 248 §10.4. Ranks and border ranks of elements of S3C3 258 §10.5. Tensors in C3⊗C3⊗C3 260 §10.6. Normal forms for C2⊗S2W 261 §10.7. Exercises on normal forms for general points on small secant varieties 262 §10.8. Limits of secant planes 262 §10.9. Limits for Veronese varieties 264 §10.10. Ranks and normal forms in σ3(Seg(PA1⊗···⊗PAn)) 267 Part 3. Applications Chapter 11. The complexity of matrix multiplication 275 §11.1. “Real world” issues 276 §11.2. Failure of the border rank version of Strassen’s conjecture 276 §11.3. Finite group approach to upper bounds 281 §11.4. R(M3,3,3) ≤ 23 283 § 5 11.5. Bl¨aser’s 2 -Theorem 283 §11.6. The Brockett-Dobkin Theorem 285 §11.7. Multiplicative complexity 287 Chapter 12. Tensor decomposition 289 §12.1. Cumulants 290 §12.2. Blind deconvolution of DS-CMDA signals 293 §12.3. Uniqueness results coming from algebraic geometry 299 §12.4. Exact decomposition algorithms 302 §12.5. Kruskal’s theorem and its proof 305 Contents ix Chapter 13. P v. NP 311 §13.1. Introduction to complexity 312 §13.2. Polynomials in complexity theory, graph theory, and statistics315 §13.3. Definitions of VP, VNP, and other algebraic complexity classes 317 § 13.4. Complexity of permn and detn 322 §13.5. Immanants and their symmetries 328 §13.6. Geometric complexity theory approach to VPws v. VNP 332 §13.7. Other complexity classes via polynomials 339 §13.8. Vectors of minors and homogeneous varieties 340 §13.9. Holographic algorithms and spinors 347 Chapter 14. Varieties of tensors in phylogenetics and quantum mechanics 357 §14.1. Tensor network states 357 §14.2. Algebraic statistics and phylogenetics 364 Part 4. Advanced topics Chapter 15. Overview of the proof of the Alexander-Hirschowitz theorem 373 §15.1. The semiclassical cases 374 §15.2. The Alexander-Hirschowitz idea for dealing with the remaining cases 377 Chapter 16. Representation theory 381 §16.1. Basic definitions 381 §16.2. Casimir eigenvalues and Kostant’s theorem 385 §16.3. Cohomology of homogeneous vector bundles 390 §16.4. Equations and inheritance in a more general context 393 Chapter 17. Weyman’s method 395 §17.1. Ideals and coordinate rings of projective varieties 396 §17.2. Koszul sequences 397 §17.3. The Kempf-Weyman method 400 §17.4. Subspace varieties 404 Hints and answers to selected exercises 409 Bibliography 415 Index 433 Preface Tensors are ubiquitous in the sciences. One reason for their ubiquity is that they provide a useful way to organize data. Geometry is a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: as a classroom textbook, a reference work for researchers, and a research manuscript. 0.1. Usage Classroom uses. Here are several possible courses one could give from this text: (1) The first part of this text is suitable for an advanced course in multilinear algebra—it provides a solid foundation for the study of tensors and contains numerous applications, exercises, and exam- ples. Such a course would cover Chapters 1–3 and parts of Chapters 4–6. (2) For a graduate course on the geometry of tensors not assuming algebraic geometry, one can cover Chapters 1, 2, and 4–8 skipping §§2.9–12, 4.6, 5.7, 6.7 (except Pieri), 7.6 and 8.6–8.
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