Tensors: geometry and applications

464 Pages·2012·3.83 MB·English

by  J. M. Landsberg

Tensors: Geometry
and Applications
J. M. Landsberg
Graduate Studies
in Mathematics
Volume 128
American Mathematical Society
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 informationand updates on this book, visit
www.ams.org/bookpages/gsm-128
Library of Congress Cataloging-in-Publication Data
Landsberg,J.M.
Tensors: geometryandapplications/J.M.Landsberg.
p.cm. —(Graduatestudiesinmathematics;v.128)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-6907-9(alk.paper)
1.Multilinearalgebra. 2.Tensorproducts. 3.Matrices. 4.Signaltheory(Telecommunica-
tion) I.Title.
QA199.5.L36 2011
512(cid:2).5—dc23
2011021758
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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 S 140
d
§6.4. Decomposing V⊗d as a GL(V)-module with the aid of S 144
d
§6.5. Decomposing Sd(A ⊗···⊗ A ) as a G = GL(A )×···×
1 n 1
GL(A )-module 149
n
§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 σ (Seg(PA×PB ×PC)) 199
4
§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 σ (Seg(PA ⊗···⊗PA )) 267
3 1 n
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(M ) ≤ 23 283
3,3,3
§11.5. Bl¨aser’s 5-Theorem 283
2
§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. Polynomialsincomplexitytheory,graphtheory,andstatistics315
§13.3. Definitions of VP, VNP, and other algebraic complexity
classes 317
§13.4. Complexity of perm and det 322
n n
§13.5. Immanants and their symmetries 328
§13.6. Geometric complexity theory approach to VP v. VNP 332
ws
§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