Tensors : geometry and applications:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2012
|
| Schriftenreihe: | Graduate studies in mathematics
128 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024140994&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024140994&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Includes bibliographical references and index |
| Umfang: | XX, 439 S. Ill., graph. Darst. |
| ISBN: | 9780821869079 0821869078 |
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| 245 | 1 | 0 | |a Tensors : geometry and applications |c J. M. Landsberg |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2012 | |
| 300 | |a XX, 439 S. |b Ill., graph. Darst. | ||
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| 490 | 1 | |a Graduate studies in mathematics |v 128 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Multilinear algebra | |
| 650 | 4 | |a Tensor products | |
| 650 | 4 | |a Matrices | |
| 650 | 4 | |a Signal theory (Telecommunication) | |
| 650 | 7 | |a Linear and multilinear algebra; matrix theory / Instructional exposition (textbooks, tutorial papers, etc.) |2 msc | |
| 650 | 7 | |a Linear and multilinear algebra; matrix theory / Basic linear algebra / Multilinear algebra, tensor products |2 msc | |
| 650 | 7 | |a Information and communication, circuits / Communication, information / Signal theory (characterization, reconstruction, filtering, etc.) |2 msc | |
| 650 | 7 | |a Information and communication, circuits / Communication, information / Detection theory |2 msc | |
| 650 | 7 | |a Statistics / Distribution theory / Characterization and structure theory |2 msc | |
| 650 | 7 | |a Algebraic geometry / Projective and enumerative geometry / Projective techniques |2 msc | |
| 650 | 4 | |a Statistik | |
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Datensatz im Suchindex
| _version_ | 1819310544155312128 |
|---|---|
| adam_text | 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.
Ρ ν. ΝΡ
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
vi
Contents
§2.6. Symmetrie
and skew-symmetric tensors
40
§2.7.
Polynomials on the space of matrices
48
§2.8.
Decomposition of Vm
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(g)Cb(g)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
Θ^
140
§6.4.
Decomposing V®d as a GL(F)-module with the aid of <5d
144
§6.5.
Decomposing Sd(Ai<S)
· · ■
® An) as
a G
=
GL(A )
χ
· ■ ·
χ
GL(A„)-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 a4(Seg(YA
xPßx 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 FV* via dual varieties
219
viu
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 parametrized
246
§10.3.
Points in C2®Cb<g>Cc
248
§10.4.
Ranks and border ranks of elements of 53C3
258
§10.5.
Tensors in
C3®C3<8>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 a^(Seg{WAi®
·■■
® FAn))
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.
К(Мздз)<23
283
§11.5.
Blaser s f-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.
Ρ ν.
NP
311
§13.1.
Introduction to complexity
312
§13.2.
Polynomials in complexity theory, graph theory, and statistics
315
§13.3.
Definitions of
VP, VNP,
and other algebraic complexity
classes
317
§13.4.
Complexity of permn and
det„
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
Tensors
are ubiquitous in the sciences. The geometry of tensors is both a powerful
tool for extracting information from data sets, and a beautiful subject in its own
right.This book has three intended uses: a classroom textbook, a reference work for
researchers in the sciences, and an account of classical and modern results in (aspects
of) the theory that will be of interest to researchers in geometry. For classroom
use, there is a modern introduction to multilinear algebra and to the geometry and
representation theory needed to study tensors, including a large number of exer¬
cises. For researchers in the sciences, there is information on tensors in table format
for easy reference and a summary of the state of the art in elementary language.
This is the first book containing many classical results regarding tensors. Particular
applications treated in the book include the complexity of matrix multiplication,
Ρ
versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers,
there is material on secant varieties, G-varieties, spaces with finitely many orbits
and how these objects arise in applications, discussions of numerous open questions
in geometry arising in applications, and expositions of advanced topics such as the
proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for
computing syzygies.
|
| any_adam_object | 1 |
| author | Landsberg, Joseph M. 1963- |
| author_GND | (DE-588)14180677X |
| author_facet | Landsberg, Joseph M. 1963- |
| author_role | aut |
| author_sort | Landsberg, Joseph M. 1963- |
| author_variant | j m l jm jml |
| building | Verbundindex |
| bvnumber | BV039122471 |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)733546583 (DE-599)BVBBV039122471 |
| dewey-full | 512/.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.5 |
| dewey-search | 512/.5 |
| dewey-sort | 3512 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV039122471 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T15:50:32Z |
| institution | BVB |
| isbn | 9780821869079 0821869078 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024140994 |
| oclc_num | 733546583 |
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| physical | XX, 439 S. Ill., graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spellingShingle | Landsberg, Joseph M. 1963- Tensors : geometry and applications Graduate studies in mathematics Multilinear algebra Tensor products Matrices Signal theory (Telecommunication) Linear and multilinear algebra; matrix theory / Instructional exposition (textbooks, tutorial papers, etc.) msc Linear and multilinear algebra; matrix theory / Basic linear algebra / Multilinear algebra, tensor products msc Information and communication, circuits / Communication, information / Signal theory (characterization, reconstruction, filtering, etc.) msc Information and communication, circuits / Communication, information / Detection theory msc Statistics / Distribution theory / Characterization and structure theory msc Algebraic geometry / Projective and enumerative geometry / Projective techniques msc Statistik Signaltheorie (DE-588)4054945-8 gnd Tensor (DE-588)4184723-4 gnd Multilineare Algebra (DE-588)4416303-4 gnd |
| subject_GND | (DE-588)4054945-8 (DE-588)4184723-4 (DE-588)4416303-4 |
| title | Tensors : geometry and applications |
| title_auth | Tensors : geometry and applications |
| title_exact_search | Tensors : geometry and applications |
| title_full | Tensors : geometry and applications J. M. Landsberg |
| title_fullStr | Tensors : geometry and applications J. M. Landsberg |
| title_full_unstemmed | Tensors : geometry and applications J. M. Landsberg |
| title_short | Tensors : geometry and applications |
| title_sort | tensors geometry and applications |
| topic | Multilinear algebra Tensor products Matrices Signal theory (Telecommunication) Linear and multilinear algebra; matrix theory / Instructional exposition (textbooks, tutorial papers, etc.) msc Linear and multilinear algebra; matrix theory / Basic linear algebra / Multilinear algebra, tensor products msc Information and communication, circuits / Communication, information / Signal theory (characterization, reconstruction, filtering, etc.) msc Information and communication, circuits / Communication, information / Detection theory msc Statistics / Distribution theory / Characterization and structure theory msc Algebraic geometry / Projective and enumerative geometry / Projective techniques msc Statistik Signaltheorie (DE-588)4054945-8 gnd Tensor (DE-588)4184723-4 gnd Multilineare Algebra (DE-588)4416303-4 gnd |
| topic_facet | Multilinear algebra Tensor products Matrices Signal theory (Telecommunication) Linear and multilinear algebra; matrix theory / Instructional exposition (textbooks, tutorial papers, etc.) Linear and multilinear algebra; matrix theory / Basic linear algebra / Multilinear algebra, tensor products Information and communication, circuits / Communication, information / Signal theory (characterization, reconstruction, filtering, etc.) Information and communication, circuits / Communication, information / Detection theory Statistics / Distribution theory / Characterization and structure theory Algebraic geometry / Projective and enumerative geometry / Projective techniques Statistik Signaltheorie Tensor Multilineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024140994&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024140994&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT landsbergjosephm tensorsgeometryandapplications |