Tensors: Geometry and Applications

Abstract

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 6 §1.3. Tensor decomposition §1.4. P v. NP and algebraic variants §1.5. Algebraic Statistics and tensor networks §1.6. Geometry and representation theory Chapter 2. Multilinear algebra §2.1. Rust removal exercises §2.2. Groups and representations §2.3. Tensor products §2.4. The rank and border rank of a tensor §2.5. Examples of invariant tensors v vi Contents §2.6. Symmetric and skew-symmetric tensors §2.7. Polynomials on the space of matrices §2.8. Decomposition of V ⊗3 §2.9. Appendix: Basic definitions from algebra §2.10. Appendix: Jordan and rational canonical form §2.11. Appendix: Wiring diagrams Chapter 3. Elementary results on rank and border rank §3.1. Ranks of tensors §3.2. Symmetric rank §3.3. Uniqueness of CP decompositions §3.4. First tests of border rank: flattenings §3.5. Symmetric border rank §3.6. Partially symmetric tensor rank and border rank §3.7. Two useful techniques for determining border rank §3.8. Strassen's equations and variants §3.9. Equations for small secant varieties §3.10. Equations for symmetric border rank §3.11. Tensors in C 2 ⊗C b ⊗C c Part 2. Geometry and Representation Theory Chapter 4. Algebraic geometry for spaces of tensors §4.1. Diagnostic test for those familiar with algebraic geometry §4.2. First definitions §4.3. Examples of algebraic varieties §4.4. Defining equations of Veronese re-embeddings §4.5. Grassmannians §4.6. Tangent and cotangent spaces to varieties §4.7. G-varieties and homogeneous varieties §4.8. Exercises on Jordan normal form and geometry §4.9. Further information regarding algebraic varieties Chapter 5. Secant varieties §5.1. Joins and secant varieties §5.2. Geometry of rank and border rank §5.3. Terracini's lemma and first consequences Contents vii §5.4. The polynomial Waring problem §5.5. Dimensions of secant varieties of Segre Varieties §5.6. Ideas of proofs of dimensions of secant varieties of triple Segre products §5.7. BRPP and conjectures of Strassen and Comon Chapter 6. Exploiting symmetry: Representation theory for spaces of tensors §6.1. Schur's lemma §6.2. Finite groups §6.3. Representations of the permutation group S d §6.4. Decomposing V ⊗d as a GL(V) module with the aid of S d §6.5. Decomposing S d (A 1 ⊗ • • • ⊗ A n) as a G = GL(A 1) × • • • × GL(A n)-module §6.6. Characters §6.7. The Littlewood-Richardson rule §6.8. Weights and weight spaces: a generalization of eigenvalues and eigenspaces §6.9. Homogeneous varieties §6.10. Ideals of homogeneous varieties §6.11. Symmetric functions Chapter 7. Tests for border rank: Equations for secant varieties §7.1. Subspace varieties and multi-linear rank §7.2. Additional auxiliary varieties §7.3. Flattenings §7.4. Inheritance §7.5. Prolongation and multi-prolongation §7.6. Strassen's equations, applications and generalizations §7.7. Equations for σ 4 (Seg(PA × PB × PC)) §7.8. Young flattenings Chapter 8. Ad...

The ideal of the trifocal variety

“…Recall that S • V * is isomorphic to the space of homogeneous polynomials on V . The representation theory of G = SL(A) × SL(B) × SL(C)-modules is well known; however, the reader may wish to consult [Lan12] or [FH91] for reference. One fact we will use is if V = A * ⊗ B * ⊗ C, then irreducible G-modules in S • V * are all of the form S λ A ⊗ S µ B ⊗ S ν C * , where λ, µ and ν are all partitions of the same nonnegative integer.…”

“…In what follows we highlight some of these properties, which are specific cases of much more general constructions. For more details, see [Lan12,Chapter 7]). 4.1.…”

Symmetric tensor rank over an infinite field

“…is the usual symmetric tensor rank of P [14]. Using Theorem 1.1, we are able to prove the following extension of Ballico [5,Theorem 1], to a non-algebraically closed field.…”

Performance estimation for tensor CP decomposition with structured factors

“…Multilinear algebra [6] provides a good framework to exploit these data [7,2] by conserving the multidimensional structure of the information. Nevertheless, generalizing matrix-based algorithms to the multilinear algebra framework is not a trivial task.…”