Theory and Computation of Complex Tensors and its Applications

Preface......Page 5
References......Page 7
Contents......Page 8
1.1 Examples for Tensors......Page 12
1.2 Basics of Tensors......Page 14
1.2.1 Basic Operations......Page 15
1.2.2 Structured Tensors......Page 16
1.3.1 Tensor Spectral Theory......Page 18
1.3.2 Real Tensor Rank-One Approximations......Page 20
1.3.3 Complex Tensor Rank-One Approximations......Page 23
References......Page 24
2.1 Preliminaries......Page 29
2.2 Pseudo-Spectrum of a Complex Tensor......Page 31
2.2.1 Definition and Properties......Page 32
2.2.2 Computational Interpretation of ε-Pseudo-Spectrum......Page 35
2.2.3 Pseudo-Spectral Abscissa and Radius......Page 42
2.3.1 Definitions and Properties......Page 45
2.3.2 Backward Errors......Page 52
2.4.1 An Example......Page 54
2.4.2 Conclusion and Remarks......Page 56
References......Page 58
3.1 Preliminaries......Page 60
3.1.1 Definitions......Page 61
3.1.2 Symmetric and Mode-Symmetric Embeddings......Page 66
3.2 Perturbation Bounds of Z- and H-Eigenvalues......Page 68
3.2.1 Properties of Eigenvalues and Z-Eigenvalues......Page 69
3.2.2 Algebraically Simple Z-Eigenvalues......Page 72
3.2.3 Algebraically Simple Eigenvalues......Page 76
3.2.4 Singular Values......Page 78
3.3.1 Tensor Generalized Eigenvalue Problem......Page 81
3.3.2 Tensor Quadratic Eigenvalue Problem......Page 85
3.3.3 Tensor Polynomial Eigenvalue Problem......Page 87
3.4.1 An Inequality for the Perron Root......Page 89
3.4.2 Perturbation of the Smallest Eigenvalue of M-Tensors......Page 93
3.5.1 Verification of Inequalities......Page 97
3.5.2 Verification of Perturbation Bounds......Page 100
References......Page 104
4.1 Preliminaries......Page 106
4.1.1 Notation and Definitions......Page 107
4.2 Lemmas and Problem Description......Page 108
4.2.1 Lemmas......Page 109
4.2.2 Problem Description......Page 110
4.3.1 Necessary Conditions......Page 111
4.3.2 Solving Problem 4.2.1......Page 114
4.3.3 Solving Problem 4.2.2......Page 117
4.4.1 Stochastic TCPs......Page 118
4.4.2 Generalized Order TCPs......Page 120
References......Page 122
5.1 Preliminaries......Page 125
5.1.1 Plane Stochastic Tensors......Page 126
5.1.2 Combinatorial Determinant and Permanent of Tensors......Page 128
5.2 Sign Nonsingular Tensors......Page 130
5.3 Properties of Plane Stochastic Tensors......Page 132
5.3.1 Totally Plane Stochastic Tensors......Page 133
5.3.2 A Self Map on Totally Plane Stochastic Tensors......Page 136
5.3.3 Relationship Between Nonnegative and Plane Stochastic Tensors......Page 139
5.4.1 Diagonal Products......Page 145
5.4.2 Finding a Positive Diagonal......Page 148
5.5 Axial N-Index Assignment Problems......Page 150
References......Page 152
6 Neural Networks......Page 155
6.1.1 Tensor Singular Values and the Rank-One Approximation......Page 156
6.1.2 Tensor Z-Eigenvalues and the Symmetric Rank-One Approximation......Page 159
6.2 Neural Networks Models......Page 160
6.2.1 Tensor Rank-One Approximation......Page 161
6.2.2 Symmetric Tensor Rank-One Approximation......Page 162
6.3.1 Tensor Rank-One Approximation......Page 165
6.3.2 Symmetric Tensor Rank-One Approximation......Page 168
6.4 Generalized Models: TCCA......Page 170
6.5 Neural Networks for Generalized Tensor Eigenvalues......Page 172
6.6 Numerical Examples......Page 176
6.6.1 Nonsymmetric Tensor Examples......Page 177
6.6.2 Symmetric Tensor Examples......Page 183
6.6.3 Restricted Singular Values......Page 188
6.6.4 Symmetric-Definite Tensor Pairs......Page 190
References......Page 193
7.1.1 US- and U-Eigenpairs......Page 195
7.2 Takagi Factorization of Complex Matrices......Page 197
7.2.1 Orthogonal Type Iteration for Complex Matrices......Page 198
7.2.2 QR Type Algorithm for Complex Matrices......Page 200
7.3 Iterative Algorithm for US-Eigenpairs......Page 201
7.3.1 Properties of US-Eigenpairs......Page 202
7.3.2 QR Algorithms for Complex Symmetric Tensors......Page 203
7.3.3 Convergence......Page 204
7.3.4 Convergence of Algorithm 7.3.3......Page 207
7.4.1 Properties of U-Eigenpairs......Page 208
7.4.2 QR Algorithms for Complex Tensors......Page 210
7.4.3 Convergence......Page 211
7.5 Special Case: Real Symmetric Tensors......Page 215
7.6 Numerical Examples......Page 216
7.7 Conclusions and Further Considerations......Page 219
References......Page 221
8.1 Preliminaries......Page 223
8.1.1 Basic Operations......Page 225
8.1.2 Tucker Decomposition......Page 226
8.1.3 Tensor Train Decomposition......Page 228
8.2.1 Randomized Algorithms for Low Multilinear Rank Approximations......Page 229
8.2.2 More Considerations......Page 234
8.3.1 Probabilistic Error Bounds......Page 235
8.3.2 Some Results for Sub-Gaussian Matrices......Page 237
8.3.3 Proof of Theorem 8.3.1......Page 238
8.4 Randomized Tensor Train Approximation......Page 240
8.5.1 Low Multilinear Rank Approximations......Page 242
8.5.2 Tensor Train Approximation......Page 246
8.6 Conclusions and Further Research......Page 251
References......Page 252
Index......Page 255