Author(s): Can Chen
Series: Synthesis Lectures on Mathematics & Statistics
Publisher: Springer
Year: 2024
Preface
Acknowledgements
Contents
Acronyms
1 Tensor Preliminaries
1.1 Overview
1.2 Tensor Products
1.2.1 Kronecker Product
1.2.2 Khatri-Rao Product
1.2.3 Outer/Inner Product
1.2.4 Tensor Matrix/Vector Product
1.2.5 Einstein Product
1.2.6 t-Product
1.3 Tensor Unfoldings
1.3.1 Tensor Matricization
1.3.2 Isomorphic Unfolding
1.4 Block Tensors
1.5 Tensor Eigenvalues
1.5.1 H-Eigenvalues
1.5.2 Z-Eigenvalues
1.5.3 U-Eigenvalues
1.6 Tensor Decompositions
1.6.1 Higher-Order Singular Value Decomposition
1.6.2 Tensor Singular Value Decomposition
1.6.3 CANDECOMP/PARAFAC Decomposition
1.6.4 Tensor Orthogonal Decomposition
1.6.5 Tensor Train Decomposition
1.6.6 Hierarchical Tucker Decomposition
1.6.7 t-Singular Value/Eigenvalue Decomposition
2 Tucker Product-Based Dynamical Systems
2.1 Overview
2.2 System Identification
2.2.1 EM Algorithm
2.2.2 Multilinear PCA/Regression
2.3 System-Theoretic Properties
2.3.1 Stability
2.3.2 Reachability
2.3.3 Observability
2.4 Applications
2.4.1 Video Dataset
2.4.2 Synthetic Data: Stability
2.4.3 Synthetic Data: Reachability and Observability
3 Einstein Product-Based Dynamical Systems
3.1 Overview
3.2 System-Theoretic Properties
3.2.1 Stability
3.2.2 Reachability
3.2.3 Observability
3.3 Model Reduction/System Identification
3.3.1 TTD-Based Balanced Truncation
3.3.2 TTD-Based Balanced Proper Orthogonal Decomposition
3.3.3 TTD-Based Eigensystem Realization Algorithm
3.4 Applications
3.4.1 Synthetic Data: Reachability and Observability
3.4.2 Synthetic Data: Stability
3.4.3 Synthetic Data: TTD-Based Balanced Truncation
3.4.4 2D Heat Equations with Control
3.4.5 Room Impulsive Response Dataset
4 Tensor Vector Product-Based Dynamical Systems
4.1 Overview
4.2 System-Theoretic Properties: Stability
4.2.1 Explicit Solutions
4.2.2 Stability
4.2.3 Discrete-Time Case
4.2.4 Constant Control
4.2.5 Generalization
4.3 System-Theoretic Properties: Controllability
4.3.1 Controllability
4.3.2 Controllability of Hypergraphs
4.4 System-Theoretic Property: Observability
4.4.1 Observability
4.4.2 Observability of Hypergraphs
4.5 Applications
4.5.1 Synthetic Data: Stability
4.5.2 Synthetic Data: Stability with Constant Control
4.5.3 Mouse Neuron Endomicroscopy Dataset
4.5.4 Chromosomal Conformation Capture Dataset
5 Contracted Product-Based Dynamical Systems
5.1 Overview
5.2 Decomposed System Representations
5.2.1 CPD
5.2.2 TD
5.2.3 TTD
5.2.4 HTD
5.3 Linearization
5.4 Feedback Linearization
5.4.1 Higher-Order Polynomials
5.4.2 CPDS Feedback Linearization
5.5 Applications
5.5.1 HVAC Systems
5.5.2 Synthetic Data: Feedback Linearization
6 t-Product-Based Dynamical Systems
6.1 Overview
6.2 System-Theoretic Properties
6.2.1 Explicit Solutions
6.2.2 Stability
6.2.3 Controllability
6.2.4 Observability
6.3 State Feedback Design
6.4 Applications
6.4.1 Synthetic Data: Stability and Controllability
6.4.2 Synthetic Data: State Feedback Design