Tensors are essential in modern day computational and data sciences. This book explores the foundations of tensor decompositions, a data analysis methodology that is ubiquitous in machine learning, signal processing, chemometrics, neuroscience, quantum computing, financial analysis, social science, business market analysis, image processing, and much more. In this self-contained mathematical, algorithmic, and computational treatment of tensor decomposition, the book emphasizes examples using real-world downloadable open-source datasets to ground the abstract concepts. Methodologies for 3-way tensors (the simplest notation) are presented before generalizing to d-way tensors (the most general but complex notation), making the book accessible to advanced undergraduate and graduate students in mathematics, computer science, statistics, engineering, and physical and life sciences. Additionally, extensive background materials in linear algebra, optimization, probability, and statistics are included as appendices.
Preface; I. Tensor Basics: 1. Tensors and their subparts; 2. Indexing and reshaping tensors; 3. Tensor operations; II. Tucker Decomposition: 4. Tucker decomposition; 5. Tucker tensor structure; 6. Tucker algorithms; 7. Tucker approximation error; 8. Tensor train decomposition; III. CP Decomposition: 9. Canonical polyacidic (CP) decomposition; 10. Kruskal tensor structure; 11. CP alternating least squares (CP-ALS) optimization; 12. CP gradient-based optimization (CP-OPT); 13. CP nonlinear least squares (CP-NLS) optimization; 14. CP algorithms for incomplete or scarce data; 15. Generalized CP (GCP) decomposition; 16. CP tensor rank and special topics; IV. Closing Observations: 17. Closing observations; V. Review Materials: A. Numerical linear algebra; B. Optimization principles and methods; C. Some statistics and probability; Bibliography; Index.
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