Tensor Decompositions for Data Science

A self-contained mathematical, algorithmic, and computational treatment of tensor decomposition, including examples using real datasets.

Grey Ballard (Author), Tamara G. Kolda (Author)

9781009471671, Cambridge University Press

Hardback, published 26 June 2025

419 pages
26 x 18.1 x 2.7 cm, 1.06 kg

'Tensors play a crucial role in numerous aspects of data science, including machine learning, computer vision, natural language processing, data compression, anomaly detection, social science, computational neuroscience, materials science, microbiology, and many others. This book provides an accessible yet thorough exploration of tensor representations for complex data. The authors meticulously cover key variants, foundational theories, and practical algorithms - making complex concepts understandable for readers at different levels of expertise.' Rebecca Willett, University of Chicago

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.

Subject Areas: Numerical analysis [PBKS]