Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separ
This book provides a broad survey of models and efficient algorithms for Nonnegative Matrix Factorization (NMF). This includes NMF's various extensions and modifications, especially Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD). NMF/NTF and their extensions are increasingly used as tools in signal and image processing, and data analysis, having garnered interest due to their capability to provide new insights and relevant information about the complex latent relationships in experimental data sets. It is suggested that NMF can provide meaningful components with physical interpretations; for example, in bioinformatics, NMF and its extensions have been successfully applied to gene expression, sequence analysis, the functional characterization of genes, clustering and text mining. As such, the authors focus on the algorithms that are most useful in practice, looking at the fastest, most robust, and suitable for large-scale models. Key features:* Acts as a single source reference guide to NMF, collating information that is widely dispersed in current literature, including the authors' own recently developed techniques in the subject area.* Uses generalized cost functions such as Bregman, Alpha and Beta divergences, to present practical implementations of several types of robust algorithms, in particular Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms.* Provides a comparative analysis of the different methods in order to identify approximation error and complexity.* Includes pseudo codes and optimized MATLAB source codes for almost all algorithms presented in the book. The increasing interest in nonnegative matrix and tensor factorizations, as well as decompositions and sparse representation of data, will ensure that this book is essential reading for engineers, scientists, researchers, industry practitioners and graduate students across signal and image processing; neuroscience; data mining and data analysis; computer science; bioinformatics; speech processing; biomedical engineering; and multimedia.
Why Read This Book
You should read this book if you want a thorough, practical and mathematically grounded treatment of nonnegative matrix and tensor factorization methods so you can apply NMF/NTF to blind source separation, audio/speech analysis and multiway signal data. It gives you both algorithmic recipes (multiplicative updates, ALS, projected gradients) and guidance on choosing divergences, constraints and regularizers that produce physically interpretable components.
Who Will Benefit
Graduate students, researchers, and signal-processing engineers working on blind source separation, audio/speech analysis, hyperspectral unmixing or any multiway data analysis task who need robust factorization algorithms and practical implementation guidance.
Level: Advanced — Prerequisites: Linear algebra (matrix/tensor algebra), basic convex/nonconvex optimization, probability/statistics and some signal processing background; familiarity with MATLAB or numerical computing is helpful.
Key Takeaways
- Implement core NMF algorithms (multiplicative updates, alternating least squares, projected-gradient) and understand their convergence behavior
- Extend matrix factorization ideas to multiway data: implement PARAFAC/CP, Tucker and nonnegative tensor factorizations
- Select and apply appropriate cost functions and constraints (Euclidean, KL, Itakura-Saito, sparsity, smoothness, orthogonality) for signal-processing use cases
- Apply NMF/NTF methods to blind source separation problems in audio, speech and hyperspectral data and interpret recovered components physically
- Diagnose practical issues (initialization, scaling/normalization, local minima, regularization) and choose remedies to improve robustness
- Integrate factorization methods with other statistical tools (ICA, sparse coding) to build hybrid processing pipelines
Topics Covered
- 1. Introduction and motivations for nonnegative factorizations
- 2. Mathematical preliminaries: linear and multilinear algebra
- 3. Basic NMF models and loss functions (LS, KL, Itakura-Saito)
- 4. Optimization algorithms for NMF (multiplicative updates, ALS, projected gradients)
- 5. Constraints and regularization: sparsity, smoothness, orthogonality, graph constraints
- 6. Advanced NMF variants: hierarchical, supervised, convolutive, shift-invariant
- 7. Tensor decompositions: CP/PARAFAC and Tucker basics
- 8. Nonnegative tensor factorizations (NTF/NTD) and multilinear algorithms
- 9. Efficient implementations and computational issues (scaling, parallelism)
- 10. Applications to blind source separation, audio/speech, image and hyperspectral unmixing
- 11. Practical considerations: initialization, convergence diagnostics, model selection
- 12. Software resources, experiments and appendices (algorithms, proofs)
Languages, Platforms & Tools
How It Compares
Covers NMF/NTF in more depth and with more applied signal-processing orientation than the classic Lee & Seung NMF papers; for more general tensor-decomposition theory and broader multilinear algebra coverage compare with Kolda & Bader's SIAM survey (2009).
