Multilinear Subspace Learning: Dimensionality Reduction of Multidimensional Data

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ISBN 9781439857243
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  • Introduces both MSL theories and practical considerations, including multilinear algebra fundamentals, multilinear projections, framework formulation, optimality criterion construction, and implementation tips
  • Provides a strong foundation for developing new MSL algorithms and exploring new MSL applications
  • Presents pseudocode for algorithms in a unifying format, with MATLAB code available on a supporting website
  • Offers examples of real-world applications in video surveillance, biometrics, and object recognition
  • Includes numerous figures that clarify and link concepts, enabling readers to easily grasp and visualize the main ideas
  • Covers mathematical background, data preprocessing, and software tools in the appendices


Due to advances in sensor, storage, and networking technologies, data is being generated on a daily basis at an ever-increasing pace in a wide range of applications, including cloud computing, mobile Internet, and medical imaging. This large multidimensional data requires more efficient dimensionality reduction schemes than the traditional techniques. Addressing this need, multilinear subspace learning (MSL) reduces the dimensionality of big data directly from its natural multidimensional representation, a tensor.

Multilinear Subspace Learning: Dimensionality Reduction of Multidimensional Data gives a comprehensive introduction to both theoretical and practical aspects of MSL for the dimensionality reduction of multidimensional data based on tensors. It covers the fundamentals, algorithms, and applications of MSL.

Emphasizing essential concepts and system-level perspectives, the authors provide a foundation for solving many of today’s most interesting and challenging problems in big multidimensional data processing. They trace the history of MSL, detail recent advances, and explore future developments and emerging applications.

The book follows a unifying MSL framework formulation to systematically derive representative MSL algorithms. It describes various applications of the algorithms, along with their pseudocode. Implementation tips help practitioners in further development, evaluation, and application. The book also provides researchers with useful theoretical information on big multidimensional data in machine learning and pattern recognition. MATLAB® source code, data, and other materials are available at

Table of Contents

Tensor Representation of Multidimensional Data
Dimensionality Reduction via Subspace Learning
Multilinear Mapping for Subspace Learning

Fundamentals and Foundations
Linear Subspace Learning for Dimensionality Reduction
Principal Component Analysis
Independent Component Analysis
Linear Discriminant Analysis
Canonical Correlation Analysis
Partial Least Squares Analysis
Unified View of PCA, LDA, CCA, and PLS
Regularization and Model Selection
Ensemble Learning

Fundamentals of Multilinear Subspace Learning
Multilinear Algebra Preliminaries
Tensor Decompositions
Multilinear Projections
Relationships among Multilinear Projections
Scatter Measures for Tensors and Scalars

Overview of Multilinear Subspace Learning
Multilinear Subspace Learning Framework
PCA-Based MSL Algorithms
LDA-Based MSL Algorithms
History and Related Works
Future Research on MSL

Algorithmic and Computational Aspects
Alternating Partial Projections for MSL
Projection Order, Termination, and Convergence
Synthetic Data for Analysis of MSL Algorithms
Feature Selection for TTP-Based MSL
Computational Aspects

(A Summary and Further Reading appear at the end of each chapter in this section.)

Algorithms and Applications
Multilinear Principal Component Analysis
Generalized PCA
Multilinear PCA
Tensor Rank-One Decomposition
Uncorrelated Multilinear PCA
Boosting with MPCA
Other Multilinear PCA Extensions

Multilinear Discriminant Analysis
Two-Dimensional LDA
Discriminant Analysis with Tensor Representation
General Tensor Discriminant Analysis
Tensor Rank-One Discriminant Analysis
Uncorrelated Multilinear Discriminant Analysis
Other Multilinear Extensions of LDA

Multilinear ICA, CCA, and PLS
Overview of Multilinear ICA Algorithms
Multilinear Modewise ICA
Overview of Multilinear CCA Algorithms
Two-Dimensional CCA
Multilinear CCA
Multilinear PLS Algorithms

Applications of Multilinear Subspace Learning
Pattern Recognition System
Face Recognition
Gait Recognition
Visual Content Analysis in Computer Vision
Brain Signal/Image Processing in Neuroscience
DNA Sequence Discovery in Bioinformatics
Music Genre Classification in Audio Signal Processing
Data Stream Monitoring in Data Mining
Other MSL Applications

Appendix A: Mathematical Background
Appendix B: Data and Preprocessing
Appendix C: Software



Author Bio(s)

Editorial Reviews

"Experimentally inclined readers will probably like this book … . Practitioners will appreciate that the presentation of the subject matter is goal oriented … The structure that this book builds can allow a neophyte to avoid much of the initial confusion and wasted effort necessary to classify unfamiliar work and distinguish between what may be useful or not to one’s intents and interests. … an exquisitely enriched literature review that is almost good enough to use as an auxiliary graduate textbook … a rich yet accessible introduction …"
Computing Reviews, October 2014