Low Rank Approximation

Matrix low-rank approximation is intimately related to data modelling; a problem that arises frequently in many different fields. Low Rank Approximation: Algorithms, Implementation, Applications is a comprehensive exposition of the theory, algorithms, and applications of structured low-rank approximation. Local optimization methods and effective suboptimal convex relaxations for Toeplitz, Hankel, and Sylvester structured problems are presented. A major part of the text is devoted to application of the theory. Applications described include:

• system and control theory: approximate realization, model reduction, output error, and errors-in-variables identification;
• signal processing: harmonic retrieval, sum-of-damped exponentials, finite impulse response modeling, and array processing;
• machine learning: multidimensional scaling and recommender system;
• computer vision: algebraic curve fitting and fundamental matrix estimation;
• bioinformatics for microarray data analysis;
• chemometrics for multivariate calibration;
• psychometrics for factor analysis; and
• computer algebra for approximate common divisor computation. Special knowledge from the respective application fields is not required. The book is complemented by a software implementation of the methods presented, which makes the theory directly applicable in practice. In particular, all numerical examples in the book are included in demonstration files and can be reproduced by the reader. This gives hands-on experience with the theory and methods detailed. In addition, exercises and MATLAB® examples will assist the reader quickly to assimilate the theory on a chapter-by-chapter basis.

Low Rank Approximation: Algorithms, Implementation, Applications is a broad survey of the theory and applications of its field which will be of direct interest to researchers in system identification, control and systemstheory, numerical linear algebra and optimization. The supplementary problems and solutions render it suitable for use in teaching graduate courses in those subjects as well.

Auteur

Dr. Ivan Markovsky completed his PhD in the Electrical Engineering Department of the Katholieke Universiteit Leuven, Belgium under the supervision of S. Van Huffel, B. De Moor, and J.C. Willems. He was a postdoctoral researcher at the same department, and since January 2007, he has been a lecturer at the School of Electronics and Computer Science of the University of Southampton. His research interests are in system identification in the behavioural setting, total least squares, errors-in-variables estimation, and data-driven control; topics on which he has published 23 journal papers and one monograph (with SIAM). Dr. Markovsky won Honorable Mention in the Alston Householder Prize for best dissertation in numerical linear algebra. He is a co-organiser of the Fourth International Workshop on Total Least Squares and Errors-in-Variables Modelling, a guest editor of Signal Processing for a special issue on total least squares, and an associate editor of the International Journal of Control.

Résumé

Data Approximation by Low-complexity Models details the theory, algorithms, and applications of structured low-rank approximation. Efficient local optimization methods and effective suboptimal convex relaxations for Toeplitz, Hankel, and Sylvester structured problems are presented. Much of the text is devoted to describing the applications of the theory including: system and control theory; signal processing; computer algebra for approximate factorization and common divisor computation; computer vision for image deblurring and segmentation; machine learning for information retrieval and clustering; bioinformatics for microarray data analysis; chemometrics for multivariate calibration; and psychometrics for factor analysis.

Software implementation of the methods is given, making the theory directly applicable in practice. All numerical examples are included in demonstration files giving hands-on experience and exercises and MATLAB® examples assist in the assimilation of the theory.

Contenu

Introduction.- From Data to Models.- Applications in System and Control Theory.- Applications in Signal Processing.- Applications in Computer Algebra.- Applications in Machine Learing.- Subspace-type Algorithms.- Algorithms Based on Local Optimization.- Data Smoothing and Filtering.- Recursive Algorithms.