# Low Rank Approximation

## Algorithms, Implementation, Applications

• Provides the reader with an analysis tool which is more generally applicable than the commonly-used total least squares

• Shows the reader solutions to the problem of data modelling by linear systems from a sweeping field of applications

• Supplementary electronic and class-based materials will aid tutors in presenting this material to their students

Book

Part of the Communications and Control Engineering book series (CCE)

1. Front Matter
Pages I-X
2. ### Linear Modeling Problems

1. Front Matter
Pages 33-33
2. Ivan Markovsky
Pages 1-32
3. ### Linear Modeling Problems

1. Front Matter
Pages 33-33
2. Ivan Markovsky
Pages 35-72
3. Ivan Markovsky
Pages 73-106
4. Ivan Markovsky
Pages 107-131
4. ### Miscellaneous Generalizations

1. Front Matter
Pages 133-133
2. Ivan Markovsky
Pages 135-177
3. Ivan Markovsky
Pages 179-197
4. Ivan Markovsky
Pages 199-226
5. Back Matter
Pages 227-256

### Introduction

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 systems theory, numerical linear algebra and optimization. The supplementary problems and solutions render it suitable for use in teaching graduate courses in those subjects as well.

### Keywords

Control Control Theory Data Approximation Hankel Linear Algebra Linear Models Low-complexity Model Numerical Algorithms OJ0061 Sylvester System Identification System Theory Time-invariant System Toeplitz

#### Authors and affiliations

1. 1.School of Electronics & Computer ScienceUniversity of SouthamptonSouthamptonUnited Kingdom

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.

### Bibliographic information

• Book Title Low Rank Approximation
• Book Subtitle Algorithms, Implementation, Applications
• Authors Ivan Markovsky
• Series Title Communications and Control Engineering
• Series Abbreviated Title Communicat., Control Engineering
• DOI https://doi.org/10.1007/978-1-4471-2227-2
• Copyright Information Springer-Verlag London Limited 2012
• Publisher Name Springer, London
• eBook Packages Engineering Engineering (R0)
• Hardcover ISBN 978-1-4471-2226-5
• Softcover ISBN 978-1-4471-5836-3
• eBook ISBN 978-1-4471-2227-2
• Series ISSN 0178-5354
• Edition Number 1
• Number of Pages X, 258
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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## Reviews

From the reviews:

“This is a carefully-elaborated monographic work on low rank approximation. It covers the state of the art in this field (key theoretical topics accompanied by the description of the associated algorithms) and discusses various classes of applications. The book provides a rigorous and self-contained material, including numerical examples implemented in MATLAB and a collection of relevant problems. The exposition corresponds to a postgraduate level.” (Octavian Pastravanu, Zentralblatt MATH, Vol. 1245, 2012)

“This book gently takes the reader from the basic ideas of LRA to the most critical concepts, with an adequate number of examples to explain things along the way. … Markovsky has presented LRA in a way that is unifying and cross-disciplinary. The pages abound with code, examples, applications, and problems, from which readers can pick according to their own interests and without the risk of losing the main thread of the book. … it is a good reference for students, practitioners, and researchers.” (Corrado Mencar, ACM Computing Reviews, December, 2012)