Low Rank Approximation_ Algorithms, Implementation, Applications [Markovsky 2011-11-19].pdf

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Communications and Control Engineering
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Ivan Markovsky
Low Rank
Approximation
Algorithms, Implementation,
Applications
Ivan Markovsky
School of Electronics & Computer Science
University of Southampton
Southampton, UK
im@ecs.soton.ac.uk
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ISSN 0178-5354 Communications and Control Engineering
ISBN 978-1-4471-2226-5
e-ISBN 978-1-4471-2227-2
DOI 10.1007/978-1-4471-2227-2
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Preface
Mathematical models are obtained from first principles (natural laws, interconnec-
tion, etc.) and experimental data. Modeling from first principles is common in
natural sciences, while modeling from data is common in engineering. In engineer-
ing, often experimental data are available and a simple approximate model is pre-
ferred to a complicated detailed one. Indeed, although optimal prediction and con-
trol of a complex (high-order, nonlinear, time-varying) system is currently difficult
to achieve, robust analysis and design methods, based on a simple (low-order, lin-
ear, time-invariant) approximate model, may achieve sufficiently high performance.
This book addresses the problem of
data approximation by low-complexity models.
A unifying theme of the book is low rank approximation: a prototypical data
modeling problem. The rank of a matrix constructed from the data corresponds to
the complexity of a linear model that fits the data exactly. The data matrix being full
rank implies that there is no exact low complexity linear model for that data. In this
case, the aim is to find an approximate model. One approach for approximate mod-
eling, considered in the book, is to find small (in some specified sense) modification
of the data that renders the modified data exact. The exact model for the modified
data is an optimal (in the specified sense) approximate model for the original data.
The corresponding computational problem is low rank approximation. It allows the
user to trade off accuracy vs. complexity by varying the rank of the approximation.
The distance measure for the data modification is a user choice that specifies the
desired approximation criterion or reflects prior knowledge about the accuracy of the
data. In addition, the user may have prior knowledge about the system that generates
the data. Such knowledge can be incorporated in the modeling problem by imposing
constraints on the model. For example, if the model is known (or postulated) to be
a linear time-invariant dynamical system, the data matrix has Hankel structure and
the approximating matrix should have the same structure. This leads to a Hankel
structured low rank approximation problem.
A tenet of the book is: the estimation accuracy of the basic low rank approx-
imation method can be improved by exploiting prior knowledge, i.e., by adding
constraints that are known to hold for the data generating system. This path of de-
velopment leads to weighted, structured, and other constrained low rank approxi-
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