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Zusatztext "The book is very well and carefully written. Every chapter starts with a page-long introduction clearly outlining its goals and how they are achieved together with possible relations to other chapters. I find the material very well explained and supported with appropriate examples. It is a pleasure to work with such a book." ---Nickolay T. Trendafilov, Foundations of Computational Mathematics Informationen zum Autor P.-A. Absil is associate professor of mathematical engineering at the Université Catholique de Louvain in Belgium. R. Mahony is reader in engineering at the Australian National University. R. Sepulchre is professor of electrical engineering and computer science at the University of Liège in Belgium. Klappentext Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists. Zusammenfassung Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It is of interest to applied mathematicians, and computer scientists. Inhaltsverzeichnis List of Algorithms xi Foreword, by Paul Van Dooren xiii Notation Conventions xv Chapter 1. Introduction 1 Chapter 2. Motivation and Applications 5 2.1 A case study: the eigenvalue problem 5 2.1.1 The eigenvalue problem as an optimization problem 7 2.1.2 Some benefits of an optimization framework 9 2.2 Research problems 10 2.2.1 Singular value problem 10 2.2.2 Matrix approximations 12 2.2.3 Independent component analysis 13 2.2.4 Pose estimation and motion recovery 14 2.3 Notes and references 16 Chapter 3. Matrix Manifolds: First-Order Geometry 17 3.1 Manifolds 18 3.1.1 Definitions: charts, atlases, manifolds 18 3.1.2 The topology of a manifold* 20 3.1.3 How to recognize a manifold 21 3.1.4 Vector spaces as manifolds 22 3.1.5 The manifolds Rn x p and Rn x p 22 3.1.6 Product manifolds 23 3.2 Differentiable functions 24 3.2.1 Immersions and submersions 24 3.3 Embedded submanifolds 25 3.3.1 General theory 25 3.3.2 The Stiefel manifold 26 3.4 Quotient manifolds 27 3.4.1 Theory of quotient manifolds 27 3.4.2 Functions on quotient manifolds 29 3.4.3 The real projective space RPn x 1 30 3.4.4 The Grassmann manifold Grass(p, ...
