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Algoritmi di ottimizzazione su collettori a matrice di P -A Absil: usati-

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Condizione: Buone condizioni: Libro che è già stato letto ma è in buone condizioni. Mostra piccolissimi danni ... Buone condizioni: Libro che è già stato letto ma è in buone condizioni. Mostra piccolissimi danni alla copertina incluse alcune rigature, ma nessun foro o strappo. È possibile che la sovraccoperta per le copertine rigide non sia inclusa. La rilegatura presenta minimi segni di usura. La maggior parte delle pagine non è danneggiata e mostra una quantità minima di piegature o strappi, sottolineature di testo a matita, nessuna evidenziazione di testo né scritte ai margini. Non ci sono pagine mancanti. Per maggiori dettagli e la descrizione di eventuali imperfezioni, consulta l'inserzione del venditore. Vedi tutte le definizioni delle condizioniviene aperta una nuova finestra o scheda

Book Title: Optimization Algorithms on Matrix Manifolds

Publication Date: 2007-12-23

Pages: 240

ISBN: 9780691132983

Publication Year: 2007

Type: Textbook

Format: Hardcover

Language: English

Publication Name: Optimization Algorithms on Matrix Manifolds

Item Height: 1in

Author: P. -A. Absil, Rodolphe Sepulchre, R. Mahony

Item Length: 9.5in

Publisher: Princeton University Press

Item Width: 6.5in

Item Weight: 16 Oz

Number of Pages: 240 Pages

Informazioni su questo prodotto

Product Information

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.

Product Identifiers

Publisher

Princeton University Press

ISBN-10

0691132984

ISBN-13

9780691132983

eBay Product ID (ePID)

60190382

Product Key Features

Author

P. -A. Absil, Rodolphe Sepulchre, R. Mahony

Publication Name

Optimization Algorithms on Matrix Manifolds

Format

Hardcover

Language

English

Publication Year

2007

Type

Textbook

Number of Pages

240 Pages

Dimensions

Item Length

9.5in

Item Height

1in

Item Width

6.5in

Item Weight

16 Oz

Additional Product Features

Lc Classification Number

Qa188

Reviews

[T]his book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first. ---Anders Linnér, American Mathematical Society, "[T]his book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first." --Anders Linnér, American Mathematical Society, "[T]his book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first." --Anders Linnr, American Mathematical Society, "This book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first." --Anders Linner, Mathematical Reviews, "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, 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, "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, "[T]his book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first."-- Anders Linnér, American Mathematical Society, "The treatment strikes an appropriate balance between mathematical, numerical, and algorithmic points of view. The quality of the writing is quite high and very readable. The topic is very timely and is certainly of interest to myself and my students." --Kyle A. Gallivan, Florida State University, "This book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first."-- Anders Linner, Mathematical Reviews, "[T]his book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first."-- Anders Linnr, American Mathematical Society, This book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first. ---Anders Linner, Mathematical Reviews, This book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first., 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, This book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first. -- Anders Linner, Mathematical Reviews, 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.

Table of Content

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 R n x p and R n 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 RP n x 1 30 3.4.4 The Grassmann manifold Grass(p, n) 30 3.5 Tangent vectors and differential maps 32 3.5.1 Tangent vectors 33 3.5.2 Tangent vectors to a vector space 35 3.5.3 Tangent bundle 36 3.5.4 Vector fields 36 3.5.5 Tangent vectors as derivations? 37 3.5.6 Differential of a mapping 38 3.5.7 Tangent vectors to embedded submanifolds 39 3.5.8 Tangent vectors to quotient manifolds 42 3.6 Riemannian metric, distance, and gradients 45 3.6.1 Riemannian submanifolds 47 3.6.2 Riemannian quotient manifolds 48 3.7 Notes and references 51 Chapter 4. Line-Search Algorithms on Manifolds 54 4.1 Retractions 54 4.1.1 Retractions on embedded submanifolds 56 4.1.2 Retractions on quotient manifolds 59 4.1.3 Retractions and local coordinates* 61 4.2 Line-search methods 62 4.3 Convergence analysis 63 4.3.1 Convergence on manifolds 63 4.3.2 A topological curiosity* 64 4.3.3 Convergence of line-search methods 65 4.4 Stability of fixed points 66 4.5 Speed of convergence 68 4.5.1 Order of convergence 68 4.5.2 Rate of convergence of line-search methods* 70 4.6 Rayleigh quotient minimization on the sphere 73 4.6.1 Cost function and gradient calculation 74 4.6.2 Critical points of the Rayleigh quotient 74 4.6.3 Armijo line search 76 4.6.4 Exact line search 78 4.6.5 Accelerated line search: locally optimal conjugate gradient 78 4.6.6 Links with the power method and inverse iteration 78 4.7 Refining eigenvector estimates 80 4.8 Brockett cost function on the Stiefel manifold 80 4.8.1 Cost function and search direction 80 4.8.2 Critical points 81 4.9 Rayleigh quotient minimization on the Grassmann manifold 83 4.9.1 Cost function and gradient calculation 83 4.9.2 Line-search algorithm 85 4.10 Notes and references 86 Chapter 5. Matrix Manifolds: Second-Order Geometry 91 5.1 Newton's method in R n 91 5.2 Affine connections 93 5.3 Riemannian connection 96 5.3.1 Symmetric connections 96 5.3.2 Definition of the Riemannian connection 97 5.3.3 Riemannian connection on Riemannian submanifolds 98 5.3.4 Riemannian connection on quotient manifolds 100 5.4 Geodesics, exponential mapping, and parallel translation 101 5.5 Riemannian Hessian operator 104 5.6 Second covariant derivative* 108 5.7 Notes and references 110 Chapter 6. Newton's Method 111 6.1 Newton's method on manifolds 111 6.2 Riemannian Newton method for real-valued functions 113 6.3 Local convergence 114 6.3.1 Calculus approach to local convergence analysis 117 6.4 Rayleigh quotient algorithms 118 6.4.1 Rayleigh quotient on the sphere 118 6.4.2 Rayleigh quotient on the Grassmann manifold 120<

2008

Topic

Engineering (General), Computer Science, Algebra / General, Applied, Optimization

Dewey Decimal

518.1

Intended Audience

College Audience

Dewey Edition

Illustrated

Yes

Genre

Computers, Technology & Engineering, Mathematics

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United States

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Numero oggetto eBay:364043760861

Ultimo aggiornamento: 14 mag 2024 20:48:25 CESTVedi tutte le revisioniVedi tutte le revisioni

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