Optimization Algorithms on Matrix Manifolds (eBook)

P.-A. Absil, Robert Mahony, Rodolphe Sepulchre (Autoren)

eBook Download: PDF

2009
Princeton University Press (Verlag)
9781400830244 (ISBN)

Infos

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.

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.

Produkt-Details

• Erscheint lt. Verlag: 11.4.2009
• Zusatzinfo: 24 line illus. 3 tables.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• Schlagworte: Affine space • algorithm • analytic function • Broyden–Fletcher–Goldfarb–Shanno algorithm • Calculation • Complex Projective Space • Computation • Computational problem • Conjugate Gradient Method • Constrained optimization • cotangent space • covariant derivative • Differentiable function • Differential structure • Directional derivative • Eigenvalue algorithm • Eigenvalues and Eigenvectors • Equation • Euclidean space • Euclidean Vector • Exponential map (Lie theory) • Factorization • Finsler manifold • Fixed-point theorem • Gauss–Newton algorithm • general linear group • geometric analysis • Geometry • Global Optimization • Gradient descent • Grassmannian • Heine–Borel theorem • Hessian Matrix • Inclusion map • Invariant subspace • Invariant subspace problem • Invertible matrix • Iteration • Iterative Method • Levenberg–Marquardt algorithm • Likelihood-ratio test • Limit of a sequence • Limit point • linear algebra • Linear Approximation • linear equation • Linear map • Linear space (geometry) • Linear subspace • line search • local convergence • Machine epsilon • manifold • Mathematical Optimization • Matrix decomposition • Matrix (mathematics) • matrix representation • Maxima and minima • Newton's method • normed vector space • Numerical analysis • Numerical error • Numerical Integration • numerical linear algebra • optimization problem • Orthonormal basis • Power iteration • Principal Component Analysis • Product topology • Projection (linear algebra) • projective space • QR decomposition • Quadratic equation • quadratic form • Quasi-Newton method • Quotient space (topology) • Rate of Convergence • Rayleigh quotient • Rayleigh quotient iteration • Real projective space • Riemannian Geometry • Riemannian manifold • Riemannian submanifold • Row and column spaces • scale invariance • Simultaneous Equations • singular value decomposition • stiefel manifold • submanifold • Submersion (mathematics) • Subset • Subspace topology • Tangent Space • Tangent vector • Taylor's theorem • Tensor • Theorem • Topological space • Topology • Vector Space
• ISBN-13: 9781400830244 / 9781400830244