Optimization algorithms on matrix manifolds

• Main Authors: Absil, P.A, Mahony,R, Sepulchre,R
• Format: eBook Book
• Language: English
• Published: Princeton Princeton University Press 2008
• Edition: 1
• Subjects: Affine space; Algorithm; Algorithms; Analytic function; Complex projective space; Computation; Computational problem; COMPUTERS / Computer Science; Conjugate gradient method; Constrained optimization; Directional derivative; Eigenvalue algorithm; Eigenvalues and eigenvectors; Equation; Euclidean vector; Finsler manifold; Fixed-point theorem; Gauss–Newton algorithm; Geometry; Global optimization; Gradient descent; Inclusion map; Invariant subspace; Invariant subspace problem; Invertible matrix; Iteration; Iterative method; Levenberg–Marquardt algorithm; Line search; Linear algebra; Linear map; Linear space (geometry); Linear subspace; Local convergence; Manifold; Mathematical optimization; MATHEMATICS; MATHEMATICS / Applied; Matrices; Matrix (mathematics); Matrix decomposition; Newton's method; Normed vector space; Numerical analysis; Numerical integration; Numerical linear algebra; Optimization problem; Principal component analysis; Product topology; Projection (linear algebra); Quadratic equation; 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; Stiefel manifold; Submanifold; Submersion (mathematics); Subset; Subspace topology; Tangent space; Tangent vector; Taylor's theorem; Technology; Technology & Engineering / Engineering (General); Theorem; Topological space; Topology; Vector space; Subset; Euclidean vector; Iterative method; Projection (linear algebra); Matrix (mathematics); Tangent vector; Gradient descent; Vector space; Eigenvalue algorithm; Eigenvalues and eigenvectors; Invertible matrix; Tangent space; Finsler manifold; Real projective space; Numerical linear algebra; Invariant subspace problem; Topology; Subspace topology; Riemannian manifold; Matrix decomposition; Algorithm; Complex projective space; Gauss–Newton algorithm; Line search; Directional derivative; Numerical analysis; Quasi-Newton method; Computation; Topological space; Submersion (mathematics); Linear algebra; Inclusion map; Normed vector space; Mathematical optimization; Rayleigh quotient; Submanifold; Invariant subspace; Product topology; Fixed-point theorem; Iteration; Quotient space (topology); Linear subspace; Computational problem; Riemannian geometry; Taylor's theorem; Manifold; Linear map; Rayleigh quotient iteration; Stiefel manifold; Theorem; Row and column spaces; Rate of convergence; Local convergence; Conjugate gradient method; Scale invariance; Numerical integration; Linear space (geometry); Levenberg–Marquardt algorithm; Equation; Analytic function; Affine space; Quadratic equation; Constrained optimization; Geometry; Simultaneous equations; Global optimization; Newton's method; Optimization problem; Riemannian submanifold; Principal component analysis
• ISBN: 9780691132983; 0691132984; 1400830249; 9781400830244
• DOI: 10.1515/9781400830244

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.