Optimization Algorithms on Matrix Manifolds



Publisher: Princeton University | Pages:240 | 2007-12-03 | ISBN 0691132984| PDF | 4 MB

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.