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.
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