Optimization Algorithm on Matrix Manifolds
Chapter One
Introduction
This book is about the design of numerical algorithms for computational
problems posed on smooth search spaces. The work is motivated by matrix
optimization problems characterized by symmetry or invariance properties
in the cost function or constraints. Such problems abound in algorithmic
questions pertaining to linear algebra, signal processing, data mining,
and statistical analysis. The approach taken here is to exploit the
special structure of these problems to develop efficient numerical
procedures.
An illustrative example is the eigenvalue problem. Because of their scale
invariance, eigenvectors are not isolated in vector spaces. Instead, each
eigendirection defines a linear subspace of eigenvectors. For numerical
computation, however, it is desirable that the solution set consist only
of isolated points in the search space. An obvious remedy is to impose a
norm equality constraint on iterates of the algorithm. The resulting
spherical search space is an embedded submanifold of the original vector
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