A New Aspect of Optimization Algorithms
on the Grassmann Manifold


新しい観点からのグラスマン多様体上の最適化アルゴリズム


In the literature on optimization algorithms on manifolds, the
Grassmann manifold is often regarded as a quotient manifold of the
matrix Euclidean space, on which several algorithms are developed.
The Grassmann manifold can be, however, realized as a submanifold of
the matrix Euclidean space, that is, the set of all orthogonal
projection matrices of constant rank. This article takes this point of
view to set up several algorithms in terms of such matrices, which
will make the algorithms more feasible.  Starting with a brief review
of materials needed for the description of optimization methods on
manifolds, this article provides a geometric setting for matrix
calculus on the Grassmann manifold to develop the steepest descent
method, Newton’s method, and the trust-region method on the Grassmann
manifold together with some applications.  In particular, it is shown
that Newton’s equation for the Rayleigh quotient minimization problem
in the proposed Newton’s method is expressed as a Lyapunov equation,
for which efficient algorithms exist, and thereby the present Newton’
s method works efficiently.