  We present an adaptive parametrization strategy for optimization problems
over the Stiefel manifold by using generalized Cayley transforms to utilize
powerful Euclidean optimization algorithms efficiently. The generalized Cayley
transform can translate an open dense subset of the Stiefel manifold into a
vector space, and the open dense subset is determined according to a tunable
parameter called a center point. With the generalized Cayley transform, we
recently proposed the naive Cayley parametrization, which reformulates the
optimization problem over the Stiefel manifold as that over the vector space.
Although this reformulation enables us to transplant powerful Euclidean
optimization algorithms, their convergences may become slow by a poor choice of
center points. To avoid such a slow convergence, in this paper, we propose to
estimate adaptively 'good' center points so that the reformulated problem can
be solved faster. We also present a unified convergence analysis, regarding the
gradient, in cases where fairly standard Euclidean optimization algorithms are
employed in the proposed adaptive parametrization strategy. Numerical
experiments demonstrate that (i) the proposed strategy succeeds in escaping
from the slow convergence observed in the naive Cayley parametrization
strategy; (ii) the proposed strategy outperforms the standard strategy which
employs a retraction.
