  A trace ratio optimization problem over the Stiefel manifold is investigated
from the perspectives of both theory and numerical computations. At least three
special cases of the problem have arisen from Fisher linear discriminant
analysis, canonical correlation analysis, and unbalanced Procrustes problem,
respectively. Necessary conditions in the form of nonlinear eigenvalue problem
with eigenvector dependency are established and a numerical method based on the
self-consistent field (SCF) iteration is designed and proved to be always
convergent. As an application to multi-view subspace learning, a new framework
and its instantiated concrete models are proposed and demonstrated on real
world data sets. Numerical results show that the efficiency of the proposed
numerical methods and effectiveness of the new multi-view subspace learning
models.
