  Optimization on Riemannian manifolds widely arises in eigenvalue computation,
density functional theory, Bose-Einstein condensates, low rank nearest
correlation, image registration, and signal processing, etc. We propose an
adaptive regularized Newton method which approximates the original objective
function by the second-order Taylor expansion in Euclidean space but keeps the
Riemannian manifold constraints. The regularization term in the objective
function of the subproblem enables us to establish a Cauchy-point like
condition as the standard trust-region method for proving global convergence.
The subproblem can be solved inexactly either by first-order methods or a
modified Riemannian Newton method. In the later case, it can further take
advantage of negative curvature directions. Both global convergence and
superlinear local convergence are guaranteed under mild conditions. Extensive
computational experiments and comparisons with other state-of-the-art methods
indicate that the proposed algorithm is very promising.
