Optimization algorithms on the Grassmann and the Stiefel manifolds

Sato Hiroyuki

30th June (Thurs) 13:30

Optimization algorithms on matrix manifolds have been developed mainly as methods for constrained problems on Euclidean spaces. In this talk, discussed are two topics about optimization algorithms on manifolds which have useful applications.

One is about a hybrid method composed of the steepest descent and Newton's methods on the Grassmann manifold. It is known that the problem of minimizing the Rayleigh quotient on the Grassmann manifold is closely related with eigenvalue problems. Results of numerical experiments will be given to demonstrate the speed of the convergence of the present method.

The other is about an algorithm on the product manifold of two Stiefel manifolds with different sizes. The solution of a certain problem on the product manifold of two Stiefel manifolds gives arbitrary numbers of dominant singular values of a matrix. After an overview of the present algorithm, numerical results and some future prospects will be given.

If time permits, the Karush-Kuhn-Tucker conditions, which are necessary for a solution to be optimal, on Riemannian manifolds will be reviewed.

References:
[1] P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ, 2008.
[2] U. Helmke, K. Huper, J. Trumpf, Newton¡Çs method on Grassmann manifolds, arXiv: 0709.2205v2, 2007.
[3] C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds. Boston, MA: Kluwer, 1994.