A New Aspect of Optimization Algorithms on the Grassmann Manifold
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.