Riemannian Newton's Method on the Grassmann Manifold Exploiting the Quotient Structure

Abstract

Abstract. In this paper, we deal with optimization problems on the Grassmann manifold, which is regarded as a quotient manifold of the Stiefel manifold by the orthogonal group. We develop Newton’s method for minimization of a general objective function. In particular, Newton’s equation is derived as a linear system of equations on a horizontal space. We also propose an efficient method for solving the linear system without explicitly using the representation matrix of the Hessian of the objective function based on the Krylov subspace method.