Numerical study of learning algorithms on Stiefel manifold

Abstract

Convex optimization methods are used for many machine learning models such as support vector machine. However, the requirement of a convex formulation can place limitations on machine learning models. In recent years, a number of machine learning methods not requiring convexity have emerged. In this paper, we study non-convex optimization problems on the Stiefel manifold in which the feasible set consists of a set of rectangular matrices with orthonormal column vectors. We present examples of non-convex optimization problems in machine learning and apply three nonlinear optimization methods for finding a local optimal solution; geometric gradient descent method, augmented Lagrangian method of multipliers, and alternating direction method of multipliers. Although the geometric gradient method is often used to solve non-convex optimization problems on the Stiefel manifold, we show that the alternating direction method of multipliers generally produces higher quality numerical solutions within a reasonable computation time.