A Riemannian Proximal Newton Method

In recent years, the proximal gradient method and its variants have been generalized to Riemannian manifolds for solving optimization problems with an additively separable structure, i.e., f + h, where f is continuously differentiable, and h may be nonsmooth but convex with computationally reasonable proximal mapping. In this paper, we generalize the proximal Newton method to embedded submanifolds for solving the type of problem with h(x) = μ |x|1. The generalization relies on the Weingarten and semismooth analysis. It is shown that the Riemannian proximal Newton method has a local quadratic convergence rate under certain reasonable assumptions. Moreover, a hybrid version is given by concatenating a Riemannian proximal gradient method and the Riemannian proximal Newton method. It is shown that if the switch parameter is chosen appropriately, then the hybrid method converges globally and also has a local quadratic convergence rate. Numerical experiments on random and synthetic data are used to demonstrate the performance of the proposed methods.