A Riemannian Proximal Newton-CG Method

Recently, a Riemannian proximal Newton method has been developed for optimizing problems in the form of $\min_{x\in\mathcal{M}} f(x) + \mu \|x\|_1$, where $\mathcal{M}$ is a compact embedded submanifold and $f(x)$ is smooth. Although this method converges superlinearly locally, global convergence is not guaranteed. The existing remedy relies on a hybrid approach: running a Riemannian proximal gradient method until the iterate is sufficiently accurate and switching to the Riemannian proximal Newton method. This existing approach is sensitive to the switching parameter. This paper proposes a Riemannian proximal Newton-CG method that merges the truncated conjugate gradient method with the Riemannian proximal Newton method. The global convergence and local superlinear convergence are proven. Numerical experiments show that the proposed method outperforms other state-of-the-art methods.