On the linear convergence rate of Riemannian proximal gradient method

Composite optimization problems on Riemannian manifolds arise in applications such as sparse principal component analysis and dictionary learning. Recently, Huang and Wei introduced a Riemannian proximal gradient method (Huang and Wei in MP 194:371-413, 2022) and an inexact Riemannian proximal gradient method (Wen and Ke in COA 85:1-32, 2023), utilizing the retraction mapping to address these challenges. They established the sublinear convergence rate of the Riemannian proximal gradient method under the retraction convexity and a geometric condition on retractions, as well as the local linear convergence rate of the inexact Riemannian proximal gradient method under the Riemannian Kurdyka-Lojasiewicz property. In this paper, we demonstrate the linear convergence rate of the Riemannian proximal gradient method and the linear convergence rate of the proximal gradient method proposed in Chen et al. (SIAM J Opt 30:210-239, 2020) under strong retraction convexity. Additionally, we provide a counterexample that violates the geometric condition on retractions, which is crucial for establishing the sublinear convergence rate of the Riemannian proximal gradient method.