Riemannian Anderson Mixing Methods for Minimizing <i>C</i>² Functions on Riemannian Manifolds

Anderson mixing (AM) method is a popular approach for accelerating fixed-point iterations by leveraging historical information from previous steps. In this paper, we introduce the Riemannian Anderson mixing (RAM) method, an extension of AM to Riemannian manifolds, and analyze its local linear convergence under reasonable assumptions. Unlike other extrapolation-based algorithms on Riemannian manifolds, RAM does not require computing the inverse retraction or inverse exponential mapping and has a lower per-iteration cost. Furthermore, we propose a variant of RAM called regularized RAM (RRAM), which establishes global convergence and exhibits similar local convergence properties to RAM. Our proof relies on careful error estimations based on the local geometry of Riemannian manifolds. Finally, we present experimental results on various manifold optimization problems that demonstrate the superior performance of our proposed methods over existing Riemannian gradient descent and limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) approaches. Funding: This research was supported by the National Key R&amp;D Program of China [Grant 2021YFA1001300] and the National Natural Science Foundation of China [Grant 12271291].