  The 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 as 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 LBFGS approaches.
