Abstract: We propose a stochastic recursive momentum method for Riemannian non-convex optimization that achieves a nearly-optimal complexity to find epsilon-approximate solution with one sample. The new algorithm requires one-sample gradient evaluations per iteration and does not require restarting with a large batch gradient, which is commonly used to obtain a faster rate. Extensive experiment results demonstrate the superiority of the proposed algorithm. Extensions to nonsmooth and constrained optimization settings are also discussed.