Riemannian stochastic quasi-Newton algorithm with variance reduction and its convergence analysis

Stochastic variance reduction algorithms have recently become popular for minimizing the average value of a large but finite number of loss functions. This paper proposes a Riemannian stochastic quasi-Newton algorithm with variance reduction (R-SQN-VR). We present convergence analyses of the R-SQN-VR on both non-convex and retraction strongly convex functions with retraction and vector transport. The proposed algorithm is tested on the Riemannian centroid computation on the symmetric positive-definite manifold and the low-rank matrix completion on the Grassmann manifold. In all cases, the proposed algorithm outperforms the state-of-the-art Riemannian batch and stochastic gradient algorithms.