In recent years, stochastic variance reduction algorithms have attracted
considerable attention for minimizing the average of a large but finite number
of loss functions. This paper proposes a novel Riemannian extension of the
Euclidean stochastic variance reduced gradient (R-SVRG) algorithm to a manifold
search space. The key challenges of averaging, adding, and subtracting multiple
gradients are addressed with retraction and vector transport. For the proposed
algorithm, we present a global convergence analysis with a decaying step size
as well as a local convergence rate analysis with a fixed step size under some
natural assumptions. In addition, the proposed algorithm is applied to the
computation problem of the Riemannian centroid on the symmetric positive
definite (SPD) manifold as well as the principal component analysis and
low-rank matrix completion problems on the Grassmann manifold. The results show
that the proposed algorithm outperforms the standard Riemannian stochastic
gradient descent algorithm in each case.

本文提出了一种新颖的 Riemann 扩展欧几里得随机方差约减算法，用于测地线搜索空间，在流形上应用于对称正定子流形上的 Riemannian 质心计算问题以及 Grassmann 流形上的主成分分析和低秩矩阵完成问题中，并证明在该问题上优于标准 Riemann 随机梯度下降算法。