Riemannian stochastic variance reduced gradient algorithm with retraction and vector transport
This work addresses optimization challenges on manifolds for applications in machine learning and data analysis, representing an incremental improvement over existing methods.
The paper tackles the problem of minimizing the average of many loss functions on Riemannian manifolds by proposing a Riemannian stochastic variance reduced gradient (R-SVRG) algorithm, which outperforms standard Riemannian stochastic gradient descent in applications like computing Riemannian centroids and solving principal component analysis and low-rank matrix completion problems.
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.