Fast Gradient Method for Low-Rank Matrix Estimation

Projected gradient descent and its Riemannian variant belong to a typical class of methods for low-rank matrix estimation. This paper proposes a new Nesterov’s Accelerated Riemannian Gradient algorithm using efficient orthographic retraction and tangent space projection. The subspace relationship between iterative and extrapolated sequences on the low-rank matrix manifold provides computational convenience. With perturbation analysis of truncated singular value decomposition and two retractions, we systematically analyze the local convergence of gradient algorithms and Nesterov’s variants in the Euclidean and Riemannian settings. Theoretically, we estimate the exact rate of local linear convergence under different parameters using the spectral radius in a closed form and give the optimal convergence rate and the corresponding momentum parameter. When the parameter is unknown, the adaptive restart scheme can avoid the oscillation problem caused by high momentum, thus approaching the optimal convergence rate. Extensive numerical experiments confirm the estimations of convergence rate and demonstrate that the proposed algorithm is competitive with first-order methods for matrix completion and matrix sensing.