Reviews: Gradient Descent Meets Shift-and-Invert Preconditioning for Eigenvector Computation

The main idea is incorporating Nesterov's accelerated gradient descent (AGD) in eigenvalue problem. The approach relies on shift-and-invert preconditioning method that reduces the non-convex objective of Rayleigh quotient to a sequence of convex programs. Shift-and-invert preconditioning improves the convergence dependency of the gradient method to the eigengap of the given matrix. The focus of this paper is using AGD method to approximately solve the convex programs and reaching an accelerated convergence rate for the convex part. Exploiting the accelerated convergence of AGD, they reach an accelerated convergence for the first-order optimization of the eigenvalue problem.