EPIC: A PROVABLE ACCELERATED EIGENSOLVER BASED ON PRECONDITIONING AND IMPLICIT CONVEXITY

This paper is concerned with the extraction of the smallest eigenvalue and its corresponding eigenvector of a symmetric positive definite matrix pencil. We reveal implicit convexity of the eigenvalue problem in Euclidean space. A provable accelerated eigensolver based on preconditioning and implicit convexity (EPIC) is proposed. Theoretical analysis shows the acceleration of EPIC with a rate of convergence resembling the conjectured rate of convergence of the well-known locally optimal preconditioned conjugate gradient. Numerical results confirm our theoretical findings of EPIC.