RANDOMIZED NYSTROM PRECONDITIONING

This paper introduces the Nystro"\m preconditioned conjugate gradient (PCG) algo-rithm for solving a symmetric positive-definite linear system. The algorithm applies the randomized Nystro"\m method to form a low-rank approximation of the matrix, which leads to an efficient pre -conditioner that can be deployed with the conjugate gradient algorithm. Theoretical analysis shows that the preconditioned system has constant condition number as soon as the rank of the approx-imation is comparable with the number of effective degrees of freedom in the matrix. The paper also develops adaptive methods that provably achieve similar performance without knowledge of the effective dimension. Numerical tests show that Nystro"\m PCG can rapidly solve large linear systems that arise in data analysis problems, and it surpasses several competing methods from the literature.