Superlinear Convergence of Randomized Block Lanczos Algorithm

The low rank approximation of a matrix is a crucial component in many data mining applications today. A competitive algorithm for this class of problems is the randomized block Lanczos algorithm - an amalgamation of the traditional block Lanczos algorithm with a randomized starting matrix. While empirically this algorithm performs well, there has been scant new theoretical insights on its convergence behavior and approximation accuracy, and those known results have been restricted to impractical settings in the all-important block-size parameter. In this paper, we present a unified singular value convergence analysis for this algorithm, for all valid choices of the block size. We present novel singular value lower bounds and demonstrate superlinear convergence in leading singular values for matrices with decaying singular values. Additionally, we provide results from numerical experiments that validate our analysis.