Superlinear Convergence of Randomized Block Lanczos Algorithm
For practitioners using low-rank matrix approximations, this work provides theoretical guarantees for the randomized block Lanczos algorithm, filling a gap in convergence theory.
The paper provides a unified singular value convergence analysis for the randomized block Lanczos algorithm, proving superlinear convergence under certain spectrum regimes, and validates the analysis with numerical experiments.
The low rank approximation of matrices 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 quite well, there has been scant new theoretical results on its convergence behavior and approximation accuracy, and past results have been restricted to certain parameter settings. In this paper, we present a unified singular value convergence analysis for this algorithm, for all valid choices of the block size parameter. We present novel results on the rate of singular value convergence and show that under certain spectrum regimes, the convergence is superlinear. Additionally, we provide results from numerical experiments that validate our analysis.