UC Berkeley

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 component appearing in the form of 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 thesis, we present a unified convergence analysis for this algorithm, for all valid choices of the block size parameter. We give an overview of how the Lanczos algorithm has developed historically and how past and adjacent results in the convergence analysis of these algorithms tie in with the current work. 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.