On a randomized small-block Lanczos method for large-scale null space computations

TL;DR

This paper introduces a randomized small-block Lanczos method for efficiently computing the null space of large sparse matrices, enabling smaller block sizes and incremental null space computation without prior nullity knowledge.

Contribution

It demonstrates how randomness allows for smaller block sizes in Lanczos methods, ensuring reliable convergence and reducing computational resources.

Findings

Random diagonal perturbation separates zero eigenvalues.

Convergence is robust even with block size d=1.

Method reduces memory and computational requirements.

Abstract

Computing the null space of a large sparse matrix $A$ is a challenging computational problem, especially if the nullity -- the dimension of the null space -- is not small. When applying a block Lanczos method to $A^\mathsf{T} A$ for this purpose, conventional wisdom suggests to use a block size $d$ that is not smaller than the nullity. In this work, we show how randomness can be utilized to allow for smaller $d$ without sacrificing convergence or reliability. Even $d = 1$, corresponding to the standard single-vector Lanczos method, becomes a safe choice. This is achieved by using a small random diagonal perturbation, which moves the zero eigenvalues of $A^\mathsf{T} A$ away from each other, and a random initial guess. We analyze the effect of the perturbation on the attainable quality of the null space and derive convergence results that establish robust convergence for $d=1$. As demonstrated by our numerical experiments, a smaller block size combined with restarting and partial reorthogonalization results in reduced memory requirements and computational effort. It also allows for the incremental computation of the null space, without requiring a priori knowledge of the nullity. Our algorithm is best suited for situations when the nullity of $A$ is moderate.