A Fast Randomized Algorithm for Computing an Approximate Null Space

TL;DR

This paper analyzes a fast randomized algorithm for approximating the null space of tall-skinny matrices, demonstrating significant speedups in practical applications through theoretical and empirical evaluation.

Contribution

It provides a theoretical analysis of a randomized algorithm for computing approximate null spaces and demonstrates its efficiency and accuracy in practical scenarios.

Findings

Algorithm achieves $O(mn\, ext{log}\,n + n^3)$ complexity, faster than traditional methods.

Numerical experiments show up to 30x speedup in AAA algorithm.

Significant speedups observed in total least squares problems.

Abstract

Randomized algorithms in numerical linear algebra can be fast, scalable and robust. This paper examines the effect of sketching on the right singular vectors corresponding to the smallest singular values of a tall-skinny matrix. We analyze a fast algorithm by Gilbert, Park and Wakin for finding the trailing right singular vectors using randomization by examining the quality of the solution using multiplicative perturbation theory. For an $m\times n$ ($m\geq n$) matrix, the algorithm runs with complexity $O(mn\log n +n^3)$ which is faster than the standard $O(mn^2)$ methods. In applications, numerical experiments show great speedups including a $30\times$ speedup for the AAA algorithm and $10\times$ speedup for the total least squares problem.