Superfast 1-Norm Estimation

TL;DR

This paper introduces novel superfast algorithms for accurate 1-norm estimation of real-world matrices, promising efficient computations with potential practical applications.

Contribution

The authors develop new superfast algorithms that reliably estimate the 1-norm of matrices, overcoming previous limitations on worst-case inputs.

Findings

Algorithms run at sublinear cost compared to matrix size

Consistently produce accurate 1-norm estimates on real-world matrices

Potential for practical adoption with further testing and refinement

Abstract

A matrix algorithm is said to be superfast (that is, runs at sublinear cost) if it involves much fewer scalars and flops than the input matrix has entries. Such algorithms have been extensively studied and widely applied in modern computations for matrices with low displacement rank and more recently for low-rank approximation of matrices, even though they are known to fail on worst-case inputs in the latter application. We devise novel superfast algorithms that consistently produce accurate 1-norm estimates for real-world matrices and discuss some promising extensions of our surprisingly simple techniques. With further testing and refinement, our algorithms can potentially be adopted in practical computations.