More on the Power of Randomized Matrix Computations

TL;DR

This paper explores the use of randomized matrix multipliers to improve fundamental matrix computations, demonstrating their effectiveness in stabilization, approximation, and tensor decomposition with both theoretical and empirical support.

Contribution

It introduces novel randomized techniques for matrix computations, showing their efficiency and extending their application to tensor decompositions and rank estimation.

Findings

Randomized multipliers improve numerical stability in Gaussian elimination.

Sparse and structured random multipliers perform comparably to Gaussian ones.

The methods effectively approximate singular spaces and low-rank matrices.

Abstract

A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination with no pivoting as well as block Gaussian elimination, approximation of the leading and trailing singular spaces of an ill conditioned matrix, associated with its largest and smallest singular values, respectively, and approximation of this matrix by low-rank matrices, with further extensions to computing numerical ranks and the approximation of tensor decomposition. We formally support the efficiency of the proposed techniques where we employ Gaussian random multipliers, but our extensive tests have consistently produced the same outcome where instead we used sparse and structured random multipliers, defined by much fewer random parameters compared to the number of their entries.