More on the Power of Randomized Matrix Computations

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.