Random Multipliers Numerically Stabilize Gaussian and Block Gaussian Elimination: Proofs and an Extension to Low-rank Approximation

TL;DR

This paper proves that Gaussian random multipliers can stabilize Gaussian elimination processes and support low-rank approximation, with empirical results confirming their effectiveness and exploring circulant and Toeplitz variants.

Contribution

It provides theoretical proofs for the stabilization and low-rank approximation capabilities of Gaussian random multipliers, extending understanding beyond traditional oversampling methods.

Findings

Gaussian multipliers stabilize elimination methods

Random circulant and Toeplitz multipliers are empirically effective

Supports low-rank approximation without oversampling

Abstract

We prove that standard Gaussian random multipliers are expected to numerically stabilize both Gaussian elimination with no pivoting and block Gaussian elimination. Moreover we prove that such a multiplier (even without the customary oversampling) is expected to support low-rank approximation of a matrix. Our test results are in good accordance with this analysis. Empirically random circulant or Toeplitz multipliers are as efficient as Gaussian ones, but their formal support is more problematic.