Optimal Smoothed Analysis and Quantitative Universality for the Smallest Singular Value of Random Matrices

TL;DR

This paper establishes a universal behavior for the smallest singular value of random matrices and demonstrates how Gaussian noise can rapidly improve matrix conditioning, providing optimal smoothed analysis results.

Contribution

It proves the first quantitative universality for the smallest singular value and condition number of random matrices, and derives optimal smoothed analysis via Gaussian approximation.

Findings

Quantitative universality for smallest singular value

Gaussian noise rapidly improves matrix conditioning

Optimal smoothed analysis for random matrices

Abstract

The smallest singular value and condition number play important roles in numerical linear algebra and the analysis of algorithms. In numerical analysis with randomness, many previous works make Gaussian assumptions, which are not general enough to reflect the arbitrariness of the input. To overcome this drawback, we prove the first quantitative universality for the smallest singular value and condition number of random matrices. Moreover, motivated by the study of smoothed analysis that random perturbation makes deterministic matrices well-conditioned, we consider an analog for random matrices. For a random matrix perturbed by independent Gaussian noise, we show that this matrix quickly becomes approximately Gaussian. In particular, we derive an optimal smoothed analysis for random matrices in terms of a sharp Gaussian approximation.