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

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.