Smooth analysis of the condition number and the least singular value

Abstract

Abstract. Let x be a complex random variable with mean zero and bounded variance. Let N n be the random matrix of size n whose entries are iid copies of x and let M be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix M + N n , generalizing an earlier result of Spielman and Teng for the case when x is gaussian.Our investigation reveals an interesting fact that the "core" matrix M does play a role on tail bounds for the least singular value of M +N n . This does not occur in Spielman-Teng studies when x is gaussian. Consequently, our general estimate involves the norm M . In the special case when M is relatively small, this estimate is nearly optimal and extends or refines existing results.

Edge universality for non-Hermitian random matrices

“…for any l ≥ 1, uniformly in |z| ≤ 2, that follows directly from [55,Theorem 3.2] without Assumption (B). Using 5makes Assumption (B) superfluous in the entire paper, albeit at the expense of a quite sophisticated proof.…”

Random matrices: tail bounds for gaps between eigenvalues

“…It is often useful to view M n as a random perturbation of the deterministic matrix F n , especially with respect to applications in data science and numerical analysis, where matrices (as data or inputs to algorithms) are often perturbed by random noise. One can consult for instance [36] where this viewpoint is discussed with respect to the least singular value problem. As an illustration for this view point, we are going to present an application in numerical analysis.…”

Ill-Posed Algebraic Systems with Noise Data

“…Observe that the spectral properties of the matrix tend to improve as the condition number of an ill-conditioned noisy system decreases with the higher noise amplitude [7]. Thus the condition number of a real-valued random matrix increases as log[ ] 1.537 N + , where N -dimension of a matrix [8].…”

“…1a the calculated condition numbers of a random matrix are displayed, from where we can observe that these numbers with a high probability do not reach critical values. According to the theoretical results of [7] the norm of the inverse noise contaminated matrix in an ill-posed problem and its condition number can be easily calculated. A an example, Fig.…”

“…Thus, we may claim the solution to be "chaotic", but in reality, it is the exact classical solution of the perturbed system. Such solution cannot satisfy us though, as according to the results [7], the system has to be well-conditioned with high probability. We will notice that the methods of smoothing do not reveal the hidden solution if they don't use a priori information about errors.…”