Smooth analysis of the condition number and the least singular value

Abstract

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