Smooth Analysis of the Condition Number and the Least Singular Value

Let x be a complex random variable with mean zero and bounded variance. Let Nn 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 +Nn, 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+Nn. 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.