Smoothed analysis of the condition number under low-rank perturbations
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Let M be an arbitrary n by n matrix of rank n-k. We study the condition number of M plus a low rank perturbation UV^T where U, V are n by k random Gaussian matrices. Under some necessary assumptions, it is shown that M+UV^T is unlikely to have a large condition number. The main advantages of this kind of perturbation over the well-studied dense Gaussian perturbation where every entry is independently perturbed is the O(nk) cost to store U,V and the O(nk) increase in time complexity for performing the matrix-vector multiplication (M+UV^T)x. This improves the Ω(n^2) space and time complexity increase required by a dense perturbation, which is especially burdensome if M is originally sparse. We experimentally validate our approach and consider generalizations to symmetric and complex settings. Lastly, we show barriers in applying our low rank model to other problems studied in the smoothed analysis framework.
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