Smoothed analysis of the condition number under low-rank perturbations
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by
Rikhav Shah, Sandeep Silwal
2021
Abstract
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. Our results also extend to
the case where U and V have rank larger than k and to symmetric and
complex settings. We also give an application to linear systems solving and
perform some numerical experiments. Lastly, barriers in applying low-rank noise
to other problems studied in the smoothed analysis framework are discussed.
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