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- W3204016023 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(n²) 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." @default.
- W3204016023 created "2021-10-11" @default.
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- W3204016023 date "2021-01-01" @default.
- W3204016023 modified "2023-09-26" @default.
- W3204016023 title "Smoothed Analysis of the Condition Number Under Low-Rank Perturbations." @default.
- W3204016023 doi "https://doi.org/10.4230/lipics.approx/random.2021.40" @default.
- W3204016023 hasPublicationYear "2021" @default.
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