Smoothed Analysis of the Condition Number Under Low-Rank Perturbations

Authors Rikhav Shah, Sandeep Silwal

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Author Details

Rikhav Shah
  • University of California at Berkeley, CA, USA
Sandeep Silwal
  • Massachusetts Institute of Technology, Cambridge, MA, USA


We thank Sushruth Reddy and Samson Zhou for helpful conversations. We also thank Piotr Indyk and Arsen Vasilyan for helpful feedback on a draft of the paper.

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Rikhav Shah and Sandeep Silwal. Smoothed Analysis of the Condition Number Under Low-Rank Perturbations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 40:1-40:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Numerical analysis
  • Theory of computation → Randomness, geometry and discrete structures
  • Smoothed analysis
  • condition number
  • low rank noise


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