License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.40
URN: urn:nbn:de:0030-drops-147332
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14733/
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Shah, Rikhav ; Silwal, Sandeep

Smoothed Analysis of the Condition Number Under Low-Rank Perturbations

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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.

BibTeX - Entry

@InProceedings{shah_et_al:LIPIcs.APPROX/RANDOM.2021.40,
  author =	{Shah, Rikhav and Silwal, Sandeep},
  title =	{{Smoothed Analysis of the Condition Number Under Low-Rank Perturbations}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{40:1--40:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14733},
  URN =		{urn:nbn:de:0030-drops-147332},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.40},
  annote =	{Keywords: Smoothed analysis, condition number, low rank noise}
}

Keywords: Smoothed analysis, condition number, low rank noise
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)
Issue Date: 2021
Date of publication: 15.09.2021

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