License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.17
URN: urn:nbn:de:0030-drops-94210
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9421/
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Levi, Amit ; Yoshida, Yuichi

Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices

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Abstract

We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.

BibTeX - Entry

@InProceedings{levi_et_al:LIPIcs:2018:9421,
  author =	{Amit Levi and Yuichi Yoshida},
  title =	{{Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices}},
  booktitle =	{Approximation, Randomization, and Combinatorial  Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{17:1--17:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9421},
  URN =		{urn:nbn:de:0030-drops-94210},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.17},
  annote =	{Keywords: Qudratic function minimization, Approximation Algorithms, Matrix spectral decomposition, Graph limits}
}

Keywords: Qudratic function minimization, Approximation Algorithms, Matrix spectral decomposition, Graph limits
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)
Issue Date: 2018
Date of publication: 13.08.2018

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