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DOI: 10.4230/LIPIcs.ITCS.2019.3
URN: urn:nbn:de:0030-drops-100966
URL: http://drops.dagstuhl.de/opus/volltexte/2018/10096/
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Andoni, Alexandr ; Krauthgamer, Robert ; Pogrow, Yosef

On Solving Linear Systems in Sublinear Time

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LIPIcs-ITCS-2019-3.pdf (0.6 MB)


Abstract

We study sublinear algorithms that solve linear systems locally. In the classical version of this problem the input is a matrix S in R^{n x n} and a vector b in R^n in the range of S, and the goal is to output x in R^n satisfying Sx=b. For the case when the matrix S is symmetric diagonally dominant (SDD), the breakthrough algorithm of Spielman and Teng [STOC 2004] approximately solves this problem in near-linear time (in the input size which is the number of non-zeros in S), and subsequent papers have further simplified, improved, and generalized the algorithms for this setting. Here we focus on computing one (or a few) coordinates of x, which potentially allows for sublinear algorithms. Formally, given an index u in [n] together with S and b as above, the goal is to output an approximation x^_u for x^*_u, where x^* is a fixed solution to Sx=b. Our results show that there is a qualitative gap between SDD matrices and the more general class of positive semidefinite (PSD) matrices. For SDD matrices, we develop an algorithm that approximates a single coordinate x_{u} in time that is polylogarithmic in n, provided that S is sparse and has a small condition number (e.g., Laplacian of an expander graph). The approximation guarantee is additive | x^_u-x^*_u | <=epsilon | x^* |_infty for accuracy parameter epsilon>0. We further prove that the condition-number assumption is necessary and tight. In contrast to the SDD matrices, we prove that for certain PSD matrices S, the running time must be at least polynomial in n (for the same additive approximation), even if S has bounded sparsity and condition number.

BibTeX - Entry

@InProceedings{andoni_et_al:LIPIcs:2018:10096,
  author =	{Alexandr Andoni and Robert Krauthgamer and Yosef Pogrow},
  title =	{{On Solving Linear Systems in Sublinear Time}},
  booktitle =	{10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
  pages =	{3:1--3:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{124},
  editor =	{Avrim Blum},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10096},
  URN =		{urn:nbn:de:0030-drops-100966},
  doi =		{10.4230/LIPIcs.ITCS.2019.3},
  annote =	{Keywords: Linear systems, Laplacian solver, Sublinear time, Randomized linear algebra}
}

Keywords: Linear systems, Laplacian solver, Sublinear time, Randomized linear algebra
Seminar: 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)
Issue Date: 2018
Date of publication: 21.12.2018


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