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.ITCS.2019.3
URN: urn:nbn:de:0030-drops-100966
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/10096/
Andoni, Alexandr ;
Krauthgamer, Robert ;
Pogrow, Yosef
On Solving Linear Systems in Sublinear Time
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}
}
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Keywords: |
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Linear systems, Laplacian solver, Sublinear time, Randomized linear algebra |
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Collection: |
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10th Innovations in Theoretical Computer Science Conference (ITCS 2019) |
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Issue Date: |
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2018 |
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Date of publication: |
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08.01.2019 |
