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 nearlinear time (in the input size which is the number of nonzeros 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^_ux^*_u  <=epsilon  x^* _infty for accuracy parameter epsilon>0. We further prove that the conditionnumber 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 = {3:13:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770958},
ISSN = {18688969},
year = {2018},
volume = {124},
editor = {Avrim Blum},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/10096},
URN = {urn:nbn:de:0030drops100966},
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 
Collection: 

10th Innovations in Theoretical Computer Science Conference (ITCS 2019) 
Issue Date: 

2018 
Date of publication: 

08.01.2019 