On Solving Linear Systems in Sublinear Time
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We study sublinear algorithms that solve linear systems locally. In the classical version of this problem the input is a matrix S∈R^n× n and a vector b∈R^n in the range of S, and the goal is to output x∈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∈ [n] together with S and b as above, the goal is to output an approximation x̂_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 | x̂_u-x^*_u | <ϵ x^* _∞ for accuracy parameter ϵ>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. This holds even when one wants to obtain the same additive approximation, and S has bounded sparsity and condition number.
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