On solving linear systems in sublinear time

We study sublinear algorithms that solve linear systems locally. In the classical version of this problem the input is a matrix S ∈ Rⁿ×ⁿ and a vector b ∈ Rⁿ in the range of S, and the goal is to output x ∈ Rⁿ 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 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 (Formula presented.) for accuracy parameter > 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.