On Solving Linear Systems in Sublinear Time

摘要

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 | 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.