On Solving Linear Systems in Sublinear Time

Abstract: We study \emph{sublinear} algorithms that solve linear systems locally. In the classical version of this problem the input is a matrix $S\in \mathbb{R}^{n\times n}$ and a vector $b\in\mathbb{R}^n$ in the range of $S$, and the goal is to output $x\in \mathbb{R}^n$ 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 $\hat{x}u$ 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 $| \hat{x}u-x^*_u | \le \epsilon | x^* |\infty$ 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 $n$. This holds even when one wants to obtain the same additive approximation, and $S$ has bounded sparsity and condition number.