Effective Resistances, Statistical Leverage, and Applications to Linear Equation Solving

TL;DR

This paper introduces a simple, practical algorithm for approximating solutions to Laplacian linear systems, connecting graph resistance and statistical leverage to improve linear equation solving methods.

Contribution

It presents a non-recursive, easy-to-implement algorithm for Laplacian systems that bridges theoretical concepts with practical applications, linking graph resistance and statistical leverage.

Findings

Algorithm runs in O(n^2 polylog(n)) time with an oracle for leverage scores

Connects graph resistance to statistical leverage in linear solvers

Provides a practical approach for approximate solutions to Laplacian systems

Abstract

Recent work in theoretical computer science and scientific computing has focused on nearly-linear-time algorithms for solving systems of linear equations. While introducing several novel theoretical perspectives, this work has yet to lead to practical algorithms. In an effort to bridge this gap, we describe in this paper two related results. Our first and main result is a simple algorithm to approximate the solution to a set of linear equations defined by a Laplacian (for a graph $G$ with $n$ nodes and $m \le n^2$ edges) constraint matrix. The algorithm is a non-recursive algorithm; even though it runs in $O(n^2 \cdot \polylog(n))$ time rather than $O(m \cdot polylog(n))$ time (given an oracle for the so-called statistical leverage scores), it is extremely simple; and it can be used to compute an approximate solution with a direct solver. In light of this result, our second result is a straightforward connection between the concept of graph resistance (which has proven useful in recent algorithms for linear equation solvers) and the concept of statistical leverage (which has proven useful in numerically-implementable randomized algorithms for large matrix problems and which has a natural data-analytic interpretation).