Approximate Gaussian Elimination for Laplacians: Fast, Sparse, and Simple

TL;DR

This paper introduces a simple, nearly linear time algorithm for approximate Gaussian elimination on Laplacian matrices, achieving sparsity and efficiency without relying on traditional graph-theoretic tools.

Contribution

It presents the first nearly linear time Laplacian solver based solely on random sampling, avoiding complex graph constructions.

Findings

Achieves sparse approximate Cholesky factorization efficiently

Provides a novel concentration bound for matrix martingales

First to use purely sampling-based approach for Laplacian solvers

Abstract

We show how to perform sparse approximate Gaussian elimination for Laplacian matrices. We present a simple, nearly linear time algorithm that approximates a Laplacian by a matrix with a sparse Cholesky factorization, the version of Gaussian elimination for symmetric matrices. This is the first nearly linear time solver for Laplacian systems that is based purely on random sampling, and does not use any graph theoretic constructions such as low-stretch trees, sparsifiers, or expanders. The crux of our analysis is a novel concentration bound for matrix martingales where the differences are sums of conditionally independent variables.