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

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 the product of a sparse lower triangular matrix with its transpose. This gives 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. Our algorithm performs a subsampled Cholesky factorization, which we analyze using matrix martingales. As part of the analysis, we give a proof of a concentration inequality for matrix martingales where the differences are sums of conditionally independent variables.