Sparsified Cholesky Solvers for SDD linear systems

TL;DR

This paper introduces sparsified Cholesky solvers for SDD matrices, enabling efficient approximate factorizations and linear system solutions with nearly linear time algorithms, including spectral vertex sparsifier construction.

Contribution

It presents the first nearly-linear work algorithms for spectral vertex sparsifiers and efficient sparse Cholesky factorizations for SDD matrices.

Findings

Linear-sized sparse Cholesky factorizations for SDD matrices.

Linear system solutions in L and L^T in O(n) work.

Nearly linear time algorithms for spectral vertex sparsifiers.

Abstract

We show that Laplacian and symmetric diagonally dominant (SDD) matrices can be well approximated by linear-sized sparse Cholesky factorizations. We show that these matrices have constant-factor approximations of the form $L L^{T}$, where $L$ is a lower-triangular matrix with a number of nonzero entries linear in its dimension. Furthermore linear systems in $L$ and $L^{T}$ can be solved in $O (n)$ work and $O(\log{n}\log^2\log{n})$ depth, where $n$ is the dimension of the matrix. We present nearly linear time algorithms that construct solvers that are almost this efficient. In doing so, we give the first nearly-linear work routine for constructing spectral vertex sparsifiers---that is, spectral approximations of Schur complements of Laplacian matrices.