Sparsified Cholesky and Multigrid Solvers for Connection Laplacians

TL;DR

This paper presents novel sparsified Cholesky and multigrid algorithms that enable nearly linear time solutions for connection Laplacian systems, generalizing efficient solvers for graph Laplacians.

Contribution

Introduction of sparsified Cholesky and multigrid methods for connection Laplacians, including the first nearly linear time algorithms and linear-sized approximate inverses.

Findings

Developed sparsified algorithms that accelerate Gaussian elimination.

Proved existence of linear-sized approximate inverses for connection Laplacians.

Achieved linear time solutions for systems in connection Laplacians.

Abstract

We introduce the sparsified Cholesky and sparsified multigrid algorithms for solving systems of linear equations. These algorithms accelerate Gaussian elimination by sparsifying the nonzero matrix entries created by the elimination process. We use these new algorithms to derive the first nearly linear time algorithms for solving systems of equations in connection Laplacians, a generalization of Laplacian matrices that arise in many problems in image and signal processing. We also prove that every connection Laplacian has a linear sized approximate inverse. This is an LU factorization with a linear number of nonzero entries that is a strong approximation of the original matrix. Using such a factorization one can solve systems of equations in a connection Laplacian in linear time. Such a factorization was unknown even for ordinary graph Laplacians.