Specifying Gaussian Markov Random Fields with Incomplete Orthogonal Factorization using Givens Rotations

TL;DR

This paper introduces a method using incomplete orthogonal factorization with Givens rotations to obtain sparser and more efficient Cholesky factors for Gaussian Markov random fields, applicable to both square and rectangular matrices.

Contribution

The paper presents a novel approach for constructing sparse incomplete Cholesky factors via Givens rotations, improving computational efficiency and stability for GMRFs.

Findings

Produces sparser Cholesky factors than standard methods

Applicable to both square and rectangular matrices

Demonstrates usefulness on various precision matrix structures

Abstract

In this paper an approach for finding a sparse incomplete Cholesky factor through an incomplete orthogonal factorization with Givens rotations is discussed and applied to Gaussian Markov random fields (GMRFs). The incomplete Cholesky factor obtained from the incomplete orthogonal factorization is usually sparser than the commonly used Cholesky factor obtained through the standard Cholesky factorization. On the computational side, this approach can provide a sparser Cholesky factor, which gives a computationally more efficient representation of GMRFs. On the theoretical side, this approach is stable and robust and always returns a sparse Cholesky factor. Since this approach applies both to square matrices and to rectangle matrices, it works well not only on precision matrices for GMRFs but also when the GMRFs are conditioned on a subset of the variables or on observed data. Some common structures for precision matrices are tested in order to illustrate the usefulness of the approach. One drawback to this approach is that the incomplete orthogonal factorization is usually slower than the standard Cholesky factorization implemented in standard libraries and currently it can be slower to build the sparse Cholesky factor.