Everything is Vecchia: Unifying low-rank and sparse inverse Cholesky approximations

TL;DR

This paper unifies low-rank and sparse inverse Cholesky approximations through a novel approach that combines partial Cholesky and Vecchia methods, revealing broad applicability and theoretical connections.

Contribution

It introduces a unified framework showing how combining partial Cholesky and Vecchia approximations results in a new, exact Vecchia approximation of the original matrix.

Findings

Vecchia approximations encompass existing matrix approximation methods.

The combined approach yields an exact Vecchia approximation with an augmented sparsity pattern.

The framework broadens the applicability of matrix approximation techniques.

Abstract

The partial pivoted Cholesky approximation accurately represents matrices that are close to being low-rank. Meanwhile, the Vecchia approximation accurately represents matrices with inverse Cholesky factors that are close to being sparse. What happens if a partial Cholesky approximation is combined with a Vecchia approximation of the residual? This paper shows how the sum is exactly a Vecchia approximation of the original matrix with an augmented sparsity pattern. Thus, Vecchia approximations subsume a class of existing matrix approximations and have broad applicability.