Low-rank approximations for large stationary covariance matrices, as used in the Bayesian and generalized-least-squares analysis of pulsar-timing data

TL;DR

This paper introduces two practical methods for approximating large stationary covariance matrices as low-rank products, significantly speeding up data analysis in applications like pulsar-timing residuals.

Contribution

The authors present novel low-rank approximation techniques for stationary covariance matrices, enabling faster computations in Bayesian and generalized-least-squares analyses.

Findings

Methods achieve high accuracy in approximating covariance matrices.

Significant reduction in computational cost for large-scale data analysis.

Applicable to various contexts involving stationary noise processes.

Abstract

Many data-analysis problems involve large dense matrices that describe the covariance of stationary noise processes; the computational cost of inverting these matrices, or equivalently of solving linear systems that contain them, is often a practical limit for the analysis. We describe two general, practical, and accurate methods to approximate stationary covariance matrices as low-rank matrix products featuring carefully chosen spectral components. These methods can be used to greatly accelerate data-analysis methods in many contexts, such as the Bayesian and generalized-least-squares analysis of pulsar-timing residuals.