# Efficient Covariance Approximations for Large Sparse Precision Matrices

## TL;DR

This paper presents a fast, memory-efficient Monte Carlo method for approximating specific elements of covariance matrices in high-dimensional sparse precision models, with applications to neuroimaging.

## Contribution

It introduces a Rao-Blackwellized Monte Carlo approach and a subdomain iteration technique for accurate covariance approximation in large sparse inverse covariance matrices.

## Key findings

- The methods provide precise variance and confidence bounds without extra costs.
- They significantly reduce approximation errors in neuroimaging data.
- The approaches require low memory, outperforming direct methods.

## Abstract

The use of sparse precision (inverse covariance) matrices has become popular because they allow for efficient algorithms for joint inference in high-dimensional models. Many applications require the computation of certain elements of the covariance matrix, such as the marginal variances, which may be non-trivial to obtain when the dimension is large. This paper introduces a fast Rao-Blackwellized Monte Carlo sampling based method for efficiently approximating selected elements of the covariance matrix. The variance and confidence bounds of the approximations can be precisely estimated without additional computational costs. Furthermore, a method that iterates over subdomains is introduced, and is shown to additionally reduce the approximation errors to practically negligible levels in an application on functional magnetic resonance imaging data. Both methods have low memory requirements, which is typically the bottleneck for competing direct methods.