Efficient Covariance Approximations for Large Sparse Precision Matrices

posted on 01.04.2019 by Per Sidén, Finn Lindgren, David Bolin, Mattias Villani

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 nontrivial to obtain when the dimension is large. This article 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.


This work was funded by Swedish Research Council (Vetenskapsrådet) grant no 2013-5229 and grant no 2016-04187. Finn Lindgren was funded by the European Union's Horizon 2020 Programme for Research and Innovation, no 640171, EUSTACE.