Efficient Covariance Approximations for Large Sparse Precision Matrices
For researchers working with high-dimensional models using sparse precision matrices, this method provides an efficient way to compute covariance elements that is faster and more memory-efficient than direct methods.
This paper introduces a fast Rao-Blackwellized Monte Carlo sampling method to approximate selected elements of the covariance matrix from large sparse precision matrices, with precisely estimable variance and confidence bounds. The method reduces approximation errors to negligible levels in fMRI data and has low memory requirements.
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.