Estimating sparse precision matrices

TL;DR

This paper introduces a method for efficiently estimating sparse precision matrices from Gaussian samples, leveraging sparsity to improve convergence rates and reduce the number of simulations needed, with applications in cosmology.

Contribution

The paper applies a novel estimation technique that exploits sparsity in precision matrices, achieving faster convergence and lower errors in cosmological data analysis.

Findings

Significant reduction in error factors compared to sample precision matrices.

Faster convergence rates with fewer simulations in large-scale structure analysis.

Method demonstrated effective on toy models and cosmological data.

Abstract

We apply a method recently introduced to the statistical literature to directly estimate the precision matrix from an ensemble of samples drawn from a corresponding Gaussian distribution. Motivated by the observation that cosmological precision matrices are often approximately sparse, the method allows one to exploit this sparsity of the precision matrix to more quickly converge to an asymptotic 1/sqrt(Nsim) rate while simultaneously providing an error model for all of the terms. Such an estimate can be used as the starting point for further regularization efforts which can improve upon the 1/sqrt(Nsim) limit above, and incorporating such additional steps is straightforward within this framework. We demonstrate the technique with toy models and with an example motivated by large-scale structure two-point analysis, showing significant improvements in the rate of convergence.For the large-scale structure example we find errors on the precision matrix which are factors of 5 smaller than for the sample precision matrix for thousands of simulations or, alternatively, convergence to the same error level with more than an order of magnitude fewer simulations.