Diagonal Nonlinear Transformations Preserve Structure in Covariance and Precision Matrices

TL;DR

This paper demonstrates that for a class of non-Gaussian distributions obtained via diagonal transformations of Gaussian variables, the structure of covariance and precision matrices still reflects independence properties, extending Gaussian insights.

Contribution

It proves that diagonal nonlinear transformations preserve the correspondence between matrix structure and independence in nonparanormal distributions.

Findings

Covariance matrices retain exact independence information.

Precision matrices approximately preserve conditional independence.

Numerical examples validate theoretical results.

Abstract

For a multivariate normal distribution, the sparsity of the covariance and precision matrices encodes complete information about independence and conditional independence properties. For general distributions, the covariance and precision matrices reveal correlations and so-called partial correlations between variables, but these do not, in general, have any correspondence with respect to independence properties. In this paper, we prove that, for a certain class of non-Gaussian distributions, these correspondences still hold, exactly for the covariance and approximately for the precision. The distributions -- sometimes referred to as "nonparanormal" -- are given by diagonal transformations of multivariate normal random variables. We provide several analytic and numerical examples illustrating these results.