Diagonal Nonlinear Transformations Preserve Structure in Covariance and Precision Matrices
This work addresses the problem of extending Gaussian graphical model interpretations to non-Gaussian data for statisticians and data scientists, providing a theoretical foundation for structure learning in high-dimensional settings.
The paper proves that for nonparanormal distributions, which are diagonal transformations of multivariate normal variables, the covariance matrix exactly preserves independence properties and the precision matrix approximately preserves conditional independence properties, unlike in general non-Gaussian distributions where such correspondences fail.
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.