# Diagonal Nonlinear Transformations Preserve Structure in Covariance and Precision Matrices

arXiv: https://arxiv.org/abs/2107.04136 · HTML: https://mlsanity.com/paper/2107.04136/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 13 references that are in the map.
- First posted on arXiv 8 Jul 2021.

## Builds on (most cited first, 6 of 6)

- [On Graphical Models via Univariate Exponential Family Distributions](https://mlsanity.com/paper/1301.4183/) (2013)
- [An Overview on the Estimation of Large Covariance and Precision Matrices](https://mlsanity.com/paper/1504.02995/) (2015)
- [Estimation of High-Dimensional Graphical Models Using Regularized Score Matching](https://mlsanity.com/paper/1507.00433/) (2015)
- [Learning non-Gaussian graphical models via Hessian scores and triangular transport](https://mlsanity.com/paper/2101.03093/) (2021)
- [Beyond normality: Learning sparse probabilistic graphical models in the non-Gaussian setting](https://mlsanity.com/paper/1711.00950/) (2017)
- [Communication-Avoiding Optimization Methods for Distributed Massive-Scale Sparse Inverse Covariance Estimation](https://mlsanity.com/paper/1710.10769/) (2017)

## Built on by (most cited first, 4 of 4)

- [A friendly introduction to triangular transport](https://mlsanity.com/paper/2503.21673/) (2025)
- [Learning local neighborhoods of non-Gaussian graphical models: A measure transport approach](https://mlsanity.com/paper/2503.13899/) (2025)
- [CARE: Large Precision Matrix Estimation for Compositional Data](https://mlsanity.com/paper/2309.06985/) (2023)
- [Conditional Independence Estimates for the Generalized Nonparanormal](https://mlsanity.com/paper/2508.11050/) (2025)

## Abstract (arXiv)

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.