Non-Linear Transformations of Gaussians and Gaussian-Mixtures with implications on Estimation and Information Theory

TL;DR

This paper develops a theoretical framework for analyzing non-linear transformations of Gaussian and Gaussian-mixture variables, enabling simplified computation of key metrics in estimation and information theory.

Contribution

It introduces general conditions for regression properties of non-linear Gaussian transformations, extending to Gaussian-mixtures, with applications in communication and estimation.

Findings

Derived conditions for regression coefficients of non-linear Gaussian transformations.

Extended properties to Gaussian-mixtures, broadening applicability.

Facilitated closed-form computation of SNR, MSE, and mutual information bounds.

Abstract

This paper investigates the statistical properties of non-linear transformations (NLT) of random variables, in order to establish useful tools for estimation and information theory. Specifically, the paper focuses on linear regression analysis of the NLT output and derives sufficient general conditions to establish when the input-output regression coefficient is equal to the \emph{partial} regression coefficient of the output with respect to a (additive) part of the input. A special case is represented by zero-mean Gaussian inputs, obtained as the sum of other zero-mean Gaussian random variables. The paper shows how this property can be generalized to the regression coefficient of non-linear transformations of Gaussian-mixtures. Due to its generality, and the wide use of Gaussians and Gaussian-mixtures to statistically model several phenomena, this theoretical framework can find applications in multiple disciplines, such as communication, estimation, and information theory, when part of the nonlinear transformation input is the quantity of interest and the other part is the noise. In particular, the paper shows how the said properties can be exploited to simplify closed-form computation of the signal-to-noise ratio (SNR), the estimation mean-squared error (MSE), and bounds on the mutual information in additive non-Gaussian (possibly non-linear) channels, also establishing relationships among them.