On Conditions for Linearity of Optimal Estimation

TL;DR

This paper investigates the precise conditions under which optimal estimators are linear, extending classical Gaussian results to more general distributions, and explores the uniqueness and asymptotic behavior of such estimators across different SNR regimes.

Contribution

It derives conditions for the existence and uniqueness of distributions that produce linear optimal estimators at a given SNR, and characterizes the special role of Gaussian distributions in this context.

Findings

Gaussian source-channel pair is unique for linear MSE estimators at multiple SNRs.

Matching source distribution must be identical to noise when variances are equal.

Asymptotic linearity occurs at low SNR for Gaussian channels and high SNR for Gaussian sources.

Abstract

When is optimal estimation linear? It is well known that, when a Gaussian source is contaminated with Gaussian noise, a linear estimator minimizes the mean square estimation error. This paper analyzes, more generally, the conditions for linearity of optimal estimators. Given a noise (or source) distribution, and a specified signal to noise ratio (SNR), we derive conditions for existence and uniqueness of a source (or noise) distribution for which the $L_p$ optimal estimator is linear. We then show that, if the noise and source variances are equal, then the matching source must be distributed identically to the noise. Moreover, we prove that the Gaussian source-channel pair is unique in the sense that it is the only source-channel pair for which the mean square error (MSE) optimal estimator is linear at more than one SNR values. Further, we show the asymptotic linearity of MSE optimal estimators for low SNR if the channel is Gaussian regardless of the source and, vice versa, for high SNR if the source is Gaussian regardless of the channel. The extension to the vector case is also considered where besides the conditions inherited from the scalar case, additional constraints must be satisfied to ensure linearity of the optimal estimator.