Wiener Filters in Gaussian Mixture Signal Estimation with Infinity-Norm Error

TL;DR

This paper analyzes the worst-case estimation error using Wiener filters for signals generated by Gaussian mixtures in noisy channels, showing asymptotic optimality for the $\,\ell_$-norm error.

Contribution

It proves that Wiener filters asymptotically minimize the worst-case error for Gaussian mixture signals under the $\,\ell_$-norm metric, extending to linear mixing systems.

Findings

Wiener filter asymptotically achieves optimal $\,\ell_$-norm error.

Largest variance Gaussian component dominates the worst-case error.

Results applicable to sparse signal models like Bernoulli-Gaussian.

Abstract

Consider the estimation of a signal ${\bf x}\in\mathbb{R}^N$ from noisy observations ${\bf r=x+z}$, where the input~${\bf x}$ is generated by an independent and identically distributed (i.i.d.) Gaussian mixture source, and ${\bf z}$ is additive white Gaussian noise (AWGN) in parallel Gaussian channels. Typically, the $\ell_2$-norm error (squared error) is used to quantify the performance of the estimation process. In contrast, we consider the $\ell_\infty$-norm error (worst case error). For this error metric, we prove that, in an asymptotic setting where the signal dimension $N\to\infty$, the $\ell_\infty$-norm error always comes from the Gaussian component that has the largest variance, and the Wiener filter asymptotically achieves the optimal expected $\ell_\infty$-norm error. The i.i.d. Gaussian mixture case is easily applicable to i.i.d. Bernoulli-Gaussian distributions, which are often used to model sparse signals. Finally, our results can be extended to linear mixing systems with i.i.d. Gaussian mixture inputs, in settings where a linear mixing system can be decoupled to parallel Gaussian channels.