Mixture Gaussian Signal Estimation with L_infty Error Metric

TL;DR

This paper investigates the problem of estimating signals from noisy measurements using the $ ext{L}_ extinfty$ error metric, demonstrating the asymptotic optimality of Wiener filtering in certain Gaussian settings and proposing practical algorithms for finite dimensions.

Contribution

It establishes the asymptotic optimality of Wiener filters for mixture Gaussian signals and introduces a method to approximate $ ext{L}_ extinfty$ error minimization with $ ext{L}_p$ error minimization for finite-dimensional signals.

Findings

Wiener filter is asymptotically optimal for i.i.d. mixture Gaussian signals in scalar channels.

Applying Wiener filter after relaxed BP is asymptotically optimal in linear mixing systems under certain conditions.

Practical algorithms can approximate $ ext{L}_ extinfty$ error minimization using $ ext{L}_p$ minimization with proper $p$.

Abstract

We consider the problem of estimating an input signal from noisy measurements in both parallel scalar Gaussian channels and linear mixing systems. The performance of the estimation process is quantified by the $\ell_\infty$ norm error metric. We first study the minimum mean $\ell_\infty$ error estimator in parallel scalar Gaussian channels, and verify that, when the input is independent and identically distributed (i.i.d.) mixture Gaussian, the Wiener filter is asymptotically optimal with probability 1. For linear mixing systems with i.i.d. sparse Gaussian or mixture Gaussian inputs, under the assumption that the relaxed belief propagation (BP) algorithm matches Tanaka's fixed point equation, applying the Wiener filter to the output of relaxed BP is also asymptotically optimal with probability 1. However, in order to solve the practical problem where the signal dimension is finite, we apply an estimation algorithm that has been proposed in our previous work, and illustrate that an $\ell_\infty$ error minimizer can be approximated by an $\ell_p$ error minimizer provided the value of $p$ is properly chosen.