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

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 $\ell2$-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.