Wiener Filters in Gaussian Mixture Signal Estimation With $\ell _\infty $ -Norm Error
Jin Tan, Dror Baron, Liyi Dai · IEEE Transactions on Information Theory · 2014
Consider the estimation of a signal${\mathbf {x}}\in \mathbb {R}^{N}$from noisy observations${{\mathbf {r}}={\mathbf {x}}+{\mathbf {z}}}$, where the input${{\mathbf x}}$is generated by an independent and identically distributed (i.i.d.) Gaussian mixture source, and${{\mathbf z}}$is additive white Gaussian noise 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 can be extended 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.