Signal estimation with low infinity-norm error by minimizing the mean p-norm error

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 ℓ_∞-norm error metric (worst case error). Our previous results have shown for independent and identically distributed (i.i.d.) Gaussian mixture input signals that, when the input signal dimension goes to infinity, the Wiener filter minimizes the ℓ_∞-norm error. However, the input signal dimension is finite in practice. In this paper, we estimate the finite dimensional input signal by minimizing the mean ℓ_p-norm error. Numerical results show that the ℓ_p-norm minimizer outperforms the Wiener filter, provided that the value of p is properly chosen. Our results further suggest that the optimal value of p increases with the signal dimension, and that for i.i.d. Bernoulli-Gaussian input signals, the optimal p increases with the percentage of nonzeros.