I. Introduction
Consider the linear inverse problem encountered in signal processing, where the aim is to recover a signal vector given the noisy observation , and the channel response matrix . As shown in Fig. 1, the observation vector can be expressed as \begin{equation*} \boldsymbol {y} = \boldsymbol {H} \boldsymbol {x}+ \boldsymbol {w},\tag{1}\end{equation*} where is an additive measurement noise, that is independent and identically distributed (i.i.d) with an unknown distribution . From a Bayesian perspective, the optimal solution to the above problem is the maximum a posteriori (MAP) estimation \begin{align*} \hat { \boldsymbol {x}}_{MAP}=&\arg \max _{ \boldsymbol {x} \in \mathcal {X}} p(\boldsymbol {x}| \boldsymbol {y}), \tag{2}\\=&\arg \max _{ \boldsymbol {x} \in \mathcal {X}} p(\boldsymbol {y}| \boldsymbol {x}) p(\boldsymbol {x}),\tag{3}\end{align*} where denotes the set of all possible signal vectors. When there is no prior knowledge on the transmitted symbols, the MAP estimate is equivalent to the maximum likelihood estimation (MLE), which can be expressed as \begin{align*} \hat { \boldsymbol {x}}_{MAP} = \hat { \boldsymbol {x}}_{MLE}=&\arg \max _{ \boldsymbol {x} \in \mathcal {X}} p(\boldsymbol {y}| \boldsymbol {x}), \tag{4}\\=&\arg \max _{ \boldsymbol {x} \in \mathcal {X}} p_{ \boldsymbol {w}}(\boldsymbol {y}- \boldsymbol {H} \boldsymbol {x}). \tag{5}\end{align*} In most of the existing works in the literature, the noise is assumed to be additive white Gaussian noise (AWGN), whose probability density function (PDF) is analytical and the associated likelihood of each possible signal vector is tractable. In this case, the MLE in (5) becomes \begin{equation*} \hat { \boldsymbol {x}}_{\text {E-MLE}} = \arg \min _{ \boldsymbol {x} \in \mathcal {X}} \| \boldsymbol {y} - \boldsymbol {H} \boldsymbol {x} \|^{2},\tag{6}\end{equation*} which aims to minimize the Euclidean distance, referred to as E-MLE. However, in practical communication scenarios, we may have little or even no statistical knowledge on the noise. In particular, the noise may present some impulsive characteristics and may be not analyzable. For example, the noise distribution becomes unknown and mainly impulsive for scenarios like long-wave, underwater communications, and multiple access systems [1]–[4]. In these cases, the performance of E-MLE will deteriorate severely [5]. In contrast to the Gaussian case, the exact PDF of impulsive noise is usually unknown and not analytical [3], [4], [6], [7], which means that the exact likelihood is computationally intractable
The system model for linear inverse problems.