Bayesian Convolutional Neural Networks for Compressed Sensing Restoration

Deep Neural Networks (DNNs) have aroused great attention in Compressed Sensing (CS) restoration. However, the working mechanism of DNNs is not explainable, thereby it is unclear that how to design an optimal DNNs for CS restoration. In this paper, we propose a novel statistical framework to explain DNNs, which proves that the hidden layers of DNNs are equivalent to Gibbs distributions and interprets DNNs as a Bayesian hierarchical model. The framework provides a Bayesian perspective to explain the working mechanism of DNNs, namely some hidden layers learn a prior distribution and other layers learn a likelihood distribution. Moreover, the framework provides insights into DNNs and reveals two inherent limitations of DNNs for CS restoration. In contrast to most previous works designing an end-to-end DNNs for CS restoration, we propose a novel DNNs to model a prior distribution only, which can circumvent the limitations of DNNs. Given the prior distribution generated from the DNNs, we design a Bayesian inference algorithm to realize CS restoration in the framework of Bayesian Compressed Sensing. Finally, extensive simulations validate the proposed theory of DNNs and demonstrate that the proposed algorithm outperforms the state-of-the-art CS restoration methods.