  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.
