Embedding Prior Knowledge Within Compressed Sensing by Neural Networks

In the compressed sensing framework, different algorithms have been proposed for sparse signal recovery from an incomplete set of linear measurements. The most known can be classified into two categories:
$\ell_{1}$
norm minimization-based algorithms and
$\ell_{0}$
pseudo-norm minimization with greedy matching pursuit algorithms. In this paper, we propose a modified matching pursuit algorithm based on the orthogonal matching pursuit (OMP). The idea is to replace the correlation step of the OMP, with a neural network. Simulation results show that in the case of random sparse signal reconstruction, the proposed method performs as well as the OMP. Complexity overhead, for training and then integrating the network in the sparse signal recovery is thus not justified in this case. However, if the signal has an added structure, it is learned and incorporated in the proposed new OMP. We consider three structures: first, the sparse signal is positive, second the positions of the non zero coefficients of the sparse signal follow a certain spatial probability density function, the third case is a combination of both. Simulation results show that, for these signals of interest, the probability of exact recovery with our modified OMP increases significantly. Comparisons with
$\ell_{1}$
based reconstructions are also performed. We thus present a framework to reconstruct sparse signals with added structure by embedding, through neural network training, additional knowledge to the decoding process in order to have better performance in the recovery of sparse signals of interest.