OMP-Net: Neural network unrolling of weighted Orthogonal Matching Pursuit

In recent years, algorithm unrolling has emerged as a promising methodology in various signal-processing applications, intending to combine the strengths of iterative algorithms and deep learning. Despite its success, unrolling greedy sparse recovery algorithms, particularly Orthogonal Matching Pursuit (OMP), has yet to receive much attention. The primary challenge is the non-differentiable nature of the argsort operator, a key component in greedy algorithms, which hinders gradient backpropagation during training. To address this, we introduce ‘OMPNet’ by utilizing softsorting to approximate the argsort operator in a differentiable manner. Our numerical and theoretical analysis shows that under certain conditions, the approximation error is minimal, and the performance of the approximated OMP, which we call ‘Soft-OMP’, closely matches the original. We then incorporate Soft-OMP into the feedforward neural network’s layers, integrating learnable weight parameters to connect our approach to weighted sparse recovery. Our numerical results demonstrate that this network is trainable and can surpass the performance of the original OMP in certain scenarios.