Data-driven Support Recovery for Sparse Signals with Non-stationary Modulation.

Estimating a sparse signal from its low-dimensional observations arises in many applications including signal demixing and compression. If each dictionary atom undergoes an unknown modulation process, this problem becomes a sparse recovery and blind demodulation problem. In this paper, we further allow the modulation process to be different for different dictionary atoms, which is known as non-stationary modulation. In the presence of noise, the sparse signal and modulation parameters cannot be recovered exactly. We propose to solve the support recovery problem with non-stationary modulation via an optimization-inspired data-driven method. Specifically, by assuming the modulating signals live in a known common subspace and applying the lifting technique, we formulate the support recovery problem as recovering a column-wise sparse matrix from linear observations, which could then be solved via a block norm regularized quadratic minimization. By unfolding the proximal gradient descent algorithm for that regularized quadratic minimization and replacing the proximal operator with a proximal network, we construct a novel recurrent neural network (RNN) to efficiently solve the support recovery problem. Experiments indicate that the proposed network is very efficient in solving the support recovery problem, can be adaptive to different sensing processes without retraining the network, and is applicable when the matrix of interest is not strictly column-wise sparse and when we only know an approximation of the sensing process.