Jointly Sparse Signal Recovery via Deep Auto-encoder and Parallel Coordinate Descent Unrolling

In this paper, combining techniques in compressed sensing, parallel optimization and deep learning, an auto-encoder-based approach is proposed to jointly design the common measurement matrix and jointly sparse signal recovery method for complex sparse signals. The encoder achieves noisy linear compression for jointly sparse signals, with a common measurement matrix. The decoder realizes jointly sparse signal recovery based on an iterative parallel-coordinate descent algorithm which is proposed to solve GROUP LASSO in a parallel manner. In particular, the decoder consists of an approximation part which unfolds (several iterations of) the proposed iterative algorithm to obtain an approximate solution of GROUP LASSO and a correction part which reduces the difference between the approximate solution and the actual jointly sparse signals. To our knowledge, this is the first time that an optimization-based jointly sparse signal recovery method is implemented using a neural network. The proposed approach achieves higher recovery accuracy with less computation time than the classic GROUP LASSO method, and the gain significantly increases in the presence of extra structures in sparse patterns. The common measurement matrix obtained by the proposed approach is also suitable for the classic GROUP LASSO method. We consider an application example, i.e., channel estimation in Multiple-Input Multiple-Output (MIMO)-based grant-free massive access for massive machine-type communications (mMTC). By numerical results, we demonstrate the substantial gains of the proposed approach over GROUP LASSO and AMP when the number of jointly sparse signals is not very large.