  In this paper, a data-driven approach is proposed to jointly design the
common sensing (measurement) matrix and jointly support recovery method for
complex signals, using a standard deep auto-encoder for real numbers. The
auto-encoder in the proposed approach includes an encoder that mimics the noisy
linear measurement process for jointly sparse signals with a common sensing
matrix, and a decoder that approximately performs jointly sparse support
recovery based on the empirical covariance matrix of noisy linear measurements.
The proposed approach can effectively utilize the feature of common support and
properties of sparsity patterns to achieve high recovery accuracy, and has
significantly shorter computation time than existing methods. We also study an
application example, i.e., device activity detection in Multiple-Input
Multiple-Output (MIMO)-based grant-free random access for massive machine type
communications (mMTC). The numerical results show that the proposed approach
can provide pilot sequences and device activity detection with better detection
accuracy and substantially shorter computation time than well-known recovery
methods.
