  The problem of sparse multichannel blind deconvolution (S-MBD) arises
frequently in many engineering applications such as radar/sonar/ultrasound
imaging. To reduce its computational and implementation cost, we propose a
compression method that enables blind recovery from much fewer measurements
with respect to the full received signal in time. The proposed compression
measures the signal through a filter followed by a subsampling, allowing for a
significant reduction in implementation cost. We derive theoretical guarantees
for the identifiability and recovery of a sparse filter from compressed
measurements. Our results allow for the design of a wide class of compression
filters. We, then, propose a data-driven unrolled learning framework to learn
the compression filter and solve the S-MBD problem. The encoder is a recurrent
inference network that maps compressed measurements into an estimate of sparse
filters. We demonstrate that our unrolled learning method is more robust to
choices of source shapes and has better recovery performance compared to
optimization-based methods. Finally, in data-limited applications (fewshot
learning), we highlight the superior generalization capability of unrolled
learning compared to conventional deep learning.
