  We propose a learned-structured unfolding neural network for the problem of
compressive sparse multichannel blind-deconvolution. In this problem, each
channel's measurements are given as convolution of a common source signal and
sparse filter. Unlike prior works where the compression is achieved either
through random projections or by applying a fixed structured compression
matrix, this paper proposes to learn the compression matrix from data. Given
the full measurements, the proposed network is trained in an unsupervised
fashion to learn the source and estimate sparse filters. Then, given the
estimated source, we learn a structured compression operator while optimizing
for signal reconstruction and sparse filter recovery. The efficient structure
of the compression allows its practical hardware implementation. The proposed
neural network is an autoencoder constructed based on an unfolding approach:
upon training, the encoder maps the compressed measurements into an estimate of
sparse filters using the compression operator and the source, and the linear
convolutional decoder reconstructs the full measurements. We demonstrate that
our method is superior to classical structured compressive sparse multichannel
blind-deconvolution methods in terms of accuracy and speed of sparse filter
recovery.
