  Compressed sensing combines the power of convex optimization techniques with
a sparsity-inducing prior on the signal space to solve an underdetermined
system of equations. For many problems, the sparsifying dictionary is not
directly given, nor its existence can be assumed. Besides, the sensing matrix
can change across different scenarios. Addressing these issues requires solving
a sparse representation learning problem, namely dictionary learning, taking
into account the epistemic uncertainty of the learned dictionaries and,
finally, jointly learning sparse representations and reconstructions under
varying sensing matrix conditions. We address both concerns by proposing a
variant of the LISTA architecture. First, we introduce Augmented Dictionary
Learning ISTA (A-DLISTA), which incorporates an augmentation module to adapt
parameters to the current measurement setup. Then, we propose to learn a
distribution over dictionaries via a variational approach, dubbed Variational
Learning ISTA (VLISTA). VLISTA exploits A-DLISTA as the likelihood model and
approximates a posterior distribution over the dictionaries as part of an
unfolded LISTA-based recovery algorithm. As a result, VLISTA provides a
probabilistic way to jointly learn the dictionary distribution and the
reconstruction algorithm with varying sensing matrices. We provide theoretical
and experimental support for our architecture and show that our model learns
calibrated uncertainties.
