  This paper addresses the problem of simultaneous signal recovery and
dictionary learning based on compressive measurements. Multiple signals are
analyzed jointly, with multiple sensing matrices, under the assumption that the
unknown signals come from a union of a small number of disjoint subspaces. This
problem is important, for instance, in image inpainting applications, in which
the multiple signals are constituted by (incomplete) image patches taken from
the overall image. This work extends standard dictionary learning and
block-sparse dictionary optimization, by considering compressive measurements,
e.g., incomplete data). Previous work on blind compressed sensing is also
generalized by using multiple sensing matrices and relaxing some of the
restrictions on the learned dictionary. Drawing on results developed in the
context of matrix completion, it is proven that both the dictionary and signals
can be recovered with high probability from compressed measurements. The
solution is unique up to block permutations and invertible linear
transformations of the dictionary atoms. The recovery is contingent on the
number of measurements per signal and the number of signals being sufficiently
large; bounds are derived for these quantities. In addition, this paper
presents a computationally practical algorithm that performs dictionary
learning and signal recovery, and establishes conditions for its convergence to
a local optimum. Experimental results for image inpainting demonstrate the
capabilities of the method.
