Blind Compressed Sensing (BCS) is an extension of Compressed Sensing (CS)
where the optimal sparsifying dictionary is assumed to be unknown and subject
to estimation (in addition to the CS sparse coefficients). Since the emergence
of BCS, dictionary learning, a.k.a. sparse coding, has been studied as a matrix
factorization problem where its sample complexity, uniqueness and
identifiability have been addressed thoroughly. However, in spite of the strong
connections between BCS and sparse coding, recent results from the sparse
coding problem area have not been exploited within the context of BCS. In
particular, prior BCS efforts have focused on learning constrained and complete
dictionaries that limit the scope and utility of these efforts. In this paper,
we develop new theoretical bounds for perfect recovery for the general
unconstrained BCS problem. These unconstrained BCS bounds cover the case of
overcomplete dictionaries, and hence, they go well beyond the existing BCS
theory. Our perfect recovery results integrate the combinatorial theories of
sparse coding with some of the recent results from low-rank matrix recovery. In
particular, we propose an efficient CS measurement scheme that results in
practical recovery bounds for BCS. Moreover, we discuss the performance of BCS
under polynomial-time sparse coding algorithms.Comment: To appear in the 53rd Annual Allerton Conference on Communication,
Control and Computing, University of Illinois at Urbana-Champaign, IL, USA,
201