Block-Sparse Recovery via Convex Optimization

Given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks, we study the problem of finding a block-sparse representation of the signal, i.e., a representation that uses the minimum number of blocks. Motivated by signal/image processing and computer vision applications, such as face recognition, we consider the block-sparse recovery problem in the case where the number of atoms in each block is arbitrary, possibly much larger than the dimension of the underlying subspace. To find a block-sparse representation of a signal, we propose two classes of non-convex optimization programs, which aim to minimize the number of nonzero coefficient blocks and the number of nonzero reconstructed vectors from the blocks, respectively. Since both classes of problems are NP-hard, we propose convex relaxations and derive conditions under which each class of the convex programs is equivalent to the original non-convex formulation. Our conditions depend on the notions of mutual and cumulative subspace coherence of a dictionary, which are natural generalizations of existing notions of mutual and cumulative coherence. We evaluate the performance of the proposed convex programs through simulations as well as real experiments on face recognition. We show that treating the face recognition problem as a block-sparse recovery problem improves the state-of-the-art results by 10% with only 25% of the training data.Comment: IEEE Transactions on Signal Processin

Block Orthonormal Overcomplete Dictionary Learning

In the field of sparse representations, the overcomplete dictionary learning problem is of crucial importance and has a growing application pool where it is used. In this paper we present an iterative dictionary learning algorithm based on the singular value decomposition that efficiently construct unions of orthonormal bases. The important innovation described in this paper, that affects positively the running time of the learning procedures, is the way in which the sparse representations are computed - data are reconstructed in a single orthonormal base, avoiding slow sparse approximation algorithms - how the bases in the union are used and updated individually and how the union itself is expanded by looking at the worst reconstructed data items. The numerical experiments show conclusively the speedup induced by our method when compared to previous works, for the same target representation error