Block-Sparse Recovery via Convex Optimization

• Published in: IEEE transactions on signal processing Vol. 60; no. 8; pp. 4094 - 4107
• Main Authors: Elhamifar, Ehsan, Vidal, René
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.08.2012; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Artificial intelligence; Block-sparse signals; Blocking; Coherence; Computer science; control theory; systems; Computer vision; convex optimization; Detection, estimation, filtering, equalization, prediction; Dictionaries; Exact sciences and technology; Face recognition; Image processing; Information, signal and communications theory; Optimization; Pattern recognition; Pattern recognition. Digital image processing. Computational geometry; principal angles; Recovery; Representations; Signal and communications theory; Signal processing; Signal, noise; Silicon; Studies; Subspaces; Telecommunications and information theory; Vectors; Biometrics; Performance evaluation; Dictionaries; State of the art; Non convex programming; Segmentation; Image processing; Convex programming; Learning; Relaxation; principal angles; Sparse representation; Computer vision; Face recognition; Block-sparse signals; Subspace method; Pattern recognition; subspaces; Simulation; Computer applications; NP hard problem; Signal processing; convex optimization; Mutual coherence; Automatic recognition
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2012.2196694

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 nonconvex 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 nonconvex 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.