Theoretical Results on Sparse Representations of Multiple-Measurement Vectors

• Published in: IEEE transactions on signal processing Vol. 54; no. 12; pp. 4634 - 4643
• Main Authors: Chen, Jie, Huo, Xiaoming
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.12.2006; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Analytical models; Applied sciences; Basis pursuit; Computational efficiency; Computational modeling; Criteria; Dictionaries; Equations; Exact sciences and technology; Greedy algorithms; Information theory; Information, signal and communications theory; Joints; Magnetic analysis; Matching pursuit algorithms; Mathematical analysis; Matrix theory; multiple-measurement vector (MMV); orthogonal matching pursuit (OMP); Predictive models; Representations; Sparse matrices; sparse representation; Studies; Telecommunications and information theory; Vectors (mathematics); Performance evaluation; Dictionaries; Random matrix; Probabilistic approach; Theoretical study; Iterative method; orthogonal matching pursuit (OMP); Basis pursuit; multiple-measurement vector (MMV); Simulation; Matching pursuit; Information processing; Greedy algorithm; Sparse representation
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2006.881263

The sparse representation of a multiple-measurement vector (MMV) is a relatively new problem in sparse representation. Efficient methods have been proposed. Although many theoretical results that are available in a simple case-single-measurement vector (SMV)-the theoretical analysis regarding MMV is lacking. In this paper, some known results of SMV are generalized to MMV. Some of these new results take advantages of additional information in the formulation of MMV. We consider the uniqueness under both an lscr 0 -norm-like criterion and an lscr 1 -norm-like criterion. The consequent equivalence between the lscr 0 -norm approach and the lscr 1 -norm approach indicates a computationally efficient way of finding the sparsest representation in a redundant dictionary. For greedy algorithms, it is proven that under certain conditions, orthogonal matching pursuit (OMP) can find the sparsest representation of an MMV with computational efficiency, just like in SMV. Simulations show that the predictions made by the proved theorems tend to be very conservative; this is consistent with some recent advances in probabilistic analysis based on random matrix theory. The connections will be discussed