Theoretical Results About Finding the Sparsest Representations of Multiple Measurement Vectors (MMV) in an Over-complete Dictionary, Using `1-Norm Minimization and Greedy Algorithms

Multiple Measurement Vectors (MMV) is a newly emerged problem in sparse over-complete representation. Efficient methods have been designed. Considering many theoretical results that are available in a simple case— Single Measure Vector (SMV)—the theoretical analysis regarding MMV is lacking. In this paper, some known results of SMV are generalized to MMVs; new results particularly for MMV are also derived. Our theoretical results show under what conditions a sparse representation is unique. Moreover, the equivalence of the solutions to both ‘minimizing the `0 norm’ problems and ‘minimizing the `1 norm’ problems indicates a computationally efficient approach of finding the sparsest representations. It is proved that under certain conditions, Orthogonal Matching Pursuit (OMP) – which is a greedy algorithm – and a modified version of OMP can find the sparsest representations for MMVs as well. Interestingly, simulations show that the predictions made by the above theorems tend to be conservative; this is consistent with some recent advances, which will be discussed.