Sparse representations for multiple measurement vectors (MMV) in an over-complete dictionary

The multiple measurement vector (MMV), a newly emerged problem in sparse representation in an over-complete dictionary motivated by a neuro-magnetic inverse problem that arises in magnetoencephalography (MEG) - a modality for imaging the possible activation regions in the brain, poses new challenges. Efficient methods have been designed to search for sparse representations; however, we have not seen substantial development in the theoretical analysis, considering what has been done in a simpler case - single measurement vector (SMV) - in which many theoretical results are known. This paper extends the known results of SMV to MMV. Our theoretical results show the fundamental limitation on when a sparse representation is unique. Moreover, the relation between the solutions of ℓ₀-norm minimization and the solutions of ℓ₁-norm minimization indicates a computationally efficient approach to find a sparse representation. Interestingly, simulations show that the predictions made by these theorems tend to be conservative.