Theoretical and Empirical Results for Recovery From Multiple Measurements

The joint-sparse recovery problem aims to recover, from sets of compressed measurements, unknown sparse matrices with nonzero entries restricted to a subset of rows. This is an extension of the single-measurement-vector (SMV) problem widely studied in compressed sensing. We study the recovery properties of two algorithms for problems with noiseless data and exact-sparse representation. First, we show that recovery using sum-of-norm minimization cannot exceed the uniform-recovery rate of sequential SMV using l ₁ minimization, and that there are problems that can be solved with one approach, but not the other. Second, we study the performance of the ReMBo algorithm (M. Mishali and Y. Eldar, ¿Reduce and boost: Recovering arbitrary sets of jointly sparse vectors,¿ IEEE Trans. Signal Process., vol. 56, no. 10, 4692-4702, Oct. 2008) in combination with l ₁ minimization, and show how recovery improves as more measurements are taken. From this analysis, it follows that having more measurements than the number of linearly independent nonzero rows does not improve the potential theoretical recovery rate.