Subspace Aware Recovery of Low Rank and Jointly Sparse Signals
We consider the recovery of a matrix <inline-formula><tex-math notation="LaTeX">${\mathbf X}$</tex-math> </inline-formula>, which is simultaneously low rank and joint sparse, from few measurements of its columns using a two-step algorithm. Each column of <inline-formula><tex-math notation="LaTeX">${\mathbf X}$</tex-math></inline-formula> is measured using a combination of two measurement matrices; one which is the same for every column, while the second measurement matrix varies from column to column. The recovery proceeds by first estimating the row subspace vectors from the measurements corresponding to the common matrix. The estimated row subspace vectors are then used to recover <inline-formula><tex-math notation="LaTeX">${\mathbf X}$</tex-math></inline-formula> from all the measurements using a convex program of joint sparsity minimization. Our main contribution is to provide sufficient conditions on the measurement matrices that guarantee the recovery of such a matrix using the above two-step algorithm. The results demonstrate quite significant savings in number of measurements when compared to the standard multiple measurement vector scheme, which assumes same time-invariant measurement pattern for all the time frames. We illustrate the impact of the sampling pattern on reconstruction quality using breath held cardiac cine MRI and cardiac perfusion MRI data, while the utility of the algorithm to accelerate the acquisition is demonstrated on MR parameter mapping.