We study the MMV (Multiple Measurement Vectors) compressive sensing setting with a specific sparse structured support. The locations of the non-zero rows in the sparse matrix are not known. All that is known is that the locations of the non-zero rows have probabilities that vary from one group of rows to another. We propose two novel greedy algorithms for the exact recovery of the sparse matrix in this structured MMV compressive sensing problem. The first algorithm models the matrix sparse structure using a shallow non- linear neural network. The input of this network is the residual matrix after the prediction and the output is the sparse matrix to be recovered. The second algorithm improves the shallow neural network prediction by using the stacking operation to form a deep stacking network. Experimental evaluation demonstrates the superior performance of both new algorithms over existing MMV methods. Among all, the algorithm using the deep stacking network for modelling the structure in MMV compressive sensing performs the best.