We propose a learning-based joint sparse recovery method for the multiple measurement vector (MMV) problem using deep unfolding. We unfold an iterative alternating direction method of multipliers (ADM) algorithm for MMV joint sparse recovery algorithm into a trainable deep network. This ADM algorithm is first obtained by modifying the squared error penalty function of an existing ADM algorithm to a back-projected squared error penalty function. Numerical results for a massive random access system show that our proposed modification to the MMV-ADM method and deep unfolding provide significant improvement in convergence and estimation performance. © 2020 IEEE.