  Greedy sparse recovery has become a popular tool in many applications,
although its complexity is still prohibitive when large sparsifying
dictionaries or sensing matrices have to be exploited. In this paper, we
formulate first a new class of sparse recovery problems that exploit
multidimensional dictionaries and the separability of the measurement matrices
that appear in certain problems. Then we develop a new algorithm, Separable
Multidimensional Orthogonal Matching Pursuit (SMOMP), which can solve this
class of problems with low complexity. Finally, we apply SMOMP to the problem
of joint localization and communication at mmWave, and numerically show its
effectiveness to provide, at a reasonable complexity, high accuracy channel and
position estimations.
