  In this paper, we consider the challenge of reconstructing jointly sparse
vectors from linear measurements. Firstly, we show that by utilizing the rank
of the output data matrix we can reduce the problem to a full column rank case.
This result reveals a reduction in the computational complexity of the original
problem and enables a simple implementation of joint sparse recovery algorithms
for full-rank setting. Secondly, we propose a new method for joint sparse
recovery in the form of a non-convex optimization problem on a non-compact
Stiefel manifold. In our numerical experiments our method outperforms the
commonly used $\ell_{2,1}$ minimization in the sense that much fewer
measurements are required for accurate sparse reconstructions. We postulate
this approach possesses the desirable rank aware property, that is, being able
to take advantage of the rank of the unknown matrix to improve the recovery.
