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In this paper, we consider the challenge of reconstructing jointly sparsevectors from linear measurements. Firstly, we show that by utilizing the rankof 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 originalproblem and enables a simple implementation of joint sparse recovery algorithmsfor full-rank setting. Secondly, we propose a new method for joint sparserecovery in the form of a non-convex optimization problem on a non-compactStiefel manifold. In our numerical experiments our method outperforms thecommonly used $\ell_{2,1}$ minimization in the sense that much fewermeasurements are required for accurate sparse reconstructions. We postulatethis approach possesses the desirable rank aware property, that is, being ableto take advantage of the rank of the unknown matrix to improve the recovery.

Reconstruction of jointly sparse vectors via manifold optimization