Reconstruction of jointly sparse vectors via manifold optimization
For researchers in compressed sensing and sparse recovery, this work offers a more efficient method for jointly sparse vectors, though it is an incremental improvement over existing approaches.
The paper tackles joint sparse recovery from linear measurements, showing that using the rank of the output data matrix reduces computational complexity and enables simpler algorithms. Their proposed manifold optimization method outperforms ℓ₂,₁ minimization, requiring fewer measurements for accurate reconstruction.
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.