The rank of the sparse signals brought by multiple measurement vectors (MMV) augments the performance of joint sparse recovery. In general, suppose the sparsity level $ k$ is less than or equal to $ [rank(\boldsymbol{X})+spark(\boldsymbol{A})-1]/2$ , the sparsest solution of the MMV problem is unique and recoverable via various methods. It is shown in this letter that the unique solution of the sparsity level $ k$ up to $ spark(\boldsymbol{A})-1$ actually exists in a measure theoretical point of view. More specifically, even when $ [rank(\boldsymbol{X})+spark(\boldsymbol{A})-1]/2\leq k< spark(\boldsymbol{A})$ , the sparsest solution to $ \boldsymbol{A}\boldsymbol{X}=\boldsymbol{Y}$ is still unique with full Lebesgue measure in every $ k$ -sparse coordinate space. This phenomenon is fully confirmed by the MMV tail- $ \ell _{2,1}$ minimization technique. Furthermore, the phenomenon that the traditional $ \ell _{2,1}$ minimization actually fails to recover $ \boldsymbol{X}$ with $ k \geq [spark(\boldsymbol{A})-1]/2$ is investigated from the same perspective of measure theory. Extensive numerical tests conducted by the MMV tail- $ \ell _{2,1}$ minimization and $ \ell _{2,1}$ minimization are demonstrated to confirm the findings. The tail minimization procedure exhibits the most prominent effectiveness for the larger sparsity levels among all known techniques.

中文翻译：

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火花级稀疏和MMV尾信号的联合稀疏恢复$\ell _{2,1}$ 最小化

多测量向量（MMV）带来的稀疏信号的秩增强了联合稀疏恢复的性能。一般来说，假设稀疏水平$千$ 小于或等于 $ [rank(\boldsymbol{X})+spark(\boldsymbol{A})-1]/2$ ，MMV 问题的最稀疏解是唯一的，并且可以通过各种方法恢复。这封信显示了稀疏度的唯一解$千$ 取决于 $ spark(\boldsymbol{A})-1$实际上存在于测度理论的观点中。更具体地说，即使当$ [rank(\boldsymbol{X})+spark(\boldsymbol{A})-1]/2\leq k< spark(\boldsymbol{A})$ ，最稀疏的解 $ \boldsymbol{A}\boldsymbol{X}=\boldsymbol{Y}$ 仍然是独一无二的，在每个方面都有完整的勒贝格度量 $千$ - 稀疏坐标空间。MMV的尾部完全证实了这一现象。 $ \ell _{2,1}$最小化技术。此外，传统的现象$ \ell _{2,1}$ 最小化实际上无法恢复 $ \boldsymbol{X}$ 和 $ k \geq [spark(\boldsymbol{A})-1]/2$从测度论的相同角度进行研究。MMV尾部进行了广泛的数值测试 $ \ell _{2,1}$ 最小化和 $ \ell _{2,1}$证明了最小化以确认结果。在所有已知技术中，尾部最小化过程对于较大的稀疏水平表现出最突出的有效性。