In this paper, we propose an alternating direction method of multipliers (ADMM)-based optimization algorithm to achieve better undersampling rate for multiple measurement vector (MMV) problem. The core is to introduce the $\ell_{2,0}$-norm sparsity constraint to describe the joint-sparsity of the MMV problem, which is different from the widely used $\ell_{2,1}$-norm constraint in the existing research. In order to illustrate the better performance of $\ell_{2,0}$-norm, first this paper proves the equivalence of the sparsity of the row support set of a matrix and its $\ell_{2,0}$-norm. Afterward, the MMV problem based on $\ell_{2,0}$-norm is proposed. Moreover, building on the Kurdyka-Lojasiewicz property, this paper establishes that the sequence generated by ADMM globally converges to the optimal point of the MMV problem. Finally, the performance of our algorithm and comparison with other algorithms under different conditions is studied by simulated examples.
翻译:在本文中,我们提出了一种基于交替方向乘子法(ADMM)的优化算法,以实现更好的多测量向量(MMV)问题的欠采样率。关键是引入$\ell_{2,0}$-范数稀疏约束来描述MMV问题的联合稀疏性,这与现有研究中广泛使用的$\ell_{2,1}$-范数约束不同。为了说明$\ell_{2,0}$-范数的更好性能,本文首先证明了矩阵行支撑集的稀疏性和其$\ell_{2,0}$-范数的等价性。之后,提出了基于$\ell_{2,0}$-范数的MMV问题。此外,本文基于Kurdyka-Lojasiewicz性质,建立了ADMM生成的序列全局收敛于MMV问题的最优点。最后,通过模拟示例研究了我们的算法的性能和在不同条件下与其他算法的比较。