Compressed Sensing (CS) is a novel technique for simultaneous signal sampling and compression based on the existence of a sparse representation of signal and a projected dictionary $\PP\D$, where $\PP\in\mathbb{R}^{m\times d}$ is the projection matrix and $\D\in\mathbb{R}^{d\times n}$ is the dictionary. To exactly recover the signal with a small number of measurements $m$, the projected dictionary $\PP\D$ is expected to be of low mutual coherence. Several previous methods attempt to find the projection $\PP$ such that the mutual coherence of $\PP\D$ can be as low as possible. However, they do not minimize the mutual coherence directly and thus their methods are far from optimal. Also the solvers they used lack of the convergence guarantee and thus there has no guarantee on the quality of their obtained solutions. This work aims to address these issues. We propose to find an optimal projection by minimizing the mutual coherence of $\PP\D$ directly. This leads to a nonconvex nonsmooth minimization problem. We then approximate it by smoothing and solve it by alternate minimization. We further prove the convergence of our algorithm. To the best of our knowledge, this is the first work which directly minimizes the mutual coherence of the projected dictionary with a convergence guarantee. Numerical experiments demonstrate that the proposed method can recover sparse signals better than existing methods.

本研究提出通过直接最小化投影词典的互不一致性来找到最优投影，并证明了算法的收敛性。该算法在实验中表现优于现有方法，能更好地恢复稀疏信号。

通过直接相互相干最小化优化压缩感知的投影