  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 $PD$, where $P\in\mathbb{R}^{m\times d}$ 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 $PD$ is expected to be of low mutual coherence. Several
previous methods attempt to find the projection $P$ such that the mutual
coherence of $PD$ 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 $PD$ 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.
