[论文解读] Optimized Projections for Compressed Sensing via Direct Mutual Coherence Minimization

Canyi Lu, Huan Li+1 more|arXiv (Cornell University)|Aug 13, 2015

Sparse and Compressive Sensing Techniques参考文献 21被引用 16

一句话总结

本文提出了一种新颖方法，通过优化投影矩阵，直接最小化压缩感知中投影字典的互相关性，采用具有收敛性证明的平滑化交替最小化算法。该方法通过直接针对互相关性这一关键因素进行优化，实现了优于现有方法的稀疏信号恢复效果。

ABSTRACT

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 $P\in\mathbb{R}^{m imes d}$ is the projection matrix and $D\in\mathbb{R}^{d imes n}$ 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.

研究动机与目标

• 解决先前方法在压缩感知投影设计中未直接最小化互相关性的局限性。
• 开发一种算法，通过优化投影矩阵以最小化投影字典 PD 的互相关性。
• 确保优化过程的收敛性，而这是先前方法所缺乏的。
• 通过直接最小化互相关性（恢复成功的关键决定因素）来提升稀疏信号恢复性能。

提出的方法

• 将问题表述为投影字典 PD 的互相关性非凸、非光滑最小化问题。
• 对非光滑的互相关性目标函数进行平滑化处理，以使其适用于优化。
• 采用交替最小化策略，迭代更新投影矩阵 P 和辅助变量。
• 使用带有收敛性保证的近端梯度法，基于强凸性和Lipschitz连续性性质。
• 引入一种平滑逼近方法，以处理互相关性不可微的特性。
• 通过证明迭代序列的下降性和有界性，证明算法的全局收敛性。

实验结果

研究问题

• RQ1 在投影字典 PD 中直接最小化互相关性是否能提升压缩感知中的稀疏信号恢复效果？
• RQ2 一种具有收敛性保证、直接优化互相关性的算法是否优于现有间接方法？
• RQ3 所提出的平滑化与交替最小化策略如何在非凸、非光滑设置下确保收敛性？
• RQ4 低互相关性对基追踪和OMP算法的恢复性能有何影响？
• RQ5 所提出的方法是否能在测量次数更少的情况下，实现优于最先进投影设计技术的重建精度？

主要发现

• 所提方法直接最小化了投影字典 PD 的互相关性，这是确保稀疏信号精确恢复的关键因素。
• 该算法具有理论保证的收敛性，而先前方法缺乏此类收敛性分析。
• 数值实验表明，所提方法在稀疏信号恢复方面比现有最先进方法更准确。
• 平滑化与交替最小化策略有效处理了互相关性最小化问题的非凸性和非光滑性。
• 该方法在各种稀疏信号恢复任务中，均表现出更低的重建误差和更高的成功率。
• 理论分析证实，迭代序列保持有界，且目标函数单调递减，支持收敛至驻点。