压缩感知专题
Topics in Compressed Sensing
AI总结：
本文综述了压缩感知的两种主要稀疏恢复方法：L1最小化（如基追踪）和贪婪算法（如ROMP和CoSaMP），并介绍了ROMP和CoSaMP在保证和速度上的优势。
摘要：
压缩感知具有广泛的应用，包括纠错、成像、雷达等。给定高维空间中的稀疏信号，我们希望从远小于其实际维数的线性测量中准确且高效地重建该信号。尽管理论上这是可能的，但难点在于构建高效的重建算法，以及确定哪种线性测量允许重建。目前有两种主要的稀疏恢复方法，各有优缺点。第一种是L1最小化方法，如基追踪，利用线性优化问题恢复信号。该方法提供了强保证和稳定性，但依赖于线性规划，其方法尚未有强多项式时间界限。第二种方法使用贪婪方法，迭代计算信号的支撑集。这些方法通常比基追踪快得多，但直到最近才能提供相同的保证。当我们开发并分析了贪婪算法正则化正交匹配追踪（ROMP）后，这两种方法之间的差距被弥合。ROMP提供了与基追踪类似的保证以及贪婪算法的速度。我们最近的算法压缩采样匹配追踪（CoSaMP）改进了这些保证，并在每个重要方面都是最优的。
英文摘要：
Compressed sensing has a wide range of applications that include error correction, imaging, radar and many more. Given a sparse signal in a high dimensional space, one wishes to reconstruct that signal accurately and efficiently from a number of linear measurements much less than its actual dimension. Although in theory it is clear that this is possible, the difficulty lies in the construction of algorithms that perform the recovery efficiently, as well as determining which kind of linear measurements allow for the reconstruction. There have been two distinct major approaches to sparse recovery that each present different benefits and shortcomings. The first, L1-minimization methods such as Basis Pursuit, use a linear optimization problem to recover the signal. This method provides strong guarantees and stability, but relies on Linear Programming, whose methods do not yet have strong polynomially bounded runtimes. The second approach uses greedy methods that compute the support of the signal iteratively. These methods are usually much faster than Basis Pursuit, but until recently had not been able to provide the same guarantees. This gap between the two approaches was bridged when we developed and analyzed the greedy algorithm Regularized Orthogonal Matching Pursuit (ROMP). ROMP provides similar guarantees to Basis Pursuit as well as the speed of a greedy algorithm. Our more recent algorithm Compressive Sampling Matching Pursuit (CoSaMP) improves upon these guarantees, and is optimal in every important aspect.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
