压缩感知邻域中的带限制性等距常数（英文）

Compressed Sensing with Restricted Isometry Property

众所周知,带限制性等距常数是压缩感知领域中的核心概念.在压缩感知理论发展的十几年历史中,几乎所有的重要理论结果都与这个概念密切相关.此文主要是总结近十余年来带限制性等距常数的若干重要结果,特别是最佳上界的发现.我们首先表明许多具有最少行数的随机矩阵满足这个性质,而一些确定性矩阵也满足这个性质.但是与随机矩阵相比,确定性矩阵的行数要明显多.其次,我们给出了刻画l_1优化模型范数最小解与最稀疏解等价性的最佳带限制性等距常数,对于l_p(0<p <1)优化模型也得到了类似结果.最后,我们延拓这些结果到低秩矩阵恢复以及在字典表示下具有稀疏信号恢复的情形.

It is well known that the restricted isometry property is in the center of com- pressed sensing. In the past decades, almost all important theories in this field are related to this notion. In this paper, we mainly review the restricted isometry property in compressed sensing in various ways, and summarize some theoretical results, in particular the latest find- ings. We first show that many random matrices satisfy RIP with fewer rows and some explicit matrices also satisfy RIP with more rows. Then for the popular models, we also give the opti- mal restricted isometry constants 5tk with arbitrary positive number t for/1-minimization and optimal restricted isometry constants 52k for gp-miuimization with 0 〈 p≤ 1. Furthermore, these results can also be extended to the low-rank matrix recovery problem and sparse recovery with redundant dictionary.