压缩感知和稀疏优化简介  

Introduction to compressive sensing and sparse optimization

作　　者：文再文[1] 印卧涛 刘歆[3] 张寅[2] 

机构地区：[1]上海交通大学数学系,自然科学研究院和科学工程计算教育部重点实验室,上海200240 [2]美国莱斯大学计算与应用数学系 [3]中国科学院数学与系统科学研究院,北京100190

出　　处：《运筹学学报》2012年第3期49-64,共16页Operations Research Transactions

摘　　要：介绍压缩感知和稀疏优化的基本概念、理论基础和算法概要,压缩感知利用原始信号的稀疏性,从远少于信号元素个数的测量出发,通过求解稀疏优化问题来恢复完整的原始稀疏信号,通过一个小例子展示这一过程,并以此说明压缩感知和稀疏优化的基本理念,接着简要介绍用以保证(?)_1凸优化恢复稀疏信号的零空间性质和RIP条件,最后介绍求解稀疏优化的几个经典算法。We briefly introduce the basic principle and theory of compressive sensing and sparse optimization. Compressive sensing is a new paradigm of signal acquisition, which senses a sparse signal by taking a set of incomplete measurements and recovers the signal by solving an optimization problem. This article first illustrates the compressive sensing paradigm through a synthetic example. Then we describe two sufficient conditions, the null space property and restricted isometry principle, for l1 convex minimization to give the sparsest solution. Finally, we summarize a few typical algorithms for solving the optimization models arising from compressive sensing.

关 键 词：压缩感知 稀疏优化 零空间性质 受限正交条件 紧缩算子 线性化近似点算法 分裂Bregman方法和交替方向增广拉格朗日函数法 Bregman方法和增广拉格朗日函数法 

分 类 号：O224[理学—运筹学与控制论] O29[理学—数学]