一种基于变量约简的稀疏优化算法

A sparse optimization algorithm based on variable reduction

压缩感知理论能够为处理大规模信号数据提供有效支持.压缩感知中信号的稀疏表示和稀疏重构问题本质是一个稀疏优化问题,该问题是要从满足欠定方程组约束的无穷多解中找到稀疏度最大的解.鉴于此,提出一种基于变量约简求解压缩感知中稀疏优化问题的算法(VRSO),变量约简从欠定方程组约束中挖掘出变量关系,将变量分为核心变量和约简变量并用核心变量表示约简变量,通过设置核心变量中元素为0,将求解整个变量解空间上的最小化问题简化为求解约简变量解空间上的最小化问题.所提出算法通过原子与观测信号的内积大小对核心变量集合进行迭代更新,并找出优化问题的1组稀疏解.实验结果表明,所提出算法的重构误差和稀疏度误差优于匹配追踪算法、正交匹配追踪算法、迭代硬阈值算法等5种所选的对比算法,所求解的信号精度更高、稀疏度更好.

Compressed sensing provides an effective support for processing large scale signal data. The problem of sparse signal representation and sparse signal reconstruction in compressed sensing is essentially a sparse optimization problem,which aims to find the sparsest solution from the infinite solutions that satisfy the constraint of underdetermined system of equations. This paper proposes an algorithm based on variable reduction to solve the sparse optimization problem in compressed sensing(VRSO). Variable reduction extracts the relationships between variables from the constraint of the underdetermined system of equations, and divides variables into core variables and reduced variables. During the calculation, the core variables are always used to represent the reduced variables. By setting the elements in the core variables to be 0, the minimization problem in the whole variable solution space is simplified to the solution space of reduced variables. This algorithm updates core variables in terms of the inner product of atoms and observation signal, so as to find a group of sparse solutions. According to the experimental results, the reconstruction error and sparsity error of VRSO are better than other comparative algorithms such as matching pursuit, orthogonal matching pursuit and iterative hard thresholding. The results show that the signal obtained by VRSO has higher precision and better sparsity.