稀疏随机矩阵有限等距性质分析

摘要

Sparse random matrices have attractive properties, such as low storage requirement, low computational complexity in both encoding and recovery, easy incremental updates, and they show great advantages in distributed applications. To make sure sparse random matrices can be used as the measurement matrix, the Restricted Isometry Property (RIP) of such matrices is proved in this paper. Firstly, it is shown that the measurement matrix satisfies RIP is equivalent to the Gram matrix of its submatrix has all of eigenvalues around 1; then it is proved that sparse random matrices satisfy RIP with high probability provided the numbers of measurements satisfy certain conditions. Simulation results show that sparse random matrices can guarantee accurate reconstruction of original signal, while greatly reduce the time of measuring and reconstruction.%稀疏随机矩阵由于具有存储容量小、编码和重构复杂度低、易于更新等优良特性而适用于分布式应用。为确保稀疏随机矩阵可作为压缩感知观测矩阵，该文证明了稀疏随机矩阵的有限等距性质(RIP)。首先，证明了测量矩阵满足有限等距性质等价于其子矩阵的格拉姆矩阵特征值分布于1附近；在此基础上，证明了当测量值个数满足特定条件时，稀疏随机矩阵以接近于1的概率满足有限等距性质。仿真实验表明，稀疏随机矩阵在保证稀疏信号精确重建的同时，大大节约了测量和重建所需的时间。