自适应梯度下降观测矩阵优化算法

Adaptive gradient descent optimization algorithm for measurement matrix

基于可以通过减小压缩感知中观测矩阵与稀疏矩阵之间的互相关性来提高信号的重构质量,结合无约束凸优化问题中梯度下降的思想,提出了一种自适应梯度下降算法(adaptive gradient descent,AGD)。首先利用等角紧框架(equiangular tight frame,ETF)收缩传感矩阵的Gram矩阵,然后通过收缩得到的Gram矩阵建立一个无约束凸优化问题,最后通过梯度下降方法求解无约束凸优化问题进而得到优化后的观测矩阵。AGD算法通过每次更新梯度下降的方向,使Gram矩阵能够在最短时间内逼近ETF。仿真实验表明,该算法不仅迭代次数少,且优化后的观测矩阵与稀疏矩阵之间的互相关性大大降低。与传统的优化算法相比,信号恢复效果更好。

Based on the mutual correlation between measurement matrix and sparse matrix in compressed sensing（CS）, it can improve the quality of reconstructed signal by reducing the correlation coefficient. Combining with the gradient descent idea of unconstrained convex optimization problem, this paper proposed an adaptive gradient descent（AGD） algorithm. First, it shrinked the Gram matrix of sensing matrix using equiangular tight frame（ETF） theory. Then, it established an unconstrained convex optimization problem over Gram matrix. Finally, it could obtain the optimized measurement matrix by the adaptive gradient descent method. By updating the direction of gradient descent during each iteration, it could make the Gram matrix approximate ETF in the shortest time. Simulation results show that this algorithm not only requires fewer iterations, but reduces the mutual correlation greatly between measurement matrix and sparse matrix. The proposed method demonstrates better performance than conventional optimization methods.