An Efficient Optimization Algorithm for Measurement Matrix Based on SVD and Improved Nesterov Accelerated Gradient

Abstract

In compressed sensing, a measurement matrix having low coherence with a specified sparse dictionary has been shown to be advantageous over a Gaussian random matrix in terms of reconstruction performance. In this paper the problem of efficiently designing the measurement matrix is addressed. The measurement matrix is designed by iteratively minimizing the difference between the Gram matrix of the sensing matrix and a target Gram matrix. A new target Gram matrix is designed by applying singular value decomposition to the sensing matrix and utilizing entry shrinking in the Gram matrix, leading to lower mutual coherence indicators. An improved Nesterov accelerated gradient algorithm is derived to update the measurement matrix, which can improve the convergence behavior. An efficient optimization algorithm for measurement matrix is proposed on the basis of alternating minimization. The experimental results and analysis show that the proposed algorithm performs well in terms of both computational complexity and reconstruction performance.