Abstract: Aiming at the problem of sparse signal reconstruction, an optimization algorithm based on Recurrent Neural Network (RNN) was proposed. Firstly, the signal sparseness was represented, and the mathematical model was transformed into an optimization problem. Then, based on the fact that the l₀-norm is a non-convex and non-differentiable function, and the optimization problem is NP-hard, under the premise that the measurement matrix A met Restricted Isometry Property (RIP), the equivalent optimization problem was proposed. Finally, the corresponding Hopfield RNN model was established to solve the equivalent optimization problem, so as to reconstruct sparse signals. The experimental results show that under different observation number m, compared the RNN algorithm and the other three algorithms, it is found that the relative error of the RNN algorithm is smaller and the observations number is smaller, and the RNN algorithm can reconstruct the sparse signals efficiently.

Key words: l₀ optimization, Recurrent Neural Network (RNN), Restricted Isometry Property (RIP), energy function

摘要： 针对稀疏信号的重构问题，提出了一种基于反馈神经网络（RNN）的优化算法。首先，需要对信号进行稀疏表示，将数学模型化为优化问题；接着，基于l₀范数是非凸且不可微的函数，并且该优化问题是NP难的，因此在测量矩阵A满足有限等距性质（RIP）的前提下，提出等价优化问题；最后，通过建立相应的Hopfield反馈神经网络模型来解决等价的优化问题，从而实现稀疏信号的重构。实验结果表明，在不同观测次数m下，对比RNN算法和其他三种算法的相对误差，发现RNN算法相对误差小，且需要的观测数也少，能够高效地重构稀疏信号。

关键词: l₀最优化, 反馈神经网络, 有限等距性, 能量函数