A Novel Projection Neural Network for Sparse Optimization With ${L_\mathrm{{1}}}$-Minimization Problem

In this paper, a novel projection neural network (PNN) for solving the
$L_{1}$
-minimization problem is proposed, which can be applied to sparse signal reconstruction and image reconstruction. First, a one-layer PNN is designed with the projection matrix and the projection operator, which is shown to be stable in the Lyapunov sense and converges globally to the optimal solution of the
$L_{1}$
-minimization problem. Then, the finite-time convergence of the proposed PNN is further investigated, with the upper bound on the convergence time given and the convergence rate analyzed. Finally, we make comparisons of our proposed PNN with the existing neural networks. Experimental results based on random Gaussian sparse signals demonstrate the effectiveness and performance of our proposed PNN. Moreover, the experiments on grayscale image reconstruction and color image reconstruction are further implemented, which sufficiently demonstrate the superiority of our proposed PNN.