Key Laboratory of Dependable Services Computing in Cyber-Physical Society (Chongqing) Ministry of Education, College of Computer Science, Chongqing University, Chongqing, 400044, China.
College of Electronic and Information Engineering, Southwest University, Chongqing, 400715, China.
Neural Netw. 2021 Aug;140:100-112. doi: 10.1016/j.neunet.2021.02.006. Epub 2021 Feb 27.
In this paper, we propose a smoothing inertial neurodynamic approach (SINA) which is used to deal with L-norm minimization problem to reconstruct sparse signals. Note that the considered optimization problem is nonsmooth, nonconvex and non-Lipschitz. First, the problem is transformed into a smooth optimization problem based on smoothing approximation method, and the Lipschitz property of gradient of the smooth objective function is discussed. Then, SINA based on Karush-Kuhn-Tucker (KKT) condition, smoothing approximation and inertial dynamical approach, is designed to handle smooth optimization problem. The existence, uniqueness, global convergence and optimality of the solution of the SINA are discussed by the Cauchy-Lipschitz-Picard theorem, energy function and KKT condition. In addition, for p=1, the SINA has a mean sublinear convergence rate O1∕t under some mild conditions. Finally, some numerical examples on sparse signal reconstruction and image restoration are given to illustrate the theoretical results and the efficiency of SINA.
在本文中，我们提出了一种平滑惯性神经动力学方法（SINA），用于处理 L-范数最小化问题以重建稀疏信号。请注意，所考虑的优化问题是非光滑、非凸和非 Lipschitz 的。首先，基于平滑逼近方法将问题转化为一个光滑优化问题，并讨论了光滑目标函数梯度的 Lipschitz 性质。然后，基于 Karush-Kuhn-Tucker（KKT）条件、平滑逼近和惯性动力方法，设计了 SINA 来处理光滑优化问题。通过柯西-黎曼-皮卡定理、能量函数和 KKT 条件，讨论了 SINA 的解的存在性、唯一性、全局收敛性和最优性。此外，对于 p=1，在一些较温和的条件下，SINA 具有均值次线性收敛速度 O1∕t。最后，给出了稀疏信号重建和图像恢复方面的一些数值示例，以验证 SINA 的理论结果和效率。
