Neurodynamic Algorithms With Finite/Fixed-Time Convergence for Sparse Optimization via ℓ1 Regularization

• Published in: IEEE transactions on systems, man, and cybernetics. Systems Vol. 54; no. 1; pp. 131 - 142
• Main Authors: Wen, Hongsong, He, Xing, Huang, Tingwen, Yu, Junzhi
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Convergence; finite-time stability (FTS); fixed-time stability (FxTS); Initial conditions; locally competitive algorithm (LCA); Minimization; Neurodynamics; Optimization; Recovery; Regularization; Signal processing algorithms; Signal reconstruction; Sliding mode control; sparse optimization; Stability analysis; Stability criteria; Upper bound; Upper bounds; ℓ₁ regularization
• ISSN: 2168-2216; 2168-2232
• DOI: 10.1109/TSMC.2023.3304850

Sparse optimization problems have been successfully applied to a wide range of research areas, and useful insights and elegant methods for proving the stability and convergence of neurodynamic algorithms have been yielded in previous work. This article develops several neurodynamic algorithms for sparse signal recovery by solving the <inline-formula> <tex-math notation="LaTeX">\ell _{1} </tex-math></inline-formula> regularization problem. First, in the framework of the locally competitive algorithm (LCA), modified LCA (MLCA) with finite-time convergence and MLCA with fixed-time convergence are designed. Then, the sliding-mode control (SMC) technique is introduced and modified, i.e., modified SMC (MSMC), which is combined with LCA to design MSMC-LCA with finite-time convergence and MSMC-LCA with fixed-time convergence. It is shown that the solutions of the proposed neurodynamic algorithms exist and are unique under the observation matrix satisfying restricted isometry property (RIP) condition, while finite-time or fixed-time convergence to the optimal points is shown via Lyapunov-based analysis. In addition, combining the notions of finite-time stability (FTS) and fixed-time stability (FxTS), upper bounds on the convergence time of the proposed neurodynamic algorithms are given, and the convergence results obtained for the MLCA and MSMC-LCA with fixed-time convergence are shown to be independent of the initial conditions. Finally, simulation experiments of signal recovery and image recovery are carried out to demonstrate the superior performance of the proposed neurodynamic algorithms.