A projection-based neurodynamic system method with convergence guarantees for 𝐿 _𝑞 -regularized (0 < 𝑞 < 1) sparse recovery problem

Abstract Projection-based neurodynamic systems have recently been used for sparse reconstruction, but most existing designs focus on convex <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> L_{1} models and may be suboptimal for highly sparse signals. This paper develops an inertial projection-based neurodynamic system (PBNS) for the nonconvex <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> L_{q} -regularized ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>q</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> 0&lt;q&lt;1 ) sparse recovery problem. By applying a standard variable splitting and introducing a smooth absolute-value regularization, we transform the original model into the minimization of a continuously differentiable objective over a closed convex set, which enables a rigorous dynamical-system formulation. We design a two-layer inertial PBNS driven by the projected gradient mapping and obtain an implementable reconstruction algorithm via time discretization. The proposed dynamics is shown to be globally well-posed (existence and uniqueness of trajectories), and its trajectory converges asymptotically to the equilibrium set under a cocoercivity-type condition, with an additional local convergence result under local cocoercivity. Extensive experiments on synthetic compressed sensing data and real fetal ECG signals demonstrate that the proposed PBNS- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> L_{q} method is stable and achieves improved reconstruction accuracy and efficiency compared with representative convex and nonconvex baselines.