Neurodynamic Flow Approach for Convex and Quasi-Convex Optimization on Riemannian Manifolds With Diagonal Metrics
研究了对角度量黎曼流形上的神经动力流方法，用于求解凸和拟凸优化问题，并证明了全局收敛性，可应用于压缩感知中的稀疏信号与图像重建。
In recent years, Riemannian geometry, as an import- ant tool for designing and analyzing continuous trajectory flows, has been widely applied to solve nonlinear programming problems due to its ability to inspire the design of a new class of approaches for solving such problems. We first review some properties of invariant Riemannian metrics, then investigate the diagonal metric defined in general on the manifold <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {R}_{+ +}^{n}, $ </tex-math></inline-formula> some geometric properties are derived, and finally a class of neurodynamic flow approaches are also proposed. The global existence and feasibility results of the produced neurodynamic flow approaches are obtained and the results can be further extended to a more universal smooth function. The asymptotical behaviors of proposed neurodynamic flow approaches are also analyzed and studied, and global convergence of the proposed neurodynamic flow approaches for seeking the minimum point of the convex and quasi-convex minimum problems on the non-negative orthant domain is developed. In addition, the derived convergence results can directly be adapted to the competitive Cohen–Grossberg neural networks as well as to image and signal processing in compressive sensing. The correctness and superiority of our proposed neurodynamic flow approaches are verified by some sparse signal and image reconstruction examples.