AI 中文总结
该研究针对闭凸集上带状态依赖矩阵度量的变分不等式投影神经网络，改进了收敛估计，证明了局部与全局指数收敛，还发现了含非可积逆度量的周期轨道环带，经计算验证了相关结论。
AI 中文摘要
我们研究闭凸集上用于变分不等式的连续时间投影神经网络。一个依赖于状态的正定矩阵作为算子的预条件，其逆定义了投影度量。现有针对该流的收敛分析涵盖了由海森矩阵生成的逆度量和状态依赖标量度量；在前一种情况中，布雷格曼距离可消除度量导数项。本文处理的是其逆不属于上述两类的矩阵度量。我们证明了投影在其自变量和度量上的联合正则性。在常见谱界下，欧氏利普希茨估计从平方界改进为界本身。对于利普希茨强单调算子和利普希茨度量，我们证明了具有显式速率和半径的局部指数收敛。在紧可行集上，显式度量变化界产生全局指数收敛。二阶连续可微、一致正定的度量与强单调线性算子生成周期轨道的环带，该环带内逆度量违反海森可积性。确定性计算验证了解析公式并量化了两个收敛证书中的松弛量。
英文摘要
We study continuous-time projection neural networks for variational inequalities on closed convex sets. A positive definite matrix that depends on the state preconditions the operator, and its inverse defines the projection metric. Existing convergence analyses of this flow cover Hessian-generated inverse metrics and state-dependent scalar metrics. In the first case, a Bregman distance eliminates metric-derivative terms. We treat a matrix metric whose inverse is of neither kind. We prove joint regularity of the projection in its argument and metric. Under common spectral bounds, the Euclidean Lipschitz estimate improves from the squared bound to the bound itself. For Lipschitz strongly monotone operators and Lipschitz metrics, we prove local exponential convergence with explicit rate and radius. On compact feasible sets, an explicit metric-variation bound yields global exponential convergence. A twice continuously differentiable, uniformly positive definite metric and a strongly monotone linear operator produce an annulus of periodic orbits. The inverse metric violates Hessian integrability throughout the annulus. Deterministic computations confirm the analytic formulas and quantify slack in both convergence certificates.
Comments31 pages
