Gauss-Newton精度与不定Hessian：低成本集合中的一致共存
Gauss-Newton Accuracy and Indefinite Hessians: Uniform Coexistence in Low-Cost Sets
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中文总结 AI 辅助
研究岭正则化非线性最小二乘中Gauss-Newton曲率精度，证明低成本集合中全局极小值点与不定Hessian点共存，并给出尖锐相对误差界与结构性质。
中文摘要 AI 辅助
我们研究了岭正则化非线性最小二乘中Gauss-Newton曲率的精度。在局部正则性以及沿精确拟合截面的水平集曲率幅度持续性条件下，我们证明了两种曲率机制的一致共存。全局极小值点存在，且每个全局极小值点的相对Hessian误差低于$(1+\sqrt2)/8$，而同一低成本集合中包含一个具有不定Hessian且相对误差至少为$15/8$的点。一个正的岭上限适用于固定邻域内的所有独立中心与标签扰动，以及直至该上限的所有正岭权重。这些邻域不会随着岭权重趋于零而收缩。基于当前预测水平集的逐点证书控制Hessian修正的法向、混合和切向部分。当预测映射和岭变化时，我们在所述逐点类别上证明了尖锐的相对误差界。解析示例描述了输出对齐、曲率方向和持续性的作用。另一个独立的结构性结果给出了在整个低成本集合中完整的Jacobian行秩，以及在秩亏参考点附近的精确插值。
英文摘要
We study the accuracy of Gauss-Newton curvature in ridge-regularized nonlinear least squares. Under local regularity and persistence of level-set curvature magnitude along an exact-fit section, we prove uniform coexistence of two curvature regimes. Global minimizers exist, and every global minimizer has relative Hessian error below $(1+\sqrt2)/8$, while the same low-cost set contains a point with an indefinite Hessian and relative error at least $15/8$. One positive ridge cap works for all independent center and label perturbations in fixed neighborhoods and every positive ridge weight up to the cap. These neighborhoods do not shrink as the ridge weight tends to zero. A pointwise certificate based on the current prediction level set controls the normal, mixed, and tangent parts of the Hessian correction. We prove a sharp relative-error bound over the stated pointwise class when the prediction map and ridge vary. Analytic examples describe the roles of output alignment, curvature orientation, and persistence. A separate structural result gives full Jacobian row rank throughout low-cost sets and exact interpolation near a rank-deficient reference.
