🤖 AI Summary
This study addresses the open question of whether high-precision Gauss–Newton curvature and indefinite Hessians can uniformly coexist within low-cost sets under ridge regularization in nonlinear least squares. By integrating local regularity analysis, level-set curvature persistence theory, and structural analysis of Jacobian row rank, this work constructs a pointwise certificate mechanism to characterize error bounds. It provides the first proof that high-accuracy curvature and indefinite Hessians coexist uniformly within such low-cost sets. Furthermore, it establishes the applicability of a single positive ridge upper bound over fixed neighborhoods that does not shrink with weight decay. A theoretical upper bound of (1+√2)/8 on the relative Hessian error is derived, while an indefinite counterexample yielding an error of at least 15/8 is constructed to verify the tightness of this bound.
📝 Abstract
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.