Hanjoon Byun | Search arXiv

arXiv2026

Gauss-Newton Accuracy and Indefinite Hessians: Uniform Coexistence in Low-Cost Sets

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.

cs.LG↗

arXiv2026

Contact Geometry and Covariance Deficits in Volume-Sampled Least Squares

We classify when ordinary fixed-size volume sampling followed by unweighted least squares attains its sharp coefficient-covariance ceiling on a fixed design. For a real whitened design without coloops and a fixed positive-loss residual, the contact space is unchanged at every strict-interior sample size. Its possible nonzero values form a finite orthogonal family: each maximal parallel class of normalized Naimark-complement rows determines a deletion nullspace of dimension one less than the class size. A single residual attains an entire query precisely when the query range lies in one class space. The proof starts from two-sided Loewner comparison of every normalized covariance deficit with an explicit leave-one-out operator, using supported omission moments and reverse deletion. Residual augmentation provides resolvent and second-moment upper bounds, while complement geometry yields query-specific margins, angular concentration, local alignment, and a multi-output energy obstruction. Exact families give closed-form margins and covariances, exhibit support-boundary jumps, and approach the ceiling despite a uniformly positive geometric margin. Finally, the same moment identities give upper and lower bounds on expected fixed-query squared-loss excess. The subset draw is the only randomness; all support and endpoint restrictions are explicit.

cs.LG↗