Tangent-Aligned Quadratic Regression for Tangent Space Estimation

Local quadratic fitting estimates tangent spaces by learning an orientation and polynomial coefficients. We study a tangent-aligned normal graph with an intercept and no normal linear term, so the optimized frame is the fitted tangent. Profiling the coefficients separates coefficient conditioning, residualized linear separation, and full local orientation curvature. We construct an exact quadratic saddle with nonsingular coefficient and linear designs but a singular profiled Hessian: predictor rotation cancels linear tilt information. A deformation gives curvature proportional to \(\delta^2\), and a perturbation bound preserves weak curvature without exact symmetry. Conversely, symmetric isotropic graphs receive a nonnegative curvature contribution. We also establish conditional noisy-anchor recovery and local optimization bounds. Experiments with observed-data safeguards cover synthetic and image neighborhoods. Matched ablations show improved noiseless quadratic reproduction from full optimization, but no uniform accuracy benefit under noise or cubic model mismatch. The results explain which orientation information survives coefficient refitting and distinguish local objective geometry from practical recovery guarantees.