A common method for estimating the Hessian operator from random samples on a
low-dimensional manifold involves locally fitting a quadratic polynomial. Although
widely used, it is unclear if this estimator introduces bias, especially in complex
manifolds with boundaries and nonuniform sampling. Rigorous theoretical guarantees
of its asymptotic behavior have been lacking. We show that, under mild conditions,
this estimator asymptotically converges to the Hessian operator, with nonuniform
sampling and curvature effects proving negligible, even near boundaries. Our
analysis framework simplifies the intensive computations required for direct
analysis.
