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.

翻译：从低维流形上的随机样本估计Hessian算子的常用方法涉及局部拟合二次多项式。尽管该方法被广泛使用，但尚不清楚该估计量是否会引入偏差，特别是在具有边界和非均匀采样的复杂流形中。其渐近行为的严格理论保证一直缺失。我们证明，在温和条件下，该估计量渐近收敛于Hessian算子，非均匀采样和曲率效应的影响可忽略不计，即使在边界附近也是如此。我们的分析框架简化了直接分析所需的大量计算。