On sharp stochastic zeroth-order Hessian estimators over Riemannian manifolds

Abstract We study Hessian estimators for functions defined over an $n$-dimensional complete analytic Riemannian manifold. introduce new stochastic zeroth-order using $O (1)$ function evaluations. show that, real-valued $f$, our estimator achieves a bias bound of order $ O ( \gamma \delta ^2 ) $, where depends on both the Levi–Civita connection and $\delta is finite difference step size. To best knowledge, results provide first that explicitly geometry underlying also downstream computations based estimators. The supremacy method evidenced by empirical