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- Title
On sharp stochastic zeroth-order Hessian estimators over Riemannian manifolds.
- Authors
Wang, Tianyu
- Abstract
We study Hessian estimators for functions defined over an |$n$| -dimensional complete analytic Riemannian manifold. We introduce new stochastic zeroth-order Hessian estimators using |$O (1)$| function evaluations. We show that, for an analytic real-valued function |$f$| , our estimator achieves a bias bound of order |$ O (\gamma \delta ^2) $| , where |$ \gamma $| depends on both the Levi–Civita connection and function |$f$| , and |$\delta $| is the finite difference step size. To the best of our knowledge, our results provide the first bias bound for Hessian estimators that explicitly depends on the geometry of the underlying Riemannian manifold. We also study downstream computations based on our Hessian estimators. The supremacy of our method is evidenced by empirical evaluations.
- Subjects
RIEMANNIAN geometry; FINITE differences; ANALYTIC functions; RIEMANNIAN manifolds
- Publication
Information & Inference: A Journal of the IMA, 2023, Vol 12, Issue 2, p787
- ISSN
2049-8764
- Publication type
Article
- DOI
10.1093/imaiai/iaac027