A diffusion-map-based algorithm for gradient computation on manifolds and applications

• URL: http://arxiv.org/abs/2108.06988v5
• Date: Mon, 5 Jun 2023 08:09:57 GMT
• Title: A diffusion-map-based algorithm for gradient computation on manifolds and applications
• Authors: Alvaro Almeida Gomez, Ant\^onio J. Silva Neto, Jorge P. Zubelli
• Abstract summary: We recover the gradient of a given function defined on interior points of a Riemannian submanifold in the Euclidean space. This approach is based on the estimates of the Laplace-Beltrami operator proposed in the diffusion-maps theory.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We recover the Riemannian gradient of a given function defined on interior points of a Riemannian submanifold in the Euclidean space based on a sample of function evaluations at points in the submanifold. This approach is based on the estimates of the Laplace-Beltrami operator proposed in the diffusion-maps theory. The Riemannian gradient estimates do not involve differential terms. Analytical convergence results of the Riemannian gradient expansion are proved. We apply the Riemannian gradient estimate in a gradient-based algorithm providing a derivative-free optimization method. We test and validate several applications, including tomographic reconstruction from an unknown random angle distribution, and the sphere packing problem in dimensions 2 and 3.

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