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

TL;DR

This paper introduces a diffusion-map-based method to estimate Riemannian gradients from sample data on manifolds, enabling derivative-free optimization with applications in tomography and sphere packing.

Contribution

It presents a novel gradient estimation technique on manifolds using diffusion maps, with proven convergence and practical applications in optimization problems.

Findings

Accurate Riemannian gradient estimates without differential terms

Successful application in tomographic reconstruction

Effective in sphere packing problems in low dimensions

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.