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

2021/08/16 by Alvaro Almeida Gomez, Gomez, Alvaro Almeida, Antônio José da Silva Neto +3
Mathematics · #49N45 #65J22 #65K05 #68T01 #68T20 #90C53 #94A08 #FOS: Computer and information sciences #Machine Learning (cs.LG) #Morphological variations and asymmetry

paper · pdf · doi:10.48550/arxiv.2108.06988

openalex publication_date 2021/08/16 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

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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