Estimativa do tensor de Ricci sobre dados discretos

Abstract

The study of geometry in high-dimensional spaces, particularly in point clouds, presents challenges in the calculation and interpretation of geometric properties, such as the metric tensor and Ricci curvature. This work proposes an approach that integrates concepts from Riemannian geometry with machine learning tools to estimate these tensors in discrete data, building upon a finite difference-based method and extending it through an adaptation employing neural networks. The methodology was experimentally validated on two-dimensional surfaces, yielding results consistent with those of the reference work—that is, accurate estimates on surfaces with positive and constant curvature, but with limitations on surfaces with negative curvature. Meanwhile, experiments with the neural network-based adaptation showed inferior results when compared to the original method, though they were comparable, in certain aspects, to other exis- ting approaches, such as Ollivier-Ricci and Forman-Ricci curvatures. These results demonstrate the feasibility of the proposed approach and highlight important challenges to be addressed in future work.