An Intrinsic Approach to Scalar-Curvature Estimation for Point Clouds

TL;DR

This paper presents an intrinsic scalar curvature estimator for point clouds that relies solely on the metric structure, demonstrating consistency and stability, and validated through experiments on synthetic manifold data.

Contribution

The paper introduces a novel intrinsic scalar curvature estimator for finite metric spaces that does not depend on embedding, with proven consistency and stability properties.

Findings

Estimator converges to true scalar curvature on sampled manifolds.

Estimator remains stable under metric perturbations and noise.

Validated effectiveness on synthetic manifold data.

Abstract

We introduce an intrinsic estimator for the scalar curvature of a data set presented as a finite metric space. Our estimator depends only on the metric structure of the data and not on an embedding in $\mathbb{R}^n$. We show that the estimator is consistent in the sense that for points sampled from a probability measure on a compact Riemannian manifold, the estimator converges to the scalar curvature as the number of points increases. To justify its use in applications, we show that the estimator is stable with respect to perturbations of the metric structure, e.g., noise in the sample or error estimating the intrinsic metric. We validate our estimator experimentally on synthetic data that is sampled from manifolds with specified curvature.