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.

翻译：我们提出了一种针对有限度量空间数据集标量曲率的内蕴估计量。该估计量仅依赖于数据的度量结构，无需嵌入$\mathbb{R}^n$空间。我们证明了该估计量的一致性：对于从紧致黎曼流形上概率测度中采样的点，随着点数量的增加，该估计量收敛于标量曲率。为论证其在实际应用中的有效性，我们证明了该估计量对度量结构扰动（如样本噪声或内蕴度量估计误差）具有稳定性。通过在具有指定曲率的流形上采样的合成数据，我们实验验证了该估计量的性能。