Let $\mathcal{M} \subseteq \mathbb{R}^d$ denote a low-dimensional manifold and let $\mathcal{X}= \{ x_1, \dots, x_n \}$ be a collection of points uniformly sampled from $\mathcal{M}$. We study the relationship between the curvature of a random geometric graph built from $\mathcal{X}$ and the curvature of the manifold $\mathcal{M}$ via continuum limits of Ollivier's discrete Ricci curvature. We prove pointwise, non-asymptotic consistency results and also show that if $\mathcal{M}$ has Ricci curvature bounded from below by a positive constant, then the random geometric graph will inherit this global structural property with high probability. We discuss applications of the global discrete curvature bounds to contraction properties of heat kernels on graphs, as well as implications for manifold learning from data clouds. In particular, we show that the consistency results allow for characterizing the intrinsic curvature of a manifold from extrinsic curvature.

翻译：令 $\mathcal{M} \subseteq \mathbb{R}^d$ 表示一个低维流形，$\mathcal{X}= \{ x_1, \dots, x_n \}$ 为从 $\mathcal{M}$ 上均匀采样的点集。我们通过Ollivier离散Ricci曲率的连续极限，研究了由 $\mathcal{X}$ 构建的随机几何图曲率与流形 $\mathcal{M}$ 曲率之间的关系。我们证明了逐点、非渐近的一致性结果，并表明若 $\mathcal{M}$ 的Ricci曲率具有正常数下界，则随机几何图将以高概率继承这一全局结构性质。我们讨论了全局离散曲率下界在图热核收缩性质中的应用，以及对数据云流形学习的启示。特别地，我们证明了一致性结果能够通过外蕴曲率刻画流形的内蕴曲率。