Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise
  consistency and global lower bounds
Nicolas Garcia Trillos Melanie Weber
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.