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Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds

Nicolas Garcia Trillos, Melanie Weber

TL;DR

This work develops nonasymptotic, pointwise consistency results that connect Ollivier's discrete Ricci curvature on random geometric graphs to the intrinsic Ricci curvature of a manifold from which data are sampled. It introduces data‑driven and geometry‑aware distance constructions $d_G$ and establishes global curvature lower bounds on graphs that mirror positive curvature on the manifold, with explicit rates and a scale‑dependent factor $s_K$. The results yield practical implications for heat kernel contraction on graphs and offer principled tools for curvature‑aware manifold learning from data clouds, including numerical demonstrations on spheres. The framework elegantly bridges discrete graph geometry with continuum Riemannian curvature, enabling intrinsic curvature estimation from extrinsic data and informing graph‑based learning and sampling dynamics. The approach relies on careful distance approximations $\hat{d}_g$, OT techniques, and a two‑scale analysis that glues local manifold geometry to the global graph structure.

Abstract

Let $M$ denote a low-dimensional manifold embedded in Euclidean space and let ${X}= \{ x_1, \dots, x_n \}$ be a collection of points uniformly sampled from it. We study the relationship between the curvature of a random geometric graph built from ${X}$ and the curvature of the manifold $M$ via continuum limits of Ollivier's discrete Ricci curvature. We prove pointwise, non-asymptotic consistency results and also show that if $M$ 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 our consistency results allow for estimating the intrinsic curvature of a manifold by first estimating concrete extrinsic quantities.

Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds

TL;DR

This work develops nonasymptotic, pointwise consistency results that connect Ollivier's discrete Ricci curvature on random geometric graphs to the intrinsic Ricci curvature of a manifold from which data are sampled. It introduces data‑driven and geometry‑aware distance constructions and establishes global curvature lower bounds on graphs that mirror positive curvature on the manifold, with explicit rates and a scale‑dependent factor . The results yield practical implications for heat kernel contraction on graphs and offer principled tools for curvature‑aware manifold learning from data clouds, including numerical demonstrations on spheres. The framework elegantly bridges discrete graph geometry with continuum Riemannian curvature, enabling intrinsic curvature estimation from extrinsic data and informing graph‑based learning and sampling dynamics. The approach relies on careful distance approximations , OT techniques, and a two‑scale analysis that glues local manifold geometry to the global graph structure.

Abstract

Let denote a low-dimensional manifold embedded in Euclidean space and let be a collection of points uniformly sampled from it. We study the relationship between the curvature of a random geometric graph built from and the curvature of the manifold via continuum limits of Ollivier's discrete Ricci curvature. We prove pointwise, non-asymptotic consistency results and also show that if 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 our consistency results allow for estimating the intrinsic curvature of a manifold by first estimating concrete extrinsic quantities.
Paper Structure (28 sections, 23 theorems, 161 equations, 3 figures, 1 table)

This paper contains 28 sections, 23 theorems, 161 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Background and Notation
  4. Notions from Differential Geometry
  5. Ollivier's Ricci curvature
  6. Curvature on Random Geometric Graphs
  7. Distance functions over $\mathcal{X}$
  8. Assumptions on $\hat{d}_g$
  9. Definitions of $d_G$
  10. Main Results and Discussion
  11. Poinwise Consistency
  12. Global Curvature Lower Bounds
  13. Numerical illustration of results
  14. Related Work
  15. Nonasymptotic Guarantees on Curvature Consistency
  16. ...and 13 more sections

Key Result

Proposition 1

Let $\varepsilon>0$ be a number smaller than $\iota_\mathcal{M}$, the injectivity radius of $\mathcal{M}$. Let $x, y \in \mathcal{M}$ be such that $d_\mathcal{M}(x,y) < \iota_\mathcal{M}$ and let $\tilde{x} \in B_\mathcal{M}(x,\varepsilon)$ and $\tilde{y} := \mathcal{P}(\tilde{x})$ with $\mathcal{P} where $v = \frac{\log_x(y)}{|\log_x(y)|}$ and $w= {\log_x(\tilde{x})}$.

Figures (3)

  • Figure 1: Levi-Civita parallelogram. All curves represent geodesics and the straight lines represent tangent vectors.
  • Figure 2: Plot of the function $\psi(t):= \frac{1}{4} (1-t)^3 + t\text{ if } 0 \leq t \leq 1t\text{ if } t >1,$ which satisfies all the required conditions in Assumption \ref{['assumpPsi']}.
  • Figure 3: Distribution of curvature values for a random geometric graph $G_\varepsilon$ constructed from $n$ data points sampled from the 2-sphere $\mathbb{S}^2 \subseteq \mathbb{R}^3$. The vertical axis is plotted in log-scale.

Theorems & Definitions (53)

  • Proposition 1: cf Proposition 6 in ollivier2009ricci
  • Definition 1: Embedded submanifold (see, e.g., boumal)
  • Definition 2: Ollivier's Ricci curvature on manifolds ollivier2009ricci
  • Theorem 2: Ollivier ollivier2009ricci
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 43 more