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Coarse Ricci curvature of weighted Riemannian manifolds

Marc Arnaudon, Xue-Mei Li, Benedikt Petko

TL;DR

The paper links coarse curvature defined via Wasserstein-1 distances of non-uniform weighted local measures to the generalized Ricci tensor on weighted Riemannian manifolds. By combining precise manifold-distance estimates (via Jacobi fields and geodesic variations) with carefully crafted approximate transport maps and density analyses, it derives an asymptotic expansion that recovers $\mathrm{Ric}_{x}(v,v) + 2\mathrm{Hess}_{x}V(v,v)$. As an application, it extends previous results on the convergence of coarse curvature for random geometric graphs from Poisson samples to the non-uniform intensity setting $e^{-V}$, establishing convergence of the graph curvature to the generalized Ricci curvature. The work thus provides a robust framework to infer smooth geometric invariants from discrete, non-uniform samples, with implications for curvature estimation in manifold learning and graph-based models.

Abstract

We show that the generalized Ricci tensor of a weighted complete Riemannian manifold can be retrieved asymptotically from a scaled metric derivative of Wasserstein 1-distances between normalized weighted local volume measures. As an application, we demonstrate that the limiting coarse curvature of random geometric graphs sampled from Poisson point process with non-uniform intensity converges to the generalized Ricci tensor.

Coarse Ricci curvature of weighted Riemannian manifolds

TL;DR

The paper links coarse curvature defined via Wasserstein-1 distances of non-uniform weighted local measures to the generalized Ricci tensor on weighted Riemannian manifolds. By combining precise manifold-distance estimates (via Jacobi fields and geodesic variations) with carefully crafted approximate transport maps and density analyses, it derives an asymptotic expansion that recovers . As an application, it extends previous results on the convergence of coarse curvature for random geometric graphs from Poisson samples to the non-uniform intensity setting , establishing convergence of the graph curvature to the generalized Ricci curvature. The work thus provides a robust framework to infer smooth geometric invariants from discrete, non-uniform samples, with implications for curvature estimation in manifold learning and graph-based models.

Abstract

We show that the generalized Ricci tensor of a weighted complete Riemannian manifold can be retrieved asymptotically from a scaled metric derivative of Wasserstein 1-distances between normalized weighted local volume measures. As an application, we demonstrate that the limiting coarse curvature of random geometric graphs sampled from Poisson point process with non-uniform intensity converges to the generalized Ricci tensor.
Paper Structure (9 sections, 39 theorems, 240 equations, 5 figures)

This paper contains 9 sections, 39 theorems, 240 equations, 5 figures.

Table of Contents

  1. Introduction and result
  2. Manifold distance estimates
  3. Pointwise transport distance estimate
  4. Projection distance estimate
  5. Wasserstein distance approximations
  6. Density estimates
  7. An approximate transport map
  8. Application to random geometric graphs
  9. Appendix

Key Result

Theorem 1.1

For any point $x_0 \in M$, vector $v \in T_{x_0}M$ with $\|v\| =1$, sufficiently small $\delta, \varepsilon > 0$ and $y := \exp_{x_0}(\delta v)$, it holds that

Figures (5)

  • Figure 1: Geodesic variation $c_1$ (with positive sectional curvature in the $v,w$-plane)
  • Figure 2: Geodesic variation $c_2$
  • Figure 3: Geodesic variation $c_3$
  • Figure 4: For small $\varepsilon>0$, the densities of the non-uniform measures $\bar{\nu}_{x_0}^\varepsilon, \bar{\nu}_y^\varepsilon$ resemble cylinders with a slant top, the slanting being described by the gradient of $V$. This motivates the choice of $T$ as a scaled difference of the gradients of the tops, together with parallel translation from $B_\varepsilon(x_0)$ to $B_\varepsilon(y)$.
  • Figure 5: Illustration of the relationship between $T_* \bar{\nu}_{x_0}^\varepsilon$ and $\bar{\nu}_y^\varepsilon$. The two measures are equivalent with mutual density of the form $1+O(\delta \varepsilon^2)$.

Theorems & Definitions (84)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.2
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Proposition 2.7
  • ...and 74 more