Denoising distances beyond the volumetric barrier

2026-03-31

AI Summary

This work addresses the reconstruction of the intrinsic geometry of a $d$-dimensional Riemannian manifold from noisy pairwise distances or random geometric graphs. To overcome the so-called volumetric barrier—traditionally limiting pointwise distance estimation accuracy to $n^{-1/d}$ due to sample spacing—the authors propose the Orthogonal Ring Distance Estimation Routine (ORDER). This method achieves, for the first time when $d > 5$, an improved estimation rate of $n^{-2/(d+5)}$, matching the Wasserstein convergence rate of the empirical measure. By integrating orthogonal ring denoising, Gromov–Wasserstein metric analysis, and random geometric graph theory, the reconstructed metric-measure space attains a Gromov–Wasserstein distance of $n^{-1/d}$ to the true manifold, thereby asymptotically matching the performance achievable with a complete distance matrix.

Abstract

We study the problem of reconstructing the latent geometry of a $d$-dimensional Riemannian manifold from a random geometric graph. While recent works have made significant progress in manifold recovery from random geometric graphs, and more generally from noisy distances, the precision of pairwise distance estimation has been fundamentally constrained by the volumetric barrier, namely the natural sample-spacing scale $n^{-1/d}$ coming from the fact that a generic point of the manifold typically lies at distance of order $n^{-1/d}$ from the nearest sampled point. In this paper, we introduce a novel approach, Orthogonal Ring Distance Estimation Routine (ORDER), which achieves a pointwise distance estimation precision of order $n^{-2/(d+5)}$ up to polylogarithmic factors in $n$ in polynomial time. This strictly beats the volumetric barrier for dimensions $d > 5$. As a consequence of obtaining pointwise precision better than $n^{-1/d}$, we prove that the Gromov--Wasserstein distance between the reconstructed metric measure space and the true latent manifold is of order $n^{-1/d}$. This matches the Wasserstein convergence rate of empirical measures, demonstrating that our reconstructed graph metric is asymptotically as good as having access to the full pairwise distance matrix of the sampled points. Our results are proven in a very general setting which includes general models of noisy pairwise distances, sparse random geometric graphs, and unknown connection probability functions.

Problem

• manifold reconstruction
• random geometric graph
• distance denoising
• volumetric barrier
• latent geometry

Innovation

• ORDER
• volumetric barrier
• manifold reconstruction
• random geometric graph
• Gromov–Wasserstein distance