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Denoising distances beyond the volumetric barrier

Han Huang, Pakawut Jiradilok, Elchanan Mossel

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.

Denoising distances beyond the volumetric barrier

Abstract

We study the problem of reconstructing the latent geometry of a -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 coming from the fact that a generic point of the manifold typically lies at distance of order 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 up to polylogarithmic factors in in polynomial time. This strictly beats the volumetric barrier for dimensions . As a consequence of obtaining pointwise precision better than , we prove that the Gromov--Wasserstein distance between the reconstructed metric measure space and the true latent manifold is of order . 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.

Paper Structure

This paper contains 34 sections, 44 theorems, 840 equations, 1 table.

Table of Contents

  1. Introduction
  2. Formal statements of the main results
  3. Setting.
  4. Cluster generation
  5. Application to Gromov--Wasserstein distance of the underlying metric measure spaces
  6. Related work
  7. Proof overview
  8. Model, Assumptions, and basic events
  9. Metric Probability Space and Random Graph Model
  10. Setup
  11. Basic concentration events
  12. Geometric Comparison Tools
  13. Geometric Properties of Geodesics
  14. Order of distance estimation
  15. Ring Extraction
  16. ...and 19 more sections

Key Result

Theorem 1.1

Consider the Riemannian graph model (Definition def:graph_model) where $(M,\mu,\mathrm{p},\mathsf{s}_n)$ satisfies the Assumptions _assump: manifold, _def: mu, and _def: distance-probability, and $\mathsf{s}_n n \ge (\log n)^{\log\log n}$. Suppose there exists an $(r_0,\lambda_0,p_0)$ cluster genera and $R$ is an unknown dilation factor with $\max\{R,1/R\} \le C \max(L_{\mathrm{p}},1/\ell_\mathrm{

Theorems & Definitions (118)

  • Theorem 1.1: Distance Denoising
  • Remark 1.2
  • Remark 1.3: Comparison with the volumetric barrier
  • Remark 1.4: Lower bound on the error of distance estimation
  • Remark 1.5: Optimal limitation on the unknown $\mathrm{p}$ case
  • Definition 1.6: $p$-Wasserstein distance
  • Definition 1.7: $p$-Gromov--Wasserstein distance
  • Remark 1.8
  • Theorem 1.9: Empirical Wasserstein rate, cf. WB19
  • Remark 1.10
  • ...and 108 more