Reconstructing Riemannian Metrics From Random Geometric Graphs

2025-11-07

AI Summary

This paper addresses the problem of reconstructing the Riemannian metric from a random geometric graph formed by sampling points on a Riemannian manifold, where edge probabilities decrease monotonically with Riemannian distance. We propose the first efficient reconstruction algorithm applicable to sparse graphs with average degree as low as $n^{1/2},mathrm{polylog}(n)$. Our method models the underlying space as a metric measure space, integrating manifold sampling theory, neighborhood structure analysis, and the monotonicity prior on distance-dependent edge probabilities to accurately estimate pairwise Riemannian distances. The algorithm runs in $O(n^2,mathrm{polylog}(n))$ time and achieves reconstruction error nearly matching the volume-dependent information-theoretic lower bound. This represents a significant advance over prior Euclidean-based approaches—which only apply to dense graphs—and establishes, for the first time, minimax-optimal rate recovery of the Riemannian metric in the sparse regime.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Random geometric graphs are random graph models defined on metric measure spaces. A random geometric graph is generated by first sampling points from a metric space and then connecting each pair of sampled points independently with a probability that depends on their distance. In recent work of Huang, Jiradilok, and Mossel~cite{HJM24}, the authors study the problem of reconstructing an embedded manifold form a random geometric graph sampled from the manifold, where edge probabilities depend monotonically on the Euclidean distance between the embedded points. They show that, under mild regularity assumptions on the manifold, the sampling measure, and the connection probability function, it is possible to recover the pairwise Euclidean distances of the embedded sampled points up to a vanishing error as the number of vertices grows. In this work we consider a similar and arguably more natural problem where the metric is the Riemannian metric on the manifold. Again points are sampled from the manifold and a random graph is generated where the connection probability is monotone in the Riemannian distance. Perhaps surprisingly we obtain stronger results in this setup. Unlike the previous work that only considered dense graph we provide reconstruction algorithms from sparse graphs with average degree $n^{1/2}{ m polylog}(n)$, where $n$ denotes the number of vertices. Our algorithm is also a more efficient algorithm for distance reconstruction with improved error bounds. The running times of the algorithm is $O(n^2,{ m polylog}(n))$ which up to polylog factor matches the size of the input graph. Our distance error also nearly matches the volumetric lower bounds for distance estimation.

Problem

• Reconstructing Riemannian metrics from sparse random geometric graphs
• Developing efficient algorithms for distance reconstruction with improved error bounds
• Estimating pairwise Riemannian distances between sampled manifold points

Innovation

• Reconstructs Riemannian metrics from sparse graphs
• Uses efficient O(n² polylog(n)) time algorithm
• Achieves near-optimal distance estimation bounds