Reconstructing Riemannian Metrics From Random Geometric Graphs

TL;DR

This paper presents improved algorithms for reconstructing Riemannian metrics from sparse random geometric graphs, achieving near-optimal error bounds and efficient computation, advancing the understanding of metric recovery on manifolds.

Contribution

It introduces a new method for reconstructing Riemannian metrics from sparse graphs with improved efficiency and accuracy, surpassing previous dense-graph approaches.

Findings

Reconstruction algorithms work with average degree $n^{1/2}{ m polylog}(n)$.

The algorithms achieve near-optimal volumetric error bounds.

Running time is $O(n^2 { m polylog}(n))$, matching input size up to polylog factors.

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 $n^{1/2}{\rm polylog}(n)$, 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 $O(n^2\,{\rm polylog}(n))$ 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.