Exact Label Recovery in Euclidean Random Graphs

TL;DR

This paper establishes the fundamental limits and proposes an efficient algorithm for exact label recovery in Euclidean random graphs, extending classical problems with a geometric perspective and demonstrating a local-to-global amplification phenomenon.

Contribution

It introduces a geometric extension of graph problems, determines the exact recovery threshold, and provides an efficient two-phase algorithm for label inference.

Findings

Exact recovery threshold characterized by Chernoff-Hellinger divergence.

Impossibility of recovery below the threshold using Cramér lower bound.

Efficient two-phase algorithm achieves exact recovery above the threshold.

Abstract

In this paper, we propose a family of label recovery problems on weighted Euclidean random graphs. The vertices of a graph are embedded in $\mathbb{R}^d$ according to a Poisson point process, and are assigned to a discrete community label. Our goal is to infer the vertex labels, given edge weights whose distributions depend on the vertex labels as well as their geometric positions. Our general model provides a geometric extension of popular graph and matrix problems, including submatrix localization and $\mathbb{Z}_2$-synchronization, and includes the Geometric Stochastic Block Model (proposed by Sankararaman and Baccelli) as a special case. We study the fundamental limits of exact recovery of the vertex labels. Under a mild distinctness of distributions assumption, we determine the information-theoretic threshold for exact label recovery, in terms of a Chernoff-Hellinger divergence criterion. Impossibility of recovery below the threshold is proven by a unified analysis using a Cram\'er lower bound. Achievability above the threshold is proven via an efficient two-phase algorithm, where the first phase computes an almost-exact labeling through a local propagation scheme, while the second phase refines the labels. The information-theoretic threshold is dictated by the performance of the so-called genie estimator, which decodes the label of a single vertex given all the other labels. This shows that our proposed models exhibit the local-to-global amplification phenomenon.