On the Estimation of Latent Distances Using Graph Distances

TL;DR

This paper analyzes how accurately latent positions can be estimated from graph distances in geometric graphs, providing error bounds and matching them with information lower bounds in specific cases.

Contribution

It derives error bounds for latent position estimation using graph distances and establishes their optimality in certain scenarios with proportional link functions.

Findings

Error bounds match information lower bounds in simple cases

Derived bounds hold under various assumptions on the link function

Provides theoretical guarantees for latent position recovery

Abstract

We are given the adjacency matrix of a geometric graph and the task of recovering the latent positions. We study one of the most popular approaches which consists in using the graph distances and derive error bounds under various assumptions on the link function. In the simplest case where the link function is proportional to an indicator function, the bound matches an information lower bound that we derive.