Random Graph Matching in Geometric Models: the Case of Complete Graphs

This paper studies the problem of matching two complete graphs with edge weights correlated through latent geometries, extending a recent line of research on random graph matching with independent edge weights to geometric models. Specifically, given a random permutation $π^*$ on $[n]$ and $n$ iid pairs of correlated Gaussian vectors $\{X_{π^*(i)}, Y_i\}$ in $\mathbb{R}^d$ with noise parameter $σ$, the edge weights are given by $A_{ij}=κ(X_i,X_j)$ and $B_{ij}=κ(Y_i,Y_j)$ for some link function $κ$. The goal is to recover the hidden vertex correspondence $π^*$ based on the observation of $A$ and $B$. We focus on the dot-product model with $κ(x,y)=\langle x, y \rangle$ and Euclidean distance model with $κ(x,y)=\|x-y\|^2$, in the low-dimensional regime of $d=o(\log n)$ wherein the underlying geometric structures are most evident. We derive an approximate maximum likelihood estimator, which provably achieves, with high probability, perfect recovery of $π^*$ when $σ=o(n^{-2/d})$ and almost perfect recovery with a vanishing fraction of errors when $σ=o(n^{-1/d})$. Furthermore, these conditions are shown to be information-theoretically optimal even when the latent coordinates $\{X_i\}$ and $\{Y_i\}$ are observed, complementing the recent results of [DCK19] and [KNW22] in geometric models of the planted bipartite matching problem. As a side discovery, we show that the celebrated spectral algorithm of [Ume88] emerges as a further approximation to the maximum likelihood in the geometric model.