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

• URL: http://arxiv.org/abs/2202.10662v2
• Date: Wed, 23 Feb 2022 20:25:07 GMT
• Title: Random Graph Matching in Geometric Models: the Case of Complete Graphs
• Authors: Haoyu Wang, Yihong Wu, Jiaming Xu, Israel Yolou
• Abstract summary: This paper studies the problem of matching two complete graphs with edge weights correlated through latent geometries. We derive an approximate maximum likelihood estimator, which provably achieves, with high probability, perfect recovery of $pi*$. 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.
• Score: 21.689343447798677
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\pi^*$ on $[n]$ and $n$ iid pairs of correlated Gaussian vectors $\{X_{\pi^*(i)}, Y_i\}$ in $\mathbb{R}^d$ with noise parameter $\sigma$, the edge weights are given by $A_{ij}=\kappa(X_i,X_j)$ and $B_{ij}=\kappa(Y_i,Y_j)$ for some link function $\kappa$. The goal is to recover the hidden vertex correspondence $\pi^*$ based on the observation of $A$ and $B$. We focus on the dot-product model with $\kappa(x,y)=\langle x, y \rangle$ and Euclidean distance model with $\kappa(x,y)=\|x-y\|^2$, 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 $\pi^*$ when $\sigma=o(n^{-2/d})$ and almost perfect recovery with a vanishing fraction of errors when $\sigma=o(n^{-1/d})$. 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.

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