Random Graph Matching in Geometric Models: the Case of Complete Graphs
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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 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)=⟨ x, y ⟩ 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.
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