The planted matching problem: Sharp threshold and infinite-order phase transition
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We study the problem of reconstructing a perfect matching M^* hidden in a randomly weighted n× n bipartite graph. The edge set includes every node pair in M^* and each of the n(n-1) node pairs not in M^* independently with probability d/n. The weight of each edge e is independently drawn from the distribution 𝒫 if e ∈ M^* and from 𝒬 if e ∉ M^*. We show that if √(d) B(𝒫,𝒬) ≤ 1, where B(𝒫,𝒬) stands for the Bhattacharyya coefficient, the reconstruction error (average fraction of misclassified edges) of the maximum likelihood estimator of M^* converges to 0 as n→∞. Conversely, if √(d) B(𝒫,𝒬) ≥ 1+ϵ for an arbitrarily small constant ϵ>0, the reconstruction error for any estimator is shown to be bounded away from 0 under both the sparse and dense model, resolving the conjecture in [Moharrami et al. 2019, Semerjian et al. 2020]. Furthermore, in the special case of complete exponentially weighted graph with d=n, 𝒫=exp(λ), and 𝒬=exp(1/n), for which the sharp threshold simplifies to λ=4, we prove that when λ≤ 4-ϵ, the optimal reconstruction error is exp( - Θ(1/√(ϵ)) ), confirming the conjectured infinite-order phase transition in [Semerjian et al. 2020].
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