The planted matching problem: Sharp threshold and infinite-order phase transition

Abstract

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 √ dB(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 √ dB(P, Q) ≥ 1 + ǫ for an arbitrarily small constant ǫ > 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 [20,24]. 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−ǫ, the optimal reconstruction error is exp (−Θ(1/ √ ǫ)), confirming the conjectured infinite-order phase transition in [24].

Strong recovery of geometric planted matchings

“…As n → ∞ the prefactor and the constant term in the exponent denominator are irrelevant, so this predicts a critical transition at n = exp(d/4σ 2 ), or σ 2 = 1 4 d log n . Per our discussion above, this is the correct strong recovery threshold for the MLE; indeed, the proof of the positive results in [18] goes by analyzing the MLE, so this is not surprising. However, the greedy algorithm applied to this model (to agree with the setting of [18], we should think of the input as the matrix W with entries distributed roughly according to P and Q, rather than the "raw" point sets {x i } and {y i }) achieves a better strong recovery threshold of σ 2 = 1 2 d log n .…”

“…The independent Gaussian limit discussed above falls in the range of models treated by previous works [18,36,42]. In particular, it was predicted in [36,42] and proved for certain models (not including the Gaussian model of P and Q above) in [18] that the strong recovery threshold in such a model should correspond to √ nB(P, Q) = 1, where B(P, Q) is the Bhattacharyya coefficient.…”

“…The limits of recovering planted matchings under independent weights are increasingly well understood. These models exhibit a phase transition in the recoverability of π , which was conjectured by [15], proved in a special case by [36], and studied in greater detail and generality by [18,42]. The approach of [36] in particular may be viewed as an extension to the planted setting of an earlier line of work studying optimal matchings under i.i.d.…”

“…However, the greedy algorithm applied to this model (to agree with the setting of [18], we should think of the input as the matrix W with entries distributed roughly according to P and Q, rather than the "raw" point sets {x i } and {y i }) achieves a better strong recovery threshold of σ 2 = 1 2 d log n . Essentially the same is noted in Remark 1 of [18], where the authors bring up a similar independent Gaussian model as an instance where the Bhattacharyya coefficient does not give a correct prediction.…”

“…Models of this type have attracted significant recent interest in the computer science and statistics communities, and precise results are now known in a number of different settings [18,36,42]. Despite this progress, however, the original problem of recovering planted geometric matchings to our knowledge has not received any attention since its proposal by [15].…”