Recovery thresholds in the sparse planted matching problem

Abstract

We consider the statistical inference problem of recovering an unknown perfect matching, hidden in a weighted random graph, by exploiting the information arising from the use of two different distributions for the weights on the edges inside and outside the planted matching. A recent work has demonstrated the existence of a phase transition, in the large size limit, between a full and a partial recovery phase for a specific form of the weights distribution on fully connected graphs. We generalize and extend this result in two directions: we obtain a criterion for the location of the phase transition for generic weights distributions and possibly sparse graphs, exploiting a technical connection with branching random walk processes, as well as a quantitatively more precise description of the critical regime around the phase transition.

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

“…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].…”

“…This problem is first proposed by [5] and motivated from tracking moving objects in a video, such as flocks of birds, motile cells, or particles in a fluid. A slight variation of Definition 1 is studied in [24] for unipartite graphs, where M * is chosen uniformly at random from the set of all perfect matchings on the complete unipartite graph K n (with even n) and the edge set of G includes all n/2 node pairs in M * and each of the n 2 − n/2 node pairs not in M * independently with probability d n ; the edge weights are still independently distributed according to P for edges in M * and Q otherwise. In this paper, we present our results and analysis for bipartite graphs; nevertheless, the proof techniques can be straightforwardly extended to the unipartite version and all conclusions hold verbatim.…”

Strong recovery of geometric planted matchings

“…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.…”

“…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].…”