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We consider the classic problem of establishing a statistical ranking of aset of n items given a set of inconsistent and incomplete pairwise comparisonsbetween such items. Instantiations of this problem occur in numerousapplications in data analysis (e.g., ranking teams in sports data), computervision, and machine learning. We formulate the above problem of ranking withincomplete noisy information as an instance of the group synchronizationproblem over the group SO(2) of planar rotations, whose usefulness has beendemonstrated in numerous applications in recent years. Its least squaressolution can be approximated by either a spectral or a semidefinite programming(SDP) relaxation, followed by a rounding procedure. We perform extensivenumerical simulations on both synthetic and real-world data sets, showing thatour proposed method compares favorably to other algorithms from the recentliterature. Existing theoretical guarantees on the group synchronizationproblem imply lower bounds on the largest amount of noise permissible in theranking data while still achieving exact recovery. We propose a similarsynchronization-based algorithm for the rank-aggregation problem, whichintegrates in a globally consistent ranking pairwise comparisons given bydifferent rating systems on the same set of items. We also discuss the problemof semi-supervised ranking when there is available information on the groundtruth rank of a subset of players, and propose an algorithm based on SDP whichrecovers the ranks of the remaining players. Finally, synchronization-basedranking, combined with a spectral technique for the densest subgraph problem,allows one to extract locally-consistent partial rankings, in other words, toidentify the rank of a small subset of players whose pairwise comparisons areless noisy than the rest of the data, which other methods are not able toidentify.

Sync-Rank: Robust Ranking, Constrained Ranking and Rank Aggregation via Eigenvector and Semidefinite Programming Synchronization