  We consider the classic problem of establishing a statistical ranking of a
set of n items given a set of inconsistent and incomplete pairwise comparisons
between such items. Instantiations of this problem occur in numerous
applications in data analysis (e.g., ranking teams in sports data), computer
vision, and machine learning. We formulate the above problem of ranking with
incomplete noisy information as an instance of the group synchronization
problem over the group SO(2) of planar rotations, whose usefulness has been
demonstrated in numerous applications in recent years. Its least squares
solution can be approximated by either a spectral or a semidefinite programming
(SDP) relaxation, followed by a rounding procedure. We perform extensive
numerical simulations on both synthetic and real-world data sets, showing that
our proposed method compares favorably to other algorithms from the recent
literature. Existing theoretical guarantees on the group synchronization
problem imply lower bounds on the largest amount of noise permissible in the
ranking data while still achieving exact recovery. We propose a similar
synchronization-based algorithm for the rank-aggregation problem, which
integrates in a globally consistent ranking pairwise comparisons given by
different rating systems on the same set of items. We also discuss the problem
of semi-supervised ranking when there is available information on the ground
truth rank of a subset of players, and propose an algorithm based on SDP which
recovers the ranks of the remaining players. Finally, synchronization-based
ranking, combined with a spectral technique for the densest subgraph problem,
allows one to extract locally-consistent partial rankings, in other words, to
identify the rank of a small subset of players whose pairwise comparisons are
less noisy than the rest of the data, which other methods are not able to
identify.
