  Pairwise comparison matrices have received substantial attention in a variety
of applications, especially in rank aggregation, the task of flattening items
into a one-dimensional (and thus transitive) ranking. However, non-transitive
preference cycles can arise in practice due to the fact that making a decision
often requires a complex evaluation of multiple factors. In some applications,
it may be important to identify and preserve information about the inherent
non-transitivity, either in the pairwise comparison data itself or in the
latent feature space. In this work, we develop structured models for
non-transitive pairwise comparison matrices that can be exploited to recover
such matrices from incomplete noisy data and thus allow the detection of
non-transitivity. Considering that individuals' tastes and items' latent
features may change over time, we formulate time-varying pairwise comparison
matrix recovery as a dynamic skew-symmetric matrix recovery problem by modeling
changes in the low-rank factors of the pairwise comparison matrix. We provide
theoretical guarantees for the recovery and numerically test the proposed
theory with both synthetic and real-world data.
