Recovery Guarantees for Time-varying Pairwise Comparison Matrices with Non-transitivity

Abstract

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.

Dynamic Ranking with the BTL Model: A Nearest Neighbor based Rank Centrality Method

“…As mentioned in Section 1, existing theoretical results for the dynamic ranking setup are limited, and the only works we are aware of are the recent results of Bong et al [2] and of Li and Wakin [15]. We now discuss these two results in more detail.…”

“…Comparison to Li and Wakin [15]. The model introduced by Li and Wakin [15] aims at recovering a pairwise comparison matrix X(T ) at a time T from noisy linear measurements of the matrices X(t) for t ∈ [T ].…”

“…Comparison to Li and Wakin [15]. The model introduced by Li and Wakin [15] aims at recovering a pairwise comparison matrix X(T ) at a time T from noisy linear measurements of the matrices X(t) for t ∈ [T ]. For each pair of items {i, j}, X ij (t) quantifies the preference of item j over item i at time t with X ij (t) > 0 if item j is preferred to item i.…”

“…A specificity of this model is that the preferences over the collection of items are supposed to depend on 2r latent factors, captured in the matrices S t ∈ R n×r and Q ∈ R n×r . The model considered in [15] assumes that…”

“…• Very recently, a couple of theoretical results were obtained for the dynamic ranking problem. Li and Wakin [15] include in their model the observed difference of scores (for example the score of a football game) which clearly contains richer information in comparison to simple binary outcomes. Bong et al [2] considered an adaptation of the BTL model to the dynamic setting similar to what has been described above.…”