Nonconvex Matrix Completion with Linearly Parameterized Factors

Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones, with broad machine learning applications including collaborative filtering, pairwise ranking, etc. In practice, additional structures are usually employed in order to improve the accuracy of matrix completion. Examples include subspace constraints formed by side information in collaborative filtering, and skew symmetry in pairwise ranking. This paper performs a unified analysis of nonconvex matrix completion with linearly parameterized factorization, which covers the aforementioned examples as special cases. Importantly, uniform upper bounds for estimation errors are established for all local minima, provided that the sampling rate satisfies certain conditions determined by the rank, condition number, and incoherence parameter of the ground-truth low rank matrix. Empirical efficiency of the proposed method is further illustrated by numerical simulations

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

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