Mixture Matrix Completion
This addresses a more accurate modeling need for recommender systems and other completion and clustering problems, representing an incremental advancement over existing low-rank and column-based mixture models.
The paper tackles the problem of matrix completion by introducing a more general model called mixture matrix completion (MMC), where each entry corresponds to one of several low-rank matrices, and shows it is theoretically possible with derived identifiability conditions and sample complexity, while providing a practical algorithm with performance comparable to state-of-the-art on synthetic and real data.
Completing a data matrix X has become an ubiquitous problem in modern data science, with applications in recommender systems, computer vision, and networks inference, to name a few. One typical assumption is that X is low-rank. A more general model assumes that each column of X corresponds to one of several low-rank matrices. This paper generalizes these models to what we call mixture matrix completion (MMC): the case where each entry of X corresponds to one of several low-rank matrices. MMC is a more accurate model for recommender systems, and brings more flexibility to other completion and clustering problems. We make four fundamental contributions about this new model. First, we show that MMC is theoretically possible (well-posed). Second, we give its precise information-theoretic identifiability conditions. Third, we derive the sample complexity of MMC. Finally, we give a practical algorithm for MMC with performance comparable to the state-of-the-art for simpler related problems, both on synthetic and real data.