Going off the Grid: Iterative Model Selection for Biclustered Matrix Completion

In the matrix completion problem, we seek to recover or estimate a matrix, when only a fraction of its entries are observed. While it is impossible to complete an arbitrary matrix using only partial observations of its entries, it may be possible to fully recover matrix entries when the matrix has an appropriate underlying structure. For example, most low-rank matrices can be completed accurately with high probability, by solving a convex optimization problem (Candés and Recht, 2009). Consequently, algorithms for lowrank matrix completion have enjoyed widespread use across many disciplines, including collaborative filtering and recommender systems (Koren et al., 2009), multi-task learning and classification (Amit et al., 2007; Argyriou et al., 2007; Wu and Lange, 2015), computer vision (Chen and Suter, 2004), statistical genetics (Chi et al., 2013), as well as remote sensing (Malek-Mohammadi et al., 2014). In this paper, we consider matrix completion under a structural assumption that is closely related to the low-rank assumption; i.e., we assume that the matrix entries vary "smoothly" with respect to a graphical organization of the rows and columns. For example, in the context of a movie recommendation system, we seek to complete a user-by-movies ratings matrix. We may have additional information about users, such as if pairs of users are friends on a social media application, as well as additional information from a movie database, such as the co-occurrence of certain film principles. We expect the entries of a movie ratings matrix to vary "smoothly" over a neighborhood of users, defined by a friendship graph, and over a neighborhood of movies, defined by a shared movie principles graph. When such local similarity structure exists, and is available, it behooves us to leverage this information to predict missing entries in a matrix.