Aligned Matrix Completion: Integrating Consistency and Independency in Multiple Domains

Matrix completion is the task of recovering a data matrix from a sample of entries, and has received significant attention in theory and practice. Normally, matrix completion considers a single matrix, which can be a noisy image or a rating matrix in recommendation. In practice however, data is often obtained from multiple domains rather than a single domain. For example, in recommendation, multiple matrices may exist as user x movie and user x book, while correlations among the multiple domains can be reasonably exploited to improve the quality of matrix completion. In this paper, we consider the problem of aligned matrix completion, where multiple matrices are recovered that correspond to different representations of the same group of objects. In the proposed model, we maintain consistency of multiple domains with a shared latent structure, while allowing independent patterns for each separate domain. In addition, we impose the low-rank structure of a matrix with a novel regularizer which provides better approximation than the standard nuclear norm relaxation.