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Matrix Completion With Covariate Information

Mao, Xiaojun, Chen, Song Xi, Wong, Raymond K. W.
Journal of the American Statistical Association 2019 v.114 no.525 pp. 198-210
algorithms, equations, models, statistics
This article investigates the problem of matrix completion from the corrupted data, when the additional covariates are available. Despite being seldomly considered in the matrix completion literature, these covariates often provide valuable information for completing the unobserved entries of the high-dimensional target matrix A₀. Given a covariate matrix X with its rows representing the row covariates of A₀, we consider a column-space-decomposition model A₀ = Xβ₀ + B₀, where β₀ is a coefficient matrix and B₀ is a low-rank matrix orthogonal to X in terms of column space. This model facilitates a clear separation between the interpretable covariate effects (Xβ₀) and the flexible hidden factor effects (B₀). Besides, our work allows the probabilities of observation to depend on the covariate matrix, and hence a missing-at-random mechanism is permitted. We propose a novel penalized estimator for A₀ by utilizing both Frobenius-norm and nuclear-norm regularizations with an efficient and scalable algorithm. Asymptotic convergence rates of the proposed estimators are studied. The empirical performance of the proposed methodology is illustrated via both numerical experiments and a real data application.