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Abstract:
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This paper investigates the problem of matrix completion from corrupted data when additional covariates are available. 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=X*beta0+B0 where beta0 is a coefficient matrix and B0 is a low-rank matrix orthogonal to X in terms of column space. This model decomposes A0 orthogonally into two components allowing separation between the interpretable covariate effects and the flexible hidden factor effects. 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, and achieve its practical computation 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.
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