Matrix Completion With Covariate Information
研究了当有额外协变量时，如何从损坏数据中补全矩阵，提出一种结合Frobenius范数和核范数惩罚的估计方法，并给出理论收敛速度与实证效果。
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.