A Note on Entrywise Consistency for Mixed-Data Matrix Completion

Abstract

This note studies matrix completion for a partially observed $n$ by $p$ data matrix involving mixed types of variables (e.g., continuous, binary, ordinal). A general family of non-linear factor models is considered, under which the matrix completion problem becomes the estimation of an $n$ by $p$ low-rank matrix ${\mathbf M}$. For existing methods in the literature, estimation consistency is established by showing $\Vert \hat {\mathbf M} - {\mathbf M}^*\Vert_F/\sqrt{np}$, the scaled Frobenius norm of the difference between the estimated and true ${\mathbf M}$ matrices, converges to zero in probability as $n$ and $p$ grow to infinity. However, this notion of consistency does not guarantee the convergence of each individual entry and, thus, may not be sufficient when specific data entries or the worst-case scenario is of interest. To address this issue, we consider the notion of entrywise consistency based on $\Vert \hat {\mathbf M} - {\mathbf M}^* \Vert_{\mbox{max}}$, the max norm of the estimation error matrix. We propose refinement procedures that turn estimators, which are consistent in the Frobenius norm sense, into entrywise estimators through a one-step refinement. Tight probabilistic error bounds are derived for the proposed estimators. The proposed methods are evaluated by simulation studies and real-data applications for collaborative filtering and large-scale educational assessment.