A Note on Entrywise Consistency for Mixed-data Matrix Completion

Yunxiao Chen; Xiaoou Li

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.

A Note on Entrywise Consistency for Mixed-data Matrix Completion

Abstract