Exact Error in Matrix Completion: Approximately Low-Rank Structures and Missing Blocks

Abstract

We study the completion of approximately low rank matrices with entries missing not at random (MNAR). In the context of typical large-dimensional statistical settings, we establish a framework for the performance analysis of the nuclear norm minimization ($\ell_1^*$) algorithm. Our framework produces \emph{exact} estimates of the worst-case residual root mean squared error and the associated phase transitions (PT), with both exhibiting remarkably simple characterizations. Our results enable to {\it precisely} quantify the impact of key system parameters, including data heterogeneity, size of the missing block, and deviation from ideal low rankness, on the accuracy of $\ell_1^*$-based matrix completion. To validate our theoretical worst-case RMSE estimates, we conduct numerical simulations, demonstrating close agreement with their numerical counterparts.