Sharp Bounds for Multiple Models in Matrix Completion

TL;DR

This paper uses advanced matrix concentration inequalities to eliminate the dimensional factor in convergence rates for matrix completion, achieving minimax optimality for key estimators in high-dimensional settings.

Contribution

It introduces a refined spectral norm analysis leveraging recent inequalities to remove dimensional factors, establishing minimax optimality of matrix completion estimators.

Findings

Eliminates dimensional factor in convergence rate bounds

Establishes minimax rate optimality of estimators

Provides a more precise spectral norm analysis

Abstract

In this paper, we demonstrate how a class of advanced matrix concentration inequalities, introduced in \cite{brailovskaya2024universality}, can be used to eliminate the dimensional factor in the convergence rate of matrix completion. This dimensional factor represents a significant gap between the upper bound and the minimax lower bound, especially in high dimension. Through a more precise spectral norm analysis, we remove the dimensional factors for three popular matrix completion estimators, thereby establishing their minimax rate optimality.