# Matrix Completion under Low-Rank Missing Mechanism

## TL;DR

This paper introduces a novel approach for matrix completion that accounts for a low-rank missing data mechanism, estimating observation probabilities via high-dimensional low-rank matrix estimation and inverse probability weighting, with proven optimal convergence rates.

## Contribution

It develops a new framework for matrix completion under low-rank missing mechanisms, extending beyond uniform missing assumptions with theoretical guarantees.

## Key findings

- Derived optimal convergence rates for probability and matrix estimators
- Proposed a high-dimensional low-rank estimation procedure for missing data
- Analyzed the effects of extreme missing probabilities on estimation accuracy

## Abstract

Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion methods often assume a simple uniform missing mechanism. In this work, we study matrix completion from corrupted data under a novel low-rank missing mechanism. The probability matrix of observation is estimated via a high dimensional low-rank matrix estimation procedure, and further used to complete the target matrix via inverse probabilities weighting. Due to both high dimensional and extreme (i.e., very small) nature of the true probability matrix, the effect of inverse probability weighting requires careful study. We derive optimal asymptotic convergence rates of the proposed estimators for both the observation probabilities and the target matrix.