Weighted matrix completion from non-random, non-uniform sampling patterns

TL;DR

This paper introduces a weighted matrix completion method tailored for deterministic, non-uniform sampling patterns, providing theoretical guarantees and demonstrating its efficiency and accuracy through numerical experiments.

Contribution

It proposes a simple debiased projection scheme for matrix recovery under non-uniform sampling, with theoretical error bounds and optimality analysis.

Findings

The method achieves near-optimal recovery error bounds.

Debiasing is crucial for non-uniform sampling accuracy.

Numerical experiments confirm computational efficiency and effectiveness.

Abstract

We study the matrix completion problem when the observation pattern is deterministic and possibly non-uniform. We propose a simple and efficient debiased projection scheme for recovery from noisy observations and analyze the error under a suitable weighted metric. We introduce a simple function of the weight matrix and the sampling pattern that governs the accuracy of the recovered matrix. We derive theoretical guarantees that upper bound the recovery error and nearly matching lower bounds that showcase optimality in several regimes. Our numerical experiments demonstrate the computational efficiency and accuracy of our approach, and show that debiasing is essential when using non-uniform sampling patterns.