Adjusting Leverage Scores by Row Weighting: A Practical Approach to Coherent Matrix Completion

TL;DR

This paper introduces a practical method for coherent matrix completion by adjusting leverage scores through row weighting, enabling accurate recovery of highly coherent matrices under uniform sampling.

Contribution

It presents the first effective approach for coherent matrix completion under standard uniform sampling by computing weighting matrices to uniformize leverage scores.

Findings

Successfully recovers highly coherent matrices with high precision

Outperforms unweighted methods on synthetic data

Provides theoretical guarantees under certain conditions

Abstract

Low-rank matrix completion is an important problem with extensive real-world applications. When observations are uniformly sampled from the underlying matrix entries, existing methods all require the matrix to be incoherent. This paper provides the first working method for coherent matrix completion under the standard uniform sampling model. Our approach is based on the weighted nuclear norm minimization idea proposed in several recent work, and our key contribution is a practical method to compute the weighting matrices so that the leverage scores become more uniform after weighting. Under suitable conditions, we are able to derive theoretical results, showing the effectiveness of our approach. Experiments on synthetic data show that our approach recovers highly coherent matrices with high precision, whereas the standard unweighted method fails even on noise-free data.