Clipped Matrix Completion: A Remedy for Ceiling Effects

TL;DR

This paper introduces a theoretical framework and practical algorithms for recovering low-rank matrices from clipped observations, addressing a common issue in scientific data analysis where traditional matrix completion fails.

Contribution

It provides the first theoretical guarantees for exact recovery in clipped matrix completion and proposes novel algorithms with tailored regularization and loss functions.

Findings

Theoretical guarantee for exact recovery of clipped matrices.

Effective algorithms using squared hinge loss and combined trace-norm regularization.

Successful experiments on synthetic and benchmark data demonstrating improved recovery.

Abstract

We consider the problem of recovering a low-rank matrix from its clipped observations. Clipping is conceivable in many scientific areas that obstructs statistical analyses. On the other hand, matrix completion (MC) methods can recover a low-rank matrix from various information deficits by using the principle of low-rank completion. However, the current theoretical guarantees for low-rank MC do not apply to clipped matrices, as the deficit depends on the underlying values. Therefore, the feasibility of clipped matrix completion (CMC) is not trivial. In this paper, we first provide a theoretical guarantee for the exact recovery of CMC by using a trace-norm minimization algorithm. Furthermore, we propose practical CMC algorithms by extending ordinary MC methods. Our extension is to use the squared hinge loss in place of the squared loss for reducing the penalty of over-estimation on clipped entries. We also propose a novel regularization term tailored for CMC. It is a combination of two trace-norm terms, and we theoretically bound the recovery error under the regularization. We demonstrate the effectiveness of the proposed methods through experiments using both synthetic and benchmark data for recommendation systems.