Adaptive and Implicit Regularization for Matrix Completion

TL;DR

This paper introduces an adaptive, implicit low-rank regularization method called AIR for matrix completion, which dynamically captures data features and improves performance, especially with non-uniform missing data.

Contribution

It proposes a novel adaptive and implicit regularization technique by parameterizing the Laplacian matrix, enhancing implicit regularization and adapting to data during training.

Findings

AIR outperforms traditional explicit regularizations in benchmark tasks.

The method is particularly effective with non-uniform missing data.

Theoretical analysis shows AIR vanishes at the end of training.

Abstract

The explicit low-rank regularization, e.g., nuclear norm regularization, has been widely used in imaging sciences. However, it has been found that implicit regularization outperforms explicit ones in various image processing tasks. Another issue is that the fixed explicit regularization limits the applicability to broad images since different images favor different features captured by different explicit regularizations. As such, this paper proposes a new adaptive and implicit low-rank regularization that captures the low-rank prior dynamically from the training data. The core of our new adaptive and implicit low-rank regularization is parameterizing the Laplacian matrix in the Dirichlet energy-based regularization, which we call the regularization AIR. Theoretically, we show that the adaptive regularization of \ReTwo{AIR} enhances the implicit regularization and vanishes at the end of training. We validate AIR's effectiveness on various benchmark tasks, indicating that the AIR is particularly favorable for the scenarios when the missing entries are non-uniform. The code can be found at https://github.com/lizhemin15/AIR-Net.