AIR-Net: Adaptive and Implicit Regularization Neural Network for Matrix Completion

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 kinds of images since different images favor different features captured by using 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. At the core of our new adaptive and implicit low-rank regularization is parameterizing the Laplacian matrix in the Dirichlet energy-based regularization with a neural network, and we call the proposed model \textit{AIR-Net}. Theoretically, we show that the adaptive regularization of AIR-Net enhances the implicit regularization and vanishes at the end of training. We validate AIR-Net's effectiveness on various benchmark tasks, indicating that the AIR-Net is particularly favorable for the scenarios when the missing entries are non-uniform. The code can be found at \href{https://github.com/lizhemin15/AIR-Net}{https://github.com/lizhemin15/AIR-Net}.