AIR-Net: Adaptive and Implicit Regularization Neural Network for Matrix Completion
This work addresses the limitation of fixed explicit regularizations in matrix completion for imaging sciences, offering an incremental improvement by adapting to different image features.
The paper tackles the problem of matrix completion for image processing by proposing an adaptive and implicit low-rank regularization method, which dynamically captures low-rank priors from training data and shows effectiveness, particularly in scenarios with non-uniform missing entries.
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 \textit{AIR}. Theoretically, we show that the adaptive regularization of 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