  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
