Learning a more compact representation for low-rank tensor completion

Transform-based tensor nuclear norm (TNN) methods have gained considerable attention for their effectiveness in addressing tensor recovery challenges. The integration of deep neural networks as nonlinear transforms has been shown to significantly enhance their performance. Minimizing transform-based TNN is equivalent to minimizing the ${\ell }_1$ norm of singular values in the transformed domain, which can be interpreted as finding a sparse representation with respect to the bases supported by singular vectors. This work aims to advance deep transform-based TNN methods by identifying a more compact representation through learnable bases, ultimately improving recovery accuracy. We specifically employ convolutional kernels as these learnable bases, demonstrating their capability to generate more compact representation, i.e., sparser coefficients of real-world tensor data compared to singular vectors. Our proposed model consists of two key components: a transform component, implemented through fully connected neural networks (FCNs), and a convolutional component that replaces traditional singular matrices. Then, this model is optimized using the ADAM algorithm directly on the incomplete tensor in a zero-shot manner, meaning all learnable parameters within the FCNs and convolution kernels are inferred solely from the observed data. Experimental results indicate that our method, with this straightforward yet effective modification, outperforms state-of-the-art approaches on video and multispectral image recovery tasks.