DELTA: Deep Low-Rank Tensor Representation for Multi-Dimensional Data Recovery

Low-rank tensor recovery methods within the tensor singular value decomposition (t-SVD) framework have demonstrated considerable success by leveraging the inherent low-dimensional structures of multi-dimensional data. However, previous approaches in this framework often rely on linear transforms or, in some cases, nonlinear transforms constructed with fully connected networks (FCNs). These methods typically promote a global low-rank structure, which may not fully exploit the nature of multiple subspaces in real-world data. In this work, we propose a nonlinear transform to capture long-range dependencies and diverse patterns across multiple subspaces of the data within the t-SVD framework. This approach provides a richer and more nuanced representation compared to the localized processing typically seen in FCN-based transforms. In the transform domain, we construct a low-rank self-representation layer that fully exploits the multi-subspace structure inherent in tensor data. Instead of merely enforcing overall low-rankness, our method minimizes the nuclear norm of a self-representation tensor, allowing for a more precise and joint characterization of multiple subspaces. This results in a more accurate representation of the data's intrinsic low-dimensional structures, leading to superior recovery performance. This new framework, termed the DEep Low-rank Tensor representAtion (DELTA), is evaluated across several typical multi-dimensional data recovery applications, including tensor completion, robust tensor completion, and spectral snapshot imaging. Experiments on various real-world multi-dimensional data illustrate the superior performance of our DELTA.