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.

Completing multidimensional color images is a fundamental challenge in image processing and computer vision. However, some tensor-based methods often treat RGB channels as independent modes, thereby neglecting their intrinsic correlations. To address this limitation, we represent RGB values as pure quaternions and organize them into a quaternion tensor for holistic modeling that preserves chromatic relationships. To better capture the nonlinear characteristics inherent in visual data and to improve the compactness of low-rank representations, we propose a nonlinear transformation within the quaternion domain. This design enables more expressive modeling compared to conventional linear approaches. In addition, we introduce two novel regularization terms that jointly encode global low-rankness and local smoothness, with the nonlinear transformation further enhancing the exploitation of structural priors. The overall model is optimized via a nonlinear alternating direction method of multipliers (ADMM), with theoretical guarantees of convergence. Extensive experiments on several datasets demonstrate that the proposed method significantly outperforms state-of-the-art low-rank tensor and quaternion tensor recovery techniques in multidimensional color image completion tasks.

In this paper, the fixed-time synchronization (FXTSYN) of unilateral coefficients quaternion-valued memristor-based neural networks (UCQVMNNs) with mixed delays is investigated. A direct analytical approach is suggested to obtain FXTSYN of UCQVMNNs utilizing one-norm smoothness in place of decomposition. When dealing with drive-response system discontinuity issues, use the set-valued map and the differential inclusion theorem. To accomplish the control objective, innovative nonlinear controllers and the Lyapunov functions are designed. Furthermore, some criteria of FXTSYN for UCQVMNNs are given using inequality techniques and the novel FXTSYN theory. And the accurate settling time is obtained explicitly. Finally, in order to show that the obtained theoretical results are accurate, useful, and applicable, numerical simulations are presented at the conclusion.