Recently, the block-term decomposition with rank- $(L_{r}, L_{r}, 1)$ (termed as LL1 decomposition), which decomposes a third-order tensor into the sum of the outer products between vector and matrix factors, has received increasing attention for high-dimensional image reconstruction. However, the fixed low-rank matrix decomposition in LL1 is restricted to third-order tensors, which hinders its development for higher-order tensor data (i.e., order $N \gt 3$ ). To address this, we propose a Block Customized Topology Term Decomposition (BCTD), which represents an $N$ th-order tensor as a sum of outer products of basis vectors and customized $(N-1)$ th-order coefficient tensors with flexible internal topological structures. The proposed BCTD enjoys two advantages: Firstly, it allows tackling higher-order tensors beyond the third-order tensor setting of LL1, which can better preserve the high-dimensional structure of the tensor. Secondly, it allows each term to have a customized topological structure beyond the fixed topological structure (i.e., low-rank matrix decomposition) in LL1, which can better explore the intrinsic high-dimensional low-rank structures of the tensor. To evaluate the performance of the proposed BCTD, we build the corresponding high-dimensional image reconstruction model and provide a theoretical generalization error bound between the recovered tensor of the proposed model and the underlying tensor. To solve the resulting optimization problem, we apply a proximal alternating minimization (PAM)-based algorithm with a theoretical convergence guarantee. Extensive experimental results on high-dimensional image completion and compression tasks using real-world datasets (color videos and light field images) demonstrate the superiority of the proposed model over other baseline models.

In real-world scenarios, complex data such as multispectral images and multi-frame videos inherently exhibit robust low-rank property. This property is vital for multi-dimensional inverse problems, such as tensor completion, spectral imaging reconstruction, and multispectral image denoising. Existing tensor singular value decomposition (t-SVD) definitions rely on hand-designed or pre-given transforms, which lack flexibility for defining tensor nuclear norm (TNN). The TNN-regularized optimization problem is solved by the singular value thresholding (SVT) operator, which leverages the t-SVD framework to obtain the low-rank tensor. However, it's quite complicated to introduce SVT into deep neural network due to the numerical instability problem in solving the derivatives of the eigenvectors. In this paper, we introduce a novel data-driven generative low-rank t-SVD model based on the learnable orthogonal transform, which can be naturally solved under its representation. Prompted by the linear algebra theorem of the Householder transformation, our learnable orthogonal transform is achieved by constructing an endogenously orthogonal matrix adaptable to neural networks, optimizing it as arbitrary orthogonal matrices. Additionally, we propose a low-rank solver as a generalization of SVT, which utilizes an efficient representation of generative networks to obtain low-rank structures. Extensive experiments highlight its significant restoration enhancements.

Recently, transform-based tensor nuclear norm (TNN) methods have received increasing attention as a powerful tool for multi-dimensional visual data (color images, videos, and multispectral images, etc.) recovery. Especially, the redundant transform-based TNN achieves satisfactory recovery results, where the redundant transform along spectral mode can remarkably enhance the low-rankness of tensors. However, it suffers from expensive computational cost induced by the redundant transform. In this paper, we propose a learnable spatial-spectral transform-based TNN model for multi-dimensional visual data recovery, which not only enjoys better low-rankness capability but also allows us to design fast algorithms accompanying it. More specifically, we first project the large-scale original tensor to the small-scale intrinsic tensor via the learnable semi-orthogonal transforms along the spatial modes. Here, the semi-orthogonal transforms, serving as the key building block, can boost the spatial low-rankness and lead to a small-scale problem, which paves the way for designing fast algorithms. Secondly, to further boost the low-rankness, we apply the learnable redundant transform along the spectral mode to the small-scale intrinsic tensor. To tackle the proposed model, we apply an efficient proximal alternating minimization-based algorithm, which enjoys a theoretical convergence guarantee. Extensive experimental results on real-world data (color images, videos, and multispectral images) demonstrate that the proposed method outperforms state-of-the-art competitors in terms of evaluation metrics and running time.

In this article, we propose a novel bilayer low-rankness measure and two models based on it to recover a low-rank (LR) tensor. The global low rankness of underlying tensor is first encoded by LR matrix factorizations (MFs) to the all-mode matricizations, which can exploit multiorientational spectral low rankness. Presumably, the factor matrices of all-mode decomposition are LR, since local low-rankness property exists in within-mode correlation. In the decomposed subspace, to describe the refined local LR structures of factor/subspace, a new low-rankness insight of subspace: a double nuclear norm scheme is designed to explore the so-called second-layer low rankness. By simultaneously representing the bilayer low rankness of the all modes of the underlying tensor, the proposed methods aim to model multiorientational correlations for arbitrary N -way ( N ≥ 3 ) tensors. A block successive upper-bound minimization (BSUM) algorithm is designed to solve the optimization problem. Subsequence convergence of our algorithms can be established, and the iterates generated by our algorithms converge to the coordinatewise minimizers in some mild conditions. Experiments on several types of public datasets show that our algorithm can recover a variety of LR tensors from significantly fewer samples than its counterparts.