Completing missing entries in multidimensional visual data is a typical ill-posed problem that requires appropriate exploitation of prior information of the underlying data. Commonly used priors can be roughly categorized into three classes: global tensor low-rankness, local properties, and nonlocal self-similarity (NSS); most existing works utilize one or two of them to implement completion. Naturally, there arises an interesting question: can one concurrently make use of multiple priors in a unified way, such that they can collaborate with each other to achieve better performance? This work gives a positive answer by formulating a novel tensor completion framework which can simultaneously take advantage of the global-local-nonlocal priors. In the proposed framework, the tensor train (TT) rank is adopted to characterize the global correlation; meanwhile, two Plug-and-Play (PnP) denoisers, including a convolutional neural network (CNN) denoiser and the color block-matching and 3 D filtering (CBM3D) denoiser, are incorporated to preserve local details and exploit NSS, respectively. Then, we design a proximal alternating minimization algorithm to efficiently solve this model under the PnP framework. Under mild conditions, we establish the convergence guarantee of the proposed algorithm. Extensive experiments show that these priors organically benefit from each other to achieve state-of-the-art performance both quantitatively and qualitatively.

Tensor analysis has received widespread attention in high-dimensional data learning. Unfortunately, the tensor data are often accompanied by arbitrary signal corruptions, including missing entries and sparse noise. How to recover the characteristics of the corrupted tensor data and make it compatible with the downstream clustering task remains a challenging problem. In this article, we study a generalized transformed tensor low-rank representation (TTLRR) model for simultaneously recovering and clustering the corrupted tensor data. The core idea is to find the latent low-rank tensor structure from the corrupted measurements using the transformed tensor singular value decomposition (SVD). Theoretically, we prove that TTLRR can recover the clean tensor data with a high probability guarantee under mild conditions. Furthermore, by using the transform adaptively learning from the data itself, the proposed TTLRR model can approximately represent and exploit the intrinsic subspace and seek out the cluster structure of the tensor data precisely. An effective algorithm is designed to solve the proposed model under the alternating direction method of multipliers (ADMMs) algorithm framework. The effectiveness and superiority of the proposed method against the compared methods are showcased over different tasks, including video/face data recovery and face/object/scene data clustering.

The mining of meaningful features and effective fusion of multisource remote sensing (RS) data have always been the challenging research problems in the joint classification of hyperspectral image (HSI) and light detection and ranging (LiDAR) data. In this article, we propose a multifrequency graph convolutional network with cross-modality mutual enhancement (MFGCN-CME) for multisource RS data classification. Specifically, we design an adaptive multifrequency graph feature learning module to capture the low- and high-frequency multiscale features of HSI and LiDAR in parallel and further adaptively aggregate them. Then, we propose a bipartite graph (BG) enhancement learning module to obtain the spatial-enhanced HSI features and spectral-enhanced LiDAR features by propagating intermodality information. To the best of our knowledge, the BG is first used to multisource RS data classification task. Furthermore, compared with traditional fusion methods, a gated fusion module is used to fully explore the complementarity of two data sources. Finally, a joint loss function combing a classification loss and a semisupervised contrastive loss is developed to improve the model robustness. Comprehensive experiments on different HSI and LiDAR datasets demonstrate that our proposed method can yield better performance compared with several state-of-the-art multisource RS data classification methods.

Recently, tensor singular value decomposition (t- SVD), based on the tensor-tensor product (t-product), has become a powerful tool for processing third-order tensor data. However, constrained by the fact that the basic element is the fiber (i.e., vector) in the t-product, higher-order tensor data (i.e., order d > 3) are usually unfolded into third-order tensors to satisfy the classical t-product setting, which leads to the destroying of high-dimensional structure. By revisiting the basic element in the t-product, we suggest a generalized t-product called elementbased tensor-tensor product (elt-product) as an alternative of the classic t-product, where the basic element is a ( d - 2)th-order tensor instead of a vector. The benefit of the elt-product is that it can better preserve high-dimensional structures and that it can explore more complex interactions via higher-order convolution instead of first-order convolution in classic t-product. Starting from the elt-product, we develop new tensor-SVD and low-rank tensor metrics (e.g., rank and nuclear norm). Equipped with the suggested metrics, we present a tensor completion model for high-order tensor data and prove the exact recovery guarantees. To harness the resulting nonconvex optimization problem, we apply an alternating direction method of the multiplier (ADMM) algorithm with a theoretical convergence guarantee. Extensive experimental results on the simulated and real-world data (color videos, light-field images, light-field videos, and traffic data) demonstrate the superiority of the proposed model against the state-of-the-art baseline models.

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.