Benefiting from the powerful tensor techniques, the tensor low-rank representation has been proposed to construct sophisticated subspace clustering models. Existing tensor low-rank representation methods predominantly rely on a single low-rank prior to reconstruct the row space, which is instrumental in determining the subspace membership of samples by the row space information. However, this strategy neglects the column space and would lead to a subspace information loss. To address this issue, we propose a Dual Tensor Low-Rank Representation method (DTLRR), the first subspace clustering framework to theoretically recover both row and column subspaces simultaneously. Particularly, not simply formulating a dual self-representation model, we instead prove the recovery of both row and column spaces via a unified theoretical framework. Then, we impose low-rank constraints on the two corresponding affinity tensors to effectively capture high-order correlations. Meanwhile, we theoretically demonstrate the existence of compact dictionary tensors within the dual self-representation framework, which effectively eliminates the null spaces of the affinity tensors and significantly reduces computational complexity. Furthermore, an efficient Alternating Direction Method of Multipliers (ADMM) algorithm is designed to solve the proposed DTLRR model with guaranteed convergence. Extensive experiments validate the superior performance of the proposed DTLRR in data clustering, hyperspectral image denoising, and hyperspectral anomaly detection.

Multi-view clustering (MVC) has emerged as a powerful approach for integrating diverse sources of information from complex datasets. Nevertheless, existing methods struggle to accurately capture the global correlations and high-order structures in the data, and employ anchor-based techniques within a single dimension, limiting their representation. To address these issues, we propose an Anchor-induced Serial Tensor Representation (ASTR) framework, which effectively harnesses serial tensor representation to capture comprehensive multi-view information while reducing approximation errors and enhancing clustering performance. Specifically, ASTR begins with projection learning to explore low-dimensional latent spaces in multi-view data. Then, we introduce multi-anchor learning, where multiple anchor configurations are generated within the latent spaces, yielding a set of corresponding bipartite graphs. Besides, we organize these bipartite graphs into a sequence of global tensors, forming the serial tensor representation that encapsulates high-order inter- and intra-view relationships. Furthermore, we introduce the Laplace function to achieve a more accurate tensor rank approximation, complemented by a thorough theoretical analysis. Finally, a one-step clustering process, guided by adaptive weights, directly fuses the learned graphs to produce the final clustering indicator matrix. Experimental results demonstrate that ASTR possesses superior clustering accuracy and comparable efficiency.

Deep unfolding network has gained significant attention for magnetic resonance imaging super-resolution (MRI SR) due to its performance and interpretability. However, 1) existing methods predominantly focus on cross-contrast correlations while neglecting high-order correlations embedded within spatially adjacent slices in volumetric MRI data. 2) Their degradation models are optimized via the proximal gradient algorithm (PGA) that relies on manually designed hyperparameters (e.g., step size), often leading to overshooting or suboptimal solutions. To solve these limitations, we propose HocMRI, a deep unfolding multi-contrast MRI SR framework, which seamlessly integrates dual-prior modeling and hyperparameter-free PGA for enhanced reconstruction. Specifically, we first design a novel degradation model based on the dual-prior mechanism: an explicit prior based on low-rank tensor factorization to capture intra- and inter-slice dependencies, and an implicit prior leveraging a Mamba-based network with a novel 3D scanning strategy to further exploit high-order correlations across slices. Then, we derive a hyperparameter-free PGA to boost the traditional PGA, which employs a hyperbolic tangent function to dynamically control the gradient descent step, eliminating manual tuning while ensuring stable convergence with theoretical proofs. Based on the hyperparameter-free PGA, we develop an efficient iterative optimization algorithm to solve the degradation model and unfold it into a multi-stage deep network. Numerous experimental results from widely used MRI datasets demonstrate that our HocMRI achieves superior performance with enhanced efficiency compared to the state-of-the-art methods.

Deep unfolding networks (DUNs), combining conventional iterative optimization algorithms and deep neural networks into a multi-stage framework, have achieved remarkable accomplishments in Image Restoration (IR), such as spectral imaging reconstruction, compressive sensing and super-resolution. It unfolds the iterative optimization steps into a stack of sequentially linked blocks. Each block consists of a Gradient Descent Module (GDM) and a Proximal Mapping Module (PMM) which is equivalent to a denoiser from a Bayesian perspective, operating on Gaussian noise with a known level. However, existing DUNs suffer from two critical limitations: 1) their PMMs share identical architectures and denoising objectives across stages, ignoring the need for stage-specific adaptation to varying noise levels; and 2) their chain of structurally repetitive blocks results in severe parameter redundancy and high memory consumption, hindering deployment in large-scale or resource-constrained scenarios. To address these challenges, we introduce generalized Deep Low-rank Adaptation (LoRA) Unfolding Networks for image restoration, named LoRun, harmonizing denoising objectives and adapting different denoising levels between stages with compressed memory usage for more efficient DUN. LoRun introduces a novel paradigm where a single pretrained base denoiser is shared across all stages, while lightweight, stage-specific LoRA adapters are injected into the PMMs to dynamically modulate denoising behavior according to the noise level at each unfolding step. This design decouples the core restoration capability from task-specific adaptation, enabling precise control over denoising intensity without duplicating full network parameters and achieving up to $N$ times parameter reduction for an $N$ -stage DUN with on-par or better performance. Extensive experiments conducted on three IR tasks validate the efficiency of our method.

Under large-scale data with missing views, fast incomplete multi-view clustering (IMVC) with anchor learning is of critical importance due to its linear complexity$\mathcal {O}(n)$. However, existing anchor-based methods only explore the column orthogonality of anchor points, where their arbitrary column orthogonal basis vectors have weak constraint relationships with real samples and significant deviations from more representative anchors, thereby impeding the precise representation of sample similarities. To solve this issue, we propose a Reliable Entropy-induced anchor learning for incomplete Multi-view subspace Clustering (REMC), which performs an entropy approximation term to learn more representative anchors, and we prove that the information entropy minimization can be relaxed into the$\ell _{2,1}$-norm paradigm. Specifically, the proposed REMC first integrates anchor learning and subspace clustering to produce multiple view-specific bipartite graphs and capture the high-order correlations by imposing these bipartite graphs with the tensor nuclear norm. Then, we fuse all the view-specific bipartite graphs to build a consensus bipartite graph with entropy approximation regularization, and hence the proposed REMC can produce a more discriminative similarity graph, preserving each non-zero element in its column close to 1, while the other elements are approaching 0. Besides, an efficient algorithm is designed to solve the proposed REMC. Numerous results show the superior performance of our method on both the complete and incomplete data.

Incomplete multi-view clustering (IMVC), excavating diversity and consistency from multiple incomplete views, has aroused widespread research enthusiasm. Nevertheless, most existing methods still encounter the following issues: 1) they generally concentrate on pair-wise instance correlation, which consumes at least a quadratic complexity and precludes them from applying at large scales; 2) they only concentrate on pair-wise instance relevance, whereas ignoring the discriminative correlation hidden across views. To overcome these drawbacks, we propose the Self-Completed Bipartite Graph Learning (SCBGL) method for fast IMVC, which adaptively learns a self-completed consensus bipartite graph with the guidance of global information. Specifically, SCBGL learns the consensus anchor matrix shared among diverse views and further constructs a consensus intra-view bipartite graph with missing instances to explore the diversity and complementarity underlying different views. Meanwhile, we concatenate all the multiple features with projection learning to learn global anchors that would be employed to construct an inter-view bipartite graph. Furthermore, SCBGL dexterously utilizes the abundant inter-view information to tutor the self-completion of the consensus intra-view bipartite graph. By devising an alternatively iterative strategy, we present an efficient algorithm, which enjoys a linear time complexity, to solve the proposed SCBGL model. Numerous experiments conducted on large-scale datasets substantiate the superior performance of the SCBGL beyond the state-of-the-arts.

To manipulate large-scale data, anchor-based multi-view clustering methods have grown in popularity owing to their linear complexity in terms of the number of samples. However, these existing approaches pay less attention to two aspects. 1) They target at learning a shared affinity matrix by using the local information from every single view, yet ignoring the global information from all views, which may weaken the ability to capture complementary information. 2) They do not consider the removal of feature redundancy, which may affect the ability to depict the real sample relationships. To this end, we propose a novel fast multi-view clustering method via pick-and-place transform learning named PPTL, which could capture insightful global features to characterize the sample relationships quickly. Specifically, PPTL first concatenates all the views along the feature direction to produce a global matrix. Considering the redundancy of the global matrix, we design a pick-and-place transform with ℓ2,p-norm regularization to abandon the poor features and consequently construct a compact global representation matrix. Thus, by conducting anchor-based subspace clustering on the compact global representation matrix, PPTL can learn a consensus skinny affinity matrix with a discriminative clustering structure. Numerous experiments performed on small-scale to large-scale datasets demonstrate that our method is not only faster but also achieves superior clustering performance over state-of-the-art methods across a majority of the datasets.

Due to its heterogeneous property, multi-view data has been widely concerned over single-view data for performance improvement. Unfortunately, some instances may be with partially available information because of some uncontrollable factors, for which the incomplete multi-view clustering (IMVC) problem is raised. IMVC aims to partition unlabeled incomplete multi-view data into their clusters by exploiting the heterogeneity of multi-view data and overcoming the difficulty of data loss. However, most existing IMVC methods like BSV, MIC, OMVC, and IVC tend to conduct basic completion processing on the input data, without taking advantage of the correlation between samples and information redundancy. To overcome the above issue, we propose one novel IMVC method named data completion-guided unified graph learning (DCUGL), which could complete the data of missing views and fuse multiple learned view-specific similarity matrices into one unified graph. Specifically, we first reduce the dimension of the input data to learn multiple view-specific similarity matrices. By stacking all view-specific similarity matrices, DCUGL constructs a third-order tensor with the low-rank constraint, such that sample correlation within and between views can be well explored. Finally, by dividing the original data into observed data and unobserved data, DCUGL can infer and complete the missing data according to the view-specific similarity matrices, and obtain a unified graph, which can be directly used for clustering. To solve the proposed model, we design an iterative algorithm, which is based on the alternating direction method of multipliers framework. The proposed model proves to be superior by benchmarking on six challenging datasets compared with state-of-the-art IMVC methods.

Incomplete multi-view clustering is gaining increased attention owing to its great success in mining underlying information from the missing views. However, the existing approaches still encounter two issues: 1) They generally do not give sufficient consideration to the robustness of incomplete multi-view data with noise; 2) They only exploit the low-rank structures in the intra-view graphs, while the low-rank priors embedded in inter-view graphs are ignored. To this end, we propose a Robust Tensor Recovery for Incomplete Multi-view Clustering (RIMC) method, which transforms the view-missing problem into the tensor graph recovery problem by manipulating the comprehensive low-rank priors. Specifically, RIMC first employs a marginalized denoising operation to construct robust graphs and further builds a tensor graph by stacking these robust graphs. Then, we develop a novel tensor completion to recover the tensor graph by performing comprehensive low-rank priors: low-rank structures in the inter-view graphs (i.e., horizontal and lateral slices); low-rank structures in the intra-view graphs (i.e., frontal slices). Meanwhile, we integrate the tensor completion and spectral clustering to learn a unified indicator matrix. Extensive experiments show the promising performance of our method.

Ensemble clustering (EC), utilizing multiple basic partitions (BPs) to yield a robust consensus clustering, has shown promising clustering performance. Nevertheless, most current algorithms suffer from two challenging hurdles: (1) a surge of EC-based methods only focus on pair-wise sample correlation while fully ignoring the high-order correlations of diverse views. (2) they deal directly with the co-association (CA) matrices generated from BPs, which are inevitably corrupted by noise and thus degrade the clustering performance. To address these issues, we propose a novel Double High-Order Correlation Preserved Robust Multi-View Ensemble Clustering (DC-RMEC) method, which preserves the high-order inter-view correlation and the high-order correlation of original data simultaneously. Specifically, DC-RMEC constructs a hypergraph from BPs to fuse high-level complementary information from different algorithms and incorporates multiple CA-based representations into a low-rank tensor to discover the high-order relevance underlying CA matrices, such that double high-order correlation of multi-view features could be dexterously uncovered. Moreover, a marginalized denoiser is invoked to gain robust view-specific CA matrices. Furthermore, we develop a unified framework to jointly optimize the representation tensor and the result matrix. An effective iterative optimization algorithm is designed to optimize our DC-RMEC model by resorting to the alternating direction method of multipliers. Extensive experiments on seven real-world multi-view datasets have demonstrated the superiority of DC-RMEC compared with several state-of-the-art multi-view ensemble clustering methods.

Due to the physical nature of color images, color image processing such as denoising and inpainting has shown extensive and versatile possibilities over grayscale image processing. The monochromatic and the concatenation model have been widely used to process color images by processing each color channel independently or concatenating three color channels as one unified one and then used existing grayscale image processing methods directly without specific operations. These above schemes, however, have some limitations: (1) they would destroy the inherent correlation among three color channels since they cannot represent color images holistically; (2) they usually focus on one specific handcrafted prior such as smoothness, low-rankness, or even deep prior and thus failing to fuse deep and handcrafted priors of color images flexibly. To conquer these limitations, we propose one unified model to integrate deep prior and low-rank quaternion prior (DLRQP) for color image processing under the plug-and-play (PnP) framework. Specifically, the quaternion representation with low-rank constraint is introduced to denote the color image in a holistic way and one advanced denoiser is adopted to explore the deep prior in an iterative process. To tightly approximate the quaternion rank, one nonconvex penalty function is further utilized. We derive an alternate iterative approach to tackle the proposed model. We empirically demonstrate that our model can achieve superior performance over existing methods on both color image denoising and inpainting tasks.

Low-rank tensor representation (LRTR) methods have attracted great interest for their powerful ability to separate backgrounds and anomalies. However, most of the current LRTR models use the popular and convex surrogate tensor nuclear norm to solve optimization problems, which results in a loose approximation and suboptimal solver for the original problem. Besides, most existing methods solve the nonconvex optimization problems case-by-case, consequently losing one unified solver. To solve the above issues, we propose the Generalized Nonconvex Low-rank Tensor Representation (GNLTR) for hyperspectral anomaly detection (HAD), a unified solver not case-by-case one of existing nonconvex optimization problems. Compared to the tensor nuclear norm, GNLTR contains many popular nonconvex penalty functions as tighter regularizers of the tensor tubal rank to constrain the low rank of the background. Moreover, theL2,1norm has been integrated into the GNLTR model for the sparse anomalies. For the optimization problem, it is handled quickly and efficiently through a well-organized alternating direction method of multipliers (ADMM). The experiments on several real-world hyperspectral data sets demonstrate the superior performance of the GNLTR model in comparison with some state-of-the-art anomaly detection models.

As a significant extension of classical clustering methods, ensemble clustering first generates multiple basic clusterings and then fuses them into one consensus partition by solving a problem concerning graph partition with respect to the co-association matrix. Although the collaborative cluster structure among basic clusterings can be well discovered by ensemble clustering, most advanced ensemble clustering utilizes the self-representation strategy with the constraint of low-rank to explore a shared consensus representation matrix in multiple views. However, they still encounter two challenges: (1) high computational cost caused by both the matrix inversion operation and singular value decomposition of large-scale square matrices; (2) less considerable attention on high-order correlation attributed to the pursue of the two-dimensional pair-wise relationship matrix. In this article, based on low-rank and sparse decomposition from both matrix and tensor perspectives, we propose two novel multi-view ensemble clustering methods, which tangibly decrease computational complexity. Specifically, our first method utilizes low-rank and sparse matrix decomposition to learn one common co-association matrix, while our last method constructs all co-association matrices into one third-order tensor to investigate the high-order correlation among multiple views by low-rank and sparse tensor decomposition. We adopt the alternating direction method of multipliers to solve two convex models by dividing them into several subproblems with closed-form solution. Experimental results on ten real-world datasets prove the effectiveness and efficiency of the proposed two multi-view ensemble clustering methods by comparing them with other advanced ensemble clustering methods.

In recent years, low-rank representation (LRR) has received increasing attention on subspace clustering. Due to inevitable matrix inversion and singular value decomposition in each iteration, however, most of existing LRR algorithms may suffer from high computational complexity, and hence can not cope with the large-scale sample data commendably. To overcome this problem, in this paper, we propose a bilateral fast low-rank representation (BFLRR), which has a linear time complexity with respect to the number of samples. Specifically, we introduce the equivalent transformation method to remove the null spaces of both the columns and rows of the coefficient matrix so that a hypercompact coefficient matrix can be learned. Furthermore, the proposed BFLRR is embedded into a distributed framework as DFC-BFLRR to make it more efficient, which utilizes a combination of the global and local projection matrices. Extensive experiments are carried out on real datasets, and the results testify that the proposed methods not only perform faster-computing speed but also obtain favorable clustering accuracy in comparison with the competing methods among large-scale sample data.

As well known, low rank representation method (LRR) has obtained promising performance for subspace clustering, and many LRR variants have been developed, which mainly solve the three problems existing in LRR: 1) Problem of mean calculation; 2) Problem of deviating from the real low rank solution; 3) Problem of high computation cost on the large-scale data. In this paper, we first propose a universal LRR method referring to the first two problems. More specifically, we introduce the ability of removing the optimal mean automatically into LRR and extend it to a Schatten$p$-norm minimization problem. Then, referring to the third problem, we reformulate the universal LRR version as an equivalent fast optimization version by removing the null space of data. More importantly, the effective theory proof is proposed to guarantee that the fast optimization method can dramatically improve the algorithmic computation efficiency on the large-scale data but without any loss of information. Finally, experimental results demonstrate the effectiveness and efficiency of the proposed method.