Tensorial Multi-view Clustering (TMC) methods have received significant attention due to their ability to capture the high-order correlation among different views. Despite their notable progress, two issues persist: 1) Most TMC methods are highly reliant on prior structure information in sorted datasets, rendering them challenging to handle the shuffled case. 2) Extremely high computational complexity arising from tensor-related operations. To address these limitations, we propose a novel framework termed Anchors Bring Stability and Efficiency: Fast Tensorial Multi-view Clustering on Shuffled Case (SE-FTMC). SE-FTMC first learns a set of anchors and a unified corresponding matrix between anchors and samples, then the low-rank tensor learning is adopted in anchor space instead of sample space to avoid reliance on the prior information hidden in sorted samples and reduce the computational complexity. Finally, SE-FTMC passes the high-order correlations learned in the anchor space into the sample space through the corresponding matrix to improve the efficiency and stability of clustering. Furthermore, SE-FTMC is solved by an efficient algorithm with linear complexity. Extensive experiments on various datasets demonstrate the effectiveness and superiority of our SE-FTMC compared with state-of-the-art methods. The code is publicly available at: https://github.com/jijintian/SE-FTMC.

Tensorial Multi-view Clustering (TMC), a prominent approach in multi-view clustering, leverages low-rank tensor learning to capture high-order correlation among views for consistent clustering structure identification. Despite its promising performance, the TMC algorithms face three key challenges: 1). The severe computational burden makes it difficult for TMC methods to handle large-scale datasets. 2). Estimation bias problem caused by the convex surrogate of the tensor rank. 3). Lack of explicit balance of consistency and complementarity. Being aware of these, we propose a basic framework Efficient and Scalable Tensorial Multi-View Subspace Clustering (ESTMC) for large-scale multi-view clustering. ESTMC integrates anchor representation learning and non-convex function-based low-rank tensor learning with a Generalized Non-convex Tensor Rank (GNTR) into a unified objective function, which enhances the efficiency of the existing subspace-based TMC framework. Furthermore, a novel model ESTMC-C with the proposed Enhanced Tensor Rank (ETR), Consistent Geometric Regularization (CGR), and Tensorial Exclusive Regularization (TER) is extended to balance the learning of consistency and complementarity among views, delivering divisible representations for the clustering task. Efficient iterative optimization algorithms are designed to solve the proposed ESTMC and ESTMC-C, which enjoy time-economical complexity and exhibit theoretical convergence. Extensive experimental results on various datasets demonstrate the superiority of the proposed algorithms as compared to state-of-the-art methods.

The Unaligned Multi-view Clustering (UMC) problem is currently receiving widespread attention, focusing on clustering unaligned multi-view data generated in real-world applications. Although some algorithms have emerged to address this issue, there still exist the following drawbacks: 1) The fully unknown correspondence of samples across views can significantly limit the exploration of consistent clustering structure. 2) The fixed representation space makes it difficult to mine the comprehensive information in the original data. 3) Unbiased tensor rank approximation is desired to capture the high-order correlation among different views. To address these issues, we proposed a novel UMC framework termed Tensorized Unaligned Multi-view Clustering with Multi-scale Representation Learning (TUMCR). Specifically, TUMCR designs a multi-scale representation learning and alignment framework, which constructs multi-scale representation spaces to comprehensively explore the unknown correspondence across views. Then, a tensorial multi-scale fusion module is proposed to fuse multi-scale representations and explore the high-order correlation hidden in different views, which utilizes the Enhanced Tensor Rank (ETR) to learn the low-rank structure. Furthermore, TUMCR is solved by an efficient algorithm with good convergence. Extensive experiments on different types of datasets demonstrate the effectiveness and superiority of our TUMCR compared with state-of-the-art methods. Our code is publicly available at: https://github.com/jijintian/TUMCR.

The tensor-based multi-view subspace clustering algorithms have received widespread attention due to the powerful ability to capture high-order correlation across views. Although such algorithms have achieved remarkable success, they still suffer from three main issues: 1)The extremely high computational complexity makes tensor-based methods difficult to handle large-scale data sets. 2)The subspace-based methods usually ignore the local geometric structure of the original data. 3)The commonly used Tensor Nuclear Norm (TNN) treats different singular values equally and under-penalizes the noise components, resulting in a sub-optimal representation tensor. Being aware of these, we propose Anchor Structure Regularitation Induced Multi-view Subspace Clustering via Enhanced Tensor Rank Minimization (ASR-ETR). Specifically, an anchor-representation tensor is constructed by using the anchor representation strategy rather than the self-representation strategy to reduce the time complexity, and the local geometric structure in the learned anchor-representation tensor is enhanced by adopting the Anchor Structure Regularization (ASR). We further devise an Enhanced Tensor Rank (ETR), which is a tighter surrogate of the tensor rank to effectively capture the multi-view high-order correlation. An efficient iterative optimization algorithm is designed to solve the ASR-ETR, which is time-economical and enjoys favorable convergence. Extensive experimental results on various data sets demonstrate the superiority of the proposed algorithm as compared to state-of-the-art methods.

Recently, tensor-based multi-view clustering methods have achieved promising results, primarily benefited from their superior ability in exploring high-order consistent information among views. Despite significant progress, these methods inevitably suffer from several drawbacks: 1) Extremely high computational complexity restricts their feasibility for large-scale data sets. 2) Prevalently adopted tensor rank approximations (e.g., Tensor Nuclear Norm (TNN)) tend to under-penalize small singular values, resulting in noise residuals. 3) Tensor structure is rarely utilized for high-order complementarity investigation. In light of this, we propose High-order Complementarity Induced Fast Multi-View Clustering with Enhanced Tensor Rank Minimization (CFMVC-ETR). Specifically, two sets of representation matrices are learned from original multi-view data via the matrix factorization mechanism with a group of base matrices, which are further reconstructed into the consistent tensor and the complementary tensor, respectively. Subsequently, a novel Enhanced Tensor Rank is imposed on the consistent tensor, which is a tighter approximation of the tensor rank and is more noisy-robust to explore the high-order consistency. Meanwhile, a tensor-level constraint termed Tensorial Exclusive Regularization is proposed on the complementary tensor to enhance the view-specific feature and well capture the high-order complementarity. Moreover, we adopt a concatenation-fusion approach to integrate these two parts, deriving a discriminative unified embedding for the clustering task. We solve CFMVC-ETR by an efficient algorithm with good convergence. Extensive experiments on nine challenging data sets demonstrate the superiority of the proposed method.