Tensorial Multi-view Clustering with Deep Anchor Graph Projection

1 Introduction

Multi-view clustering (MVC) improves clustering performance by exploiting the consistent and complementary information from multiple perspectives to uncover the intrinsic data clustering structure [Fu et al., 2020], which receives significant attention in various applications like bioinformatics [Yang et al., 2022; Gao et al., 2020b]. MVC methods can be roughly categorized into subspace learning methods, graph-based methods, co-training methods, and multi-kernel learning methods. Among them, subspace learning techniques project multi-view data into a common latent subspace for clustering [Zheng et al., 2023; Liu et al., 2021]; graphbased methods construct similarity graphs to represent sample relationships to capture the geometric structure and then apply spectral clustering on the graph [Wei et al., 2017]; cotraining approaches iteratively train models on each view and exchange information to enhance clustering results, making them particularly effective when views provide complementary information [Jiang et al., 2013]; multi-kernel learning methods integrate diverse views by constructing a unified kernel space, enabling flexible representation learning and improving clustering accuracy [Tzortzis and Likas, 2012].

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However, traditional approaches face several challenges, including inefficiency in handling high-dimensional or largescale data [Li et al., 2020b]. Moreover, most methods achieve view fusion by minimizing the differences between individual view embeddings and a shared global embedding, which often fails to fully leverage the complementary information across different views [Gao et al., 2020b]. To address efficiency issues, several works introduce the anchor technique [Li et al., 2015; Li et al., 2024c; Qin et al., 2024], which selects a subset of representative samples as anchors and constructs an n ×m anchor graph for clustering instead of the raw data, where m is anchor number [Wang et al., 2021b]. In this way, these methods significantly improve model efficiency. Some methods further represent multi-view data as high-order tensors to capture complementary information across views, which well preserves spatial structures and inter-view dependencies [Guo et al., 2022]. The above methods generally rely on post-processing to obtain the final clustering labels, which increases computational complexity and may limit clustering performance. Some efforts have focused on directly projecting anchor graphs into the label space via bipartite graphs to avoid post-processing [Lei et al., 2024; Zhao et al., 2024]. Among them, AGFS-OMVC [Zhao et al., 2024] combines feature selection with anchor graph construction by using sparse constraints to remove noisy anchors and projecting the anchors directly into the label space, and obtains clustering results without post-processing. However, this method adopts single-layer projection, which is not feasible for extracting the hierarchical structures within multiview data.

To mitigate these problems, we propose a novel tensorized MVC method, i.e., Tensorial Multi-view Clustering with Deep Anchor Graph Projection (TMVC-DAGP). The proposed method guarantees efficiency by utilizing anchor graphs and addresses the deep feature extraction problem by introducing a deep multi-layer projection strategy. Specifically, inspired by the MVC method based on deep nonnegative matrix factorization (NMF) [Zhao et al., 2017], we design a multi-layer projection structure that is capable of extracting the deep and hierarchical latent features. Besides, it incorporates a sparsity constraint on projection matrices to enhance robustness by focusing on significant features. Additionally, tensor-based fusion with Schatten p-norm regularization is employed to capture inter-view complementary information, ensuring effective use of correlations between views. The main contributions of this work are as follows:

• We develop a novel deep multi-layer anchor graph projection method that directly maps anchor graphs into the label space via deep projection, effectively extracting deep information from multi-view data.
• This method employs a ℓ2,p-norm based sparse regularization on the projection matrices, which eliminates the negative impact of noises and redundancy and thus does not rely on post-processing for accurate clustering.
• We conduct extensive experimental validation on multiple multi-view datasets, demonstrating the superiority of TMVC-DAGP over existing approaches in clustering performance and computational efficiency.

2 RelatedWork

2.1 Tensor-BasedMulti-viewClustering

Most of the existing MVC methods perform view fusion by minimizing the difference between view embeddings and global embeddings, and this fusion strategy does not fully utilize the view complementary information of different views of multi-view data [Zhao et al., 2024]. To overcome this limitation, tensor-based MVC methods have gained increasing attention due to their ability to capture structural dependencies and complementary information across views [Gao et al., 2020b]. Tensor nuclear norm minimization, particularly when combined with tensor singular value decomposition (tSVD), has emerged as a powerful tool to model higher-order correlations and preserve the spatial structure of multi-view data [Xia et al., 2021]. For instance, low-rank tensor-based proximity learning jointly optimizes multi-view affinity matrices and consensus graphs within a unified framework, effectively capturing deeper cross-view relationships [Chen et al., 2022]. EMVC-NTLC applies low-rank constraints to anchor maps of different views using tensor sp norm to extract complementary information from multi-view data [Li et al., 2023]. Orth-NTF extends the matrix factorization to the tensor factorization and proposes a nonnegative tensor factorization method for MVC, which directly constructs multi-view data into tensors for decomposition and effectively improves clustering results [Li et al., 2024b].

2.2 Multi-view Projection for Clustering

Feature projection aims to map high-dimensional data into lower-dimensional spaces, effectively reducing noise and redundancy while preserving essential structure [Li et al., 2024a]. Traditional projection-based clustering methods primarily focus on dimensionality reduction and feature selection to enhance clustering performance. Gao et al. [2020a]

proposed a framework that combines dimensionality reduction with manifold learning, ensuring an effective balance between data representation and computational efficiency. Similarly, Wang et al. [2021a] introduced sparse regularization based on ℓ2,1-norm to improve the constructed graph, providing enhanced noise resistance and clustering accuracy. To further improve clustering quality and address the limitations of traditional approaches, recent studies have explored tensorbased projection techniques. Li et al. [2024a] proposed a label learning method based on tensor projection (LLMTP), which projects the anchor graph directly into the label space using an orthogonal projection matrix, thus enabling one-step clustering without the need for post-processing. By extending the projection from matrices to tensors, the approach effectively captures the spatial structure and complementary information across views. Additionally, the incorporation of the tensor Schatten p-norm regularization helps ensure consistency across views. Despite these advancements, existing projection-based methods often adopts single-layer projection and thus neglect the deep latent information within each individual view, leaving room for further improvements in representation learning and clustering robustness.

3 Methodology

3.1 Motivations and Problem Statement

MVC aims to obtain an indicator matrix that uncovers the underlying cluster structure by leveraging multiple data representations. However, current methods generally confronts the following challenges: (1) insufficient deep information extraction, as shallow structures fail to capture complex hierarchical patterns; (2) reliance on post-processing, which increases complexity and introduces potential information loss; (3) ineffective exploitation of inter-view complementarity, limiting the integration of diverse information; and (4) sensitivity to noise, which affects clustering robustness. Therefore, a more effective framework is needed to extract deep features, minimize post-processing, and enhance multi-view information integration while ensuring robustness.

Problem Formulation: Given multi-view dataset X = {Xv }(1 ≤ v ≤ V ), where Xv ∈ Rn×dv denotes the data matrix for the v-th view, n is sample number, and dv is the dimension of the v-th view data. Anchor-based methods reduce computational complexity by constructing an anchor matrix Sv ∈ Rn×m and employing Sv for clustering instead of Xv , where m is the anchor number. Our objective is to derive an indicator matrix Hv ∈ Rn×k , where k is cluster number,

3.2 Objective

To fully exploit the deep and hidden information within each view, we design a multi-layer projection structure. This structure performs layer-by-layer projection on the anchor graph, capturing complex patterns and relationships in the data. Each layer in this deep projection structure extracts features at varying layers, enhancing the overall data representation and directly projecting the data into the labeling space, resulting in more accurate clustering. The objective function for this deep projection structure is:

where Sv ∈ Rn×m is the anchor graph for the v-th view constructed via the method illustrated in [Li et al., 2024b]. Gvi ∈ Rdi-1×di is the projection matrix for the i-th layer of the v-th view, where di is the dimension of the i-th layer, l is the set number of layers, and dl = k, d0 = m. Hvl ∈ Rn×k represents the clustering indicator matrix for the l-th layer, with k being the number of clusters. αv is the weight assigned to the v-th view.

Tensorial Fusion: Eq. (1) only provides the view-specific clustering results, and the final result can be obtained with a regularization term ∑ ∥H-Hvl ∥2F , as adopted in [Zhao et al., 2024]. However, this regularization is too strong and may hinder exploiting complete information of different views [Li et al., 2023]. Following the tensored low-rank MVC methods, we introduce a weighted tensor Schatten p-norm regularization as formulated below:

where H, constructed based on the approach specified in the [Li et al., 2024b], is the tensor formed by all the indicator matrices Hvl , β is a parameter balancing the tensor norm constraint, and ∥ · ∥ω,Sp represents the weighted tensor Schatten Sp-norm. This formulation allows us to capture the complementary information among views, further enhancing clustering performance.

Sparse Anchor Projection: To ensure robust feature selection and enhance resistance to noise, we incorporate a sparsity constraint using the ℓ2,p-norm on the projection matrix. This constraint filters out noise and redundancy, enhancing clustering accuracy and eliminating the need of postprocessing for accurate clustering. The objective function with the sparsity constraint is expressed as:

where λ is used to adjust the impact of the sparsity constraint.

3.3 Optimization

We use the Augmented Lagrange Multiplier (ALM) to optimize the objective Eq. (2). Specifically, we introduce three auxiliary variablesQvi =Gvi , J =H, andMv =Hvl where Mv ≥ 0. let Bvl =Gv1Gv2 , ...,Gvl , the objective function can be optimized by solving the following problem:

where Z ∈ Rn×k×v , Yv ∈ Rn×k , and Wv i ∈ Rdi-1×di are Lagrange multipliers; ρi are the penalty parameters. We adopt alterative direction minimization method to optimize Eq.(4).

Optimizing Qvi : Fixing other variables, Eq.(4) becomes:

where Uvi = Gvi + Wv i ρ1 . According to [Zhao et al., 2024], it can be solved with the Generalized Soft-Thresholding (GST) algorithm on Uvi :

where Qvi,j: andUvi,j: are the j-th row vectors of Qvi andUvi , respectively. σ is obtained by solving:

Optimizing Gvi : Fixing other variables, Eq.(4) becomes:

Taking the derivative of Eq.(8) with respect to Gvi and setting it to zero, we have:

Thus, the optimal solution for Gvi is:

Optimizing Hvi (i < l): Fixing other variables, the optimization problem becomes:

To find the optimal Hvi , we directly set the optimization problem to zero. After that, we get:

Optimizing Hvl : Fixing other variables, the optimization problem becomes:

This leads to the equivalent trace maximization problem:

where the auxiliary variable Dv is defined as Dv = 2 αv SvBvl +ρ2 (Mv - Yv ρ2 ) + ρ3 (J v - Zv ρ3 )

To solve this problem, perform a Singular Value Decomposition (SVD) on the matrix Dv : Dv = UΣV⊤. The optimal solution for Hv∗ l is then given by:

This iterative procedure updates the matrix Hv∗ l for each view v, ensuring the optimization of clustering results across all views.

Optimizing J : Fixing other variables, the optimization problem becomes:

According to [Zhao et al., 2024], the optimal solution for Eq.(16) is

The term Γ β ρ3 is a generalized shrinkage operator that enforces the Schatten Sp-norm regularization on the tensor H+ Z ρ3 . It controls the rank of J , enhancing solution robustness. The detailed optimization process is provided in the appendix for further reference.

Optimizing Mv : The optimization problem for Mv can be formulated as follows:

The optimal solution to this problem is given by:

Optimization of αv: Fixing other variables, the optimization problem becomes:

The solution is:

where τv = ∥SvBvl -Hvl ∥2F + λ ∑li=1 ∥Gvi ∥2,p

Update Other Variables: Update the Lagrange multipliers by

The penalty parameters ρi = min(ϕρρi, ρmax), where ϕρ and ρmax are constants.

3.4 Complexity Analysis

Our method primarily consists of two main parts: the initialization of the anchor graph Sv and the iterative parameter updates. The computational complexity of initializing the anchor graph Sv is consistent with previous works, which is O(vnmd + vnmlog(m)), where v, m, and n represent the number of views, anchors, and samples, respectively. Here, d = ∑vv=1 dv is the total dimensionality across all views And assume that the number of iterations of the training process is t. The computational complexity of the update process for Qvi , Hvi , Hvl ,Gvi , J ,Mv are: O(vnmklt), O(vnmklt), O((nvm2 + vnmk)t), O((nvm2 + vm3 )lt), O((vnklog(vm) + v2nk)t), O(vmkt). Here, l, c, and v are small constants, with n usually the largest. Despite optimizing several variables, the limited number of layers (2-3) keeps the overall computational complexity, O(vnmd + vnm2 lt), reasonable in practice.

4 Experiments

4.1 Dataset

We evaluate the performance of the proposed method on eight widely adapted multi-view learning benchmark datasets, which are Yale [Yale University, 2001], BBCSport [Greene and Cunningham, 2006], Sonar [Sejnowski and Gorman, ], Vehicle Sensor [Duarte and Hu, 2004], NGs [Hussain et al., 2010],WebKB [Blum and Mitchell, 1998],MSRC [Winn and Jojic, 2005] and SentencesNYU v2 (RGB-D) [Silberman et al., 2012]. Detailed information on dataset specifications is provided in Table 1.

4.2 ComparisonMethods

In this paper, we compare our proposed method with multiple state-of-the-art multi-view clustering algorithms. A detailed description of each comparison method is provided in the Appendix. The comparison methods include GMC [Wang et al., 2019] , UDBGL [Fang et al., 2023], DiMSC [Cao et al., 2015], MvLRSSC [Brbi´c and Kopriva, 2018], RMSL [Li et al., 2019], FastMICE [Huang et al., 2023], MvDGNMF [Li et al., 2020a], MVC-DMF-PA [Zhang et al., 2021], AGFSOMVC [Zhao et al., 2024], Orth-NTF [Li et al., 2024b], MVC-DNTF [Feng et al., 2024].

4.3 Experimental Setup

All experiments were executed on a desktop with an Intel(R) Core(TM) i5-13400 CPU and 32 GB of RAM, using MATLAB 2023a. Data normalization was performed as a preprocessing step for all datasets to ensure consistent input quality. We assessed the clustering quality using Accuracy (ACC), Normalized Mutual Information (NMI), and Purity (PUR).

For the deep anchor graph projection approach, the layer size configurations were determined based on the complexity of the dataset. Specifically, for a two-layer projection, sizes [d1, k] were used, where d1 varied among [4k, 5k, 6k]. For a three-layer configuration, sizes [d1, d2, k] were used, with d1 in the range of [8k, 10k, 12k] and d2 between [4k, 5k, 6k]. These configurations allowed us to explore the impact of depth in the deep projection on the clustering results.

4.4 Clustering Performance

The experimental results presented in Tables 2 and 3 demonstrate the clustering performance of our proposed method cross multiple datasets in comparison with several benchmark algorithms. The best results are highlighted in bold, while the second-best results are underlined.

The results indicate that the deep projection structure in TMVC-DAGP significantly enhances clustering accuracy by capturing complex data relationships more effectively. Compared with AGFS-OMVC, which also employs a projectionbased approach, and Orth-NTF, which leverages tensor decomposition, TMVC-DAGP consistently outperforms them. For instance, on the RGB-D dataset, TMVC-DAGP achieved an ACC of 72.25%, surpassing the best-performing baselines. This highlights the advantage of our deep structure in extracting meaningful information from multi-view data.

Furthermore, the direct projection of anchor graphs into the label space contributes to the robustness and precision of TMVC-DAGP. In contrast to MVC-DMF-PA and MVCDNTF, which utilize deep non-negative matrix factorization, our approach demonstrates superior performance. Notably, TMVC-DAGP achieved an ACC of 98.78% on the Yale dataset and superior clustering results to other methods aon the Sonar and Vehicle Sensor datasets. The integration of tensor-based view fusion further strengthens clustering performance by effectively leveraging complementary information across different views.

4.5 Convergence

To validate the convergence of TMVC-DAGP, we conducted comprehensive experiments on multiple datasets by tracking the changes in objective function values, specifically focusing on the differences between Hv and Mv , as well as H and J . The convergence curves for the BBCSport, Sonar, MSRC, and Yale datasets are presented in Figure 1. The results demonstrate a consistent decrease in objective function values as iterations increase, reflecting the stability and robustness of the method. For instance, the BBCSport dataset achieves convergence after approximately 100 iterations, while Sonar, MSRC, and Yale datasets converge within 80 iterations. This rapid and consistent convergence across diverse datasets underscores the effectiveness of TMVC-DAGP in practice. This is a sentence in 7pt font size.

Additionally, to validate the clustering process and evaluate how well the data structures align during iterations, we employed t-SNE visualizations. These visualizations demonstrate the evolution of cluster separability at different stages of the iterative process, as shown in Figure 2.

Overall, the convergence analysis and visualizations confirm that TMVC-DAGP achieves both numerical stability and enhanced clustering performance in multi-view scenarios.

4.6 Parameter Analysis

We study how different parameters affect the clustering results. First, we analyze the impact of the anchor rate on clustering performance. Figure 3 illustrates ACC under different anchor rates on BBCSport, Yale, MSRC, and Sonar datasets. It can be observed that choosing an appropriate anchor rate is important for improving clustering performance, as too high or too low anchor rates may lead to decreased performance.

Next, we explore the effects of the regularization parameters λ, β, p of ℓ2,p norm and p of Schatten p norm in conjunction with clustering performance. Figures 4 and 5 illustrate the impact of different combinations of these parameters across several datasets. This demonstrates the robustness of TMVC-DAGP, as it maintains effective clustering performance even when parameters are varied.

4.7 Ablation Study

We conducted ablation experiments to evaluate the effectiveness of each component. First, we compared the performance of the algorithm when the Schatten p-norm (case1) and ℓ2,pnorm (case2) constraints were retained individually. Table 4 shows the clustering accuracy under different settings across four benchmark datasets. The results indicate that removing any of these constraints leads to a drop in performance, especially on complex datasets, showing that these constraints are crucial for maintaining stability and improving accuracy.

We also studied the effect of the layer number in the deep projection structure, as shown in Table 5 and Table 6. It demonstrates that increasing the layer number can effectively improve clustering performance but will not introduce much computational burden, as deeper structures better capture the intrinsic structure of the data. Considering the improvement of performance, the extra computational cost is acceptable.

5 Conclusion

In this paper, we introduced a novel Tensorial Multi-view Clustering with Deep Anchor Graph Projection (TMVCDAGP), which effectively captures the complex structures via projecting the anchor graph into label space. Together with tensorized schatten p-norm, TMVC-DAGP exploits complementary information across various views. Besides, a sparsity constraint using the ℓ2,p-norm enhances robustness and achieves one-step clustering, particularly in high-dimensional and noisy environments. Extensive experiments and comparisons on several multi-view datasets illustrate the effectiveness and superiority of our method.

Acknowledgments

This work is supported by This work is supported by the National Science Foundation of China (No. 62037001), the Key Research and Development Project in Shaanxi Province (2024PT-ZCK-89), the Fundamental Research Funds for the Central Universities (ZYTS25267, QTZX25004), and the Science and Technology Project of Xi'an (Grant 2022JHJSYF-0009), Open Project of Anhui Provincial Key Laboratory of Multimodal Cognitive Computation, Anhui University (No. MMC202416), Selected Support Project for Scientific and Technological Activities of Returned Overseas Chinese Scholars in Shaanxi Province 2023-02, and the Xidian Innovation Fund (Project NoYJSJ25007).