A dual-tensor multi-view subspace clustering method with robust self-representation

Abstract

The application discloses a double-tensor multi-view subspace clustering method with robust self-expression, obtains multi-view data, and constructs a multi-view data set; first-stage self-expression is performed on the multi-view data set to obtain an initial self-expression matrix, and tensor stacking is performed to obtain a first-stage tensor; subspace decomposition and second-stage self-expression are performed on the stacked first-stage tensor to obtain a second self-expression matrix, and second tensor stacking is performed to obtain a second-stage tensor; based on the second-stage tensor, cross-view difference constraint is introduced to strengthen inter-view complementarity, a graph regularization term is introduced to maintain data local geometric structure, and a final target function is constructed; based on the final target function, an ALM augmented Lagrangian method is used to construct a Lagrangian function, an ADMM alternating direction multiplier method is used to iteratively optimize and solve the Lagrangian function, and finally, a final affinity matrix is obtained; and the final affinity matrix is input into spectral clustering to complete spectral clustering and realize final spatial clustering.

Description

Technical Field

[0001] This invention relates to the field of machine learning technology, and in particular to a dual tensor multi-view subspace clustering method with robust self-representation. Background Technology

[0002] Clustering, as one of the core tasks of unsupervised learning, aims to uncover the inherent similarity structure of data and divide the data into different clusters based on different levels of similarity. Multi-view clustering technology uncovers cross-view similarity... Figure 1 Consistency and complementarity integrate dispersed heterogeneous features into a unified and consistent representation, which can more accurately reveal the essential structure of data and significantly improve clustering robustness. Although single-view subspace clustering methods have achieved good results in various tasks, they rely solely on information from a single perspective and are insufficient to fully characterize the potential relationships between complex samples. While multi-view subspace clustering methods have made significant progress in integrating multi-source information, matrix-based modeling still has limitations in characterizing higher-order correlations between views. Furthermore, the representation matrix using tensors is not sufficiently clean and contains noise due to the inherent noise in multi-view data. Additionally, tensor stacking often ignores the feature differences between views in multi-view data and the local geometric structure of the data in the view network, resulting in poor model robustness and stability. Summary of the Invention

[0003] Purpose of the invention: In order to overcome the shortcomings of the existing technology, the present invention provides a robust self-representation dual-tensor multi-view subspace clustering method. It improves the noise resistance by stacking dual tensors and removing redundant data, and enhances the clustering discriminativeness by combining cross-view constraints and local structure preservation, so as to achieve the final spatial clustering.

[0004] Technical Solution: To achieve the above objectives, this invention provides a robust self-representation-based dual-tensor multi-view subspace clustering method. The method acquires multi-view data and constructs a multi-view dataset. A first-stage self-representation is performed on the multi-view dataset to obtain an initial self-representation matrix. Tensor stacking is then performed on the initial self-representation matrix to obtain a first-stage tensor. Subspace decomposition and a second-stage self-representation are performed on the stacked first-stage tensor to obtain a second self-representation matrix. A second tensor stacking is then performed on the second self-representation matrix to obtain a second-stage tensor. Simultaneously, redundancy is calculated on the second-stage tensor after the second stacking. The data undergoes secondary sparsity removal; based on the second-stage tensor, cross-view difference constraints are introduced to enhance the complementarity between views, and graph regularization terms are introduced to maintain the local geometric structure of the data, thus constructing the final objective function; based on the final objective function, the ALM augmented Lagrangian method is used to construct the Lagrangian function, and the ADMM alternating direction multiplier method is used to iteratively optimize and solve the Lagrangian function to obtain the final robust self-representation matrix; the final robust self-representation matrix is ​​symmetricized to obtain the final affinity matrix, and the final affinity matrix is ​​input into spectral clustering to complete spectral clustering and obtain the clustering results, thus achieving the final spatial clustering.

[0005] Furthermore, the multi-view dataset undergoes a first-stage self-representation to obtain an initial self-representation matrix. This initial self-representation matrix is ​​then stacked into tensors to obtain the first-stage tensor Z. Introducing the concept of tensors, the self-representation matrices obtained from the decomposition of each view data in the multi-view dataset are stacked into tensors, and constraints are applied to the resulting tensor and the decomposition error terms. The calculation process is shown below:

[0006] ;

[0007] ;

[0008] ;

[0009] ;

[0010] In the formula, Z is the first-stage tensor. Let V be the v-th view data in the multi-view dataset, where V is the number of view data in the multi-view dataset; The self-representation matrix is ​​obtained from the self-representation of the v-th view data in the first stage.

[0011] Furthermore, the first-stage tensor after stacking is subjected to subspace decomposition and second-stage self-representation to obtain the second self-representation matrix. The second self-representation matrix is ​​then subjected to a second tensor stacking to obtain the second-stage tensor S. The calculation process is shown below:

[0012] ;

[0013] ;

[0014] In the formula, S is the second-stage tensor. This represents the v-th view data in the multi-view dataset.

[0015] Furthermore, based on the second-stage tensor, cross-view difference constraints are introduced to enhance the complementarity between views, and graph regularization terms are introduced to preserve the local geometric structure of the data, thus constructing the final objective function; the calculation process is shown below:

[0016] ;

[0017] ;

[0018] In the formula, Let v be the self-representation matrix obtained from the self-representation of the v-th view data in the second stage. For the v-th view data in the multi-view dataset, , , , , All are hyperparameters; The self-representation matrix is ​​obtained from the self-representation of the v-th view data in the first stage.

[0019] Furthermore, auxiliary variables are introduced into the final objective function. , , The Lagrangian function is constructed using the ALM augmented Lagrangian method; the calculation process is shown below:

[0020] ;

[0021] In the formula, , , , , All are Lagrange multipliers; For penalty parameters, and All are first-stage tensors Auxiliary variables, For the second stage tensor Auxiliary variables.

[0022] Furthermore, the Alternating Direction Multiplier Method (ADMM) is used to iteratively optimize the Lagrangian function, yielding the final robust self-representation matrix. During the iterative optimization process, the Lagrangian function is optimized in each iteration. , E, S, J, K, W, Lagrange multipliers and penalty parameters Update; including The update calculation process is as follows:

[0023] ;

[0024] ;

[0025] in , , ;

[0026] right The update calculation process is as follows:

[0027] ;

[0028] ;

[0029] In the formula, It is an identity matrix.

[0030] Furthermore, the calculation process for updating E is as follows:

[0031] ;

[0032] ;

[0033] The calculation process for updating S is shown below:

[0034] ;

[0035] ;

[0036] In the formula, It is an identity matrix.

[0037] Furthermore, the calculation process for updating J is as follows:

[0038] ;

[0039] The calculation process for updating K is shown below:

[0040] ;

[0041] The calculation process for updating W is shown below:

[0042] ;

[0043] ;

[0044] In the formula, It is an identity matrix.

[0045] Beneficial Effects: This invention provides a robust self-representation dual-tensor multi-view subspace clustering method. First, an initial self-representation matrix is ​​obtained from each view. Then, a second subspace decomposition and tensor stacking are performed to further remove noise and enhance the quality of the affinity matrix, thereby improving the robustness and stability of the model. To mitigate feature differences between views, cross-view difference constraints are introduced to promote the effective fusion of complementary information, ensuring that each view can capture unique features, thus improving the discriminative power of clustering. Simultaneously, after the second tensor stacking, a graph regularization term is introduced to preserve the local geometric structure within the view-specific network. Finally, iterative optimization is performed using an ADMM-based iterative optimization method. By enhancing noise resistance through dual tensor stacking and redundant data removal mechanisms, and combining cross-view constraints and local structure preservation to enhance cluster discriminative power, the final spatial clustering is achieved. Extensive experiments on seven benchmark datasets demonstrate that this method outperforms existing state-of-the-art multi-view clustering models in multiple aspects. Attached Figure Description

[0046] Figure 1 This is a flowchart of a dual-tensor multi-view subspace clustering method with robust self-representation.

[0047] Figure 2 This is a schematic diagram of a dual-tensor multi-view subspace clustering method with robust self-representation. Detailed Implementation

[0048] The invention will now be further described with reference to the accompanying drawings.

[0049] like Figure 1As shown, a robust self-representation dual-tensor multi-view subspace clustering method is proposed. The method acquires multi-view data and constructs a multi-view dataset. A first-stage self-representation is performed on the multi-view dataset to obtain an initial self-representation matrix. Tensor stacking is then applied to the initial self-representation matrix to obtain a first-stage tensor. Subspace decomposition and a second-stage self-representation are performed on the stacked first-stage tensor to obtain a second self-representation matrix. A second-stage tensor stacking is then applied to the second self-representation matrix to obtain a second-stage tensor. Simultaneously, secondary sparsification of redundant data is performed on the second-stage tensor after the second stacking. The process involves several steps: First, based on the second-stage tensor, cross-view difference constraints are introduced to enhance inter-view complementarity. A graph regularization term is introduced to preserve the local geometric structure of the data, leading to the construction of the final objective function. Then, an ALM augmented Lagrangian method is used to construct the Lagrangian function, which is iteratively optimized using the ADMM alternating direction multiplier method to obtain the final robust self-representation matrix. This matrix is ​​then symmetricized to obtain the final affinity matrix, which is input into spectral clustering to achieve the final spatial clustering. This leads to a dual-tensor multi-view subspace clustering method with robust self-representation (DT-MSCRS).

[0050] like Figure 2 As shown, the general formula for multi-view subspace clustering based on self-representation is as follows:

[0051] ;

[0052] ;

[0053] In the formula, The regularization term is a constraint on the Z self-representation matrix, including various regularization constraint methods; The loss function term is used to constrain the error of the self-representation method decomposition.

[0054] The first-stage self-representation of the multi-view dataset is performed to obtain an initial self-representation matrix. This initial self-representation matrix is ​​then stacked into tensors to obtain the first-stage tensor Z. To compensate for the shortcomings of this method, such as loss of structural information and insufficient cross-view correlation mining, the concept of tensors is introduced. The self-representation matrices obtained from the decomposition of each view data in the multi-view dataset are stacked into tensors, and constraints are applied to the resulting tensor and the error terms obtained from the decomposition. Simultaneously, during the first-stage self-representation, noise in the first-stage tensor is sparsified to remove redundant data. The calculation process is shown below:

[0055] ;

[0056] ;

[0057] ;

[0058] ;

[0059] In the formula, Z is the first-stage tensor. Let v be the view data in the multi-view dataset. The multi-view dataset is... V represents the number of view data in the multi-view dataset; The self-representation matrix obtained from the self-representation of the v-th view data in the first stage is the initial self-representation matrix of the v-th view data. Let v be the reconstruction error matrix of the v-th view data in the first stage of the self-representation process. The overall error matrix is ​​formed by splicing together the reconstruction error matrices of all view data during the first stage of self-representation. (·) is the tensor stacking operator.

[0060] The first-stage tensor after stacking is subjected to subspace decomposition and second-stage self-representation to obtain the second self-representation matrix. The second self-representation matrix is ​​then subjected to a second tensor stacking to obtain the second-stage tensor S. Simultaneously, during the second-stage self-representation, noise in the second-stage tensor is sparsified to remove redundant data. The calculation process is shown below:

[0061] ;

[0062] ;

[0063] In the formula, S is the second-stage tensor. For the v-th view data in the multi-view dataset, The self-representation matrix is ​​obtained from the self-representation of the v-th view data in the first stage; Let v be the reconstruction error matrix of the v-th view data in the first stage of the self-representation process. The overall error matrix is ​​formed by splicing together the reconstruction error matrices of all view data during the first stage of self-representation. The overall error matrix is ​​formed by splicing together the reconstruction error matrices of all view data during the second-stage self-representation process; (·) is the tensor stacking operator.

[0064] According to the optimization objective function, the first-stage self-representation uses an error term to sparsely constrain noise and outliers in the original data, reducing the impact of redundant information on the first-stage self-representation result. The second-stage self-representation further uses an error term to sparsify the residual noise and redundant connections after subspace decomposition, obtaining a more sparse and discriminative subspace representation, thus enhancing the discriminative ability of the second-stage self-representation result. Therefore, both stages of self-representation employ sparsification, but at different levels: the first stage focuses on noise suppression at the original data level, while the second stage focuses on removing redundant connections at the subspace representation level.

[0065] To address the common oversight of inter-view feature differences and local geometric structure in multi-view data when stacking tensors, this paper introduces cross-view difference constraints to enhance inter-view complementarity and graph regularization to preserve local geometric structure, based on the second-stage tensor. The final objective function is then constructed. The calculation process is shown below:

[0066] ;

[0067] ;

[0068] In the formula, The self-representation matrix obtained in the second stage of the self-representation of the v-th view data is the second self-representation matrix of the v-th view data; The self-representation matrix obtained in the second stage of the self-representation of the u-th view data is the second self-representation matrix of the u-th view data; For the v-th view data in the multi-view dataset, , , , , All are hyperparameters; The self-representation matrix obtained from the self-representation of the v-th view data in the first stage is the initial self-representation matrix of the v-th view data. The reconstruction error matrix of the v-th view data in the first stage of the self-representation process; The overall error matrix is ​​formed by splicing the reconstruction error matrices of all view data in the first stage of self-representation, and is used to characterize the noise and redundant data filtered out in the first stage of self-representation. The overall error matrix is ​​formed by splicing the reconstruction error matrices of all view data in the second-stage self-representation process. It is used to characterize the residual noise and redundant data filtered out in the second-stage self-representation. (·) is a tensor stacking operator used to stack the self-representation matrices of multiple views along the third dimension; This is the graph Laplacian matrix corresponding to the v-th view data, used to preserve the local geometric structure of the data; (·) represents the trace operation of a matrix, which is the sum of the elements on the main diagonal of the matrix; T represents the transpose operation.

[0069] Introducing auxiliary variables into the final objective function , , The ALM augmented Lagrangian method is used to construct the Lagrangian function; auxiliary variables are introduced to decompose the original joint optimization problem into multiple subproblems, thereby achieving independent optimization and alternating updates of each constraint term within the augmented Lagrangian framework; the calculation process is shown below:

[0070] ;

[0071] In the formula, , , , , All are Lagrange multipliers; For penalty parameters, and All are first-stage tensors Auxiliary variables, For the second stage tensor Auxiliary variables.

[0072] In the optimization framework based on the Augmented Lagrangian method (ALM), to decompose the original complex joint optimization problem into multiple independently solvable subproblems, this paper introduces auxiliary variables J, W, and K to decouple the original variables. J and W are both auxiliary variables for the first-stage tensor Z, introduced to separate low-rank constraints from other constraints, allowing subproblems involving low-rank constraints and reconstruction error and consistency constraints to be optimized independently, improving solution efficiency and convergence stability. K is an auxiliary variable for the second-stage tensor variable S, separating the second-stage low-rank constraints from other regularization terms, enabling the second-stage optimization problem to be solved efficiently using a split variable approach. Therefore, J, W, and K are not new independent variables, but rather equivalent copies of the original variables Z and S in different constrained subproblems, used to achieve variable splitting and subproblem decoupling in the optimization problem—a standard modeling strategy in the Augmented Lagrangian method.

[0073] The Alternating Directional Multiplier Method (ADMM) is used to iteratively optimize the Lagrangian function, yielding the final robust self-representation matrix. During the iterative optimization process, the Lagrangian function is optimized in each iteration. , E, S, J, K, W, Lagrange multipliers and penalty parameters Update; obtain by organizing the Lagrange function. , E, S, J, K, W, Lagrange multipliers and penalty parameters The update calculation formula is as follows: According to the corresponding subproblems in the Lagrange function, the variables, Lagrange multipliers, and penalty parameter μ are updated alternately; among them, for... The update calculation process is as follows:

[0074] ;

[0075] ;

[0076] in , , ;

[0077] right The update calculation process is as follows:

[0078] ;

[0079] ;

[0080] In the formula, It is the identity matrix. For the first stage tensor The auxiliary variable corresponding to the v-th view data in the middle, For the first stage tensor The auxiliary variable corresponding to the v-th view data.

[0081] The calculation process for updating E is shown below:

[0082] ;

[0083] ;

[0084] Similarly, the solution for E is the same as... Similarly, the same solution method is used to solve the problem; the above function is used to update E.

[0085] The calculation process for updating S is shown below:

[0086] ;

[0087] ;

[0088] In the formula, Let S be the identity matrix. The solution method for S is the same as... Similarly, the equation is solved after differentiating the relevant variables and finding that it is 0; the above function is used to update S.

[0089] The calculation process for updating J is as follows:

[0090] ;

[0091] The calculation process for updating K is shown below:

[0092] ;

[0093] The calculation process for updating W is shown below:

[0094] ;

[0095] ;

[0096] In the formula, It is the identity matrix. and All are first-stage tensors Auxiliary variables, For the second stage tensor Auxiliary variables.

[0097] Among them, the Lagrange multipliers and the penalty parameter The update calculation process is as follows:

[0098] ; ;

[0099] ; ;

[0100] ; ;

[0101] In the formula, This indicates taking the minimum value among the values ​​within the parentheses. This is the growth factor of the penalty parameter, used to control the update speed of the penalty parameter; This is the maximum value of the penalty parameter, used to prevent it from increasing indefinitely during iteration; , , , , All are updated Lagrange multipliers. The updated penalty parameters; The reconstruction error matrix for the v-th view data in the second-stage self-representation process; For the second stage tensor The auxiliary variable corresponding to the v-th view data.

[0102] The Alternating Directional Multiplier Method (ADMM) is used to iteratively optimize the Lagrangian function, yielding the final robust self-representation matrix. The final affinity matrix is ​​further constructed based on this matrix, specifically by symmetrizing it. This final affinity matrix is ​​then input into spectral clustering as its input, further constructing the degree matrix and graph Laplacian matrix. Eigenvectors are extracted through eigenvalue decomposition, ultimately completing the spectral clustering. The resulting clustering results represent the final spatial clustering. Spectral clustering is a clustering algorithm based on graph theory and linear algebra. Its core idea is to construct an undirected weighted graph from all data points, transforming the clustering problem into an optimal graph partitioning problem, increasing the similarity within subgraphs and decreasing the similarity between subgraphs, ultimately achieving spatial clustering.

[0103] Example 1: Using the MSRCV1 dataset, ORL dataset, YaleA dataset, BBC dataset, BBC-Sport dataset, Caltech101-20 dataset, and HW2sources dataset, spatial clustering was performed on these seven datasets using the DT-MSCRS method of this invention and other model methods. The clustering results were evaluated, and it can be seen that the DT-MSCRS method of this invention has better clustering performance than other methods overall.

[0104] The MSRCV1 dataset contains 240 images from 8 target categories; this paper selects 7 categories and extracts 6 different types of features, resulting in 210 samples. The ORL dataset contains 400 face images of 40 individuals, each corresponding to 10 different expressions or poses, and is a widely used dataset in the field of face recognition. The YaleA dataset contains 165 face images of 15 individuals under different lighting conditions, facial expressions, and poses, with some samples exhibiting partial occlusion. The BBC dataset consists of news text from four different views, divided into 5 topic categories, totaling 685 samples. The BBC-Sport dataset is a subset of the BBC dataset, containing 5 sports news categories, totaling 544 samples, and is commonly used for text clustering tasks. The Caltech101-20 dataset is a subset of 20 target categories selected from Caltech101, with each category containing 40 to 800 images, totaling 2,386 samples. The HW2sources dataset is a handwritten digit dataset containing the digits 0-9. It is composed of two data sources, USPS and MNIST, and contains a total of 2,000 samples.

[0105] On the MSRCV1 dataset, the DT-MSCRS method of this invention achieved near-perfect clustering results. Compared with the classic t-SVD-MVC method, the ACC accuracy was improved by approximately 24.25%, indicating that the DT-MSCRS method of this invention can effectively eliminate redundant information in multiple views and accurately capture the inherent clustering structure. This performance improvement is mainly attributed to the proposed two-stage tensor construction mechanism, which effectively preserves key local and global structures while suppressing noise and ensuring cross-view accuracy. Figure 1 Consistency. On the BBC-Sport dataset, the DT-MSCRS method of this invention achieved an ACC accuracy of 0.9982, an NMI normalized mutual information of 0.9927, and an F-score of 0.9955, with a standard deviation of zero. This fully demonstrates the robustness of the DT-MSCRS method in terms of noise suppression and preservation of key information, which is mainly attributed to the synergistic effect of cross-view discriminative constraints and local geometric regularization. On the HW2sources dataset, the DT-MSCRS method of this invention also significantly outperforms the comparative methods, achieving an ACC accuracy of 99.2%, with all indicators approaching the theoretical optimum, verifying the advantages of adaptive self-representation in characterizing the discriminative subspace structure. Furthermore, on the larger Caltech101-20 dataset, despite the more challenging task, the DT-MSCRS method of this invention still exhibits strong adaptability, achieving a PUR purity of 0.8035 and an NMI normalized mutual information of 0.6245, significantly outperforming other methods. The experimental results from seven datasets show that the DT-MSCRS method of this invention excels in redundant information elimination, key cluster structure preservation, robustness enhancement, and cross-view and local performance. Figure 1 It has stable advantages in consistent modeling; these performance improvements are mainly due to the dual tensor modeling framework, the introduction of structured regularization terms, and the efficient ADMM optimization strategy, indicating that the DT-MSCRS method of this invention has good versatility and effectiveness in various multi-view clustering tasks.

[0106] Example 2: To verify the effectiveness of the dual tensor stacking and redundant information elimination mechanism proposed in the DT-MSCRS method of this invention, three sets of ablation experiments were designed and conducted. The three sets of ablation experiments were as follows: The first set removed the second tensor stacking process, applied cross-view difference constraints and graph regularization terms only to the self-representation matrix, and then performed spectral clustering; the second set retained the second tensor stacking process but removed the nuclear norm constraint acting on the second tensor; the third set retained the second tensor stacking process but skipped the subsequent step of filtering out the second redundant information. The three sets of ablation experiments were evaluated using ACC accuracy and NMI normalized mutual information; the ACC accuracy evaluation table for the three sets of ablation experiments is shown in Table 1 below; and the NMI normalized mutual information evaluation table for the three sets of ablation experiments is shown in Table 2 below.

[0107] ACC accuracy evaluation table for three groups of ablation experiments

[0108]

[0109] NMI Normalized Mutual Information Evaluation Table for Three Ablation Experiments

[0110]

[0111] As shown in the two evaluation tables above, it can be observed that all ablation experiments resulted in varying degrees of performance degradation in clustering. The first and third groups showed the most significant performance degradation: on datasets with complex noise and redundancy, such as the ORL dataset, the performance degradation across various evaluation metrics reached up to 60%; while on the HW2sources dataset, the performance degradation of the first ablation experiment was approximately 40%. These results clearly demonstrate that the second tensor stacking process and its accompanying redundancy elimination mechanism are key factors in improving the performance of the DT-MSCRS method of this invention. This mechanism significantly enhances the robustness and accuracy of multi-view clustering in complex noisy environments by effectively removing redundant information and preserving discriminative feature representations.

[0112] In summary, the DT-MSCRS method proposed in this invention is a robust self-representation dual-tensor multi-view subspace clustering method. By introducing two rounds of tensor stacking on top of the subspace self-representation and removing sparse noise twice, it effectively reduces the impact of redundant and noisy data, constructs a more robust model, and provides a cleaner affinity matrix for the clustering task. Experimental results on seven datasets show that the DT-MSCRS method of this invention consistently outperforms most existing multi-view subspace clustering methods, including tensor-based correlation methods, and achieves near-perfect clustering results on datasets such as MSRCV1, HW2sources, and BBC-Sport.

[0113] The above description is merely a preferred embodiment of the present invention. Those skilled in the art can make several modifications and optimizations based on the above disclosure without departing from the basic principles described above. These modifications and optimizations should be considered within the scope of protection as understood by the present invention.

Claims

1. A robust self-representation method for clustering dual-tensor multi-view subspaces, characterized in that: Acquire multi-view data and construct a multi-view dataset; perform a first-stage self-representation on the multi-view dataset to obtain an initial self-representation matrix; stack tensors on the initial self-representation matrix to obtain a first-stage tensor; perform subspace decomposition and a second-stage self-representation on the stacked first-stage tensor to obtain a second self-representation matrix; and perform a second-stage tensor stack on the second self-representation matrix to obtain a second-stage tensor. Simultaneously, redundant data is removed from the second-stage tensor after the second stacking. Based on the second-stage tensor, cross-view difference constraints are introduced to enhance the complementarity between views, and graph regularization terms are introduced to maintain the local geometric structure of the data, thus constructing the final objective function. Based on the final objective function, the ALM augmented Lagrangian method is used to construct the Lagrangian function, and the ADMM alternating direction multiplier method is used to iteratively optimize and solve the Lagrangian function to obtain the final robust self-representation matrix. The final robust self-representation matrix is ​​symmetricized to obtain the final affinity matrix. The final affinity matrix is ​​then input into spectral clustering to complete the spectral clustering and obtain the clustering result, thus achieving the final spatial clustering.

2. The dual-tensor multi-view subspace clustering method with robust self-representation according to claim 1, characterized in that: The first stage of self-representation is performed on the multi-view dataset to obtain an initial self-representation matrix. Tensor stacking is then performed on this initial self-representation matrix to obtain the first-stage tensor Z. Introducing the concept of tensors, the self-representation matrices obtained from the decomposition of each view data in the multi-view dataset are stacked as tensors, and constraints are applied to the resulting tensor and the decomposition error terms. The calculation process is shown below: ； ； ； ； In the formula, Z is the first-stage tensor. Let V be the v-th view data in the multi-view dataset, where V is the number of view data in the multi-view dataset; The self-representation matrix is ​​obtained from the self-representation of the v-th view data in the first stage.

3. The dual-tensor multi-view subspace clustering method with robust self-representation according to claim 1, characterized in that: The first-stage tensor after stacking is subjected to subspace decomposition and second-stage self-representation to obtain the second self-representation matrix. The second self-representation matrix is ​​then subjected to a second tensor stacking to obtain the second-stage tensor S. The calculation process is as follows: ； ； In the formula, S is the second-stage tensor. This represents the v-th view data in the multi-view dataset.

4. The dual-tensor multi-view subspace clustering method with robust self-representation according to claim 1, characterized in that: Based on the second-stage tensor, cross-view difference constraints are introduced to enhance the complementarity between views, and graph regularization terms are introduced to preserve the local geometric structure of the data, thus constructing the final objective function; the calculation process is shown below: ； ； In the formula, Let v be the self-representation matrix obtained from the self-representation of the v-th view data in the second stage. For the v-th view data in the multi-view dataset, , , , , All are hyperparameters; The self-representation matrix is ​​obtained from the self-representation of the v-th view data in the first stage.

5. The dual-tensor multi-view subspace clustering method with robust self-representation according to claim 1, characterized in that: Introducing auxiliary variables into the final objective function , , The Lagrangian function is constructed using the ALM augmented Lagrangian method; the calculation process is shown below: ； In the formula, , , , , All are Lagrange multipliers; For penalty parameters, and All are first-stage tensors Auxiliary variables, For the second stage tensor Auxiliary variables.

6. The dual-tensor multi-view subspace clustering method with robust self-representation according to claim 1, characterized in that: The Alternating Directional Multiplier Method (ADMM) is used to iteratively optimize the Lagrangian function, yielding the final robust self-representation matrix. During the iterative optimization process, the Lagrangian function is optimized in each iteration. , E, S, J, K, W, Lagrange multipliers and penalty parameters Update; including The update calculation process is as follows: ； ； in , , ; right The update calculation process is as follows: ； ； In the formula, It is an identity matrix.

7. A dual-tensor multi-view subspace clustering method with robust self-representation according to claim 6, characterized in that: The calculation process for updating E is shown below: ； ； The calculation process for updating S is shown below: ； ； In the formula, It is an identity matrix.

8. A dual-tensor multi-view subspace clustering method with robust self-representation according to claim 6, characterized in that: The calculation process for updating J is as follows: ； The calculation process for updating K is shown below: ； The calculation process for updating W is shown below: ； ； In the formula, It is an identity matrix.

Images