Incomplete multi-view clustering method for aligning missing view recovery and subspace representation
By constructing a unified framework for collaborative view restoration and Zhang quantum space learning, the problems of low view restoration fidelity and insufficient module collaboration in incomplete multi-view clustering are solved, achieving more efficient clustering results, which are applicable to fields such as image recognition, text classification, and bioinformatics.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-01
- Publication Date
- 2026-04-07
AI Technical Summary
Existing incomplete multi-view clustering methods suffer from low view recovery fidelity and insufficient module collaboration, which affects the clustering results.
A unified framework for collaborative view reconstruction and Zhang quantum space learning is constructed. By alternately updating variables through the ADMM algorithm, the alignment of view reconstruction and subspace representation is achieved, thereby improving clustering performance.
It significantly improves the accuracy and stability of incomplete multi-view clustering, and is suitable for unsupervised learning scenarios such as image recognition, text classification, and bioinformatics.
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Figure CN121808443A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to an incomplete multi-view clustering method, and more particularly to an incomplete multi-view clustering method that aligns missing view recovery with subspace representation. Background Technology
[0002] In recent years, the rapid development of information technology has led to increasingly complex data formats, including text, images, and audio. Extracting usable information from multi-view data collected from different sources has become a major challenge. Clustering, as a key unsupervised learning technique, has seen research expand from single-view clustering to multi-view clustering, aiming to leverage the consistency and complementarity among multi-view data to divide unlabeled multi-view data into clusters.
[0003] (1) Multi-view subspace clustering
[0004] Among various multi-view clustering methods, multi-view subspace clustering has attracted considerable attention due to its significant performance advantages and robustness. Subspace clustering typically assumes that high-dimensional data is distributed in the union of several low-dimensional subspaces, with each subspace corresponding to a category or cluster. Multi-view subspace clustering extends this idea to the multi-view scenario by establishing consistent or complementary low-dimensional representations across views, mining shared potential structural information across different views, and reducing redundancy and noise across multiple views, thereby obtaining more accurate and robust clustering results. The core objective of this type of method is to achieve synergy in subspace representations across different views while maintaining the differences in features among them, improving robustness to noise, redundant information, and view imbalance. When consistency and view diversity are fully explored and well balanced, clustering performance can be significantly improved.
[0005] (2) Incomplete multi-view clustering
[0006] In real-world scenarios, due to sensor malfunctions, data corruption, or missing annotations, it is often difficult to obtain complete data for all views. Reducing the impact of multi-view incompleteness on clustering results is a major challenge currently facing multi-view clustering. Therefore, Incomplete Multi-View Clustering (IMVC) methods have been widely proposed and are divided into two categories: imputation-based methods and imputation-free methods. Imputation-based IMVC methods follow a process of first restoring the views and then exploring the cluster structure; imputation-free IMVC methods focus only on a subset of observable views to explore clustering information, avoiding the computational cost of completing missing views. Although imputation-free IMVC methods are simpler, their discriminative ability to extract clustering information is limited by the number of available views, especially when the missing rate is high.
[0007] Imputation-based IMVC methods utilize heuristics or learnable strategies to impute missing views, providing a strong foundation for exploring clustering information and enhancing interpretability. Several existing methods jointly recover missing views and learn clustering representations within a unified framework, significantly improving clustering quality in cases of incomplete views. Despite these achievements, most existing imputation-based IMVC methods still face two key challenges. First, the recovered or completed views often suffer from limited structural fidelity and insufficient reconstruction of diversity and consistency, which significantly impact the results of multi-view clustering. More importantly, when exploring the complementarity and consistency of multiple views, view recovery and subspace representation learning lack explicit alignment and collaborative interaction. Summary of the Invention
[0008] This invention aims to provide an incomplete multi-view clustering method that aligns missing view recovery with subspace representation, in order to address the shortcomings of existing incomplete multi-view clustering methods, namely "low view recovery fidelity" and "insufficient module collaboration".
[0009] To achieve the above objectives, the specific plan is as follows:
[0010] An incomplete multi-view clustering method that aligns missing view recovery with subspace representation includes the following steps:
[0011] S1, Establish the model objective function: The model objective function includes the Collaborative View Recovery (CVR) module and the Zhang Quantum Space Learning (TSL) module;
[0012] S2, Unified Framework Objective Function: Integrates CVR and TSL modules, introducing hyperparameters. By controlling the weights of the HSIC terms, the final objective function of ARSL-IMVC can be obtained;
[0013] S3, Optimize the objective function: Introduce auxiliary variables and An augmented Lagrangian function is constructed, and the variables are updated alternately using the ADMM algorithm until the convergence condition is met.
[0014] S4, Generate clustering results: Iterate and update all initial variables until the stopping condition is met; obtain the subspace representation. Then, construct an affinity matrix. Used for spectral clustering.
[0015] In summary, the present invention has the following advantages over the prior art:
[0016] The ARSL-IMVC method proposed in this invention addresses the shortcomings of existing incomplete multi-view clustering methods, namely "low view recovery fidelity" and "insufficient module collaboration," by constructing a unified framework of "collaborative view reconstruction - Zhang quantum space learning." Experiments show that this method exhibits excellent clustering performance and stability on various datasets with different missing rates, and can be widely applied to unsupervised learning scenarios that rely on multi-view data, such as image recognition, text classification, and bioinformatics. Attached Figure Description
[0017] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings:
[0018] Figure 1 To perform incomplete multi-view clustering using the ARSL-IMVC method;
[0019] Figure 2 Clustering results for all methods on Yale, NGs, 100leaves, and Scene-15;
[0020] Figure 3 for and Parameter sensitivity analysis on Yale and 100 leaves with a missing rate of 0.1;
[0021] Figure 4 t-SNE visualization on BBCSport and HandWritten with a missing rate of 0.1. Detailed Implementation
[0022] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.
[0023] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments of the present invention. As used herein, the singular form may also include the plural form unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.
[0024] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values of the components and steps set forth in these embodiments do not limit the scope of the invention. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be further discussed in subsequent figures.
[0025] The ARSL-IMVC (Incomplete Multi-View Clustering Based on Aligned Collaborative View Reconstruction and Tensor Quantum Space Learning) method proposed in this invention is based on constructing a unified framework of "Collaborative View Reconstruction (CVR) - Tensor Quantum Space Learning (TSL)" as a dual module. It uses the Alternating Multiplier Method (ADMM) to achieve iterative optimization of variables, ultimately outputting high-precision clustering results. Its overall process is as follows: Figure 1 As shown, the CVR module is responsible for completing incomplete views and preserving cross-view diversity and consistency, while the TSL module transforms the subspace representation into a low-rank tensor to capture high-order cross-view associations. The two modules achieve semantic alignment and collaborative interaction by sharing the latent representation H, which significantly improves the clustering performance and stability of incomplete multi-view data.
[0026] See Figures 1 to 4 As shown, this invention provides an incomplete multi-view clustering method for missing view recovery and subspace representation alignment, including the following steps:
[0027] S1, Establish the model objective function: The model objective function includes a Collaborative View Restoration (CVR) module and a Tensor Quantum Space Learning (TSL) module, wherein: the Collaborative View Restoration (CVR) module is used to balance cross-view while completing missing views. Figure 1 Consistency (ensuring the fidelity of observable sample reconstruction) and diversity (avoiding feature redundancy across different views); the Tensor Quantum Space Learning (TSL) module is used to transform shared and view-specific subspace representations into low-rank tensors, capture high-order cross-view semantic associations, and achieve collaborative optimization of global and local clustering structures.
[0028] (1) Collaborative View Restoration Module (CVR)
[0029] For a One sample and An incomplete multi-view dataset consisting of several views Considering the complexity of the view missing patterns, a diagonal indicator matrix is introduced. ( The function is used to retrieve observable samples (i.e., the i-th sample is observable in the v-th view, otherwise it is 0).
[0030] In multi-view clustering, multi-view data is typically embedded in a shared latent space. Through the inverse assumption, the projection matrix can be utilized. From potential representation Linear inference of missing views. To increase the degrees of freedom in view reconstruction, view-specific feature estimators are introduced. The basic form of view reconstruction is:
[0031]
[0032] To ensure that the recovered views retain sufficient complementary information, the Hilbert-Schmidt Independence Criterion (HSIC) is introduced as an arbitrary estimator. and The diversity regularization term between them, where the empirical HSIC term is defined as:
[0033]
[0034] In the formula: and They represent the first With the Feature estimates for each view; and They are respectively and The inner product kernel matrix; For a centered matrix ( Let n be an n-order identity matrix, where n is the number of samples. (This indicates a transpose operation).
[0035] The HSIC constraint penalizes dependencies between different reconstructed views, encouraging their diversity at the feature level. To explore consistency and diversity and ensure the quality of reconstructed views, the complete CVR module is represented as follows:
[0036]
[0037]
[0038] In the formula: , This represents non-missing data, and the equality constraints are used to ensure the reconstruction fidelity of the observable samples. .
[0039] (2) Zhang Quantum Space Learning Module (TSL)
[0040] Subspace representation learning is an effective method for exploring clustering semantics in low-dimensional embedding spaces, especially for self-representation-based methods. By leveraging shared latent representations and recovered view-specific features, both global semantics and local clustering structures can be effectively characterized.
[0041]
[0042] In the formula: This represents a view sharing a subspace representation; This represents the specific subspace representation of the v-th view; and This indicates the error term.
[0043] To explore cross-view consistency and complementarity at the subspace representation level, both shared and specific subspace representations are transformed into unified low-rank tensors. The TSL module is composed as follows:
[0044]
[0045]
[0046]
[0047] (5)
[0048] In the formula: It is a tensor constructor that represents a subspace. Stack them up and rotate them to form a subspace representing the tensor. In the low-rank tensor space, higher-order cross-view correlations at different levels are effectively captured, which is beneficial for semantic alignment across different views and enables collaborative interaction of local and global structural information.
[0049] S2, Unified Framework Objective Function: Integrates CVR and TSL modules, introducing hyperparameters. By controlling the weights of the HSIC terms, the final objective function of ARSL-IMVC can be obtained.
[0050] The ultimate goal of ARSL-IMVC can be stated as:
[0051]
[0052]
[0053]
[0054]
[0055] (6)
[0056] In the formula: It is a set of unknown variables. and It is a hyperparameter that controls the contribution of different regularization terms in the objective function.
[0057] In equation (6), the shared latent representation This not only facilitates joint optimization and collaborative interaction between view restoration and subspace learning, but also serves as a semantic anchor, aligning the reconstructed view with the corresponding subspace representation to capture cross-view complementarity and consistency. Information flow propagation within the ARSL-IMVC framework enables consistent exploration of cross-view semantic associations, improving the clustering quality of complex and incomplete multi-view clustering tasks.
[0058] S3, Optimize the objective function: Introduce auxiliary variables (Replacing tensor Z to simplify nuclear norm optimization) and (alternative) Using the decomposed constraints, an augmented Lagrangian function is constructed, and the variables are alternately updated using the ADMM algorithm until the convergence condition is met.
[0059] (I) Construction of Augmented Lagrange Function
[0060]
[0061] In the formula: It is a Lagrange multiplier. It is a penalty parameter. In the above formula, each variable is easy to solve. By fixing irrelevant variables, all variables are optimized alternately in an iterative manner.
[0062] (ii) Alternating update of variables
[0063] (1) Update the projection matrix
[0064] Keep other variables fixed. The optimization problem is transformed into a linear least squares problem under orthogonal constraints:
[0065]
[0066] The problem in equation (8) can be further transformed into:
[0067]
[0068] Its optimal closed-form solution can be obtained. ,in and yes The left singular vector and the right singular vector.
[0069] (2) Update auxiliary variables
[0070] With other variables fixed, update by minimizing the following objective. :
[0071]
[0072] Get and By setting the relevant derivative to zero, we can obtain The optimal solution:
[0073]
[0074] In the formula: , ,as well as .
[0075] (3) Update the shared latent representation
[0076] With other variables fixed, update by minimizing the following objective. :
[0077]
[0078] Get and By setting the relevant derivative to zero, we can obtain The optimal solution:
[0079]
[0080] In the formula: , , Equation (12) is a Sylvester equation. In practical problems, to maintain the stability of the solution, the matrix... Strictly defined as ( It is a very small positive scalar.
[0081] (4) Update the view-specific subspace representation
[0082] With other variables fixed, update by minimizing the following objective. :
[0083]
[0084] In the formula: , , get and By setting the relevant derivative to zero, we can obtain The optimal solution:
[0085]
[0086] (5) Update the view shared subspace representation
[0087] With other variables fixed, update by minimizing the following objective. :
[0088]
[0089] In the formula: , , get and By setting the relevant derivative to zero, we can obtain The optimal solution:
[0090]
[0091] (6) Update view-specific errors
[0092] With other variables fixed, update by minimizing the following objective. :
[0093]
[0094] In the formula: the HSIC term uses an inner product kernel, that is... Obtaining and By setting the relevant derivative to zero, we can obtain The optimal solution:
[0095]
[0096] In the formula: .
[0097] (7) Update subspace error
[0098] With other variables fixed, update by minimizing the following objective. :
[0099]
[0100] The problem in equation (20) can be further transformed into:
[0101]
[0102] In the formula: ,use Minimize the threshold operation by solving column by column:
[0103]
[0104] (8) Update subspace error
[0105] With other variables fixed, update by minimizing the following objective. :
[0106]
[0107] Similar to , The solution, through Minimize the threshold operation by solving column by column:
[0108]
[0109] In the formula: .
[0110] (9) Update auxiliary variables
[0111] With other variables fixed, update by minimizing the following objective. :
[0112]
[0113] The problem in equation (25) can be further transformed into:
[0114]
[0115] This is a classic tensor kernel norm minimization problem, whose closed-form solution can achieve the optimization objective based on t-SVT decomposition.
[0116] (10) Update the Lagrange multipliers , , and penalty parameters
[0117]
[0118]
[0119] In the formula: and These correspond to the upper limit of the multiplier coefficient and the positive coefficient, respectively.
[0120] S4, Generate Clustering Results: Iterate and update all initial variables until the stopping condition is met. This yields the subspace representation. Then, construct an affinity matrix. Used for spectral clustering, i.e.
[0121] .
[0122] The overall process of the incomplete multi-view clustering framework using the ARSL-IMVC method is as follows: Figure 1 As shown, it consists of two parts: CVR and TSL.
[0123] Example 1:
[0124] To evaluate the effectiveness of the proposed ARSL-IMVC, we will detail the experimental setup and analyze the experimental results in detail.
[0125] I. Clustering Indicators
[0126] To comprehensively evaluate the ARSL-IMVC method, three metrics were used to verify clustering performance: accuracy (ACC), normalized mutual information (NMI), and purity. The metrics range from 0 to 1, with higher values indicating better clustering performance.
[0127] II. Dataset Description
[0128] The experiments were conducted on seven benchmark datasets:
[0129] (1) BBCSport is a news dataset consisting of 544 news articles from the BBC News website. The experiment selected two text views, containing five different categories (athletics, cricket, football, rugby, and tennis).
[0130] (2) HandWritten is a handwritten digit dataset containing 10 digits, namely 0~9, with 200 handwritten images for each digit, and 6 views were extracted from each sample.
[0131] (3) BDGP is a dataset containing a large number of Drosophila embryo microscopic images. The dataset consists of 2,500 samples, divided into 5 classes, including image and text modalities, with each modality representing a separate view.
[0132] (4) NGs is a subset of the 20Newsgroups dataset, containing 500 samples, divided into 5 different news topic categories, and has 3 views.
[0133] (5) Yale is a face image dataset containing 165 images from 15 different individuals. Each individual has variations in facial expressions, lighting conditions, etc., forming 3 views.
[0134] (6) 100leaves is a dataset of plant leaf images, containing 1600 samples belonging to 100 plant categories. Each sample is represented by three features: texture histogram, contour shape, and shape descriptor, corresponding to three views respectively.
[0135] (7) Scene-15 is a scene image dataset containing 4485 images, divided into 15 different scenes. Each sample contains 3 visual feature representations, forming 3 views.
[0136] III. Constructing an Incomplete Dataset
[0137] To more closely approximate the incompleteness of real-world scenarios, we randomly remove samples from each view while ensuring that each sample appears in at least one view. On the BBCSport, HandWritten, and BDGP datasets, the missing rates are {0.1, 0.3, 0.5}, while on the NGs, Yale, 100leaves, and Scene-15 datasets, the missing rates range from 0.1 to 0.8 with a step size of 0.1.
[0138] IV. Comparison Methods
[0139] To demonstrate that the proposed method is effective for incomplete multi-view data, it is compared with several representative methods, including BSV, Concat, IMSC-AGL, DAIMC, UEAF, HCP-IMSC, HCLS-CGL, BWIC-TIMC, and RMoGL.
[0140] V. Experimental Results and Analysis
[0141] Tables 1, 2, and 3 present the clustering results on the BBCSport, HandWritten, and BDGP datasets, respectively. Figure 2 Experimental results with a wide range of missing rates are presented on the Yale, NGs, 100leaves, and Scene-15 datasets, from which the following conclusions can be drawn:
[0142] Table 1: Clustering results (%) of all methods on the BBCSport dataset
[0143]
[0144] (1) The proposed ARSL-IMVC method outperforms other existing IMVC methods in most cases. For example, in terms of the ACC metric, ARSL-IMVC improves performance by 4.60%, 8.31%, and 5.41% respectively on the BBCSport, HandWritten, and BDGP datasets when the missing data rate is 0.1. Figure 2As shown, most IMVC methods exhibit significant performance degradation with increasing missing rate, while ARSL-IMVC maintains higher stability. This fully demonstrates the superiority of the proposed ARSL-IMVC in complex IMVC tasks.
[0145] (2) Compared with interpolation-free IMVC methods (DAIMC, IMSC-AGL, HCLS-CGL), the proposed ARSL-IMVC typically achieves superior clustering performance, demonstrating the effectiveness of the proposed collaborative view restoration strategy. Furthermore, it emphasizes the reconstruction of cross-view structures at the feature level. Figure 1 Consistency and diversity are indeed more conducive to view restoration.
[0146] (3) The proposed ARSL-IMVC method also outperforms interpolation-based methods (i.e., UEAF, HCP-IMSC, BWIC-TIMC, RMoGL) because its learned latent representation serves as the semantic basis for both view retrieval and subspace learning, enabling them to capture cross-view... Figure 1 Alignment is achieved in terms of consistency and complementarity. This explicit information flow facilitates deep interaction between the CVR and TSL modules, improving the ability to accurately recover missing views and the ability to discriminate clustering semantics.
[0147] Table 2: Clustering results (%) of all methods on the HandWritten dataset
[0148]
[0149] Table 3: Clustering results (%) of all methods on the BDGP dataset
[0150]
[0151] VI. Ablation Experiment
[0152] To verify the latent representation The effectiveness of aligned view restoration and subspace learning is improved by removing... To further explore subspace learning, an ablation variant, ARSL-IMVC-1, was designed. Ablation experiments were conducted on BBCSport, HandWritten, Yale, NGs, and 100leaves with a missing value of 0.1. The results are shown in Table 4. ARSL-IMVC outperformed the ablation variant ARSL-IMVC-1 in all cases. This indicates that aligning view recovery and subspace learning in complex association exploration promotes semantic consistency between them and contributes to obtaining clearer clustering results.
[0153] Table 4: Ablation experiment results of ARSL-IMVC on 5 datasets (%)
[0154]
[0155] VII. Parameter Sensitivity Analysis
[0156] To evaluate the sensitivity of ARSL-IMVC to parameters, different parameters were used when the missing rate was 0.1. and The ACC results on the Yale and 100leaves datasets are as follows: Figure 3 As shown. The experimental results clearly show that when... When fixed, the performance of ARSL-IMVC is affected by The impact is not significant, and at the same time When fixed, the model is subject to The impact is also relatively small. This demonstrates that ARSL-IMVC has good robustness when the parameters vary within a reasonable range.
[0157] VIII. Visual Analysis
[0158] To visually illustrate the clustering performance of ARSL-IMVC, spectral embeddings on the BBCSport and HandWritten datasets with a missing rate of 0.1 are visualized using t-SNE. Figure 4 As shown, the spectral embeddings obtained by ARSL-IMVC have a relatively clear clustering structure, and the samples of different clusters are clearly separated, which verifies its distinguishability in clustering semantic exploration.
[0159] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. An incomplete multi-view clustering method for missing view recovery and subspace representation alignment, characterized in that, Including the following steps: S1, Establish the model objective function: The model objective function includes the Collaborative View Recovery (CVR) module and the Zhang Quantum Space Learning (TSL) module; S2, Unified Framework Objective Function: Integrates CVR and TSL modules, introducing hyperparameters. By controlling the weights of the HSIC terms, the final objective function of ARSL-IMVC can be obtained; S3, Optimize the objective function: Introduce auxiliary variables and An augmented Lagrangian function is constructed, and the variables are updated alternately using the ADMM algorithm until the convergence condition is met, thus obtaining the optimal solution for each variable. S4, Generate clustering results: Iterate and update all initial variables until the stopping condition is met; Obtaining the subspace representation Then, construct an affinity matrix. Used for spectral clustering.
2. The incomplete multi-view clustering method for missing view recovery and subspace representation alignment according to claim 1, characterized in that, In step S1, the CVR module is represented as follows: ； ； In the formula: , This represents non-missing data, and the equality constraints are used to ensure the fidelity of the reconstruction of the observable samples. .
3. The incomplete multi-view clustering method for missing view recovery and subspace representation alignment according to claim 1, characterized in that, In step S1, the TSL module is composed of: ； ； ； ； In the formula: It is a tensor constructor that represents a subspace. Stack them up and rotate them to form a subspace representing the tensor. .
4. The incomplete multi-view clustering method for missing view recovery and subspace representation alignment according to claim 1, characterized in that, In step S2, the ultimate goal of ARSL-IMVC can be expressed as: ； ； ； ； ； In the formula: It is a set of unknown variables. and It is a hyperparameter that controls the contribution of different regularization terms in the objective function.
5. The incomplete multi-view clustering method for missing view recovery and subspace representation alignment according to claim 1, characterized in that, In step S3, the augmented Lagrange function is: ； In the formula: It is a Lagrange multiplier. It is a penalty parameter. .
6. The incomplete multi-view clustering method for missing view recovery and subspace representation alignment according to claim 1, characterized in that, In step S3, the updated variables include: the projection matrix. Auxiliary variables Shared potential representation View-specific subspace representation View shared subspace representation View-specific errors Subspace error and Auxiliary variables and Lagrange multipliers , , and penalty parameters .
7. The incomplete multi-view clustering method for missing view recovery and subspace representation alignment according to claim 6, characterized in that, Projection matrix The optimal closed-form solution is: ； In the formula: and yes The left singular vector and the right singular vector; Auxiliary variables The optimal solution is: ； In the formula: , ,as well as ; Shared latent representation The optimal solution is: ； In the formula: , , ; View-specific subspace representation The optimal solution is: ； View shared subspace representation The optimal solution is: ； View-specific error The optimal solution is: ； In the formula: ; Subspace error and The optimal solution is: ； ； In the formula: .
8. The incomplete multi-view clustering method for missing view recovery and subspace representation alignment according to claim 6, characterized in that, Auxiliary variables Represented as:
9. The incomplete multi-view clustering method for missing view recovery and subspace representation alignment according to claim 6, characterized in that, Lagrange multipliers , , and penalty parameters Represented as: ； ； In the formula: and These correspond to the upper limit of the multiplier coefficient and the positive coefficient, respectively.
10. The incomplete multi-view clustering method for missing view recovery and subspace representation alignment according to claim 1, characterized in that, In step S4, the affinity matrix for: 。