Multi-view subspace clustering method based on non-convex tensor nuclear norm

Abstract

The invention provides a multi-view subspace clustering method (TNMVSC) based on a non-convex tensor nuclear norm. According to the method, the low-rank representation theory (LRR), the graph regularization technology and the tensor decomposition theory are combined, and the global structure and the local attribute of each view can be captured at the same time. Besides, the TNMVSC decomposes each low-rank representation sparse matrix into three factor matrixes through a matrix three-factor decomposition theory so as to realize alignment of the representation matrixes, thereby ensuring that the generated core matrix can effectively retain key information. Meanwhile, a non-convex low-rank tensor nuclear norm is adopted to capture high-order correlation among a plurality of core matrixes. In the aspect of algorithm optimization, on the basis of the established optimization model, an alternating direction multiplier method (ADMM) is adopted to carry out optimization solution on the representation matrix of each view. And then, performing angle correction on the fused representation matrix to obtain a similarity matrix among the samples, and clustering the similarity matrix by using a spectral clustering method to realize clustering of the samples. In order to verify the effectiveness of the TNMVSC, tests are performed on reference data sets in multiple fields, including computer vision, bioinformatics, social media analysis and the like. A large number of simulation experiment results show that the TNMVSC is excellent in clustering precision and robustness, and the performance of multi-view clustering can be effectively improved.

Description

Technical Field

[0001] The present invention relates to the cross - field of mathematics and computer, and specifically proposes a multi - view data clustering analysis method based on data feature mining and modeling. Background Art

[0002] With the rapid development of artificial intelligence, machine learning, and big data technologies, data in many practical application scenarios often comes from multiple sources. How to effectively extract valuable information from this data has become the core issue in the research of various fields. As an unsupervised learning method, clustering analysis can automatically reveal the potential structures and patterns in data, providing important support for data analysis and decision - making.

[0003] With the continuous increase in data diversity and complexity, single - view data often fails to comprehensively reveal the potential structure of samples, thus limiting the performance of clustering methods. In contrast, multi - view clustering methods can make full use of complementary information from different perspectives, overcome the limitations of single - view methods, and provide a more accurate and comprehensive data description by integrating complementary information from multiple views, thereby improving the clustering effect. Multi - view clustering methods have been widely applied in many fields such as bioinformatics, text analysis, image processing, and social network analysis. In these fields, data can usually be expressed from different angles or modalities (such as text, image, speech, etc.), and each view reflects different features of the data. The core goal of multi - view clustering methods is to integrate the information in each view and then reveal the potential patterns and structures in the data. Due to its unique advantages and wide applicability, multi - view clustering algorithms have important application prospects and practical values in the fields of data mining and machine learning.

[0004] With the continuous increase of multi-view data, the complexity and diversity of the data are also constantly rising. How to efficiently mine multi-view data and design effective clustering methods has become an important challenge at present. In recent years, researchers at home and abroad have proposed a variety of multi-view clustering algorithms. Among them, the graph-based learning method has been widely applied to various multi-view learning tasks because it can effectively reveal the latent structure of the data. However, when constructing the similarity graph, due to factors such as data scale, neighborhood size, similarity measure selection, noise, and outliers, the accuracy of the similarity measure faces great challenges. Huang et al. proposed an innovative multi-view clustering method (MVCSK). This method can directly divide the optimal graph into connected components corresponding to the number of clusters by simultaneously performing multi-view clustering and similarity learning in the kernel space. In addition, this method can automatically assign appropriate weights to each view without additional parameter adjustment. Matrix factorization technology reveals the underlying structure of the data by decomposing the data matrix into several low-rank matrices. The RC_MSC method parameterizes the low-rank structure of all self-representation coefficient matrices through matrix three-factor decomposition and orthogonal constraints. This constraint ensures that the self-representation coefficient matrices of different views have the same rank, thus effectively promoting the structural consistency across views. This method can not only learn the consistent structure between different views but also fully mine the complementary information provided by each view.

[0005] The main goal of the multi-view subspace (MVSC) clustering algorithm is to reveal the underlying subspace structure, accurately group a set of data points, and obtain the corresponding categories. However, the success of MVSC largely depends on the quality of the similarity matrix. Existing methods usually adopt an optimization and symmetric two-step separation process, but this often fails to guarantee the symmetry and adaptive locality of the similarity matrix. To solve this problem, Ma et al. proposed a new symmetric multi-view subspace clustering method based on automatic neighbor discovery (SMSC-AND). This method aims to integrate the symmetrization and localization processes of the ideal similarity matrix into a unified framework. Theoretical and experimental results show that SMSC-AND can directly generate a refined symmetric similarity matrix without relying on traditional post-processing steps. In addition, SMSC-AND also proposes an automatic neighbor discovery strategy, avoiding the fixed neighbor size or rank constraint adopted in previous methods, thus further improving the clustering effect. In addition, scholars such as Zhang proposed a latent multi-view subspace clustering method (LMSC). This method clusters by using the latent representation of data points and explores the complementary information in multiple views. LMSC finds the latent representation of data in multiple views and reconstructs the data based on this, thus establishing a more complete representation and significantly improving the accuracy and robustness of the subspace representation.

[0006] However, due to the high time complexity, existing multi-view clustering algorithms are difficult to apply to large-scale datasets. Some scholars have reduced the computational complexity by introducing anchor points, but the heuristic sampling and the selection of anchor points in the clustering process will have a great impact on the results. At the same time, the anchor graphs of different views are constructed independently, and the complementary information between views is not fully utilized. For this reason, scholars such as Guo have proposed a new scalable multi-bipartite multi-view clustering method (SMCMB). This method effectively processes large-scale datasets by jointly learning multiple bipartite graphs. Different from the single anchor graph view method, SMCMB generates multiple anchor graphs on each view, performs subspace representation learning based on these anchor graphs, and effectively partitions the bipartite graph, and finally generates a unified bipartite graph for clustering. In recent years, multi-view clustering based on low-rank tensors has received extensive attention because it can effectively capture the complex relationships and high-order information between different views. A common method is to stack the similarity matrices of multiple views into a three-dimensional tensor and impose a low-rank constraint on this basis to reveal the underlying structure of the data and improve the clustering accuracy. Based on this advantage, Wu et al. proposed a new type of multi-view clustering method (UGLTL). This method obtains the similarity matrix of each view through projection graph learning and reconstructs these matrices into tensor form. Then, using graph learning and low-rank tensor decomposition techniques, the high-order correlations between different views are further mined to improve the clustering performance.

[0007] Although the methods in the above-mentioned literature have made some progress in multi-view data clustering, with the continuous increase of multi-view data, the complexity and diversity of the data are also increasing day by day. How to efficiently cluster these multi-view data has become a key challenge in multi-view data mining. At present, there is still room for further optimization in the accuracy and efficiency of multi-view data-based clustering methods. Summary of the Invention

[0008] The present invention proposes a multi-view subspace clustering method based on non-convex tensor nuclear norm, aiming to effectively analyze multi-view data by combining multi-view matrix low-rank representation, matrix three-factor decomposition, non-convex tensor nuclear norm and graph regularization techniques. This method can fuse the unique properties of different view data and capture the high-order correlations between views, so as to achieve efficient clustering of data samples.

[0009] The technical solution of the present invention includes the following steps:

[0010] S1: Based on the collected multi-view data, define a mathematical optimization model that combines non-convex tensor nuclear norm and local regularization term. This model can effectively characterize the high-order correlations between different view data and the local attributes between samples, and the specific description is as follows:

[0011]

[0012] Among them, X v represents the v-th view data matrix in the multi-view dataset; E v represents the error term corresponding to the v-th view, which is used to characterize the noise in the data; Z v represents the representation matrix under the v-th view, C v represents the core matrix under the v-th view, P v and Q v represent the orthogonal matrix under the v-th view; represents the tensor of the non-convex nuclear norm; represents the similarity between samples i and j under the v-th view; λ1 and λ2 represent regularization parameters;

[0013] S2: Based on the optimization model, construct an augmented Lagrangian function, and use the alternating direction method of multipliers (ADMM) to perform separate iterative optimization on each variable. Through the iterative process, finally obtain the optimized representation matrix Z v ;

[0014] S3: Sum the representation matrices under each view and then perform angular correction on the matrix The specific steps are as follows: First, perform singular value decomposition (SVD) on the matrix Z~ to obtain the matrix A = U∑ 1 / 2 ; Then, obtain the similarity matrix W by normalizing the matrix A, where

[0015] S4: Use the spectral clustering method to perform clustering analysis on the obtained similarity matrix W, so as to realize the classification of samples.

[0016] Preferably, the augmented Lagrangian function is specifically described as:

[0017]

[0018] Among them, is the introduced auxiliary variable, and the initial value is the zero matrix; is the Laplacian matrix, where and represent the Lagrange multipliers; λ1 and λ2 represent regularization parameters, which are used to prevent overfitting, and μ is a positive penalty scalar.

[0019] Preferably, the specific steps of using ADMM to optimize the objective function in step S2 are as follows:

[0020] S201: The update of tensor Y~ can be divided into n sub-problems. The j-th sub-problem is expressed as:

[0021]

[0022] where FFT(·) represents the fast Fourier transform; and θ represent the i-th largest singular value and the weight of the singular value respectively; Ω(·) represents a non-convex tensor function;

[0023] S202: Update Z by solving the following problem v :

[0024]

[0025] Take the partial derivative of L(Z v ) with respect to Z v , and set its partial derivative to 0 to obtain the update formula of Z v as follows:

[0026]

[0027] S203: Update C through the following formula v :

[0028]

[0029] where

[0030] S204: Update P by solving the following problem v :

[0031]

[0032] where, the closed-form solution of the above optimization problem is:

[0033]

[0034] where, M v and N v are the left and right singular value matrices after performing SVD decomposition on respectively;

[0035] S205: Similarly, the optimization problem for solving Q v can be obtained as

[0036]

[0037] where, the optimal solution of Q v is Among them, M' v and N' v are respectively the left and right singular value matrices obtained by performing SVD decomposition on ;

[0038] S206: Update E through the following formula v :

[0039]

[0040] where Θ represents the singular value threshold operator;

[0041] S207: Update and μ through the following formulas:

[0042]

[0043] where μ, ρ, and μ max are all given constants;

[0044] S208: Iterate according to the parameter values updated in S201 - S207, and calculate the iteration error. The specific calculation formula is as follows:

[0045]

[0046] Terminate the iteration when the number of iterations meets the set maximum iteration steps or the error value Error_value is less than the set value, and obtain the optimized Z v .

[0047] Preferably, the specific steps of step S3 are as follows:

[0048] S301: Sum the optimized representation matrices Z under each view to obtain the composite matrix v ;

[0049] S302: Perform singular value decomposition on the composite representation matrix ;

[0050] S303: Calculate A = U∑ 1 / 2 ;

[0051] S304: Construct the similarity matrix W:

[0052] Preferably, the spectral clustering method used in step S4 to cluster the similarity matrix W includes the following specific steps:

[0053] S401: Construct the normalized Laplacian matrix L = D -1 / 2WD -1 / 2 , where D is the diagonal matrix d ii = ∑ j w ij ;

[0054] S402: Calculate the eigenvalues of matrix L to obtain the eigenvectors V = [v i , v2,..., v k ;

[0055] S403: Normalize V using the L2 norm

[0056] S404: Use the k-means method to cluster the normalized matrix V to obtain k classes, and each class corresponds to a sample type.

[0057] Preferably, the maximum number of iterations is 30, and the set value of the error in Error_value is 1e-2.

[0058] The beneficial effects of the present invention are as follows:

[0059] (1) Compared with the prior art, the method of the present invention effectively extracts the core features under different views through matrix three-factor decomposition on the basis of fully considering the global structural features of multi-view data, and uses local graph regularization to capture the local attributes between data, thereby significantly improving the clustering effect;

[0060] (2) The present invention uses non-convex tensor nuclear norm to describe the high-order correlations in all view data, which can effectively reduce the influence of factors such as noise, breaking through the limitation that the traditional nuclear norm can only be processed at the single-view level;

[0061] (3) Compared with the existing methods, the method of the present invention has obvious advantages in clustering effect, can effectively cluster complex multi-view data, helps to reveal the potential structure and pattern in the data, and provides strong support for data analysis and decision-making. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required in the embodiments. Obviously, the drawings are only partial embodiments of the present application, and those of ordinary skill in the art can still obtain other possible drawings without creative work.

[0063] Figure 1 It is the flow chart of the method of the present invention.

[0064] Figure 2Visual comparison of the clustering results of the method of the present invention (TNMVSC) and three other clustering methods (RC_MSC, SM2SC, FMR) on three groups of multi-view test datasets - Inhouse (A), Dermatology (B) and Mfeat (C). In the figure, each point represents a sample, and samples of the same class are represented by the same marker symbol. Detailed implementation manners

[0065] To more clearly understand the purpose, features and advantages of the present invention, the present invention will be described in detail below in conjunction with the accompanying drawings and specific implementation manners. Through the elaboration of specific examples, the beneficial effects of the present invention are further illustrated, aiming to help readers better understand the essence of the present invention, but it does not constitute any limitation to the implementation and protection scope of the present invention.

[0066] The present invention provides a multi-view clustering method based on non-convex tensor nuclear norm. This method uses non-convex tensor nuclear norm to capture the high-order correlations between views, and combines low-rank representation theory and local graph model to characterize the global structure and local attributes of data, thus proposing a new multi-view clustering method. The main steps of this method include:

[0067] (1) Based on the collected multi-view data, define a mathematical optimization model that combines non-convex tensor nuclear norm and local regularization term;

[0068] (2) Based on the optimization model, construct an augmented Lagrangian function, and use the alternating direction method of multipliers (ADMM) to perform separate iterative optimization on each variable. Through the iterative process, the optimized representation matrix under each view is finally obtained;

[0069] (3) Sum the representation matrices under each view Then for the matrix perform angular correction. The specific steps are as follows: First, perform singular value decomposition on the matrix to obtain the matrix A = U∑ / 1 / 2 , and then obtain the similarity matrix W by normalizing the matrix A.

[0070] (4) Use the spectral clustering method to perform clustering analysis on the obtained similarity matrix W, so as to realize the classification of samples.

[0071] To test the effectiveness of the method of the present invention, multi-view test data provided in the literature is used. The descriptions and data sources of the seven groups of multi-view data specifically used for testing are as follows:

[0072] (1) COIL-20: This dataset contains 1440 images of 20 categories. Each category contains 72 images with a black background, taken from different positions. The dataset contains dimensionality information of three extracted views, namely: intensity (4096 dimensions), Local Binary Patterns (LBP) (3304 dimensions), and Gabor (6750 dimensions).

[0073] (2) MSRCV1: This dataset contains 210 images from Microsoft Research Cambridge, divided into seven categories: trees, buildings, airplanes, cows, faces, cars, and bicycles. It includes six different views: CENT (1302 dimensions), CMT (48 dimensions), GIST (512 dimensions), HOG (100 dimensions), LBP (256 dimensions), and SIFT (210 dimensions).

[0074] (3) ORL: This dataset contains 400 face images of 40 individuals. In our experiment, three feature sets were selected for testing: Gabor features of 6750 dimensions, LBP (Local Binary Patterns) features of 3304 dimensions, and intensity features of 4096 dimensions.

[0075] (4) 100leaves: It contains 1600 samples from one hundred plant species. For each sample, shape descriptors, fine-scale edges, and texture histograms are given.

[0076] (5) Mfeat: This handwritten digit (0 - 9) dataset is from the UCI repository. The dataset contains 2000 samples. Each sample is represented by six types of features.

[0077] (6) Dermatology: This dataset is a dermatology dataset, including two views, namely 12 clinical attributes and 22 histopathological attributes, and a total of 366 samples. The samples are divided into six different categories: psoriasis, seborrheic dermatitis, lichen planus, pityriasis rosea, chronic dermatitis, and pityriasis rubra pilaris. Due to 8 missing values in the age feature, the relevant instances were deleted, and the final number of valid samples is 358.

[0078] (7) Inhouse: The Inhouse dataset contains CITE-seq data, measuring matched transcriptome and surface protein data of 1182 samples from six categories.

[0079] Table 1. Statistical information of multi-view data used in the experiment

[0080]

[0081] First, for these collected multi-view data, assuming there are V different views, each view is a non-negative matrix X v , X v The rows of which correspond to samples and the columns correspond to features. For the multi-view data X v Perform clustering, and the specific steps are as follows:

[0082] S1: Based on the collected multi-view data, define a mathematical optimization model that combines a joint non-convex tensor nuclear norm and a local regularization term. This model can effectively characterize the high-order correlation between different view data and the local attributes between samples, and the specific description is as follows:

[0083]

[0084] Among them, X v represents the v-th view data matrix in the multi-view dataset; E v represents the error term corresponding to the v-th view, which is used to characterize the noise in the data; Z v represents the representation matrix under the v-th view, C v represents the core matrix under the v-th view, P v and Q v represent orthogonal matrices under the v-th view; represents the non-convex nuclear norm of the tensor ; represents the similarity between samples i and j under the v-th view; λ1 and λ2 represent regularization parameters. In the embodiments of the present invention, the values of λ1 and λ2 are 10 and 10 respectively.

[0085] S2: Based on the optimization model, establish an augmented Lagrangian function, and the specific description of the augmented Lagrangian function is as follows:

[0086]

[0087] Among them, is the introduced auxiliary variable, and its initial value is a zero matrix; is the Laplacian matrix, where and represent Lagrange multipliers; λ1 and λ2 represent regularization parameters to prevent overfitting, and μ is a positive penalty scalar. Using the alternating direction method of multipliers (ADMM), fixing other variables, each variable is optimized separately, an iterative format is established, and the optimized representation matrix Z v under each view is obtained through iterative loops, and the specific steps are as follows;

[0088] S201: The update of the tensor can be divided into n sub-problems, and the j-th sub-problem is expressed as:

[0089]

[0090] wherein FFT(·) represents the fast Fourier transform; and θ respectively represent the i-th largest singular value of and the weight of the singular value; Ω(·) represents a non-convex tensor function;

[0091] S202: Update Z by solving the following problem v :

[0092]

[0093] Take the partial derivative of L(Z v ) with respect to Z v , and set its partial derivative to 0 to obtain the update formula for Z v as follows:

[0094]

[0095] S203: Update C by the following formula v :

[0096]

[0097] wherein

[0098] S204: Update P by solving the following problem v :

[0099]

[0100] wherein, the closed-form solution of the above optimization problem is:

[0101]

[0102] wherein, M v and N v are respectively the left and right singular value matrices after performing SVD decomposition on ;

[0103] S205: Similarly, the optimization problem for solving Q v can be obtained as

[0104]

[0105] wherein, the optimal solution of Q v is wherein, M' v and N' v are respectively for The left and right singular value matrices obtained by performing SVD decomposition;

[0106] S206: Update E through the following formula v :

[0107]

[0108] where Θ represents the singular value threshold operator;

[0109] S207: Update and μ through the following formula:

[0110]

[0111] where μ, ρ, and μ max are all given constants. In the embodiments of the present invention, ρ, μ, and μ max are respectively set to 2.7, 1e - 2, and 1e - 8;

[0112] S208: Iterate according to the parameter values updated in S201 - S207, and calculate the iteration error. The specific calculation formula is as follows:

[0113]

[0114] Terminate the iteration when the number of iterations meets the set maximum number of iteration steps or the error value Error_value is less than the set value, and obtain the optimized Z v . In the embodiments of the present invention, the maximum number of iteration steps is 30, and the set value of the error value Error_value is 1e - 2;

[0115] S3: Sum the representation matrices under each view Then perform angular correction on the matrix . The specific steps are as follows:

[0116] S301: Sum the optimized representation matrices Z v under each view to obtain the composite matrix

[0117] S302: Perform singular value decomposition on the composite representation matrix

[0118] S303: Calculate A = U∑ 1 / 2 ;

[0119] S304: Construct the similarity matrix W:

[0120] ​S4: Based on the similarity matrix W obtained in step S3 and the number of known classes provided in the test data, use the spectral clustering method to cluster the similarity matrix W, thereby obtaining the clustering results of the corresponding samples. The specific steps are as follows:

[0121] S401: Construct a normalized Laplacian matrix L = D -1 / 2 WD -1 / 2 , where D is a diagonal matrix d ii = ∑ j w ij ;

[0122] S402: Calculate the eigenvalues of matrix L to obtain the eigenvectors V = [v i , v2,..., v k corresponding to the k smallest eigenvalues;

[0123] S403: Normalize V using the L2 norm

[0124] S404: Use the k-means method to cluster the normalized matrix V to obtain k classes, and each class corresponds to a sample type.

[0125] The method of the present invention can be summarized into four steps. First, based on the multi-view data matrix, construct a mathematical model based on non-convex tensor nuclear norm and local graph regularization. Next, use the alternating direction multiplier method (ADMM) to construct an augmented Lagrangian function L based on the established optimization model, and each time select a variable to perform separate optimization and solution on the selected variable while fixing other variables, thereby obtaining an iterative update algorithm for solving the optimization problem and solving the characterization matrix Z v for each view. In addition, by performing angular correction on the synthesized characterization matrix, the similarity matrix W between samples is further obtained. Finally, use the spectral clustering method to cluster the similarity matrix W to obtain the clustering labels of the samples.

[0126] Usually, in order to evaluate the effect of the clustering method, accuracy (ACC), normalized mutual information (NMI), and Rand index (ARI) are often used for measurement. The larger these three metrics are, the better the clustering effect of the method.

[0127] The definition of ACC is as follows:

[0128]

[0129] where t i is the true class label of the i-th sample, and p iis the predicted class label, n is the number of samples; δ(a, b) is an indicator function, which takes the value 1 when a = b and 0 otherwise; map(·) is a mapping function. To achieve the best class label matching, the Hungarian algorithm is used to implement it.

[0130] Assume the true clustering label T and the predicted clustering label Y. The definition of NMI is as follows:

[0131]

[0132] Among them, MI(T, Y) is the mutual information between the clustering labels T and Y, H(Y) and H(T) are the entropies corresponding to the clustering labels Y and T respectively, p(t, y) represents the joint probability distribution of t and y, and p(t) and p(y) represent the marginal probabilities of t and y respectively.

[0133] The definition of ARI is as follows:

[0134]

[0135] Among them, a ty represents the number of point pairs that belong to the same class in both the true class and the predicted class; a t represents the number of point pairs that belong to the same class in the true class but not in the same class in the predicted clustering; a y represents the number of point pairs that belong to the same class in the predicted class but not in the same class in the true class; a represents the number of point pairs that do not belong to the same class in both the true class and the predicted class.

[0136] To evaluate the effectiveness of the method of the present invention, it is compared with the latest representative methods in the literature (FMR (R. Li, C. Zhang, Q. Hu, P. Zhu, and Z. Wang, Flexible multi-view representation learning for subspace clustering, in Proc. IJCAI, vol. 2019, pp. 2916--2922, Aug. 2019.), LMVSC (Z. Kang, W. Zhou, Z. Zhao, J. Shao, M. Han, and Z. Xu, Large-scale multi-view subspace clustering in linear time, Proceedings of the AAAI Conference on Artificial Intelligence, vol. 34, no. 4, pp. 4412--4419, Apr. 2020.), UDBGL (S. Fang, D. Huang, X. Cai, C. Wang, C. He, and Y. Tang, Efficient multi-view clustering via unified and discrete bipartite graph learning, IEEE Transactions on Neural Networks and Learning Systems, vol. 35, no. 8, pp. 11436--11447, Aug. 2024.), SM2SC (Z. Yang, Q. Xu, W. Zhang, X. Cao, and Q. Huang, Split Multiplicative Multi-View Subspace Clustering, IEEE Transactions on Image Processing, vol. 28, no. 10, pp. 5147--5160, Oct. 2019.), SMCMB (J. Lao, D. Huang, C. D. Wang, and J. H. Lai, Towards scalable multi-view clustering via joint learning of many bipartite graphs, IEEE Transactions on Big Data, vol. 10, no. 1, pp. 71--91, Oct. 2023.), RC_MSC (J. Guo, Y. Sun, J. Gao, Y.Hu, and B. Yin, "Rank consistency induced multiview subspace clustering via low-rank matrix factorization," IEEE Trans. Neural Netw. Learn. Syst., vol. 33, no. 7, pp. 3157--3170, Dec. 2021.), GMC (H. Wang, Y. Yang, and B. Liu, "GMC: Graph-based multi-view clustering," IEEE Transactions on Knowledge and Data Engineering, vol. 32, no. 6, pp. 1116--1129, Mar. 2019.)) were compared on existing multi-view sequencing data, and three metrics, ACC, NMI, and ARI, were used to evaluate the clustering performance, so as to compare the advantages and disadvantages of each method.

[0137] A. Comparison of ACC metric based on clustering results

[0138] Table 1 shows the ACC metric of the clustering results of the method of the present invention under seven groups of test multi-view data and compares it with the clustering results of seven other methods. The larger the ACC value, the better the clustering effect. It can be seen from Table 1 that the clustering effect of the method of the present invention is significantly better than other methods.

[0139] Table 1. Comparison of ACC of clustering results between the newly invented method and several other clustering methods under multi-view data tests

[0140]

[0141] B. Comparison of NMI metric based on clustering results

[0142] The results in Table 2 show the comparison of the NMI metric of the clustering results between the method of the present invention and seven other clustering methods under seven groups of test multi-view data. It can be seen from the table that the clustering results of the method of the present invention on all datasets are better than those of several other clustering methods.

[0143] Table 2. Comparison of NMI of clustering results between the newly invented method and several other clustering methods under multi-view data tests

[0144]

[0145] C. Comparison of ARI metric based on clustering results

[0146] Table 3 lists the ARI metrics of the clustering results of the method of the present invention under multi-view data in seven groups of tests, and compares them with seven other methods. The larger the ARI value, the better the clustering effect. It can be seen from Table 3 that the method of the present invention is significantly superior to other methods in terms of clustering effect.

[0147] Table 3. Comparison of ARI of clustering results between the newly invented method and several other clustering methods under multi-view data tests

[0148]

[0149] Compared with the prior art, the embodiments of the present invention effectively characterize the high-order correlation and local features of data by introducing non-convex tensor nuclear norm and local graph regularization. The numerical experimental results show that the present invention can significantly improve the accuracy of multi-view clustering and has obvious advantages compared with the prior methods. In addition, the method of the present invention has good universality and can be widely applied to different types of multi-view data analysis and processing, with a wide application range and high accuracy.

[0150] The above description is only for the description of the embodiments of the present invention and does not limit the protection scope of the present invention. Without departing from the design spirit of the present invention, those of ordinary skill in the art can make various deformations and improvements to the technical solutions of the present invention, and these deformations and improvements should be included in the protection scope defined by the claims of the present invention.

Claims

1. A multi-view subspace clustering method based on non-convex tensor nuclear norm, comprising the following steps: S1: According to the collected multi-view data, construct a mathematical optimization model that combines non-convex tensor nuclear norm and local regularization terms. This model can effectively characterize the high-order correlation between different view data and the local attributes between samples, and the specific description is as follows: Among them, X v represents the v-th view data matrix in the multi-view dataset; E v represents the error term of the v-th view, which is used to characterize the noise in the data; Z v represents the representation matrix of the v-th view; C v represents the core matrix of the v-th view, P v and Q v represent the orthogonal matrix of the v-th view; represents the tensor of the non-convex nuclear norm; represents the similarity between samples i and j under the v-th view; λ1 and λ2 represent regularization parameters; S2: Based on the optimization model, construct the augmented Lagrangian function and use the Alternating Direction Method of Multipliers (ADMM) to perform separate iterative optimization on each variable. Through the iterative process, finally obtain the optimized representation matrix Z under each view v ; S3: Sum the representation matrices of each view Then for the matrix Perform angle correction. The specific steps are as follows: First, perform singular value decomposition on the matrix to obtain the matrix A = U∑ ; Then, by normalizing the matrix A, obtain the similarity matrix W, where 1 / 2 ; Then, by normalizing the matrix A, obtain the similarity matrix W, where S4: Use the spectral clustering method to perform clustering analysis on the obtained similarity matrix W, so as to realize the classification of samples.

2. The method according to claim 1, characterized in that The specific description of the augmented Lagrangian function is as follows: Among them, is an introduced auxiliary variable, and its initial value is set to a zero matrix; is the Laplacian matrix, where and represent Lagrange multipliers; λ1 and λ2 represent regularization parameters used to prevent overfitting, and μ is a positive penalty scalar.

3. The method according to claim 2, wherein: The specific steps of using ADMM to optimize the objective function value in step S2 include: S201: Tensor The update of can be divided into n sub-problems, and the j-th sub-problem is expressed as: Among them FFT(·) represents the fast Fourier transform; and θ respectively represent the i-th largest singular value of and the weight of the singular value; Ω(·) represents a non-convex tensor function; S202: Update Z by solving the following problem v : For L(Z v ) with respect to Z v Take the partial derivative and set it to 0 to obtain the update formula for Z v as follows: S203: Update C according to the following formula v : Among them S204: Update P by solving the following problems v : Among them, The closed-form solution of the above optimization problem is: Among them, M v and N v are respectively the left and right singular value matrices after performing SVD decomposition on ​ S205: Similarly, the optimization problem of solving Q v is as follows Among them, Q v The optimal solution of where M' v and N' v are respectively the left and right singular value matrices obtained by performing SVD decomposition on S206: Update E using the following formula v : Among them, Θ represents the singular value threshold operator; S207: Update by the following formula and μ: where μ, ρ, and μ max are all given constants; S208: Iterate according to the update method provided by S201 - S207, and calculate the iteration error. The specific calculation formula is as follows: Terminate the iteration when the number of iterations meets the set maximum number of iteration steps or the error value Error_value is less than the set value, and obtain the optimized Z v .

4. The method according to claim 3, wherein: The matrix angle correction method adopted in step S3 includes the following specific steps: S301: Sum the optimized representation matrices Z under each view to obtain a composite matrix v ​ S302: Perform singular value decomposition on the synthetic characterization matrix ​ S303: Calculate A = U∑ 1 / 2 ; S304: Construct similarity matrix W:

5. The method according to claim 4, characterized in that: The specific steps of using the spectral clustering method to cluster the similarity matrix W mentioned in step S4 include: S401: Construct the normalized Laplacian matrix L = D -1 / 2 WD -1 / 2 , where D is the diagonal matrix d ii = ∑ j w ij ; S402: Calculate the eigenvalues of matrix L to obtain the eigenvectors V = [v i , v2,..., v k corresponding to the k smallest eigenvalues; S403: Normalize matrix V using the L2 norm S404: Use the k-means method to cluster the normalized matrix V to obtain k classes, and each class corresponds to a sample type.

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