Multi-view subspace clustering method based on kernel low-rank consistency

Abstract

The invention provides a multi-view subspace clustering method based on kernel low rank consistency. According to the method, nonlinear data are mapped to a high-dimensional linear feature space by using a kernel function to mine nonlinear features in the data, and meanwhile, a low-rank representation (LRR) theory is adopted to obtain a low-rank representation matrix to capture global features of the linear space data. And further decomposing the low-rank representation sparse matrix into three matrixes through a matrix three-factor decomposition theory to realize the consistency among different views. In order to effectively reduce the influence of noise, a common representation matrix of data under different views is constrained by adopting a Schatten-p norm. Based on the established optimization model, an alternating direction multiplier method (ADMM) is adopted to carry out optimization solution to obtain a representation matrix under each view, then angle correction is carried out on the fused representation matrix to obtain an inter-sample similarity matrix, and finally a spectral clustering method is adopted to cluster the similarity matrix, so that clustering of the samples is realized. A large number of simulation experiment results show that the method can significantly improve the clustering effect of the multi-view data, and a new effective method is provided for multi-view data analysis.

Description

Technical Field

[0001] The present invention relates to the field of interdisciplinary research in mathematics and computers, and in particular to a method for clustering analysis of multi-view data through data analysis modeling. Background Art

[0002] With the rapid development of information technology, massive, high-dimensional, and multi-source data has been generated. How to extract valuable information from these data has become a research hotspot in various fields. Cluster analysis, as an unsupervised learning method, can automatically discover the potential structures and patterns in the data and provide support for data analysis and decision-making.

[0003] Traditional single-view clustering can only mine features from one aspect of the data to cluster samples. Since the single-view clustering method only uses information from one aspect of the data, there are often problems of information loss and inability to obtain global information of the samples. Multi-view clustering can overcome the limitations of single-view clustering. It is a method that uses complementary and consistent information from multiple views to provide a more comprehensive and accurate data description, thereby improving clustering performance.

[0004] The application scenarios of multi-view clustering algorithms are very wide, such as in bioinformatics, text analysis, image recognition, social network analysis and other fields. In these applications, data can be observed and described from multiple perspectives or modalities (such as text, image, sound, etc.), and each view contains part of the information of the data entity. The goal of the algorithm is to integrate this information from different views to discover the potential structure and pattern in the data. Due to its unique advantages and wide range of application scenarios, multi-view clustering algorithms have important significance and application value in the fields of data mining and machine learning.

[0005] However, with the increasing abundance of multi-view data, the complexity and diversity of data are constantly increasing. How to effectively mine multi-view data and design effective clustering methods is a huge challenge currently faced. In recent years, scholars at home and abroad have proposed many algorithms for multi-view clustering. For example, Zhang et al. proposed a latent multi-view subspace clustering method (LMSC), which clusters data points using the latent representation, and explores complementary information in multiple perspectives. This method finds the latent representation of data in multiple views, reconstructs the data on this basis, establishes a complete representation of the data, and improves the accuracy and robustness of the subspace representation. In order to avoid the existing methods that ignore the intrinsic association between views by reconstructing data based on each individual view, Li et al. proposed a subspace clustering method FMR based on flexible multi-view representation. This method can effectively avoid using partial information for data reconstruction, and by introducing the kernel dependency metric, the latent representation can flexibly encode complementary information in different perspectives and explore the nonlinear high-order correlation between these perspectives. Numerical results show that this method can effectively improve the clustering effect of multi-view data. In this paper, scholars such as M.et al. proposed a multi-view subspace clustering method based on low-rank and sparse constraints, which learns the joint subspace representation by constructing a common similarity matrix between all views. The alternating direction multiplier method is used to solve the relevant low-rank and sparse constraint optimization problems for each view. In addition, by solving the corresponding problems in the reproducing kernel Hilbert space, the method is extended to the clustering of data extracted from nonlinear subspaces. Lan et al. proposed a new multi-view subspace clustering algorithm LSGMC based on the consistent decomposition of the matrix. This method establishes a multi-view subspace clustering algorithm based on different views. Figure 1 The algorithm uses a mathematical model that is consistent with low-rank structure and symmetry constraints, and uses a fusion mechanism and Shatten-p norm to obtain a low-rank representation matrix considering the influence of angle information and noise. It uses the graph structure to capture the relationship between data points and achieves a more accurate clustering effect by jointly optimizing the graph representation of multiple views.

[0006] Considering that the existing multi-view clustering algorithms have high time complexity and are difficult to apply to actual large-scale data scenarios, some scholars have proposed to establish anchor points to reduce the computational complexity. However, the separation in the heuristic sampling and clustering process leads to a great influence of the selection of anchor points on the results. In addition, since the graph is constructed separately based on the anchor points of different views, the complementary information between views is not fully utilized. Sun et al. proposed a scalable multi-view subspace clustering algorithm (SMVSC) based on a unified anchor set. This method combines anchor point selection and graph construction into a unified optimization framework. Therefore, the obtained anchor points can more accurately represent the data distribution and obtain a more discriminative clustering structure. In addition, this method has linear time complexity and can be applied to large-scale data. Guo et al. proposed a new scalable multi-bipartite graph (SMCMB) multi-view clustering method, which can jointly learn and fuse multiple bipartite graphs from multiple views while maintaining high efficiency for very large data sets. Different from the single anchor set view paradigm, SMCMB first generates multiple different anchor sets on each view, thereby obtaining multiple anchor sets on multiple views, based on which anchor-based subspace representation learning is performed, and multiple bipartite graphs are learned simultaneously. These bipartite graphs are then effectively segmented to generate base clusters, which are reformulated as unified bipartite graphs for final clustering.

[0007] Although the methods in the above literature have done some work on clustering multi-view data, with the increasing abundance of multi-view data, the complexity and diversity of data are constantly increasing. How to effectively cluster multi-view data is the key to multi-view data pattern mining. At present, the accuracy and efficiency of clustering methods based on multi-view data still need to be further improved. Summary of the invention

[0008] The present invention aims to provide a new method for multi-view data clustering based on kernel low-rank consistency, which realizes effective data analysis of multi-view data, integrates the unique properties of different view data and the complementary features between views based on the low-rank representation of multi-view matrices, kernel function mapping, matrix consistency decomposition and regularization, and thus realizes effective clustering of data samples.

[0009] The technical solution adopted by the present invention is: a method for multi-view data clustering based on kernel low-rank consistency, comprising the following steps:

[0010] S1: Based on the collected multi-view data, a mathematical optimization model combining kernel low-rank representation and matrix consistent decomposition is defined. The model can effectively characterize the nonlinear characteristics of the data. The specific description is:

[0011]

[0012]

[0013] Among them, X v represents a matrix containing v view data; E v represents the error term corresponding to the vth view, which is used to characterize data noise; Z v represents the representation matrix under the vth view, C represents the common feature matrix under different views, and R v and L v represents the orthogonal matrix under the v-th view; λ1, λ2 and λ3 represent regularization parameters; represents the Schatten-p norm of the matrix C; ||B v || * Represents the matrix B v The nuclear norm of ; φ(·) is the kernel function, which maps the original data to the kernel Hilbert space; is a positive semidefinite kernel Gram matrix; Representation Matrix The value of the element at position (i,j) of

[0014] S2: Based on the optimization model, an augmented Lagrangian function is established, and the alternating direction multiplier method, i.e. ADMM, is used to optimize each variable separately, and an iterative format is established to obtain the representation matrix Z of each view after the iterative cycle optimization. v ;

[0015] S3: Sum the representation matrix under each view Then for Z * The matrix performs angle correction, specifically by adjusting the matrix Z * Perform SVD decomposition Z * =U * ∑ * (V * ) T , we get the matrix M = U * (∑ * ) 12 , and finally the similarity matrix W is obtained by normalizing the M matrix.

[0016] S4: Cluster the similarity matrix W using a spectral clustering method to obtain corresponding sample classifications.

[0017] Preferably, the augmented Lagrangian function is specifically described as:

[0018]

[0019]

[0020] Among them, Jv Represents the introduced variables, the initial value is 0 matrix; and represents the Lagrange multiplier; λ1, λ2, λ3 and μ represent regularization parameters to prevent overfitting; represents the Schatten-p norm of the matrix C; ||B v || * Represents the matrix B v The nuclear norm of .

[0021] Preferably, the step S2 of optimizing the objective function value by using the alternating direction multiplier method includes the following specific steps:

[0022] S201: The update of matrix C requires optimizing and solving the following problems:

[0023]

[0024] in, In order to solve C, the generalized soft threshold (GST (Generalized Soft-Thresholding)) algorithm is used for solution. The specific iterative method is shown in Algorithm 1;

[0025]

[0026] S202: Update B by solving the following problem:

[0027]

[0028] in,

[0029] The closed-form solution to the above optimization problem is:

[0030]

[0031] Among them, Γ * and V are both The SVD decomposition of the matrix gives, express SVD decomposition of Γ * is a diagonal matrix, element σ i for The i-th singular value of ;

[0032] S203: Update Z by the following formula v :

[0033]

[0034] in,

[0035] S204: Update J by the following formula v :

[0036]

[0037] in,

[0038] S205: Update L by solving the following problem v :

[0039]

[0040] in,

[0041] The closed-form solution to the above optimization problem is:

[0042] L v =M v N vT

[0043] Among them, M v and N v L v The left and right singular value matrices after SVD decomposition;

[0044] S206: Similarly, we can get the solution for R v The optimization problem is:

[0045]

[0046] in,

[0047] R v The optimal solution is

[0048] Among them, M v' and Respectively The left and right singular value matrices obtained by SVD decomposition;

[0049] S207: Update E by the following formula:

[0050]

[0051] Where, Θ represents the singular value threshold operator;

[0052] S208: Update Q1, Q2, Q3 and μ respectively by the following formulas:

[0053]

[0054] Among them, ρ,μ max All are given constants;

[0055] S209: Iterate according to the parameter values ​​updated in S201-S208 to calculate the iteration error. The specific calculation formula is as follows:

[0056]

[0057] When the number of iterations meets the set maximum number of iterations or the error value Error_value is less than the set value, the iteration is terminated and the optimized Z is obtained. v .

[0058] Preferably, the specific steps of step S3 are as follows:

[0059] S301: Optimize the representation matrix Z for each view v Sum them up and get the composite matrix

[0060] S302: Perform singular value decomposition Z on the synthetic representation matrix Z* * =U * ∑ * (V * ) T ;

[0061] S303: Calculate M=U * (∑ * ) 12 ;

[0062] S304: Constructing a similarity matrix

[0063] Preferably, the step S4 of clustering the similarity matrix W using the spectral clustering method comprises the following specific steps:

[0064] S401: Constructing the normalized Laplace matrix L=D -12 WD -12 , where D is a diagonal matrix

[0065] S402: Calculate the eigenvalues ​​of the matrix L, and obtain the eigenvectors V=[v1,v2,…v k ];

[0066] S403: Normalize V using L2 norm

[0067] S404: Use the Kmeans method to cluster the normalized matrix V to obtain k classes, each class corresponding to a sample type.

[0068] Preferably, the maximum number of iteration steps is 30, and the error is set to 1e-2 at the Error_value setting.

[0069] The beneficial effects of the present invention are:

[0070] (1) Compared with the existing technology, the method of the present invention effectively uses the kernel function to mine the nonlinear structure in the data based on the global structural characteristics of multi-view data, and decomposes the features under different views into common features of all views through a matrix for effective characterization. This method can effectively improve the clustering effect;

[0071] (2) The present invention uses the Shatten-p norm to characterize the "common part" of all view data, which can effectively reduce the influence of factors such as noise. Compared with the original nuclear norm, it has obvious advantages in characterizing the global structural features of data;

[0072] (3) The method of the present invention has obvious advantages over existing methods in terms of clustering effect. It can effectively cluster complex multi-view data, help discover potential structures and patterns in the data, and provide support for data analysis and decision-making. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0074] Figure 1 The figure is a flow chart of the method of the present invention.

[0075] Figure 2 This is a visual comparison of the clustering results of the method of the present invention (LRKCMS) and the other three clustering methods (RC_MSC, LSGMC, FMR) on three sets of multi-view data: test data Inhouse (A), MSRCV (B) and Mfeat (C). The points in the figure represent samples, and samples clustered into the same category are marked with a symbol. DETAILED DESCRIPTION

[0076] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. The beneficial effects of the present invention will be described in detail below in conjunction with examples, in order to help readers better understand the essence of the present invention, but it does not constitute any limitation on the implementation and protection scope of the present invention.

[0077] The present invention provides a multi-view clustering method based on a kernel low-rank consistency model and Figure 1 The mathematical model that considers the nonlinear characteristics of multi-view data and the complementary information between views is established by consistent decomposition, and a new multi-view clustering method is proposed. The main steps of this method include:

[0078] (1) Based on the collected multi-view data, a mathematical model combining kernel low-rank representation and matrix consistent decomposition is defined;

[0079] (2) Based on the established mathematical model, an augmented Lagrangian function is constructed, and the alternating direction multiplier method (ADMM) is used to optimize the objective function value to obtain the representation matrix Z of each view after iterative optimization. v ;

[0080] (3) Sum the representation matrix under each view Then for Z * Matrix SVD decomposition Z*=U*∑*(V*) T , we get the matrix M = U * (∑ * ) 12 , then the similarity matrix W is obtained by normalizing the M matrix,

[0081] (4) The similarity matrix W is clustered using a spectral clustering method to obtain corresponding sample class labels.

[0082] In order to test the effectiveness of the method of the present invention, the multi-view test data provided in the literature is used. The description and data sources of the 7 sets of multi-view data used for the test are as follows:

[0083] (1) COIL-20: This dataset contains 1440 images from 20 categories. Each category contains 72 images with a black background, taken from different positions. The dataset contains three extracted view dimensions: intensity (4096 dimensions), local binary pattern (LBP) (3304 dimensions), and Gabor (6750 dimensions).

[0084] (2) MSRCV1: This dataset contains 210 images from Microsoft Research Cambridge, divided into seven categories: tree, building, airplane, cow, face, car, and bicycle. It includes six different views: CENT (1302 dimensions), CMT (48 dimensions), GIST (512 dimensions), HOG (100 dimensions), LBP (256 dimensions), and SIFT (210 dimensions).

[0085] (3) 3Sources: This dataset contains 169 articles, which are manually classified into six groups. Each article comes from three different online news media (BBC, Reuters, and The Guardian), and each news media is used as an independent view of the article.

[0086] (4) 100leaves: It contains 1600 samples from one hundred plants. For each sample, shape descriptors, fine-scale edges and texture histograms are given.

[0087] (5) Mfeat: This handwritten digit (0-9) dataset comes from the UCI repository. The dataset contains 2000 samples. Each sample is represented by six types of features.

[0088] (6) Hdigit: This dataset contains 10,000 images of handwritten digits from two datasets: MNIST and USPS handwritten digits, divided into 10 categories. In our experiments, we use two views with dimensions of 784 and 256 respectively.

[0089] (7) Inhouse: The Inhouse dataset contains CITE-seq data, measuring matched transcriptome and surface protein data of 1,182 samples from six categories.

[0090] Table 1. Statistics of multi-view data used in the experiments

[0091]

[0092] First, these collected multi-view data are assumed to have V different views, each of which is a non-negative matrix X v , X v The rows correspond to samples and the columns correspond to features. v To perform clustering, the specific steps are as follows:

[0093] S1: Based on the collected multi-view data, a mathematical optimization model combining kernel low-rank representation and matrix consistent decomposition is defined. The model can effectively characterize the nonlinear characteristics of the data. The specific description is:

[0094]

[0095]

[0096] Among them, X v represents a matrix containing v view data; E v represents the error term corresponding to the vth view, which is used to characterize data noise; Z v represents the representation matrix under the vth view, C represents the common feature matrix under different views, and R v and L v represents the orthogonal matrix under the v-th view; λ1, λ2 and λ3 represent regularization parameters; represents the Schatten-p norm of the matrix C; ||B v ||* Represents the matrix B v The nuclear norm of ; φ(·) is the kernel function, which maps the original data to the kernel Hilbert space; is a positive semidefinite kernel Gram matrix; Representation Matrix The value of the element at position (i,j) of

[0097] In the embodiment of the present invention, the values ​​of λ1, λ2 and λ3 are 100, 0.03 and 10 respectively, a=11, b=2.

[0098] S2: Establishing an augmented Lagrangian function based on the optimization model, wherein the augmented Lagrangian function is specifically described as:

[0099]

[0100]

[0101] Among them, J v Represents the introduced variables, the initial value is 0 matrix; and represents the Lagrange multiplier; λ1, λ2, λ3 and μ represent regularization parameters to prevent overfitting; represents the Schatten-p norm of the matrix C; ||B v || * Represents the matrix B v The nuclear norm of .

[0102] Using the alternating direction multiplier method (ADMM), fix other variables, optimize each variable separately, establish an iterative format, and obtain the representation matrix Z of each view after the iterative cycle optimization. v , the specific steps are as follows;

[0103] S201: The update of matrix C requires optimizing and solving the following problems:

[0104]

[0105] in, In order to solve C, the generalized soft threshold (GST (Generalized Soft-Thresholding)) algorithm is used for solution. The specific iterative method is shown in Algorithm 1;

[0106]

[0107] In the embodiment of the present invention, the value of the input parameter p of Algorithm 1 is 0.9.

[0108] S202: Update B by solving the following problem:

[0109]

[0110] in, The closed-form solution to the above optimization problem is:

[0111] B v* =Γ * V T

[0112] Among them, Γ * and V are both The SVD decomposition of the matrix gives, express SVD decomposition of Γ * is a diagonal matrix, element σ i for The i-th singular value of ;

[0113] S203: Update Z by the following formula v :

[0114]

[0115] in,

[0116] S204: Update J by the following formula v :

[0117]

[0118] in,

[0119] S205: Update L by solving the following problem v :

[0120]

[0121] in,

[0122] The closed-form solution to the above optimization problem is:

[0123]

[0124] Among them, M v and N v L v The left and right singular value matrices after SVD decomposition;

[0125] S206: Similarly, we can get the solution for R v The optimization problem is

[0126]

[0127] in,

[0128] R v The optimal solution is Among them, M v 'and Respectively The left and right singular value matrices obtained by SVD decomposition;

[0129] S207: Update E by the following formula:

[0130]

[0131] Where, Θ represents the singular value threshold operator;

[0132] S208: Update Q1, Q2, Q3 and μ respectively by the following formulas:

[0133]

[0134] Among them, ρ,μ max are all given constants; in the embodiment of the present invention, ρ, μ and μ max The values ​​are 2.7, 0.01, and 1e-8 respectively.

[0135] S209: Iterate according to the parameter values ​​updated in S201-S208 to calculate the iteration error. The specific calculation formula is as follows:

[0136]

[0137] When the number of iterations meets the set maximum number of iterations or the error value Error_value is less than the set value, the iteration is terminated and the optimized Z is obtained. v In the embodiment of the present invention, the maximum number of iteration steps is 30, and the setting value of the error value Error_value is 1e-2.

[0138] S3: The representation matrix Z of each view obtained in step S2 is v Sum and perform angle correction. The specific steps are as follows:

[0139] S301: Optimize the representation matrix Z for each view v Sum them up and get the composite matrix

[0140] S302: Perform singular value decomposition Z on the synthetic representation matrix Z* * =U * ∑ *(V * ) T ;

[0141] S303: Calculate M=U * (∑ * ) 12 ;

[0142] S304: Constructing a similarity matrix

[0143] S4: Based on the similarity matrix W obtained in step S3 and the number of known classes provided in the test data, the similarity matrix W is clustered using a spectral clustering method to obtain the clustering results of the corresponding samples. The specific steps are as follows:

[0144] S401: Constructing the normalized Laplace matrix L=D -12 WD -12 , where D is a diagonal matrix

[0145] S402: Calculate the eigenvalues ​​of the matrix L, and obtain the eigenvectors V=[v1,v2,…v k ];

[0146] S403: Normalize V using L2 norm

[0147] S404: Use the Kmeans method to cluster the normalized matrix V to obtain k classes, each class corresponding to a sample type.

[0148] The method of the present invention can be summarized into four steps. Based on the multi-view data matrix, a mathematical model based on kernel low-rank representation and matrix consistency decomposition is established; then, the alternating direction multiplier method is used to construct an augmented Lagrangian function L based on the established optimization model. One variable is selected each time, and then the selected variable is optimized and solved separately while other variables are fixed, so as to obtain an iterative update algorithm for solving the optimization problem, and solve the representation matrix Z under each view. v ; Further, the similarity matrix W between samples is obtained by performing angle correction on the synthesized representation matrix; finally, the spectral clustering method is used to cluster the similarity matrix W to obtain the clustering labels for the samples.

[0149] Usually, in order to evaluate the quality of clustering methods, accuracy (ACC), normalized mutual information (NMI) and Rand index (ARI) are used to measure. The larger the three metrics are, the better the clustering effect of the method is.

[0150] The definition of accuracy ACC is as follows:

[0151]

[0152] Among them, t i is the true class label of the i-th sample, y i is the predicted class label, n is the number of samples, δ(a,b) is the indicator function that takes the value of 1 when a=b and 0 otherwise, and map(·) is the mapping function. In order to achieve the best class label matching, the Hungarian algorithm is used.

[0153] Assuming the true cluster label T, and the predicted cluster label Y, NMI is defined as follows:

[0154]

[0155]

[0156] Among them, MI(T,Y) is the mutual information between cluster labels T and Y, H(Y) and H(T) are the entropies corresponding to cluster 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.

[0157] The definition of ARI is as follows:

[0158]

[0159] Among them, a ty Indicates the number of point pairs that belong to the same class in both the true class and the predicted class, a t Indicates the number of points that belong to the same class in the true class but do not belong to the same class in the predicted cluster, a y It represents the number of point pairs that belong to the same class in the predicted class but do not belong to the same class in the true class, and a represents the number of point pairs that do not belong to the same class in both the true class and the predicted class.

[0160] One of the most interesting and emotionally charged patients in the world is the LSGMC(W.Lan,T.Ya ng, Q. Chen, S. Zhang, Y. Dong, H. Zhou, and Y. Pan, Multiview subspaceclustering via low-rank symmetric affinity graph,IEEE Trans.NeuralNetw.Learn.Syst.,pp.1--14,Mar.2023.); clustering,inProc.IJCAI,vol.2019,pp.2916--2922,Aug.2019.)、RC_MSC(J.Guo,Y.Sun,J.Gao,Y.Hu,and B.Yin,Rank consistency induced multiview subspace clustering via low-rankmatrix factorization,IEEE Trans.Neural Netw.Learn.Syst.,vol.33,no.7,pp.3157--3170,Dec.2021.)、LMVSC(Z.Kang,W.Zhou,Z.Zhao,J.Shao,M.Han,and Z.Xu,Large-scalemulti-view subspace clustering in linear time,Proceedings ofthe AAAIConference on Artificial Intelligence},vol.34,no.4,pp.4412--4419,Apr.2020.)、SMVSC(Chen,P.,Liu,L.,Ma,Z.and Kang,Z.,2021.Smooth multi-view subspaceclustering.In Neural Computing for AdvancedApplications:Second InternationalConference,NCAA2021,Guangzhou,China,August27-30,2021,Proceedings 2(pp.128-140).Springer Singapore.) and 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.)) were compared with the existing multi-view sequencing data, and the clustering effect was evaluated using three metrics, ACC, NMI and ARI, to compare the advantages and disadvantages of various methods. .

[0161] A. Comparison of NMI indicators based on clustering results

[0162] The results in Table 1 show that the NMI index of the clustering results of the method of the present invention is compared with that of other 6 clustering methods under 7 groups of multi-view data tested. The clustering results of the method of the present invention under all data are better than those of the other clustering methods.

[0163] Table 1. Comparison of NMI of clustering results between the new invented method and several other clustering methods on the test multi-view data.

[0164]

[0165] B. Comparison based on ARI indicators

[0166] Table 2 compares the ARI index of the clustering results of the method of the present invention under 7 groups of multi-view data tested with the clustering results of the other 6 methods. The larger the ARI, the better the clustering effect of the method. It can be seen from Table 2 that the clustering effect of the method of the present invention is significantly better than that of other methods.

[0167] Table 2. Comparison of clustering results ARI between the new invented method and several other clustering methods on the test multi-view data.

[0168]

[0169] Table 3 is a comparison of the ACC index of the clustering results of the method of the present invention under 7 groups of multi-view data tested with the clustering results of the other 6 methods. The larger the ACC, the better the clustering effect of the method. It can be seen from Table 3 that the clustering effect of the method of the present invention is significantly better than that of other methods.

[0170] Table 3. Comparison of clustering results ACC between the new invented method and several other clustering methods on the test multi-view data.

[0171]

[0172] Compared with the existing technology, the embodiment of the present invention characterizes the global structure and nonlinear characteristics of the data through kernel low rank and matrix consistent decomposition. The comparison results of numerical experiments show that the embodiment of the present invention can effectively improve the accuracy of multi-view clustering and has obvious advantages over the existing methods. In addition, the method of the present invention has good universality and can be applied to different types of multi-view data analysis and processing, with a wide range of applications and high accuracy.

[0173] The above description is only a description of the implementation mode of the present invention, and does not limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solution of the present invention by ordinary technicians in this field should fall within the protection scope determined by the claims of the present invention.

Claims

1. A multi-view subspace clustering method based on kernel low-rank consistency, characterized in that: The following steps are involved: S1: Based on the collected multi-view data, a mathematical optimization model combining kernel low-rank representation and matrix consistent decomposition is defined. The model can effectively characterize the nonlinear characteristics of the data. The specific description is: Among them, X v represents a matrix containing v view data; E v represents the error term corresponding to the vth view, which is used to characterize data noise; Z v represents the representation matrix under the vth view, C represents the common feature matrix under different views, and R v and L v represents the orthogonal matrix under the v-th view; λ1, λ2 and λ3 represent regularization parameters; represents the Schatten-p norm of the matrix C; ||B v || * Represents the matrix B v The nuclear norm of ; φ(·) is the kernel function, which maps the original data to the kernel Hilbert space; is a positive semidefinite kernel Gram matrix; Representation Matrix The value of the element at position (i,j) of S2: Based on the optimization model, an augmented Lagrangian function is established, and the alternating direction multiplier method, i.e. ADMM, is used to optimize each variable separately, and an iterative format is established to obtain the representation matrix Z of each view after the iterative cycle optimization. v ; S3: Sum the representation matrix under each view Then for Z * The matrix performs angle correction, specifically by adjusting the matrix Z * Perform SVD decomposition Z * =U * ∑ * (V * ) T , we get the matrix M = U * (∑ * ) 1 / 2 , and finally the similarity matrix W is obtained by normalizing the M matrix. S4: Clustering the similarity matrix W using a spectral clustering method to obtain corresponding sample classifications.

2. The method according to claim 1, characterized in that: The augmented Lagrangian function is specifically described as: Among them, J v Represents the introduced variables, the initial value is 0 matrix; and represents the Lagrange multiplier; λ1, λ2, λ3 and μ represent regularization parameters to prevent overfitting; represents the Schatten-p norm of the matrix C; ||B v || * Represents the matrix B v The nuclear norm of .

3. The method according to claim 2, characterized in that: The step S2 of using the alternating direction multiplier method (ADMM) to optimize the objective function value includes the following specific steps: S201: The update of matrix C requires optimizing and solving the following problems: in, β=1 / Vμ. In order to solve C, the generalized soft threshold (GST (Generalized Soft-Thresholding)) algorithm is used for solution. The specific iterative method is shown in Algorithm 1; S202: Update B by solving the following problem: in, The closed-form solution to the above optimization problem is: B v* =C * V T Among them, Γ * and V are both The SVD decomposition of the matrix gives, express SVD decomposition of Γ * is a diagonal matrix, element σ i for The i-th singular value of ; S203: Update Z by the following formula v : in, S204: Update J by the following formula v : in, S205: Update L by solving the following problem v : in, The closed-form solution to the above optimization problem is: Among them, M v and N v L v The left and right singular value matrices after SVD decomposition; S206: Similarly, we can get the solution for R v The optimization problem is in, R v The optimal solution is Among them, M v 'and Respectively The left and right singular value matrices obtained by SVD decomposition; S207: Update E by the following formula: Where, Θ represents the singular value threshold operator; S208: Update Q1, Q2, Q3 and μ respectively by the following formulas: Among them, ρ,μ max All are given constants; S209: Iterate according to the parameter values ​​updated in S201-S208 to calculate the iteration error. The specific calculation formula is as follows: When the number of iterations meets the set maximum number of iterations or the error value Error_value is less than the set value, the iteration is terminated and the optimized Z is obtained. v .

4. The method according to claim 3, characterized in that: The method for performing angle correction on the matrix described in step S3 includes the following specific steps: S301: Optimize the representation matrix Z for each view v Sum them up and get the composite matrix S302: Perform singular value decomposition Z on the synthetic representation matrix Z* * =U * ∑ * (V * ) T ; S303: Calculate M=U * (∑ * ) 12 ; S304: Construct similarity matrix W:

5. The method according to claim 4, characterized in that: The step S4 of clustering the similarity matrix W using the spectral clustering method comprises the following specific steps: S401: Constructing the normalized Laplace matrix L=D -12 WD -12 , where D is a diagonal matrix S402: Calculate the eigenvalues ​​of the matrix L, and obtain the eigenvectors V=[v1,v2,…v k ]; S403: Normalize V using L2 norm S404: Use the Kmeans method to cluster the normalized matrix V to obtain k classes, each class corresponding to a sample type.

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