An Incomplete Multi-View Clustering Method and System Based on Co-Regularized Spectral Clustering

Abstract

The present invention discloses an incomplete multi-view clustering method and system based on co-regularized spectral clustering, which relates to the technical fields of computer vision and pattern recognition, and includes: obtaining an incomplete multi-view data set and converting each view into a data matrix; performing dimensionality reduction processing on the data matrix to obtain a coefficient matrix; constructing a similarity matrix based on the self-representation characteristics of the coefficient matrix, and further calculating a Laplacian matrix; obtaining a clustering indicator matrix for each view based on the Laplacian matrix; performing kernel alignment on the clustering indicator matrix to obtain a consistent clustering indicator matrix; finally constructing an objective function for incomplete multi-view clustering and optimizing and solving it to obtain an optimal consistent clustering indicator matrix; and inputting the optimal consistent clustering indicator matrix into an existing classification algorithm to obtain the clustering result of the incomplete multi-view. The present invention deeply mines the complementary and consistent information of view data in the case of incomplete multi-views, and improves the clustering effect of incomplete multi-views.

Description

Technical Field

[0001] The present invention relates to the technical field of computer vision and pattern recognition, and more specifically, to an incomplete multi-view clustering method and system based on co-regularized spectral clustering. Background Art

[0002] As a fundamental technology in pattern recognition, computer vision, and machine learning, clustering divides a set of data points into their respective categories by establishing the similarity between them. In the past few decades, many classic clustering algorithms have been proposed, such as K-means clustering, spectral clustering, etc. However, these algorithms are mainly used to solve single-view clustering problems and it is difficult to find suitable clustering results for multi-view data. Directly using these methods to cluster multi-view data usually cannot obtain ideal results. In the real world, images can be represented by various descriptors, such as SIFT, Local Binary Pattern (LBP), HOG, etc. Web pages can be described by images, links, texts, etc.; the values of blood tests and magnetic resonance images can be used as two references for disease diagnosis. The above phenomena indicate that almost all things can be represented from different perspectives / views. For example, multi-view image data can be used for face recognition, biological species classification, retrograde detection, image retrieval, and automatic album categorization, etc. For multi-view data, different views often have different statistical characteristics; comprehensively using the complementary information of multiple views can produce better clustering performance. In recent years, many multi-view clustering methods have emerged, such as multi-view k-means clustering, methods based on Canonical Correlation Analysis (CCA), co-regularized multi-view spectral clustering, methods based on low-rank tensors, methods based on deep matrix factorization, etc. These methods are all based on the condition that multi-views are complete, but in fact, this condition is very difficult to achieve. Due to the temporary failure of data collectors or limitations caused by factors such as light, target distance, resolution, especially in the field of medical images, patients may skip some examinations, resulting in the lack of data in some views, and thus the obtained multi-view data is incomplete. Most of the existing multi-view clustering algorithms are designed based on complete data and cannot directly process incomplete multi-view data. Therefore, there is a study on incomplete multi-view clustering, aiming to reduce the impact of missing data and at the same time use the consistency and complementary information of multiple views to improve the clustering effect. The existing incomplete multi-view clustering methods generally have the following problems: (1) ignoring the common relationship between data points in all views; (2) not making full use of the complementary multi-view information represented by the original data; (3) being unable to handle arbitrary incomplete scenarios or data containing negative entries.

[0003] The prior art discloses an image clustering method based on multi-view K multi-means with adaptive weights, which uses multiple sub-cluster centers to capture the data distribution of each cluster in each view, and can adaptively assign weights to each view, thereby reasonably fusing complementary information and compatible information of different views to construct a shared bipartite graph. Finally, a Laplace rank constraint is imposed on the shared bipartite graph to divide it into C clusters, thereby achieving clustering of image samples and further used to solve multi-view image recognition and detection problems. However, this method is based on the condition that the view data is complete, and clustering cannot be achieved under the condition that the view data is incomplete. Summary of the invention

[0004] In order to overcome the defect that the above-mentioned existing multi-view clustering methods cannot achieve accurate clustering under the condition of incomplete view data, the present invention provides an incomplete multi-view clustering method and system based on co-regularized spectral clustering, which can deeply mine the complementarity and consistency information of view data when the view data is incomplete, thereby improving the clustering effect of incomplete multi-views.

[0005] In order to solve the above technical problems, the technical solution of the present invention is as follows:

[0006] The present invention provides an incomplete multi-view clustering method based on co-regularized spectral clustering, comprising:

[0007] S1: Obtain an incomplete multi-view dataset and convert each view in the dataset into a data matrix;

[0008] S2: Perform dimensionality reduction processing on the data matrix of each view to obtain the coefficient matrix of each view;

[0009] S3: construct a similarity matrix for each view based on the self-representation property of the coefficient matrix of each view;

[0010] S4: Calculate the Laplacian matrix of each view according to the similarity matrix of each view, and obtain the clustering indicator matrix of each view based on the Laplacian matrix;

[0011] S5: performing kernel alignment on the clustering indicator matrix of each view to obtain a consistent clustering indicator matrix;

[0012] S6: constructing an objective function of incomplete multi-view clustering based on the consistent clustering indicator matrix, optimizing and solving the objective function using an alternating method, and obtaining an optimal consistent clustering indicator matrix;

[0013] S7: Input the optimal consistent clustering indicator matrix into the existing classification algorithm to obtain the clustering result of the incomplete multi-view.

[0014] Preferably, in the step S1, the data matrix of each view contains the non-missing data of the view. Denote the data matrix of the v-th view as:

[0015]

[0016] where represents the n-th v non-missing data element in the v-th view, d v represents the feature dimension of the non-missing data in the v-th view, n v represents the number of non-missing data in the v-th view; V represents the number of views in the incomplete multi-view dataset, and v = 1, 2, …, V.

[0017] Preferably, in the step S2, perform semi-nonnegative matrix factorization on the data matrix of each view to achieve dimensionality reduction, and obtain the coefficient matrix of each view. The expression is:

[0018]

[0019] s.t. V (v) ≥0

[0020] In the formula, U (v) represents the basis matrix of the v-th view, V (v) represents the coefficient matrix of the v-th view, c represents the number of clusters; represents calculating the F norm.

[0021] Preferably, in the step S3, the expression for constructing the similarity matrix of each view is:

[0022]

[0023]

[0024] In the formula, Z (v) represents the similarity matrix of the v-th view, represents the element in the i-th row and j-th column of Z (v) , represents calculating the F norm.

[0025] In order to utilize the intra-view similarity structure and the potential representation subspace structure, use the self-representation property of the coefficient matrix to construct a graph in the potential representation subspace, and obtain the self-representation matrix of each view, that is, the similarity matrix. The similarity matrix can reflect the similarity structure of the multi-view, and each element in it reflects the similarity weight.

[0026] Preferably, in the step S4, the specific method for calculating the Laplacian matrix of each view according to the similarity matrix of each view is as follows:

[0027] According to the similarity matrix Z of each view (v) calculate the corresponding adjacency matrix W (v) :

[0028]

[0029] where |*| represents the modulo operation, and |Z (v) | T represents the transpose of |Z (v) |;

[0030] According to the adjacency matrix W (v) calculate the corresponding weighted degree matrix D (v) :

[0031]

[0032] where represents the element in the i-th row and i-th column of the weighted degree matrix D (v) ; represents the element in the i-th row and j-th column of the adjacency matrix W (v) which represents the weight of the element in the i-th row and j-th column; if there is no edge connection between two vertices, the corresponding element in the adjacency matrix is 0; calculate each diagonal element in the same way and fill to obtain the weighted degree matrix D (v) ;

[0033] According to the adjacency matrix W (v) and the weighted degree matrix D (v) calculate the Laplacian matrix of the similarity matrix Z (v) :

[0034]

[0035] where represents the Laplacian matrix of the v-th view.

[0036] Preferably, in the step S5, the specific method for obtaining the clustering indicator matrix of each view based on the Laplacian matrix is as follows:

[0037] Set the index matrix G according to the data matrix (v) , where:

[0038]

[0039] Use spectral clustering technology to calculate the clustering indicator matrix of each view:

[0040]

[0041] In the formula, F (v) represents the clustering indicator matrix of the v-th view; represents the transpose matrix of the index matrix H (v) of, represents F (v) of the transpose matrix, and Tr(*) represents calculating the trace of the matrix.

[0042] Since the missing data of each view is different, the dimensions of the Laplacian matrices obtained from the similarity matrices will also be different, and the inconsistency of dimensions cannot be used for fusion learning; to overcome this problem, an index matrix is introduced, and the elements at the positions of the missing data are set to 0, reducing the negative impact of the uncertain similarity information corresponding to the incomplete multi-view on the subsequent clustering. The local clustering process normalizes all feature vectors to unit length and forms the clustering indicator matrix F (v) by stacking the feature vectors in columns. Each row of the clustering indicator matrix is a new low-dimensional representation, containing more discriminative local clustering information.

[0043] Preferably, in the step S5, the specific method for aligning the clustering indicator matrices of each view to obtain a consistent clustering indicator matrix is as follows:

[0044]

[0045] In the formula, S represents the consistent clustering indicator matrix; K S represents the linear kernel matrix of S, K S = SS T ; represents the linear kernel matrix of F (v) of,

[0046] Based on rewrite the above formula as:

[0047]

[0048] In the formula, Γ(*) represents the alignment operation, and c represents the number of clusters.

[0049] To utilize the potential common information, co-regularized spectral clustering is used to integrate the consistent information and complementary information between different views. The alignment method effectively fuses the complementary information and consistent information of the multi-view and obtains a consistent clustering indicator matrix S containing the global information of all views.

[0050] Preferably, in the step S6, the objective function of the incomplete multi-view clustering constructed based on the consistent clustering indicator matrix is expressed as:

[0051]

[0052]

[0053] Wherein, λ 1 represents the first penalty parameter, and λ 2 represents the second penalty parameter, and I represents the identity matrix.

[0054] Preferably, in the step S6, the specific method for optimizing and solving the objective function by using the alternating method to obtain the optimal consistent clustering indicator matrix is as follows:

[0055] S6.1: Based on the objective function of incomplete multi-view clustering, introduce an auxiliary variable matrix A (v) , and transform the objective function into an augmented Lagrangian equation:

[0056]

[0057] Wherein, represents the Laplacian matrix of the auxiliary variable matrix A (v) , represents the first Lagrange multiplier, represents the second Lagrange multiplier, and μ represents the third penalty parameter; in the objective function, c is a constant and is ignored during optimization;

[0058] S6.2: Use the alternating direction method of multipliers ADMM to iteratively optimize each term in the augmented Lagrangian equation. For the six variable matrices U (v) , V (v) , Z (v) , F (v) , A (v) , S, fix the other five variable matrices, and optimize U (v) , V (v) , Z (v) , A (v) , F (v) , S in turn to obtain the objective function value of the current round; specifically:

[0059] (1) Fix other variables and optimize U (v) :

[0060]

[0061] Since the views are independent of each other, the basis matrix U (v) of each view can be solved separately. Therefore, the minimization problem of U (v) is rewritten as:

[0062]

[0063] For U(v) Taking the partial derivative and setting it equal to 0, we can obtain:

[0064]

[0065] Furthermore, the optimal closed-form solution is obtained:

[0066]

[0067] (2) Fixing other variables, minimize V (v) The optimization problem is simplified to:

[0068]

[0069] s.t. V (v) ≥ 0

[0070] Update V according to the update strategy in semi-nonnegative matrix factorization (v) , since the semi-nonnegative matrix factorization and the subspace learning process are independent of each other, the partial derivative with respect to V (v) is written as:

[0071]

[0072] According to the optimization strategy of semi-NMF and KKT conditions, we can obtain:

[0073]

[0074] where, represents the element in the i-th row and c-th column of matrix V.

[0075] P (v) = I - Z (v) - Z (v)T + Z (v) Z (v)T , note that there is U (v)T U (v) = (U (v)T U (v) ) + - (U (v)T U (v) ) - ,

[0076] Therefore, we can obtain the following update strategy for V (v) :

[0077]

[0078] where, using and to represent the positive and negative elements of matrix M, they have the following properties:

[0079] (3) Fix other variables and optimize Z (v) :

[0080]

[0081] Take its partial derivative with respect to Z (v) and set it equal to 0 to obtain

[0082]

[0083] where Let Q (v) = V (v)T V (v) , we get

[0084]

[0085] (4) Fix other variables and optimize A (v) :

[0086]

[0087] Define P (v) = G (v) F (v) , The above equation is equivalent to the following optimization problem:

[0088]

[0089] where and represent the i-th and j-th row vectors of matrix P (v) . Obviously, each row in the above matrix is independent. Let Then the above problem can be simplified to:

[0090]

[0091] where is 's i-th element. The optimal solution to the above problem is as follows:

[0092]

[0093] By setting A (v) = max(A (v) , 0) to force all elements of matrix A (v) to be not less than 0. If there are elements less than 0, force them to be 0 and keep other elements. According to the constraint The Lagrange multiplier is updated as follows:

[0094]

[0095] (5) Fix other variables and optimize F (v) :

[0096]

[0097] The above problem can be solved by eigenvalue decomposition. Among them, the c-th eigenvector corresponds to the c-th largest eigenvalue of the matrix and is also selected as the optimal solution of the variable F (v) ;

[0098] (6) Fix other variables and optimize S:

[0099]

[0100] The above can also be obtained by simple eigenvalue decomposition. The optimal solution of the variable S is the set of eigenvectors corresponding to the c-th largest eigenvalue of the matrix ;

[0101] S6.3: Update the first Lagrange multiplier and the third penalty parameter μ;

[0102]

[0103] μ = min(ρμ, μ 0 )

[0104] where ρ and μ 0 are both constants;

[0105] S6.4: Determine whether the objective function value of the current round satisfies the preset iteration optimization termination condition; if it is satisfied, stop the iteration and proceed to step S6.5; otherwise, return to step S6.2; the iteration optimization termination condition is:

[0106] (obj (t-1) - obj (t) ) / obj (t) ≤ ε 0

[0107] where obj (t) , obj (t-1) represent the objective function values of the t-th and (t - 1)-th rounds of iteration respectively, and ε 0 is the preset precision;

[0108] S6.5: Compare the magnitudes of the objective function values of all rounds before the iteration optimization termination, and use the consensus clustering indicator matrix corresponding to the smallest objective function value as the optimal consensus clustering indicator matrix.

[0109] Preferably, in step S7, the optimal consensus clustering indicator matrix is input into an existing k-means algorithm to obtain the clustering result of the incomplete multi-view.

[0110] The present invention also provides an incomplete multi-view clustering system based on co-regularized spectral clustering for implementing the above-mentioned incomplete multi-view clustering method based on co-regularized spectral clustering. The system includes:

[0111] A data acquisition and transformation module for acquiring an incomplete multi-view data set and transforming each view in the data set into a data matrix;

[0112] A coefficient matrix calculation module for performing dimensionality reduction processing on the data matrix of each view to obtain the coefficient matrix of each view;

[0113] A similarity matrix calculation module for constructing the similarity matrix of each view based on the self-representation characteristics of the coefficient matrix of each view;

[0114] A clustering indicator matrix calculation module for calculating the Laplacian matrix of each view according to the similarity matrix of each view and obtaining the clustering indicator matrix of each view based on the Laplacian matrix;

[0115] A consensus clustering indicator matrix calculation module for performing co-alignment on the clustering indicator matrix of each view to obtain a consensus clustering indicator matrix;

[0116] A target function construction and solution module for constructing the target function of incomplete multi-view clustering based on the consensus clustering indicator matrix and optimizing and solving the target function by an alternating method to obtain an optimal consensus clustering indicator matrix;

[0117] A clustering module for inputting the optimal consensus clustering indicator matrix into an existing classification algorithm to obtain the clustering result of the incomplete multi-view.

[0118] Compared with the prior art, the beneficial effects of the technical solution of the present invention are:

[0119] The present invention first obtains an incomplete multi-view data set and converts each view into a data matrix; then performs dimensionality reduction processing on the data matrix to obtain a coefficient matrix for each view, and uses the self-representation characteristics of the coefficient matrix to construct a similarity matrix for each view, which can reflect the similarity structure of multiple views; then calculates the Laplacian matrix of each view, and obtains a clustering indicator matrix for each view based on the Laplacian matrix, which contains more discriminative local clustering information in the view; performs kernel alignment on the clustering indicator matrix of each view, and uses co-regularized spectral clustering to integrate consistent information and complementary information between different views. The kernel alignment method effectively integrates the complementary information and consistency information of multiple views, and obtains a consistent clustering indicator matrix containing global information of all views; finally, based on the objective function of the incomplete multi-view clustering constructed by the consistent clustering indicator matrix, the alternating method is used to optimize and solve the objective function, and the optimal consistent clustering indicator matrix is ​​obtained, which is used as the input of the classification algorithm to obtain an incomplete multi-view clustering method. The present invention can deeply mine the complementarity and consistency information of view data when the view data is incomplete, and improves the clustering effect of incomplete multi-views. BRIEF DESCRIPTION OF THE DRAWINGS

[0120] Figure 1 This is a flow chart of an incomplete multi-view clustering method based on co-regularized spectral clustering described in Example 1;

[0121] Figure 2 A curve diagram showing the relationship between the objective function value and the number of iterations when 30% of the data for each view is missing as described in Example 2;

[0122] Figure 3 This is a schematic structural diagram of an incomplete multi-view clustering system based on co-regularized spectral clustering described in Example 3. DETAILED DESCRIPTION

[0123] The drawings are for illustrative purposes only and should not be construed as limiting the present patent;

[0124] In order to better illustrate the present embodiment, some parts in the drawings may be omitted, enlarged or reduced, and do not represent the size of the actual product;

[0125] It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.

[0126] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0127] Example 1

[0128] This embodiment provides an incomplete multi-view clustering method based on co-regularized spectral clustering, such as Figure 1 As shown, including:

[0129] S1: Obtain an incomplete multi-view dataset, and convert each view in the dataset into a data matrix;

[0130] S2: Perform dimensionality reduction on the data matrix of each view to obtain the coefficient matrix of each view;

[0131] S3: Based on the self-representation characteristics of the coefficient matrix of each view, construct the similarity matrix of each view;

[0132] S4: Calculate the Laplacian matrix of each view according to the similarity matrix of each view, and obtain the clustering indicator matrix of each view based on the Laplacian matrix;

[0133] S5: Perform co-alignment on the clustering indicator matrix of each view to obtain a consistent clustering indicator matrix;

[0134] S6: Construct an objective function for incomplete multi-view clustering based on the consistent clustering indicator matrix, and use an alternating method to optimize and solve the objective function to obtain an optimal consistent clustering indicator matrix;

[0135] S7: Input the optimal consistent clustering indicator matrix into an existing classification algorithm to obtain the clustering result of the incomplete multi-view.

[0136] In the specific implementation process, in this embodiment, first, an incomplete multi-view dataset is obtained, and each view is converted into a data matrix; then, dimensionality reduction is performed on the data matrix to obtain the coefficient matrix of each view, and the self-representation characteristics of the coefficient matrix are used to construct the similarity matrix of each view, and the similarity matrix can reflect the similarity structure of the multi-view; then, the Laplacian matrix of each view is calculated, and the clustering indicator matrix of each view is obtained based on the Laplacian matrix, and the clustering indicator matrix contains more discriminative local clustering information in the view; co-alignment is performed on the clustering indicator matrix of each view, and co-regularized spectral clustering is used to integrate the consistent information and complementary information between different views, and the co-alignment method effectively fuses the complementary information and consistent information of the multi-view to obtain a consistent clustering indicator matrix containing the global information of all views; finally, an objective function for incomplete multi-view clustering constructed based on the consistent clustering indicator matrix is used, and an alternating method is used to optimize and solve the objective function to obtain an optimal consistent clustering indicator matrix, which is used as the input of the classification algorithm to obtain an incomplete multi-view clustering method. The present invention can deeply mine the complementary and consistent information of view data in the case of incomplete view data, and improve the clustering effect of incomplete multi-views.

[0137] Embodiment 2

[0138] This embodiment provides an incomplete multi-view clustering method based on co-regularized spectral clustering, including:

[0139] S1: Obtain an incomplete multi-view dataset, and convert each view in the dataset into a data matrix;

[0140] The data matrix of each view contains the data that is not missing in that view. Denote the data matrix of the v-th view as:

[0141]

[0142] Where, represents the n-th non-missing data element in the v-th view, d v represents the feature dimension of the non-missing data in the v-th view, n v represents the number of non-missing data in the v-th view; V represents the number of views in the incomplete multi-view dataset, v = 1, 2, …, V; v

[0143] S2: Perform dimensionality reduction on the data matrix of each view to obtain the coefficient matrix of each view;

[0144] Perform semi-nonnegative matrix factorization on the data matrix of each view to achieve dimensionality reduction, and obtain the coefficient matrix of each view. The expression is:

[0145]

[0146] s.t. V (v) ≥ 0

[0147] U (v) represents the basis matrix of the v-th view, V (v) represents the coefficient matrix of the v-th view, c represents the number of clusters; represents calculating the Frobenius norm;

[0148] S3: Based on the self-representation property of the coefficient matrix of each view, construct the similarity matrix of each view;

[0149] In order to utilize the intra-view similarity structure and the latent representation subspace structure, use the self-representation property of the coefficient matrix to construct a graph in the latent representation subspace to obtain the self-representation matrix of each view, that is, the similarity matrix. The similarity matrix can reflect the similarity structure of the multi-view, and each element in it reflects the similarity weight. The expression is:

[0150]

[0151]

[0152] In the formula, Z (v) represents the similarity matrix of the v-th view, Denote Z (v) The element in the i-th row and j-th column of Denote calculating the F norm.

[0153] S4: Calculate the Laplacian matrix of each view based on the similarity matrix of each view, and obtain the clustering indicator matrix of each view based on the Laplacian matrix;

[0154] The specific method for calculating the Laplacian matrix of each view is as follows:

[0155] Based on the similarity matrix Z of each view (v) Calculate the corresponding adjacency matrix W (v) :

[0156]

[0157] In the formula, |*| represents the modulus operation, |Z (v) | T Denote |Z (v) |'s transpose;

[0158] Based on the adjacency matrix W (v) Calculate the corresponding weighted degree matrix D (v) :

[0159]

[0160] In the formula, Denote the element in the i-th row and i-th column of the weighted degree matrix D (v) ; Denote the element in the i-th row and j-th column of the adjacency matrix W (v) Representing the weight of the element in the i-th row and j-th column; if there is no edge connection between two vertices, the corresponding element in the adjacency matrix is 0; calculate each diagonal element in the same way and fill it to obtain the weighted degree matrix D (v) ;

[0161] Based on the adjacency matrix W (v) and the weighted degree matrix D (v) Calculate the Laplacian matrix of the similarity matrix Z (v)

[0162]

[0163] In the formula, Denote the Laplacian matrix of the v-th view;

[0164] Set the index matrix G according to the data matrix (v) , where:

[0165] ​

[0166] Calculate the clustering indicator matrix for each view using spectral clustering technology:

[0167]

[0168] In the formula, F (v) represents the clustering indicator matrix of the v-th view; represents the transpose matrix of the index matrix G (v) , represents the transpose matrix of F )v) , and Tr(*) represents taking the trace of the matrix.

[0169] Since the missing data for each view is different, the dimensions of the Laplacian matrix obtained from the similarity matrix will also be different, and the inconsistency in dimensions cannot be used for fusion learning; to overcome this problem, an index matrix is introduced, and the elements at the missing data positions are set to 0 to reduce the negative impact of the uncertain similarity information corresponding to the incomplete multi-views on subsequent clustering. The local clustering process normalizes all feature vectors to unit length and forms the clustering indicator matrix F (v) by stacking the feature vectors in columns. Each row of the clustering indicator matrix is a new low-dimensional representation, containing more discriminative local clustering information.

[0170] S5: Check and align the clustering indicator matrices of each view to obtain a consistent clustering indicator matrix;

[0171] To utilize the potential common information, co-regularized spectral clustering is used to integrate the consistent information and complementary information between different views. The check and alignment method effectively fuses the complementary information and consistent information of multi-views to obtain a consistent clustering indicator matrix S that contains the global information of all views:

[0172]

[0173] In the formula, S represents the consistent clustering indicator matrix; K S represents the linear kernel matrix of S, K S = SS T ; represents the linear kernel matrix of F (v) ,

[0174] Based on rewrite the above formula as:

[0175]

[0176] In the formula, Γ(*) represents the check and alignment operation, and c represents the number of clusters.

[0177] S6: Construct the objective function for incomplete multi-view clustering based on the consensus clustering indicator matrix, and use the alternating method to optimize and solve the objective function to obtain the optimal consensus clustering indicator matrix;

[0178] The objective function is expressed as:

[0179]

[0180]

[0181] In the formula, λ 1 represents the first penalty parameter, λ 2 represents the second penalty parameter, I represents the identity matrix;

[0182] The specific method for using the alternating method to optimize and solve the objective function to obtain the optimal consensus clustering indicator matrix is as follows:

[0183] Based on the objective function of incomplete multi-view clustering, introduce the auxiliary variable matrix A (v) , and transform the objective function into an augmented Lagrangian equation:

[0184]

[0185] In the formula, represents the Laplacian matrix of the auxiliary variable matrix A (v) , represents the first Lagrange multiplier, represents the second Lagrange multiplier, μ represents the third penalty parameter; in the objective function, c is a constant and is ignored during optimization;

[0186] Use the alternating direction method of multipliers ADMM to iteratively optimize each term in the augmented Lagrangian equation. For the six variable matrices U (v) , V (v) , Z (v) , F (v) , A (v) , S, fix the other five variable matrices, and optimize U (v) , V (v) , Z (v) , A (v) , F (v) , S in turn to obtain the objective function value of the current round; specifically:

[0187] (1) Fix other variables and optimize U (v) :

[0188]

[0189] Since the views are independent of each other, the basis matrix U of each view (v)can be solved separately, so the minimization problem of U (v) is rewritten as:

[0190]

[0191] Taking the partial derivative of U (v) and setting it equal to 0, we get:

[0192]

[0193] Furthermore, the optimal closed-form solution is obtained:

[0194]

[0195] (2) Fixing other variables, minimize V (v) The optimization problem is simplified to:

[0196]

[0197] s.t. V (v) ≥0

[0198] Update V according to the update strategy in semi-nonnegative matrix factorization (v) , because the semi-nonnegative matrix factorization and the subspace learning process are independent of each other, so the partial derivative with respect to V (v) is written as:

[0199]

[0200] According to the optimization strategy of semi-NMF and KKT conditions, we can get:

[0201]

[0202] where represents the element in the i-th row and c-th column of matrix V.

[0203] P (v) = I - Z (v) - Z (v)T + Z (v) Z (v)T , note that there is U (v)T U (v) =(U (v)T U (v) ) + -(U (v)T U (v) ) - ,

[0204] Therefore, we can obtain the following update strategy for V (v) :

[0205]

[0206] Among them, use and to represent the positive and negative elements of matrix M, and they have the following properties:

[0207] (3) Fix other variables and optimize Z (v) :

[0208]

[0209] Take its partial derivative with respect to Z (v) and set it equal to 0 to obtain

[0210]

[0211] where Let Q (v) = V (v)T V (v) , we get

[0212]

[0213] (4) Fix other variables and optimize A (v) :

[0214]

[0215] Define P (v) = G (v) F (v) , The above equation is equivalent to the following optimization problem:

[0216]

[0217] where and represent the i-th and j-th row vectors of matrix P (v) . Obviously, each row in the above matrix is independent. Let Then the above problem can be simplified to:

[0218]

[0219] where is 's i-th element. The optimal solution to the above problem is as follows:

[0220]

[0221] By setting A (v) = max(A (v) , 0) to force matrix A (v)All elements are not less than 0. If there are elements less than 0, they are forced to be 0 and other elements are retained. According to the constraint Lagrange multiplier is updated as follows:

[0222]

[0223] (5) Fix other variables and optimize F (v) :

[0224]

[0225] The above problem can be solved by eigenvalue decomposition. Among them, the c-th eigenvector corresponds to the c-th largest eigenvalue of the matrix and is also selected as the optimal solution of the variable F (v) ;

[0226] (6) Fix other variables and optimize S:

[0227]

[0228] The above can also be obtained by simple eigenvalue decomposition. The optimal solution of the variable S is the set of eigenvectors corresponding to the c-th largest eigenvalue of the matrix ;

[0229] Update the first Lagrange multiplier and the third penalty parameter μ;

[0230]

[0231] μ = min(ρμ, μ 0 )

[0232] where ρ and μ 0 are both constants;

[0233] Judge whether the objective function value of the current round meets the preset iteration optimization termination condition; if not, perform the next round of iteration; if so, stop the iteration, compare the sizes of the objective function values of all rounds before the iteration optimization termination, and use the consensus clustering indicator matrix corresponding to the smallest objective function value as the optimal consensus clustering indicator matrix;

[0234] The iteration optimization termination condition is:

[0235] (obj (t-1) -obj (t) ) / obj (t) ≤ ε 0

[0236] where obj (t) and obj(t-1) represent the objective function values of the t-th and (t - 1)-th iterations respectively, and ε 0 is a preset precision.

[0237] In this embodiment, the parameters are set as λ 1 = 1, λ 2 = 0.01, μ 0 = 0.01, ρ = 1.1.

[0238] After experiments, as Figure 2 described, when 30% of the data in each view is missing, it is a curve graph of the objective function value versus the number of iterations. The vertical coordinate in the graph represents the objective function value, and the horizontal coordinate is the number of iterations. It can be seen that the method provided in this embodiment converges at the 20th iteration.

[0239] S7: Input the optimal consensus clustering indicator matrix into the existing k-means algorithm to obtain the clustering results of the incomplete multi-view.

[0240] In the specific implementation process, the method proposed in this embodiment is experimented on the Caltech7 dataset. The Caltech7 dataset is a subset of the Caltech101 dataset, consisting of 1474 samples, including seven categories of "faces", "motorbikes", "dollar bill", "garfield", "snoopy", "stop sign", and "windsor chair" and six views of "Gabor", "wavelet moments", "Cenhist", "Hog", "Gist", and "LBP"; on the Caltech7 dataset, a certain proportion of the data on each view is randomly lost, and the loss proportions are selected as 10%, 30%, and 50%, and are compared with several common incomplete multi-view clustering methods, including the BSV algorithm, Concat algorithm, MIC algorithm, OMVC algorithm, DAIMC algorithm, UEAF algorithm, IMSC_AGL algorithm, GIMC_FLSD algorithm; these algorithms together with the method (Ours) proposed in this embodiment are randomly lost 5 times on the Caltech7 dataset, and the mean and standard deviation of running 5 times are given. The comparison results include the clustering accuracy (ACC), normalized mutual information (NMI), and clustering purity (Purity); the experimental results are shown in the following table:

[0241]

[0242]

[0243] As can be seen from the table, on the Caltech7 dataset, when the missing rates are 10%, 30%, and 50%, both the normalized mutual information (NMI) and clustering purity (Purity) of the method proposed in this embodiment are the highest; the clustering accuracy (ACC) is the highest when the missing rates are 30% and 50%, and is the second highest when the missing rate is 10%. The experimental results reflect that the clustering effect of the method proposed in this embodiment is significantly higher than that of the existing algorithms, greatly improving the clustering effect of incomplete multi-view data.

[0244] Embodiment 3

[0245] This embodiment provides an incomplete multi-view clustering system based on co-regularized spectral clustering for implementing the incomplete multi-view clustering method based on co-regularized spectral clustering described in Embodiment 1 or 2, as Figure 3 shown, the system includes:

[0246] A data acquisition and conversion module, configured to acquire an incomplete multi-view dataset and convert each view in the dataset into a data matrix;

[0247] A coefficient matrix calculation module, configured to perform dimensionality reduction processing on the data matrix of each view to obtain a coefficient matrix for each view;

[0248] A similarity matrix calculation module, configured to construct a similarity matrix for each view based on the self-representation characteristics of the coefficient matrix of each view;

[0249] A clustering indicator matrix calculation module, configured to calculate the Laplacian matrix for each view according to the similarity matrix of each view, and obtain a clustering indicator matrix for each view based on the Laplacian matrix;

[0250] A consistent clustering indicator matrix calculation module, configured to perform co-alignment on the clustering indicator matrix of each view to obtain a consistent clustering indicator matrix;

[0251] A target function construction and solution module, configured to construct a target function for incomplete multi-view clustering based on the consistent clustering indicator matrix, and optimize and solve the target function using an alternating method to obtain an optimal consistent clustering indicator matrix;

[0252] A clustering module, configured to input the optimal consistent clustering indicator matrix into an existing classification algorithm to obtain a clustering result of the incomplete multi-view data.

[0253] The same or similar reference numerals correspond to the same or similar components;

[0254] The terms describing the positional relationship in the drawings are only for illustrative purposes and should not be construed as a limitation of this patent;

[0255] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, rather than limitations on the implementation manners of the present invention. For those of ordinary skill in the art, other different forms of changes or variations can be made based on the above description. It is not necessary and impossible to enumerate all implementation manners here. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the claims of the present invention.

Claims

1. An incomplete multi-view clustering method based on co-regularized spectral clustering, characterized in that, it includes: S1: Obtain an incomplete multi-view data set, and convert each view in the data set into a data matrix; S2: Perform dimensionality reduction processing on the data matrix of each view to obtain the coefficient matrix of each view; S3: Based on the self-representation characteristics of the coefficient matrix of each view, construct the similarity matrix of each view; S4: Calculate the Laplacian matrix of each view according to the similarity matrix of each view, and obtain the clustering indicator matrix of each view based on the Laplacian matrix; S5: Perform kernel alignment on the clustering indicator matrix of each view to obtain a consistent clustering indicator matrix; S6: Construct an objective function for incomplete multi-view clustering based on the consistent clustering indicator matrix, and use an alternating method to optimize and solve the objective function to obtain an optimal consistent clustering indicator matrix. The specific method is: S6.1: Based on the objective function of incomplete multi-view clustering, introduce the auxiliary variable matrix A (v) , and transform the objective function into an augmented Lagrangian equation: In the formula, represents the Laplacian matrix of the auxiliary variable matrix A (v) , represents the first Lagrange multiplier, represents the second Lagrange multiplier, and μ represents the third penalty parameter; U (v) represents the basis matrix of the v-th view, V (v) represents the coefficient matrix of the v-th view, Z (v) represents the similarity matrix of the v-th view, F (v) represents the clustering indicator matrix of the v-th view, Y (v) represents the data matrix of the v-th view, G (v) represents the index matrix of the v-th view, S represents the consistent clustering indicator matrix, and λ 1 represents the first penalty parameter, λ 2 represents the second penalty parameter, Tr(*) represents taking the trace of a matrix, represents taking the F-norm; S6.2: Use the Alternating Direction Method of Multipliers (ADMM) to iteratively optimize each term in the augmented Lagrangian equation. For the six variable matrices U (v) , V (v) , Z (v) , F (v) , A (v) , S, fix the other five variable matrices, and optimize U (v) , V (v) , Z (v) , A (v) , F (v) , S in turn to obtain the objective function value for the current iteration; S6.3: Update the first Lagrange multiplier and the third penalty parameter μ; S6.4: Determine whether the objective function value in the current round satisfies the preset iterative optimization termination condition; if it is satisfied, stop the iteration and proceed to step S6.5; otherwise, return to step S6.2; S6.5: Compare the magnitudes of the objective function values in all rounds before the iterative optimization termination, and use the consistent clustering indicator matrix corresponding to the smallest objective function value as the optimal consistent clustering indicator matrix; S7: Input the optimal consistent clustering indicator matrix into an existing classification algorithm to obtain the clustering result of the incomplete multi-view.

2. The incomplete multi-view clustering method based on co-regularized spectral clustering according to claim 1, characterized in that, in the step S1, the data matrix of each view contains the data that is not missing in that view. Denote the data matrix of the v-th view as: Among them, represents the nth v non-missing data element in the vth view, and d v represents the feature dimension of the non-missing data in the vth view, and n v represents the number of non-missing data in the vth view; V represents the number of views in the incomplete multi-view dataset, and v = 1, 2, …, V.

3. The incomplete multi-view clustering method based on co-regularized spectral clustering according to claim 2, characterized in that, in the step S2, perform semi-nonnegative matrix factorization on the data matrix of each view to achieve dimensionality reduction operation, and obtain the coefficient matrix of each view. The expression is: s.t.V (v) ≥0 where U (v) represents the basis matrix of the v-th view, V (v) represents the coefficient matrix of the v-th view, c represents the number of clusters; denotes the calculation of the F-norm.

4. The incomplete multi-view clustering method based on co-regularized spectral clustering according to claim 3, characterized in that, in the step S3, the expression for constructing the similarity matrix of each view is: where, Z (v) represents the similarity matrix of the v-th view, represents Z (v) the element at the i-th row and j-th column in denotes taking the F-norm.

5. The incomplete multi-view clustering method based on co-regularized spectral clustering according to claim 4, characterized in that, in the step S4, the specific method for calculating the Laplacian matrix of each view according to the similarity matrix of each view is: According to the similarity matrix Z of each view (v) calculate the corresponding adjacency matrix W (v) : wherein, |*| represents the modulo operation, and |Z(v)| T represents the transpose of |Z (v) |; According to the adjacency matrix W (v) calculate the corresponding weighted degree matrix D (v) : In the formula, represents the weighted degree matrix D (v) the element in the i-th row and i-th column; represents the adjacency matrix W (v) the element in the i-th row and j-th column, representing the weight of the element in the i-th row and j-th column; According to the adjacency matrix W (v) and the weighted degree matrix D (v) calculate the similarity matrix Z (v) of the Laplacian matrix In the formula, represents the Laplacian matrix of the v-th view.

6. The incomplete multi-view clustering method based on co-regularized spectral clustering according to claim 5, characterized in that, in the step S5, the specific method for obtaining the clustering indicator matrix of each view based on the Laplacian matrix is: Set the index matrix G according to the data matrix (v) , where: Use spectral clustering technology to calculate the clustering indicator matrix of each view: where, F (v) represents the clustering indication matrix of the v-th view; represents the index matrix G (v) of the transposed matrix, represents F (v) of the transposed matrix, Tr(*) represents the trace of the matrix.

7. The incomplete multi-view clustering method based on co-regularized spectral clustering according to claim 6, characterized in that, in the step S5, the specific method for performing kernel alignment on the clustering indicator matrix of each view to obtain a consistent clustering indicator matrix is: Wherein, S represents a consensus clustering indicator matrix; K S represents the linear kernel matrix of S, K S = SS T ; represents F (v) 's linear kernel matrix, Based on Rewrite the above equation as: where Γ(*) represents the kernel alignment operation, and c represents the number of clusters.

8. The incomplete multi-view clustering method based on co-regularized spectral clustering according to claim 7, It is characterized in that In step S6, the objective function of the incomplete multi-view clustering constructed based on the consensus clustering indicator matrix is expressed as: where λ 1 represents the first penalty parameter, λ 2 represents the second penalty parameter, and I represents the identity matrix.

9. An incomplete multi-view clustering system based on co-regularized spectral clustering, which is used to implement the incomplete multi-view clustering method based on co-regularized spectral clustering according to any one of claims 1-8 It is characterized in that The system includes: A data acquisition and transformation module, which is used to acquire an incomplete multi-view data set and transform each view in the data set into a data matrix; A coefficient matrix calculation module, which is used to perform dimensionality reduction processing on the data matrix of each view to obtain the coefficient matrix of each view; A similarity matrix calculation module, which is used to construct the similarity matrix of each view based on the self-representation characteristics of the coefficient matrix of each view; A clustering indicator matrix calculation module, which is used to calculate the Laplacian matrix of each view according to the similarity matrix of each view, and obtain the clustering indicator matrix of each view based on the Laplacian matrix; A consensus clustering indicator matrix calculation module, which is used to perform check alignment on the clustering indicator matrix of each view to obtain a consensus clustering indicator matrix: An objective function construction and solution module, which is used to construct the objective function of the incomplete multi-view clustering based on the consensus clustering indicator matrix, and use the alternating method to optimize and solve the objective function to obtain the optimal consensus clustering indicator matrix, including: Based on the objective function of incomplete multi-view clustering, introduce the auxiliary variable matrix A (v) , and transform the objective function into an augmented Lagrangian equation: In the formula, represents the Laplacian matrix of the auxiliary variable matrix A (v) , represents the first Lagrange multiplier, represents the second Lagrange multiplier, and μ represents the third penalty parameter; U (v) represents the basis matrix of the v-th view, V (v) represents the coefficient matrix of the v-th view, Z (v) represents the similarity matrix of the v-th view, F (v) represents the clustering indicator matrix of the v-th view, Y (v) represents the data matrix of the v-th view, G (v) represents the index matrix of the v-th view, S represents the consistent clustering indicator matrix, and λ 1 represents the first penalty parameter, λ 2 represents the second penalty parameter, Tr(*) represents the trace of a matrix, represents the calculation of the F-norm; The alternating direction method of multipliers (ADMM) is used to iteratively optimize each term in the augmented Lagrangian equation. For the six variable matrices U (v) , V (v) , Z (v) , F (v) , A (v) , S, fix the other five variable matrices, and optimize U (v) , V (v) , Z (v) , A (v) , F (v) , S in turn to obtain the objective function value for the current round; Update the first Lagrange multiplier and the third penalty parameter μ; Judging whether the objective function value of the current round meets the preset iteration optimization termination condition; if it meets, stop the iteration, compare the sizes of the objective function values of all rounds before the iteration optimization termination, and use the consensus clustering indicator matrix corresponding to the smallest objective function value as the optimal consensus clustering indicator matrix; Otherwise, return to the step of iteratively optimizing each item in the augmented Lagrangian equation by using the alternating direction multiplier method ADMM until the objective function value of the current round meets the preset iteration optimization termination condition; A clustering module, which is used to input the optimal consensus clustering indicator matrix into an existing classification algorithm to obtain the clustering result of the incomplete multi-view.

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