A Fast Image Clustering Method Based on Bipartite Graph Embedding and Discriminant Information

Abstract

The present invention relates to a fast image clustering method based on bipartite graph embedding and discriminant information. Aiming at the problems of high spatio-temporal complexity required for eigenvalue decomposition and similarity matrix construction in spectral clustering algorithms, and the independence between low-dimensional feature space embedding and label learning and the lack of discriminant information, the present invention proposes a fast image clustering method based on bipartite graph embedding and discriminant information. A bipartite graph is constructed in combination with anchor points, and the coordinate descent optimization method is used to replace eigenvalue decomposition and post-processing to reduce the difficulty of data storage and improve the data processing speed. In addition, the present invention simultaneously performs subspace learning and label learning, which can preserve the local features of data and make full use of discriminant information. Therefore, fast and efficient clustering analysis of large-scale and high-dimensional image data can be realized.

Description

Technical Field

[0001] The present invention belongs to the field of image analysis and data mining, and relates to a fast image clustering method based on bipartite graph embedding and discriminant information. Background Art

[0002] Clustering analysis is an important research topic in the field of data mining. Its task is to mine the label information of data according to the spatial distribution characteristics of the data, so that the similarity between data with the same label is as large as possible, while the similarity between data with different labels is as small as possible. As one of the classical clustering methods, the spectral clustering method embeds the original data into a low-dimensional feature space by constructing a similarity matrix and eigenvalue decomposition and completes the clustering task in this subspace. However, the development of new media technology has led to a geometric increase in the number of samples and the number of features of image data. The spectral clustering method cannot be applied to these larger-scale and higher-dimensional image data.

[0003] Li Kang, Xu Jindong, Zhao Tianyu, etc. ("A Fuzzy Spectral Clustering Algorithm for Hyperspectral Image Classification"), Chinese Science Bulletin, 2021, 16(7): 743-747.) uses fuzzy similarity measure to replace the traditional Gaussian kernel function method to construct a similarity matrix and introduces an anchor graph structure to reduce the computational complexity of constructing the similarity matrix. Liu Zhongmin, Li Zhanming, Li Bohao, etc. ("Spectral Clustering Image Segmentation Algorithm Based on Sparse Matrix". Journal of Jilin University (Engineering and Technology Edition), 2017, 47(4): 1308-1313.) effectively reduces the computational complexity of spectral clustering by constructing a sparse similarity matrix. Although these spectral clustering optimization algorithms can alleviate the pressure on data storage and processing devices to a certain extent, the acquisition of data labels requires K-means clustering as post-processing. Therefore, in the algorithm implementation process, the embedding of the low-dimensional feature space and label learning are independent of each other and do not fully utilize the discriminant information of the data. The expression of data in the low-dimensional feature space is not always conducive to label learning, and the clustering accuracy still needs to be improved. Summary of the Invention

[0004] Technical Problems to be Solved

[0005] In order to avoid the deficiencies of the prior art, the present invention proposes a fast image clustering method based on bipartite graph embedding and discriminant information.

[0006] Technical Solution

[0007] A fast image clustering method based on bipartite graph embedding and discriminant information, characterized by the following steps:

[0008] Step 1, construction of an anchor-based sparse bipartite graph and construction of an intrinsic graph: For a sample matrix with n samples and a feature dimension of d Among them, the feature dimension corresponds to the image resolution, and the row vector corresponds to the information of a single image;

[0009] The constructed sparse bipartite graph:

[0010] where γ is an adaptive parameter, 1 m is a vector of all 1s with dimension m×1;

[0011] The similarity between the i-th sample and the j-th anchor point:

[0012] where and the row vector e i The elements are arranged in ascending order;

[0013] The intrinsic graph:

[0014] where, 1 n+m is a vector of all 1s with dimension (n+m)×1;

[0015] Step 2: Integrate the constructed bipartite graph B and the intrinsic graph H n+m into the framework of spectral learning, and set up the objective function:

[0016]

[0017] The corresponding constraint conditions are: F1 c =1 n , F≥0; G1 c =1 m , G≥0;

[0018] Among them, the optimization variables are the sample point label matrix and the anchor point label matrix 1 c is a vector of all 1s with dimension c×1, 1 n is a vector of all 1s with dimension n×1, the Laplacian matrix L S =D - S, the similarity matrix and B is the bipartite graph matrix obtained in Step 1, the matrix is the degree matrix and The label matrix can be regarded as the feature distribution of data points in the original space in the subspace. λ>0 is a coefficient that needs to be manually assigned;

[0019] Step 3: Alternately iterate and optimize F and G by the coordinate descent method until the objective function gradually converges;

[0020] Step 4: Directly obtain the final sample labels according to the sample point label matrix F obtained in Step 3. When f ik is the sample point label vector fi When it reaches the maximum value among them, the \(i\)-th sample point is assigned to the \(k\)-th data cluster.

[0021] The steps of alternately iteratively optimizing \(F\) and \(G\) are as follows:

[0022] Step (1): Fix the anchor point label matrix \(G\) and update the sample point label matrix \(F\): \(f\) i They are independent of each other. The optimization problem of the label matrix \(F\) is transformed into a step-by-step optimization problem of the label vector \(f\) i , \(i = 1, 2, \cdots, n\). The problem to be optimized is in the following form:

[0023]

[0024] Substitute into the above formula to obtain the final optimization function: When \(\eta>0\), this optimization problem can be calculated to obtain a closed-form solution according to the KKT constraint conditions by the Lagrange multiplier method; when \(\eta < 0\), the original optimization problem can be transformed into the following form:

[0025] Step (2): Fix the anchor point label matrix \(F\) and update the sample point label matrix \(G\): \(g\) j They are independent of each other. The optimization problem of the label matrix \(G\) is transformed into a step-by-step optimization problem of the label vector \(g\) j , \(j = 1, 2, \cdots, m\). The original problem is equivalently transformed into the following form:

[0026]

[0027]

[0028] Substitute into the above formula to obtain the final optimization function;

[0029] When \(\eta>0\), this optimization problem can be calculated to obtain a closed-form solution according to the KKT constraint conditions by the Lagrange multiplier method; when \(\eta < 0\), the original optimization problem is transformed into the following form:

[0030] Repeat step (2) and step (3) until the algorithm converges, that is, the change in the objective function value is less than \(1\times10\) -1 .

[0031] Beneficial effects

[0032] A fast image clustering method based on bipartite graph embedding and discriminant information proposed by the present invention aims at the problems of high spatio-temporal complexity required for eigenvalue decomposition and similarity matrix construction in spectral clustering algorithms, and the independence between low-dimensional feature space embedding and label learning and the lack of discriminant information. The present invention proposes a fast image clustering method based on bipartite graph embedding and discriminant information, constructs a bipartite graph by combining anchor points and adopts a coordinate descent optimization method to replace eigenvalue decomposition and post-processing to reduce the difficulty of data storage and improve the data processing speed. In addition, the present invention simultaneously performs subspace learning and label learning, can preserve local features of data and make full use of discriminant information. Therefore, fast and efficient clustering analysis of large-scale and high-dimensional image data can be realized.

[0033] The beneficial effects of adopting the method of the present invention mainly include:

[0034] (1) Compared with constructing a similarity matrix and adopting an eigenvalue decomposition method in traditional spectral clustering algorithms, the present invention greatly reduces the spatio-temporal complexity of the algorithm by constructing an anchor point-based bipartite graph matrix and adopting a coordinate descent method. At the same time, the introduction of a probability model makes the algorithm more in line with the actual application scenario.

[0035] (2) The present invention can collaboratively complete subspace learning and label learning tasks. In addition, by introducing a global divergence term, the present invention fully excavates data discriminant information to obtain a subspace most beneficial to label extraction, and improves the clustering accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 is a flowchart for establishing a comprehensive model.

[0037] Figure 2 is a flowchart for solving the model.

[0038] Figure 3 is a graph showing the change trend of the label prediction accuracy of the algorithm with the number of algorithm iterations on the Umist dataset. As Figure 3 can be seen, with the iteration of the algorithm, the label prediction accuracy gradually improves and tends to be stable after about 40 iterations. The final label prediction accuracy reaches 63.82%, and the time used is 53.739 seconds, significantly improving the efficiency of the spectral clustering algorithm.

[0039] Figure 4 is a numerical visualization result of the elements in the sample point label matrix optimized by the algorithm on the classical machine learning dataset Two-moon. The Two-moon dataset is often used to detect the clustering effect of algorithms, and it contains 200 sample points. The total number of label types of all sample points is two categories, and the number of sample points contained in each sub-dataset is the same, which is 100. As Figure 4It can be seen that in the sample point label matrix obtained through optimization, most of the label vectors follow a 0-1 discrete distribution, and the extremely few label vectors that do not follow the discrete distribution can be approximately regarded as meeting the distribution requirements. Thus, it can be seen that the optimized sample point label matrix is closer to the distribution characteristics of the potential data clusters in the original data. By setting the number of anchor points to 80, the L0 norm of the row vectors in the bipartite graph matrix to 6, and the global divergence term coefficient λ to 0.2, the label prediction accuracy of this invention on this dataset reaches 100%. Specific implementation mode

[0040] The present invention will be further described below in conjunction with embodiments and drawings:

[0041] The present invention is realized through the following technical solutions. A fast image clustering method based on bipartite graph embedding and discriminant information, and its specific steps are as follows:

[0042] Step 1: Construction of a sparse bipartite graph based on anchor points and construction of an intrinsic graph

[0043] For a sample matrix with n samples and a feature dimension of d where the feature dimension corresponds to the image resolution, and the row vectors correspond to the information of a single image. Based on the sample data, the anchor point data can be obtained using the k-means++ algorithm The correlation relationship between each sample point and each anchor point can be described by a bipartite graph matrix B ∈ R n×m The row vectors of the bipartite graph matrix represent the similarity vector composed of the similarities between the i-th sample point and each anchor point, and b ij represents the similarity between the i-th sample and the j-th anchor point. To learn the local characteristics of the data distribution, the correlation relationship between the sample point and the neighboring anchor points should be stronger than that with other anchor points. At the same time, to reduce the computational complexity, we control the L0 norm k0 (k0 < m) of the row vectors in the bipartite graph matrix to achieve sparse construction of the bipartite graph. Thus, the construction process of the sparse bipartite graph can be equivalent to solving the following problem:

[0044]

[0045]

[0046] where γ is an adaptive parameter, 1 m is a vector of all 1s with a dimension of m × 1. The constraints of the problem indicate that the similarity relationship between the sample point and the anchor point represents the similarity under the probability model. Using the Lagrange multiplier method under the KKT inequality constraint conditions to solve the problem, the following initialization results are obtained:

[0047]

[0048] Among them and the row vector e i The elements are arranged in ascending order.

[0049] At this time, we can obtain the bipartite graph constructed by sparsification. However, this bipartite graph only describes the neighbor relationship between the anchor points and the sample points, that is, the local features of the original data. To learn the global features of the data distribution to extract discriminant information, we continue to construct the intrinsic graph where, 1 n+m is a vector of all 1s with dimension (n + m)×1.

[0050] Step 2: Integrate the constructed bipartite graph B and the intrinsic graph H n+m into the framework of spectral learning, and set up the objective function. The specific process is as follows:

[0051] Set up the objective function as: The corresponding constraint conditions are: F1 c = 1 n , F ≥ 0; G1 c = 1 m , G ≥ 0

[0052] where the optimization variables are the sample point label matrix and the anchor point label matrix 1 c is a vector of all 1s with dimension c×1, 1 n is a vector of all 1s with dimension n×1, the Laplacian matrix L S = D - S, the similarity matrix and B is the bipartite graph matrix obtained in Step 1, the matrix is the degree matrix and The label matrix can be regarded as the feature distribution of the data points in the original space in the subspace. λ > 0 is a coefficient that needs to be manually assigned.

[0053] The first term of the objective function is to learn a feature distribution in the subspace composed of the sample point label matrix F and the anchor point label matrix G that can better represent the neighbor relationship between the sample points and the anchor points, and can effectively preserve the local features of the data distribution in the original space; the second term of the objective function is the global divergence of the distribution of the sample points and the anchor points in the subspace. By optimizing the objective function, the separability of the data clusters in the subspace can be enhanced. The continuous constraint conditions are to make the row sums of F and G equal to 1, that is, the sum of the membership degrees of the sample points or anchor points to different data clusters is 1, which is convenient for subsequent optimization and solution.

[0054] The first term and the second term of the objective function can be respectively transformed into the following forms:

[0055]

[0056]

[0057] Substituting it into the objective function, we can further express the objective function in the following form:

[0058]

[0059] Since the construction of the bipartite graph is sparse, that is, for the row vector b i most of the elements b ij = 0. If the second term in the objective function is missing, the optimization of the objective function cannot avoid the appearance of abnormal membership distributions. This abnormal membership distribution refers to the situation where when b ij = 0, that is, when there is no neighbor relationship between the anchor point and the sample point in the original space, the corresponding sample point and the anchor point in the latent space completely coincide, that is and thus loses its meaning. By incorporating the second term in the objective function into the model, when and only when the optimization of the objective function will make the distance between the corresponding f i and g j in the subspace as close as possible; when the optimization of the objective function will make the distance between the corresponding f i and g j as far as possible. Thus, by optimizing this objective function, sample points with similar features can be grouped into the same data cluster, and the distance between different data clusters is as far as possible.

[0060] It should be noted that when the value of λ is too large, the key neighbor relationship between the sample points and the anchor points in the original space will be destroyed. Specifically, when b ij > 0, it means that the similarity between the anchor point and the sample point is relatively high. A too large value of λ will make the distance between the sample point and the anchor point in the corresponding latent space increase with the optimization of the objective function, which does not conform to the distribution characteristics of the sample points and the anchor points in the original space. When the value of λ is too small, the model fails to fully explore the discriminative information and thus it is difficult to obtain the subspace feature distribution that is most beneficial to label learning.

[0061] Step 3: Alternately iterate and optimize F and G by the coordinate descent method until the objective function gradually converges. In the objective function established in Step 2, the variables to be optimized are the sample point label matrix F and the anchor point label matrix G. Before optimizing F and G, they first need to be randomly initialized, that is, under the constraint of the probability model, randomly assign values to the elements of the label matrix F and the label matrix G so that the elements of the label matrix are non - negative and the row sum is one. The specific steps of alternately iterating and optimizing F and G are as follows:

[0062] (1) Fix the anchor point label matrix G and update the sample point label matrix F. Since f i are independent of each other, the optimization problem of the label matrix F can be transformed into a step-by-step optimization problem of the label vectors f i , i = 1, 2,..., n. At this time, taking the optimization of the sample point label vector f i as an example, the problem to be optimized can be expressed in the following form:

[0063]

[0064] Substitute into the above formula to obtain the final optimization function:

[0065]

[0066] i. When η > 0, this optimization problem can be calculated to obtain a closed-form solution by the Lagrange multiplier method according to the KKT constraint conditions.

[0067] ii. When η < 0, the original optimization problem can be transformed into the following form:

[0068]

[0069] (2) Fix the anchor point label matrix F and update the sample point label matrix G. Since g j are independent of each other, the optimization problem of the label matrix G can be transformed into a step-by-step optimization problem of the label vectors g j , j = 1, 2,..., m. At this time, taking the optimization of the anchor point label vector g j as an example, the original problem is equivalently transformed into the following form:

[0070]

[0071] Substitute into the above formula to obtain the final optimization function:

[0072]

[0073] i. When η > 0, this optimization problem can be calculated to obtain a closed-form solution by the Lagrange multiplier method according to the KKT constraint conditions.

[0074] ii. When η < 0, the original optimization problem can be transformed into the following form:

[0075]

[0076] (3) Repeat steps (2) and (3) until the algorithm converges (the change in the objective function value is less than 1×10 -1 ).

[0077] Step 4: Directly obtain the final sample labels based on the sample point label matrix F obtained in Step 3. From the constraint conditions of the objective function in Step 2, it can be known that F does not conform to the discrete distribution of the indicator matrix, that is, its row vectors do not meet the requirement that only one element is 1 and the rest of the elements are all 0. By optimizing the objective function, the present invention can ensure that most (≥95%) of the row vectors in the sample point label matrix F satisfy the discrete distribution (see Figure 4 ). For a small number of sample point label vectors that do not satisfy the discrete distribution, the final predicted labels can be directly obtained according to the numerical magnitudes of the elements in the label vectors. Specifically, when f ik is the maximum value in the sample point label vector f i , the i-th sample point is classified into the k-th data cluster. Specific embodiments:

[0079] The specific implementation manner includes the following steps:

[0080] The following combines the instance data set in the field of machine learning to illustrate the specific implementation manner of the present invention, but the technical content of the present invention is not limited to the described scope.

[0081] The present invention proposes a fast image clustering method based on bipartite graph embedding and discriminant information, including the following steps:

[0082] Step 1: Initialize matrices B, H n+m , F and G, and initialize the parameter λ; Step 2: Update the label matrix F; Step 3: Update the label matrix G; Step 4: Repeat Steps 2 and 3 until the objective function value converges; Step 5: Obtain the sample point labels according to the label matrix F.

[0083] Step 1. For the given image data matrix with the number of samples being n, the feature dimension being d, and the number of true data labels being c. In practical applications, the image feature dimension is the image resolution. Select the Umist face image data set as the clustering example, which has 574 samples, a resolution of 112×92, and 20 true data labels. According to experience, set the number of anchor points m to 210, and use the k-means++ algorithm to select the anchor points The L0 norm of the row vectors in the bipartite graph matrix is 3, and the construction of the sparse bipartite graph can be completed by solving the following problem:

[0084]

[0085]

[0086] The optimal solution to this problem is as follows:

[0087]

[0088] Among them And the row vector e i The elements are arranged in ascending order. Construction of the intrinsic graph Randomly initialize the label matrices F and G, and randomly assign values to the elements of the label vector and the label matrix under the constraint of the probability model, such that the elements of the label matrix are non-negative and the row sums are one. Initialize the global divergence term coefficient λ to be 0.2 according to experience.

[0089] Step 2: Fix the label matrix G and update the label matrix F. Since f i are independent of each other, the optimization problem of the label matrix F can be transformed into a step-by-step optimization problem of the label vectors f i , i = 1, 2,..., n. At this time, taking the optimization of the label vector f i as an example, the original problem is equivalently transformed into the following form:

[0090]

[0091] Substitute into the above formula to obtain the final optimization function:

[0092]

[0093] i. When η > 0, the optimization direction is not affected by the magnitude of the square term coefficient, so its value can be fixed as

[0094] This optimization problem can be transformed into the following form according to the Lagrange multiplier method under the KKT inequality constraint:

[0095]

[0096] Among them, the vectors ω and μ ≥ 0 are both Lagrange multipliers, and there are f i T 1 - 1 = 0 and μ T f i = 0. Taking the derivative of f i gives:

[0097]

[0098] Substitute into f i T 1 - 1 = 0 to get:

[0099]

[0100] Substitute into to get:

[0101]

[0102] Substituting gives the following further:

[0103]

[0104] From f i , μ≥0 and μ T f i = 0, it can be obtained that f i , μ are orthogonal vectors, that is, for any k (1 ≤ k ≤ c), the following is satisfied

[0105] satisfies f ik μ k = 0. Substituting gives:

[0106]

[0107] where (x) + = max(x, 0). At this time, v ik is fixed, and the optimal solution of f ik depends on

[0108] Substituting into gives:

[0109]

[0110] At this time, solving the optimal value can be equivalent to solving the zero point of the following function:

[0111]

[0112] where and this function problem is a linear piecewise convex problem. Using the following Newton iteration method can obtain the solution of (when the change in the function value does not exceed 1×10 -10 , it can be approximately considered that is the solution of this function and generally the number of iterations does not exceed 100):

[0113]

[0114] ii. When η < 0, the original optimization problem can be transformed into the following form:

[0115]

[0116] It is easy to see that when f iWhen there is exactly one element equal to 1 and the rest are all 0, f i T f i achieves the maximum value and the maximum value is 1. On this basis, when f ik = 1 and w ik is the maximum value in the vector w i , f i T w i achieves the maximum value. At this time, the f i that satisfies this condition is the optimal solution to this problem.

[0117] Step 3: Fix the label matrix F and update the label matrix G. Since g j are independent of each other, the optimization problem of the label matrix G can be transformed into a step-by-step optimization problem of the label vector g j , j = 1, 2,..., m. At this time, taking the optimization of the anchor label vector g j as an example, the original problem is equivalently transformed into the following form:

[0118]

[0119] Substituting into the above formula, the final optimization function can be obtained:

[0120]

[0121] i. When η > 0, the optimization direction is not affected by the magnitude of the square term coefficient, so its value can be fixed as

[0122] This optimization problem can be transformed into the following form according to the Lagrange multiplier method under the KKT inequality constraint.

[0123]

[0124] Among them, the vector ω and the vector μ ≥ 0 are both Lagrange multipliers, and there are and μ T g j = 0. Taking the derivative of g j gives:

[0125]

[0126] Substituting into gives:

[0127]

[0128] Substituting into gives:

[0129]

[0130] Substituting in, we can further obtain:

[0131]

[0132] From g j , μ≥0 and μ T g j = 0, it can be obtained that g j , μ are orthogonal vectors, that is, for any k (1≤k≤c), all

[0133] satisfy g jk μ k = 0. Substituting in, we can obtain:

[0134]

[0135] where (x) + = max(x, 0). At this time, the optimal solution of g jk depends on

[0136] Substituting into we can obtain:

[0137]

[0138] At this time, solving the optimal value can be equivalent to solving the zero point of the following function:

[0139]

[0140] where, and this function problem is a linear piecewise convex problem. Using the following Newton iteration method, the solution of can be obtained (when the change in the function value does not exceed 1×10 -10 , it can be approximately considered that is the solution of this function and generally the number of iterations does not exceed 100):

[0141]

[0142] ii. When η < 0, the original optimization problem can be transformed into the following form:

[0143]

[0144] It is easy to know that when g jwhen there is exactly one element equal to 1 and the rest are all 0, the maximum value is obtained and the maximum value is 1. On this basis, when g jk = 1 and w jk is the maximum value in the vector w j then the maximum value is obtained. At this time, the g satisfying this condition j is the optimal solution to this problem.

[0145] Step 4. Repeat Step 2 and Step 3 until the algorithm converges (the change in the objective function value is less than 1×10 -1 ).

[0146] Step 5. Obtain the final predicted label according to the numerical values of the elements in the label vector Specifically, when f ik is the maximum value in the sample point label vector f i the label of the i-th sample point can be obtained as k.

[0147] By comparing the true label and the predicted label The clustering accuracy (ACC) of the present invention reaches 63.82% and the clustering normalized mutual information (NMI) reaches 79.19% on the Umist dataset, and the running time is 53.739 s, significantly improving the clustering efficiency and clustering accuracy of face image data.

[0148] Therefore, compared with other spectral clustering related algorithms, the subspace learning process in this invention is also the learning process of sample labels, without the need to use the K-means clustering algorithm as post-processing.

Claims

1. A fast image clustering method based on bipartite graph embedding and discriminant information, characterized in that The steps are as follows: Step 1. Construction of Sparse Bipartite Graph and Intrinsic Graph Based on Anchor Points: For a sample matrix with n samples and a feature dimension of d where the feature dimension Corresponding to the image resolution, the row vector corresponds to the information of a single image; The sparsified constructed bipartite graph: where γ is an adaptive parameter, 1 m is a vector of all 1s with dimension m×1; The similarity between the i-th sample and the j-th anchor point: wherein and the row vector e i has elements arranged in ascending order; The inherent graph: where 1 n+m is a vector of all 1s with dimension (n + m)×1; Step 2: Integrate the constructed bipartite graph B and the inherent graph H n+m into the framework of spectral learning and set up the objective function: The corresponding constraint is: F1 c = 1 n , F ≥ 0; G1 c = 1 m , G ≥ 0; Among them, the optimization variables are the sample point label matrix and the anchor point label matrix 1 c is a vector of all 1s with dimension c×1, 1 n is a vector of all 1s with dimension n×1, the Laplacian matrix L S = D - S, the similarity matrix and B is the bipartite graph matrix obtained in step 1, the matrix is the degree matrix and The label matrix can be regarded as the feature distribution of data points in the original space in the subspace, λ > 0 is a coefficient that needs to be manually assigned; Step 3: Alternately iterate and optimize F and G by the coordinate descent method until the objective function gradually converges; Step 4: Directly obtain the final sample labels based on the sample point label matrix F obtained in Step 3. When f ik is the maximum value in the sample point label vector f i then divide the i-th sample point into the k-th data cluster.

2. The fast image clustering method based on bipartite graph embedding and discriminant information according to claim 1, characterized in that: The steps of the alternate iterative optimization of F and G are as follows: Step (1) Fix the anchor point label matrix G and update the sample point label matrix F: f i are independent of each other, and the optimization problem of the label matrix F is transformed into the step-by-step optimization problem of the label vector f i , i = 1, 2,..., n, and the problem to be optimized is in the following form: Substitute into the above formula to obtain the final optimized function: When η > 0, this optimization problem can be calculated to obtain a closed-form solution by the Lagrange multiplier method according to the KKT constraint conditions; When η < 0, the original optimization problem can be transformed into the following form: Step (2) Fix the anchor label matrix F and update the sample point label matrix G: g j are independent of each other, and the optimization problem of the label matrix G is transformed into the step-by-step optimization problem of the label vector g j , j = 1, 2,..., m, and the original problem is equivalently transformed into the following form: Substitute into the above formula to obtain the final optimized function; When η > 0, this optimization problem can be calculated to obtain a closed-form solution by the Lagrange multiplier method according to the KKT constraint conditions; When η < 0, the original optimization problem is transformed into the following form: Repeat steps (2) and (3) until the algorithm converges, i.e., the change in the objective function value is less than 1×10 -1 .

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