CN111310813A - A subspace clustering method and apparatus for latent low-rank representation - Google Patents

Abstract

The invention discloses a subspace clustering method and a subspace clustering device for potential low-rank representation, which are used for obtaining a characteristic matrix by acquiring data and preprocessing the data; representing subspace clustering by taking the unobserved potential low-rank of the data samples into consideration, and replacing a rank function by using a Schatten-p norm as a regular term to convert the problem that NP is difficult to solve into a problem which can be solved; introduction of l_pThe norm constrains the error term to construct an optimized objective function of potential low-rank representation subspace clustering; then, solving an optimization objective function to obtain a low-rank representation matrix; computing an affinity matrix based on the low-rank representation matrix; and calculating and dividing the affinity matrix by using a spectral clustering algorithm to realize the potential low-rank expression subspace clustering of the data. The method solves the problems of insufficient low-rank representation samples and difficulty in solving the rank function, and enhances the potentialThe robustness of the low-rank representation subspace clustering improves the performance of potential low-rank representation subspace clustering.

Description

A subspace clustering method and apparatus for latent low-rank representation

technical field

The present invention relates to the technical field of pattern recognition computing, and in particular, to a subspace clustering method and device for potential low-rank representation.

Background technique

With the progress of science and the development of artificial intelligence, pattern recognition processes and analyzes various forms of information that characterize things or phenomena, so as to describe, identify, classify and explain things or phenomena. Subspace clustering is widely used in many application domains, such as images, videos, texts, etc.

The importance of subspaces naturally leads to the conundrum of subspace segmentation, where the goal is to partition (or group) the data into each cluster corresponding to the subspace. The main challenge for subspace segmentation is how to effectively deal with the coupling between noise correction and data segmentation. As a subspace segmentation algorithm, latent low-rank representation subspace clustering can be regarded as an enhanced version of low-rank representation, so as to obtain more accurate segmentation results. Significant features can be automatically extracted from corrupted data to generate effective features for classification, which has attracted extensive attention and great attention in related technical fields. The latent low-rank representation subspace can take into account the unobserved data in the data, thus improving the performance of clustering. In order to solve the problem of insufficient samples, related technologies generally use the nuclear norm to constrain the regular term. However, the related art only considers the approximate constraint of using the nuclear norm as the rank function, but when the singular value of the matrix is large, from the definition of the rank function and the nuclear norm, the latter is too relaxed to be more accurate. The rank of the matrix is estimated, so that the clustering performance is reduced, the accuracy is not high, and the robustness is not strong.

SUMMARY OF THE INVENTION

In order to solve the problems of insufficient low-rank representation samples, weak robustness and insufficient performance of potential low-rank representation subspace clustering in the existing subspace clustering, the present invention provides a subspace clustering with potential low-rank representation. Class methods and devices.

In order to achieve the above purpose of the invention, the technical means adopted are:

A subspace clustering method for latent low-rank representations, including the following steps:

S1. Acquire data and preprocess it to obtain a feature matrix;

S2. Based on the feature matrix, use the Schatten- _p norm as a regular term to replace the rank function, and use the lp norm as a constraint function of the error term to construct a potential low rank to represent the optimization objective function of subspace clustering;

S3. Solve the optimization objective function to obtain a low-rank representation matrix;

S4. Calculate the affinity matrix based on the low-rank representation matrix;

S5. Use spectral clustering algorithm to calculate and segment the affinity matrix to realize subspace clustering of potential low-rank representation of the data.

In the above scheme, based on the low-rank representation subspace clustering, the potential low-rank representation subspace clustering considering the unobserved data samples is used, and the Schatten-p norm is used as a regular term to replace the rank function, The NP-hard problem is transformed into a solvable problem, and the _lp norm constraint error term is introduced, which solves the problem of insufficient low-rank representation samples and intractable rank functions, and enhances the potential low-rank representation subspace aggregation. The robustness of classes improves the performance of subspace clustering of latent low-rank representations.

Preferably, after the feature matrix is obtained in step S1, it further includes normalizing each feature point in the feature matrix. In this preferred solution, subsequent data processing can be facilitated after normalization.

Preferably, the optimization objective function of the potential low-rank representation subspace clustering described in step S2 is specifically:

s.t.X=XZ+XL+E;

为Schatten-p范数，定义为

为l_p范数，定义为

where Z is the subspace low-rank representation matrix, L is the subspace sparse representation matrix, X is the feature matrix, E is the reconstruction error matrix, and λ is the hyperparameter that controls the loss penalty;

is the Schatten-p norm, defined as

is the _lp norm, defined as

Preferably, the step S3 specifically includes the following steps:

Introduce auxiliary variables J and S to the optimization objective function, where Z=J, L=S:

s.t.X=XZ+XL+E, Z=J, L=S

The constraints are transformed into the augmented Lagrangian function using the Lagrangian multiplier method, and then the alternating method is used to iteratively optimize the various variables in the augmented Lagrangian function until convergence to obtain a low-rank representation matrix .

Preferably, the step S5 specifically includes the following steps:

Calculate the degree matrix of the affinity matrix using the following formula:

Wherein the degree matrix is a square matrix, D _{i, i} is the element of the ith row of the degree matrix, S _{i, j} is the element of the ith row and the jth column of the affinity matrix;

计算归一化的拉普拉斯矩阵L：use

Compute the normalized Laplacian matrix L:

where D is the degree matrix, S is the affinity matrix, and I is the identity matrix;

Calculate the eigenvectors of the Laplacian matrix L, take the vectors with the largest eigenvalues of the first k and arrange them in columns as a column matrix X=[x ₁ , x ₂ ,..., x _k ]∈R ^n*k ;

Convert the row vector of the column matrix into a unit vector to get the target matrix;

The target matrix is clustered by K-means clustering method, and K clustering results are obtained, thereby realizing the subspace clustering of potential low-rank representation.

The present invention also provides a subspace clustering device represented by a potential low rank, including:

The data preprocessing module is used to obtain the data and preprocess it to obtain the feature matrix;

The optimization objective function building module, based on the feature matrix, uses the Schatten- _p norm as the regular term to replace the rank function, and uses the lp norm as the constraint function of the error term to construct the optimization objective of the potential low-rank representation subspace clustering function;

a subspace representation matrix calculation module, used for solving the optimization objective function to obtain a low-rank representation matrix;

an affinity matrix calculation module for calculating an affinity matrix based on the low-rank representation matrix;

The subspace clustering module is used to calculate and segment the affinity matrix by using the spectral clustering algorithm, so as to realize the subspace clustering of the potential low-rank representation of the data.

In the above scheme, in the optimization objective function building module, based on the low rank representation subspace clustering, the subspace clustering is represented by the potential low rank considering the unobserved data samples, and the Schatten-p norm is used as the regularity. term to replace the rank function, transforming the NP-hard problem into a solvable problem, and introducing the _lp norm constraint error term to solve the problem of insufficient low-rank representation samples and difficult rank functions to solve, enhancing the potential The robustness of low-rank representation subspace clustering improves the performance of potentially low-rank representation subspace clustering.

Preferably, the data preprocessing module is further configured to normalize each feature point in the feature matrix after obtaining the feature matrix.

Preferably, the optimization objective function of the potential low-rank representation subspace clustering constructed by the optimization objective function building module is specifically:

s.t.X=XZ+XL+E;

为Schatten-p范数，定义为

0＜p≤∞；

为l_p范数，定义为

where Z is the subspace low-rank representation matrix, L is the subspace sparse representation matrix, X is the feature matrix, E is the reconstruction error matrix, and λ is the hyperparameter that controls the loss penalty;

is the Schatten-p norm, defined as

0<p≤∞;

is the _lp norm, defined as

Preferably, the subspace representation matrix calculation module is further used for:

Introduce auxiliary variables J and S to the optimization objective function, where Z=J, L=S:

s.t.X=XZ+XL+E, Z=J, L=S

The constraints are transformed into the augmented Lagrangian function using the Lagrangian multiplier method, and then the alternating method is used to iteratively optimize various variables in the augmented Lagrangian function until convergence, so as to obtain a low-rank representation matrix.

Preferably, the subspace clustering module is further used for:

Calculate the degree matrix of the affinity matrix using the following formula:

Wherein the degree matrix is a square matrix, D _{i, i} is the element of the ith row of the degree matrix, S _{i, j} is the element of the ith row and the jth column of the affinity matrix;

计算归一化的拉普拉斯矩阵L：use

Compute the normalized Laplacian matrix L:

where D is the degree matrix, S is the affinity matrix, and I is the identity matrix;

Calculate the eigenvectors of the Laplacian matrix, and take the vectors with the largest eigenvalues of the first k and arrange them in columns as a column matrix X=[x ₁ , x ₂ ,..., x _k ]∈R ^n*k ;

Convert the row vector of the column matrix into a unit vector to get the target matrix;

The target matrix is clustered by K-means clustering method, and K clustering results are obtained, thereby realizing the subspace clustering of potential low-rank representation.

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

The invention uses the subspace clustering represented by the potential low rank to include unobserved data samples, and solves the problem of insufficient low-rank representation samples. The Schatten-p norm is used to replace the rank function, and the Schatten-p norm is higher than the nuclear norm. Better approximation effect, transform the NP-hard problem into a solvable problem, and introduce the _lp norm constraint error term to construct the optimization objective function of potential low-rank representation subspace clustering; the invention improves the algorithm The robustness and clustering performance of the existing subspace clustering solves the problems of insufficient low-rank representation samples, weak robustness and insufficient performance of potential low-rank representation subspace clustering in the existing subspace clustering.

In addition, the present invention also provides a corresponding implementation device for the subspace clustering method based on the latent low rank representation, further making the method more practical, and the device has corresponding advantages.

Description of drawings

Fig. 1 is the flow chart of the method of the present invention.

FIG. 2 is a block diagram of the device of the present invention.

Detailed ways

The accompanying drawings are for illustrative purposes only, and should not be construed as limitations on this patent;

In order to better illustrate this embodiment, some parts of the drawings are omitted, enlarged or reduced, which do not represent the size of the actual product;

It will be understood by those skilled in the art that some well-known structures and their descriptions may be omitted from the drawings.

The technical solutions of the present invention will be further described below with reference to the accompanying drawings and embodiments.

Example 1

The present embodiment 1 provides a subspace clustering method with a potential low-rank representation, as shown in FIG. 1 , including the following steps:

S1. Acquire data and preprocess it to obtain a feature matrix;

This preprocessing step can be done by common means known in the art. For example, the preprocessing of image data is to normalize and grayscale the target image, remove noise, and then extract edges, regions or textures from the features of the target image. As experimental features, for example, Gabor features are extracted from face image data, and HOG features are extracted from handwritten data sets; the obtained feature matrix X ⁱ =[x ₁ , x ₂ ,...,x _N ]∈R ^D*N is the data vector The formed feature matrix, each column vector in the feature matrix corresponds to the feature vector of a feature point, where D is the dimension of the feature space, and N is the number of feature points;

In order to facilitate subsequent data processing, the following formulas are used to normalize each feature point in the feature matrix:

In the formula, x′ _i is the normalized value of the ith feature point, and x _i is the value of the ith feature point before normalization.

S2. Based on the feature matrix, the Schatten- _p norm is used as a regular term to replace the rank function, and the lp norm is used as a constraint function of the error term to construct a potential low-rank optimization objective function representing subspace clustering:

To solve the objective function minimization problem, in this step, a potential low-rank representation subspace clustering objective function of the rank minimization problem is first constructed:

stX=[X ₀ , X _H ]Z+E;

In the formula, rank( ) represents the rank of the matrix, Z is the low-rank representation matrix of the subspace, X is the feature matrix, X ₀ is the observed data sample matrix, X _H is the unobserved data sample matrix, E is the reconstructed error matrix, λ is the hyperparameter that controls the loss penalty;

However, the kernel norm is usually used as the best approximation norm for the rank function, so as to obtain a low-rank matrix. In order to fully consider the observed data, that is, to consider finding the optimal low-rank representation matrix, in this embodiment, the Schatten- _p norm of the matrix is used as the regular term to estimate the rank function, and the lp norm is used as the constraint function of the error term, combined with The potential low-rank representation constructs the objective function;

That is, in this step, the optimization objective function of the potential low-rank representation subspace clustering is constructed:

s.t.X=XZ+XL+E;

为Schatten-p范数，定义为

0＜p≤∞，用来约束矩阵的低秩；秩是矩阵非0奇异值的个数，秩为非凸的，因此为一个NP难得问题，Schatten-p范数是凸的，Schatten-p范数是秩的凸近似，用Schatten-p范数最小化来近似实现低秩约束。

为l_p范数，定义为

In the formula, Z is the subspace low-rank representation matrix, L is the subspace sparse representation matrix, X is the feature matrix, and λ is the hyperparameter that controls the loss penalty;

is the Schatten-p norm, defined as

0<p≤∞, used to constrain the low rank of the matrix; the rank is the number of non-zero singular values of the matrix, the rank is non-convex, so it is a rare NP problem, the Schatten-p norm is convex, and the Schatten-p The norm is a convex approximation of the rank, and the Schatten-p norm minimization is used to approximate the low-rank constraint.

is the _lp norm, defined as

S3. Solve the optimization objective function to obtain a low-rank representation matrix;

In this embodiment, by converting the optimization objective function into a convex optimization problem, auxiliary variables J and S are introduced, where Z=J, L=S:

s.t.X=XZ+XL+E, Z=J, L=S

The constraints are transformed into the augmented Lagrangian function using the Lagrangian multiplier method, and then the alternating method is used to iteratively optimize the various variables in the augmented Lagrangian function until convergence to obtain a low-rank representation matrix . The solution of the augmented Lagrangian function specifically includes the following steps:

A1. Set parameters and initialize Z=J=0, L=S=0, E=0, _Y1 =0, _Y2 =0, _Y3 =0, μ=10 ⁻⁶ , max _u =10 ⁶ , ρ =1.1, and ε=10 ^-6 , Y ₁ , Y ₂ , Y ₃ , Z, L are multiplier terms, μ and max _u are the penalty item parameters, ρ is the update coefficient of the penalty parameter, and ε is the convergence threshold;

其中，G＝Z+Y₂/μA2. Update J:

Among them, G=Z+Y ₂ /μ

其中，Q_G、

分别代表G的左奇异值和右奇异值，Δ为一个对角矩阵，通过以下公式求解：Specifically, the optimal solution of J is

Among them, Q _G ,

Represent the left singular value and right singular value of G respectively, and Δ is a diagonal matrix, which is solved by the following formula:

其中δ_i和σ_i是矩阵J和G的第i个奇异值，通过以下步骤对公式

进行求解：

where δ _i and σ _i are the ith singular values of matrices J and G, and the formula is

To solve:

最优解x^*分为两种情况：define constants

The optimal solution x ^* is divided into two cases:

1) When δ _i is less than or equal to v ₁ , x ^* = 0; 2) When δ _i is greater than v ₁ , x ^* is calculated by iteration x ⁽ⁱ⁺¹⁾ = δ _i -λp(x ⁽ⁱ⁾ ) ^p-1 get the optimal solution;

Construct the diagonal matrix Δ according to the obtained x ^* , and finally obtain the optimal solution of J

其中，M＝L+Y₃/μA3. Update S:

Wherein, M=L+Y ₃ /μ

其中，Q_M、

分别代表M的左奇异值和右奇异值。Specifically, it is consistent with the solution method of A1, where the optimal solution is

Among them, Q _M ,

represent the left and right singular values of M, respectively.

A4. Update Z: Z=(I+X ^T X) ^-1 (X ^T (X-LX-E)+J+X ^T Y ₁ -Y ₂ /μ)

A5. Update L: L=((X-XZ-E)X ^T +S+(Y ₁ X ^T -Y ₃ )/μ)(I+XX ^T ) ^-1

其中，N＝X-XZ-LX+Y₁/μA6. Update E:

Among them, N=X-XZ-LX+Y ₁ /μ

Specifically, it is consistent with the solutions of A1 and A2, wherein the optimal solution x ^* is divided into three cases: 1) when δ _i is less than v ₁ , x ^* =0; 2) when δ _i is equal to v ₁ , x ^* =υ; 3) When δ _i is greater than v ₁ , x ^* obtains the optimal solution by iterative calculation x ⁽ⁱ⁺¹⁾ =δ _i -λp(x ⁽ⁱ⁾ ) ^p-1 ;

A7. Update multipliers: Y ₁ =Y ₁ +μ(X-XZ-LX-E), Y ₂ =Y ₂ +μ(ZJ), Y ₃ =Y ₃ +μ(LS)

A8. Update parameters: μ=min(ρμ, max _u ).

It should be noted that, in this step, other methods may also be used to solve the optimization problem of the optimization objective function, and this embodiment only exemplifies one of the calculation methods.

S4. Calculate the affinity matrix based on the low-rank representation matrix;

The affinity matrix S can be calculated by using S=||Z ^T Z‖ ₂ , and Z is the representation matrix.

It should be noted that, in addition to using the above method to calculate the affinity matrix, other methods may also be used for calculation in this step, and this embodiment only exemplifies one of the calculation methods, which is not limited in this application.

S5. Use spectral clustering algorithm to calculate and segment the affinity matrix to realize the subspace clustering of the potential low-rank representation of the data:

Calculate the degree matrix of the affinity matrix using the following formula:

Wherein the degree matrix is a square matrix, D _{i, i} is the element of the ith row of the degree matrix, S _{i, j} is the element of the ith row and the jth column of the affinity matrix;

计算归一化的拉普拉斯矩阵L：use

Compute the normalized Laplacian matrix L:

where D is the degree matrix, S is the affinity matrix, and I is the identity matrix;

Calculate the eigenvectors of the Laplacian matrix L, take the vectors with the largest eigenvalues of the first k and arrange them in columns as a column matrix X=[x ₁ , x ₂ ,..., x _k ]∈R ^n*k ;

Convert the row vector of the column matrix into a unit vector to get the target matrix;

The target matrix is clustered by K-means clustering method, and K clustering results are obtained, thereby realizing the subspace clustering of potential low-rank representation.

Example 2

This Embodiment 2 provides a corresponding implementation device for the subspace clustering method of potential low-rank representation provided in Embodiment 1, which further makes the method more practical. The subspace clustering apparatus for potential low-rank representation provided in this embodiment is introduced below. The subspace clustering apparatus for potential low-rank representation described below may correspond to the subspace clustering method for potential low-rank representation described above. Reference.

As shown in Figure 2, the device includes:

The data preprocessing module is used to obtain and preprocess the data to obtain a feature matrix, and normalize each feature point in the feature matrix;

The optimization objective function building module is based on the feature matrix to construct a potential low-rank objective function representing subspace clustering, and the Schatten- _p norm is used as the regular term to replace the rank function, and the lp norm is used as the constraint function of the error term, so that Constructing an optimization objective function that represents subspace clustering with potential low rank;

a subspace representation matrix calculation module, used for solving the optimization objective function to obtain a low-rank representation matrix;

an affinity matrix calculation module for calculating an affinity matrix based on the low-rank representation matrix;

The subspace clustering module is used to calculate and segment the affinity matrix by using the spectral clustering algorithm, so as to realize the subspace clustering of the potential low-rank representation of the data.

The functions of each functional module of the subspace clustering apparatus for potential low-rank representation provided in this embodiment of the present invention may be specifically implemented according to the method in the foregoing method embodiment 1, and the specific implementation process may refer to the relevant description of the foregoing method embodiment 1 , and will not be repeated here.

The terms describing the positional relationship in the accompanying drawings are only used for exemplary illustration, and should not be construed as a limitation on this patent;

Obviously, the above-mentioned embodiments of the present invention are only examples for clearly illustrating the present invention, rather than limiting the embodiments of the present invention. For those of ordinary skill in the art, changes or modifications in other different forms can also be made on the basis of the above description. There is no need and cannot be exhaustive of all implementations here. Any modifications, equivalent replacements and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the claims of the present invention.

Claims (10)

1. a subspace clustering method represented by a potential low rank, is characterized in that, comprises the following steps: S1. Acquire data and preprocess it to obtain a feature matrix; S2. Based on the feature matrix, use the Schatten- _p norm as a regular term to replace the rank function, and use the lp norm as a constraint function of the error term to construct a potential low rank to represent the optimization objective function of subspace clustering; S3. Solve the optimization objective function to obtain a low-rank representation matrix; S4. Calculate the affinity matrix based on the low-rank representation matrix; S5. Use spectral clustering algorithm to calculate and segment the affinity matrix to realize subspace clustering of potential low-rank representation of the data. 2 . The subspace clustering method of potential low-rank representation according to claim 1 , wherein after obtaining the feature matrix in step S1 , the method further comprises normalizing each feature point in the feature matrix. 3 . 3. The subspace clustering method of potential low-rank representation according to claim 1, wherein the optimization objective function of the potential low-rank representation subspace clustering described in step S2 is specifically:

s.t.X=XZ+XL+E; where Z is the subspace low-rank representation matrix, L is the subspace sparse representation matrix, X is the feature matrix, E is the reconstruction error matrix, and λ is the hyperparameter that controls the loss penalty;

is the Schatten-p norm, defined as

is the _lp norm, defined as

4. The subspace clustering method of potential low-rank representation according to claim 3, wherein the step S3 specifically comprises the following steps: Introduce auxiliary variables J and S to the optimization objective function, where Z=J, L=S:

s.t.X=XZ+XL+E, Z=J, L=S The constraints are transformed into the augmented Lagrangian function using the Lagrangian multiplier method, and then the alternating method is used to iteratively optimize the various variables in the augmented Lagrangian function until convergence to obtain a low-rank representation matrix . 5. The subspace clustering method of latent low-rank representation according to claim 4, wherein the step S5 specifically comprises the following steps: Calculate the degree matrix of the affinity matrix using the following formula:

The degree matrix is a square matrix, D _i,i is the element of the ith row of the degree matrix, and S _i,j is the element of the ith row and the jth column of the affinity matrix; use

Compute the normalized Laplacian matrix L: where D is the degree matrix, S is the affinity matrix, and I is the identity matrix; Calculate the eigenvectors of the Laplacian matrix L, take the vectors with the largest eigenvalues of the first k and arrange them in columns as a column matrix X=[x ₁ ,x ₂ ,...,x _k ]∈R ^n*k ; Convert the row vector of the column matrix into a unit vector to get the target matrix; The target matrix is clustered by K-means clustering method, and K clustering results are obtained, thereby realizing the subspace clustering of potential low-rank representation. 6. A subspace clustering device represented by a potential low rank, comprising: The data preprocessing module is used to obtain the data and preprocess it to obtain the feature matrix; The optimization objective function building module is based on the feature matrix to construct a potential low-rank objective function representing subspace clustering, and the Schatten- _p norm is used as the regular term to replace the rank function, and the lp norm is used as the constraint function of the error term, so that Constructing an optimization objective function that represents subspace clustering with potential low rank; a subspace representation matrix calculation module, used for solving the optimization objective function to obtain a low-rank representation matrix; an affinity matrix calculation module for calculating an affinity matrix based on the low-rank representation matrix; The subspace clustering module is used to calculate and segment the affinity matrix by using the spectral clustering algorithm, so as to realize the subspace clustering of the potential low-rank representation of the data. 7 . The subspace clustering device of potential low-rank representation according to claim 6 , wherein the data preprocessing module is further configured to normalize each feature point in the feature matrix after obtaining the feature matrix. 8 . processing. 8. The subspace clustering device of potential low-rank representation according to claim 6, wherein the optimization objective function of the potential low-rank representation subspace clustering constructed by the optimization objective function building module is specifically:

s.t.X=XZ+XL+E; where Z is the subspace low-rank representation matrix, L is the subspace sparse representation matrix, X is the feature matrix, E is the reconstruction error matrix, and λ is the hyperparameter that controls the loss penalty;

is the Schatten-p norm, defined as

is the _lp norm, defined as

9. The subspace clustering device of latent low-rank representation according to claim 8, wherein the subspace representation matrix calculation module is further used for: Introduce auxiliary variables J and S to the optimization objective function, where Z=J, L=S:

s.t.X=XZ+XL+E, Z=J, L=S The constraints are transformed into the augmented Lagrangian function using the Lagrangian multiplier method, and then the alternating method is used to iteratively optimize various variables in the augmented Lagrangian function until convergence, so as to obtain a low-rank representation matrix. 10. The subspace clustering device of latent low-rank representation according to claim 9, wherein the subspace clustering module is further used for: Calculate the degree matrix of the affinity matrix using the following formula:

The degree matrix is a square matrix, D _i,i is the element of the ith row of the degree matrix, and S _i,j is the element of the ith row and the jth column of the affinity matrix; use

Compute the normalized Laplacian matrix L: where D is the degree matrix, S is the affinity matrix, and I is the identity matrix; Calculate the eigenvectors of the Laplacian matrix L, take the vectors with the largest eigenvalues of the first k and arrange them in columns as a column matrix X=[x ₁ ,x ₂ ,...,x _k ]∈R ^n*k ; Convert the row vector of the column matrix into a unit vector to get the target matrix; The target matrix is clustered by K-means clustering method, and K clustering results are obtained, thereby realizing the subspace clustering of potential low-rank representation.

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