Compressible subspace clustering method for large-scale high-dimensional image data set
By constructing a small-scale dictionary and learning the optimal bipartite graph on a large-scale high-dimensional dataset, the high computational complexity and information redundancy of traditional subspace clustering algorithms are solved, and efficient clustering results are obtained.
Patent Information
- Application Number
- CN202510875639.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-06-27
- Publication Date
- 2025-10-17
AI Technical Summary
Traditional subspace clustering algorithms have high computational complexity and information redundancy when processing large-scale, high-dimensional data, leading to increased time costs and decreased clustering accuracy.
Principal component analysis and a fast hierarchical clustering algorithm are used to select a subset of samples to construct a small-scale dictionary. Combined with optimal bipartite graph learning and Laplacian matrix rank constraints, the clustering results are obtained directly through dictionary representation learning under low-rank constraints and bipartite graph construction, avoiding post-processing.
It effectively reduces computational costs, minimizes the impact of dictionary redundancy, improves clustering efficiency and accuracy, and directly yields clustering results without post-processing.
Smart Images
Figure CN120807985A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application belongs to the technical field of machine learning and data mining, and particularly relates to a compressible subspace clustering method for large-scale high-dimensional image data sets, which mainly aims to solve the problems of high time cost and low clustering accuracy of traditional clustering algorithms in processing large-scale high-dimensional data sets, and can be effectively applied to large-scale image data mining scenes. BACKGROUND
[0002] As an unsupervised learning technique, the subspace clustering method (SC) has shown good processing performance in the field of high-dimensional data mining and has been more and more widely concerned in recent years. The subspace clustering algorithm assumes that the data from the high-dimensional space is distributed in multiple low-dimensional subspaces, so its main working principle is to learn the low-dimensional subspace of the same class of samples in the high-dimensional data, which plays a crucial role in the high-dimensional clustering problem. Based on different clustering mechanisms, the subspace clustering algorithm can be roughly divided into four categories, including matrix factorization method, statistical learning method, algebraic method and spectral analysis method.
[0003] The present application mainly focuses on the subspace clustering method based on spectral analysis. The method based on spectral analysis has been more and more concerned in the past decade due to its simplicity and superior performance. Generally speaking, this method is a combination of dictionary representation and graph learning method. First, the method learns the optimal coefficient matrix through dictionary representation to construct a similarity graph, and then uses spectral clustering for post-processing to obtain the clustering result. There are two methods for learning the optimal coefficient matrix, namely sparse learning such as sparse subspace learning method and low-rank representation such as robust subspace segmentation algorithm based on low-rank representation. Under these two typical methods, many different variants have been developed in recent years, such as scalable sparse subspace learning algorithm and implicit low-rank representation method. In addition, in order to avoid the two-stage clustering process in the above-mentioned methods, recent related work proposes the idea of combining dictionary representation and graph learning into a general framework, such as subspace clustering method based on block diagonal representation. In addition, structural graph learning (SGL) is also introduced to obtain better graph structure to directly obtain the clustering result.
[0004] Structured graph learning has also attracted much attention in recent years. In order to find the optimal graph with clear structure, the related work proposes to use the special relationship between the similarity graph and its Laplacian matrix to construct the optimal graph learning framework. Further, by introducing the relationship between the graph matrix and its Laplacian matrix into the co-clustering method, a subspace clustering algorithm based on structured optimal bipartite graph learning can be proposed. It can be seen that introducing structured graph learning is undoubtedly an effective means to solve the post-processing process of traditional subspace clustering methods. For example, the subspace clustering method based on block diagonal representation is based on the block diagonal regularization term to learn the structured graph in the subspace problem. In addition, an optimal graph learning method based on dictionary update is proposed, which not only introduces the structured graph learning method, but also aims to obtain the optimal dictionary to reduce the influence of redundant information and abnormal noise samples.
[0005] Therefore, it is a good idea to combine dictionary representation and structured graph learning to solve the subspace clustering problem, which can obtain better performance than the current two-stage clustering algorithm. However, when dealing with large-scale high-dimensional data sets, or even in a noisy data environment, the above method may increase the time cost of the algorithm. Most of the current traditional subspace clustering methods use the entire data as a dictionary for self-representation in the representation learning process. This leads to the fact that when dealing with large-scale high-dimensional data sets, the learned coefficient matrix in the dictionary representation will also become too large due to the large dictionary. Since the construction of the similarity graph requires the learned coefficient matrix, the traditional algorithm cannot obtain the desired similarity graph in the ideal time when dealing with large-scale high-dimensional data sets. In addition, the original data in real situations usually contains a large amount of noise and redundant information, which also reduces the performance of the method in the process of learning the structure graph. Therefore, for traditional subspace clustering methods, how to deal with large-scale data sets is a problem that needs to be solved at present. SUMMARY
[0006] (1) Technical problems to be solved
[0007] The technical problem to be solved by the present application is how to solve the high computational complexity and information redundancy problem encountered by traditional subspace clustering algorithms when dealing with large-scale high-dimensional data.
[0008] (2) Technical solutions
[0009] To solve the above technical problems, the present application provides a compressible subspace clustering method for large-scale high-dimensional image data sets, which comprises the following steps:
[0010] Step 1: Given a large-scale high-dimensional image dataset, the principal component analysis technique and normalization method are used to preprocess the original data, so as to eliminate the noise features in the data and further improve the consistency of the data feature expression.
[0011] Step 2: According to the input original data sample X = [x1, x2,..., xn] ∈ Rn×m, 30% of the samples are selected as the dictionary, and the remaining samples form the new input data; the fast hierarchical clustering algorithm (BKHK) is used to select the corresponding representative samples from the original data according to the dictionary size to form a small-scale data set to construct the dictionary X2 ∈ Rn×m; the remaining data samples form a new input large-scale data set X1 ∈ Rn×m, where N = n + m. N d×N d×m d×n
[0012] Step 3: The constructed dictionary and new input data are applied to the compressible subspace clustering method based on optimal bipartite graph learning, and the coefficient representation matrix of the input data under the constructed dictionary is obtained through dictionary representation learning under low rank constraint; based on the idea of joint clustering, the learned coefficient representation matrix is used to construct a bipartite graph to fully consider the correlation between the samples in the dictionary and the input data samples, so as to realize the clustering of all data samples; in addition, the constructed bipartite graph is structured under the rank constraint of the Laplacian matrix of the graph to obtain the optimal bipartite graph; under the learned optimal bipartite graph, the clustering results of all data samples can be directly obtained from the graph without post-processing.
[0013] Step 3 includes:
[0014] Step 31: Construct a compressible subspace clustering model based on optimal bipartite graph learning, and the optimization problem is as follows:
[0015]
[0016] Here, Tr(·) represents the trace function; the first term is the dictionary representation loss, which uses the l 2,1 norm as a measure of loss; in addition, the model introduces the kernel norm as a low-rank constraint to optimize the coefficient matrix Z globally, as shown in the second term; G S is a bipartite graph constructed by the subgraph S, and its construction formula is:
[0017]
[0018] For the above optimization problem (1), the optimal bipartite graph G S , the specific algorithm process is as follows:
[0019] Step 32: low rank dictionary representation learning; in this process, the model learns an optimal coefficient matrix Z under the fixed variables S and F; therefore, the optimization problem (1) can be simplified as,
[0020]
[0021] By introducing an intermediate variable, and combining the definition of the augmented Lagrange function, the above problem can be further equivalent to the following optimization problem,
[0022]
[0023] Here, J is the intermediate variable matrix, E is the error matrix variable, μ represents the penalty parameter, and matrices Y1, Y2∈R m×n Indicates the Lagrange multiplier;
[0024] The step 32 proposes an iterative optimization strategy based on the optimization problem (4) to optimize the solution; the specific is as follows:
[0025] Step 321: the sub-problem of optimization problem (4) about the optimization solving variable J can be rewritten as,
[0026]
[0027] The related literature has provided an efficient solution method to obtain the analytical solution of the above problem;
[0028] Step 322: the sub-problem of optimization problem (4) about the optimization solving variable Z is a convex problem, so the optimal solution can be obtained by taking the derivative about Z and setting it to zero. Based on this, let the matrix I represent the unit matrix, and the calculation formula about the variable Z can be written as,
[0029]
[0030] Step 323: the sub-problem of problem (4) about the optimization solving variable E can be equivalent to,
[0031]
[0032] The related literature has designed an optimization algorithm to directly obtain the optimal solution of the above problem;
[0033] Step 324: to update the parameter μ, the given learning rate ρ and the upper threshold max μ The formula μ=max(ρμ,max μ) to calculate; for Lagrange multipliers Y1, Y2, the updating calculation formula is Y1=Y1+μ(X1-X2Z-E) and Y2=Y2+μ(Z-J);
[0034] Step 33: optimal bipartite graph learning; after learning the optimal coefficient matrix Z, the optimal bipartite graph learning process is to learn a subgraph S with an optimal structure under the rank constraint of the Laplacian matrix of the graph; therefore, the original optimization problem (1) can be simplified as the following optimization problem,
[0035]
[0036] Step 331: the subproblem of optimization problem (8) about the optimization solving variable S can be simplified as,
[0037]
[0038] A solving method is proposed in the related literature to obtain the optimal solution of the above optimization problem;
[0039] Step 332: the subproblem of optimization problem (8) about the optimization solving variable F can be rewritten as,
[0040]
[0041] It can be known that the optimal solution F of the above problem is composed of the eigenvectors corresponding to the first k smallest eigenvalues of the Laplacian matrix ;
[0042] Step 34: the calculation process in steps 32 and 33 is repeated through alternating iteration until the objective value of optimization problem (1) converges, so that the bipartite graph G S with an optimal block diagonal structure can be obtained, and then the final clustering results of all data, including the newly input data samples and the samples constituting the dictionary, can be directly obtained from the graph G S , without any post-processing technology.
[0043] (Three) beneficial effects
[0044] The application designs a dictionary representation model based on a block mechanism to select part of samples to construct a small-scale dictionary to replace the entire original data; then a bipartite graph learning method is introduced to solve the similarity graph construction problem. Finally, under the combination of the rank constraint of the Laplacian matrix of the graph, the application proposes a compressible subspace clustering method based on optimal bipartite graph learning. In addition, based on the augmented Lagrange multiplier method, the application designs an efficient optimization algorithm based on alternating iteration to solve the method.
[0045] Compared with the prior art, the main contribution of the present application is that: due to the design of a dictionary representation method based on the block mechanism, the small-scale dictionary is constructed by selecting part of the samples, thereby greatly reducing the calculation cost of the algorithm and avoiding the influence of dictionary redundancy. In addition, the present application combines the idea of joint clustering and introduces a bipartite graph construction method, which fully utilizes the ability to measure the duality between samples and features to realize clustering of all data. In order to avoid post-processing, the present application introduces the rank constraint of the Laplacian matrix of the graph, so that the bipartite graph constructed can learn the optimal block diagonal structure, thereby directly obtaining the clustering result. BRIEF DESCRIPTION OF DRAWINGS
[0046] Figure 1 FIG. 1 is a schematic diagram of a compressible subspace clustering algorithm based on optimal bipartite graph learning in the process of clustering high-dimensional image data.
[0047] The figure includes the subspace clustering process for high-dimensional data. The proposed compressible subspace clustering algorithm first uses the fast hierarchical clustering method (BKHK) to divide the original data into large-scale data as new input data and small-scale data as constructed dictionary, thereby performing dictionary representation learning based on low-rank constraint. After obtaining the coefficient representation matrix, the structured graph learning process is used to obtain the optimal fast diagonal structure of the bipartite graph, thereby realizing efficient clustering of the original data without the need for post-processing of the learned bipartite graph.
[0048] Figure 2 It is a technical solution flowchart. DETAILED DESCRIPTION
[0049] In order to make the purpose, content, and advantages of the present application clearer, the specific embodiments of the present application are described in further detail below in combination with the drawings and examples.
[0050] To solve the above technical problems, the present application provides a compressible subspace clustering method for large-scale high-dimensional image data sets, which comprises the following steps:
[0051] Step 1: Under the given large-scale high-dimensional image data set, the principal component analysis technology and normalization method are used to preprocess the original data, so as to eliminate the noise features in the data and further improve the consistency of the data feature expression;
[0052] Step 2: According to the input original data sample X=[x1,x2,...,x N ]∈R d×N, 30% of the samples are selected as the dictionary, and the remaining samples form the new input data; the fast hierarchical clustering algorithm (BKHK) is used to select the corresponding representative samples from the original data according to the size of the dictionary to form a small-scale data set to construct the dictionary X2∈R d×m , and the remaining data samples form a large-scale data set X1∈R d×n , where N=n+m;
[0053] Step 3: Apply the constructed dictionary and new input data to the compressible subspace clustering method based on optimal bipartite graph learning, and obtain the coefficient representation matrix of the input data under the constructed dictionary through dictionary representation learning under low rank constraint; based on the idea of joint clustering, use the learned coefficient representation matrix to construct a bipartite graph to fully consider the correlation between the samples in the dictionary and the input data samples, thereby realizing the clustering of all data samples; in addition, under the rank constraint of the Laplacian matrix of the graph, the constructed bipartite graph is structured to learn the optimal bipartite graph; under the learned optimal bipartite graph, the clustering results of all data samples can be directly obtained from the graph without post-processing.
[0054] The step 3 includes:
[0055] Step 31: Construct a compressible subspace clustering model based on optimal bipartite graph learning, and the optimization problem is as follows:
[0056]
[0057] Here, Tr(·) represents the trace function; the first term is the dictionary representation loss, which uses the l 2,1 norm as a measure of loss; in addition, the model introduces a kernel norm as a low-rank constraint to optimize the coefficient matrix Z globally, as shown in the second term; G S is a bipartite graph constructed for the subgraph S, and its construction formula is:
[0058]
[0059] For the above optimization problem (1), the optimal bipartite graph G S with optimal block diagonal structure is learned according to the optimization algorithm of the two steps of low-rank dictionary representation learning and optimal bipartite graph learning, and the specific algorithm process is as follows:
[0060] Step 32: Low-rank dictionary representation learning; in this process, the model learns an optimal coefficient matrix Z under the fixed variables S and F; therefore, the optimization problem (1) can be simplified as,
[0061]
[0062] By introducing an intermediate variable and combining the definition of the augmented Lagrangian function, the above problem can be further equivalent to the following optimization problem,
[0063]
[0064] Here, J is an intermediate variable matrix, E is an error matrix variable, μ represents a penalty parameter, and matrices Y1, Y2∈R m×n denotes the Lagrange multiplier;
[0065] The strategy proposed in step 32 can be used to optimize the solution of optimization problem (4); the specific steps are as follows:
[0066] Step 321: The sub-problem of optimization problem (4) about the optimization solution variable J can be rewritten as,
[0067]
[0068] The relevant literature has provided an efficient solution method to obtain the analytical solution of the above problem;
[0069] Step 322: The sub-problem of optimization problem (4) about the optimization solution variable Z is a convex problem, so the optimal solution can be obtained by taking the derivative with respect to Z and setting it to zero. Based on this, the matrix I is expressed as the unit matrix, and the calculation formula with respect to variable Z can be written as,
[0070]
[0071] Step 323: The sub-problem of problem (4) about the optimization solution variable E can be equivalent to,
[0072]
[0073] The relevant literature has designed an optimization algorithm to directly obtain the optimal solution of the above problem;
[0074] Step 324: To update the parameter μ, given the learning rate ρ and the upper threshold max μ The formula μ = max(ρμ, max μ ) is used for calculation; for the Lagrange multipliers Y1 and Y2, the update calculation formulas are Y1 = Y1 + μ(X1-X2Z-E) and Y2 = Y2 + μ(Z-J);
[0075] Step 33: Optimal bipartite graph learning; after learning the optimal coefficient matrix Z, the optimal bipartite graph learning process is to learn a subgraph S with the optimal structure under the constraint of the Laplacian matrix rank of the graph. Therefore, the original optimization problem (1) can be simplified as the following optimization problem,
[0076]
[0077] Step 331: The sub-problem of optimization problem (8) about optimizing the solution variable S can be simplified as,
[0078]
[0079] The related literature has proposed a solution method to obtain the optimal solution of the above optimization problem;
[0080] Step 332: The sub-problem of optimization problem (8) about optimizing the solution variable F can be rewritten as,
[0081]
[0082] It can be known that the optimal solution F of the above problem is composed of the eigenvectors corresponding to the first k smallest eigenvalues of the Laplacian matrix ;
[0083] Step 34: By repeating the calculation process in steps 32 and 33 through alternating iteration until the objective value of optimization problem (1) converges, the bipartite graph G with optimal block diagonal structure can be obtained S , and further, the final clustering results of all data, including new input data samples and samples constituting the dictionary, can be directly obtained from the graph G S without any post-processing technology.
[0084] Embodiment 1
[0085] For large-scale high-dimensional data clustering problems, this embodiment proposes a compressible subspace clustering method based on optimal bipartite graph learning, and the main working mechanism is as follows:
[0086] Step 1: First, under the given large-scale high-dimensional image data set, the principal component analysis technology and the normalization method are used to preprocess the original data, so as to initially eliminate noise features and improve the consistency of data feature expression.
[0087] Step 2: According to the number of input data samples, 30% of the samples are selected as the dictionary, and the remaining samples are represented learning. In the process of selecting the dictionary samples, the fast hierarchical clustering method (BKHK) is used to select the corresponding representative samples from the original data according to the size of the dictionary to construct, and the remaining samples constitute new input data.
[0088] Step 3: Apply the constructed dictionary and new input data to the proposed compressible subspace clustering method based on optimal bipartite graph learning. The coefficient representation matrix of the input data under the constructed dictionary is obtained through dictionary representation learning under low-rank constraints. Based on the idea of joint clustering, the learned coefficient representation matrix can be used to construct a bipartite graph to fully consider the correlation between the samples in the dictionary and the input data samples, thereby achieving clustering of all data samples. Furthermore, under the Laplacian matrix rank constraint of the graph, the constructed bipartite graph can be structured learned to obtain the optimal bipartite graph. Under the learned optimal bipartite graph, the clustering results of all data samples can be directly obtained from the graph without the need for post-processing.
[0089] Example 2
[0090] This paper proposes a compressible subspace clustering model based on optimal bipartite graph learning. The main implementation process is as follows:
[0091] 1. Compressible Subspace Clustering Model Based on Optimal Bipartite Graph Learning
[0092] Given a high-dimensional image dataset X=[x1,x2,...,x N ]∈R d×N , dictionary representation learning assumes that the samples are distributed in a linear Euclidean space. Therefore, in the given dictionary matrix A=[a1,a2,...,a m ]∈R d×m Next, we perform a linear representation on each sample in the data set, that is,
[0093] X=AZ=[Az1,Az2,...,Az N ]. (1)
[0094] Here, Z represents the system matrix, each column vector z i All correspond to sample x i Linear combination under dictionary A. Although the traditional method uses the entire data set X as the dictionary, that is, X = XZ, and achieves good performance under small-scale data sets. However, this method often has problems of information redundancy and high time cost when facing large-scale data sets. Therefore, a method based on anchor selection
[11] is proposed to solve the above problem, which randomly extracts m samples from the original data set to form a small-scale dictionary A, where m << N. Based on this method, the present invention adopts a "divide and conquer" strategy to improve the efficiency of dictionary representation, specifically using preprocessing techniques, such as the fast hierarchical clustering algorithm (BKHK), to convert the original data X∈R d×N Split into a large-scale dataset X1∈R d×n As new input data, and a small-scale data X2∈Rd×m As a dictionary, here N = n + m. Based on this, the application proposes a dictionary representation learning model based on a block mechanism, which has the following mathematical form:
[0095]
[0096] For problem (2), the first term is the dictionary representation loss, and γ1 is the penalty parameter. Here, the l 2,1 norm is used as a measure to effectively improve the robustness of the model to noise in the dictionary representation process. In order to learn the optimal coefficient matrix Z ∈ R m×n , the application introduces the nuclear norm as a low-rank constraint to globally optimize the learning of the coefficient matrix Z.
[0097] After obtaining the optimal representation Z, it is necessary to use it to construct a similarity graph. However, since Z is a non-square matrix, it cannot be directly used to construct a similarity graph, and it is difficult to obtain the clustering results of all samples (N = n + m) from the coefficient matrix Z ∈ R m×n . Benefiting from the idea of joint clustering method, the application designs a bipartite graph construction method to evaluate the class cluster relationship between all samples in the new input data X1 and the dictionary X2. According to the bipartite graph construction formula, the coefficient matrix Z ∈ R m×n is used to construct a bipartite graph G Z ∈ R N×N , which has the following mathematical form:
[0098]
[0099] In order to ensure the effectiveness of the graph, it is necessary to ensure that all elements in Z are non-negative. Therefore, the joint problem (2) and equation (3) can realize the clustering process for large-scale high-dimensional data. However, due to the large size of the bipartite graph G Z , it will cause greater computational cost in the post-processing process. Therefore, further processing of the obtained bipartite graph is needed to avoid excessive computational cost. Literature
[13] gives an important relationship between the bipartite graph G Z and its corresponding Laplacian matrix , that is, when the rank of the matrix is N-k, the graph G Z has k connected components. Therefore, the joint problem (2) and equation (3) can obtain a compressible subspace clustering model based on optimal graph learning as follows,
[0100]
[0101] where an intermediate variable S is introduced to construct the bipartite graph G S, where 1 denotes a column vector with all elements being 1. Define matrix The i-th smallest eigenvalue of Since is positive semi-definite, we have and its rank is also equal to the number of non-zero eigenvalues. Therefore, when minimizing the top k smallest eigenvalues of matrix , the constraint is also satisfied. Furthermore, according to the Ky Fan theorem
[14] , we have
[0102]
[0103] Here, Tr(·) denotes the trace function. By combining the above analysis, the joint equation (5) and the proposed model (4), given a large enough parameter γ3, the final optimization problem proposed in this invention can be converted into the following equivalent problem,
[0104]
[0105] Therefore, problem (6) is the final optimization model proposed in this invention. It can be seen that, compared with problem (4), the optimization problem (6) only has constraints on variables S and F, and therefore is easier to optimize and solve.
[0106] 2. Solution algorithm design
[0107] To solve the proposed optimization problem (6), this invention combines the augmented Lagrange multiplier method and designs an optimization algorithm based on alternating iteration to optimize and solve the three variables in the model in turn. According to the solving process, the optimization process of problem (6) can be divided into two steps, including low-rank dictionary representation learning and optimal bipartite graph learning.
[0108] 2.1 Low-rank dictionary representation learning
[0109] In the dictionary representation learning process, the goal is to learn an optimal coefficient matrix Z under the fixed variables S and F. Therefore, the original problem (6) can be simplified as follows,
[0110]
[0111] In order to better optimize the above problem, this invention introduces a low-rank matrix variable J to replace the variable Z imposed by the nuclear norm low-rank constraint, and an error matrix variable E to replace the dictionary representation loss X1-X2Z in the first item of the problem. Thus, problem (7) can be equivalently converted into,
[0112]
[0113] According to the definition of the augmented Lagrangian function, the augmented Lagrangian function of problem (8) can be written as
[0114]
[0115] Here, μ denotes the penalty parameter, and matrices Y1, Y2∈R m×n denote the Lagrange multipliers. Thus, an iterative optimization based strategy can be used to solve problem (9).
[0116] Fixing other variables, problem (9) with respect to optimization variable J can be transformed as
[0117]
[0118] The solution of the optimal solution of optimization problem (10) is given in
[15] .
[0119] Fixing other variables, the optimization subproblem of problem (9) with respect to variable Z can be written as
[0120]
[0121] Optimization problem (11) with respect to variable Z is a convex problem, thus the optimal solution can be obtained by taking the derivative with respect to Z and setting it to zero. Based on this, the calculation formula with respect to variable Z can be written as
[0122]
[0123] Here, matrix I denotes the identity matrix.
[0124] Fixing other variables, solving the optimal solution E of problem (9) is equivalent to optimizing the following problem
[0125]
[0126] For optimization problem (13), an optimization algorithm is designed in
[16] to directly obtain its optimal solution.
[0127] To update parameter μ, at a given learning rate ρ and upper threshold max μ The update is performed using the calculation formula μ = max(ρμ, max μ ). For Lagrange multipliers Y1, Y2, their update formulas are Y1 = Y1 + μ(X1 - X2Z - E) and Y2 = Y2 + μ(Z - J).
[0128] 2.2 Optimal bipartite graph learning
[0129] After learning the optimal coefficient matrix Z, the optimal bipartite graph learning process is to learn a subgraph S with clear structure under the constraint of the Laplacian matrix rank of the graph. Therefore, the original problem (6) can be simplified as the following optimization problem,
[0130]
[0131] Under the given coefficient matrix Z and fixed variable F, the problem of optimization variable S can be rewritten as,
[0132]
[0133] For the above problem, the literature
[10] gives an efficient solution algorithm, which converts problem (15) into n independent vector optimization problems to solve.
[0134] Under the fixed variable S, the optimization problem (14) about F can be simplified as:
[0135]
[0136] It can be seen that the optimal solution F of problem (16) is composed of the eigenvectors corresponding to the first k smallest eigenvalues of the Laplacian matrix .
[0137] Comprehensive of the above low-rank dictionary representation learning and optimal bipartite graph learning steps, the present application designs the above algorithm flow to iteratively solve the optimization problem (6) until convergence, so that the bipartite graph G S with optimal block diagonal structure can be obtained. S Further, the final clustering results of all data, including new input data samples and samples in the dictionary, can be directly obtained from the graph G S , without any post-processing technology. Therefore, the method proposed in the present application can effectively improve the clustering efficiency when processing large-scale high-dimensional data sets, and reduce the problem of performance degradation caused by information redundancy in the dictionary.
[0138] The above only describes the preferred embodiments of the present application, and it should be noted that for those skilled in the art, without departing from the technical principles of the present application, a number of improvements and modifications can be made, and these improvements and modifications should also be considered as the protection scope of the present application.
Claims
1. A compressible subspace clustering method for large-scale high-dimensional image datasets, characterized by: The method comprises the following steps: Step 1: Given a large-scale high-dimensional image dataset, use principal component analysis and normalization methods to preprocess the raw data to remove noise features in the data and further improve the consistency of data feature expression; Step 2: According to the input original data sample X=[x1,x2,...,x N ]∈R d×N , select 30% of the samples as the dictionary, and the remaining samples constitute the new input data; use the fast hierarchical clustering algorithm to select the corresponding representative samples from the original data according to the dictionary size to form a small-scale data set to construct the dictionary X2∈R d×m , the remaining data samples constitute the new input large-scale data set X1∈R d×n , where N = n + m; Step 3: Apply the constructed dictionary and new input data to the compressible subspace clustering method based on optimal bipartite graph learning, and obtain the coefficient representation matrix of the input data under the constructed dictionary through dictionary representation learning under low-rank constraints; based on the idea of joint clustering, use the learned coefficient representation matrix to construct a bipartite graph to fully consider the correlation between the samples in the dictionary and the input data samples, thereby realizing the clustering of all data samples; in addition, under the rank constraint of the Laplace matrix of the graph, the constructed bipartite graph is structuredly learned to obtain the optimal bipartite graph; under the learned optimal bipartite graph, the clustering results of all data samples can be obtained directly from the graph without the need for post-processing.
2. The compressible subspace clustering method for large-scale high-dimensional image datasets according to claim 1, characterized in that: The step 3 comprises: Step 31: Construct a compressible subspace clustering model based on optimal bipartite graph learning. The optimization problem is as follows: Here, Tr(·) represents the trace function; the first term is the dictionary representing the loss, where l is used. 2,1 norm as a metric loss; in addition, the model introduces the nuclear norm as a low-rank constraint to optimize the coefficient matrix Z globally, as shown in the second term; G S The bipartite graph constructed for subgraph S is constructed as follows: For the optimization problem (1) mentioned above, we learn the bipartite graph G with the optimal block diagonal structure by using the two-step optimization algorithm of low-rank dictionary representation learning and optimal bipartite graph learning. S ,The specific algorithm process is as follows; Step 32: Low-rank dictionary representation learning; in this process, the proposed model learns an optimal coefficient matrix Z under fixed variables S and F; therefore, the optimization problem (1) can be simplified to, By introducing intermediate variables and combining the definition of augmented Lagrangian function, the above problem can be further transformed into the following optimization problem: Here, J is the intermediate variable matrix, E is the error matrix variable, μ represents the penalty parameter, and the matrix Y1,Y2∈R m×n represents the Lagrange multiplier; In step 32, a strategy based on iterative optimization is proposed to optimize and solve the optimization problem (4); specifically, the strategy is as follows: Step 321: The sub-problem of optimizing the variable J in the optimization problem (4) can be rewritten as: Step 322: Optimization problem (4) The subproblem of optimizing the variable Z is a convex problem, so the optimal solution can be obtained by taking its derivative with respect to Z and setting it to zero. Based on this, let the matrix I be represented as the identity matrix, and the calculation formula for the variable Z can be written as, Step 323: The sub-problem of problem (4) regarding the optimization solution of variable E can be equivalent to, Step 324: To update the parameter μ, at a given learning rate ρ and upper threshold max μ Next, use the formula μ=max(ρμ,max μ ) for calculation; for the Lagrange multipliers Y1, Y2, the update calculation formulas are Y1 = Y1 + μ (X1 - X2Z - E) and Y2 = Y2 + μ (ZJ); Step 33: Optimal bipartite graph learning; After learning the optimal coefficient matrix Z, the optimal bipartite graph learning process is to learn a subgraph S with the optimal structure under the Laplace matrix rank constraint of the graph; therefore, the original optimization problem (1) can be simplified to the following optimization problem, Step 331: The sub-problem of optimizing the variable S in the optimization problem (8) can be simplified to: Step 332: The sub-problem of optimizing the variable F in the optimization problem (8) can be rewritten as: It can be seen that the optimal solution F of the above problem is given by the Laplace matrix The eigenvectors corresponding to the first k smallest eigenvalues of Step 34: Repeat the calculation process in step 32 and step 33 by alternating iteration until the target value of the optimization problem (1) converges, so that the bipartite graph G with the optimal block diagonal structure can be obtained. S , and then from graph G S The final clustering results of all data, including the newly input data samples and the samples that make up the dictionary, can be directly obtained without any post-processing technology.
3. The compressible subspace clustering method for large-scale high-dimensional image datasets according to claim 2, characterized in that: In formula (1), Tr(·) represents the trace function.
4. The compressible subspace clustering method for large-scale high-dimensional image datasets according to claim 2, characterized in that: In formula (1), the first term is the dictionary representation loss, where l is used 2,1 norm is used as a metric for loss.
5. The compressible subspace clustering method for large-scale high-dimensional image datasets according to claim 2, wherein: In formula (1), the nuclear norm is introduced as a low-rank constraint to optimize the coefficient matrix Z globally, as shown in the second term.
6. The compressible subspace clustering method for large-scale high-dimensional image datasets according to claim 2, characterized in that: In formula (1), G S A bipartite graph constructed for subgraph S.
7. The compressible subspace clustering method for large-scale high-dimensional image datasets according to claim 2, characterized in that: In formula (4), J is the intermediate variable matrix.
8. The compressible subspace clustering method for large-scale high-dimensional image datasets according to claim 2, characterized in that: In formula (4), E is the error matrix variable.
9. The compressible subspace clustering method for large-scale high-dimensional image datasets according to claim 2, characterized in that: In formula (4), μ represents the penalty parameter, and the matrix Y1,Y2∈R m×n represents the Lagrange multiplier.
10. The compressible subspace clustering method for large-scale high-dimensional image datasets according to claim 2, characterized in that: The method designs a dictionary representation learning model based on a blocking mechanism to select some samples to construct a small-scale dictionary to replace the entire original data. Then, using the idea of joint clustering, it cleverly introduces a bipartite graph construction method, which not only solves the problem that the bipartite graph cannot be directly constructed due to non-square coefficient matrices, but also fully considers the correlation between dictionary samples and new input data samples. Under the Laplace matrix rank constraint of the combined graph, the proposed method can directly learn an optimal structured bipartite graph, which can directly obtain the final clustering result without any post-processing.