A Multi-View Subspace Clustering Method, Device, Equipment and Storage Medium

Abstract

The present invention discloses a multi-view subspace clustering method, apparatus, device and storage medium, which relates to the technical field of multi-view subspace clustering, and includes the following steps: mapping multi-view data through a plurality of feature mapping matrices to obtain a plurality of embedded anchor graphs; decomposing the plurality of embedded anchor graphs to obtain a plurality of intrinsic anchor graphs; stacking the plurality of intrinsic anchor graphs to obtain a third-order tensor with a tensor nuclear norm constraint and rotating it, applying a tensor nuclear norm constraint based on tensor singular value decomposition to the rotated third-order tensor to obtain an objective function; solving the objective function to obtain a plurality of iteratively optimized intrinsic anchor graphs; fusing the plurality of iteratively optimized intrinsic anchor graphs to obtain a similar anchor graph; applying tensor singular value decomposition to the similar anchor graph to obtain a clustering result. The present invention captures higher-quality anchor points, explores the high-order relationships between different views, and improves the accuracy of the data cluster division result.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-view subspace clustering, and particularly to a multi-view subspace clustering method, apparatus, device and storage medium. Background Art

[0002] In the current era of increasingly developed information technology, data collection is multi-faceted. Data can be obtained from different sources and represented in different forms. Such data is called multi-view data. Therefore, exploring how to efficiently integrate valuable information in different views is a crucial challenge. For this reason, multi-view clustering that can aggregate the diversity and complementarity between views has become a very popular topic in the fields of machine learning and data mining. Among them, multi-view subspace clustering (MVSC) is a very popular unsupervised technique.

[0003] Based on the idea of self-representation theory, MVSC uses the data set as a dictionary to find a unified latent subspace and learns a similarity graph that can reflect the underlying structure of the views from the fused samples. For example, Chen et al. proposed using tensor singular value decomposition based on tensor nuclear norm and popularity regularization simultaneously to capture cross-view relationships and local structures respectively. Chen et al. integrated the latent embedding space, global structure and clustering metric matrix into a unified framework for learning. Kang et al. adopted an optimized fusion of discriminative-level information, starting from the idea of making the affinity graph more separable. Tang et al. proposed integrating adaptive graph learning and spectral clustering into an optimized structure and adding a late fusion mechanism to generate an optimal clustering partition.

[0004] Although existing multi-view subspace clustering algorithms have achieved remarkable success in clustering performance and applications, due to the high time complexity of most multi-view subspace clustering, it limits its efficiency in processing high-dimensional data and large-scale samples. Its high time complexity comes from the construction of the first graph and spectral clustering. In the process of constructing the first graph, the equation can be decomposed into n convex quadratic programming (QP) sub-problems, which requires time consumption. In the spectral clustering process, singular value decomposition (SVD) is used to cluster the similarity graph, which requires The time complexity. Therefore, studying scalable MVSC algorithms for real-world applications is an urgent problem to be solved. To address this issue, a multi-view clustering method based on matrix factorization is proposed, which aims to decompose the original data matrix into a basis matrix and a coefficient matrix with smaller dimensions. One such method is non-negative orthogonal factorization, which decomposes the high-dimensional data matrix into three low-dimensional matrices, making it more suitable for processing large-scale datasets. Cai et al. proposed an algorithm that can effectively integrate heterogeneous information in massive data. Nie et al. introduced the relaxed K-means technique and the fast multi-view matrix triple factorization technique, which can cluster the rows and columns of the input data matrix simultaneously.

[0005] In recent years, an emerging anchor-based MVSC method has been proposed. They perform independent sampling in the samples using K-means clustering or random sampling, select k basic anchor points, and then construct an anchor graph by learning the similarity between the base points and the samples. The size of the anchor graph is n×k, thus replacing the consensus graph of size n×n in the original clustering method. The anchor-based MVSC method can reduce the time complexity to It can play a great advantage in actual large-scale applications. For example, Li et al. used the local manifold fusion method and the bipartite graph to construct a fusion graph. What Kang et al. did was to combine the anchor point technique and the subspace clustering method. Wang et al. proposed to unify the selection of anchor points and the construction of the graph to explore the consensus anchor graph.

[0006] Although the above-mentioned anchor-based methods have achieved great achievements, there are still some problems that need to be further improved. Due to the complexity of the original data, there are a large number of noises and outliers in the original space, which will make the selection of anchor points unclean, so higher-quality anchor points cannot be captured. At the same time, these existing methods ignore the exploration of the high-order relationships between different views, so they cannot fully explore the internal structure of the data, thus greatly reducing the accuracy of the final data cluster division result. Summary of the Invention

[0007] Based on the defects of the above-mentioned existing technologies, the present invention provides a multi-view subspace clustering method, device, equipment and storage medium, which solves the problem that there are a large number of noises and outliers in the original space in the existing anchor-based methods, which will make the selection of anchor points unclean, so higher-quality anchor points cannot be captured. At the same time, these existing methods ignore the exploration of the high-order relationships between different views, so they cannot fully explore the internal structure of the data, thus greatly reducing the accuracy of the final data cluster division result.

[0008] The present invention adopts the following technical solutions:

[0009] First aspect, the present invention provides a multi-view subspace clustering method, including the following steps:

[0010] Based on multi-view data, obtain corresponding multiple feature mapping matrices, and map the multi-view data through the multiple feature mapping matrices to obtain multiple embedded anchor graphs;

[0011] Decompose the multiple embedded anchor graphs to obtain multiple intrinsic anchor graphs;

[0012] Stack the multiple intrinsic anchor graphs to obtain a third-order tensor with tensor nuclear norm constraint, rotate the third-order tensor, and apply the tensor nuclear norm constraint based on tensor singular value decomposition to the rotated third-order tensor to obtain an objective function;

[0013] Solve the objective function through an alternating strategy to obtain the iteratively optimized multiple intrinsic anchor graphs;

[0014] Fuse the iteratively optimized multiple intrinsic anchor graphs to obtain a similar anchor graph; apply tensor singular value decomposition to the similar anchor graph to obtain a clustering result.

[0015] Preferably, the mapping of the multi-view data through the multiple feature mapping matrices specifically includes:

[0016] Construct a mapping function based on the feature mapping matrix;

[0017] Map the multi-view data through the mapping function;

[0018] The mapping function is specifically as follows:

[0019]

[0020] In the formula, M t is the feature mapping matrix of the t-th view, Α t is the anchor matrix of the t-th view, Z t is the embedded anchor graph of the t-th view, α t is the trade-off factor of the t-th view, X t is the t-th view, is the Frobenius norm, V is the total number of views, I is the identity matrix, T is the transpose symbol, and 1 is a matrix of all 1s.

[0021] Preferably, the decomposition of the multiple embedded anchor graphs specifically includes:

[0022] Decompose the embedded anchor graph into an intrinsic anchor graph and a corresponding orthogonal matrix based on the decomposition formula;

[0023] The decomposition formula is specifically as follows:

[0024] Z t= W t N t ;

[0025] s.t. (W t ) T W t = I;

[0026] In the formula, W t is an orthogonal matrix, and N t is the learned intrinsic anchor graph.

[0027] Preferably, the objective function is specifically as follows:

[0028]

[0029] In the formula, λ1 and λ2 are balance parameters, is a third-order tensor, is the tensor nuclear norm, N 1 is the first intrinsic anchor graph, N 2 is the second intrinsic anchor graph, N t is the t-th intrinsic anchor graph, fold(·) represents stacking all the anchor graphs into a third-order tensor, and rotate(·) represents rotating the tensor.

[0030] Preferably, solving the objective function by an alternating strategy specifically includes the following steps:

[0031] Introduce auxiliary variables to rewrite the objective function;

[0032] Iterate the rewritten objective function until the objective function converges. Among them, in each iteration, when updating any variable in the rewritten objective function, fix the remaining variables and update it through the optimization method corresponding to each variable.

[0033] Preferably, fuse the multiple intrinsic anchor graphs after iterative optimization to obtain a similar anchor graph, where the similar anchor graph is specifically as follows:

[0034]

[0035] Among them, is the similar anchor graph.

[0036] Preferably, apply tensor singular value decomposition to the similar anchor graph to obtain a clustering result, where the clustering result is specifically as follows:

[0037]

[0038] In the formula, L is the left singular value component of the similar anchor graph, M Tis the transpose expression of the right singular value component of the similar anchor graph, and Σ is a diagonal matrix.

[0039] In a second aspect, the present invention provides a multi-view subspace clustering device, including:

[0040] An embedding module, configured to obtain corresponding multiple feature mapping matrices based on multi-view data, map the multi-view data through the multiple feature mapping matrices to obtain multiple embedded anchor graphs;

[0041] A decomposition module, configured to decompose the multiple embedded anchor graphs to obtain multiple intrinsic anchor graphs;

[0042] A stacking module, configured to stack the multiple intrinsic anchor graphs to obtain a third-order tensor with a tensor nuclear norm constraint, rotate the third-order tensor, and apply a tensor nuclear norm constraint based on tensor singular value decomposition to the rotated third-order tensor to obtain an objective function;

[0043] A solving module, configured to solve the objective function through an alternating strategy to obtain multiple iteratively optimized intrinsic anchor graphs;

[0044] A clustering module, configured to fuse the multiple iteratively optimized intrinsic anchor graphs to obtain a similar anchor graph; apply tensor singular value decomposition to the similar anchor graph to obtain a clustering result.

[0045] In a third aspect, the present invention provides a computer device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, the above multi-view subspace clustering method is implemented.

[0046] In a fourth aspect, the present invention provides a computer-readable storage medium, where the storage medium stores a computer program, and when the computer program is executed by a processor, the above multi-view subspace clustering method is implemented.

[0047] Compared with the prior art, at least one of the above technical solutions adopted by the present invention can achieve the following beneficial effects:

[0048] The present invention first introduces a feature mapping matrix to map multi-view data, adaptively completes the learning of anchor points and the construction of embedded anchor graphs, and obtains a plurality of embedded anchor graphs. The capture of anchor points is transferred from the original space with a large amount of noise and outliers in the traditional method to a clean feature transfer space, so that the learned anchor points and embedded anchor graphs have higher quality. Then, the plurality of embedded anchor graphs are decomposed to obtain a plurality of intrinsic anchor graphs. The present invention stacks the plurality of intrinsic anchor graphs into a third-order tensor and applies a tensor nuclear norm to obtain an objective function, which can fully reflect the high-order relationship between views. Finally, by solving the objective function, similar anchor graphs are obtained based on the solution results; tensor singular value decomposition is applied to the similar anchor graphs to obtain a clustering result. By capturing higher-quality anchor points and exploring the high-order relationship between different views, the present invention improves the accuracy of the data cluster division result. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required in the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0050] Figure 1 It is a flowchart of a multi-view subspace clustering method of the present invention;

[0051] Figure 2 It is a schematic diagram of the running time of all algorithms in the embodiments of the present invention on seven datasets;

[0052] Figure 3 It is a schematic diagram of the analysis of the number of anchor points to be selected in the embodiments of the present invention;

[0053] Figure 4 It is a schematic diagram of setting the number of selected anchor points to 20 and adjusting two balance parameters in the objective function in the Caltech101-all dataset in the embodiments of the present invention;

[0054] Among them, Figure 4 (a) of it: a schematic diagram of the accurate value of the clustering evaluation index, Figure 4 (b) of it: a schematic diagram of the normalized mutual information of the clustering evaluation index;

[0055] Figure 5 It is a schematic diagram of setting the number of selected anchor points to 62 and adjusting two balance parameters in the objective function in the NUSWIDEOBJ dataset in the embodiments of the present invention;

[0056] Among them, Figure 5 (a) of it: a schematic diagram of the accurate value of the clustering evaluation index,Figure 4 (b): Schematic diagram of the normalized mutual information of the clustering evaluation index;

[0057] Figure 6 Schematic diagram for experimentally verifying the convergence of the embodiments of the present invention on 6 data sets in the embodiments of the present invention;

[0058] Among them, Figure 6 (a): Convergence is achieved within 10 iterations on the Caltech101-7 data set, Figure 6 (b): Convergence is achieved within 10 iterations on the Caltech101-20 data set, Figure 6 (c): Convergence is achieved within 10 iterations on the Caltech101-all data set, Figure 6 (d): Convergence is achieved within 10 iterations on the NUSWIDEOBJ data set, Figure 6 (e): Convergence is achieved within 10 iterations on the SUNRGBD data set, Figure 6 (f): Convergence is achieved within 10 iterations on the AwA data set. Detailed implementation manners

[0059] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0060] As one of the most successful extensions of multi-view clustering, subspace clustering is an effective way to cluster high-dimensional data. It generally adopts a self-representation strategy, assuming that each data point is a linear combination of other data points. Based on this view, given a set of multi-view data There are n samples and d view features in the t-th view. The basic framework of multi-view subspace clustering is represented as follows:

[0061]

[0062] Where λ1, λ2 > 0 are balance parameters, Z t is the self-representation coefficient matrix of the t-th view, S is the affinity matrix learned about all views, Ω(·) is a regularization term that imposes different constraints on Z, θ(·) is the graph fusion process, jointly learning the self-representation coefficient matrix and the affinity matrix, and finally performing spectral clustering on the learned consensus graph.

[0063] Based on this strategy, a large number of multi-view subspace clustering models have been derived. Although these methods have achieved ideal results in terms of clustering performance, no matter how the fusion mechanism is improved, an n*n graph construction process needs to be executed, so a large amount of computing time is required. And spatial storage

[0064] I. Explanation of the embodiment. In order to enable those skilled in the art to fully understand how the present invention is specifically implemented, this part is an explanatory embodiment that expands on the technical solution of the claims.

[0065] Figure 1 It is a schematic flowchart of a multi-view subspace clustering method provided by an embodiment of the present invention. That is, the embedded anchor coupled low-rank tensor learning (ALTMSC) for in-multi-view subspace clustering. Specifically, ALTMSC introduces a mapping matrix to complete the feature transfer of the original data, and performs anchor point selection and embedded anchor graph learning in this low-dimensional latent space. After eliminating noise and outliers, the present invention can obtain a higher-quality anchor graph. Then, the present invention uses a rank-preserving decomposition technique to learn the intrinsic anchor graph from the initial embedded anchor graph, which can not only not destroy the intrinsic clustering structure of the view, but also remove view-specific information, thereby promoting global consistency. The present invention assembles the learned intrinsic anchor graphs into a third-order tensor for optimization. Applying the tensor nuclear norm (TNN) constraint based on tensor singular value decomposition (t-SVD) to the target tensor can fully capture the high-order relationships between views and enhance cross-view Figure 1 consistency, thereby restoring the global low rank of the embedded anchor graph. The following will be combined with Figure 1 Detailed introduction to the method provided by the embodiment of the present invention, which specifically includes the following steps:

[0066] S1: Introduce a projection matrix to map the multi-view data to the embedded space to complete the feature transfer, and adaptively complete the learning of anchor points and the construction of the embedded anchor graph in the embedded space.

[0067] Current multi-view subspace clustering generally adopts the framework of self-representation learning, representing each sample in the self-representation space with data samples in the original space, and has been widely used. Although the subspace clustering method can well capture heterogeneous information and explore the global structure, because the subsequent process is related to the global graph, it requires high time consumption and spatial storage, thus limiting its performance in real big data applications. In addition, a small number of data points can be used to describe a view, and it is relatively redundant and unnecessary to use all samples to reconstruct the latent subspace. Therefore, the present invention introduces an anchor strategy, using a small group of samples as anchor points or landmarks to reconstruct the underlying subspace and capture the manifold structure.

[0068] Large-scale application data from the real world often contains noise and incorrect values. Therefore, existing anchor-based MVSC will have problems due to errors and heterogeneous information in the original data. To solve the above problems, the present invention introduces a projection matrix M t , projects the original data into the embedding space, and imposes an orthogonality constraint on the feature space, which can not only complete the feature transfer of the data but also obtain clean data. This method transfers the adaptive learning of anchor points from the original space to the latent space, reducing the influence of noise and error information. Based on this view, the formula proposed by the present invention is as follows:

[0069]

[0070] where, the t-th view in the dataset, d t and n represent the view dimension and the number of samples respectively. is the feature mapping matrix of the t-th view, d is the dimension of the latent space, and an orthogonality constraint is imposed on it so that (M t ) T M t = 1. is defined as an anchor matrix with d latent space dimensions and k anchor points. The number of anchor points used in the present invention is m ∈ {k, 2k, 3k}, which is the same as the number of clusters and can improve the time efficiency. An orthogonality constraint (A t ) T A t = I is imposed on the embedded anchor matrix, which can make the learned anchors contain less redundant information, so that Α t can exhibit more distinguishable properties. is the embedded anchor graph learned from multiple views in the feature transfer space. According to the self-representation learning idea of the multi-view subspace, the original data matrix is replaced with an anchor matrix with fewer sample points . Based on this, a new embedded anchor graph can be learned. α t is a trade-off factor that adaptively selects the importance of different views.

[0071] S2: Use the rank-preserving decomposition technique to decompose the embedded anchor graph to obtain multiple intrinsic anchor graphs.

[0072] Different views have different statistical characteristics, and this view-specific information may even be contradictory. If the original embedded anchor graph is directly used to construct the target tensor and the constraint of the tensor nuclear norm is imposed, this will greatly increase the difficulty of forming global consistency. Therefore, before constructing the target tensor, without destroying the inherent clustering structure of the data, the view-specific information of each view is partitioned, so as to achieve the removal of view-specific information. To achieve this goal, the present invention adopts the rank-preserving decomposition technique to decompose the embedded anchor graph Z t , and the proposed formula is as follows:

[0073] Z t =W t N t ;

[0074] s.t.(W t ) T W t =I;

[0075] Among them, is a mapping matrix, is the learned intrinsic anchor graph. Since W t is an orthogonal matrix, the rank of Z t is equal to the rank of N t , so N t retains the clustering property and W t contains the view-specific information. Then, use to replace to construct the target tensor

[0076] S3: Stack multiple intrinsic anchor graphs to obtain a third-order tensor with a tensor nuclear norm constraint.

[0077] In the existing methods, although anchor point learning and graph construction have been integrated into a unified framework, due to the insufficient exploration of high-order relationships, a more discriminative anchor graph cannot be obtained.

[0078] A third-order tensor is represented as The present invention uses Matlab symbols to define the elements in the tensor . Specifically, and respectively represent the horizontal, lateral, and frontal slices of the third-order tensor . and respectively represent column, row, and tube fibers, and are also respectively called mode-1, mode-2, and mode-3 fibers. For the third-order tensor represents the fast Fourier transform (FFT) along mode-3. Similarly, the present invention can use the inverse fast Fourier transform (IFFT) to obtain That is The present invention gives the following several block operators related to t-SVD.

[0079] Block circulant matrix:

[0080] Block diagonal matrix:

[0081] Block vectorization: Its inverse operation is

[0082] Identity tensor The first front slice of satisfies the identity matrix of n1×n1, and all other front slices are zero matrices.

[0083] Orthogonal tensor can be expressed as

[0084] To further understand the tensor singular value decomposition (t-SVD), the present invention introduces the following tensor nuclear norm (TNN) and other definitions.

[0085] Definition 1 (Tensor product) Given tensors and tensor Their product is defined as That is:

[0086]

[0087] Definition 2 (f-diagonal tensor) If each frontal slice of a tensor is a diagonal matrix, then the tensor is called an f-diagonal tensor.

[0088] Definition 3 (Tensor transpose) By transposing each frontal slice of tensor the transposed tensor

[0089] Definition 4 (t-SVD) The t-SVD of a third-order tensor can be expressed as:

[0090]

[0091] where and are orthogonal tensors, represents an f-diagonal tensor of size n1×n2×n3, and * represents the tensor product.

[0092] Definition 5 (Tensor nuclear norm) Given tensor The tensor nuclear norm based on t-SVD is defined as the sum of the singular values of all frontal slices of, that is:

[0093]

[0094] TNN is the tightest convex relaxation of the tensor multi-rank, which can provide accurate low-rank properties.

[0095] After learning multiple view-specific anchor graphs, the present invention successively forms T intrinsic anchor graphs into a tensor along the third dimension of the tensor It can more comprehensively reflect the complementary information between different views. At the same time, a rotation operation is performed on this tensor, and the rotated tensor is Its k-th frontal slice encompasses the global relationship between the k-th anchor point and n different samples from multiple views. Imposing the TNN constraint based on t-SVD on the rotated tensor can obtain accurate low-rank properties, so that the recovered anchor graph has global low-rankness.

[0096]

[0097] Among them, fold(·) represents stacking all anchor graphs into a third-order tensor, and rotate(·) rotates this tensor and transforms it into

[0098] S4: Solve the objective function through an alternating strategy, and iteratively optimize the variables in the objective function.

[0099] It is very difficult to directly solve the above formula because it is a non-conjoint convex coupling problem. The present invention designs an alternating iterative solution mechanism to effectively solve the optimization problem of the ALTMSC method. At the same time, an auxiliary variable is introduced Rewrite it as:

[0100]

[0101] Through the alternating strategy, the equation can be decomposed into the following steps.

[0102] Step 1: Update M t .

[0103] Fix other variables: A t , Z t , W t , N t , and α t , the optimization strategy for M t can be rewritten as:

[0104]

[0105] For M that is independent on each corresponding view tFor this invention, the F-norm can be extended by a tracking method, and the above equation can be rewritten as:

[0106]

[0107] Since the solution of the above formula only relates to M t Therefore, the above formula can be simplified by eliminating the terms unrelated to M t to:

[0108]

[0109] where To effectively optimize the above equation, the invention adopts singular value decomposition according to the fact that it is an orthogonal "Prox" problem. Assume that the SVD decomposition of B t is Then the optimal M can be obtained by calculating t .

[0110] Step 2: Update A t .

[0111] By fixing P t , Z t , W t , N t , and α t , the optimization process of A t can be described by the following sub-problem:

[0112]

[0113] In the same way as updating M t , the above equation is changed to the following expression:

[0114]

[0115] where In the same way, the optimal solution of the singular value matrix of is calculated by SVD decomposition, so as to obtain the optimal

[0116] Step 3: Update Z t .

[0117] Once the other variables are fixed, the optimization solution for Z t can be updated to:

[0118]

[0119] To more conveniently solve the above problem, the present invention transforms it into a more common Quadratic Programming (QP) problem:

[0120]

[0121] Among them, On this basis, by calculating the QP problem of each column of Z t to complete the optimization process of Z t .

[0122] Step 4: Update W t .

[0123] Fix other variables and update W t is transformed into:

[0124]

[0125] Similar to the way of updating M t , the above equation is changed to the following expression:

[0126]

[0127] Among them, D t = λ1Z t (N t ). Similarly, assume that the SVD decomposition of D T is t Then the optimal solution of W t t can be obtained by calculating .

[0128] Step 5: Update N t .

[0129] Fix other variables and update N t is transformed into:

[0130]

[0131] Similar to the way of updating M t , the above expression can be rewritten as the following form:

[0132]

[0133] Among them, E t = λ1W t Z t . Similarly, the optimal value of N t can be obtained through the SVD decomposition technology.

[0134] Step 6: Update

[0135] When other variables are fixed and updated It is transformed into:

[0136]

[0137] It can be seen that the above formula is a minimization problem of t-TNN. To solve this problem, the following Theorem 1 is used for effective solution.

[0138] Theorem 1: Given a third-order tensor and and a scalar τ > 0, the following problem:

[0139]

[0140] The global optimal solution can be obtained by using the tensor tubal-shrinkage operator:

[0141]

[0142] where the tensor is an f-diagonal tensor, and its diagonal elements in the Fourier domain are

[0143] Step 7: Update α t .

[0144] The optimization problem with respect to α t is:

[0145]

[0146] According to the Cauchy-Buniakowsky-Schwarz inequality, the present invention gives h t = ||W t X t - A t Z t || F , then there is Therefore, the optimization result of α t can be obtained by calculating , where

[0147] Step 8: Update μ, y.

[0148] The parameters and Lagrange multipliers can be updated in the following way:

[0149]

[0150] During each iteration, the convergence of the ALTMSC method is judged by the following formula:

[0151]

[0152] where ε = 10 -7 .

[0153] Finally, the present invention summarizes all the optimization steps of the proposed ALTMSC method in Algorithm 1.

[0154]

[0155] S5: Obtain the final similar anchor graph through the fusion technology, and apply tensor singular value decomposition to it to obtain the left singular value components of the similar anchor graph, thereby obtaining the final clustering result.

[0156] For the self-representation based multi-view subspace clustering algorithm, the final clustering goal is to fuse t self-representation coefficient matrices to obtain the final similarity graph, and then perform spectral clustering. However, since the size of the anchor graph is no longer n×n, spectral clustering cannot be used for it. If the traditional clustering strategy is followed to restore the anchor graph to an n×n graph G, and then the fusion graph is obtained using the following expression

[0157]

[0158] Performing spectral clustering on the fusion graph yields:[[]]

[0159]

[0160] According to Proposition 1, the present invention combines t intrinsic anchor graphs into and solves its k left singular value components to obtain the final clustering result.

[0161]

[0162] where is the final similar anchor graph obtained through the fusion technology. Since mt << n, using to replace can significantly reduce the time consumption.

[0163] Proposition 1: Given a set of affinity matrices S t , each of which can be expressed as Let Assume its singular value decomposition is where LL T = I and MM T= I. The following expressions can be obtained:

[0164]

[0165] Proof 1: It can be found in the above formula that inserting it into the equation can obtain the following derivation:

[0166]

[0167] Therefore,

[0168]

[0169] So, The left singular value component of is the same as the eigencomponent of

[0170] Generally speaking, the main contributions of the present invention are in the following three aspects:

[0171] 1. The learning of anchor points and the construction of an embedded anchor graph are adaptively completed in a clean embedding space, which can eliminate the damage of noise and error information to the anchor points, thereby obtaining a higher-quality anchor graph.

[0172] 2. Since different views have their own different statistical attributes, and even contradict each other, and these information greatly hinder the mining of global consistency, the present invention performs a rank-preserving decomposition technique on the initial embedded anchor graph to eliminate view-specific information in pursuit of global consistency.

[0173] 3. Stack the learned intrinsic anchor graphs into a third-order tensor and apply a tensor nuclear norm (t-TNN) constraint based on tensor singular value decomposition (t-SVD). This measure can effectively explore high-order relationships and restore the global low rank of the embedded anchor graph.

[0174] 4. An effective alternating optimization algorithm is designed to solve the optimization problem of the objective function and its provable convergence. A large number of experimental results on 7 datasets show that this algorithm is superior to existing advanced methods. Especially, it shows superiority and effectiveness on large-scale datasets.

[0175] II. Evidence of the relevant effects of the embodiments. The embodiments of the present invention have achieved some positive effects during the research and development or use process, and indeed have great advantages compared with the prior art. The following content is described in combination with the data and charts in the test process.

[0176] Embodiment 1:

[0177] The computational complexity of Algorithm 1 consists of four main parts. When updating M t it is necessary to perform operations on B tPerform the SVD operation, and the loss time complexity is It is also necessary to perform matrix multiplication to obtain the optimal M t For updating Α t , the process is the same as that of M t and also consumes and for the SVD process and matrix multiplication. The present invention converts the update of Z t into a QP problem, and this process consumes For updating W t and N t , the total consumption is Stack and rotate the anchor graph to obtain a tensor whose dimension is n×V×m. Therefore, updating the third-order tensor will use FFT, inverse FFT, and t-SVD operations in each iteration process. The first two consume SVD requires The present invention only needs to consume to update α t . Therefore, the total computational complexity for Algorithm 1 is Since d << n, s << n, m << n, V << n, the computational complexity of Algorithm 1 can be approximated as

[0178] In Table 1, the time complexity and the required space cost of the present invention and other comparison algorithms are listed. Most traditional multi-view clustering algorithms require a space cost of The space cost required by the latest anchor-based clustering algorithm and the present invention is reduced to

[0179] Table 1 Complexity analysis of the present invention and existing comparison algorithms

[0180]

[0181]

[0182] Example 2

[0183] The present invention conducts experiments on 7 publicly used benchmark datasets, compares it with 5 state-of-the-art multi-view clustering methods and 4 large-scale algorithms, and comprehensively evaluates the efficiency and clustering quality of the proposed ALTMSC method from aspects such as clustering performance, running efficiency, parameter sensitivity, and convergence.

[0184] Experimental evaluations of the proposed ALTMSC method were carried out on the 7 large-scale datasets shown in Table 2: Caltech101-7, Caltech101-20, Caltech101-all, Handwritten, SUNRGBD, NUSWIDEOBJ, and AwA.

[0185] Table 2 Information of large-scale datasets

[0186] Dataset Number of Views Number of Clusters Number of Samples Number of Features Caltech101-7 6 7 1474 48,40,254,1984,512,928 Handwritten 6 10 2000 216,76,64,6,240,47 Caltech101-20 6 20 2386 48,40,254,1984,512,928 Caltech101-all 6 102 9144 48,40,254,1984,512,928 SUNRGBD 2 45 10335 4096,4096 NUSWIDEOBJ 5 31 30000 65,226,145,74,129 AwA 6 50 30475 2688,2000,252,20000,2000,2000

[0187] The proposed method of the present invention was compared with the following multiple state-of-the-art multi-view clustering algorithms.

[0188] Multi-view Subspace Clustering (MVSC): It was first proposed to use the subspace learning strategy to complete multi-view clustering, and its effectiveness was verified compared with single-view clustering.

[0189] Latent Multi-view Subspace Clustering (LMSC): Based on multi-view features, it learns the latent representation and further completes data reconstruction.

[0190] Partition-level Multi-view Subspace Clustering (PMSC): It constructs a unified algorithm model, integrating three parts: graph learning of each view, generating basic partitions, and fusing consensus partitions.

[0191] Multi-view Clustering in Latent Embedding Space (MLES): It learns multiple samples in the latent embedding space and synchronously learns the global structure and clustering metric matrix.

[0192] Parameter-free Auto-weighted Multi-graph Learning (AMGL): It automatically assigns the best weight to each view on a parameter-free basis to obtain optimal performance.

[0193] Multi-view k-means Clustering on Big Data (RMKM): By jointly using the heterogeneous information of large-scale data, it can complete large-scale visual data clustering.

[0194] Large-scale Multi-view Subspace Clustering in Linear Time (LMVSC): Inspired by the anchor graph strategy, it constructs a smaller anchor graph through anchor points and realizes large-scale clustering with only linear time complexity.

[0195] Scalable Multi-view Subspace Clustering with Unified Anchor (SMVSC): It integrates the anchor point strategy and the reconstruction of the consensus graph into a jointly optimized model, enabling the two to promote each other, thus obtaining a clearer data structure.

[0196] Fast Parameter-free Multi-view Subspace Clustering with Consensus Anchor Guidance (FPMVS-CAG): It automatically learns the optimal anchor graph without generating any additional hyperparameters.

[0197] According to the parameters involved in the experiment, the present invention needs to adjust the parameters λ1 and λ2 to balance the anchor learning and the high-order similarity learning. According to experience, the present invention sets the adjustment range thereof at {10 -3 ,..., 10 3}. At the same time, it is necessary to set the Lagrangian penalty parameter μ, initialize it to 0.001, and set the maximum value to μ max = 10 12 . For fairness, other comparison methods complete the experiment according to the parameter settings in the original literature. Four commonly used different evaluation indicators are adopted to comprehensively measure the clustering results, namely accuracy (ACC), normalized mutual information (NMI), purity, and F-score.

[0198] The present invention compares the algorithm of the present invention with other 9 clustering algorithms on 7 data sets, and the experimental results are shown in Table 3-9.

[0199] Table 3 Experimental results of different algorithms on the Caltech101-7 data set

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[0202] Table 4 Experimental results of different algorithms on the Handwritten data set

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[0204] Table 5 Experimental results of different algorithms on the Caltech101-20 data set

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[0206] Table 6 Experimental results of different algorithms on the Caltech101-all data set

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[0209] Table 7 Experimental results of different algorithms on the SUNRGBD data set

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[0211] Table 8 Experimental results of different algorithms on the NUSWIDEOBJ data set

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[0213] Experimental Results of Different Algorithms on the AwA Dataset

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[0215] In Table 3-9, "-" indicates that the method has insufficient memory or time issues during operation. The optimal value of each evaluation metric is shown in bold, and the sub-optimal value is marked in blue. From Table 3-9, the present invention can draw the following conclusions:

[0216] 1. Compared with other baselines, the ALTMSC proposed by the present invention obtains better clustering results on most datasets. Especially in terms of NMI and Purity, the results of ALTMSC are optimal on 7 datasets. For example, on the image dataset Caltech101-all, the NMI achieved is 16.56% higher than that of the second-best SMVSC, and the Purity is also 6.41% higher than that of the second-best AMGL. Additionally, in terms of ACC, ALTMSC is only in the second-best position on two datasets, Caltech101-7 and Caltech101-all. Considering these results comprehensively, the ALTMSC of the present invention can exhibit good performance when dealing with multi-view clustering data.

[0217] 2. By analyzing Tables 3 - 9, the present invention can learn that traditional MVSC methods (such as MVSC, LMSC, PMSC, MCLES) cannot obtain results even after spending tens of thousands of time on large-scale datasets, so "-" is used to indicate insufficient memory or time issues. In contrast, anchor-based methods such as LMVSC, SMVSC, FPMVS-CAG, and the ALTMSC of the present invention have achieved better performance on large-scale datasets and can be better applied to large-scale and high-dimensional real-world scenarios.

[0218] 3. Compared with the methods (SMVSC, FPMVS-CAG) that introduce projection in anchor learning, the clustering performance of the method (LMVSC) that completes anchor learning in the original space is damaged by adverse information in the data, so the clustering effect is relatively low. In addition, the clustering indices of SMVSC and FPMVS-CAG are slightly worse, which may be because they only introduce a predefined mapping matrix to map the learned unified anchor to the original space, and this method cannot filter out noise and outliers in the original space well. Compared with these methods, the ALTMSC proposed by the present invention learns the embedded anchor graph in a clean embedding space, which can effectively eliminate noise and error information in the original data, thereby obtaining a cleaner and higher-quality anchor graph.

[0219] 4. Compared with other anchor-based methods, the method of the present invention achieves optimal or sub-optimal results in all clustering metrics. This is because the method of the present invention enhances the global consistency across views and completes the exploration of high-order relationships between views, and can restore the global low-rankness of the anchor graph. On the dataset AwA with a sample size exceeding 30,000, the method ALTMSC of the present invention is 0.9%, 9.3%, 2.2%, and 5.1% higher than FPMVS-CAG in terms of ACC, NMI, Purity, and F-score respectively.

[0220] To further reasonably evaluate the running efficiency of the proposed method, the present invention records the computing time of each algorithm and the method of the present invention on each real dataset, and uses a logarithmic form on the vertical axis to narrow the comparison gap. Analysis Figure 2 It can be concluded that compared with the traditional multi-view methods MVSC, LMSC, PMSC, MCLES, and AMGL, the method ALTMSC of the present invention consumes less time on the benchmark dataset, which proves that the method has good running efficiency. At the same time, through theoretical analysis and experimental verification, it is proved that ALTMSC is time-effective, can efficiently complete the processing process of large-scale multi-view data, and shows a running complexity comparable to the most advanced large-scale data processing methods. In addition, ALTMSC completes the feature transfer of data in the latent space, learns anchor points and anchor graphs in this space, and uses the tensor stacking of the anchor graph to further effectively learn the high-order relationships between views, significantly improving the clustering performance.

[0221] In the objective function proposed by the present invention, there are a total of three parameters to be adjusted: the number of anchor points m and two hyperparameters λ1 and λ2. Since a sufficient number of anchor points can effectively represent the distribution of all original data, in order to deeply explore the influence of the number of anchor points m on the clustering effect of ALTMSC, the present invention sets m to vary in m ∈ {k, 2k, 3k}, where k is the number of clusters. The influence of the change of the number of anchor points m on the clustering effect is as Figure 3 shown. It can be observed from the present invention that although different numbers of anchor points will bring different clustering performances, it does not mean that the more anchor points, the better the effect.

[0222] Next, taking the Caltech101-all and NUSWIDEOBJ datasets as examples, fix m = 20 for the Caltech101-all dataset and fix m = 62 for the NUSWIDEBJ dataset. Then, adjust the parameters λ1 and λ2 within the range of {10 -3 ,..., 10 3}. Figure 4 and Figure 5It shows that selecting appropriate parameters for different datasets will obtain the best clustering results. For example, for the Caltech10 - all dataset, when the parameter λ1 = 1000 and the parameter λ2 varies within the range of {10, 10 2 , 10 3}, the best value will be obtained. In addition, the performance of ALTMSC is relatively stable and easy to adjust within a suitable parameter range.

[0223] The objective function of the ALTMSC method proposed by the present invention is decomposed into six convex sub - problems, and a minimization method using ADMM is proposed to solve it, successfully solving the optimization problem. Starting from the theory, the algorithm of the present invention can converge to the local optimum. From the perspective of experimental verification, according to the convergence condition, the error in each iteration is defined in the present invention. Figure 6 It shows the convergence of the objective values of the algorithm of the present invention on six datasets: Caltech101 - 7, Caltech101 - 20, Caltech101 - all, NUSWIDEOBJ, SUNRGBD, and AwA. As Figure 6 shown, as the number of iterations increases, the error value monotonically decreases and can basically converge within less than 10 iterations. Both theory and experiment verify the convergence of ALTMSC proposed by the present invention.

[0224] In the present invention, a new method named Embedded Anchor - Coupled Low - Rank Tensor Learning for Multi - view Intra - subspace Clustering is proposed. This method first introduces a mapping matrix to project the data from the original space to the feature transfer space, and adaptively completes the learning of anchor points and the construction of the embedded anchor graph in the clean low - dimensional latent space. This makes the learned anchor points and anchor graph of higher quality. In addition, to strengthen the cross - view global consistency, the present invention learns multiple intrinsic anchor graphs through rank - preserving decomposition to weaken the adverse effects of view - specific statistical characteristics on consistency and promote global consistency. At the same time, the anchor graphs are stacked into a third - order tensor and the tensor nuclear norm is imposed, so that it can fully reflect the high - order relationships between views and restore the low - rank property of the embedded anchor graph. Since ALTMSC has fast running time, it can adapt to the realistic large - scale application data scenarios. Compared with the existing traditional and large - scale multi - view subspace clustering methods, a large number of experiments on multiple datasets confirm the superiority of the proposed ALTMSC method.

[0225] Based on the same concept, the present invention also provides a multi - view subspace clustering device, including an embedding module, a decomposition module, a stacking module, a solving module, and a clustering module.

[0226] The embedding module is used to obtain corresponding multiple feature mapping matrices based on multi - view data, and map the multi - view data through the multiple feature mapping matrices to obtain multiple embedded anchor graphs.

[0227] A decomposition module for decomposing a plurality of embedded anchor graphs to obtain a plurality of intrinsic anchor graphs.

[0228] A stacking module for stacking a plurality of intrinsic anchor graphs to obtain a third-order tensor with tensor nuclear norm constraint, rotating the third-order tensor, and applying a tensor nuclear norm constraint based on tensor singular value decomposition to the rotated third-order tensor to obtain an objective function.

[0229] A solving module for solving the objective function through an alternating strategy to obtain a plurality of iteratively optimized intrinsic anchor graphs.

[0230] A clustering module for fusing a plurality of iteratively optimized intrinsic anchor graphs to obtain similar anchor graphs; applying tensor singular value decomposition to the similar anchor graphs to obtain a clustering result.

[0231] The present invention also provides a computer device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, the above multi-view subspace clustering method is implemented.

[0232] The present invention also provides a computer-readable storage medium storing a computer program, and when the computer program is executed by a processor, the above multi-view subspace clustering method is implemented.

[0233] Although the preferred embodiments of the present invention have been described, those skilled in the art can make additional changes and modifications once they learn the basic creative concept. Therefore, the appended claims are intended to be construed to include the preferred embodiments as well as all changes and modifications falling within the scope of the present invention.

[0234] Obviously, those skilled in the art can make various changes and deformations to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and deformations of the present invention fall within the scope of the claims of the present invention and their equivalent technologies, the present invention also intends to include these changes and deformations.

Claims

1. A multi-view subspace clustering method, characterized in that It includes the following steps: Based on multi-view data, obtain corresponding multiple feature mapping matrices, and map the multi-view data through the multiple feature mapping matrices to obtain multiple embedded anchor graphs; Decompose the multiple embedded anchor graphs to obtain multiple intrinsic anchor graphs; Stack the multiple intrinsic anchor graphs to obtain a third-order tensor with tensor nuclear norm constraint, rotate the third-order tensor, and impose a tensor nuclear norm constraint based on tensor singular value decomposition on the rotated third-order tensor to obtain an objective function; Solve the objective function through an alternating strategy to obtain multiple iteratively optimized intrinsic anchor graphs; Fuse the multiple iteratively optimized intrinsic anchor graphs to obtain similar anchor graphs; impose tensor singular value decomposition on the similar anchor graphs to obtain a clustering result.

2. The multi-view subspace clustering method according to claim 1, wherein The mapping of the multi-view data through the multiple feature mapping matrices specifically includes: Construct a mapping function based on the feature mapping matrix; Map the multi-view data through the mapping function; The specific form of the mapping function is as follows: Where, M t is the feature mapping matrix of the t-th view, Α t is the anchor matrix of the t-th view, Z t is the embedded anchor graph of the t-th view, α t is the trade-off factor of the t-th view, X t is the t-th view, is the Frobenius norm, V is the total number of views, I is the identity matrix, T is the transpose symbol, and 1 is the matrix of all 1s.

3. A multi-view subspace clustering method according to claim 2, characterized in that, The decomposition of the multiple embedded anchor graphs specifically includes: Decompose the embedded anchor graph into an intrinsic anchor graph and a corresponding orthogonal matrix based on a decomposition formula; The specific form of the decomposition formula is as follows: Z t = W t N t ; s.t.(W t ) T W t =I; where, W t is an orthogonal matrix, and N t is the learned intrinsic anchor graph.

4. The multi-view subspace clustering method according to claim 3, wherein The specific form of the objective function is as follows: s.t.(M t ) T M t =I; (A t ) T A t = I; (W t ) T W t = I; Z t ＞0； (Z t ) T 1＝1； where λ1 and λ2 are balance parameters, is a third-order tensor, is the tensor nuclear norm, N 1 is the first intrinsic anchor graph, N 2 is the second intrinsic anchor graph, N t is the t-th intrinsic anchor graph, fold(·) represents stacking all anchor graphs into a third-order tensor, and rotate(·) represents rotating the tensor.

5. A multi-view subspace clustering method according to claim 4, characterized in that, The solution of the objective function through the alternating strategy specifically includes the following steps: Introduce auxiliary variables to rewrite the objective function; Iterate the rewritten objective function until the objective function converges. Among them, in each iteration, when updating any variable in the rewritten objective function, fix the remaining variables and update them through the optimization method corresponding to each variable.

6. The multi-view subspace clustering method according to claim 5, wherein The fusion of the multiple iteratively optimized intrinsic anchor graphs to obtain similar anchor graphs, where the specific form of the similar anchor graph is as follows: Among them, is a similar anchor graph.

7. A multi-view subspace clustering method according to claim 6, characterized in that, The imposition of tensor singular value decomposition on the similar anchor graph to obtain a clustering result, where the specific form of the clustering result is as follows: where L is the left singular value component of the similar anchor graph, i.e., the clustering result, and M T is the transposed expression of the right singular value component of the similar anchor graph, and ∑ is a diagonal matrix.

8. A multi-view subspace clustering device, characterized in that It includes: An embedding module for obtaining corresponding multiple feature mapping matrices based on multi-view data, and mapping the multi-view data through the multiple feature mapping matrices to obtain multiple embedded anchor graphs; A decomposition module for decomposing the multiple embedded anchor graphs to obtain multiple intrinsic anchor graphs; A stacking module for stacking the multiple intrinsic anchor graphs to obtain a third-order tensor with tensor nuclear norm constraint, rotating the third-order tensor, and imposing a tensor nuclear norm constraint based on tensor singular value decomposition on the rotated third-order tensor to obtain an objective function; A solution module for solving the objective function through an alternating strategy to obtain multiple iteratively optimized intrinsic anchor graphs; A clustering module for fusing the multiple iteratively optimized intrinsic anchor graphs to obtain similar anchor graphs; Imposing tensor singular value decomposition on the similar anchor graph to obtain a clustering result.

9. A computer device, characterized in that, It includes a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, it implements the multi-view subspace clustering method according to any one of claims 1-7 above.

10. A computer-readable storage medium, characterized in that, The storage medium stores a computer program, and when the computer program is executed by the processor, it implements the multi-view subspace clustering method according to any one of claims 1-7 above.

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