A multi-view clustering method based on unified tensor constrained non-negative embedding and spectral embedding

Abstract

The application discloses a multi-view clustering method based on uniform tensor constraint non-negative embedding and spectral embedding, and belongs to the technical field of multi-view clustering, and comprises the following steps: S1, constructing a multi-view clustering model; S2, adjusting the multi-view clustering model by using an adaptive weighting coefficient; S3, adding constraint to the adjusted multi-view clustering model by using a third-order tensor, and obtaining a multi-view clustering model based on uniform tensor constraint non-negative embedding and spectral embedding; and S4, outputting a final clustering result according to the multi-view clustering model based on uniform tensor constraint non-negative embedding and spectral embedding. The application combines NMF and graph learning, and proposes a new multi-view clustering framework; the NMF and the graph learning are combined in a unified optimization framework, and consensus representation and view-specific complementary representation are successfully captured. In addition, low-rank tensor constraint effectively maintains high-order correlation and structural consistency between multiple views.

Description

Technical Field

[0001] This invention belongs to the field of multi-view clustering technology, specifically relating to a multi-view clustering method based on unified tensor constrained nonnegative embedding and spectral embedding. Background Technology

[0002] As one of the fundamental techniques in machine learning and data mining, clustering has achieved remarkable success in areas such as face clustering, community detection, and text mining. The goal of clustering is to divide a set of samples into distinct subgroups based on inherent similarity, such that samples within the same group have high similarity, while samples from different groups have low similarity. Among various clustering techniques, nonnegative matrix factorization (NMF) has attracted widespread attention due to its interpretability and ability to generate part-based representations. By decomposing a nonnegative data matrix into two nonnegative matrices, NMF naturally produces sparse, low-dimensional, and interpretable decompositions. Recent research has theoretically analyzed the relationship between NMF, spectral clustering, and K-means. Despite its widespread adoption, NMF has certain limitations, prompting the development of several variants. For example, semi-NMF allows for potentially wider applications due to its relaxed restrictions on data and factor matrices. Convex NMF can clearly interpret basis vectors, facilitating the utilization of the nonlinear structure of the data in the kernel space. Although these extensions effectively address some of the limitations of NMF, two major challenging tasks remain to be solved. First, while NMF methods have proven effective, they are limited to single-view clustering. In the real world, data is typically collected from multiple sources or feature extractors, forming multi-view data that provides complementary information. Second, most existing methods rely on post-processing to obtain discrete labels, thus introducing additional randomness and complexity.

[0003] Existing techniques include: 1. Nonnegative matrix factorization (NMF). NMF was initially developed as a dimensionality reduction technique. 2. Symmetric nonnegative matrix factorization (SymNMF). Symmetric nonnegative matrix factorization (SymNMF) is a typical variant of NMF.

[0004] These two techniques are mainly applied to single-view clustering tasks and are not suitable for multi-view clustering tasks, while in the real world, data is usually multi-view data composed of multiple sources.

[0005] The third existing technique is Nonnegative Embedding and Spectral Embedding (NESE). NMF and SymNMF are mainly designed for single-view clustering. This technique does not take into account higher-order structural relationships across views. Summary of the Invention

[0006] To address the above problems, this invention proposes a multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding.

[0007] The technical solution of this invention is: a multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding, comprising the following steps:

[0008] S1. Construct a multi-view clustering model;

[0009] S2. Adjust the multi-view clustering model using adaptive weighting coefficients;

[0010] S3. Use the third-order tensor to add constraints to the adjusted multi-view clustering model to obtain a multi-view clustering model based on unified tensor constraint non-negative embedding and spectral embedding.

[0011] S4. Output the final clustering results based on the multi-view clustering model based on unified tensor-constrained nonnegative embedding and spectral embedding.

[0012] Furthermore, in S1, the expression for the multi-view clustering model is:

[0013] ;

[0014] ;

[0015] in, Indicate each The corresponding nonnegative embedding, This represents a non-negative consistent embedding matrix shared across views. Indicate each The corresponding spectral embedding, Indicates the number of views. Indicates the first Similarity graph of views, Indicates matrix transpose. Indicates the first equilibrium parameter. Represents norm operations.

[0016] Furthermore, in S2, the expression for adjusting the multi-view clustering model using adaptive weighting coefficients is as follows:

[0017] ;

[0018] ;

[0019] in, Indicate each The corresponding nonnegative embedding, This represents a non-negative consistent embedding matrix shared across views. Indicate each The corresponding spectral embedding, Indicates the first The weight of each view, Indicates the number of views. Indicates the first Similarity graph of views, Indicates matrix transpose. Indicates the first equilibrium parameter. Represents norm operations.

[0020] Furthermore, in S3, the expression for the multi-view clustering model based on unified tensor-constrained nonnegative embedding and spectral embedding is:

[0021] ;

[0022] ;

[0023] in, Indicate each The corresponding nonnegative embedding, This represents a non-negative consistent embedding matrix shared across views. Indicate each The corresponding spectral embedding, Indicates the first The weight of each view, Indicates the number of views. Indicates the first Similarity graph of views, Indicates matrix transpose. Indicates the first equilibrium parameter. Represents the tensor nuclear norm. This represents the second equilibrium parameter. Represents a third-order tensor. This indicates that the input matrices are stacked in a given order to form a front slice of a tensor. This indicates the non-negative embedding of the first view. This indicates the non-negative embedding of the second view.

[0024] Furthermore, S4 includes the following sub-steps:

[0025] S41. Initialize parameters;

[0026] S42. Alternately update the initialized parameters;

[0027] S43. Determine whether the alternating update result has converged or reached the maximum number of iterations. If so, terminate the alternating update and proceed to S44. Otherwise, increment the iteration count by 1 and return to S42.

[0028] S44. Merge the non-negative embeddings corresponding to the similarity maps of each view after alternating updates and the non-negative consistent embedding matrix shared across views to obtain the fused non-negative embeddings.

[0029] S45. Based on the fused nonnegative embedding, output the final clustering result.

[0030] Furthermore, in S41, the initialization parameters include the first... Similarity graph of views Each Corresponding nonnegative embedding Non-negative consistent embedding matrix shared across views Each Corresponding spectral embedding Augmented Lagrange auxiliary variables , No. Weight of each view Augmented Lagrange multipliers And step size factor к.

[0031] Furthermore, in S45, the column index corresponding to the maximum value of each row of the fused non-negative embedding is used as the label to output the final clustering result.

[0032] The beneficial effects of this invention are as follows: This invention combines NMF with graph learning to propose a novel multi-view clustering framework; by combining NMF and graph learning within a unified optimization framework, it successfully captures consensus representations and view-specific complementary representations. Furthermore, low-rank tensor constraints effectively maintain high-order correlations and structural consistency among multiple views. Attached Figure Description

[0033] Figure 1 The flowchart shows a multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding.

[0034] Figure 2 A schematic diagram illustrating the t-SNE visualization of UTC-NESE on NGs and uci-digit datasets;

[0035] Figure 3 A schematic diagram comparing the clustering performance of UTC-NESE with 12 state-of-the-art baseline methods;

[0036] Figure 4This is a schematic diagram illustrating the parameter sensitivity of UTC-NESE on the BBC dataset.

[0037] Figure 5 Example plot of variable sequence convergence for UTC-NESE on NGs and scenes;

[0038] Figure 6 This is a schematic diagram comparing the performance of UTC-NESE with multiple ablation variants on four datasets. Detailed Implementation

[0039] The embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0040] like Figure 1 As shown, this invention provides a multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding, comprising the following steps:

[0041] S1. Construct a multi-view clustering model;

[0042] S2. Adjust the multi-view clustering model using adaptive weighting coefficients;

[0043] S3. Use the third-order tensor to add constraints to the adjusted multi-view clustering model to obtain a multi-view clustering model based on unified tensor constraint non-negative embedding and spectral embedding.

[0044] S4. Output the final clustering results based on the multi-view clustering model based on unified tensor-constrained nonnegative embedding and spectral embedding.

[0045] While classic methods such as MultiNMF and NESE achieve effective clustering results, they fail to fully utilize the complementary information between different views in multi-view data. Specifically, MultiNMF and NESE use the original data and a pre-constructed similarity map as inputs, respectively, focusing on constructing a unified consensus representation matrix while ignoring complementary information across views. To address this issue, this invention proposes a unified framework that leverages both consensus and complementary information from multi-view data.

[0046] This invention combines the advantages of NMF and graph learning methods, proposing a novel multi-view clustering framework that allows for the simultaneous learning of consensus and complementary representations of multi-view data within a unified optimization framework. The invention introduces a low-rank tensor constraint, where the tensor is formed by stacking consensus and view-specific complementary representations. This strategy helps capture high-order correlations across views and consensus, and enhances structural consistency. Extensive experiments were conducted on various benchmark datasets. Experimental results demonstrate the effectiveness of the invention.

[0047] In this embodiment of the invention, specifically in S1, It is a view-specific basis matrix with column orthogonality, ensuring that the embedding is represented by independent basis vectors. Meanwhile... It is a non-negative consistent embedding matrix shared across views, where each element This reflects the activation intensity of sample i in the j latent dimensions, and the nonnegativity constraint enhances the interpretability of the latent structure. To balance the contributions of consensus representation and view-specific representation, a balancing parameter is introduced, and the model is reformulated.

[0048] The expression for the multi-view clustering model is:

[0049] ;

[0050] ;

[0051] in, Indicate each The corresponding nonnegative embedding, This represents a non-negative consistent embedding matrix shared across views. Indicate each The corresponding spectral embedding, Indicates the number of views. Indicates the first Similarity graph of views, Indicates matrix transpose. Indicates the first equilibrium parameter. Represents norm operations.

[0052] In this embodiment of the invention, in S2, in order to balance the contributions of different views in learning the consensus representation, an adaptive weighting coefficient is introduced into the consensus term.

[0053] The expression for adjusting the multi-view clustering model using adaptive weighting coefficients is as follows:

[0054] ;

[0055] ;

[0056] in, Indicate each The corresponding nonnegative embedding, This represents a non-negative consistent embedding matrix shared across views. Indicate each The corresponding spectral embedding, Indicates the first The weight of each view, Indicates the number of views. Indicates the first Similarity graph of views, Indicates matrix transpose. Indicates the first equilibrium parameter. Represents norm operations.

[0057] In this embodiment of the invention, in S3, the invention merges them into a third-order tensor by stacking. Specifically, definition Then, regarding A low-rank constraint is imposed to preserve the underlying intrinsic structure of both representations. (By...) Low-order constraints are imposed to explicitly enforce that: (1) all view-specific and consensus representations share a potential low-dimensional subspace, reducing redundancy between embeddings; (2) nonnegative embeddings are jointly optimized through high-order tensor-low-order structures; and (3) the model automatically discovers indistinguishable cross-view structural patterns. This holistic constraint enhances the structural consistency between consensus and view-specific nonnegative embeddings, leading to more robust and interpretable representations. The above model is referred to as multi-view clustering based on unified tensor-constrained nonnegative embeddings and spectral embeddings.

[0058] The expression for the multi-view clustering model based on unified tensor-constrained nonnegative embedding and spectral embedding is:

[0059] ;

[0060] ;

[0061] in, Indicate each The corresponding nonnegative embedding, This represents a non-negative consistent embedding matrix shared across views. Indicate each The corresponding spectral embedding, Indicates the first The weight of each view, Indicates the number of views. Indicates the first Similarity graph of views, Indicates matrix transpose. Indicates the first equilibrium parameter. Represents the tensor nuclear norm. This represents the second equilibrium parameter. Represents a third-order tensor. This indicates that the input matrices are stacked in a given order to form a front slice of a tensor. This indicates the non-negative embedding of the first view. This indicates the non-negative embedding of the second view.

[0062] In this embodiment of the invention, S4 includes the following sub-steps:

[0063] S41. Initialize parameters;

[0064] S42. Alternately update the initialized parameters;

[0065] S43. Determine whether the alternating update result has converged or reached the maximum number of iterations. If so, terminate the alternating update and proceed to S44. Otherwise, increment the iteration count by 1 and return to S42.

[0066] S44. Merge the non-negative embeddings corresponding to the similarity maps of each view after alternating updates and the non-negative consistent embedding matrix shared across views to obtain the fused non-negative embeddings.

[0067] S45. Based on the fused nonnegative embedding, output the final clustering result.

[0068] In this embodiment of the invention, in S41, the initialized parameters include the first... Similarity graph of views Each Corresponding nonnegative embedding Non-negative consistent embedding matrix shared across views Each Corresponding spectral embedding Augmented Lagrange auxiliary variables , No. Weight of each view Augmented Lagrange multipliers And step size factor к.

[0069] In this embodiment of the invention, in step S45, the column index corresponding to the maximum value of each row of the fused non-negative embedding is used as a label, and the final clustering result is output.

[0070] In embodiments of the present invention, such as Figure 2 As shown, after the original view is visualized by t-SNE, the samples belonging to different clusters appear mixed and scattered. However, after clustering by the present invention, the samples belonging to different clusters can be clearly separated, proving the effectiveness of the present invention in revealing the potential cluster structure of multi-view data.

[0071] In embodiments of the present invention, such as Figure 3 As shown, this invention demonstrates a comparison of performance with 12 baseline methods on four clustering metrics: accuracy (ACC), normalized mutual information (NMI), purity, and F-score. The top three methods, ranked from best to worst, are marked in red, blue, and orange, respectively. It can be seen that this invention achieves the best performance on 8 different datasets, which confirms the superior clustering performance of this invention.

[0072] In embodiments of the present invention, such as Figure 4As shown, the present invention can achieve good clustering results on a wide range of parameters on the BBC dataset, demonstrating its robustness and practical value in real-world scenarios.

[0073] In embodiments of the present invention, such as Figure 5 As shown, this invention constructs an error sequence, the value of which at each iteration is defined as... Where t represents the number of iterations, The weight vector is represented by Error, which typically converges effectively within one hundred iterations on the NG and scene datasets, verifying the convergence and practical value of this invention.

[0074] In embodiments of the present invention, such as Figure 6 As shown, three variants of the invention are introduced, wherein variant 1 (Variant 1) ignores the consistency embedding. Variant 2 removes the learning of low-rank tensors, and Variant 3 removes the weights. The three variants all show a certain gap in clustering effect compared to the present invention, which strongly proves the effectiveness and necessity of the key components of the present invention.

[0075] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding, characterized in that, Includes the following steps: S1. Construct a multi-view clustering model; S2. Adjust the multi-view clustering model using adaptive weighting coefficients; S3. Use the third-order tensor to add constraints to the adjusted multi-view clustering model to obtain a multi-view clustering model based on unified tensor constraint non-negative embedding and spectral embedding. S4. Output the final clustering results based on the multi-view clustering model based on unified tensor-constrained nonnegative embedding and spectral embedding.

2. The multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding according to claim 1, characterized in that, In S1, the expression for the multi-view clustering model is: ； ； in, Indicate each The corresponding nonnegative embedding, This represents a non-negative consistent embedding matrix shared across views. Indicate each The corresponding spectral embedding, Indicates the number of views. Indicates the first Similarity graph of views, Indicates matrix transpose. Indicates the first equilibrium parameter. Represents norm operations.

3. The multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding according to claim 1, characterized in that, In S2, the expression for adjusting the multi-view clustering model using adaptive weighting coefficients is as follows: ； ； in, Indicate each The corresponding nonnegative embedding, This represents a non-negative consistent embedding matrix shared across views. Indicate each The corresponding spectral embedding, Indicates the first The weight of each view, Indicates the number of views. Indicates the first Similarity graph of views, Indicates matrix transpose. Indicates the first equilibrium parameter. Represents norm operations.

4. The multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding according to claim 1, characterized in that, In S3, the expression for the multi-view clustering model based on unified tensor-constrained nonnegative embedding and spectral embedding is: ； ； in, Indicate each The corresponding nonnegative embedding, This represents a non-negative consistent embedding matrix shared across views. Indicate each The corresponding spectral embedding, Indicates the first The weight of each view, Indicates the number of views. Indicates the first Similarity graph of views, Indicates matrix transpose. Indicates the first equilibrium parameter. Represents the tensor nuclear norm. This represents the second equilibrium parameter. Represents a third-order tensor. This indicates that the input matrices are stacked in a given order to form a front slice of a tensor. This indicates the non-negative embedding of the first view. This indicates the non-negative embedding of the second view.

5. The multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding according to claim 1, characterized in that, S4 includes the following sub-steps: S41. Initialize parameters; S42. Alternately update the initialized parameters; S43. Determine whether the alternating update result has converged or reached the maximum number of iterations. If so, terminate the alternating update and proceed to S44. Otherwise, increment the iteration count by 1 and return to S42. S44. Merge the non-negative embeddings corresponding to the similarity maps of each view after alternating updates and the non-negative consistent embedding matrix shared across views to obtain the fused non-negative embeddings. S45. Based on the fused nonnegative embedding, output the final clustering result.

6. The multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding according to claim 5, characterized in that, In step S41, the initialization parameters include the first... Similarity graph of views Each Corresponding nonnegative embedding Non-negative consistent embedding matrix shared across views Each Corresponding spectral embedding Augmented Lagrange auxiliary variables , No. Weight of each view Augmented Lagrange multipliers And step size factor к.

7. The multi-view clustering method based on unified tensor-constrained nonnegative embedding and spectral embedding according to claim 5, characterized in that, In step S45, the column index corresponding to the maximum value of each row of the fused non-negative embedding is used as the label, and the final clustering result is output.

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