Multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization
By jointly learning the collaborative regularization method of anchor points and anchor graphs, the problems of high time complexity and insufficient representativeness of anchor points in multi-view clustering are solved, and efficient and stable multi-view clustering results are achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-01
- Publication Date
- 2026-04-07
AI Technical Summary
Existing multi-view clustering methods suffer from high time complexity when processing large-scale data, and the separation of anchor point selection and anchor map construction processes leads to insufficient representativeness of anchor points, affecting clustering performance.
A multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization is proposed. By jointly learning consistent anchor points and anchor graphs, local smoothing terms and column sparsity constraints are introduced, and combined with bipartite graph Laplace rank constraints, the collaborative structure of anchor points and anchor graphs is optimized to form a two-level collaborative modeling framework.
It significantly improves clustering performance, reduces redundant connections, enhances the consistency of clustering structure between anchor points and samples, and achieves efficient clustering with low time and space complexity, making it suitable for large-scale data.
Smart Images
Figure CN121811083A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of multi-view clustering technology, and in particular to a multi-view clustering method based on anchor point to anchor graph structure co-regularization. Background Technology
[0002] With the advent of the big data era, the scale of multi-view data is becoming increasingly massive, and its internal structure is becoming increasingly complex. Multi-view data refers to a collection of data collected from multiple different angles, sources, or modalities to describe the same group of objects or entities, such as images described with different features. Multi-view clustering can leverage the consistency and complementarity information of multi-view data to divide samples into unrelated clusters, thereby uncovering the potential distribution within the data.
[0003] (1) Multi-view subspace clustering
[0004] In multi-view clustering, multi-view subspace clustering is widely used because it can map high-dimensional multi-view data to low-dimensional subspaces and integrate multi-view information. Multi-view subspace clustering methods assume that the view data is distributed across multiple low-dimensional subspaces, and that within the same subspace, a sample can be linearly represented by other samples. The method first learns the subspace representation matrix for each view, then merges these matrices to obtain a consistent representation matrix, then uses the representation matrices to construct a similarity matrix, and finally performs spectral clustering to obtain cluster labels. However, the final spectral clustering requires further analysis of the data. Singular value decomposition (SVD) is performed on the Laplace matrix, which costs [amount missing]. The time complexity of multi-view subspace clustering methods makes it difficult to handle large-scale data.
[0005] (2) Anchor-based multi-view clustering
[0006] To address the aforementioned high time complexity issue, anchor-based multi-view clustering methods have been widely proposed. These methods select a small number of representative sample points (anchor points) to represent the entire set of sample points, and construct a similarity matrix (anchor map) between the anchor points and the sample points to replace the representation matrix among all sample points. The number of anchor points selected is... Much smaller than the number of samples When performing clustering, only need to SVD is performed on the anchor map, and this computational step only takes [time / cost]. The time complexity is much smaller than that of multi-view subspace clustering methods. Time complexity.
[0007] Clustering performance is highly dependent on anchor quality; therefore, selecting representative anchors is the core challenge of anchor-based multi-view clustering methods. Based on anchor selection strategies, anchor-based multi-view clustering methods can be divided into heuristic and learning-based methods. Heuristic methods utilize prior knowledge or manual strategies such as k-means or random sampling to select anchors, then construct an anchor graph for clustering. For example, the Linear-Time Large-Scale Multi-View Subspace Clustering (LMVSC) method uses k-means to select anchors, then learns the anchor graph through data reconstruction, concatenates the anchor graphs from each view into a single anchor graph, and finally performs SVD on the concatenated anchor graph and k-means on the singular vectors to obtain the final clustering result. Because the anchor selection and anchor graph construction processes are separated, their mutual promotion is hindered, thus failing to produce truly representative anchors and anchor graphs.
[0008] Learning-based methods incorporate anchor selection and anchor graph construction into a unified framework for optimization. For example, the Fast, Parametric-Free Multi-View Subspace Clustering (FPMVS) method, guided by consistent anchors, learns consistent anchors and anchor graphs in the shared latent subspaces of all views; the One-Step Multi-View Subspace Clustering (EOMSC) method, based on consensus anchors, introduces a bipartite graph into the consensus anchor graph. Connectivity relationships can be established, and graph embeddings for clustering can be obtained. The Multi-View Subspace Clustering (MVSC-HFD) method based on hierarchical feature descent performs hierarchical dimensionality reduction clustering on the same subspace, effectively solving the problem of differences between multiple views.
[0009] While these methods have achieved considerable performance, they all focus on imposing various structural constraints on anchor graphs. They often neglect the optimization of the anchor points themselves, imposing only simple orthogonal constraints on them and ignoring the local structural information between anchor points. This is insufficient to ensure that anchor points fully capture the intrinsic feature distribution of the data and may disrupt its clustering structure. Furthermore, these methods are prone to structural mismatches and inconsistencies between anchor points and the graph, inevitably reducing clustering performance. Summary of the Invention
[0010] The present invention aims to provide a multi-view clustering method based on anchor point to anchor graph structure collaborative regularization to solve the problems mentioned in the background art.
[0011] To achieve the above objectives, the present invention provides the following technical solution:
[0012] A multi-view clustering method based on anchor point-to-anchor graph structure co-regularization includes the following steps:
[0013] S1. Establishment of the model objective function;
[0014] S2. Objective function optimization;
[0015] S3. Algorithm complexity analysis.
[0016] Furthermore, the specific steps for establishing the model objective function in S1 include:
[0017] Taking a multi-feature image as an example, the image data described by different features are: ,in The dimension representing each feature. Indicates the number of samples. Indicates the number of views;
[0018] Multi-view data is typically assumed to be embedded in a latent space, therefore the data for each view... It can be done through the projection matrix Mapped into a shared low-dimensional latent space, where Represents the potential spatial dimension;
[0019] Consistent anchor points and anchor graphs are jointly learned from the latent space:
[0020]
[0021] In the formula, Anchor point matrix, For anchor plots, This represents the number of anchor points.
[0022] Furthermore, the quality of anchor points has a crucial impact on the discriminative power of anchor graphs; therefore, learning highly representative anchor points is essential for improving overall clustering performance. Highly representative anchor points should semantically be consistent with the logical structure of the sample points; therefore, it can be assumed that anchor points are distributed in a structural space reflecting the underlying data topology.
[0023]
[0024] In the formula, Indicates the first Anchor points, Indicates the first The anchor point and the first The similarity between anchor points. The graph smoothing term in Equation (2) encourages anchor points from the same cluster to have similar representations in the feature space.
[0025] Furthermore, to enhance the discriminative power of anchor charts, column sparsity is introduced for the anchor charts. This encourages each sample point to connect to as few anchor points as possible, thereby reducing redundant connections between anchor points and sample points. Furthermore, excessive sparsity can lead to all sample points clustering on the same anchor point; to avoid such trivial solutions, the learned anchor graph is optimized. Apply the Laplace rank constraint to the bipartite graph, i.e. ,in Consensus Divide The normalized Laplace matrix, Let be the corresponding degree matrix, where and ;
[0026] The Laplace rank constraint on a bipartite graph can force the anchor graph to have ideal connectivity, that is, force the anchor points and sample points to be assigned to... Clusters enhance the clustering structure of anchor graphs.
[0027] Furthermore, the final objective function of the model is as follows:
[0028]
[0029] in, For a set of variables, and To balance the hyperparameters;
[0030] Equation (3) establishes a collaborative structure regularization of anchor points to graphs, which makes anchor point construction and anchor point graph learning mutually influential; the column sparsity and bipartite graph Laplace rank constraints on the anchor graph further enhance the structure of the anchor graph.
[0031] Furthermore, the objective function optimization implementation method in S2 is as follows:
[0032] Bipartite graph Lap rank constraint It is discrete and non-convex, making direct optimization difficult. Because... If it is positive semidefinite, then The smallest eigenvalue is 0, therefore the constraint It's a restriction The 0 eigenvalue multiplicity is According to Ky Fan's theorem:
[0033]
[0034] in, express The former Small eigenvalues, For the spectral embedding representation matrix, and Clustering indicator matrices for samples and anchors are respectively; minimize That is to let The former Small eigenvalues should be as small as possible, so equation (4) can be used to approximate the rank constraint. ;
[0035] The objective function (3) contains multiple variables and is jointly nonconvex, so it can be optimized using the alternating direction multiplier method (ADMM); to facilitate optimization, auxiliary variables are introduced. ,Will The 2,1 norm transition The augmented Lagrangian function that needs optimization is as follows:
[0036]
[0037] in, It is a balance parameter. For Lagrange multipliers, The penalty coefficient is non-negative.
[0038] In the above formula, each variable is easy to solve when other variables are fixed; therefore, all variables are optimized alternately in an iterative manner.
[0039] Furthermore, all variables are optimized iteratively in an alternating manner as follows:
[0040] 1) Update
[0041] Keep other variables fixed. The subproblems are as follows:
[0042]
[0043] Expanding the norm terms into trace terms and removing irrelevant terms, equation (6) can be equivalently represented as follows:
[0044]
[0045] in Equation (7) is an orthogonal Procrustes problem, which can be solved by... Solve using SVD, i.e. , The closed-form solution is .
[0046] 2) Update
[0047] By fixing other variables, we can obtain The subproblems are as follows:
[0048]
[0049] Question (8) can be solved by... Taking the derivative and setting it to zero, we finally obtain... The closed-form solution is:
[0050]
[0051] 3) Update
[0052] With other variables fixed, we get The subproblems are as follows:
[0053]
[0054] Simplifying it, we get:
[0055]
[0056] Considering ,therefore Then problem (11) is equivalent to the following:
[0057]
[0058] A well-known theorem: The following optimization problems:
[0059]
[0060] in, Given variables, and The variables to be solved. and The optimal solution is:
[0061]
[0062] in and They are respectively forward The left and right singular vectors corresponding to large singular values;
[0063] According to well-known theorems, The optimal solutions are respectively times The former The left and right singular vectors corresponding to the large singular values;
[0064] 4) Update
[0065] Keep other variables fixed. The subproblems are as follows:
[0066]
[0067] The optimization of problem (15) can first ignore the constraints. ,right Taking the derivative and setting it to zero, we obtain the unconstrained closed-form solution:
[0068]
[0069] in Then, by projecting the unconstrained closed-form solution onto the nonnegative normalized simplex, we can obtain the solution. The optimal solution;
[0070] 5) Update
[0071] Fix other variables and optimize variables The subproblems are as follows:
[0072]
[0073] Its solution can be obtained through The soft threshold operator is used to minimize the threshold column by column, i.e.:
[0074]
[0075] in , ;
[0076] 6) Update the Lagrange multipliers and penalty parameters
[0077]
[0078]
[0079] in These are parameters used to accelerate convergence. Given all initialized variables, each variable is updated alternately until the convergence requirement is met.
[0080] Furthermore, the algorithm complexity analysis specifically includes analyzing the time complexity and space complexity of the AGSCR-MVC algorithm respectively.
[0081] Furthermore, the time complexity includes: the algorithm's runtime is mainly consumed in updating various variables, for Its updates mainly involve matrix multiplication and SVD, with a time complexity of O(n log n). ,(in (representing the sum of all dimensions); for Its main expenses The time complexity is used for matrix multiplication and inversion; update It will cost The complexity of updating; need The complexity; for It will cost The complexity of performing a soft threshold operation on each column is reduced; considering the large-scale data... Therefore, the time complexity of the AGSCR-MVC algorithm is O(n). , with sample size The relationship is linear.
[0082] Furthermore, the space complexity includes: the algorithm needs to store all variables: Therefore, the space complexity of the algorithm is O(n). It is also linear with the number of samples.
[0083] Compared with the prior art, the present invention has the following beneficial effects:
[0084] This invention proposes a multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization (AGSCR-MVC); this method combines anchor point construction and anchor graph learning into a unified framework:
[0085] First, implicit anchor adjacency relationships are constructed from the anchor graph, and local smoothing terms are introduced on the anchors to guide anchor learning. Second, column sparsity is enforced on the anchor graph, encouraging each sample to connect to only a few anchors, thereby reducing redundant connections. Finally, Laplace rank constraints are applied to the anchor graph to improve its clustering structure. AGSCR-MVC uses the anchor graph as a bridge representation, simultaneously guiding the local structure learning and structure regularization of anchors, to achieve a two-level collaborative modeling framework between the anchor space and the graph structure. This enhances the consistency of the clustering structure between anchors and samples, significantly improving clustering performance. Attached Figure Description
[0086] Figure 1 A flowchart for multi-view clustering using the AGSCR-MVC method;
[0087] Figure 2 This is a graph showing the comparison of runtime results. Detailed Implementation
[0088] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments:
[0089] like Figure 1 As shown, this invention provides a multi-view clustering method (AGSCR-MVC) based on anchor point-to-anchor graph structure collaborative regularization, which includes the following steps:
[0090] 1. Establishment of the model objective function
[0091] Taking a multi-feature image as an example, the image data described by different features are: ,in The dimension representing each feature. Indicates the number of samples. This represents the number of views. Multi-view data is typically assumed to be embedded in a latent space, therefore the data for each view... It can be done through the projection matrix Mapped into a shared low-dimensional latent space ( (Representing the latent space dimension). Consistent anchor points and anchor maps are jointly learned from the latent space:
[0092]
[0093] In the formula, Anchor point matrix, For anchor plots, Number of anchor points;
[0094] The quality of anchor points has a crucial impact on the discriminative power of anchor graphs; therefore, learning highly representative anchor points is essential for improving overall clustering performance. Highly representative anchor points should semantically be consistent with the logical structure of the sample points; therefore, it can be assumed that anchor points are distributed in a structural space that reflects the underlying data topology.
[0095]
[0096] In the formula, Indicates the first Anchor points, Indicates the first The anchor point and the first The similarity between anchor points. The graph smoothing term in Equation (2) encourages anchor points from the same cluster to have similar representations in the feature space.
[0097] In typical anchor point multi-view clustering, the following method is usually used. To construct a similarity matrix between sample points: if two samples have high association weights at the same anchor point, then the two samples have a high similarity. Similarly, the prior anchor point similarity matrix... You can also do it through To construct this: if the weight distributions of two anchor points across multiple samples are highly similar, it means there is a strong correlation between the two anchor points. The Laplace matrix of the anchor points. ,in For the corresponding degree matrix, the values on its diagonal are: This approach utilizes anchor graphs as a bridge to achieve implicit semantic alignment between the clustering structure of the anchor points and the clustering structure represented by the samples reflected in the anchor graph. Furthermore, it establishes a collaborative optimization between anchor points and anchor graphs: on the one hand, anchor points are guided by the structural information encoded in the anchor graph; on the other hand, the anchor graph is adaptively refined based on the updated anchor points, achieving mutual promotion and enhanced clustering performance.
[0098] To further enhance the discriminative power of anchor graphs, this invention introduces column sparsity in anchor graphs. This encourages each sample point to connect to as few anchor points as possible, thereby reducing redundant connections between anchor points and sample points. Furthermore, excessive sparsity can lead to all sample points clustering on the same anchor point. To avoid such trivial solutions, this invention optimizes the learned anchor graph. Apply the Laplace rank constraint to the bipartite graph, i.e. ,in For consensus bipartite graph The normalized Laplace matrix, Let be the corresponding degree matrix, where and The Laplacian rank constraint on a bipartite graph can force the anchor graph to have ideal connectivity, i.e., force the anchor points and sample points to be assigned to... Clusters enhance the clustering structure of anchor graphs.
[0099] In summary, the final objective function of the model is as follows:
[0100]
[0101] in For a set of variables, and To balance the hyperparameters, Equation (3) establishes a collaborative structure regularization for anchor points to the graph, which allows anchor point construction and anchor point graph learning to influence each other. This not only improves the semantic structure of the anchor points themselves but also ensures that anchor points and samples remain consistent within the inherent clustering structure. Furthermore, column sparsity and bipartite graph Laplacian rank constraints on the anchor graph further enhance its structure.
[0102] 2. Objective function optimization
[0103] Bipartite graph Lap rank constraint It is discrete and non-convex, making direct optimization difficult. Because... If it is positive semidefinite, then The smallest eigenvalue is 0, therefore the constraint It's a restriction The 0 eigenvalue multiplicity is According to Ky Fan's theorem:
[0104]
[0105] in express The former Small eigenvalues, For the spectral embedding representation matrix, and These are the clustering indicator matrices for samples and anchor points, respectively. Minimize That is to let The former Small eigenvalues should be as small as possible (close to 0), so equation (4) can be used to approximate the rank constraint. .
[0106] The objective function (3) contains multiple variables and is jointly nonconvex, so it can be optimized using the alternating direction multiplier method (ADMM). To facilitate optimization, auxiliary variables are introduced. ,Will The 2,1 norm transition The augmented Lagrangian function that needs optimization is as follows:
[0107]
[0108] in, It is a balance parameter. For Lagrange multipliers, is the non-negative penalty coefficient. In the above formula, each variable is easily solved when other variables are fixed. Therefore, all variables are optimized alternately in an iterative manner.
[0109] 1) Update
[0110] Keep other variables fixed. The subproblems are as follows:
[0111]
[0112] Expanding the norm terms into trace terms and removing irrelevant terms, equation (6) can be equivalently represented as follows:
[0113]
[0114] in Equation (7) is an orthogonal Procrustes problem, which can be solved by... Solve using SVD, i.e. , The closed-form solution is .
[0115] 2) Update
[0116] By fixing other variables, we can obtain The subproblems are as follows:
[0117]
[0118] Question (8) can be solved by... Taking the derivative and setting it to zero, we finally obtain... The closed-form solution is:
[0119]
[0120] 3) Update
[0121] With other variables fixed, we get The subproblems are as follows:
[0122]
[0123] Simplifying it, we get:
[0124]
[0125] Considering ,therefore Then problem (11) is equivalent to the following:
[0126]
[0127] To solve the problem of equation (12), the following well-known theorem in the field is introduced:
[0128] A well-known theorem: The following optimization problems:
[0129]
[0130] in, Given variables, and The variables to be solved. and The optimal solution is:
[0131]
[0132] in and They are respectively forward Large singular values correspond to left and right singular vectors.
[0133] According to well-known theorems, The optimal solutions are respectively times The former The left and right singular vectors corresponding to the large singular values.
[0134] 4) Update
[0135] Keep other variables fixed. The subproblems are as follows:
[0136]
[0137] The optimization of problem (15) can first ignore the constraints. ,right Taking the derivative and setting it to zero, we obtain the unconstrained closed-form solution:
[0138]
[0139] in Then, by projecting the unconstrained closed-form solution onto the nonnegative normalized simplex, we can obtain the solution. The optimal solution.
[0140] 5) Update
[0141] Fix other variables and optimize variables The subproblems are as follows:
[0142]
[0143] Its solution can be obtained through The soft threshold operator is used to minimize the threshold column by column, i.e.:
[0144]
[0145] in , .
[0146] 6) Update the Lagrange multipliers and penalty parameters
[0147]
[0148]
[0149] in These are parameters used to accelerate convergence. Given all initialized variables, each variable is updated alternately until the convergence requirement is met.
[0150] 3. Algorithm Complexity Analysis
[0151] To illustrate the efficiency of the AGSCR-MVC method in processing large-scale data, the time and space complexity of the AGSCR-MVC algorithm are analyzed here:
[0152] Time complexity: The algorithm's runtime is primarily consumed by updating various variables. Its updates mainly involve matrix multiplication and SVD, with a time complexity of O(n log n). ,(in (representing the sum of all dimensions); for Its main expenses The time complexity is used for matrix multiplication and inversion; update It will cost The complexity of updating; need The complexity; for It will cost The complexity is used to perform a soft thresholding operation on each column. This is especially important considering the large-scale data... Therefore, the time complexity of the AGSCR-MVC algorithm is O(n). , with sample size The relationship is linear.
[0153] Space complexity: The algorithm needs to store all variables. Therefore, the space complexity of the algorithm is O(n). It is also linear with the number of samples.
[0154] In summary, the AGSCR-MVC algorithm's time and space complexity are both linear with the number of samples, thus effectively avoiding the computational and storage bottlenecks that general methods encounter on large-scale datasets, thereby ensuring the algorithm's efficient operation in massive data environments. This invention not only improves clustering performance but also features low time overhead and low space consumption, demonstrating good scalability and engineering application value.
[0155] The specific implementation process is as follows:
[0156] To evaluate the effectiveness of the proposed AGSCR-MVC, we compare its performance and runtime with currently popular multi-view clustering methods. The experimental setup and results are described in detail below.
[0157] 1. Dataset Description
[0158] To provide a more comprehensive evaluation, we selected several popular multi-view datasets, ranging from small to large sample sizes, covering text, images, and videos, to ensure that the proposed solution demonstrates stability and generalization performance across different modalities and data scales. The datasets include: BBC (text), BDGP (images), CCV (videos), Caltech101 (images), AWA (images), and MNIST (images). Detailed information about the datasets is shown in Table 1.
[0159] Table 1: Dataset Details
[0160]
[0161] 2. Comparison Methods
[0162] The experiment selected popular large-scale multi-view clustering methods from recent years as the baseline methods, including:
[0163] LMVSC; Scalable Multi-View Subspace Clustering Method with Unified Anchors (SMVSC): Learns consistent anchors and anchor points in the latent space, while using anchor graph norm squared regularization; One-pass Large-Scale Multi-View Clustering Method (OPMC): Directly obtains cluster labels based on matrix factorization; Efficient Orthogonal Multi-View Subspace Clustering Method (OMSC): Learns consistent anchors and anchor points in the latent space, while using anchor graphs for matrix factorization to obtain cluster labels; Scalable Parametric Bipartite Graph Fusion Multi-View Clustering Method (SFMC): Uses heuristics to select anchor points and construct anchor graphs, then automatically weights and fuses the anchor graphs, also incorporating bipartite graph Laplace rank constraints to constrain the clustering structure; FPMVS; EOMSC;
[0164] Automatically Weighted Large-Scale Multi-View Clustering (AWMVC): Learns coefficient matrices in the corresponding foundation matrices and then fuses them to obtain the optimal consensus matrix; Scalable and Flexible Multi-View Clustering with Anchor Graph Fusion (FDAGF): Learns anchor lattices of different scales and automatically weights them while fusing anchor graphs of different sizes; MVSC-HFD; Structured Anchor Graph Learning Multi-View Clustering (SAGL): Learns consistent latent anchors using distance and also incorporates bipartite graph Laplacian rank constraints.
[0165] 3. Experimental Setup
[0166] For the baseline method, we downloaded the source code from its official code repository and tuned the parameters according to the settings in its paper to obtain optimal performance. For the proposed AGSCR-MVC method, the latent space dimension... Set as , Fixed at 1, and exist Adjust within the range, from Choose from.
[0167] Four commonly used clustering metrics were selected for clustering performance: Accuracy (ACC), Normalized Mutual Information (NMI), Purity, and F-score. Higher metrics indicate better performance. All experiments were conducted in MATLAB R2024b (64-bit) on a Windows 11 computer equipped with an Intel i7-12800HX processor and 32 GB of memory.
[0168] 4. Performance Comparison Experiment
[0169] The results of the performance comparison experiments are shown in Table 2, with the best performance highlighted in bold and the second-best performance underlined. As can be seen from the table, the proposed AGSCR-MVC method achieves best performance on all datasets except for a few metrics on the AWA and Caltech101 datasets. Specifically, on the BDGP, BBC, Caltech101, MSRC, and CCV datasets, the ACC metric of AGSCR-MVC is 8.24%, 4.97%, 4.65%, 4.29%, and 2.33% higher than the second-best performing baseline method, respectively. Compared to existing methods, this invention significantly improves clustering performance. The AGSCR-MVC method demonstrates stable performance on datasets of various sizes and types, proving its broad generalization ability and significant technical advantages.
[0170] Compared to most related anchor-based multi-view clustering methods (such as SMVSC, FPMVS, FDAGF, and MVSC-HFD), the proposed AGSCR-MVC method still exhibits significant performance advantages. Taking the FPMVS method as an example, it only employs consistent anchor points and anchor graph learning in the latent space and does not introduce other structure regularization terms. On the BBC, BDGP, CCV, Caltech101, AWA, and MNIST datasets, the ACC index of AGSCR-MVC is 26.19%, 62.78%, 17.04%, 3.50%, 14.75%, 1.34%, and 0.29% higher than that of FPMVS, respectively. These results clearly verify the effectiveness of the anchor graph structure collaborative regularization mechanism introduced in this invention. This mechanism not only enhances the structural clarity of the anchor graph but also strengthens the local structural consistency between anchor points and anchor graphs, thereby significantly improving the representativeness of anchor points and the clustering discriminative ability of the anchor graph.
[0171] Table 2: Performance Comparison Experiment Results
[0172]
[0173] 5. Running Time Comparison Experiment
[0174] Besides performance advantages, computational efficiency is also crucial. To verify the efficiency of the AGSCR-MVC method, this invention performed a full runtime comparison with all baseline methods on all datasets. For fairness, we measured the entire runtime from the input raw data to the output of the clustering performance metric, and performed a logarithmic transformation on the runtime to narrow the gap between different methods. The results were then visualized (see...). Figure 2As shown in the figure, AGSCR-MVC requires a relatively short runtime, maintaining an acceptable computational cost even on large-scale datasets. Overall, AGSCR-MVC maintains a moderate level in terms of runtime, placing it on the same order of magnitude as most methods. Notably, AGSCR-MVC exhibits a more significant runtime advantage on relatively high-dimensional datasets such as BBC, Caltech101, and AWA, demonstrating its efficiency in handling complex data. Although AGSCR-MVC's runtime is not as long as methods like LMVSC and OMSC, considering the performance improvements it achieves, this invention achieves a better balance between performance and efficiency.
[0175] The above descriptions are merely embodiments of the present invention, and common knowledge such as specific technical solutions and / or characteristics are not described in detail here. It should be noted that those skilled in the art can make various modifications and improvements without departing from the technical solutions of the present invention, and these should also be considered within the scope of protection of the present invention. These modifications and improvements will not affect the effectiveness of the implementation of the present invention or the practicality of the patent. The scope of protection claimed in this application should be determined by the content of its claims, and the specific embodiments described in the specification can be used to interpret the content of the claims.
Claims
1. A multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization, characterized in that, The method includes the following steps: S1. Establishment of the model objective function; S2. Objective function optimization; S3. Algorithm complexity analysis.
2. The multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization according to claim 1, characterized in that, The specific steps for establishing the objective function of the model in S1 include: Taking a multi-feature image as an example, the image data described by different features are: ,in The dimension representing each feature. Indicates the number of samples. Indicates the number of views; Multi-view data is typically assumed to be embedded in a latent space, therefore the data for each view... It can be done through the projection matrix Mapped into a shared low-dimensional latent space, where Represents the potential spatial dimension; Consistent anchor points and anchor graphs are jointly learned from the latent space: ； In the formula, Anchor point matrix, For anchor plots, This represents the number of anchor points.
3. The multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization according to claim 2, characterized in that: The quality of anchor points has a crucial impact on the discriminative power of anchor graphs; therefore, learning highly representative anchor points is essential for improving overall clustering performance. Highly representative anchor points should semantically be consistent with the logical structure of the sample points; therefore, it can be assumed that anchor points are distributed in a structural space that reflects the underlying data topology. ； In the formula, Indicates the first Anchor points, Indicates the first The anchor point and the first The similarity between anchor points; the graph smoothing term in equation (2) encourages anchor points from the same cluster to have similar representations in the feature space.
4. The multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization according to claim 3, characterized in that: To further enhance the discriminative power of anchor charts, column sparsity is introduced for the anchor charts. This encourages each sample point to connect to as few anchor points as possible, thereby reducing redundant connections between anchor points and sample points. Furthermore, excessive sparsity can lead to all sample points clustering on the same anchor point; to avoid such trivial solutions, the learned anchor graph... Apply the Laplace rank constraint to the bipartite graph, i.e. ,in Consensus Divide The normalized Laplace matrix, Let be the corresponding degree matrix, where and ; The Laplace rank constraint on a bipartite graph can force the anchor graph to have ideal connectivity, that is, force the anchor points and sample points to be assigned to... Clusters enhance the clustering structure of anchor graphs.
5. A multi-view clustering method based on anchor point-to-anchor graph structure co-regularization according to claim 4, characterized in that, The final objective function of the model is as follows: ； in, For a set of variables, and To balance the hyperparameters; Equation (3) establishes a collaborative structure regularization of anchor points to graphs, which makes anchor point construction and anchor point graph learning mutually influential; the column sparsity and bipartite graph Laplace rank constraints on the anchor graph further enhance the structure of the anchor graph.
6. The multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization according to claim 1, characterized in that, The objective function optimization implementation method in S2 is as follows: Bipartite graph Lap rank constraint It is discrete and non-convex, making direct optimization difficult. Because... If it is positive semidefinite, then The smallest eigenvalue is 0, therefore the constraint It's a restriction The 0 eigenvalue multiplicity is According to Ky Fan's theorem: ； in, express The former Small eigenvalues, For the spectral embedding representation matrix, and Clustering indicator matrices for samples and anchors are respectively; minimize That is to let The former Small eigenvalues should be as small as possible, so equation (4) can be used to approximate the rank constraint. ; The objective function (3) contains multiple variables and is jointly nonconvex, so it can be optimized using the Alternating Direction Multiplier Method (ADMM). To facilitate optimization, auxiliary variables are introduced. Transform the 2,1 norm of Z to The augmented Lagrangian function that needs optimization is as follows: ； in, It is a balance parameter. For Lagrange multipliers, The penalty coefficient is non-negative. In the above formula, each variable is easy to solve when other variables are fixed; therefore, all variables are optimized alternately in an iterative manner.
7. A multi-view clustering method based on anchor point-to-anchor graph structure co-regularization according to claim 6, characterized in that, All variables are optimized alternately in an iterative manner as follows: 1) Update ; Keep other variables fixed. The subproblems are as follows: ； Expanding the norm terms into trace terms and removing irrelevant terms, equation (6) can be equivalently represented as follows: ； in Equation (7) is an orthogonal Procrustes problem, which can be solved by... Solve using SVD, i.e. , The closed-form solution is ; 2) Update ; By fixing other variables, we can obtain The subproblems are as follows: ； Question (8) can be solved by... Taking the derivative and setting it to zero, we finally obtain... The closed-form solution is: ； 3) Update ; With other variables fixed, we get The subproblems are as follows: ； Simplifying it, we get: ； Considering ,therefore Then problem (11) is equivalent to the following: ； To solve the problem of equation (12), the following well-known theorem in the field is introduced: The following optimization issues: ； in, Given variables, and The variables to be solved. and The optimal solution is: ； in and They are respectively forward The left and right singular vectors corresponding to large singular values; According to well-known theorems, The optimal solutions are respectively times The former The left and right singular vectors corresponding to the large singular values; 4) Update ; Keep other variables fixed. The subproblems are as follows: ； The optimization of problem (15) can first ignore the constraints. ,right Taking the derivative and setting it to zero, we obtain the unconstrained closed-form solution: ； in Then, by projecting the unconstrained closed-form solution onto the nonnegative normalized simplex, we can obtain the solution. The optimal solution; 5) Update ; Fix other variables and optimize variables The subproblems are as follows: ； Its solution can be found through The soft threshold operator is used to minimize the threshold column by column, i.e.: ； in , ; 6) Update the Lagrange multipliers and penalty parameters ; ； ； in These are parameters used to accelerate convergence. Given all initialized variables, each variable is updated alternately until the convergence requirement is met.
8. A multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization according to claim 1, characterized in that, The algorithm complexity analysis specifically includes analyzing the time complexity and space complexity of the AGSCR-MVC algorithm.
9. A multi-view clustering method based on anchor point-to-anchor graph structure collaborative regularization according to claim 8, characterized in that, The time complexity includes: the algorithm's runtime is mainly consumed in updating various variables, for Its updates mainly involve matrix multiplication and SVD, with a time complexity of O(n log n). ,(in (representing the sum of all dimensions); for Its main expenses The time complexity is used for matrix multiplication and inversion; update It will cost The complexity of updating; need The complexity; for It will cost The complexity of performing a soft threshold operation on each column is reduced; considering the large-scale data... Therefore, the time complexity of the AGSCR-MVC algorithm is O(n). , with sample size The relationship is linear.
10. A multi-view clustering method based on anchor point-to-anchor graph structure co-regularization according to claim 8, characterized in that, The space complexity includes: the algorithm needs to store all variables: Therefore, the space complexity of the algorithm is O(n). It is also linear with the number of samples.