Clustering method based on spectral embedding and granularity sphere partitioning and application

Abstract

This invention discloses a clustering method and application based on spectral embedding and granular sphere partitioning, belonging to the fields of data mining and machine learning. The method includes: randomly sampling to obtain a fitted subset; performing spectral embedding on the fitted subset and projecting it to a low-dimensional space; performing KMeans clustering in the spectral embedding space, determining the optimal sphere radius for each cluster based on a weighted geometric mean of three criteria: specificity, coverage, and homogeneity, thereby dividing the samples into core points, boundary points, and noise points; and using a K-nearest neighbor classifier to propagate labels from the core points to all unlabeled samples to obtain the final clustering result. This invention, through a two-stage architecture of spectral embedding followed by sphere coverage, maintains the ability to recognize nonlinear structures while solving the problems of high computational complexity, reliance on a preset number of clusters, and lack of noise rejection mechanisms in traditional spectral clustering. It can be widely applied to scenarios such as image segmentation, text mining, and bioinformatics.

Description

Technical Field

[0001] This invention belongs to the fields of image processing, data mining and machine learning technology. Specifically, it is a clustering method and application based on spectral embedding and granular sphere partitioning. It is particularly suitable for efficient and robust unsupervised clustering analysis of large-scale, high-dimensional datasets with nonlinear structures. It can be widely applied to scenarios such as image segmentation, text mining, bioinformatics, recommender systems and video surveillance. Background Technology

[0002] Clustering analysis is a core task of unsupervised learning, aiming to divide data samples into several groups, such that samples within a group are highly similar and samples between groups are significantly different. With the development of information technology, high-dimensional data such as images, text, and bioinformatics are growing exponentially, placing higher demands on the scalability, accuracy, and robustness of clustering algorithms.

[0003] Spectral clustering is a classic clustering method that projects the original high-dimensional data into a low-dimensional spectral embedding space by constructing a similarity graph of data points and performing eigenvalue decomposition on its Laplacian matrix. This effectively captures the nonlinear manifold structure in the data. Compared to traditional methods such as K-means, spectral clustering has significant advantages in handling data distributions with complex shapes.

[0004] However, traditional spectral clustering methods face significant challenges in large-scale data scenarios. Firstly, the computational complexity of kernel matrix construction and eigenvalue decomposition is typically as high as [missing information - likely a number]. even First, the algorithm is difficult to deploy in practice on datasets containing hundreds of thousands or even millions of samples. Second, distance metrics in high-dimensional spaces are prone to performance degradation, and similarity matrices based on Euclidean distance cannot accurately characterize the true relationships between samples. Furthermore, spectral clustering is extremely sensitive to hyperparameters such as kernel function type, bandwidth parameter, and number of nearest neighbors, and the optimal parameter range varies significantly across different datasets, lacking a unified configuration criterion. Moreover, the number of clusters usually needs to be pre-specified, and subjective setting in scenarios lacking prior knowledge can directly lead to performance bias. Finally, after embedding, spectral clustering still relies on hard partitioning methods such as K-means for final clustering, lacking an explicit rejection mechanism for boundary points and noise points, affecting the robustness and interpretability of the clustering results. To overcome these shortcomings, researchers have proposed various improvement schemes in recent years. For example, existing methods combine particle generation with spectral clustering, but this method first divides the data into particles in the original feature space, and then uses the particles as nodes for spectral clustering. This technical approach aims to simplify the input size of spectral clustering by using spheres, but it fails to solve the problems of spectral clustering being sensitive to hyperparameters and requiring a preset number of clusters. Furthermore, since spheres are generated in the original space, they are difficult to fully reflect the nonlinear manifold structure of the data.

[0005] In summary, current technologies lack a clustering scheme that organically integrates the ability of spectral embedding to identify nonlinear manifold structures with the adaptive coverage and noise rejection capabilities of particle sphere computation. Specifically, existing schemes all follow a "particle sphere partitioning first, then spectral clustering" approach, where particles are generated in the original space, and spectral clustering serves only as the final clustering tool. The two functions are separate and their order is fixed, failing to achieve synergistic enhancement from data space optimization to particle sphere structure learning. Therefore, a new clustering method is urgently needed that can optimize the particle sphere generation space using spectral embedding without relying on a pre-set number of clusters, while maintaining computational efficiency and robustness for large-scale data, effectively addressing the numerous problems existing in the current technologies. Summary of the Invention

[0006] In view of this, the purpose of this invention is to provide a clustering method and application based on spectral embedding and granular sphere partitioning. Through a two-stage architecture of first spectral embedding and then granular sphere covering, it solves the problems of high computational complexity, reliance on preset cluster numbers, sensitivity to hyperparameters, and lack of noise rejection mechanism in existing spectral clustering. While maintaining the ability to identify nonlinear structures, it achieves high efficiency, adaptability, and robustness in large-scale data clustering.

[0007] To achieve the above objectives, the present invention provides the following technical solution: This invention first proposes a clustering method based on spectral embedding and granular sphere partitioning, comprising the following steps: S1, Random subset sampling: Obtaining samples containing... Input dataset for each sample ; Determine the dataset Number of samples Does it exceed the preset threshold? If so, then random sampling is performed from the input dataset. Selected from The representative points constitute the fitting subset. The remaining samples are then assigned to the remaining subset. If not, then all samples will be used as the fitted subset. ; S2, Spectral embedding: for the fitted subset Performing a spectral embedding operation projects the data from the original high-dimensional space to a low-dimensional spectral embedding space, resulting in a spectral embedding matrix. ; S3, Initial Clustering and Granularity Optimization: Perform KMeans clustering in the spectral embedding space to obtain initial cluster labels. For each cluster, adaptive sphere covering is performed. Using the cluster center as the sphere center, candidate radii are traversed, and the optimal sphere radius for each cluster is determined by a weighted geometrical average based on three criteria: specificity, coverage, and homogeneity. Samples are covered and classified according to their optimal sphere radii. Samples covered by only a single sphere are marked as core points and assigned the corresponding cluster label; samples covered by multiple spheres are marked as boundary points; and samples not covered by any spheres are marked as noise points. Thus, a labeled dataset is constructed in the spectral embedding space. Compared with unlabeled datasets ; S4, KNN Label Expansion: Using a K-nearest neighbor classifier, labels are propagated from the core point to all unlabeled samples, including the boundary points, the noise points, and the remaining subset. We obtain the final clustering results for all the data in the dataset.

[0008] Furthermore, in step S2, the method steps for spectral embedding are as follows: S21: Construct a similarity matrix :

[0009] in: For the sample and The similarity between them; For kernel functions; and Each is a fitted subset Any two samples in the dataset; S22: Calculate the degree matrix :

[0010] in: Degree matrix The One diagonal element; S23: Calculate the Laplace matrix Then, the normalized Laplace matrix is ​​calculated. :

[0011] in: It is the identity matrix; S24: Solve for the normalized Laplace matrix. The former The eigenvectors corresponding to the smallest eigenvalues ​​are concatenated to form a spectral embedding matrix. ,in: This represents the number of clusters.

[0012] Furthermore, in step S3, the specificity, coverage, and homogeneity are respectively expressed as:

[0013]

[0014]

[0015] in: Indicates specificity; Indicates coverage; Indicates homogeneity; Candidate radius; This represents the number of samples currently covered by the spheres; This represents the total number of samples in the current cluster. The Shannon entropy is the distribution of sample labels within a sphere. This represents the number of clusters.

[0016] Furthermore, the three criteria are integrated into a comprehensive score using a weighted geometric mean:

[0017] in: This indicates the overall score; , and These are the weighting coefficients for specificity, coverage, and homogeneity, respectively. The radius with the highest overall score is taken as the optimal particle radius for each cluster.

[0018] Furthermore, in step S3, the boundary points and the noise points are marked as -1 to form an explicit rejection mechanism, which enables the clustering results to distinguish between the core cluster structure and uncertain samples.

[0019] Furthermore, in step S4, for the sample to be classified Its predicted label is The nearest neighbor weighted voting decision is made, where the voting weight of each nearest sample is proportional to the voting weight of that nearest sample relative to the sample to be classified. The number of neighboring samples is inversely proportional to the Euclidean distance; default ,in: For the labeled dataset The number of samples.

[0020] Furthermore, before or during steps S1 to S4, a three-stage parameter scan is performed to determine at least one hyperparameter involved in the spectral embedding, the particle optimization, and / or the KNN label expansion; the three-stage parameter scan includes: The first stage, coarse scanning: scanning the parameter space within a preset large step size range to locate the candidate parameter interval; The second stage, fine scanning: scanning is performed in small steps within the candidate parameter range to determine the region near the optimal parameter value; The third stage is boundary extension and convergence: the scanning range is gradually extended along both sides of the optimal parameter value until the clustering performance stops improving, so as to determine the final optimal parameters.

[0021] This invention also proposes an application of the clustering method based on spectral embedding and granular sphere partitioning as described above, wherein the method performs clustering tasks in at least one of the following practical application scenarios: Image segmentation scenario: pixel-level clustering and segmentation of high-resolution remote sensing images or medical images; Text mining scenario: Topic clustering of Internet documents, news, patent documents, or scientific documents; Bioinformatics scenario: Cell type identification and cluster analysis of single-cell RNA sequencing data; Recommendation system scenarios: building user profiles and segmenting user groups based on user behavior data; Video surveillance scenario: Scene clustering and behavior pattern recognition of massive video frame images; Network security scenario: Anomaly detection and classification of network traffic data; Industrial IoT scenario: Performing device status segmentation and cluster analysis on sensor data; Social network scenario: Community discovery and group identification of social network users.

[0022] The beneficial effects of this invention are as follows: This invention is a clustering method based on spectral embedding and granular sphere partitioning. Through a four-step process of random subset sampling, spectral embedding, granular sphere covering, and KNN label expansion, it achieves the following technical effects. (1) The computational efficiency is significantly improved. Traditional spectral clustering requires constructing a database on the full dataset. Similarity matrix and perform Eigenvalue decomposition is computationally extremely costly. This invention constrains spectral decomposition to a specific value through random subset sampling. At a representative point, the complexity of spectral decomposition is reduced to... The complexity of the KNN label expansion stage is... The overall complexity is... ;when much smaller Time degenerates into It grows approximately linearly, enabling the algorithm to efficiently process large-scale datasets of millions of records.

[0023] (2) High clustering accuracy, especially good at nonlinear structures. Experimental results show that the GSC algorithm of this invention achieves 100% perfect clustering accuracy on the TwoCircles nonlinear synthetic dataset, while the ARI of the five comparative algorithms, namely K-means, GMM, hierarchical clustering, DPC, and TorqueClustering, are close to 0, indicating complete failure. This directly proves that the technical route of "spectral embedding followed by particle covering" can effectively reveal the nonlinear manifold structure of the data, enabling the particles to adaptively cover the data in the optimized embedding space.

[0024] (3) Adaptive clustering without pre-setting the number of clusters. Traditional spectral clustering requires pre-specifying the number of clusters, and subjective setting without prior knowledge will directly lead to performance deviation. This invention adaptively determines the optimal sphere radius of each cluster through a weighted geometric mean of three criteria: specificity, coverage, and homogeneity. The sphere coverage process automatically determines the cluster structure, eliminating the need to pre-set the number of clusters and significantly enhancing the robustness to hyperparameters.

[0025] (4) Explicit rejection mechanism to improve robustness. This invention explicitly divides samples into three categories: core points, boundary points and noise points through particle coverage. Boundary points and noise points do not participate in subsequent clustering, which effectively solves the problem that traditional spectral clustering lacks an explicit rejection mechanism for boundary points and noise points.

[0026] (5) Excellent generalization performance. On 15 publicly available benchmark datasets, the GSC algorithm of this invention achieves the best ARI results on 9 datasets; Friedman test... Its average ranking (1.47) is confirmed to be significantly better than five comparison algorithms, including K-means, GMM, hierarchical clustering, DPC, and TorqueClustering, which proves the universality and stability of the present invention on diverse data types. Attached Figure Description

[0027] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration: Figure 1 This is a flowchart of the clustering method based on spectral embedding and granular sphere partitioning of the present invention; Figure 2 A scatter plot showing the ARI ranking of GSC and the comparison algorithm across 15 datasets; Figure 3 A heatmap of clustering performance; Figure 4 Nemenyi critical difference plot; Figure 5 Performance comparison of GSC and GBSC on 15 datasets; (a) ARI; (b) NMI; (c) AMI. Detailed Implementation

[0028] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0029] This invention aims to propose a clustering method (Granular Spectral Clustering, GSC) based on spectral embedding and granular sphere partitioning, and its application, to address the issue that existing spectral clustering methods suffer from high computational complexity in kernel matrix construction and eigenvalue decomposition on large-scale datasets. However, existing clustering methods often suffer from difficulties in practical deployment, and are generally sensitive to hyperparameters, rely on a preset number of clusters, and lack explicit mechanisms for handling boundary points and noise points. This invention addresses these issues through a two-stage architecture design, using random sampling to constrain spectral decomposition to a subset of samples, thereby reducing the overall complexity to... The particle-sphere covering mechanism adaptively determines the optimal particle-sphere radius and explicitly identifies boundary points and noise points. KNN extension propagates the core point labels to the entire dataset. While maintaining the ability of spectral clustering to identify nonlinear manifold structures, it improves the scalability of the algorithm on large-scale data and its robustness to hyperparameters, without the need to pre-specify the number of clusters.

[0030] The following provides a detailed description of the specific implementation methods and applications of the clustering method based on spectral embedding and granular sphere partitioning of this invention. Specifically, for example... Figure 1 As shown, this embodiment of the clustering method based on spectral embedding and granular sphere partitioning includes the following steps.

[0031] S1, Random subset sampling: Obtaining samples containing... Input dataset for each sample ; Determine the dataset Number of samples Does it exceed the preset threshold? If so, then random sampling is performed from the input dataset. Selected from The representative points constitute the fitting subset. The remaining samples are then assigned to the remaining subset. If not, then all samples will be used as the fitted subset. In this embodiment, a preset threshold is used. The default value is 1500, which can be adjusted according to the actual dataset size and computing resources.

[0032] Random sampling requires no additional density calculation overhead, is simple to implement, and is theoretically unbiased. It can maintain the global topology of the manifold while keeping the complexity of spectral decomposition within a certain limit. Magnitude. For those containing A dataset of images of samples, when At that time, the computational cost of spectral decomposition increased from full computation. The magnitude dropped to Magnitude.

[0033] S2, Spectral embedding: for the fitted subset Performing spectral embedding operations projects data from the original high-dimensional space to a low-dimensional spectral embedding space, causing similar samples to cluster together and dissimilar samples to separate, resulting in a spectral embedding matrix. This step includes the following sub-steps.

[0034] S21: Construct a similarity matrix For the fitted subset In Given a sample, calculate any two samples. and Similarity between them:

[0035] in: The kernel function can be an RBF kernel or a nearest neighbor kernel. In this embodiment, a nearest neighbor kernel is preferred:

[0036] in: Indicates sample of There are several nearest neighbor sets; the similarity is 1 when two samples are mutually identical or one is in the nearest neighbor set of the other, otherwise it is 0. The similarity matrix constructed in this way is a sparse binary matrix.

[0037] S22: Calculate the degree matrix Degree matrix It is a diagonal matrix, and its diagonal elements are the similarity matrix. Sum of elements in each row:

[0038] in: Degree matrix The Diagonal elements. Degree matrix. It reflects the local density information of each sample.

[0039] S23: Calculate the Laplacian matrix. First, calculate the non-normalized graphical Laplacian matrix. Then, the normalized Laplace matrix is ​​calculated. :

[0040] in: The identity matrix. Normalized Laplace matrix. It has good spectral properties, and its eigenvalues ​​are distributed in Within the range.

[0041] S24: Eigenvalue decomposition and spectral embedding. Solving for the normalized Laplacian matrix. The former The eigenvectors corresponding to the smallest eigenvalues ​​are concatenated column-wise to obtain the spectral embedding matrix. ,in: This represents the number of clusters. In this embodiment, It can be preset according to the actual application scenario, or automatically determined through subsequent particle optimization steps.

[0042] After the above spectral embedding transformation, the nonlinear structure of the original data becomes an approximately linear structure in the embedding space, which facilitates subsequent clustering operations.

[0043] S3, Initial Clustering and Granularity Optimization: Perform KMeans clustering in the spectral embedding space to obtain initial cluster labels. For each cluster, adaptive sphere covering is performed. Using the cluster center as the sphere center, candidate radii are traversed, and the optimal sphere radius for each cluster is determined by a weighted geometrical average based on three criteria: specificity, coverage, and homogeneity. Samples are covered and classified according to their optimal sphere radii. Samples covered by only a single sphere are marked as core points and assigned the corresponding cluster label; samples covered by multiple spheres are marked as boundary points; and samples not covered by any spheres are marked as noise points. Thus, a labeled dataset is constructed in the spectral embedding space. Compared with unlabeled datasets .

[0044] Specifically, in this step, the candidate radii are traversed with the cluster center as the sphere center. For each candidate radius, three evaluation criteria are calculated.

[0045] (1) Specificity : Reflects the compactness of the particle radius, defined as follows:

[0046] in: Indicates specificity; The radius represents the candidate radius. The smaller the radius, the higher the specificity, indicating a more compact granule.

[0047] (2) Coverage : Reflects the proportion of the sample covered by granules:

[0048] in: Indicates coverage; This represents the number of samples currently covered by the spheres; This represents the total number of samples in the current cluster. Higher coverage indicates better coverage of the samples by the spheres.

[0049] (3) Homogeneity : Reflects the consistency of the internal labeling of the pellets:

[0050] in: Indicates homogeneity; The Shannon entropy is the distribution of sample labels within a sphere. This represents the number of clusters. Higher homogeneity indicates more consistent labels within the clusters.

[0051] The three criteria are combined into a comprehensive score using a weighted geometric mean:

[0052] in: For comprehensive scoring; , and These are the weighting coefficients for specificity, coverage, and homogeneity, respectively. The radius with the highest overall score in each cluster is taken as the optimal sphere radius for that cluster.

[0053] Based on the optimal sphere radius of each cluster, samples are covered and classified: samples covered by only a single sphere are marked as core points and assigned the corresponding cluster label; samples covered by multiple spheres are marked as boundary points and labeled -1; samples not covered by any spheres are marked as noise points and labeled -1. Through this labeling, a labeled dataset is constructed in the spectral embedding space. Compared with unlabeled datasets Boundary points and noise points are uniformly marked as -1, forming an explicit rejection mechanism, which enables the clustering results to clearly distinguish the core cluster structure from uncertain samples.

[0054] S4, KNN Label Expansion: Using a K-nearest neighbor classifier, labels are propagated from the core point to all unlabeled samples, including the boundary points, the noise points, and the remaining subset. We obtain the final clustering results for all the data in the dataset.

[0055] Specifically, for the samples to be classified Its predicted label is The nearest neighbor weighted voting decision is made, where the voting weight of each nearest sample is proportional to the voting weight of that nearest sample relative to the sample to be classified. It is inversely proportional to the Euclidean distance:

[0056] in: For the sample to be classified The final predicted label; For the first Nearest neighbor samples Known labels; , for the first The voting weight of each neighboring sample; For the first One nearby sample; For indicator functions; Indexed by category. Number of neighboring samples. default ,in: For labeled datasets The number of samples can also be adjusted via hyperparameter scanning.

[0057] The computational complexity of the KNN label expansion stage is ,when Much larger At that time, the overall computational complexity decreased from the spectral decomposition stage. It remains dominant and maintains an approximately linear level.

[0058] In this embodiment, a three-stage parameter scan may be performed before or during steps S1 to S4 to determine hyperparameters. The three-stage parameter scan includes: The first stage, coarse scanning: This involves scanning the parameter space within a preset large step size to locate candidate parameter intervals. For example, for bandwidth parameters... ,exist Within the range, in step size Perform a scan.

[0059] The second stage, fine scanning: Scanning is performed within the candidate parameter range in small steps to determine the region near the optimal parameter value. For example, if the coarse scan determines the optimal... exist Within the interval, the step size is used. Perform a detailed scan.

[0060] The third stage is boundary extension and convergence: the scanning range is gradually extended along both sides of the optimal parameter value until the clustering performance stops improving, so as to determine the final optimal parameters.

[0061] This three-stage progressive strategy ensures the integrity of search coverage while limiting the ineffective consumption of computing resources to local areas of the parameter space.

[0062] The following section provides a detailed explanation of the specific implementation of the clustering method based on spectral embedding and granular sphere partitioning in this embodiment, using concrete experiments as examples.

[0063] To verify the effectiveness of this invention, comparative experiments were conducted on 15 publicly available benchmark datasets. The datasets cover synthetic data (Two Circles, Two Moons), classic small-scale data (Iris, Wine, Breast Cancer), medium-scale data (Digits, Pendigits, Satimage), high-dimensional multi-class data (Isolet, Semeion, USPS), image / sensor data (Texture, HAR, Waveform), and biological data (PBMC3K).

[0064] The comparison algorithms included K-means, GMM (Gaussian Mixture Model), hierarchical clustering (Ward connectivity), DPC (Density Peak Clustering), and TorqueClustering. All experimental results were subjected to 10-fold cross-validation, with ARI (Adjusted Rand Index), NMI (Normalized Mutual Information), and AMI (Adjusted Mutual Information) as the main evaluation metrics. All results are expressed as mean ± standard deviation of 10-fold cross-validation.

[0065] The experimental results are shown in Table 1. GSC achieved the best ARI results on 9 out of 15 datasets. On the TwoCircles nonlinear synthetic dataset, GSC achieved a perfect clustering accuracy of 100%, while the ARI of all comparison algorithms was close to 0, indicating complete failure. This directly verifies the effectiveness of the "spectral embedding followed by particle-sphere covering" technique in this embodiment for identifying nonlinear manifold structures. On datasets such as Wine, Breast Cancer, Digits, Isolet, Satimage, HAR, and USPS, GSC also achieved the best results, with a lead of 3 to 15 percentage points.

[0066] Friedman test The post-hoc test with Nemenyi (CD=1.95) confirmed that the average ranking of GSC (1.47) was significantly better than that of DPC (5.27) and TorqueClustering (5.20), demonstrating the versatility and statistical significance of the method in this embodiment across diverse data types.

[0067] Table 1: Comparative Experiment Results on 15 Public Benchmark Datasets

[0068] Figure 2This is a scatter plot showing the ARI rankings of GSC and five comparison algorithms across 15 datasets. Figure 2 As can be seen, GSC achieves 100% success on Two Circles, while all comparison algorithms fail. GSC leads and maintains stable performance on most datasets. Figure 3 This is a heatmap of clustering performance, where rows represent datasets, columns represent algorithms, and color intensity indicates ARI (Advanced Clustering Index) levels. Figure 3 It can be seen that GSC achieves the best performance on 9 datasets, with the darkest overall color in its column, and outperforms the comparison algorithms. Figure 4 The Nemenyi critical difference plot (CD=1.95). (From...) Figure 4 It can be seen that GSC's average ranking of 1.47 is significantly better than DPC (5.27) and TorqueClustering (5.20), and the difference is statistically significant.

[0069] To further verify the superiority of the clustering method based on spectral embedding and granular sphere partitioning in this embodiment compared to the existing "granular sphere-first, spectral clustering" technique, this embodiment also conducts a comparative experiment with the granular sphere-based spectral clustering method (GBSC). The GBSC technical route is: original feature space → PCA dimensionality reduction to 2D → granular sphere generation (weighted DM splitting) → granular sphere center construction of affinity matrix → spectral clustering → label backpropagation. After granular sphere generation, it only retains the center and radius, discarding the internal structural information of the granular spheres, and lacks an explicit rejection mechanism. The core idea of ​​GBSC is to use the granular sphere centers to replace the original data points to construct a similarity matrix, thereby reducing the matrix size. However, GBSC only retains the center and radius after granular sphere generation, discarding the internal structural information of the granular spheres, and lacks an explicit rejection mechanism and a secondary refined clustering stage. GSC is lossless and can handle boundary / noise, while GBSC sacrifices a large amount of structural information and lacks a rejection mechanism. The core differences between the GSC and GBSC technical solutions are shown in Table 2.

[0070] Table 2: Comparison of GSC and GBSC technical solutions

[0071] The experiment used the same 15 publicly available benchmark datasets and a 10-fold cross-validation protocol (with the same random seed and data partitioning). ARI, NMI, and AMI were used as evaluation metrics to directly compare GSC and GBSC. The hyperparameters of GSC were determined according to the three-stage parameter scan described in this embodiment; GBSC used the official default parameters (σ=0.1) without additional tuning. The results are as follows: Figure 5 As shown in Table 3.

[0072] Table 3: Performance comparison of GSC and GBSC on 15 datasets

[0073] Table 3 shows that GSC outperforms GBSC in ARI, NMI, and AMI on 14 out of 15 datasets. The average ARI improvement is 0.2523 (from 0.4055 to 0.6579), the average NMI improvement is 0.2080, and the average AMI improvement is 0.2109. The largest improvements occur on high-dimensional or non-linearly structured datasets such as Waveform (ARI +0.5735), Digits (+0.5343), Breast Cancer (+0.4496), and Isolet (+0.3846). Only the Two Moons dataset shows slightly lower NMI and AMI than GBSC (difference of approximately 0.01), but GSC still outperforms GBSC in ARI. Both datasets achieve perfect clustering on the Two Circles dataset.

[0074] Furthermore, this embodiment also references the publicly available experimental results of GBSC on multiple real-world datasets in the MDMSC paper (Xu et al., 2024, arXiv:2408.10493), which provides an independent comparative evaluation of GBSC. The results are shown in Table 4.

[0075] Table 4: Performance of GBSC on representative datasets (Data source: MDMSC 2024)

[0076] As shown in Table 4, GBSC exhibits extremely low clustering accuracy on several datasets with complex manifold structures or high-dimensional features, even showing negative ARI values ​​(e.g., car, -3.08%), indicating that its particle-spheric pre-sorting strategy struggles to effectively fit non-convex or non-spherical distributions in the original space. Notably, in Table 4, GBSC achieves an ARI of 42.62% on pendigits, while GSC in this embodiment reaches 64.83% on the same dataset (see Table 2), representing a relative improvement of over 50 percentage points. GBSC's ARI on optical_test is 28.70%, while GSC achieves significantly better results on similarly sized datasets such as USPS and Digits.

[0077] Based on the comparative experiments (Table 3) and independent third-party evaluations (Table 4), the following conclusions can be drawn: Firstly, GBSC's core advantage lies in its computational efficiency. Complexity, While GBSC has good particle count and noise resistance, its accuracy is extremely unstable on complex structured data. The fundamental reason is that GBSC directly generates spherical particles in the original space, which cannot adapt to non-convex manifold structures; its splitting rules only target compactness, the similarity measure uses Euclidean distance, and it lacks a secondary refinement clustering stage.

[0078] Secondly, in this embodiment, GSC first maps the data to a space with a clear low-dimensional manifold structure through spectral embedding, then performs particle-sphere adaptive covering, and through KNN secondary refinement and an explicit rejection mechanism (marking boundary points / noise points as -1), enables the particles to fit the data distribution more accurately in the embedding space. Therefore, GSC achieves significantly better clustering accuracy than GBSC on the vast majority of datasets.

[0079] The above comparison fully demonstrates the advanced nature and effectiveness of the "spectral embedding followed by particle-sphere coverage" technical approach of this invention. That is, by organically integrating the ability of spectral embedding to identify nonlinear manifold structures with the adaptive coverage capability of particle-sphere computation, the defect of unstable accuracy of the existing "particle-sphere clustering followed by spectral clustering" scheme on complex structural data is fundamentally overcome.

[0080] This embodiment also proposes an application of the clustering method based on spectral embedding and granular sphere partitioning as described above. This method performs clustering tasks in at least one of the following practical application scenarios. Specifically, the two-stage architecture of the spectral embedding and granular sphere partitioning (GSC) clustering method has natural advantages in large-scale data clustering scenarios, and is particularly suitable for the following practical scenarios.

[0081] In the field of image segmentation, pixel-level clustering is used to segment high-resolution remote sensing or medical images. Specifically, high-resolution remote sensing and medical images often contain millions of pixels of features, which traditional spectral clustering methods cannot fully address. The complexity of traditional methods cannot be directly applied. GSC constrains the computation to a representative subset of pixels through random sampling, making real-time segmentation of large images possible.

[0082] In the field of text mining, topic clustering is performed on internet documents, news, patent documents, or scientific literature. Specifically, internet document clustering systems face the challenges of massive corpora and diverse topic densities. GSC can maintain the ability to identify sparse categories and long-tail topics while reducing computational overhead, making it suitable for tasks such as news aggregation, patent classification, and literature reviews.

[0083] In the field of bioinformatics, cell type identification and cluster analysis are performed on single-cell RNA sequencing data. Specifically, single-cell RNA sequencing data is characterized by a large number of cells, high gene dimensionality, and extremely unbalanced population density distribution. The random sampling of spectral scrambling (GSC) can reduce computational overhead while keeping the cost of spectral decomposition to a constant level.

[0084] In recommender systems, user behavior data is used to construct user profiles and segment user groups. Specifically, user behavior data exhibits a highly non-linear manifold structure, and GSC can handle the complex interaction relationships between hundreds of millions of users and products without sacrificing clustering quality, supporting large-scale user profile construction and accurate recommendations.

[0085] In the field of video surveillance, scene clustering and behavior pattern recognition are performed on massive amounts of video frame images. Specifically, GSC can be used for scene clustering and behavior pattern recognition by randomly sampling massive amounts of frame images to extract representative scenes and achieve video content summarization over long periods of time.

[0086] Network security scenario: Anomaly detection and classification of network traffic data.

[0087] Industrial IoT scenario: Performing device status segmentation and cluster analysis on sensor data; Social network scenario: Community discovery and group identification of social network users.

[0088] GSC leverages its combined strengths in nonlinear structure identification and large-scale data processing to provide a universal clustering solution that balances accuracy and efficiency for practical industrial deployments.

[0089] In summary, this embodiment of the clustering method and application based on spectral embedding and granular sphere partitioning has the following technical highlights.

[0090] (1) Two-stage architecture. The computationally intensive spectral decomposition is constrained to a representative subset of randomly sampled samples, and the complexity of spectral decomposition is reduced from the global complexity. Down to The clustering results are extended to the full dataset using the KNN classifier; the complexity of the extension stage is O(n). The two-stage separation keeps the overall computational complexity at a near-linear level, avoiding the computational bottleneck caused by traditional spectral clustering in constructing kernel matrices and eigenvalue decomposition on the full dataset.

[0091] (2) Automatic Particle-Sphere Clustering. Adaptive particle-sphere coverage is performed in the spectral embedding space. Candidate radii are traversed using the center of each cluster as the sphere center. The optimal particle-sphere radius range for each cluster is determined by a weighted geometric average of three-dimensional criteria: specificity, coverage, and homogeneity. Particle-sphere coverage divides samples into three types: core points (covered by a single particle-sphere and assigned a corresponding cluster label), boundary points (covered by multiple particles simultaneously and marked as -1), and noise points (not covered by particles and marked as -1). This introduces an interpretable cluster structure description while preserving local topological information of the data. The adaptive determination mechanism of particle-sphere radius eliminates the need for a pre-set number of clusters, enabling the handling of real-world data distributions with uneven cluster sizes and densities.

[0092] (3) Third-order parameter scanning. A three-stage progressive strategy is adopted: coarse scanning, fine scanning, and boundary extension convergence. The coarse scanning locates the candidate parameter interval within a large step size. The fine scanning determines the region near the optimal value within the candidate interval with a small step size. The boundary extension convergence gradually extends along both sides of the optimal value until the performance stops improving. The three-stage mechanism ensures the integrity of the search coverage while controlling the ineffective consumption of computing resources to a local region of the parameter space.

[0093] (4) Cross-domain clustering performance. Datasets: 15 publicly available benchmark datasets, covering synthetic data (TwoCircles, Two Moons), classic datasets (Iris, Wine, Breast Cancer), medium-sized datasets (Digits, Pendigits, Satimage), high-dimensional multi-class datasets (Isolet, Semeion, USPS), image / sensor datasets (Texture, HAR, Waveform), and biological datasets (PBMC3K). For more than... The sample dataset is first subjected to stratified random sampling, and then 10-fold cross-validation is performed.

[0094] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A clustering method based on spectral embedding and granular sphere partitioning, characterized in that: Includes the following steps: S1, Random subset sampling: Obtaining samples containing... Input dataset for each sample ; Determine the dataset Number of samples Does it exceed the preset threshold? If so, then random sampling is performed from the input dataset. Selected from The representative points constitute the fitting subset. The remaining samples are then assigned to the remaining subset. If not, then all samples will be used as the fitted subset. ; S2, Spectral embedding: for the fitted subset Performing a spectral embedding operation projects the data from the original high-dimensional space to a low-dimensional spectral embedding space, resulting in a spectral embedding matrix. ; S3, Initial Clustering and Granularity Optimization: Perform KMeans clustering in the spectral embedding space to obtain initial cluster labels. For each cluster, adaptive sphere covering is performed. Using the cluster center as the sphere center, candidate radii are traversed, and the optimal sphere radius for each cluster is determined by a weighted geometrical average based on three criteria: specificity, coverage, and homogeneity. Samples are covered and classified according to their optimal sphere radii. Samples covered by only a single sphere are marked as core points and assigned the corresponding cluster label; samples covered by multiple spheres are marked as boundary points; and samples not covered by any spheres are marked as noise points. Thus, a labeled dataset is constructed in the spectral embedding space. Compared with unlabeled datasets ; S4, KNN Label Expansion: Using a K-nearest neighbor classifier, labels are propagated from the core point to all unlabeled samples, including the boundary points, the noise points, and the remaining subset. We obtain the final clustering results for all the data in the dataset.

2. The clustering method based on spectral embedding and granular sphere partitioning according to claim 1, characterized in that: In step S2, the method steps for spectral embedding are as follows: S21: Construct a similarity matrix : in: For the sample and The similarity between them; For kernel functions; and respectively fitted subsets Any two samples in the dataset; S22: Calculate the degree matrix : in: Degree matrix The One diagonal element; S23: Calculate the Laplace matrix Then, the normalized Laplace matrix is ​​calculated. : in: It is the identity matrix; S24: Solve for the normalized Laplace matrix The former The eigenvectors corresponding to the smallest eigenvalues ​​are concatenated to form a spectral embedding matrix. ,in: This represents the number of clusters.

3. The clustering method based on spectral embedding and granular sphere partitioning according to claim 1, characterized in that: In step S3, the specificity, coverage, and homogeneity are respectively represented as: in: Indicates specificity; Indicates coverage; Indicates homogeneity; Candidate radius; This represents the number of samples currently covered by the spheres; This represents the total number of samples in the current cluster. The Shannon entropy is the distribution of sample labels within a sphere. This represents the number of clusters.

4. The clustering method based on spectral embedding and granular sphere partitioning according to claim 3, characterized in that: The three criteria are combined into a comprehensive score using a weighted geometric mean: in: This indicates the overall score; , and These are the weighting coefficients for specificity, coverage, and homogeneity, respectively. The radius with the highest overall score is taken as the optimal particle radius for each cluster.

5. The clustering method based on spectral embedding and granular sphere partitioning according to claim 1, characterized in that: In step S3, the boundary points and the noise points are marked as -1 to form an explicit rejection mechanism, which enables the clustering results to distinguish between the core cluster structure and uncertain samples.

6. The clustering method based on spectral embedding and granular sphere partitioning according to claim 1, characterized in that: In step S4, for the sample to be classified Its predicted label is The nearest neighbor weighted voting decision is made, where the voting weight of each nearest sample is proportional to the voting weight of that nearest sample relative to the sample to be classified. The number of neighboring samples is inversely proportional to the Euclidean distance; default ,in: For the labeled dataset The number of samples.

7. The clustering method based on spectral embedding and granular sphere partitioning according to claim 1, characterized in that: Before or during steps S1 to S4, a three-stage parameter scan is also performed to determine at least one hyperparameter involved in the spectral embedding, the particle optimization, and / or the KNN label expansion. The three-stage parameter scan includes: The first stage, coarse scanning: scanning the parameter space within a preset large step size range to locate the candidate parameter interval; The second stage, fine scanning: scanning is performed in small steps within the candidate parameter range to determine the region near the optimal parameter value; The third stage is boundary extension and convergence: the scanning range is gradually extended along both sides of the optimal parameter value until the clustering performance stops improving, so as to determine the final optimal parameters.

8. An application of the clustering method based on spectral embedding and granular sphere partitioning as described in any one of claims 1-7, characterized in that: The method performs clustering tasks in at least one of the following practical application scenarios: Image segmentation scenario: pixel-level clustering and segmentation of high-resolution remote sensing images or medical images; Text mining scenario: Topic clustering of internet documents, news, patent documents, or scientific literature; Bioinformatics scenario: Cell type identification and cluster analysis of single-cell RNA sequencing data; Recommendation system scenarios: building user profiles and segmenting user groups based on user behavior data; Video surveillance scenario: Scene clustering and behavior pattern recognition of massive video frame images; Network security scenario: Anomaly detection and classification of network traffic data; Industrial IoT scenario: Performing device status segmentation and cluster analysis on sensor data; Social network scenario: Community discovery and group identification of social network users.

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