Boundary stripping clustering method and system based on space vector decomposition

Abstract

The present invention provides a boundary-stripping clustering method and system based on spatial vector decomposition. The boundary-stripping clustering method comprises: step 1, input step: inputting the data set #imgabs0#; step 2, boundary-stripping step: discriminating the object type based on the distribution characteristics of the object neighborhood to strip boundary points; step 3, cluster skeleton construction step: determining the cluster skeleton based on the transitive closure relationship; step 4, boundary association step: establishing the affiliation relationship between the boundary and the cluster skeleton to complete clustering; step 5, output step: outputting the clustering results. The beneficial effect of the present invention is that the present invention not only solves the limitation of existing boundary-stripping clustering that tends to distribute uniformly and spherically, but also improves the performance of the present invention under complex distribution and high-dimensional data.

Description

Technical Field

[0001] The present invention relates to the field of computer technology, and in particular to a boundary stripping clustering method and system based on space vector decomposition. Background Art

[0002] Cluster analysis is one of the key technologies in data mining and knowledge discovery. It aims to divide potential data into several non-overlapping sets based on data affinity to achieve high intra-group similarity and low inter-group similarity[1]. It is widely used in artificial intelligence scenarios such as image segmentation, target detection, recommendation systems, and massive data annotation[2][3].

[0003] At present, distance partitioning and density reachability are the two most popular clustering paradigms. Partitioning clustering represented by K-means

[35] generally first randomly specifies the cluster center and then achieves the best clustering by converging the objective function. It is often used as the underlying operator in the field of target detection and image segmentation. However, this type of algorithm is limited by the convex objective function and the nearest center partitioning strategy, and cannot effectively handle non-convex and high-dimensional data. In contrast, density clustering does not require the number of clusters to identify clusters of any shape. As the most classic algorithm, DBSCAN[4] uses the threshold Eps and the sampling radius MinPts to globally search for core objects to form the skeleton of the cluster, and classifies the objects on its density reachable path into one category. Although some variants have successively improved performance in parameter adaptation

[36] , complexity reduction, and density estimation

[37] , parameter tuning is still required to ensure the effect when facing high-dimensional data.

[0004] Rodriguez et al. [8] used density peaks as the backbone of clusters and proposed density peak clustering (DPC). Compared with partitioning clustering, DPC does not require iteration, and the cluster centers selected by the decision graph are more reasonable. Compared with DBSCAN, DPC classifies objects based on density peaks, eliminating the need for multi-layer nested traversal. As a new density clustering framework, scholars have proposed a series of optimized versions to address the shortcomings of the original DPC. Given the advantages of high visualization, EC

[11] , DPC-KNN-PCA

[18] , RECOME

[19] , and DenMune

[38] introduced methods such as k-nearest neighbors, mutual nearest neighbors, shared nearest neighbors, and fuzzy kernels to optimize the composition of the original decision graph. To ensure the continuity of the clustering flow, CFDPC

[22] , DPC-RDE

[16] , DP-MD-FN

[39] , and DPC-LG

[40] algorithms quantify the cluster center eigenvalues by designing a scoring function (γ), but the number of clusters still needs to be specified in advance. In addition, RA-Clust

[10] , CCFDP

[12] , CH-CCFDAC

[41] , and CA-CAM

[26] use statistical models (linear regression, residual analysis, 5 sigma criteria, and normal distribution curves) to fit the data distribution, thus automating the clustering process. However, they are more sensitive to hyperparameters. In general, density peak clustering has a common challenge, which is that the requirements for cluster centers are too strict, and it is not easy to ensure the correctness of cluster centers in terms of quality and quantity in complex scenarios. In detail, the membership of all objects is determined only by a small number of cluster centers. If the number of clusters is not correctly estimated or the density measurement has a certain deviation, the obtained cluster centers will lack representativeness, which can easily lead to a domino effect of incorrect object division. Some algorithms

[22]

[17]

[41]

[20] choose to sacrifice continuity to ensure clustering stability and assist clustering by specifying the number of clusters or manually selecting them. However, in actual scenarios, there is often a lack of prior knowledge of the data and it is impossible to give a correct estimate.

[0005] Considering that density peak clustering generally requires expert participation and is relatively strict on the quality of cluster centers, some scholars choose to use rich boundary information to extract a group of backbone objects to represent the skeleton of the cluster, and complete clustering by establishing the transitive closure between boundaries and boundaries, and boundaries and backbone objects, which is called boundary stripping clustering. In actual scenarios, boundary stripping clustering can not only complete data classification, but also extract boundary pattern information with certain value for subsequent decision analysis. Based on this, scholars focus on boundary stripping rather than cluster center selection. For example, Spinver

[25] uses the Hopkins statistic to identify boundary objects, CA-CSM

[26] and C-USB

[27] use the third-order moment to describe the spatial distribution of data, DCUBI

[29] observes the consistency of the distribution of the neighbors of the boundary object on its tangent plane; 3W-DPET

[28] uses the trust function in evidence theory to describe the boundary; Lever[5] uses the lever principle to model the distribution of boundary objects. Due to the inherent design limitations of the statistics, the above algorithms are difficult to effectively process non-convex and high-dimensional data. For example, the Hopkins statistic only focuses on the data distribution in two-dimensional space, and the skewness is used to describe the distribution in the overall data space rather than the local neighborhood. Recently, Averbuch et al. [7] believed that clusters are layered enclosed areas composed of some core layers and boundary layers, and proposed a boundary stripping clustering algorithm BP on PAMI. As the latest representative, BP uses density differences to remove external boundary objects layer by layer, and at the same time unidirectionally binds boundary points to the nearest unstripped objects until the core area of the cluster is effectively separated. Finally, DBSCAN is used to classify the core area. Unlike previous boundary clustering, the BP algorithm pioneered an iterative boundary stripping method, providing a new idea for boundary stripping clustering. However, the number of parameters in the BP algorithm is much higher than that of other algorithms, and the boundary judgment criterion is single: the boundary is determined only by density differences. When faced with unevenly distributed data, it is difficult for the algorithm to find the optimal partitioning structure, and it is very easy to cause over-division. Summary of the Invention

[0006] The present invention provides a boundary stripping clustering method based on space vector decomposition, comprising the following steps:

[0007] Step 1, input step: input data set X = {x1, x2, x3, ..., x n}∈R m×n , the data in the data set can be any high-dimensional or complex distributed data;

[0008] Step 2, boundary stripping step: identify the object type based on the object neighborhood distribution characteristics to strip boundary points;

[0009] Step 3, cluster skeleton construction step: determine the cluster skeleton based on the transitive closure relationship;

[0010] Step 4, boundary association step: establish the affiliation relationship between the boundary and the cluster skeleton to complete clustering;

[0011] Step 5, output step: output clustering results.

[0012] As a further improvement of the present invention, step 2 includes the following steps:

[0013] Step 21: Calculate the density ρ i ;

[0014] Step 22: Calculate the norm of the nearest neighbor vector ||v i || p ;

[0015] Step 23: Calculate Boundary Confidence

[0016] As a further improvement of the present invention, in step 21, the density ρ is calculated according to formula (7) i ;

[0017]

[0018] As a further improvement of the present invention, in step 22, the norm of the nearest neighbor vector || v is calculated according to formula (6). i || p ;

[0019]

[0020] As a further improvement of the present invention, in step 23, the boundary confidence BC is calculated according to formula (8): i ;

[0021]

[0022] As a further improvement of the present invention, step 3 includes the following steps:

[0023] Step 31: Divide the data set X into boundary sets X according to formula (9) B and core set X Core ,

[0024]

[0025] Where τ∈[0,1] is the boundary weight, It is a descending queue;

[0026] Step 32: According to formula (10), a directed connectivity graph is established between objects, the matrix is traversed, and the connected area is searched.

[0027]

[0028] As a further improvement of the present invention, in step 4, according to formula (11), the boundary objects are divided and the association between the boundary and the cluster skeleton is established.

[0029]

[0030] φ i Specifies the boundary object x i ∈X B Affiliation information.

[0031] The present invention also provides a boundary stripping clustering system based on space vector decomposition, comprising: a memory, a processor, and a computer program stored on the memory, wherein the computer program is configured to implement the steps of the boundary stripping clustering method of the present invention when called by the processor.

[0032] The present invention further provides a computer-readable storage medium storing a computer program, wherein the computer program is configured to implement the steps of the boundary stripping clustering method of the present invention when called by a processor.

[0033] The beneficial effects of the present invention are as follows: the present invention not only solves the limitation of the existing boundary stripping clustering that tends to have uniform distribution and spherical clusters, but also improves the performance of the present invention under complex distribution and high-dimensional data. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 is a flow chart of the method of the present invention;

[0035] Figure 2 is a schematic diagram of the boundary confidence measurement process, Figure 2 include Figure 2 (a) to Figure 2 (f);

[0036] Figure 3 This is a schematic diagram of the clustering visualization results of the synthetic dataset. DETAILED DESCRIPTION

[0037] 1. Introduction of the present invention:

[0038] Inspired by subspace clustering

[32]

[33] and space vector decomposition theory

[34] , this paper proposes a boundary peeling clustering method based on space vector decomposition (BP-SVD). This method projects the spatial relationship between data points and their neighbors from the original high-dimensional space to the low-dimensional subspace, and uses the distribution skewness (symmetry) of the boundary (core object) and its neighbors in the projected subspace and the distribution sparsity (density) in the original space to represent the discrete features, thereby achieving effective boundary peeling, and then guiding the object division through the clustering skeleton and density priority strategy. The work of this paper is as follows:

[0039] (1) Considering the objective distribution characteristics, we strengthen the feature representation of the boundary from the perspectives of the original space and the projected subspace, and propose a non-iterative boundary discrimination method;

[0040] (2) By utilizing the closure property of the cluster core area, a two-stage association strategy is proposed to reduce the redundant cascades between objects;

[0041] (3) A boundary-stripping clustering method based on space vector decomposition is proposed. Extensive experiments and in-depth analysis verify the effectiveness of the method in high-dimensional clustering and boundary pattern information extraction.

[0042] The boundary peeling clustering method based on space vector decomposition (BP-SVD) proposed in the present invention uses cluster skeleton as a benchmark to jointly guide object attribution, avoiding the chain reaction of division errors caused by one or several unreasonable peak points and improving parameter robustness.

[0043] The present invention does not require iterative optimization, and the judgment of boundary points no longer relies solely on density, but is based on objective distribution characteristics. It avoids the influence caused by unreasonable density measurement and difficulty in selecting neighborhood parameters, can effectively identify complex distributions and high-dimensional clusters, and has high clustering accuracy.

[0044] 2. The boundary peeling clustering method based on space vector decomposition (BP-SVD) proposed in this invention:

[0045] 2.1 Boundary Stripping:

[0046] In the present invention, we perform an affine transformation on the original data space, migrate the target object to the origin, and utilize the topological invariance of the affine transformation to more conveniently observe and analyze the positional relationship and distribution between the target object and its neighbors while maintaining the original structural information. From the perspective of multidimensional space, the skew characteristics of the boundary point in the original space can be manifested as the skewed distribution of its neighborhood objects around the boundary point in the direction of the basis vector, while for objects within the cluster, the uniformity in its neighborhood space is reflected in the approximate central symmetric distribution in the direction of the basis vector, that is, multidimensional symmetry. Based on this observation, we introduce the space vector decomposition theorem into the data space where the object is located, convert it into a vector space, and regard the pairwise relationships between objects as independent space vectors. Therefore, the local distribution estimate of the object is converted into the positional relationship judgment of the space vector, which is described as follows:

[0047] Assume that the dataset X={x1,x2,x3,...,x n}∈R m×n For any data point X, according to the space vector decomposition theorem, in the m-dimensional data space R mIn the , there exists a unique ordered real number λ1,λ2,λ3,...,λ m , so that x i =λ1e1+λ2e2+λ3e3+...+λ m e m , where e1,e2,e3,...,e m For space R m For any data point x in X i , and the data point x j The space vector h formed i,j It is expressed as follows:

[0048] h i,j =(λ i,1 -λ j,1 ,...,λ i,d -λ j,d ,...,λ i,m -λ j,m )(e1,e2,e3,...,e m ) T (Formula 1)

[0049] where λ i,d -λ j,d Indicates h i,j In the basis vector e d Considering that the feature differences between boundary objects and core objects mainly lie in their respective neighborhood spaces, we use k-nearest neighbors to extract neighborhood information. The neighborhood space formed by k-nearest neighbors contains rich neighborhood information.

[0050] Based on this, any x i ∈X and its k nearest neighbors form k space vectors. As the most direct and effective basic operation to describe the position relationship of vectors, the nearest neighbor vector V i Defined as x i The sum of the vectors formed by its k nearest neighbors is as follows:

[0051]

[0052] in, Represents x i The nearest neighbor vector of the basis vector e dThe projection coordinates on . According to the previous analysis, the neighbors of the objects within the cluster are relatively evenly distributed around them. After space vector decomposition, the above distribution pattern is directly manifested in the direction of each basis vector. The projection of the neighbor vector always shows strong central symmetry about the core object. In contrast, the projection of the neighbor vector of the boundary object has strong isotropy. Therefore, compared with the core object, the length of the neighbor vector of the boundary object is significantly longer. In other words, we can use the vector p-norm to identify the object. The following is a detailed derivation:

[0053] inference:

[0054] Assume h i,j For object x i ∈R m With object x j ∈R m The space vector composed of given one-dimensional feature space v d (1≤d≤m), vector h i,j The projection on it is where e d v d The basis vectors of .

[0055] prove:

[0056] The space vector decomposition theorem is used here as a lemma.

[0057] The statement is as follows: If S is R m The subspace of e′1,e'2,e'3,e'4,...,e' p is the basis vector of S. For any a∈S, there exists a=y1e1'+y2e2'+y3e3'+y4e4'+...+y p e p '. Let matrix A=[e1',e2',e3',e4',...,e p '] m×p , then a=Ay,y∈R p For the space vector h i,j ,Proj s h i,j Represented as h i,j Projection on the subspace S. According to the space decomposition theorem described above, Proj s h i,j ∈S and Proj s h i,j =Ay,y∈R p .

[0058] Let Proj s ⊥h i,j Indicates h i,jThe projection on the orthogonal complement of subspace S satisfies: h i,j =Proj s ⊥h i,j +Proj s h i,j . It is known that the subspace S is equal to the column space of matrix A, then the orthogonal complement of subspace S is equal to A T The null space of . For this, there is the following derivation:

[0059]

[0060] Without loss of generality, we assume that the subspace S is instantiated as a one-dimensional feature space s d , then the projection matrix A is transformed into a column vector e d (the basis vectors of this one-dimensional space), the projection is formalized as The proof is complete.

[0061] Orthogonality is a special vector relationship. Orthogonal decomposition is a common mode for analyzing space vectors. Its main contribution is to transform complex space vector operations into simple algebraic operations in mutually perpendicular directions based on the Cartesian rectangular coordinate system. With the help of the theoretical viewpoint of orthogonal decomposition, we use the standard orthogonal basis To instantiate the feature matrix A in formula 3, the object x i ∈R n The projection of the space vector formed by its neighbors on m orthogonal bases can be expressed as:

[0062]

[0063] Therefore, Equation 2 can be rewritten as:

[0064]

[0065] Furthermore, the p-norm of the neighbor vector is as follows:

[0066]

[0067] Considering the computational cost, the L1 norm becomes the first choice. In addition to the above-mentioned nearest neighbor projection norm, the closeness of the object neighborhood is also used as a judgment basis.

[0068] Definition 1. (Local Density) Density is actually an estimate of the closeness of the neighborhood of a data object. The exponential discretization of the sum of squared residuals between the object and its neighbors is used as the local density, which is defined as follows:

[0069]

[0070] The objects inside the cluster are densely distributed, and the value of the above formula is large. The objects at the boundary are relatively sparsely distributed, and the value is small.

[0071] Definition 2. (Boundary Confidence) Boundary confidence characterizes the degree to which a data point belongs to the boundary. i ∈R n ||v i || p＝1 and density ρ i The ratio of the point x i Boundary confidence BC i , which is expressed as follows:

[0072]

[0073] Boundary Confidence BC i The larger the value, the greater the data x i The greater the probability of being at the edge of the cluster. Since the judgment of boundary points is no longer based solely on density, but on objective distribution characteristics, it avoids the impact caused by density measurement imbalance and difficulty in determining neighborhood parameters.

[0074] Taking the Gaussian dataset as an example, Figure 2 The boundary confidence measurement process is shown, where data points x1 and x2 represent the boundary and core objects respectively; Figure 2 (b) Figure 2 (c) shows the local neighborhood of the above object after affine transformation. Figure 2 (d) and Figure 2 (e) shows the projection after space vector decomposition. It can be seen that the projections of x1 in two directions are mostly in the same direction, while those of x2 are symmetric; Figure 2 (f) The boundary confidence of the dataset is visualized, and it can be seen that its measurement effect is consistent with the actual situation.

[0075] 2.2 Cluster skeleton construction and boundary association:

[0076] According to the definition of boundary confidence, after the data points are sorted in descending order, the objects inside the cluster will be concentrated in the descending queue. The data set X can be divided into the boundary set X B and core set X Core ,as follows:

[0077]

[0078] Where τ∈[0,1] is the boundary weight.

[0079] Similar to DBSCAN, the latest boundary-peeling clustering methods such as BP[7] and RoBP

[31] establish the transitive closure between boundary points while identifying the boundaries, and finally associate the transitive closure of boundary points to the core area of the cluster according to the minimum distance principle. Due to the complexity of the area where the boundary points are located, the transitive closure between boundaries may cause the boundaries that originally belonged to different clusters to establish incorrect associations, which in turn causes error accumulation and cluster merging. Therefore, the cascade between boundaries is often redundant.

[0080] To this end, this paper proposes a two-stage association strategy to reduce redundant associations between objects. After boundary stripping, the remaining objects have good closure properties, and core clusters can be formed by merging the reachable neighbors of the core points.

[0081] Definition 3. (Reachable Neighbors) Any core point x i and x j Has a reachable neighbor relationship if and only if there is a chain relationship (x i ,...x r ,x r+1 ,...x j ), where any adjacent point pair (x r ,x r+1 ) satisfies d(x r ,x r+1 )<max(d(x r+1 ,{N k (x r )}). Formally, we use k-nearest neighbors to model the reachable neighbors between objects into a directed connected graph, represented by the relationship matrix M:

[0082]

[0083] The connected region in the asymmetric matrix M is the skeleton of the cluster. Assuming that there are K connected regions in the matrix M, the core set X core It can be expressed as Represents the backbone object of cluster t. Boundary points x i The association rules are as follows:

[0084]

[0085] φ i Specifies the boundary object x i ∈X B In general, the association criterion in the first stage of object partitioning is transitive reachable neighbors, which only focuses on the establishment of cluster skeletons, while the criterion in the second stage is density priority, which aims to establish the association between boundary points and skeletons.

[0086] In summary, if Figure 1As shown in FIG, the boundary stripping clustering method based on space vector decomposition includes the following steps:

[0087] Step 1, input step: input data set X = {x1, x2, x3, ..., x n}∈R m×n ;

[0088] Step 2, boundary stripping step: identify the object type based on the object neighborhood distribution characteristics to strip boundary points;

[0089] Step 3, cluster skeleton construction step: determine the cluster skeleton based on the transitive closure relationship;

[0090] Step 4, boundary association step: establish the affiliation relationship between the boundary and the cluster skeleton to complete clustering;

[0091] Step 5, output step: output clustering results.

[0092] 3 Experiments:

[0093] In order to fully verify the effectiveness of the BP-SVD proposed in this paper, in addition to K-means

[35] and DPC[8], we use the latest and representative four clustering algorithms (BP[7], EC

[11] , GB-DPC[9], SNN-DPC

[41] ) as baselines.

[0094] For K-means, we give the correct cluster number parameter; for DPC, we select the correct number of peak points as cluster centers; for other adjustable parameters involved in the algorithm, we set reasonable parameter value ranges for BP, DPC, SNN-DPC, EC, and GB-DPC. The detailed settings are shown in Table 1.

[0095] In addition to the nine common synthetic datasets, we selected the UCI dataset and image dataset, which cover data features such as complex distribution, high dimension, large sample size, and multiple clusters. See Table 2 for detailed introduction.

[0096] In order to comprehensively evaluate the performance of the algorithm, the present invention uses five evaluation indicators (Accuracy: ACC, Purify, Normalized Mutual Information: Adjusted Rand Index: ARI, Fowlkes-Mallows index: FMI, Jaccard Coefficient: JC) to evaluate the clustering results.

[0097] Table 1 Algorithm parameter settings

[0098]

[0099] Table 2 Dataset introduction

[0100]

[0101] 3.1 Synthetic Dataset

[0102] Based on the density peak concept, DPC, SNN-DPC, GB-DPC, and EC are significantly better than traditional algorithms. However, the original DPC and some variants use global density metrics, and even if the correct parameter values are given in the experiment, they still cannot effectively handle multi-density distributions. Figure 3 As shown in (6), DPC identified the correct number of clusters on the Compound dataset, but two of the cluster centers were located in the same cluster, completely ignoring the adjacent low-density areas. Secondly, the parameter dc is relatively sensitive and difficult to determine the appropriate value. Even a small change can easily cause large fluctuations in the clustering results, such as Figure 3 (18) and Flame in Table 3, when dc is determined to be 0.93 based on the empirical rule proposed in the original paper, DPC assigns some objects in the bottom cluster to the top region. However, when dc = 1.07 (only a 0.14 difference), the clustering result is very ideal (ACC = 1). In fact, the actual performance of DPC may be lower than the experimental data in Table 3, because decision diagrams in real scenarios tend to produce visual errors.

[0103] In contrast, SNN-DPC performs more balanced. This is due to the fact that SNN-DPC not only considers distance information when clustering, but also integrates neighbor information and shared neighbor information, and uses this structural information to propose a new similarity metric. As a result, the algorithm can produce more competitive results, especially on the Aggregation and Spiral datasets. However, the algorithm involves complex multiple nested loops, and the high computational cost limits its application scenarios.

[0104] As the latest improvement of DPC, GB-DPC and EC have realized process automation to ensure the continuity of cluster flow. EC uses the uniqueness of the extreme density points in the active neighborhood to automatically screen the cluster center, while GB-DPC automatically locates the boundary between the cluster center and other objects based on the maximum interval after the statistic γ is in descending order. Figure 3 (7)(27)(31) and Table 3 show that the two algorithms are not stable because both EC and GB-DPC use the density metric of the original DPC. When faced with complex distributions, unreasonable density estimation often leads to failure in cluster center selection (low representativeness, too many, too few). For example, for the Jain dataset with two clusters, the parameter value dc = 1.35 is the same as DPC, but GB-DPC identifies the dataset as 8 clusters, and the accuracy (ACC = 0.35) is much lower than DPC, SNN-DPC, and EC; similarly, EC clusters the Flame dataset into 3 clusters, while there are actually 2 clusters.

[0105] For variable density and non-convex data distributions such as Jain, Compound, Aggregation, the BP algorithm that relies only on density features for boundary stripping does not accurately identify the basic shape of the cluster, such as Figure 3 (5), (9). There are two main factors that cause this problem. First, the discrimination method based on Gaussian kernel density is too simplistic. Especially when facing unevenly distributed clusters, the BP algorithm often misjudges sparse areas as boundaries, which leads to the evolution of sparse clusters into multiple subclusters, that is, over-division. For example, BP classifies the Parabolic dataset, which originally contained only two manifold clusters, into 17 microclusters. Second, the layer-by-layer approximation boundary recognition method often causes error accumulation. Since subsequent peeling must be based on the current peeling results, when some objects are misclassified, the subsequent peeling process will be misled, resulting in a domino effect of error transmission, such as the spiral dataset.

[0106] Combine Figure 3 As shown in Table 3, some algorithms actually have good clustering performance, but their values for some metrics are very low, or even zero. For example, regarding the JC metric, K-means achieves very low values on 4k2-far, DPC on R15, and GB-DPC on Parabolic. This is due to the biased nature of the metric's evaluation criteria: clustering is evaluated by comparing whether the numerical value of each data object's label is strictly identical to the ground truth, while the evaluation of clustering results is primarily based on the overall consistency of objects within a cluster. Taking Jain as an example (two manifold clusters: 1-97: Label 1; 98-373: Label 2), BP-SVD classifies objects 1-97 into cluster 2 and objects 98-373 into cluster 1. Therefore, JC = (0+0) / 373 = 0. Overall, our BP-SVD algorithm achieves ideal clustering on datasets with different distribution characteristics.

[0107] Table 3 Clustering results of synthetic datasets

[0108]

[0109]

[0110] 3.2 UCI Dataset

[0111] Table 4 shows the clustering results of the algorithm on the UCI dataset.

[0112] SNN-DPC ensures stability when used with real data by integrating various nearest neighbor information. Furthermore, experimental data from SNN-DPC demonstrates that nearest neighbor sampling is more effective than a fixed radius for density estimation. For example, SNN improves the accuracy of original DPC by 0.353 and 0.146 on the Spectheart and Bank datasets, respectively. However, SNN-DPC requires specifying the actual number of clusters, and obtaining prior data information is often difficult.

[0113] Theoretically, the parameter ranges of DPC:dc, EC:dc, and GB-DPC:dc all lie in the space of positive real numbers. For low-dimensional data, the algorithms can roughly estimate the values based on their intuitive distribution. However, for high-dimensional data with sparse distributions that are difficult to visualize, there is a lack of effective visual reference for hyperparameter selection. Given the relatively sparse distribution of high-dimensional data, crude metrics like the cutoff distance dc are unable to describe density differences among objects, let alone the cluster center discrimination criterion based on this distance. Consequently, the clustering results of the three algorithms are unstable. For example, DPC performs worst on Parkinson, Wine, and Optical Recognition. GB-DPC achieves higher clustering accuracy than DPC on nine datasets, but also falls short on three datasets. Because EC's cluster center search strategy strictly relies on a threshold parameter, density imbalances can amplify this negative effect, resulting in overall performance inferior to GB-DPC, such as on the German and Hepatitics datasets. However, it is undeniable that GB-DPC and EC are significantly more applicable in real-world scenarios than DPC and SNN-DPC.

[0114] The BP algorithm has six hyperparameters: the maximum number of iterations T, the boundary judgment threshold b, the number of neighbors k, the outlier judgment threshold λ, the iteration stop threshold ε, and the threshold confidence C. It can be seen that in most cases, the BP algorithm performs slightly better than or equivalent to DPC, SNN-DPC, and GB-DPC, which indirectly verifies that the boundary peeling algorithm is effective in high-dimensional situations.

[0115] However, from the perspective of the algorithm itself, the boundary stripping process is a time-consuming iterative process, and ultimately requires calling a simplified version of DBSCAN to complete the core object division, which is relatively inefficient.

[0116] Table 4 Clustering results of the UCI dataset

[0117]

[0118]

[0119] As an essential component of density clustering, boundary points play a crucial role in guiding cluster convergence and improving pattern recognition capabilities. Boundary-peeling clustering, represented by the BP algorithm, leverages boundary information to ensure spatial isolation of cluster cores, thereby improving the representativeness of cluster skeletons and guiding boundary affiliation. However, existing boundary-peeling clustering methods are generally subject to limitations such as imperfect discrimination criteria, a single discrimination model, and multiple iterations. To address this, this paper proposes a boundary-peeling clustering method using space vector decomposition (BP-SVD). This method first leverages space vector decomposition theory to enhance boundary feature representation in both the projected subspace and the original space. It then leverages the strong closedness between internal objects to establish a cluster skeleton. Finally, a two-stage association strategy drives the direct affiliation between the cluster skeleton and the boundary, completing the clustering task. Experimental results on 33 synthetic, UCI, and image datasets, as well as in-depth analysis from four perspectives, validate the effectiveness of the proposed method. Compared to six state-of-the-art algorithms, the proposed method demonstrates superior overall performance in both clustering and boundary recognition.

[0120] 4 Conclusion

[0121] The present invention peels off the cluster boundaries by integrating the dimensional skewness (symmetry) and spatial sparsity (compactness) of boundary points (core points), and proposes a boundary peeling clustering method BP-SVD based on spatial vector decomposition. It not only solves the limitation of existing boundary peeling clustering that tends to have uniform distribution and spherical clusters, but also improves the performance of the present invention under complex distribution and high-dimensional data. The effectiveness of the present invention can be attributed to the following factors. First, the judgment of boundary points no longer relies solely on density. The present invention strengthens the feature representation of boundaries from two perspectives: the original data space and the projected subspace. Second, the proposed two-stage object association strategy avoids redundant cascades between boundaries, thereby reducing the closure transmission of object partitioning errors. In addition, the BP-SVD algorithm does not require iterative optimization, and its hyperparameters are fewer than those of similar algorithms and are easy to determine. Finally, the present invention compares the clustering performance of the proposed algorithm with EC, DPC, BP, K-means, GB-DPC, and SNN-DPC from theoretical analysis and experimental data. The experimental results of 33 data sets (synthetic, UCI, image) and in-depth analysis of five dimensions (robustness, complexity, statistical ranking, performance, and boundary pattern recognition) demonstrate the effectiveness of the present invention in high-dimensional clustering and boundary pattern information extraction.

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[0166] The above is a further detailed description of the present invention in conjunction with specific preferred embodiments, and the specific implementation of the present invention should not be considered to be limited to these descriptions. For those skilled in the art of the present invention, without departing from the concept of the present invention, several simple deductions or substitutions can be made, which should be considered to fall within the scope of protection of the present invention.

Claims

1. A boundary stripping clustering method based on space vector decomposition, characterized in that: The steps include: Step 1, input step: input data set X = {x1, x2, x3, ..., x n }∈R m×n ; The input data set includes a picture data set; Step 2, boundary stripping step: identify the object type based on the object neighborhood distribution characteristics to strip boundary points; Step 3, cluster skeleton construction step: determine the cluster skeleton based on the transitive closure relationship; Step 4, boundary association step: establish the affiliation relationship between the boundary and the cluster skeleton to complete clustering; Step 5, output step: output clustering results; The step 3 comprises the following steps: Step 31: Divide the data set X into boundary sets X according to formula (9) B and core set X Core , Where τ∈[0,1] is the boundary weight, is a descending queue; ξ(x i ) is an indicator variable, if ξ(x i ) is equal to 1, indicating that the object is a boundary object. If ξ(x i ) is 0, then the object is a core object; BC i represents the boundary confidence of the object; n represents the number of objects in the dataset; Step 32: According to formula (10), a directed connectivity graph is established between objects, the matrix is traversed, and the connected area is searched. In step 4, according to formula (11), the boundary objects are divided and the association between the boundary and the cluster skeleton is established. φ i Specifies the boundary object x i ∈X B Affiliation information.

2. The boundary stripping clustering method according to claim 1, characterized in that: The step 2 comprises the following steps: Step 21: Calculate the density ρ i ; Step 22: Calculate the norm of the nearest neighbor vector ||v i || p ; Step 23: Calculate Boundary Confidence 3. The boundary stripping clustering method according to claim 2, characterized in that: In step 21, the density ρ is calculated according to formula (7) i ; k represents the neighbor parameter, N k (x i ) represents the object x i The k nearest neighbor set of d(x i ,x j ) represents the object x i and x j The L2 distance between them.

4. The boundary stripping clustering method according to claim 3, characterized in that: In step 22, the norm of the nearest neighbor vector ||v is calculated according to formula (6) i || p ; x id 、x jd Represents object x respectively i and x i The value on dimension d, that is, the d-th attribute value; P represents the P paradigm; x j Represents object j.

5. The boundary stripping clustering method according to claim 4, characterized in that: In step 23, the boundary confidence BC is calculated according to formula (8): i ; 6. A boundary-peeling clustering system based on space vector decomposition, characterized in that: include: A memory, a processor, and a computer program stored in the memory, wherein the computer program is configured to implement the steps of the boundary stripping clustering method according to any one of claims 1 to 5 when called by the processor.

7. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program, and the computer program is configured to implement the steps of the boundary stripping clustering method according to any one of claims 1 to 5 when called by a processor.

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