DK-GBMKKM: Dynamic Kernel-Space Granular-Ball Multiple Kernel $k$-Means Clustering

# Dynamic Kernel-Space Granular-Ball Multiple Kernel 𝑘-Means Clustering
Source: [https://arxiv.org/html/2609.00647](https://arxiv.org/html/2609.00647)
## DK\-GBMKKM: Dynamic Kernel\-Space Granular\-Ball Multiple Kernelkk\-Means Clustering Thanks:X\. Lian, Y\. Zhang, S\. Xia, S\. Zhong & X\. Xiang are with the Chongqing Key Laboratory of Computational Intelligence, Key Laboratory of Cyberspace Big Data Intelligent Security, Ministry of Education, Sichuan\-Chongqing Co\-construction Key Laboratory of Digital Economy Intelligence and Key Laboratory of Big Data Intelligent Computing, Chongqing University of Posts and Telecommunications, 400065, Chongqing, China\.

1stXiaoyu LianAffiliation:Chongqing University of Posts and Telecommunications Chongqing, China, lianxiaoyu724@qq\.com2ndYuchao ZhangAffiliation:Chongqing University of Posts and Telecommunications Chongqing Open University Chongqing, China, weiyanshiai@qq\.com3rdShuyin Xia\*Affiliation:Chongqing University of Posts and Telecommunications Chongqing, China, xiasy@cqupt\.edu\.cn4thSiqi ZhongAffiliation:Chongqing University of Posts and Telecommunications Chongqing, China, 3572288058@qq\.com5thZhaoxu XiangAffiliation:Chongqing University of Posts and Telecommunications Chongqing, China, 2130453184@qq\.com

###### Abstract

Multiple kernelkk\-means integrates complementary nonlinear similarities by learning a combination of base kernels\. Its pointwise optimization, however, is sensitive to noisy and boundary samples and repeatedly operates on sample\-scale kernel matrices\. Granular\-ball representations organize local sample groups into mesoscopic units, but granular balls generated once in the input space may be inconsistent with the fused\-kernel geometry that evolves during multiple kernel learning\. We propose dynamic kernel\-space granular\-ball multiple kernelkk\-means \(DK\-GBMKKM\)\. The method generates granular balls in the current fused kernel space and alternates kernel\-weight learning with granular\-ball membership updates, allowing the representation to adapt to changes in the fused\-kernel geometry\. A sample\-size\-weighted granular\-ball kernel is further constructed to preserve the contributions of balls of different sizes, and its positive semidefiniteness and related equivalence properties are established\. Experiments on 12 public datasets demonstrate the strong overall clustering performance of DK\-GBMKKM\. The code has been open\-sourced for reproducibility: https://github\.com/lianxiaoyu724/DK\-GBMKKM\.

###### Index Terms:

Multiple kernel clustering, multiple kernel

kk\-means, granular\-ball computing, kernel space, granular ball\.

## IIntroduction

Classicalkk\-means clustering is widely used in image analysis, text mining, bioinformatics, and multimedia processing because of its simplicity, computational efficiency, and ease of implementation\[[1](https://arxiv.org/html/2609.00647#bib.bib1),[2](https://arxiv.org/html/2609.00647#bib.bib3),[3](https://arxiv.org/html/2609.00647#bib.bib2)\]\. To overcome the limitations of Euclidean distance in the input space, spectral clustering constructs a sample\-similarity graph, reformulates clustering as graph partitioning, and derives a low\-dimensional embedding from eigenvectors of the graph Laplacian\[[4](https://arxiv.org/html/2609.00647#bib.bib4),[5](https://arxiv.org/html/2609.00647#bib.bib5),[6](https://arxiv.org/html/2609.00647#bib.bib6)\]\. Kernelkk\-means instead maps samples implicitly into a reproducing kernel Hilbert space \(RKHS\) and clusters in that space, improving the representation of nonlinear structures\[[7](https://arxiv.org/html/2609.00647#bib.bib7),[8](https://arxiv.org/html/2609.00647#bib.bib8)\]\.

Although kernel clustering can capture nonlinear structures, the expressive power of a single kernel is limited\. Multiple kernel clustering combines several base kernels to learn task\-adaptive similarities\[[9](https://arxiv.org/html/2609.00647#bib.bib10),[10](https://arxiv.org/html/2609.00647#bib.bib11)\]\. Multiple kernelkk\-means \(MKKM\) is a representative framework that jointly optimizes kernel weights and cluster assignments\[[11](https://arxiv.org/html/2609.00647#bib.bib9)\], with subsequent extensions improving robustness, structural modeling, and scalability\[[12](https://arxiv.org/html/2609.00647#bib.bib12),[13](https://arxiv.org/html/2609.00647#bib.bib13)\]\. Nevertheless, most existing methods remain sample\-centric, making them sensitive to noisy, boundary, and outlying samples and computationally expensive for large kernel matrices\. A stable and efficient mesoscopic representation is therefore highly desirable\.

Granular\-ball computing \(GBC\) has recently emerged as an adaptive multigranularity representation\. Rather than learning directly from individual samples, GBC approximates an arbitrary data distribution with granular balls characterized by centers, radii, and sample coverage\. It thus replaces many samples with fewer and more stable mesoscopic units\[[14](https://arxiv.org/html/2609.00647#bib.bib14),[15](https://arxiv.org/html/2609.00647#bib.bib15)\]\. GBC offers inherent advantages in efficiency, robustness, and interpretability and has been integrated with classifiers, rough sets, fuzzy sets, and graph learning to build stable multigranularity learning frameworks\[[16](https://arxiv.org/html/2609.00647#bib.bib16),[17](https://arxiv.org/html/2609.00647#bib.bib17),[18](https://arxiv.org/html/2609.00647#bib.bib18),[19](https://arxiv.org/html/2609.00647#bib.bib19),[20](https://arxiv.org/html/2609.00647#bib.bib20)\]\. It has also been used in clustering to reduce computational complexity and improve robustness to noise\[[21](https://arxiv.org/html/2609.00647#bib.bib38),[22](https://arxiv.org/html/2609.00647#bib.bib21),[23](https://arxiv.org/html/2609.00647#bib.bib23),[24](https://arxiv.org/html/2609.00647#bib.bib29),[25](https://arxiv.org/html/2609.00647#bib.bib25),[26](https://arxiv.org/html/2609.00647#bib.bib28),[27](https://arxiv.org/html/2609.00647#bib.bib24),[28](https://arxiv.org/html/2609.00647#bib.bib26),[29](https://arxiv.org/html/2609.00647#bib.bib27)\]\. Introducing GBC into multiple kernel clustering is therefore a natural direction\. Granular\-ball\-induced multiple kernel k\-means \(GB\-MKKM\)\[[30](https://arxiv.org/html/2609.00647#bib.bib22)\]first embedded granular balls into MKKM\. By constructing balls in the input space and compressing the sample set, it improved efficiency and robustness and demonstrated the feasibility of mesoscopic units for multiple kernel clustering\.

GB\-MKKM and related methods nevertheless construct granular balls in the input space\. This design implicitly assumes that samples close in the input space remain close in the high\-dimensional feature spaces induced by the kernels\. The assumption is reasonable only for approximately order\-preserving mappings, such as a linear kernel\. Under commonly used nonlinear kernels, including Gaussian and polynomial kernels, neighborhood relations can change substantially after mapping\. Input\-space ball boundaries may then fail to reflect the actual local density and cluster structure in kernel space\. In addition, a fixed granular\-ball partition cannot adapt as the kernel weights and fused\-kernel geometry evolve\. To address these limitations, we propose dynamic kernel\-space granular\-ball multiple kernelkk\-means \(DK\-GBMKKM\), which aligns granular\-ball construction with the space in which multiple kernel clustering is optimized\. The main contributions are as follows:

- •We propose a fused\-kernel\-driven dynamic granular\-ball generation mechanism, where granular balls are constructed and updated directly in the current fused kernel space while keeping the ball number fixed\.
- •We develop an alternating optimization framework that jointly updates the granular\-ball structure, spectral representation, and kernel weights in the compressed ball space\.
- •Extensive experiments on 12 public datasets show that DK\-GBMKKM outperforms seven representative methods in terms of average performance and overall ranking across four clustering metrics\.

Rather than applying multiple kernel clustering directly to sample\-level kernel matrices, we construct multiple granular\-ball kernels and perform clustering on mesoscopic units that encode local structure\.

### II\-AMotivation

Existing granular\-ball multiple kernel clustering methods typically construct granular balls in the input space and then build ball\-level kernel representations\. This causes a space mismatch, as the ball structure follows input\-space geometry while clustering is optimized in the fused kernel space\. Moreover, fixed granular\-ball partitions cannot adapt to changing kernel weights and may become inconsistent with the evolving fused\-kernel geometry\. DK\-GBMKKM addresses both issues, as illustrated in Fig\.[1](https://arxiv.org/html/2609.00647#S2.F1)\. It first builds a fused kernel matrixKKfrom the current kernel weights and generates granular balls in the induced kernel space\. Ball\-level kernel matrices are then computed from within\- and between\-ball kernel relations, followed by spectral embedding and kernel\-weight learning at the ball level\. After the weights are updated, the fused kernel is reconstructed and the granular\-ball memberships are adjusted\. The representation and the multiple kernel model therefore evolve together\.

![Refer to caption](https://arxiv.org/html/2609.00647v1/DK_GBMKKM.png)Fig\. 1:Framework of DK\-GBMKKM\.
### II\-BProblem Formulation and Base Model

Let𝒳=\{xi\}i=1n\\mathcal\{X\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{n\}be a dataset withcctarget clusters\. GivenPPpositive semidefinite base kernel matrices\{K\(p\)\}p=1P\\\{K^\{\(p\)\}\\\}\_\{p=1\}^\{P\}, each of which has been symmetrized, centered, and diagonal\-normalized, letK\(p\)∈ℝn×nK^\{\(p\)\}\\in\\mathbb\{R\}^\{n\\times n\}\. The kernel\-weight vectorσ=\[σ1,…,σP\]⊤\\sigma=\[\\sigma\_\{1\},\\ldots,\\sigma\_\{P\}\]^\{\\top\}satisfies

\{σ∣σp≥0,∑p=1Pσp=1\}\.\\left\\\{\\sigma\\mid\\sigma\_\{p\}\\geq 0,\\ \\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}=1\\right\\\}\.\(1\)We use the squared\-weight kernel combination

Kσ=∑p=1Pσp2​K\(p\)\.K\_\{\\sigma\}=\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}K^\{\(p\)\}\.\(2\)
Kernel\-space granular ball:In the RKHSℋ\\mathcal\{H\}induced by the current fused kernelKσK\_\{\\sigma\}, theℓ\\ellth granular ball is denoted byBℓ=\(ℐℓ,Cℓ,Rℓ,CCMℓ\)B\_\{\\ell\}=\(\\mathcal\{I\}\_\{\\ell\},C\_\{\\ell\},R\_\{\\ell\},\\text\{CCM\}\_\{\\ell\}\)\. Here,ℐℓ\\mathcal\{I\}\_\{\\ell\}is the set of samples covered by the ball,nℓ=\|ℐℓ\|n\_\{\\ell\}=\|\\mathcal\{I\}\_\{\\ell\}\|is its size,CℓC\_\{\\ell\}andRℓR\_\{\\ell\}are its center and radius in kernel space, andCCMℓ\\text\{CCM\}\_\{\\ell\}is its kernel\-space center\-consistency measure\. The corresponding derivation is provided in Appendix[B](https://arxiv.org/html/2609.00647#A2)\.

The kernel\-space center is the mean of the mapped samples in the ball:

Cℓ=1nℓ​∑xi∈ℐℓϕ⁡\(xi\),C\_\{\\ell\}=\\frac\{1\}\{n\_\{\\ell\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{\\ell\}\}\\phi\(x\_\{i\}\),\(3\)whereϕ⁡\(xi\)\\phi\(x\_\{i\}\)is the implicit mapping ofxix\_\{i\}into the RKHS induced byKσK\_\{\\sigma\}\. For any samplexx, its squared distance toCℓC\_\{\\ell\}is

dℋ2​\(x,Cℓ\)=‖ϕ⁡\(x\)−Cℓ‖ℋ2\\displaystyle d\_\{\\mathcal\{H\}\}^\{2\}\(x,C\_\{\\ell\}\)=\\left\\\|\\phi\(x\)\-C\_\{\\ell\}\\right\\\|\_\{\\mathcal\{H\}\}^\{2\}\(4\)=Kσ​\(x,x\)−2nℓ​∑xi∈ℐℓKσ​\(x,xi\)\+1nℓ2​∑xi∈ℐℓ∑xj∈ℐℓKσ​\(xi,xj\)\.\\displaystyle=K\_\{\\sigma\}\(x,x\)\-\\frac\{2\}\{n\_\{\\ell\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{\\ell\}\}K\_\{\\sigma\}\(x,x\_\{i\}\)\+\\frac\{1\}\{n\_\{\\ell\}^\{2\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{\\ell\}\}\\sum\_\{x\_\{j\}\\in\\mathcal\{I\}\_\{\\ell\}\}K\_\{\\sigma\}\(x\_\{i\},x\_\{j\}\)\.Thus, the distance is obtained entirely fromKσK\_\{\\sigma\}without explicitly computing a center vector\. The maximum and mean radii ofBℓB\_\{\\ell\}are

Rℓ=maxxi∈ℐℓ⁡dℋ​\(xi,Cℓ\),R¯ℓ=1nℓ​∑xi∈ℐℓdℋ​\(xi,Cℓ\)\.R\_\{\\ell\}=\\max\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{\\ell\}\}d\_\{\\mathcal\{H\}\}\(x\_\{i\},C\_\{\\ell\}\),\\qquad\\bar\{R\}\_\{\\ell\}=\\frac\{1\}\{n\_\{\\ell\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{\\ell\}\}d\_\{\\mathcal\{H\}\}\(x\_\{i\},C\_\{\\ell\}\)\.\(5\)
Because label purity is unavailable in unsupervised clustering, we adapt the center\-consistency measure\[[22](https://arxiv.org/html/2609.00647#bib.bib21)\]by replacing all sample distances with kernel\-space distances\. Letχℓ=\{xi∈ℐℓ∣dℋ​\(xi,Cℓ\)≤R¯ℓ\}\\chi\_\{\\ell\}=\\\{x\_\{i\}\\in\\mathcal\{I\}\_\{\\ell\}\\mid d\_\{\\mathcal\{H\}\}\(x\_\{i\},C\_\{\\ell\}\)\\leq\\bar\{R\}\_\{\\ell\}\\\}denote the samples within the mean radius\. The dimension of a kernel space is generally unavailable explicitly, so we use radius\-normalized densities and thereby avoid a direct dependence on dimensionality\. The consistency of a singleton or zero\-radius ball is set to 1\. The densities within the mean and maximum radii are

ρℓave=\|χℓ\|R¯ℓ,ρℓmax=nℓRℓ\.\\rho\_\{\\ell\}^\{\\text\{ave\}\}=\\frac\{\|\\chi\_\{\\ell\}\|\}\{\\bar\{R\}\_\{\\ell\}\},\\qquad\\rho\_\{\\ell\}^\{\\max\}=\\frac\{n\_\{\\ell\}\}\{R\_\{\\ell\}\}\.\(6\)These ratios quantify compactness by relating the number of covered samples to the corresponding radius\. The kernel\-space center\-consistency measure is then

CCMℓ=min⁡\(ρℓave,ρℓmax\)max⁡\(ρℓave,ρℓmax\)∈\(0,1\]\.\\text\{CCM\}\_\{\\ell\}=\\frac\{\\min\(\\rho\_\{\\ell\}^\{\\text\{ave\}\},\\,\\rho\_\{\\ell\}^\{\\max\}\)\}\{\\max\(\\rho\_\{\\ell\}^\{\\text\{ave\}\},\\,\\rho\_\{\\ell\}^\{\\max\}\)\}\\in\(0,1\]\.\(7\)A value close to 1 indicates a uniform and stable kernel\-space distribution, whereas a smaller value suggests that the ball should be refined\. The granular balls are generated in the current fused kernel space by combining this measure with kernel 2\-means in the GBC procedure\[[22](https://arxiv.org/html/2609.00647#bib.bib21)\]\.

Granular\-ball kernel construction:The current fused kernelKσK\_\{\\sigma\}induces a single granular\-ball partition shared by all base kernels\. SinceKσK\_\{\\sigma\}is a weighted combination of the base kernels, each fused\-space ball center admits a consistent decomposition in the corresponding base\-kernel spaces; see Appendix[B](https://arxiv.org/html/2609.00647#A2)\. Therefore, only one partition is generated, and the same index sets\{ℐb\}b=1M\\\{\\mathcal\{I\}\_\{b\}\\\}\_\{b=1\}^\{M\}are used for all base kernels\.

For two ballsBaB\_\{a\}andBbB\_\{b\}, their similarity in theppth base\-kernel space is defined as the inner product between their centers:

K¯B\(p\)​\(a,b\)\\displaystyle\\overline\{K\}\_\{B\}^\{\(p\)\}\(a,b\)=⟨Ca\(p\),Cb\(p\)⟩=1na​nb​∑xi∈ℐa∑xj∈ℐbK\(p\)​\(xi,xj\)\.\\displaystyle=\\left\\langle C\_\{a\}^\{\(p\)\},C\_\{b\}^\{\(p\)\}\\right\\rangle=\\frac\{1\}\{n\_\{a\}n\_\{b\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{a\}\}\\sum\_\{x\_\{j\}\\in\\mathcal\{I\}\_\{b\}\}K^\{\(p\)\}\(x\_\{i\},x\_\{j\}\)\.\(8\)This kernel\-space inner product is computed solely from a sample\-level kernel matrix; the base\-space centers need not be formed explicitly\.

Define the sample\-to\-ball indicator matrixGGby

Gi​b=\{1,xi∈Bb,0,otherwise,∑b=1MGi​b=1,i=1,…,n\.G\_\{ib\}=\\begin\{cases\}1,&x\_\{i\}\\in B\_\{b\},\\\\ 0,&\\text\{otherwise\},\\end\{cases\}\\qquad\\sum\_\{b=1\}^\{M\}G\_\{ib\}=1,\\quad i=1,\\ldots,n\.\(9\)Thus,G∈\{0,1\}n×MG\\in\\\{0,1\\\}^\{n\\times M\}\. Define the ball\-size matrix as

D=G⊤​G=diag⁡\(n1,…,nM\)\.D=G^\{\\top\}G=\\diag\(n\_\{1\},\\ldots,n\_\{M\}\)\.\(10\)By Eq\. \([8](https://arxiv.org/html/2609.00647#S2.E8)\), the average granular\-ball kernel for base kernelppis

K¯B\(p\)=D−1​G⊤​K\(p\)​G​D−1,\\overline\{K\}\_\{B\}^\{\(p\)\}=D^\{\-1\}G^\{\\top\}K^\{\(p\)\}GD^\{\-1\},\(11\)whose the\(a,b\)\(a,b\)entry is⟨Ca\(p\),Cb\(p\)⟩\\langle C\_\{a\}^\{\(p\)\},C\_\{b\}^\{\(p\)\}\\rangle\.

Eq\. \([11](https://arxiv.org/html/2609.00647#S2.E11)\) measures kernel\-space similarity between ball centers but ignores ball sizes, which may underweight larger balls\. We therefore scale the average kernel symmetrically by the square roots of the ball sizes, omitting the common factor1/n1/nsince the total sample size is fixed\.

K^B\(p\)\\displaystyle\\widehat\{K\}\_\{B\}^\{\(p\)\}=D12​K¯B\(p\)​D12=D−12​G⊤​K\(p\)​G​D−12=Q⊤​K\(p\)​Q\.\\displaystyle=D^\{\\frac\{1\}\{2\}\}\\overline\{K\}\_\{B\}^\{\(p\)\}D^\{\\frac\{1\}\{2\}\}=D^\{\-\\frac\{1\}\{2\}\}G^\{\\top\}K^\{\(p\)\}GD^\{\-\\frac\{1\}\{2\}\}=Q^\{\\top\}K^\{\(p\)\}Q\.\(12\)Its\(a,b\)\(a,b\)th entry satisfies

K^B\(p\)​\(a,b\)=na​nb​⟨Ca\(p\),Cb\(p\)⟩,\\displaystyle\\widehat\{K\}\_\{B\}^\{\(p\)\}\(a,b\)=\\sqrt\{n\_\{a\}n\_\{b\}\}\\left\\langle C\_\{a\}^\{\(p\)\},C\_\{b\}^\{\(p\)\}\\right\\rangle,\(13\)Q=GD−12,Q⊤Q=IM\.\\displaystyle Q=GD^\{\-\\frac\{1\}\{2\}\},\\qquad Q^\{\\top\}Q=I\_\{M\}\.The weighted kernel retains the similarity between ball centers while encoding the number of samples represented by each ball\.

Consistent with the sample\-level kernel combination, the fused granular\-ball kernel is

K^B,σ=∑p=1Pσp2​K^B\(p\)\.\\widehat\{K\}\_\{B,\\sigma\}=\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}\\widehat\{K\}\_\{B\}^\{\(p\)\}\.\(14\)The originaln×nn\\times nmultiple kernel representation is thereby converted into anM×MM\\times Mrepresentation\.

###### Proposition 1

IfK\(p\)⪰0K^\{\(p\)\}\\succeq 0, thenK^B\(p\)⪰0\\widehat\{K\}\_\{B\}^\{\(p\)\}\\succeq 0\. Moreover, for anyσ\\sigmasatisfying Eq\. \([1](https://arxiv.org/html/2609.00647#S2.E1)\), the fused granular\-ball kernelK^B,σ\\widehat\{K\}\_\{B,\\sigma\}is positive semidefinite\.

This property ensures that the constructed matrices are valid kernels and can be used directly for kernel clustering and multiple kernel learning\. The proof is given in Appendix[C\-A](https://arxiv.org/html/2609.00647#A3.SS1)\.

###### Proposition 2

IfZ⊤​Z=IcZ^\{\\top\}Z=I\_\{c\}andH=QH=Q, thenH⊤​H=IcH^\{\\top\}H=I\_\{c\}, and for any base kernelK\(p\)K^\{\(p\)\},

tr⁡\(H⊤​K\(p\)​H\)=tr⁡\(Z⊤​K^B\(p\)​Z\)\.\\tr\(H^\{\\top\}K^\{\(p\)\}H\)=\\tr\(Z^\{\\top\}\\widehat\{K\}\_\{B\}^\{\(p\)\}Z\)\.\(15\)

As shown in Appendix[C\-B](https://arxiv.org/html/2609.00647#A3.SS2), ball\-level spectral optimization is the projection of sample\-level spectral optimization onto the subspace induced by the granular\-ball partition\. Samples in the same ball share an embedding vector scaled bynb−1/2n\_\{b\}^\{\-1/2\}\. Dynamic ball updates therefore modify the feasible subspace of the sample embedding and, in turn, the optimized ball\-level spectral representation; they do not introduce an independent free parameter outside the multiple kernel clustering model\.

### II\-CObjective and Optimization of DK\-GBMKKM

LetZ∈ℝM×cZ\\in\\mathbb\{R\}^\{M\\times c\}be the ball\-level spectral embedding withZ⊤​Z=IcZ^\{\\top\}Z=I\_\{c\}\. For the current granular\-ball partition, DK\-GBMKKM solves

minZ,σ\\displaystyle\\min\_\{Z,\\sigma\}JB​\(Z,σ\)=∑p=1Pσp2​Lp,\\displaystyle J\_\{B\}\(Z,\\sigma\)=\\sum\_\{p=1\}^\{P\}\\sigma^\{2\}\_\{p\}L\_\{p\},\(16\)s\.t\.\\displaystyle\\mathrm\{s\.t\.\}Z⊤Z=Ic,σp≥0,∑p=1Pσp=1,\\displaystyle Z^\{\\top\}Z=I\_\{c\},\\qquad\\sigma\_\{p\}\\geq 0,\\qquad\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}=1,whereLp=tr⁡\(K^B\(p\)\)−tr⁡\(Z⊤​K^B\(p\)​Z\)L\_\{p\}=\\tr\(\\widehat\{K\}^\{\(p\)\}\_\{B\}\)\-\\tr\(Z^\{\\top\}\\widehat\{K\}^\{\(p\)\}\_\{B\}Z\)\. LetZ\(t\)Z^\{\(t\)\},σ\(t\)\\sigma^\{\(t\)\},B\(t\)B^\{\(t\)\}andJ\(t\)J^\{\(t\)\}be the results after the synchronized update at iterationtt\. For fixed kernel weightsσ\(t\)\\sigma^\{\(t\)\}, Eq\. \([16](https://arxiv.org/html/2609.00647#S2.E16)\) is equivalent, with respect toZ\(t\)Z^\{\(t\)\}, to

maxZ\(t\)⊤​Z\(t\)=Ic⁡tr⁡\(Z\(t\)⊤​K^B\(t\),σ\(t\)​Z\(t\)\)\.\\max\_\{\{Z^\{\(t\)\}\}^\{\\top\}Z^\{\(t\)\}=I\_\{c\}\}\\tr\(\{Z^\{\(t\)\}\}^\{\\top\}\\widehat\{K\}\_\{B^\{\(t\)\},\\sigma^\{\(t\)\}\}Z^\{\(t\)\}\)\.\(17\)Accordingly,Z\(t\)Z^\{\(t\)\}consists of the orthonormal eigenvectors associated with thecclargest eigenvalues ofK^B\(t\),σ\(t\)\\widehat\{K\}\_\{B^\{\(t\)\},\\sigma^\{\(t\)\}\}:

Z\(t\)=Spec⁡\(K^B\(t\),σ\(t\),c\)\.Z^\{\(t\)\}=\\operatorname\{Spec\}\(\\widehat\{K\}\_\{B^\{\(t\)\},\\sigma^\{\(t\)\}\},c\)\.\(18\)
For fixedZ\(t\)Z^\{\(t\)\}, the weight subproblem is a convex quadratic program over the probability simplex\. To avoid numerical instability when a residual approaches zero, defineL~p=max⁡\(Lp,e−12\)\\widetilde\{L\}\_\{p\}=\\max\(L\_\{p\},e^\{\-12\}\)\. The kernel weights are updated as

σp\(t\+1\)=1/L~p∑q=1P1/L~q\.\\sigma\_\{p\}^\{\(t\+1\)\}=\\frac\{1/\\widetilde\{L\}\_\{p\}\}\{\\sum\_\{q=1\}^\{P\}1/\\widetilde\{L\}\_\{q\}\}\.\(19\)The derivation is provided in Appendix[D](https://arxiv.org/html/2609.00647#A4)\.

Each dynamic iteration performs one synchronized block update\. The ball\-level spectral embedding and the residualLpL\_\{p\}of each base kernel are first computed from the current weights\. Eq\. \([19](https://arxiv.org/html/2609.00647#S2.E19)\) then updates the weights, after whichZZand the objective value are recomputed using the newσ\\sigma\. The granular\-ball partition, kernel weights, spectral embedding, and objective value therefore describe the same state\.

The change in the objective is

Δ\(t\)=\|J\(t\)−J\(t−1\)\|\.\\Delta^\{\(t\)\}=\|J^\{\(t\)\}\-J^\{\(t\-1\)\}\|\.\(20\)The dynamic optimization terminates whenΔ\(t\)≤ε\\Delta^\{\(t\)\}\\leq\\varepsilonor when the maximum number of iterations is reached\. Convergence is evaluated before the next membership update, so the returned partition, weights, embedding, and objective value remain synchronized\.

If the stopping condition is not met, the latest weights define the sample\-level fused kernel

Kσ\(t\+1\)=∑p=1P\(σp\(t\+1\)\)2​K\(p\)\.K\_\{\\sigma^\{\(t\+1\)\}\}=\\sum\_\{p=1\}^\{P\}\(\\sigma\_\{p\}^\{\(t\+1\)\}\)^\{2\}K^\{\(p\)\}\.\(21\)Using theMMcenters computed from the current partition in this fused kernel space, all samples undergo one fixed\-MMkernelkk\-means assignment update:

Bl\(t\+1\)=argmin1≤b≤M⁡‖ϕσ\(t\+1\)​\(xi\)−Cb\(t\+1\)‖ℋ2,xi∈𝒳\.B\_\{l\}^\{\(t\+1\)\}=\\argmin\_\{1\\leq b\\leq M\}\\left\\\|\\phi\_\{\\sigma^\{\(t\+1\)\}\}\(x\_\{i\}\)\-C\_\{b\}^\{\(t\+1\)\}\\right\\\|\_\{\\mathcal\{H\}\}^\{2\},\\quad x\_\{i\}\\in\\mathcal\{X\}\.\(22\)This step updates only sample memberships while keepingMMfixed, without repeating center\-consistency splitting\. Empty balls are repaired by reassigning samples with large assignment distances\. The updated memberships are then used to rebuildGG,DD, and allK^B\(t\+1\)\(p\)\\widehat\{K\}\_\{B^\{\(t\+1\)\}\}^\{\(p\)\}for the next ball\-level MKKM update\. Alternating kernel\-weight learning with fixed\-count reassignment enables the partition to track the evolving fused kernel space\.

After convergence, the sample\-level embedding is recovered from the final ball\-level embedding as

H=GD−1/2Z\.H=GD^\{\-1/2\}Z\.\(23\)Each row ofHHisℓ2\\ell\_\{2\}\-normalized, and Euclideankk\-means withccclusters is applied to the normalized embedding\. The complete procedure is listed in Algorithm[1](https://arxiv.org/html/2609.00647#alg1)in the Appendix\.

The algorithm begins with uniform kernel weights and constructs the initial granular\-ball partition using GBCT in the corresponding fused kernel space\. It then fixes the number of balls and builds the sample\-to\-ball indicator, the ball\-size matrix, and the weighted granular\-ball kernel for each base kernel\. During dynamic optimization, one synchronized spectral\-and\-weight update is followed, when necessary, by one kernel\-space membership update at a fixed ball count\. After convergence, the final ball\-level embedding is lifted to the sample level and clustered\. Thus, DK\-GBMKKM couples ball\-level multiple kernel learning with dynamic kernel\-space reassignment while retaining a compact spectral problem\. Detailed algorithmic procedures and time\-complexity analysis are provided in the Appendix\.

## IIIExperimental Design and Results

We compare DK\-GBMKKM with recent multiple kernel clustering baselines on 12 public datasets and examine convergence through changes in its objective value\. All experiments were conducted in MATLAB R2025b\.

### III\-ASetup

Datasets:Experiments are conducted on 12 public datasets covering gene expression, object, speech, and face data\[[31](https://arxiv.org/html/2609.00647#bib.bib39),[32](https://arxiv.org/html/2609.00647#bib.bib36),[33](https://arxiv.org/html/2609.00647#bib.bib37),[34](https://arxiv.org/html/2609.00647#bib.bib40),[35](https://arxiv.org/html/2609.00647#bib.bib41),[36](https://arxiv.org/html/2609.00647#bib.bib42)\], including several high\-dimensional datasets from Feature Selection @ ASU\[[36](https://arxiv.org/html/2609.00647#bib.bib42)\]\. Ground\-truth labels are used only to determine the number of clusters and compute evaluation metrics\. Missing values are set to zero, followed by z\-score standardization and sample\-wiseℓ2\\ell\_\{2\}normalization\. Dataset statistics are provided in Table[II](https://arxiv.org/html/2609.00647#A1.T2)of the Appendix\.

Base kernels:We construct 12 base kernels: seven radial basis function \(RBF\) kernels, four polynomial kernels, and one cosine kernel\. The RBF scale parameters are set tot∈\{0\.01,0\.05,0\.1,1,10,50,100\}t\\in\\\{0\.01,0\.05,0\.1,1,10,50,100\\\}\. The polynomial kernels are\(xi⊤​xj\+a\)b\(x\_\{i\}^\{\\top\}x\_\{j\}\+a\)^\{b\}, wherea∈\{0,1\}a\\in\\\{0,1\\\}andb∈\{2,4\}b\\in\\\{2,4\\\}\. After row normalization, the cosine kernel isxi⊤​xjx\_\{i\}^\{\\top\}x\_\{j\}\. To ensure identical inputs across methods, every base kernel is cleaned of nonfinite entries, symmetrized, centered, and diagonal\-normalized\.

Compared methods and parameter settings:We compare DK\-GBMKKM with seven representative baselines: SMKC\[[37](https://arxiv.org/html/2609.00647#bib.bib35)\], AASC\[[38](https://arxiv.org/html/2609.00647#bib.bib30)\], MKKM\[[39](https://arxiv.org/html/2609.00647#bib.bib32)\], SimpleMKKM\[[40](https://arxiv.org/html/2609.00647#bib.bib31)\], RMKKM\[[41](https://arxiv.org/html/2609.00647#bib.bib33)\], MKKM\-SR\[[42](https://arxiv.org/html/2609.00647#bib.bib34)\], and GB\-MKKM\[[30](https://arxiv.org/html/2609.00647#bib.bib22)\]\. All methods use the same preprocessing, 12 base kernels, and number of clusters\. Baseline parameters follow the recommended settings in the corresponding papers or public implementations\.

Evaluation metrics:Clustering quality is evaluated using clustering accuracy \(ACC\), normalized mutual information \(NMI\), Purity, and the adjusted Rand index \(ARI\)\. Higher values indicate better performance for all four metrics\.

### III\-BClustering Performance

Table[I](https://arxiv.org/html/2609.00647#S3.T1)summarizes the average results over 12 datasets, with detailed comparisons shown in Fig\.[2](https://arxiv.org/html/2609.00647#S3.F2)and Appendix[G](https://arxiv.org/html/2609.00647#A7)\. DK\-GBMKKM ranks first on all four average metrics, achieving 0\.6717 ACC, 0\.5929 NMI, 0\.6848 Purity, and 0\.4647 ARI, with gains of 0\.0574, 0\.0521, 0\.0522, and 0\.0590 over the second\-best averages\. It also records the most dataset\-level wins, confirming that the improvement is consistent rather than dataset\-specific\.

TABLE I:Average clustering performance on 12 datasets\.Fig\. 2:ACC of eight methods on 12 datasets\.DK\-GBMKKM achieves the best results on all four metrics for orlraws10P, pixraw10P, warpAR10P, and warpPIE10P\. On warpPIE10P, it improves ACC, NMI, Purity, and ARI over the runner\-up MKKM\-SR by 0\.2558, 0\.1916, 0\.2028, and 0\.2402, respectively\. This result suggests that adapting ball boundaries as the fused\-kernel geometry changes reduces the mismatch between a fixed input\-space partition and the current kernel\-space structure\.

DK\-GBMKKM is not uniformly superior on every dataset\. RMKKM obtains the highest NMI and Purity on COIL20, and MKKM\-SR remains competitive on lung\_discrete and ORL\. When the local structure in the input space is already stable or spectral rotation sufficiently aligns the embedding with discrete labels, dynamic ball updates may provide limited additional benefit\. Nevertheless, the leading averages and consistent gains across most datasets support the effectiveness of DK\-GBMKKM while also revealing its dependence on data geometry\.

### III\-CConvergence Analysis

To examine whether dynamic membership updates induce persistent oscillations, we run DK\-GBMKKM once on each dataset with early stopping disabled and record 30 consecutive objective values\. Fig\.[3](https://arxiv.org/html/2609.00647#S3.F3)plots the changeΔ​J\(t\)\\Delta J^\{\(t\)\}over the first 30 iterations on all 12 datasets\. A symmetric logarithmic scale displays both positive and negative changes, including fluctuations near machine precision\.

All curves approach zero within the first few iterations\. After iteration 20,Δ​J\(t\)\\Delta J^\{\(t\)\}remains below5×10−145\\times 10^\{\-14\}on every dataset, with no sustained oscillation above this scale\. Under the current datasets and parameter settings, dynamic granular\-ball reassignment therefore causes no observable late\-stage oscillation of the objective\.

Fig\. 3:Signed change in the DK\-GBMKKM objective over the first 30 iterations on 12 datasets\.

## IVConclusion

This paper proposed DK\-GBMKKM, which constructs and dynamically updates granular balls in the fused kernel space to reduce the mismatch between granular\-ball representation and multiple kernel learning\. By coupling ball\-level MKKM optimization with kernel\-space sample reassignment, the method enables adaptive partitioning under evolving kernel geometry\. Experimental results verify its effectiveness\. Future work will focus on low\-rank kernel approximation and adaptive granularity for large\-scale clustering\.

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## Appendix AExperimental Dataset Details

TABLE II:Information on the 12 experimental datasets\.
## Appendix BTheoretical Properties of Granular\-Ball Representations in the Fused Kernel Space

This section establishes the relationship between a granular\-ball center in the fused kernel space and its counterparts in the base\-kernel spaces\. From Eq\. \([2](https://arxiv.org/html/2609.00647#S2.E2)\),

Kσ=∑p=1Pσp2​K\(p\)\.K\_\{\\sigma\}=\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}K^\{\(p\)\}\.\(24\)Letϕp​\(x\)\\phi\_\{p\}\(x\)be an implicit feature map for theppth base kernel\. An equivalent feature map for the fused kernel is

ϕσ​\(x\)=\[σ1​ϕ1​\(x\)⊤,…,σP​ϕP​\(x\)⊤\]⊤\.\\phi\_\{\\sigma\}\(x\)=\\left\[\\sigma\_\{1\}\\phi\_\{1\}\(x\)^\{\\top\},\\ldots,\\sigma\_\{P\}\\phi\_\{P\}\(x\)^\{\\top\}\\right\]^\{\\top\}\.\(25\)Indeed,

⟨ϕσ​\(xi\),ϕσ​\(xj\)⟩\\displaystyle\\left\\langle\\phi\_\{\\sigma\}\(x\_\{i\}\),\\phi\_\{\\sigma\}\(x\_\{j\}\)\\right\\rangle=∑p=1Pσp2​⟨ϕp​\(xi\),ϕp​\(xj\)⟩\\displaystyle=\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}\\left\\langle\\phi\_\{p\}\(x\_\{i\}\),\\phi\_\{p\}\(x\_\{j\}\)\\right\\rangle\(26\)=∑p=1Pσp2​K\(p\)​\(xi,xj\)\\displaystyle=\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}K^\{\(p\)\}\(x\_\{i\},x\_\{j\}\)=Kσ​\(xi,xj\),\\displaystyle=K\_\{\\sigma\}\(x\_\{i\},x\_\{j\}\),which verifies Eq\. \([25](https://arxiv.org/html/2609.00647#A2.E25)\)\.

For a granular ballBbB\_\{b\}generated in the fused kernel space, letℐb\\mathcal\{I\}\_\{b\}be its sample index set andnbn\_\{b\}its size\. From Eq\. \([3](https://arxiv.org/html/2609.00647#S2.E3)\), its center is

Cb=1nb​∑xi∈ℐbϕσ​\(xi\)\.C\_\{b\}=\\frac\{1\}\{n\_\{b\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{b\}\}\\phi\_\{\\sigma\}\(x\_\{i\}\)\.\(27\)Substituting Eq\. \([25](https://arxiv.org/html/2609.00647#A2.E25)\) gives

Cb\\displaystyle C\_\{b\}=1nb​∑xi∈ℐb\[σ1​ϕ1​\(xi\)⊤,…,σP​ϕP​\(xi\)⊤\]⊤\\displaystyle=\\frac\{1\}\{n\_\{b\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{b\}\}\\left\[\\sigma\_\{1\}\\phi\_\{1\}\(x\_\{i\}\)^\{\\top\},\\ldots,\\sigma\_\{P\}\\phi\_\{P\}\(x\_\{i\}\)^\{\\top\}\\right\]^\{\\top\}\(28\)=\[σ1​Cb\(1\)⊤,…,σP​Cb\(P\)⊤\]⊤,\\displaystyle=\\left\[\\sigma\_\{1\}C\_\{b\}^\{\(1\)\\top\},\\ldots,\\sigma\_\{P\}C\_\{b\}^\{\(P\)\\top\}\\right\]^\{\\top\},where

Cb\(p\)=1nb​∑xi∈ℐbϕp​\(xi\)C\_\{b\}^\{\(p\)\}=\\frac\{1\}\{n\_\{b\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{b\}\}\\phi\_\{p\}\(x\_\{i\}\)\(29\)is the center ofBbB\_\{b\}in theppth base\-kernel space\. Therefore, for any two ballsBaB\_\{a\}andBbB\_\{b\},

⟨Ca,Cb⟩\\displaystyle\\left\\langle C\_\{a\},C\_\{b\}\\right\\rangle=⟨\[σ1Ca\(1\)⊤,…,σPCa\(P\)⊤\]⊤,\\displaystyle=\\left\\langle\\left\[\\sigma\_\{1\}C\_\{a\}^\{\(1\)\\top\},\\ldots,\\sigma\_\{P\}C\_\{a\}^\{\(P\)\\top\}\\right\]^\{\\top\},\\right\.\(30\)\[σ1Cb\(1\)⊤,…,σPCb\(P\)⊤\]⊤⟩\\displaystyle\\left\.\\left\[\\sigma\_\{1\}C\_\{b\}^\{\(1\)\\top\},\\ldots,\\sigma\_\{P\}C\_\{b\}^\{\(P\)\\top\}\\right\]^\{\\top\}\\right\\rangle=∑p=1Pσp2​⟨Ca\(p\),Cb\(p\)⟩\.\\displaystyle=\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}\\left\\langle C\_\{a\}^\{\(p\)\},C\_\{b\}^\{\(p\)\}\\right\\rangle\.Eq\. \([30](https://arxiv.org/html/2609.00647#A2.E30)\) shows that center relations in the fused kernel space are weighted combinations of the corresponding center relations under the same partition in the base\-kernel spaces\. A single partition can therefore be generated in the current fused kernel space and shared across all base kernels\.

Since

⟨Ca\(p\),Cb\(p\)⟩=K¯B\(p\)​\(a,b\),\\left\\langle C\_\{a\}^\{\(p\)\},C\_\{b\}^\{\(p\)\}\\right\\rangle=\\overline\{K\}\_\{B\}^\{\(p\)\}\(a,b\),\(31\)we further have

⟨Ca,Cb⟩=∑p=1Pσp2​K¯B\(p\)​\(a,b\)\.\\left\\langle C\_\{a\},C\_\{b\}\\right\\rangle=\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}\\overline\{K\}\_\{B\}^\{\(p\)\}\(a,b\)\.\(32\)For a fixed granular\-ball partition, the similarities between fused\-space ball centers are thus equivalent to the weighted combination of the corresponding base\-space center similarities\.

## Appendix CProperties of the Weighted Granular\-Ball Kernel

### C\-AProof of Proposition 1

From Eq\. \([3](https://arxiv.org/html/2609.00647#S2.E3)\), the inner product between the centers of ballsBaB\_\{a\}andBbB\_\{b\}in theppth base\-kernel space is

⟨Ca\(p\),Cb\(p\)⟩\\displaystyle\\left\\langle C\_\{a\}^\{\(p\)\},C\_\{b\}^\{\(p\)\}\\right\\rangle=⟨1na​∑xi∈ℐaϕp​\(xi\),1nb​∑xj∈ℐbϕp​\(xj\)⟩\\displaystyle=\\left\\langle\\frac\{1\}\{n\_\{a\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{a\}\}\\phi\_\{p\}\(x\_\{i\}\),\\frac\{1\}\{n\_\{b\}\}\\sum\_\{x\_\{j\}\\in\\mathcal\{I\}\_\{b\}\}\\phi\_\{p\}\(x\_\{j\}\)\\right\\rangle\(33\)=1na​nb​∑xi∈ℐa∑xj∈ℐbK\(p\)​\(xi,xj\)\.\\displaystyle=\\frac\{1\}\{n\_\{a\}n\_\{b\}\}\\sum\_\{x\_\{i\}\\in\\mathcal\{I\}\_\{a\}\}\\sum\_\{x\_\{j\}\\in\\mathcal\{I\}\_\{b\}\}K^\{\(p\)\}\(x\_\{i\},x\_\{j\}\)\.Collecting all pairwise center inner products gives

K¯B\(p\)=D−1​G⊤​K\(p\)​G​D−1\.\\overline\{K\}\_\{B\}^\{\(p\)\}=D^\{\-1\}G^\{\\top\}K^\{\(p\)\}GD^\{\-1\}\.\(34\)Weighting this average kernel by the ball sizes yields

K^B\(p\)\\displaystyle\\widehat\{K\}\_\{B\}^\{\(p\)\}=D1/2​K¯B\(p\)​D1/2\\displaystyle=D^\{1/2\}\\overline\{K\}\_\{B\}^\{\(p\)\}D^\{1/2\}\(35\)=D−1/2G⊤K\(p\)GD−1/2\.\\displaystyle=D^\{\-1/2\}G^\{\\top\}K^\{\(p\)\}GD^\{\-1/2\}\.Consequently,

K^B\(p\)​\(a,b\)=na​nb​⟨Ca\(p\),Cb\(p\)⟩\.\\widehat\{K\}\_\{B\}^\{\(p\)\}\(a,b\)=\\sqrt\{n\_\{a\}n\_\{b\}\}\\left\\langle C\_\{a\}^\{\(p\)\},C\_\{b\}^\{\(p\)\}\\right\\rangle\.\(36\)
LetQ=GD−1/2Q=GD^\{\-1/2\}\. Then

K^B\(p\)=Q⊤​K\(p\)​Q\.\\widehat\{K\}\_\{B\}^\{\(p\)\}=Q^\{\\top\}K^\{\(p\)\}Q\.\(37\)For any𝐳∈ℝM\\mathbf\{z\}\\in\\mathbb\{R\}^\{M\}, ifK\(p\)⪰0K^\{\(p\)\}\\succeq 0, then

𝐳⊤​K^B\(p\)​𝐳\\displaystyle\\mathbf\{z\}^\{\\top\}\\widehat\{K\}\_\{B\}^\{\(p\)\}\\mathbf\{z\}=𝐳⊤​Q⊤​K\(p\)​Q​𝐳\\displaystyle=\\mathbf\{z\}^\{\\top\}Q^\{\\top\}K^\{\(p\)\}Q\\mathbf\{z\}\(38\)=\(Q​𝐳\)⊤​K\(p\)​\(Q​𝐳\)≥0\.\\displaystyle=\(Q\\mathbf\{z\}\)^\{\\top\}K^\{\(p\)\}\(Q\\mathbf\{z\}\)\\geq 0\.HenceK^B\(p\)⪰0\\widehat\{K\}\_\{B\}^\{\(p\)\}\\succeq 0\. Becauseσp2≥0\\sigma\_\{p\}^\{2\}\\geq 0,

K^B,σ=∑p=1Pσp2​K^B\(p\)⪰0\.\\widehat\{K\}\_\{B,\\sigma\}=\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}\\widehat\{K\}\_\{B\}^\{\(p\)\}\\succeq 0\.\(39\)

### C\-BProof of Proposition 2

UsingH=Q​ZH=QZ,Q⊤​Q=IMQ^\{\\top\}Q=I\_\{M\}, andZ⊤​Z=IcZ^\{\\top\}Z=I\_\{c\}, we obtain

H⊤​H\\displaystyle H^\{\\top\}H=Z⊤​Q⊤​Q​Z=Z⊤​Z=Ic\.\\displaystyle=Z^\{\\top\}Q^\{\\top\}QZ=Z^\{\\top\}Z=I\_\{c\}\.\(40\)For any base kernelK\(p\)K^\{\(p\)\},

tr⁡\(H⊤​K\(p\)​H\)\\displaystyle\\tr\(H^\{\\top\}K^\{\(p\)\}H\)=tr⁡\(Z⊤​Q⊤​K\(p\)​Q​Z\)\\displaystyle=\\tr\\\!\\left\(Z^\{\\top\}Q^\{\\top\}K^\{\(p\)\}QZ\\right\)\(41\)=tr⁡\(Z⊤​K^B\(p\)​Z\),\\displaystyle=\\tr\\\!\\left\(Z^\{\\top\}\\widehat\{K\}\_\{B\}^\{\(p\)\}Z\\right\),which proves Eq\. \([15](https://arxiv.org/html/2609.00647#S2.E15)\)\.

## Appendix DDerivation of the Kernel\-Weight Update

For a fixed ball\-level spectral embeddingZZ, the kernel\-weight subproblem in Eq\. \([16](https://arxiv.org/html/2609.00647#S2.E16)\) is

min⁡∑p=1Pσ⁡σp2​Lp,s\.t\.σp≥0,∑p=1Pσp=1,\\min\_\{\\sigma\}\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}L\_\{p\},\\quad\\mathrm\{s\.t\.\}\\quad\\sigma\_\{p\}\\geq 0,\\quad\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}=1,\(42\)where

Lp=Tr⁡\(K^B\(p\)\)−Tr⁡\(Z⊤​K^B\(p\)​Z\)L\_\{p\}=\\operatorname\{Tr\}\(\\widehat\{K\}\_\{B\}^\{\(p\)\}\)\-\\operatorname\{Tr\}\(Z^\{\\top\}\\widehat\{K\}\_\{B\}^\{\(p\)\}Z\)\(43\)is the clustering residual associated with theppth base granular\-ball kernel\. SinceLp≥0L\_\{p\}\\geq 0, the problem is a convex quadratic program over the probability simplex\.

The Lagrangian is

ℒ⁡\(σ,λ\)=∑p=1Pσp2​Lp−λ⁡\(∑p=1Pσp−1\),\\mathcal\{L\}\(\\sigma,\\lambda\)=\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}^\{2\}L\_\{p\}\-\\lambda\\left\(\\sum\_\{p=1\}^\{P\}\\sigma\_\{p\}\-1\\right\),\(44\)whereλ\\lambdais the multiplier for the equality constraint\. Setting the derivative with respect toσp\\sigma\_\{p\}to zero gives

∂ℒ∂σp=2​σp​Lp−λ=0,\\frac\{\\partial\\mathcal\{L\}\}\{\\partial\\sigma\_\{p\}\}=2\\sigma\_\{p\}L\_\{p\}\-\\lambda=0,\(45\)and therefore

σp=λ2​Lp\.\\sigma\_\{p\}=\\frac\{\\lambda\}\{2L\_\{p\}\}\.\(46\)The normalization constraint yields

λ2​∑p=1P1Lp=1,\\frac\{\\lambda\}\{2\}\\sum\_\{p=1\}^\{P\}\\frac\{1\}\{L\_\{p\}\}=1,\(47\)so that

λ2=1∑q=1P1/Lq\.\\frac\{\\lambda\}\{2\}=\\frac\{1\}\{\\sum\_\{q=1\}^\{P\}1/L\_\{q\}\}\.\(48\)Substitution gives the optimal weight

σp=1/Lp∑q=1P1/Lq\.\\sigma\_\{p\}=\\frac\{1/L\_\{p\}\}\{\\sum\_\{q=1\}^\{P\}1/L\_\{q\}\}\.\(49\)For numerical stability when a residual approaches zero, we use

L~p=max⁡\(Lp,ϵ\),\\widetilde\{L\}\_\{p\}=\\max\(L\_\{p\},\\epsilon\),\(50\)whereϵ\\epsilonis a small positive constant\. The resulting update is

σp\+=1/L~p∑q=1P1/L~q,\\sigma\_\{p\}^\{\+\}=\\frac\{1/\\widetilde\{L\}\_\{p\}\}\{\\sum\_\{q=1\}^\{P\}1/\\widetilde\{L\}\_\{q\}\},\(51\)which is Eq\. \([19](https://arxiv.org/html/2609.00647#S2.E19)\)\.

## Appendix EDK\-GBMKKM Optimization Procedure

Algorithm 1DK\-GBMKKM0:Base kernels

\{K\(p\)\}p=1P\\\{K^\{\(p\)\}\\\}\_\{p=1\}^\{P\}, number of clusters

cc, consistency coefficient

η\\eta, minimum child\-ball size

nminn\_\{\\min\}, maximum number of ball\-refinement rounds

TsplitT\_\{\\mathrm\{split\}\}, tolerance

ε\\varepsilon, and maximum number of iterations

TT\.

0:Cluster labels

𝐲\\mathbf\{y\}, kernel weights

σ\\sigma, and granular\-ball set

ℬ\\mathcal\{B\}\.

1:Initialize

σp\(0\)=1/P\\sigma\_\{p\}^\{\(0\)\}=1/Pand construct

Kσ\(0\)K\_\{\\sigma^\{\(0\)\}\}\.

2:Run GBCT on the initial fused kernel to obtain a complete granular\-ball partition\.

3:Fix the resulting number of balls

MMand construct

GG,

DD, and

\{K^B\(p\)\}p=1P\\\{\\widehat\{K\}\_\{B\}^\{\(p\)\}\\\}\_\{p=1\}^\{P\}\.

4:for

t=1,2,…,Tt=1,2,\\ldots,Tdo

5:Construct

K^B\(t\),σ\(t\)\\widehat\{K\}\_\{B^\{\(t\)\},\\sigma^\{\(t\)\}\}from the current

σ\(t\)\\sigma^\{\(t\)\}\.

6:Compute

Z\(t\)=Spec⁡\(K^B\(t\),σ\(t\),c\)Z^\{\(t\)\}=\\operatorname\{Spec\}\(\\widehat\{K\}\_\{B^\{\(t\)\},\\sigma^\{\(t\)\}\},c\)and the residual

LpL\_\{p\}of each base kernel\.

7:Update the kernel weights with Eq\. \([19](https://arxiv.org/html/2609.00647#S2.E19)\) to obtain

σ\(t\+1\)\\sigma^\{\(t\+1\)\}\.

8:Compute objective

J\(t\)J^\{\(t\)\}\.

9:if

t\>1t\>1and

\|J\(t\)−J\(t−1\)\|≤ε\|J^\{\(t\)\}\-J^\{\(t\-1\)\}\|\\leq\\varepsilonthen

10:break

11:endif

12:if

t<Tt<Tthen

13:Construct the latest sample\-level fused kernel

Kσ\(t\+1\)K\_\{\\sigma^\{\(t\+1\)\}\}\.

14:Apply one fixed\-

MMkernel

kk\-means assignment update to all samples using Eq\. \([22](https://arxiv.org/html/2609.00647#S2.E22)\), and repair any empty balls\.

15:Reconstruct

GG,

DD, and

\{K^Bt\+1\(p\)\}p=1P\\\{\\widehat\{K\}\_\{B^\{t\+1\}\}^\{\(p\)\}\\\}\_\{p=1\}^\{P\}from the new memberships\.

16:endif

17:endfor

18:Recover

H=GD−1/2ZH=GD^\{\-1/2\}Zusing Eq\. \([23](https://arxiv.org/html/2609.00647#S2.E23)\)\.

19:Row\-normalize

HHin the

ℓ2\\ell\_\{2\}norm and run Euclidean

kk\-means with

ccclusters to obtain

𝐲\\mathbf\{y\}\.

20:return

𝐲,σ,ℬ\\mathbf\{y\},\\sigma,\\mathcal\{B\}\.

## Appendix FTime and Space Complexity

LetMMbe the final number of granular balls andTTthe number of iterations\. GivenPPbase kernels of sizen×nn\\times n, constructing all weighted ball kernels requiresG⊤​K\(p\)​GG^\{\\top\}K^\{\(p\)\}G\. A dense implementation costsO⁡\(P​n2\)O\(Pn^\{2\}\)time andO⁡\(P​M2\)O\(PM^\{2\}\)memory for the ball kernels\. Each synchronized MKKM update performs two eigendecompositions of anM×MM\\times Mmatrix\. A full decomposition costsO⁡\(M3\)O\(M^\{3\}\), whereas iterative extraction of the leadingcceigenvectors typically costs approximatelyO⁡\(M2​c\)O\(M^\{2\}c\)\. One fixed\-count reassignment computes distances fromnnsamples toMMcenters in the fused sample kernel and costsO⁡\(n2\+n​M\)O\(n^\{2\}\+nM\)in a dense implementation\.

Ignoring initialization constants, the dynamic phase has time complexityO⁡\(T⁡\(P​n2\+M3\+n2\+n​M\)\)O\\\!\\left\(T\(Pn^\{2\}\+M^\{3\}\+n^\{2\}\+nM\)\\right\);M3M^\{3\}can be replaced byM2​cM^\{2\}cwhen a partial eigensolver is used\. If all base kernels remain in memory, the space complexity isO⁡\(P​n2\+P​M2\+n​M\)O\(Pn^\{2\}\+PM^\{2\}\+nM\)\. The current implementation therefore moves repeated spectral computations from matrices of ordernnto matrices of orderMM\. Because it retains full sample kernels and reaggregates them in each iteration, it is not a linear\-memory method\.

## Appendix GAdditional Dataset\-Level Results

Fig\.[4](https://arxiv.org/html/2609.00647#A7.F4)provide the dataset\-level NMI, Purity, and ARI results that complement the ACC comparison in Fig\.[2](https://arxiv.org/html/2609.00647#S3.F2)\.

Fig\. 4:Comparison of eight methods on 12 datasets: NMI \(top\), Purity \(middle\), and ARI \(bottom\)\.
## Appendix HStatistical Tests and Significance Analysis

To avoid drawing conclusions from mean values alone, we conduct nonparametric tests on the eight methods over the 12 datasets\. Within each dataset, methods are ranked in descending order of metric value, with tied results assigned their average rank\. A Friedman test first evaluates the null hypothesis that all eight methods have equal overall performance\. Its statistic is

χF2=12​Nk⁡\(k\+1\)​\[∑j=1kR¯j2−k​\(k\+1\)24\],\\chi\_\{F\}^\{2\}=\\frac\{12N\}\{k\(k\+1\)\}\\left\[\\sum\_\{j=1\}^\{k\}\\bar\{R\}\_\{j\}^\{2\}\-\\frac\{k\(k\+1\)^\{2\}\}\{4\}\\right\],\(52\)whereN=12N=12,k=8k=8, andR¯j\\bar\{R\}\_\{j\}is the average rank of methodjj\. The Iman–Davenport statistic

FF=\(N−1\)​χF2N⁡\(k−1\)−χF2​f​i​g:c​o​m​b​i​n​e​dc​dF\_\{F\}=\\frac\{\(N\-1\)\\chi\_\{F\}^\{2\}\}\{N\(k\-1\)\-\\chi\_\{F\}^\{2\}\}fig:combined\_\{c\}d\(53\)provides a finite\-sample correction\. If the global null hypothesis is rejected, a Nemenyi post hoc test is applied\. At significance levelα=0\.05\\alpha=0\.05, the critical difference is

CD=q0\.05​k⁡\(k\+1\)6​N=3\.031\.\\mathrm\{CD\}=q\_\{0\.05\}\\sqrt\{\\frac\{k\(k\+1\)\}\{6N\}\}=3\.031\.\(54\)
As shown in Table[III](https://arxiv.org/html/2609.00647#A8.T3), the Iman–Davenport tests produce very smallpp\-values for all four metrics:3\.71×10−133\.71\\times 10^\{\-13\}for ACC,6\.27×10−146\.27\\times 10^\{\-14\}for NMI,7\.20×10−117\.20\\times 10^\{\-11\}for Purity, and1\.10×10−131\.10\\times 10^\{\-13\}for ARI\. All are well below 0\.05, so the null hypothesis of equal performance is rejected for every metric\. DK\-GBMKKM also obtains the best average rank among the eight methods: 1\.42 for ACC, 1\.67 for NMI, 1\.58 for Purity, and 1\.42 for ARI\. Its advantage in Table[I](https://arxiv.org/html/2609.00647#S3.T1)is therefore consistent across datasets and is not an artifact of individual datasets or metric scales\.

TABLE III:Friedman and Iman–Davenport test results\.\(a\)ACC\(b\)ARI\(c\)Purity\(d\)NMI
Fig\. 5:Nemenyi critical\-difference diagrams for four metrics\. Lower average ranks indicate better performance; thick lines connect algorithms whose differences are not significant\.Fig\.[5](https://arxiv.org/html/2609.00647#A8.F5)shows the critical\-difference diagrams for ACC, ARI, Purity, and NMI\. All four useCD=3\.031\\mathrm\{CD\}=3\.031for 12 datasets and eight methods\. DK\-GBMKKM attains the lowest average rank for every metric, confirming its best overall performance across evaluation criteria\.

The rank gaps between DK\-GBMKKM and MKKM, GB\-MKKM, and AASC exceed the critical difference for all four metrics, indicating a significant advantage over these conventional multiple kernel and granular\-ball multiple kernel clustering methods\. Relative to RMKKM, the difference is significant only for ACC\. The gaps from SMKC, MKKM\-SR, and SimpleMKKM are not significant for most metrics, showing that these strong baselines remain competitive\. Nevertheless, DK\-GBMKKM ranks first on all four metrics, supporting the effectiveness of dynamic granular\-ball adjustment and kernel\-space structural modeling\.