# Ising-Based Kernel Clustering +
## 6 Conclusions  
Ising-based combinatorial clustering is receiving attention due to its quality of clustering. However, conventional Ising-based clustering methods cannot handle data with irregular distributions. To overcome this problem, this paper proposes an Ising-based combinatorial clustering method using a kernel method. The proposed method uses a kernel trick to convert various data onto high-dimensional data that can be clustered. This paper also resolves the problem of a high computation cost for the process of determining the coefficients of the kernel, Gram, and QUBO matrices by extracting the potential of highly parallelized numerical libraries.  
First, the proposed method is compared with Euclidean distance-based combinatorial clustering as the conventional method. The experimental results show that the quality of the clustering results of the proposed method is significantly higher than that of the conventional method on irregular data. Moreover, the proposed method can obtain high-quality clustering results by selecting the appropriate parameter. Second, the proposed implementation using numerical libraries is compared with the naive implementation, where kernel, Gram, and QUBO coefficients are determined iteratively. Experimental results show that the proposed method is at most 12.4 million times faster than the conventional method. The experiments have also revealed the potential to solve large-scale Ising-based kernel clustering problems. Third, comparisons between the Ising-based kernel clustering and the kernel K-means show that the Ising-based kernel clustering is effective in the situation where higher-quality clustering results are required at the expense of a bit more time. Finally, experiments on real-world datasets suggest that selecting the appropriate kernel is more important than parameter tuning for only one kernel. Even though the proposed method has the potential to minimize the given objective function of kernel clustering, it is needed to give the appropriate kernel to ensure maximum performance.  
In future work, combining methods for ensemble clustering [42; 43] that combine multiple clustering results or multiple kernels [33] to improve the accuracy and robustness of clustering will be discussed.  
Conceptualization, M.K. and K.K.; methodology, M.K. and K.K.; software, M.K.; validation, M.K.; formal analysis, M.K. and K.K.; investigation, M.K. and K.K.; resources, M.K.; data curation, M.K.; writing--original draft preparation, M.K.; writing--review and editing, M.K., K.K., M.S. and H.K.; visualization, M.K.; supervision, K.K., M.S. and H.K.; project administration, M.K., K.K., and H.K.; funding acquisition, M.K., K.K. and H.K. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was partially supported by MEXT Next Generation High-Performance Computing Infrastructures and Applications R&D Program, entitled \"R&D of A Quantum-Annealing-Assisted Next Generation HPC Infrastructure and its Applications,\" Grants-in-Aid for Scientific Research (A) #19H01095, Grants-in-Aid for Scientific Research (C) #20K11838, Grants-in-Aid for JSPS Fellows #22J22908, and WISE Program for AI Electronics, Tohoku University.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** The authors would like to thank [PERSON] of NEC Corporation for his fruitful discussions and variable comments.  
**Conflicts of Interest:** The authors declare no conflict of interest.