# Ising-Based Kernel Clustering +
## 4 Evaluation
### Quality  
Figure 2 shows the clustering results of the conventional method. Figure 2a represents a dataset containing three clusters with equal variances, while Figure 2b depicts a dataset with clusters distributed anisotropically. Figure 2c shows a dataset comprising two half circles that interleave, while Figure 2d represents a dataset with a small circle inside a larger circle. In this paper, the datasets depicted in Figure 2a-d are referred to as Blobs, Aniso, Moons, and Circles, respectively, and the data points within different clusters are indicated with different colors. Figure 2a illustrates that the conventional method is capable of clustering data with equal variances accurately. However, it fails to cluster data that are not equally distributed from the center, as depicted in Figure 2b-d. The conventional method tries to partition the data in a single linear line, making it impossible to achieve the desired clustering result. Figures 3 shows the clustering results of the proposed method. Figure 2: Clustering results of the conventional method. The \(\sigma\) parameters in Equation (8) for Figure 2(a)-d are set to 3.5, 0.55, 0.2, and 0.4, respectively, as these values produce the highest ARI. The clustering results in Figure 2(a) demonstrate that the proposed method can accurately cluster center-aggregated data, similar to the conventional method. Additionally, the proposed method can effectively cluster anisotropic and irregular data, as demonstrated in Figure 2(b)-d. This is due to the transformation of the data into linearly separable high-dimensional data through the kernel method. Figure 4 shows ARI for each dataset to further evaluate the difference in quality between the proposed and conventional methods. The clustering results for the Blobs dataset show that the conventional method achieves a higher ARI than the proposed method. This is because the Blobs dataset can be linearly partitioned based on the Euclidean distance without any transformation. However, the pro  
Figure 4: The qualities of the proposed kernel-based and conventional Euclidean-based methods for different datasets. Figure 3: Clustering results of the proposed method. Data points having the same color belong to the same cluster. posed method still achieves a relatively high ARI of 0.91 even by using the kernel method. On the other hand, for the Aniso, Moons, and Circles datasets, the proposed method outperforms the conventional method in terms of ARI. This is because the data transformation by the kernel method is effective for irregularly distributed data. In order to achieve superior clustering results in combinatorial clustering using the kernel method, it is imperative to select the optimal parameter \(\sigma\). Figure 5 vividly demonstrates the impact of varying \(\sigma\) on the clustering quality. The results presented in Figure 5 are striking, demonstrating the crucial importance of selecting the correct parameter \(\sigma\) in the Ising-based combinatorial clustering using the kernel method. For instance, it is clear from Figure 5 that the Blobs dataset achieves remarkably high ARI for several \(\sigma\), showcasing the versatility and flexibility of the proposed method in transforming the data to provide multiple decision boundaries. In contrast, Figure 5b-d show that Aniso, Moons, and Circles have a small range of maximum ARI, indicating a limited number of decision boundaries that can be achieved with the given parameter range. These findings underline the significance of selecting the optimal \(\sigma\) parameter to achieve high-quality results of Ising-based kernel clustering. Next, Figure 6 shows the clustering results with the highest ARI of the proposed method and the conventional method on the dataset with six circles to evaluate the capability of the proposed method to handle any two or more clusters. Figure 5: ARI of combinatorial clustering using kernel method when \(\sigma\) is varied.