# Ising-Based Kernel Clustering +
## 4 Evaluation  
### Experimental Environments  
The proposed method is compared with Euclidean distance-based combinatorial clustering as the conventional method. Evaluations are conducted in terms of solution quality and execution time.  
Both proposed and conventional combinatorial clustering based on the Ising model use an externally defined one-hot constraint [17; 18]. This method is preferred over the internally defined one-hot constraint in the QUBO matrix, resulting in higher-quality clustering. The reason for this is that in the internally constrained method, the Lagrange multiplier is significantly larger than the coefficients of the objective function. As a result, the constraint function has a significant influence, and solutions that satisfy the one-hot constraint can be obtained. However, as the influence of the objective function is small, the quality of clustering results decreases [18]. To address this problem, this paper utilizes externally defining the one-hot constraint of Equation (12) while only minimizing the objective function of Equation (11). By using the externally-defined one-hot constraint method, combinatorial clustering with the QUBO function containing only the objective function and the externally defined one-hot constraint can achieve higher quality results than the method that has both the objective and constraint function in the QUBO function. The Ising-based kernel clustering method utilizes the coefficients of the objective function as the elements of the Gram matrix. Let us conduct the experiments by minimizing the Hamiltonian of Equation (16) while externally defining the one-hot constraint.  
In the case of this paper, SA is used to search for the solution that minimizes the QUBO function for Ising-based kernel clustering. The SA algorithm is executed on a computing platform called SX-Aurora TSUBASA (Aurora) [23], which consists of a CPU (Intel Xeon 6126) and a VE (NEC Vector Engine Type 20B) [24]. In the SA algorithm on Aurora, the one-hot constraints are defined externally to the QUBO function, and the SA algorithm searches for solutions that allow multiple-bit flips [25]. Initially, SA searches for solutions with a single-bit flip, and once the one-hot constraint is satisfied, it then searches for only the solutions that satisfy the constraint using multiple-bit flips.  
Parallel Tempering (PT) [26] is used in addition to SA on Aurora. PT assigns different temperatures to multiple SA processes, and each process independently executes SA in parallel. The purpose of this method is to improve the exploration of the solution space by allowing solutions to escape from local minima. During the PT search, solutions are probabilistically exchanged between different SA processes based on the temperature of each process. The number of parallel temperatures for PT is set to 8. The number of sweeps, which is a parameter indicating the number of times to search for a solution, is set to 100. The inverse temperature \(\frac{1}{T}\) starts from 0.1 at the beginning of the search and gradually increases to 50 at the end. NumPy v0.22.1 [27] is used as the numerical library to perform matrix-matrix calculations shown in Equations (7), (9) and (20).  
The experiment uses four datasets generated through scikit-learn v1.0.2 [28]'s artificial data generation functions, and each dataset has 64 data points.  
The clustering results are evaluated based on the Adjusted Rand Index (ARI), which measures the similarity between the clustering result and the correct label. ARI has a maximum value of 1 when all labels match, and the average value is calculated from 100 experiments.