# Ising-Based Kernel Clustering +
## 4 Evaluation
### Execution Time
Figure 10 also shows that _Proposed_ is much faster than _Conventional_. This experimental result reveals that the matrix-matrix calculations is accelerated by the numerical library. In calculating the coefficients of the Gram matrix, the acceleration ratio of the proposed method to the conventional method is at most 392.6 million. Since the process of generating the Gram matrix has multiple matrix-matrix calculations, the parallel calculation by the numerical library is effective. Figures 11 and 12 show the execution times for determining the elements of the externally-constrained and internally-constrained QUBO matrices. Figure 10: The execution time of determining the Gram coefficients. Figure 9: The execution time of determining the Gaussian kernel coefficients. When the number of data points is eight in Figures 11 and 12, matrix-matrix calculations of the externally-constrained and internally-constrained QUBO matrix take longer than the conventional method. This is because multiple numerical functions of matrix-matrix calculations take constant overhead. Since the conventional methods do not have the overhead due to determining the QUBO coefficients, the conventional methods are faster than the proposed methods when the number of data points is small. As the numbers of data points increase, the differences between the proposed and conventional methods increase in both cases of Figures 11 and 12. The proposed methods for the externally-constrained and internally-constrained QUBO coefficients can achieve times that are 12.9 and 14.5 faster than the conventional methods for the externally-constrained and internally-constrained QUBO coefficients, respectively. Here, the acceleration ratio of generating the externally-constrained QUBO coefficients is smaller than that of generating the internally-constrained QUBO coefficients. This is because the internally-constrained QUBO coefficients require calculating both the coefficients of the Gram matrix and those of the Lagrange multiplier, while the externally-constrained QUBO coefficients require calculating only the coefficients of the Gram matrix. Since the conventional method for internally-constrained QUBO coefficients requires more complicated preprocesses, the proposed method for the internally-constrained QUBO coefficients is more effective than that for the externally-constrained QUBO coefficients. Thus, the speedup ratio for determining methods of the internally-constrained QUBO coefficients is higher than that for determining methods of the externally-constrained QUBO coefficients. Figure 11: The execution time of determining the elements of the externally-constrained QUBO matrix. Figure 12: The execution time of determining the elements of the internally-constrained QUBO matrix. These discussions clarify that it is useful for the preprocess of annealing to use numerical libraries for the matrix-matrix calculations. The most effective part of the proposed method is determining the coefficients of the Gram matrix. The Gram coefficients are also used for the state-of-the-art kernel clustering methods other than Ising-based kernel clustering methods. While these state-of-the-art kernel clustering methods calculate the Gram coefficients among centroids and the other actual data points, Ising-based kernel clustering methods calculate the Gram coefficients among only the actual data points. Since all the Gram coefficients among the actual data points are calculated simultaneously in advance, calculating the Gram matrix for Ising-based kernel clustering methods can obtain a high level of parallelism by using the numerical libraries. The experiments clarify that utilizing matrix-matrix calculations for the Gram matrix is effective. Because the kernel methods are also useful in the field of other machine learning methods, such as linear regression, support vector machine, and K-nearest neighbor, the proposed matrix-matrix calculating method has the potential to accelerate these methods based on the Ising model.