# Ising-Based Kernel Clustering +
## 1 Introduction  
Ising machineshave been developed to solve combinatorial optimization problems in the past few years. D-Wave Systems, Inc., (Burnaby, Canada) is currently working on a Quantum Annealing (QA) machine that can solve the Ising model [1]. Additionally, conventional digital computers are also being utilized to perform special-purpose computations using the Ising model, such as Simulated Annealing [2], the simulated bifurcation algorithm [3], and so on [4, 5]. In particular, Ising machines are expected to be applied to a wide range of fields such as vehicle routing problems [6], traffic flow optimization [7, 8], air traffic management [9], control of automated guided vehicles [10], planning space debris removal missions [11], and machine learning [12, 13, 14]. One of the applications utilizing the Ising machines is combinatorial clustering. Clustering is a machine learning technique that involves grouping similar objects together based on their characteristics. Clustering aims to identify natural groupings in the data, which can help with tasks such as data compression, anomaly detection, and pattern recognition. There are two main types of clustering: hierarchical and non-hierarchical approaches. In hierarchical clustering, the data are organized into a tree-like structure, with the mostsimilar objects grouped together at the bottom of the tree and progressively more dissimilar groups at higher levels. On the other hand, non-hierarchical clustering involves partitioning the data into a fixed number of clusters. This can be achieved through methods such as k-means clustering, where the data are partitioned into k clusters by minimizing the distance between each point and its assigned cluster centroid. Although non-hierarchical clustering is a powerful approach when the number of clusters is known, conventional non-hierarchical clustering makes it hard to find the globally optimum solution. This is because state-of-the-art clustering performs approximation by calculating the distance between the data point and the cluster centroid instead of the distance among data points. Combinatorial clustering is a method that enables the calculation of the distance among data points [15]. Since Ising machines have the potential to accelerate such a time-consuming calculation, combinatorial clustering based on the Ising model is drawing attention because of its ability to obtain high-quality clustering results [16; 17; 18; 19]. Ising-based combinatorial clustering typically employs the Euclidean distance as a measure of similarity. Nevertheless, numerous practical data possess irregular distributions that render clustering via Euclidean distance challenging. To improve the quality and robustness of clustering, some different approaches have been developed. One is a clustering algorithm combining deviation-sparse fuzzy C-Means with neighbor information constraints [20]. The algorithm uses a deviation-sparsity regularization term to encourage sparsity in the clustering results by incorporating spatial information about neighboring data points. Such techniques can help feature selection and dimensionality reduction beyond performing only clustering. Viewpoint-based kernel fuzzy clustering with weight information granules [21] is another novel fuzzy clustering algorithm that combines kernel clustering with viewpoint-based clustering. The algorithm uses viewpoint-based clustering to identify initial clusters and then applies kernel clustering to refine them. Weight information granules are introduced to better capture the distribution of the data, and a regularization term is included to encourage sparsity in the clustering results. However, such clustering algorithms require minimization of objective functions and are still based entirely upon the approximation of minimizing them to save the computational complexity.