# Spherical Tree-Structured SOM and Its Application to Hierarchical Clustering
## 1 Introduction  
Because of the recent developments in data science, various types of data are now analyzed in many fields. When analyzing data, if the structure of the data is unknown in advance, it is necessary to first understand the structure of the data and then select the analysis methods according to the task. If the data to be analyzed consist of few elements, it is easy to understand the structure of the data. However, in the case of high-dimensional data with many elements, it is difficult for humans to grasp the structure of the data. Therefore, data visualization that maps high-dimensional data onto low-dimensional space with two or three dimensions is often performed. By visualizing these data, humans can intuitively understand the structure of the data in the high-dimensional space. Various methods have been proposed for data visualization, and a self-organizing map (SOM) is one such data visualization method [1; 2]. The SOM maps high-dimensional data onto a low-dimensional map and expresses the similarity between data points using distance on the map. Similar data are placed close to each other on the map, and dissimilar data are placed far from each other on the map. A brief description of the SOM algorithm is as follows. First, the SOM arranges nodes that have reference vectors initialized with random values in a two-dimensional grid. In the learning phase, a SOM determines the winner node, which is the reference vector closest to the input vector, and updates the winner node's reference vector and its neighbors' reference vectors so that they are closer to the input vector. Learning in the SOM is performed by repeatedly determining the winner node and updating the reference vectors of the winner node and its neighbors. In recent years, large amounts of data have accumulated because of the development of various web services, and hence the number of data to be analyzed has increased. Generally, when visualizing a large number of data using a SOM, the number of nodes in the map is set to a large number. However, as the number of nodes increases, the SOM needs more time to search for the winner nodes. In addition, because the SOM usually arranges nodesin a two-dimensional grid, there are edges to the map. Therefore, according to the position of winner nodes, e.g., the center or edge of the map, the update area sizes are different. This means that learning is not uniform but depends on the positions of the winner nodes. To solve both these problems, we propose a spherical tree-structured SOM (S-TS-SOM). The S-TS-SOM arranges nodes on a spherical surface and applies a tree search method to speed up the search for the winner nodes while eliminating the edges of the map. We compare the effectiveness of the S-TS-SOM with a conventional spherical SOM (S-SOM) using the benchmark data MNIST [3]. The S-TS-SOM builds a tree structure and determines the winner nodes using a tree search method. In addition to speeding up the search, the advantage of the tree structure is that we can obtain different granularities of clustering in the tree structure. In this paper, we examine whether the granularity of clustering can be found in the tree structure of the S-TS-SOM. The main contributions of this paper can be summarized as follows. 1. We propose the S-TS-SOM, which applies a tree search method to the S-SOM, to speed up the search for winner nodes and eliminate the edges of the map. 2. We examine the effectiveness of the S-TS-SOM by comparing it with the S-SOM using a benchmark dataset. 3. We examine whether the granularity of clustering can be determined using the tree structure of the S-TS-SOM. The remainder of this paper is organized as follows. Section 2.1 introduces work related to this study. Section 2.2 presents the proposed method, the S-TS-SOM. In Sections 3.1 and 3.2, we evaluate the effectiveness of the S-TS-SOM by comparing it with the S-SOM using the benchmark dataset MNIST. Section 3.3 examines whether the granularity of clustering can be found using the tree structure of the S-TS-SOM.