Spherical self-organizing map using efficient indexed geodesic data structure

Abstract

The two-dimensional (2D) Self-Organizing Map (SOM) has a well-known “border effect”. Several spherical SOMs which use lattices of the tessellated icosahedron have been proposed to solve this problem. However, existing data structures for such SOMs are either not space efficient or are time consuming when searching the neighborhood. We introduce a 2D rectangular grid data structure to store the icosahedron-based geodesic dome. Vertices relationships are maintained by their positions in the data structure rather than by immediate neighbor pointers or an adjacency list. Increasing the number of neurons can be done efficiently because the overhead caused by pointer updates is reduced. Experiments show that the spherical SOM using our data structure, called a GeoSOM, runs with comparable speed to the conventional 2D SOM. The GeoSOM also reduces data distortion due to removal of the boundaries. Furthermore, we developed an interface to project the GeoSOM onto the 2D plane using a cartographic approach, which gives users a global view of the spherical data map. Users can change the center of the 2D data map interactively. In the end, we compare the GeoSOM to the other spherical SOMs by space complexity and time complexity.

Introduction

The Self-Organizing Map introduced by Kohonen (2001) is a popular tool for high-dimensional data analysis and visualization. Its applications can be found in various research areas such as social science (Lin, White, & Buzydlowski, 2003), web-document mining (Lagus, Kaski, & Kohonen, 2004) and robotics (Ritter, Martinetz, & Schulten, 1992). In a conventional SOM, the neighborhood relationship is defined by a two-dimensional rectangular or hexagonal lattice. During the training phase, all neurons compete with each other for the input signals. The winner and its neighbors update their connection weights. Ideally, at the end of the training, all samples are equally represented by neurons and the neighboring units in the grid tend to model similar regions of the data space. However, the grid units at the boundary of the SOM have fewer neighbors than the units inside the map, so they have fewer chances of being updated. This often leads to the “border effect” — weight vectors of these units “collapse to the center of the input space” (Sarle, 2002). Several approaches have been suggested to solve the problem, such as the heuristic weighting rule method by Kohonen (2001) and local-linear smoothing by Wand and Jones (1995). Aside from these mathematical solutions, a straightforward method is to remove the boundaries from the grid by implementing the SOM on a torus (Ito et al., 2000, Li et al., 1993) or a sphere (Boudjemaï et al., 2003, Hirokazu et al., 2005, Nakatsuka and Oyabu, 2003, Sangole and Knopf, 2003).

Compared to a toroidal SOM, the spherical SOM is more visually effective. Firstly, the area associated with each neuron varies significantly on the surface of a torus: larger around the outer circle and compressed near the inner circle. For a SOM, it is desirable to have all neurons receive equal geometrical treatment. Secondly, the toroidal SOM fails to provide an intuitively readable map. People are usually more familiar with maps generated from a sphere, such as the world maps, than they are with maps based on a torus.

Several spherical SOMs have been implemented and applied to different types of data sets. Using a tessellated platonic polyhedron as a lattice was first suggested by Ritter (1999). He also pointed out that the spherical SOM would be very suitable for data with underlying directional structures. In Sangole et al.’s work (Sangole & Knopf, 2003), a spherical SOM was employed in three-dimensional (3D) immersive virtual reality environments for interactive data analysis. Boudjemaï et al. (2003) applied the spherical SOM in 3D object modeling. While these applications have been successful, existing data structures for geodesic domes are either not space efficient or are time consuming when finding the neighbors. In this paper, we present a 2D data structure to store the spherical lattice. It reduces the overhead in maintaining the spherical grid structure. Finding the immediate neighbors of a vertex (neuron) is efficient because it is indexing in a 2D array. It also supports fast dome tessellation (increasing the number of neurons). We call a spherical SOM using this data structure a GeoSOM.

The remainder of this paper is organized as follows. Section 2 reviews existing techniques for spherical SOMs using lattices of geodesic domes. Details of our data structure are presented in Section 3. Section 4 describes the interface that we developed to visualize the GeoSOM. In Section 5, we test our data structure on two data sets. Results are compared with the corresponding 2D SOMs. In Section 6, GeoSOM is compared analytically with the existing spherical SOMs. Conclusions and future work are presented in Section 7.