Optimization of Geodesic Self-Organizing Map

The Geodesic Self-Organizing Map (GeoSOM) is a variation of traditional SOM, which uses an icosahedron-based tessellation as spherical lattice to eliminate the border effect to minimize the distortion in the reduction of high-dimensional spaces. Border effect is a problem intrinsic of low-dimensional neural grid, where neurons in the border have a less possibility to have its synaptic weights updated. The almost perfect regularity of a tessellated icosahedron projection onto a sphere solves this problem, reducing in two thirds of the distortion of 2D SOM. However, two problems appear resulting from complex shape of this Platonic polyhedron. First, the growth curve of lattice sizing follows a strong upward tendency that means a loss of control over the lattice sizing. Second, an overall visualization of topographic map is only possible with a geodesic projection of prototype vector positions from the surface of the sphere to a 2D plane that results in a elliptical map, flat on top and bottom, that avoids an orthogonal alignment of the data in the left and right sides, causing some distortion in the presentation of results and avoid an intuitive visualization of the map. This work proposes a geodesic self-organizing map, called 4HSOM, which uses a tessellated tetrahedron as lattice to eliminate the border effect, maximizing the control over the lattice sizing, with an easier overall visualization of topographic map without any geodetic projection, resulting from the minimalist geometric structure of the tetrahedron, although the tessellated tetrahedron has the greatest irregularity among the Platonic polyhedra. The work presents a comparative analysis between the results achieved by 4HSOM and GeoSOM.