Bearing Similarity Measures for Self-organizing Feature Maps

The neural representation of space in rats has inspired many navigation systems for robots. In particular, Self-Organizing (Feature) Maps (SOM) are often used to give a sense of location to robots by mapping sensor information to a low-dimensional grid. For example, a robot equipped with a panoramic camera can build a 2D SOM from vectors of landmark bearings. If there are four landmarks in the robot's environment, then the 2D SOM is embedded in a 2D manifold lying in a 4D space. In general, the set of observable sensor vectors form a low-dimensional Riemannian manifold in a high-dimensional space. In a landmark bearing sensor space, the manifold can have a large curvature in some regions (when the robot is near a landmark for example), making the Eulidian distance a very poor approximation of the Riemannian metric. In this paper, we present and compare three methods for measuring the similarity between vectors of landmark bearings. We also discuss a method to equip SOM with a good approximation of the Riemannian metric. Although we illustrate the techniques with a landmark bearing problem, our approach is applicable to other types of data sets