Ellipsoid ART/ARTMAP Category Regions for the Choice-by- Difference Category Choice Function

In the recent past category regions have been introduced as new geometrical concepts and provide a visualization tool that facilitates significant insight into the nature of the competition among categories during both the training and performance phase of Fuzzy ART (FA) and Fuzzy ARTMAP (FAM). These regions are defined as the geometric interpretation of the Vigilance Test and the competition of each category with an uncommitted F2-layer node for a specific input pattern (Commitment Test). In this paper we show how the notion of category regions can be naturally extended to Ellipsoid ART (EA) and Ellipsoid ARTMAP (EAM) and focus on the regions’ theoretical properties, when considering the Choice-by-Difference category choice function. Based on these properties we state three theoretical results applicable to both EA and EAM. Specifically, if U and a denote the vigilance and the choice parameter respectively, we show that in certain areas of the (a,U) plane the result of EA/EAM training is independent of the specific value of either U or Z (parameter of the activation function value for an uncommitted F2-layer node). Finally, we provide a refined upper bound on the size of categories created in EA/EAM during training. All the results are immediately applicable to FA/FAM as well.