The Modelling Abilities of the Binary CMAC

The Albus CMAC (Cerebellar Model Articulation Controller), has been widely used for signal processing and adaptive control tasks, although it is only recently that the learning rules have been properly understood and convergence results derived. The CMAC is a multi-dimensional tabular storage scheme which generates piecewise constant models. However, despite its simplicity the modelling capabilities of this network are very complex and many incorrect theories have recently been proposed. The foundations for a correct theory were recently proposed by the authors, and this paper extends these results by showing that the modelling capibilities of different CMACs (varying the generalisation parameter or the overlay displacement vector) are different. It is also shown that the binary CMAC is able to model any additive mapping exactly, but is unable to reproduce a multiplicative function. This is achieved using the consistency equations and orthogonal functions which define the space of functions which a binary CMAC can and cannot model respectively, and they can also be used to generate a lower bound for the network's modelling error