The CMAC and a theorem of Kolmogorov

Abstract

This paper shows that the Cerebellar Model Articulation Controller (CMAC) is structurally similar to networks derived from a theorem of Kolmogorov. As a foundation for this comparison, we review of a proof of Kolmogorov's theorem. From this proof and an analysis of the CMAC we derive two lemmas describing functions that cannot be modeled by a CMAC. The first lemma states that such functions have zero average value over response regions of CMAC association cells. The second lemma states that such functions have local oscillations exceeding a quantifiable percentage of the global maximum absolute value of error. This second lemma gives bounds on errors caused by hash tables used as association cells in the CMAC. We present three examples illustrating the lemmas.