Stability analysis of hypercomplex-valued discrete-time Hopfield-type neural networks
Fidelis Zanetti de Castro · 2018
Hopfield neural networks are one of the most important recurrent neural networks, initially conceived for the storage and recall of bipolar patterns.In this doctoral thesis, we analyze the stability of some of its generalizations in complex and quaternionic domains.We introduce two complex-valued models and one quaternion-valued model, which always settle down to a stationary state assuming an asynchronous update mode and the usual conditions on the synaptic weights matrix, that is, a Hermitian matrix with non-negative diagonal elements.Specifically regarding quaternion-valued models, we implement autoassociative memories for storage and recall of synthetic patterns as well as real color images, and study their storage capacity and noise tolerance.In addition, we define novel hypercomplex number systems and a broad family of activation functions assuming values in these systems.Using these family of functions, we introduce a broad class of hypercomplex-valued Hopfield-type neural networks.These networks always settle down to a stationary state, given any initial state.Theoretical results are obtained and proved in this thesis, which provides a starting point to the development of other hypercomplex-valued models and its use in several applications.