A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as Hopfield-type hypercomplex number systems. Hopfield-type hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called Hopfield-type activation functions. Broadly speaking, a Hopfield-type activation function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras.


page 1

page 2

page 3

page 4


Non-constant bounded holomorphic functions of hyperbolic numbers - Candidates for hyperbolic activation functions

The Liouville theorem states that bounded holomorphic complex functions ...

Complex-valued Neural Networks with Non-parametric Activation Functions

Complex-valued neural networks (CVNNs) are a powerful modeling tool for ...

Universal approximation with complex-valued deep narrow neural networks

We study the universality of complex-valued neural networks with bounded...

TSSR: A Truncated and Signed Square Root Activation Function for Neural Networks

Activation functions are essential components of neural networks. In thi...

Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multipliers

In this paper, we analyze the stability of feedback interconnections of ...

A Survey of Complex-Valued Neural Networks

Artificial neural networks (ANNs) based machine learning models and espe...

On the Stability and Generalization of Learning with Kernel Activation Functions

In this brief we investigate the generalization properties of a recently...

Please sign up or login with your details

Forgot password? Click here to reset