Continuous-Valued Octonionic Hopfield Neural Network

Abstract

In this paper, we generalize the famous Hopfield neural network to unit octonions. In the proposed model, referred to as the continuous-valued octonionic Hopfield neural network (CV-OHNN), the next state of a neuron is obtained by setting its octonionic activation potential to length one. We show that, like the traditional Hopfield network, a CV-OHNN operating in an asynchronous update mode always settles down to an equilibrium state under mild conditions on the octonionic synaptic weights.

A General Framework for Hypercomplex-valued Extreme Learning Machines

“…Note that Φ L (A) is a real matrix of size (n + 1)M × (n + 1)L while ϕ(B) is a real matrix of size (n + 1)L × N . The real-valued matrix ϕ(C) ∈ R (n+1)M ×N is defined analogously to (15). Furthermore, the hypercomplex matrix C ∈ A M ×N can be obtained by rearranging the elements of ϕ(C).…”

“…In a few words, HvNNs possess a similar architecture to real-valued model they derive from but the adjustable parameters as well as their inputs and outputs are elements of a hypercomplex algebra. Known examples of HvNNs are the complex-valued [7,8], hyperbolic-valued [9][10][11], quaternion-valued [12,13], and octonion-valued neural networks [14,15]. These architectures are well adapted to multi-signal processing, meaning they can cope appropriately with phase information and rotations, for instance.…”

A broad class of discrete-time hypercomplex-valued Hopfield neural networks

“…We also introduce a broad family of hypercomplex-valued activation functions and provide an important theorem concerning the stability (in the sense of Lyapunov) for discrete-time hypercomplex-valued Hopfield-type neural networks. In fact, the theorem presented in this paper can be applied for the stability analysis of many discrete-time HHNNs from the literature, including complex-valued (Jankowski et al, 1996;Zhou & Zurada, 2014;Kobayashi, 2017b), hyperbolic-valued (Kobayashi, 2013(Kobayashi, , 2016b, dual-numbered (Kobayashi, 2018a), tessarine-valued (Isokawa et al, 2010;Kobayashi, 2018c), quaternion-valued (Isokawa et al, 2008a;Valle & de Castro, 2018), and octonion-valued Hopfield neural networks (de Castro & Valle, 2018a).…”

“…Apart from the extensions of the discrete-time HNN using complex numbers and quaternions, HHNNs on hyperbolic and dual numbers have been proposed and investigated by Kobayashi (2013Kobayashi ( , 2016bKobayashi ( ,a, 2018b. Also, de Castro & Valle (2018a) introduced a discrete-time continuous-valued octonionic HHNN. HHNNs on tessarines, which are also referred to as commutative quaternions, have been investigated by Isokawa et al (2010) and, more recently, by Kobayashi (2018c).…”

“…We would like to point out that the identity (11) has been used implicitly for the stability analysis of many HHNN models (Jankowski et al, 1996;Isokawa et al, 2013;Valle, 2014). Furthermore, this property is used explicitly when the product is not associative, for instance, in the stability analysis of octonion-valued Hopfield-type neural networks (Kuroe & Iima, 2016;de Castro & Valle, 2018a). Finally, since the reverseinvolution plays an important role for the stability analysis of HHNNs, it has been included in the definition of a real-part associative hypercomplex number system.…”