Hyperbolic Hopfield neural networks with four‐state neurons

Abstract

In recent years, applications of neural networks with Clifford algebra have become widespread. Clifford algebra is also referred to as geometric algebra and is useful in dealing with geometric objects. Hopfield neural networks with Clifford algebra, such as complex numbers and quaternions, have been proposed. However, it has been difficult to construct Hopfield neural networks by Clifford algebra with positive part of the signature, such as hyperbolic numbers. Hyperbolic numbers are useful algebra to deal with hyperbolic geometry. Kuroe proposed hyperbolic Hopfield neural networks and provided their continuous activation functions and stability conditions. However, the learning algorithm has not been provided. In this paper, we provide two quantized activation functions and the primitive learning algorithm satisfying the stability condition. We also perform computer simulations and compare the activation functions.

Dual‐numbered Hopfield neural networks

“…The stability condition for HHNNs is symmetric connections w ij = w ji . Therefore, we obtain u ij = u ji and v ij = v ji .…”

“…Kobayashi proposed several models of hyperbolic HNN (HHNN), i.e. multistate and split HHNNs . The stability condition for the CHNNs and multistate HHNNs is that the mutual connections are conjugate.…”

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).…”

Complex-Valued Recurrent Correlation Neural Networks

“…Neural networks based on hyperbolic numbers constitute an active topic of research since the earlier 2000s [20,21,22]. Hyperbolic-valued Hopfield neural networks, in particular, have been extensively investigated in the last decade by Kobayashi, Kuroe, and collaborators [43,45,81,82,46,47,48]. Hyperbolicvalued Hopfield neural networks are usually synthesized using either Hebbian learning [45,48] or the projection rule [47].…”

“…On the other hand, the projection rule may fails to satisfy the stability conditions imposed on the synaptic weights [47]. Examples of the activation function employed on hyperbolic-valued Hopfield neural networks include the split-sign function [81] and the directional multistate activation function [46]. In the following, we address the stability of hyperbolic-valued RCNNs with the directional multistate activation function, which corresponds to the csgn function given by (30).…”