Reducibilities of hyperbolic neural networks

Clifford algebra includes the real and complex numbers. The hyperbolic numbers also belong to Clifford algebra. In recent years, neural networks (NNs) are extended using Clifford algebra, and hyperbolic NNs have been proposed. Since the hyperbolic numbers have zero divisors, it is difficult to analyze the hyperbolic NNs. Thus, the reducibilities of hyperbolic NNs have never been revealed. In this work, the reducibilities of hyperbolic NNs are studied. The reducibilities are tightly related to learning process. In the real-valued and complex-valued NNs, there exist three types of reducibilities. In the hyperbolic NNs, there exists another type of reducibilities, and it has been difficult to determine all the reducibilities of hyperbolic NNs. It is proved that hyperbolic NNs have another reducibility, referred to as hyperbola-reducibility, and all the reducibities of hyperbolic NNs are determined. In addition, the inherent singularities of hyperbolic NNs are revealed. These facts are expected to improve the learning process of hyperbolic NNs in future. (C) 2019 Elsevier B.V. All rights reserved.