Octonion-valued neural networks

Abstract

Neural networks with values in multidimensional domains have been intensively studied over the last few years. This paper introduces octonion-valued neural networks, for which the inputs, outputs, weights and biases are all octonions. They represent a generalization of the complex- and quaternion-valued neural networks, that do not fall into the category of Clifford-valued neural networks, because, unlike Clifford algebras, the octonion algebra is not associative. The full deduction of the gradient descent algorithm for training octonion-valued feedforward neural networks is presented. Testing of the proposed network is done using two synthetic function approximation problems and a time series prediction application.