Neural Networks in Multidimensional Domains

摘要

In this paper a new structure of multi-layer per-ceptron, based on quaternion algebra, is introduced in order to deal with multidimensional signals. A learning algorithm for the quaternion-valued MLP (QMLP) is also derived. The idea arises from the fact that, when multidimensional signals are taken into consideration, a real-valued MLP requires a number of real parameters which grows heavily with the input-output dimension. The introduction of quaternion algebra allows multidimensional signals to be managed directly without using a redundant number of real parameters, and, at the same time, avoiding the possibility of poor learning. As a consequence, such a neural network allows functions of three or four variables to be interpolated with a smaller number of connections with respect to the corresponding real-valued MLP. The simple applications reported in the paper aim to outline the QMLP capability to solve multidimensional function approximation, and also to perform efficient separation of three or more categories.