Multilayer Perceptrons to Approximate Quaternion Valued Functions

In this paper a new type of multilayer feedforward neural network is introduced. Such a structure, called hypercomplex mu[tilayerperceptron (HMLP), is deve[opedin quaterniona[gebra anda[[ows quaternionic input and output signals to be akalt with, requiring a lower number of neurons than the real MLP, thus providing a reduced computational complexity. The structure introduced represents a generalization of the multilayer perception in the complex space (CA4LP) reported in the literature. Thefundamental result reported in the paper is a new density theorem whichmakes HA4LPs universal interpolators of quaternion valued continuousfactions. Moreover theproof of the akmsity theorem can be restricted in orab to formulate a almsity theorem in the complex space. Due to the iaimtity between the quaternion and thefour-dimensional real space, such a structure is also useful to approximate multidimensional real valuedfunctions with a lower number of real parameters, &creasing the probability of being trapped in local minima during the iearning phase. A numerical example is also reported in order to show the eficiency of the proposed structure.