Neurocomputing with High Dimensional Parameters - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 39

ing account of task domains has shown their superior computational power in wide
spectrum of tasks. Thus, neural network models dealing with N signals as one cluster
are desired under the powerful framework of N-dimensional vector neuron. An N-
dimensional vector neuron is a natural extension of the 3-dimensional vector neuron
whose vital applicability has been presented in Chap. 6 . The famous neuroscience
researcher T Nitta (2007) proposed an efficient solution for the N -bit parity problem
with a single N-dimensional vector-valued neuron [ 22 ] considering the orthogonal
decision boundary. It reveals the potent computational power of N-dimensional vec-
tor neurons because this problem cannot be solved with a single usual real-valued
neuron. It is reasonable to emphasize it as a new directionality for enhancing the
capability of neural networks, and therefore worth researching the neural networks
with N-dimensional vector neuron.

2.6.1 Properties of Vectors in R N

The vectors in space can be directly extended to vectors in N-space. A vector in
N-space is represented by an ordered N-tuples of real numbers and

same for a point in N-space, R N . All the listed axioms in Definition 2.6 including
two operations (vector addition and scalar multiplication) holds for any three vectors
v 1 ,

[ x 1 , x 2 ,..., x N ]

V in N-space ( R N ), therefore V is called a vector space over the real
numeric's R .

v 2 , v 3 ∈

2.6.2 N-Dimensional Vector Based Neural Networks

The structure of N-dimensional vector neuron, which can deal with N signals in one
cluster, can be given by extending the structure of 3-dimensional vector-valued neu-
ron. In an N-dimensional vector-valued neuron all the input-output signals, thresholds
are N-D real-valued vectors and the weights are N-dimensional orthogonal matri-
ces. Additional restrictions imposed on the N-dimensional orthogonal matrix (e.g.,
it can be regular, symmetric, or orthogonal etc.) will also influence the behavioral
characteristics of the neuron. The net potential of a N-dimensional neuron can be
given as:

L

Y = W l X l + ʸ (2.15)

l = 1

T is l th input signal, W l is the N-dimensional
orthogonal weight matrix for the l th input signal and

where input signal X l =[ x 1 , x 2 ,..., x N ]

T is the
threshold value. It is also important to mention here that the N-dimensional vector
neuron presented here may be considered with the traditional activation functions.
The output of the neuron will also be a N-dimensional real-valued vector. Similar to

ʸ =[ ʸ 1 ,ʸ 2 ,...,ʸ N ]