Neurocomputing with High Dimensional Parameters

Abstract

Neurocomputing has established its identity for robustness toward ill-defined and noisy problems in science and engineering. This is due to the fact that artificial neural networks have good ability of learning, generalization, and association. In recent past, different kinds of neural networks are proposed and successfully applied for various applications concerning single dimension parameters. Some of the important variants are radial basis neural network, multilayer perceptron, support vector machines, functional link networks, and higher order neural network. These variants with single dimension parameters have been employed for various machine learning problems in single and high dimensions. A single neuron can take only real value as its input, therefore a network should be configured so that conventionally use as many neurons as the dimensions (parameters) in high dimensional data for accepting each input. This type of configuration is sometimes unnatural and also may not achieve satisfactory performance for high dimensional problems. It has been revealed by extensive research work done in recent past that neural networks with high dimension parameters have several advantages and better learning capability for high dimensional problems over conventional one. Moreover, they have surprising ability to learn and generalize phase information among the different components simultaneously with magnitude, which is not possible with the conventional neural network. There are two approaches to naturally extend the dimensionality of data elements as single entity in high dimensional neural networks. In first line of attack the number field is extended from real number (single dimension) to complex number (two dimension), to quaternion (four dimension), to octanion (eight dimension). The second tactic is to extend the dimensionality of data element using high dimensional vector with scalar components, i.e., three dimension and N-dimension real-valued vectors. Applications of these numbers and vectors to neural networks have been extensively investigated in this chapter.