Abstract
Most pattern recognition problems may be categorized as parametric and nonparametric ones on the bases of our knowledge concerning the conditional densities of the input patterns. In addition, the learning machines can further be classified into two types, supervised and nonsupervised ones.
This paper presents a mathematical model of a nonsupervised learning machine using a first principal component. Previous works related to the classifying method using the first principal component, such as those by Cooper and Cooper, and Takanuki and Morishita, have led to a nonsupervised classifiers with parametric learning methods.
In this paper, we discuss a nonparametric case. Under the assumption that the patterns of each category are clustered, the separating hyperplane should contain the mean vector Zs of all patterns used and should be perpendicular to the line governed by the first principal component WsTX. Therefore, the decision rule is given by decide: X∈{C1 if WsX>θs C2 otherwise θs=(Ws, Zs). The learning algorithm employed in the classifier is as follows: {Wk+1'=Wk+ak{(Xk-Zk, Wk)(Xk-Zk)-|Xk-Zk|2Wk}Wk+1=Wk+1'/|Wk+1'|Zk+1=Zk+ak(Xk-Zk)θk+1=(Wk+1, Zk+1), where Wk is the weight vector, Zk is the mean vector and ak is the correction increment at the k-th iteration.
Also we have made a computer simulation to study the learning performance of the classifier for the Gaussian data. The results of the experimental study show that the probability of error of the classifier proposed is slightly larger than the Bayes' minimum error.
