The use of an adaptive threshold element to design a linear optimal pattern classifier

This paper develops a relationship between two traditional statistical methods of pattern classifier design, and an adaption technique involving minimization of the mean-square error in the output of a linear threshold device. It is shown that the two-category classifier derived by least-mean-square-error adaption using an equal number of sample patterns from each category is equivalent to the optimal statistical classifier if the patterns are multivariate Gaussian random variables having the same covariance matrix for both pattern categories. It is also shown that the classifier is always equivalent to the classifier derived by R. A. Fisher. A simple modification of the least-mean-square-error adaption procedure enables the adaptive structure to converge to a nearly-optimal classifier, even though the numbers of sample patterns are not equal for the two categories. The use of minimization of mean-square error as a technique for designing classifiers has the added advantage that it leads to the optimal classifier for patterns even when the covariance matrix is singular.