Minimum Distance Pattern Classifiers Based On A New Distance Metric

A method for the design of minimum distance bipolar pattern classifiers based on a new distance metric between the bipolar patterns is described. The new distance metric is defined and its properties are demonstrated. Neural Networks with the Nearest-neighbor recall mechanism [2], trained according to this method, distinguish the images of a prototype pattern X ^kfrom the images of the opponent prototype functions X ^q, taking into account not only the erroneous pattern elements but also their distinguishing ability. A Neural pattern classifier can be defined by R linear discriminant functions [1] {F ₁(X),..,FR(X)} with \(F_k(X)=X*W^k=x_1*W^k_1+x_2*W^k_2+...+x_n*W^k_n\) where, x _i are bipolar variables \(F_k(X)=X*W^k=x_1*W^k_1+x_2*W^k_2+...+x_n*W^k_n\) and the weights W _i ^k are real numbers and -1 ≤ W _i ^k ≤ +1. In the proposed neural net architecture, the discriminant function F _k(X) is then, followed by an output function T _k (F) defined as is the new distance metric defined next. Let \(X^k=(x^k_1,x^k_2,...,x^k_n)\) be a prototype pattern with k = 1,2,...,R and X = (x ₁,…,x _n) be an unknown pattern. Their Hamming distance D(X^k,X) can then be obtained from the Hamming Discriminant Junction \(H F_k(X)=x_1\ast x^k_1+x_2\ast x^k_2+...+x_n\ast x^k_n\) with \(W^k_i=x^k_i\) , i = 1,...,n, as \(D(X^k,X)=(H F_k(X_k)-H F_k(X))/2\). Then, in the case of a finite set of R prototype patterns X ^k , k = 1,., R, a new Discriminant Function \(F_k(X)=x_1\ast W^k_1+...+x_n\ast W^k_n\) can be defined for every pattern X^k, k = 1,., R with respect to its opponent patterns X^q, q = 1,., R and q ≠ k as follows: The weights W^k ₁ i = 1,…, n of the Discriminant function F _k(X) are calculated according to the formula Erom the above formula we can see that the minimum value of a weight W _i ^k will be equal to W _i ^k = 0, in the case where x _i ^k = x _i ^q for every q= 1,…, R with q ≠ k and the maximum value of the weight W _i ^k when x _i ^k = -x _i ^k for every q, will be equal to \( W_{i}^{k}=(2*x_{i}^{k})*\left \{\sum_{q=k}1/\left(2\ * R*D\left(\textup{X}^{k},\textup{X}^{q}\right)\right)\right\} \). The properties of the new Discriminant Junction can be demonstrated using the above formula of the weights W _i ^k written as \(W_{i}^{k}=(x_{i}^{k})*\left \{\sum_{q=k}1/\left(2\ * R*D\left(\textup{X}^{k},\textup{X}^{q}\right)\right)\right\}-\left \{\sum_{q=k}(x_{i}^{q})/\left(2\ * R*D\left(\textup{X}^{k},\textup{X}^{q}\right)\right)\right\}\).