Classification based on prototypes with spheres of influence

We present a family of binary classifiers and analyse their performance. Each classifier is determined by a set of `prototypes', whose labels are given; and the classification of any other point depends on the labels of the prototypes to which it is sufficiently close, and on how close it is to these prototypes. More precisely, the classification of a given point is determined through the sign of a discriminant function. For each prototype, its sphere of influence is the largest sphere centred on it that contains no prototypes of opposite label, and, given a point to be classified, there is a contribution to the discriminant function at that point from precisely those prototypes whose spheres of influence contain the point, this contribution being positive from positive prototypes and negative from negative prototypes. Furthermore, these contributions are larger in absolute value the closer the point is (relative to the sphere's radius) to the prototype. We quantify the generalization error of such classifiers in a standard probabilistic learning model, and we do so in a way that involves the values of the discriminant function on the points of the random training sample