Learning curves of polynomial kernel classifiers

The generalization properties of polynomial kernel classifiers are examined. Since a kernel classifier nonlinearly maps an input vector to a vector in a high-dimensional feature space and linearly discriminates it there, it has a similar learning curve to a linear dichotomy that has an average generalization error proportional to the dimension of the input space and inversely proportional to the number of given examples in the asymptotic limit. This paper shows that the asymptotic average generalization error depends on the relationship between the subset in the feature space on which the feature vectors lie and the true separating hyperplane, more specifically, the essential dimension of the feature space in the neighborhood of their intersection.