15 Intrinsic dimensionality extraction

Publisher Summary This chapter discusses the computation of the intrinsic dimensionality in the context of representation, that is, the structure of a data distribution was to be preserved by the mapping and intrinsic dimensionality in the context of classification. In classification, it is well known that the Bayes classifier is the best classifier for given distributions. The resulting classification error, the Bayes error, is the minimum probability of error. As for criteria, the Bayes error is the best. However, since the Bayes error is not easily expressed in an explicit form that may be readily manipulated, many alternatives have been proposed. These include the asymptotic nearest neighbor error, the equivocation, the Chernoff bound, the Bhattacharyya bound and so on. These criteria give upper bounds for the Bayes error. However, the optimization of these criteria does not give the posterior probability functions as the solutions.