Many important classification problems are polychotomies i.e., the data are organized into K classes with K > 2. The problem of finding an approximation of an unknown function F representing a polychotomy is much easier if it is first decomposed into a set of smaller subproblems called dichotomies, i.e., classification problems of two classes. Given a training set of an unknown K-partition of , an algorithm learning this polychotomy by decomposition into L dichotomies produces an approximation of the following form:
with , where is a function selected in the set of hypotheses of the method to solve subproblem l, and is a mapping associating to each set of L answers of the classifiers, a set of K confidences of membership to each class.
Two different types of decomposition/reconstruction schemes can be distinguished: a priori schemes are determined independently of the data, while a posteriori schemes depend on the training data. This work consists of a comparative study of different methods for a posteriori reconstruction.
The decomposition scheme can be specified by a decomposition matrix such that if class k is associated with z by , if class k is ignored by .
The reconstruction basically consists in the function arg max. The determination a posteriori of the mapping m is based on a dataset distinct from the one used to select the and distinct from the one use to test the system. In this framework, one can forget about the decomposition and consider the classification problem given by to be solved by a model of the form arg max. In what follows, the task is represented by the two matrices and , with , and , if and only if . Two different ways of doing this reconstruction are studied: linear and non-linear methods.