In the training stage, supervision is directly integrated into measuring the clustering error err S (h, p) of a partition function h by exploiting the feedback given by the true partition p. The goal of training is to find a partition function h * in a hypothesis space H that minimizes the error.
given a new data set S in the application stage, under some standard learning theory assumptions, the hope is that the learned partition function can generalize well and achieve small error as well.