Conditions of Optimal Classification for Piecewise Affine Regression

We consider regression problems with piecewise affine maps. In particular, we focus on the sub-problem of classifying the datapoints, i.e. correctly attributing a datapoint to the affine submodel that most likely generated it. Then, we analyze the regression algorithm proposed by Ferrari-Trecate et. al (2003) and show that, under suitable assumptions on the dataset and the weights of the classification procedure, optimal classification can be guaranteed in presence of bounded noise. We also relax such assumptions by introducing and characterizing the property of weakly optimal classification. Finally, by elaborating on these concepts, we propose a procedure for detecting, a posteriori, misclassified datapoints.