Single-Linkage Clustering for Optimal Classification in Piecewise Affine Regression

Abstract When performing regression with piecewise affine maps, the most challenging task is to classify the data points, i.e. to correctly attribute a data point to the affine submodel that most likely generated it. In this paper, we consider a regression scheme similar to the one proposed in (Ferrari-Trecate et al ., 2001; Ferrari-Trecate et al ., 2003) that reduces the classification step to a clustering problem in presence of outliers. However instead of the K-means procedure adopted in (Ferrari-Trecate et al ., 2001; Ferrari-Trecate et al ., 2003), we propose the use of single-linkage clustering that estimates automatically the number of submodels composing the piecewise affine map. Moreover we prove that, under mild assunlptions on the data set, single-linkage clustering can guarantee optimal classification in presence of bounded noise.