A supervised learning paradigm is investigated, in which the data are represented by labeled regions of the input space. This learning model is motivated by real-world applications, such as problems of medical diagnosis and image categorization. The associated optimization framework entails the minimization of a functional obtained by introducing a loss function that involves the labeled regions. A regularization term expressed via differential operators, modeling smoothness properties of the desired input/output relationship, is included. It is shown that the optimization problem associated to supervised learning from regions has a unique solution, represented as a linear combination of kernel functions determined by the differential operators together with the regions themselves. The case of regions given by multi-dimensional intervals (i.e., "boxes") is investigated as an interesting instance of learning from regions, which models prior knowledge expressed by logical propositions. The proposed approach covers as a particular case the classical learning context, which corresponds to the situation where regions degenerate to single points. Applications and numerical examples are discussed.
Supervised learning; Kernel machines; Infinite-dimensional optimization; Constrained variational calculus; Representer theorems.