@inproceedings{eprints3133,
            year = {2014},
       booktitle = {Book of abstracts of the 44th Conference of Italian Operational Research Society (AIRO 2014)},
           title = {Supervised Learning from Regions and Box Kernels},
          author = {Giorgio Gnecco and Marco Gori and Stefano Melacci and Marcello Sanguineti},
           pages = {67},
             url = {http://eprints.imtlucca.it/3133/},
        abstract = {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.},
        keywords = {Supervised learning; Kernel machines; Infinite-dimensional optimization; Constrained variational
calculus; Representer theorems.}
}