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.

Gnecco, G., Gori, M., Melacci, S., Sanguineti, M. (2014). Supervised learning from regions and box kernels. In Book of abstracts of the 44th Conference of the Italian Operational Research Society (AIRO 2014) (pp.67-67).

Supervised learning from regions and box kernels

GORI, MARCO;MELACCI, STEFANO;
2014-01-01

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.
2014
Gnecco, G., Gori, M., Melacci, S., Sanguineti, M. (2014). Supervised learning from regions and box kernels. In Book of abstracts of the 44th Conference of the Italian Operational Research Society (AIRO 2014) (pp.67-67).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/974360
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