TY  - JOUR
SN  -  0925-2312
Y1  - 2014/04//
N1  - Available online 25 October 2013
JF  - Neurocomputing
N2  - Supervised learning is investigated, when the data are represented not only by labeled points but also labeled regions of the input space. In the limit case, such
regions degenerate to single points and the proposed approach changes back to the classical learning context. The adopted framework entails the minimization
of a functional obtained by introducing a loss function that involves such regions. An additive regularization term is expressed via differential operators that model
the smoothness properties of the desired input/output relationship. Representer
theorems are given, proving that the optimization problem associated to learning
from labeled regions has a unique solution, which takes on the form of a linear
combination of kernel functions determined by the differential operators together
with the regions themselves. As a relevant situation, the case of regions given
by multi-dimensional intervals (i.e., ?boxes?) is investigated, which models prior
knowledge expressed by logical propositions.
EP  - 32
VL  - 129
PB  - Elsevier
SP  - 25
A1  - Gnecco, Giorgio
A1  - Gori, Marco
A1  - Melacci, Stefano
A1  - Sanguineti, Marcello
AV  - public
ID  - eprints1794
TI  - A Theoretical Framework for Supervised Learning from Regions
UR  - http://www.sciencedirect.com/science/article/pii/S0925231213009879
KW  - supervised learning; kernel machines; propositional rules;
variational calculus; infinite-dimensional optimization; representer theorems
ER  -