  A particularly interesting instance of supervised learning with kernels is
when each training example is associated with two objects, as in pairwise
classification (Brunner et al., 2012), and in supervised learning of preference
relations (Herbrich et al., 1998). In these cases, one may want to embed
additional prior knowledge into the optimization problem associated with the
training of the learning machine, modeled, respectively, by the symmetry of its
optimal solution with respect to an exchange of order between the two objects,
and by its antisymmetry. Extending the approach proposed in (Brunner et al.,
2012) (where the only symmetric case was considered), we show, focusing on
support vector binary classification, how such embedding is possible through
the choice of a suitable pairwise kernel, which takes as inputs the individual
feature vectors and also the group feature vectors associated with the two
objects. We also prove that the symmetry/antisymmetry constraints still hold
when considering the sequence of suboptimal solutions generated by one version
of the Sequential Minimal Optimization (SMO) algorithm, and we present
numerical results supporting the theoretical findings. We conclude discussing
extensions of the main results to support vector regression, to transductive
support vector machines, and to several kinds of graph kernels, including
diffusion kernels.
