Vector-Valued Support Vector Regression

A vector-valued extension of the support vector regression problem is presented here. The vector-valued variant is developed by extending the notions of the estimator, loss function and regularization functional from the scalar-valued case. A particular emphasis is placed on the class of loss functions chosen which apply the epsiv-insensitive loss function to the p-norm of the error. The primal and dual optimization problems are derived and the KKT conditions are developed. The general case for the p-norm is specialized for the 1-, 2- and p-norms. It is shown that the vector-valued variant is a true extension of the scalar-valued case. It is then shown that the vector-valued approach results in sparse representations in terms of support vectors as compared to aggregated scalar-valued learning.